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the star formation history of field stars in the large magellanic cloud ( lmc ) contains much information about the formation and dynamics of our closest galactic neighbor . studies by stryker ( 1984 ) , bertelli _ et al . _ ( 1992 ) , westerlund _ et al . _ ( 1995 ) , and vallenari _ et al . _ ( 1996a , b ) , among others , conclude that the lmc field contains a majority of young to intermediate age stars overlying a minority old population . et al . _ ( 1992 ) favor a star formation history in which the star formation rate , initially at a constant low level , increases by a factor of ten in the last few billion years . this star formation history produces a stellar population reminiscent of the bimodal age distribution of lmc globular clusters ( van den bergh 1991 ; girardi _ et al . _ _ ( 1996b ) find tentative evidence that the onset of this increase in star formation is correlated with position in the lmc , and suggest that this correlation might arise if star formation is triggered by tidal interactions with the small magellanic cloud . et al . _ ( 1997 ) present _ hst _ observations of a field in the bar of the lmc and find evidence for an additional younger population of stars which is not observed in the outer regions of the lmc field . a clearer understanding of stellar populations throughout the lmc should provide clues about the age and formation history of the lmc as well as about the mechanisms which trigger star formation in this galaxy . we have observed three fields in the lmc with the wide field planetary camera 2 on the _ hubble space telescope _ to determine whether these regions share a similar formation history . the fields are all located in the outer regions of the lmc at roughly the same radial distance from the lmc bar . these observations extend several magnitudes below the main sequence turnoff and provide a significant improvement over ground - based studies . gallagher _ et al . _ ( 1996 ) present the color - magnitude diagram for one of these fields , and find that the width of the upper main sequence is consistent with a star formation rate which is roughly constant for the last few billion years . in addition , they suggest that a small burst of star formation occurred 2 gyr ago , leading to a distinct subgiant branch seen one magnitude brighter than the faintest main sequence turnoff . the lack of evidence in the _ hst _ data for a strong star formation burst is in apparent contrast to previously determined star formation histories and results which indicate the metallicity of the lmc has nearly doubled in the past 2 gyr ( dopita _ et al . _ 1997 ) . holtzman _ et al . _ ( 1997 ) analyze the luminosity function of the same _ hst _ field to constrain its initial mass function and star formation history . by comparing the luminosity function of the lower main sequence to stellar models , they constrain the imf slope , @xmath0 ( @xmath1 ) , to @xmath2 in the mass range @xmath3 . assuming a salpeter imf ( @xmath4 ) , they derive a star formation history from the entire observed luminosity function . they favor a star formation history in which the star formation rate is roughly constant for 10 gyr and then increases by a factor of three for the past 2 gyr , resulting in a stellar population with comparable numbers of stars older and younger than 4 gyr . this is in contrast to bertelli _ et al . _ ( 1992 ) , whose preferred star formation history produces a predominantly young ( @xmath5 gyr ) stellar population . _ find that a predominantly young population fits the _ hst _ observations only if the imf slope is steeper , with @xmath6 . in @xmath7 , we present _ hst _ observations of the three lmc fields . in @xmath8 , we quantitatively compare the stellar populations in these fields and show that they are statistically indistinguishable . in @xmath9 , we compare our observations with several possible star formation histories , using the r - method of bertelli _ et al . _ ( 1992 ) in combination with comparisons between the model and observed luminosity functions . our derivation of the star formation history is an improvement over previous _ determined formation histories as we use ground - based data to supplement observations at the brightest magnitudes . observations were made with the wide field planetary camera 2 of the _ hubble space telescope _ between may 1994 and december 1995 through the f555w ( @xmath10 ) and f814w ( @xmath11 ) filters . total exposure times were 4000s , 2500s , and 1000s in each filter for fields 1 , 2 , and 3 , respectively . observations through each filter were split into three or more separate exposures to allow identification and removal of cosmic ray events . in figure 1 , an image of the lmc shows the approximate positions of the three fields . a previous analysis of field 1 has been presented by gallagher _ et al . _ ( 1996 ) and holtzman _ et al . _ ( 1997 ) . the positions and exposure times for each field are listed in table 1 . the data were processed using standard reduction techniques described in holtzman _ et al . _ ( 1995a ) . this process includes a small correction for analog - to - digital errors , overscan and bias subtraction , dark subtraction , a small shutter shading correction , and flat fielding . in each filter , the images were combined and cosmic ray events were removed based on the expected variance from photon statistics and read noise . a combination of profile - fitting and aperture photometry was chosen to give good photometry at both bright and faint signal levels . since the f555w and f814w images were roughly equally deep , stars were found for each field on the summed frame of these two images . due to structure in the point spread function ( psf ) , objects found in the area surrounding the peak of bright stars were rejected . using this star list , profile - fitting photometry was performed on each frame . the model psfs were the same as those used by holtzman _ et al . _ ( 1997 ) . profile - fitting results were then used to subtract all stars from the images . final magnitudes were determined by adding each star individually back into the subtracted frame and performing aperture photometry with a 2 pixel radius aperture . aperture corrections to a 0.5 arcsecond radius aperture were individually determined for the four wfpc2 chips from bright , isolated stars . we estimate the maximum error of this correction to be a few hundredths of a magnitude . instrumental magnitudes were transformed into v and i magnitudes using the transformations given by holtzman _ et al . _ ( 1995b ) . to convert into absolute magnitudes , we adopt a distance modulus of 18.5 derived by panagia _ et al . _ ( 1991 ) from sn1987a . a re - evaluation of the cepheid distance calibration , using hipparcos parallaxes , suggests a slight upward revision in the lmc distance modulus to @xmath12 ( madore & freedman 1997 ) , however , our conclusions are insensitive to errors of this order in the distance modulus . schwering & israel ( 1991 ) determine a foreground reddening of @xmath13 towards field 1 and variations less than 0.02 in @xmath14 between the three fields . allowing for a small amount of internal extinction , we adopt @xmath15 with a corresponding extinction of @xmath16 . to estimate completeness , a set of artificial stars tests was performed . at a series of different brightnesses , artificial stars were added to each frame in an equally spaced grid and the frames were run through the photometry routine described above . the grid spacing was chosen so that artificial stars did not add significantly to crowding in the field ; 121 stars were placed on the pc and 529 stars on each of the wfs . the resulting photometry list was compared to the input list and the completeness level was determined as described by holtzman _ et al . _ ( 1997 ) . we estimate the 90% completeness level to be at @xmath17 in fields 1 and 2 , and @xmath18 in field 3 due to the lower exposure time . we restrict our analysis to stars brighter than these limits . the fraction of detected artificial stars and their associated errors were tabulated as a function of magnitude and are used in @xmath19 to simulate observed stellar populations . these errors include systematic errors due to crowding and random errors from poisson statistics . observational errors for our simulations ( discussed below ) are determined by randomly sampling from these error distributions . the stellar density in these regions of the lmc is low and the fields are not crowded ( see holtzman _ et al . _ 1997 for an image of field 1 ) . for stars brighter than @xmath20 , our observations are not representative due to saturation and the small wfpc2 field of view . in @xmath19 , we correct for small number statistics at the brightest magnitudes using ground based data taken with the mount stromlo and siding spring observatories 1 m telescope ( stappers _ et al . _ 1997 ) . three fields adjacent to and including field 1 were observed in v and i. the exposure time for each field was 1000s in each filter . the total area on the sky , excluding a region around the cluster ngc 1866 and a second smaller cluster , was 622 arcmin@xmath21 . reduction and photometry were done using standard iraf tasks . these data are complete to an apparent magnitude of @xmath22 . photometric consistency between the ground - based and wfpc2 data was checked by comparing 37 stars common to both ; differences in v and i were @xmath23 magnitudes . color - magnitude diagrams ( cmds ) for the three wfpc2 fields are shown in figure 2 . the faintest main sequence turnoff for the three fields occurs at @xmath24 , and a clear main sequence extends roughly four magnitudes fainter . the number of stars in each field is roughly comparable . error bars plotted in figure 2 are average one @xmath25 errors as determined by the aperture photometry routine . larger errors in field 3 are a result of the shorter exposure time . major features in these cmds , such as the main sequence , the main sequence turnoff , and the red giant branch , occur at the same magnitude and color , suggesting similar stellar populations . we first compare stellar distributions across the lower main sequence as these are sensitive to metallicity variations between the three populations . we then compare the distributions across the upper main sequence as these probe variations between the recent star formation histories of the three fields . as a statistical test of comparison , we use the one dimensional kolmogorov - smirnov ( ks ) test . this test is used to compare the three fields , as well as to compare the observed luminosity functions to simulated stellar populations . the ks test gives the probability ( p ) that the deviations between two distributions are the same as would be observed if they were drawn from the same population . two distributions are considered different if the probability that they are drawn from the same parent distribution can be ruled out at a confidence level greater than 95% ( @xmath26 ) . if two distributions can not be proved different , we infer that the populations are similiar , although the ks test does not imply that these distributions are the same . we estimate the sensitivity of this test to minor differences in the star formation history using simple simulations described below . as seen in figure 2 , stars appear to be concentrated towards the blue side of the lower main sequence . the lower main sequence of field 1 appears the most concentrated towards the blue , field 3 appears the least concentrated . for field 3 , this is likely the result of a lower exposure time , but for field 2 may reflect a real difference with the stellar population in field 1 . the histograms of figure 3 plot the color distributions for fields 1 and 2 in several magnitude bins across the lower main sequence . the total number of stars in each histogram has been normalized to the number in field 1 . we used the ks test to statistically compare the distribution of stars across the lower main sequence in these fields . in figure 3 , the relatively high values of the ks probability , p= 0.2 , 0.3 and 0.9 , indicate that we can not demonstrate that the two samples are drawn from different populations . the signal - to - noise in field 3 is lower due to a shorter exposure time . in order to compare lower main sequence distributions , we add gaussian noise to the higher signal - to - noise field 1 using the average errors from field 3 and perform the ks test . in figure 4 , the distribution of lower main sequence stars in field 3 is compared to the field 1 plus noise distribution . again , we can not prove that the distributions are different at a high confidence level . the application of the ks test to the lower main sequence color distributions is weakened by the small number of stars in each comparison , however , it does suggest similar stellar populations in the three field . although the distribution of stars will also be influenced by age variations and the presence of binaries , the similarity in the mean colors of the lower main sequences suggest that the three fields have similar mean metallicities . to estimate the sensitivity of our tests to differences in metallicity , we use stellar models described below to simulate stellar populations with identical star formation histories but different metallicity distributions . we find that the ks test is sensitive to metallicity differences if , between the two populations , at least 25% of the stars have a factor of four difference in metallicity . this fraction decreases to 20% of stars if the metallicity differs by a factor of ten . these results are robust for several different assumed star formation histories . from the width of the lower main sequence , we can rule out the possibility that the observed stellar populations have a single metallicity . the standard deviations of the observed lower main sequence distributions shown in figure 3 are @xmath27 for the three magnitude bins . field 3 is not included due to the higher noise . we compare this width with that of a simulated single metallicity population , assuming a constant star formation rate from 12 gyr to the present and a 50% binary fraction . observational errors are simulated by randomly sampling the error distributions determined from artificial star tests , and include both systematic and random errors . independent of assumed metallicity , the standard deviation of the lower main sequence was @xmath28 for the same magnitude bins . we find that the difference between the observed and model variances is significant based on a f - test . populations having a smaller age range or a lower assumed binary fraction will have narrower distributions . therefore , the observed lower main sequence is wider than is expected for a single metallicity population . since this is a differential comparison , we expect it to hold regardless of the adopted stellar models . the derivation of the absolute metallicity is discussed in @xmath29 and may be model dependent . the distribution of stars across the upper main sequence is shown in figure 5 . the width of the main sequence is broader than would be expected from a single age population . from stellar evolution models , stars brighter than @xmath30 evolve steadily redward during their main sequence lifetimes . thus , a uniform distribution across the upper main sequence suggests no intense star formation bursts have occurred during the lifetimes of these more massive stars . _ ( 1996 ) show that the distribution of upper main sequence stars in field 1 is roughly consistent with a star formation rate constant for the past 3 gyr . the ks test gives no evidence that the distributions of stars across the upper main sequence in the three fields are different . this test is not sensitive to variations in the star formation history more recent than 0.1 gyr , due to small number statistics at the brightest magnitudes . a second distinct main sequence turnoff discussed by gallagher _ et al . _ ( 1996 ) and seen near @xmath31 in figure 2a is not observed in the remaining two fields . _ interpret this turnoff and associated excess of stars at @xmath32 in the hertzsprung gap as signatures of a short star formation burst occurring @xmath33 gyr ago . although many stars populate this region of the cmds in field 2 and 3 , the lack of a single distinct subgiant branch in these fields argues against a short ( @xmath34 gyr ) , global 2 gyr burst throughout the lmc . the subgiant excess observed in field 1 may arise from a statistical fluke , the remnants of a localized star formation burst , or a dissolved cluster . the observed , uncorrected , differential luminosity functions are shown in figure 6 . we perform the ks test between fields 1 and 2 over the range @xmath35 and between field 1 and field 3 over the range @xmath36 . the resulting ks probabilities between fields are @xmath37 and @xmath38 , thus we are not able to show that the three luminosity functions are different . we conclude , from analysis of the luminosity function and distribution of stars across the main sequence , that the three observed regions in the lmc field contain statistically indistinguishable stellar populations . in addition , we have calculated r - ratios as defined by bertelli _ _ ( 1992 ) and discussed below ( @xmath19 ) , which compare the number of stars in different evolutionary phases ; these are shown in table 2 . these ratios are also the same between the three fields , within the errors determined by number statistics . we can not prove that the star formation histories in the three fields are different ; however , this does not imply that they are identical . to estimate the sensitivity of our tests to variations in star formation history , we simulate a simple star formation history with a constant star formation rate . we compare this to models in which star formation is turned off completely for short lengths of time at different epochs . we find that our statistical tests are unable to conclusively distinguish between such models for variations in star formation rate over periods shorter than 1 gyr anytime in the past 4 gyr , or over periods shorter than 2 gyr anytime before 4 gyr ago . we next compare our observations to stellar models using similar statistical tests in order to place constraints on possible star formation histories . as we have shown that our methods can not distinguish between the three stellar populations , we combine the observations of the three fields to improve number statistics . to determine the star formation history in the three fields , we compare our observations to simulations made using the stellar evolution models published by the padua group ( bertelli _ et al . _ 1994 and references therein ) . these isochrones range in metallicity from @xmath39 ( -1.7@xmath40\leq$]0.4 ) and are calculated for stellar masses down to @xmath41 . depending on the metallicity , this corresponds to an absolute magnitude of @xmath42 , roughly the magnitude limit of our observations . the padua models are calculated with mild convective overshoot and the most recent livermore group radiative opacities ( iglesias _ et al . _ ubvri magnitudes for these models have been calculated by bertelli _ et al . _ ( 1994 ) . there is evidence for nonsolar abundance ratios in the lmc , such that the @xmath0-element are enhanced relative to the solar ratio ( luck & lambert 1992 ) . although @xmath0-enhanced models are not currently available , salaris _ et al . _ ( 1993 ) have found that under some conditions , @xmath0-enhanced isochrones are well mimicked by scaled solar metallicity isochrones . the effect of @xmath0-element enhancement is to shift a solar abundance isochrone towards the red , which would lead to an overestimate of our derived metallicities . comparing the observed cmds to single age isochrones , we find the blue edge of the upper main sequence is best matched by an isochrone of metallicity z=0.008 , whereas the red giant branch can be fit by either an old to intermediate age , metal poor isochrone ( @xmath43 gyr and z=0.0004 ) or a young , higher metallicity isochrone ( @xmath44 gyr and z=0.001 ) . this point is well illustrated in figure 4 of holtzman _ et al . _ ( 1997 ) . problems with the stellar atmospheres at lower temperatures and/or the evolutionary models of giant stars may be responsible for the apparent mismatch . alternatively , it may be related to our use of solar abundance ratio isochrones . because of these possible problems , we do not use the color of the giant branch to derive stellar population parameters . however , some of the derived parameters , in particular metallicities , are sensitive to the model colors of main sequence stars . we note that our constraints on such parameters are derived assuming that these stellar models are perfect ; we allow for random errors in the observations but not for systematic errors in the models . the presence of a bright main sequence , seen in figure 2 , suggests recent star formation activity , whereas the faintest main sequence turnoff at @xmath24 implies an older star formation epoch . we therefore simulate cmds for a mixture of simple stellar populations . throughout the simulations , we assume that 50% of the stars are binaries with uncorrelated masses ( reid 1991 ) . in order to preserve isochrone shape during age interpolation , each isochrone is resampled into 100 equally spaced mass points within each of 9 evolutionary epochs . no interpolation is made in metallicity . the mass and age of each star are chosen according to an input initial mass function and star formation history . the absolute magnitude and color of each star can be determined for any desired age by interpolating point by point over the resampled isochrones . an apparent magnitude is determined based on the extinction and distance modulus . the star is considered detected or rejected according to the completeness histograms calculated during the artificial star tests described above . an observational error is given to each detected star by randomly sampling the error distribution appropriate for the star s magnitude determined from the artificial star tests ( see @xmath45 ) . these errors include both random and systematic effects . the number of free parameters in determining a star formation history is large . our observations do not provide enough constraints to justify an exhaustive search of parameter space . instead , we choose to discuss six representative formation histories : constant star formation throughout the history of the lmc , the two preferred histories of bertelli _ et al . _ ( 1992 ) and vallenari _ et al . _ 1996b , two proposed histories of holtzman et al . _ ( 1997 ) , and a formation history motivated by the observed age distribution of lmc globular clusters . in all cases , we assume the age of the oldest lmc stars to be 12 gyr based on age estimates of the oldest lmc globular clusters ( van den bergh 1991 ) . the parameters used in each simulation are shown graphically in figure 7 . to compare observed and theoretical stellar populations , we use a combination of two methods . first , we compare observed and model luminosity functions using the 1-d ks test as described in @xmath46 . the three fields are combined to create the observed luminosity function and are compared with models over a range @xmath47 . a star formation history is considered acceptable if the probability , p , that the luminosity function is drawn from the same population is greater than 5% . the sensitivity of this test to variation in the star formation history is the same as that discussed in @xmath48 . in comparing luminosity functions without regard to color , some star formation history information is lost , especially for stars brighter than the main sequence turnoff . thus , in addition to luminosity function fitting , we use the r - method described in detail by bertelli _ et al . _ ( 1992 ) for @xmath49 . briefly , this method defines three stellar number ratios , each sensitive to different parameters in the star formation history . the first ratio is defined as : @xmath50 the separation between main sequence and red giant stars is determined by the lines shown in figure 2 , and is consistent throughout the analysis . the next two ratios compare the number of bright to faint stars on the main sequence and the red giant branch . the magnitude separating the upper and lower regions is defined at @xmath51 , chosen to be below the red clump . the ratios are defined as : @xmath52 @xmath53 the r - ratios depend , in different ways , on the slope of the initial mass function and the relative number of young and old stars ; see bertelli _ et al . _ ( 1992 ) for a more extended discussion of these dependencies . the r - ratios for the three observed fields , as well as the ratios resulting from combining the three fields , are shown in table 2 . the errors in this table are one @xmath25 errors as determined from number statistics . the wfpc2 fields do not provide representative numbers of stars at the brightest magnitudes due to saturation and the small field of view . we corrected for this by combining the _ hst _ data with ground - based data covering a significantly larger field and including field 1 . a more detailed analysis of these data was presented by stappers _ et al . _ ( 1997 ) . to combine star counts from ground and space - based data directly , we applied a scale factor determined by the ratio of observed areas . we use ground - based star counts for magnitudes brighter than @xmath51 to recalculate the r - ratios of the combined field . as compared to the uncorrected ratios ( table 2 , second to last row ) , the use of ground - based data results in an average 10% corrections to the r - ratios ( table 2 , last row ) . constraints from the r - ratios , particularly @xmath54 , may be less secure than those based on the main sequence because of the larger uncertainties in modelling later phases of stellar evolution . we compare the observed luminosity function and r - ratios with those calculated for the six star formation histories shown in figure 7 . the observed and computed r - ratios , as well as the ks probability resulting from a comparison of the observed and model luminosity functions , are given in table 3 . the observed and simulated luminosity functions for each model are shown in figure 8 . the simulated luminosity functions are normalized to match the observations at @xmath55 . the simplest star formation history tested assumes a salpeter imf and a constant star formation rate since the formation of the lmc 12 gyr ago ( figure 7a ) . in examining the parameter @xmath56 , this simple formation scenario does not produce enough bright main sequence stars by more than a factor of two relative to the number of observed evolved stars . this deficiency has motivated most modelers to include an enhancement in the recent star formation rate . bertelli _ et al . _ ( 1992 ) observed a field @xmath57 southwest of field 1 . using the r - method , they derive a star formation history in which the star formation rate was initially low and then increased by a factor of ten 4 gyr ago , as shown in figure 7b . although this model reproduces the observed r - ratios reasonably well , it does not match the observed luminosity function . as shown in figure 8b , this formation scenario produces too many bright ( @xmath58 ) stars relative to faint stars . we note that the 1992 padua stellar models used to derive this history allow for more convective overshoot than the 1994 models used in this paper . the use of more recent models decreases the inferred time when the star formation increase rate began to approximately 2 gyr ago ( bertelli , private communication ) . such a star formation history was used by vallenari _ _ ( 1996b ) to match observations in several of their fields and its luminosity function is shown in figure 8c . it produces even more bright stars relative to the number of observed faint stars , and is ruled out by our observed luminosity function . in their simulations , et al . _ allow for interpolation in metallicity , whereas our models use discrete metallicity isochrones . we find that the luminosity functions of the three individual metallicities which contribute to the final formation scenario are inconsistent with the observations in the same direction as the composite vallenari _ _ luminosity function . therefore , this simplification is not the source of discrepancy between the vallenari _ _ scenario and our observations . holtzman _ et al . _ ( 1997 ) find that a steeper imf slope is necessary in order for a 4 gyr , ten - fold star formation rate increase to match the observed luminosity function . for an imf slope @xmath59 , this star formation history ( figure 7d ) is consistent with the observed luminosity function , but not with the r - ratios observed in our fields . of published star formation histories for the lmc field , the only one which reproduces both the luminosity function and r - ratios of our observation is that of holtzman _ et al . _ ( 1997 ) . in this formation scenario , the star formation rate remains constant for a majority of the lmc history and is increased by a factor of three from 2 gyr ago to the present ; the star formation rate is also slightly higher for the oldest stars ( figure 7e ) . in contrast to the previous three scenarios which produce primarily young stellar populations , this star formation history produces a population with roughly equal number of stars older and younger than 4 gyr . the simulated cmd for this star formation history is shown in the right panel of figure 9 . a star formation history based on the age distribution of lmc globular clusters can not fit the observed field population . the age distribution of lmc clusters is bimodal ( figure 7e ) with @xmath60 of clusters formed between @xmath61 years and 3 gyr ago , and @xmath62 of clusters having ages between 10 and 12 gyr ( van den bergh 1991 ) . there are almost no known intermediate age ( 3 - 10 gyr ) clusters in the lmc ( girardi _ et al . _ 1995 ; however see sarajedini _ et al . _ 1995 ) , however , an intermediate population is necessary to reproduce our observations . if no clusters have been destroyed , we conclude that the star formation history of lmc globular clusters is not mimicked by the field population . it is possible to predict the chemical history of the lmc from its star formation history using the simple closed box model of chemical evolution . we assume a one - zone evolution model with no infall or outflow , zero initial metal content and instantaneous recycling ( searle & sargent 1972 ) . this model has successfully predicted the relationship between metallicity and current gas fraction in magellanic irregular galaxies , although it has less success predicting this relationship in larger spiral systems ( binney & tremaine 1987 ) . we assume a present day gas to total lmc mass ratio of @xmath63 ( cohen _ et al . _ 1988 ) and an effective yield of @xmath64 , chosen so that the present day metallicity matches that inferred from the upper main sequence ( @xmath65 ) . we compare the predicted chemical evolution for two star formation histories in figure 10 . in the upper panel of figure 10 , the holtzman _ et al . _ ( 1997 ) star formation history suggests that the metallicity in the lmc has doubled in the past 2 gyr . the vallenari _ et al . _ ( 1996 ) formation history ( bottom panel ) implies a factor of five metallicity increase in the past 2 gyr . the chemical evolution predicted by the holtzman _ et al . _ model is consistent with planetary nebula observations by dopita _ et al . _ ( 1997 ) which suggest the metallicity of the lmc has almost doubled in the last 2 gyr . we also note that a significant metal poor population is predicted by the closed box model , regardless of the details of the star formation history . for an effective yield of @xmath64 , the closed box model predicts 22% of stars have metallicities less than z=0.001 . this fraction is inversely proportional to the assumed yield . simulated lower main sequences suggest a similar fraction of lmc field stars are metal poor . lower main sequence cross sections are shown for two star formation histories in figure 11 . in the left column , vallenari _ et al . _ s model assumes a metallicity range z=0.008 - 0.001 . these distributions are displaced to the red of the observations by as much as a tenth of a magnitude and are significantly narrower . although some redward evolution occurs for low mass stars during their main sequence lifetime , changing the star formation history alone is not enough to explain this color shift . the holtzman _ et al . _ model is better able to fit the lower main sequence , as it includes an old , metal poor component as shown in the right column of figure 11 . in this simulated population , 20% of stars have a metallicity z=0.0004 ( [ fe / h]= -1.7 ) . none of the models considered here , however , include chemical evolution in a fully self - consistent manner . direct measurements of metallicity in the lmc field have been limited to bright stars , but possibly suggest a similar fraction of metal poor stars . olszewski ( 1993 ) spectroscopically determined metallicities for 36 red giant stars in an outer lmc field near ngc 2257 and found that 8 or 9 ( @xmath66 ) of these stars had metallicities below z=0.001 ( [ fe / h]= more metallicity observations are needed to determine the size of a metal poor component in the lmc field ( suntzeff 1997 ) . we present deep wfpc2 observations of three fields in the outer disk lmc . we find no conclusive evidence for variation in the stellar populations between the three fields based on the morphologies of the color - magnitude diagrams , the luminosity functions , and the relative numbers of stars in different evolutionary stages . in apparent contrast to our results , vallenari _ et al . _ ( 1996b ) find significant variations in the star formation history correlated with azimuthal angle in the lmc field . a direct comparison with their results , however , is difficult . the r - ratios of our field 1 agree with those calculated for the nearly overlapping vallenari _ et al . _ ngc 1866 field . field 3 is located reasonably close to field 1 , therefore the similarity of this field to field 1 provides no direct contradiction with the vallenari _ _ results . a direct discrepancy comes from field 2 , which is located @xmath67 from the vallenari _ _ field lmc-61 . we find significant disagreement in the ratio @xmath68 between these two fields . this discrepancy may be due to the small difference in location or to some systematic error in one or both of the samples . a much larger survey is required to determine whether a correlation exits between the star formation history and position in the lmc field ( zaritsky , harris & thompson 1997 ) . other evidence that the star formation history varies within the lmc comes from elson _ et al . _ ( 1997 ) , who present evidence that the stellar population in the lmc bar is different from those presented in this paper . they analyze _ hst _ observations for a field in the bar of the lmc and identify additional peaks in the color distribution between @xmath69 not associated with the red giant branch or the most recent epoch of star formation . they attribute these peaks to a burst of star formation between 1 and 2 gyr depending on the assumed metallicity . we do not find evidence for this population in our observed color distribution . _ associate this population with the formation of the lmc bar . we have compared our observations with stellar models to place constraints on possible star formation histories in the three fields . these constraints are an improvement over previous results as they incorporate both _ hst _ and ground - based data , allowing measurements of the deep main sequence luminosity function , the distribution of stars in the upper main sequence band , and the relative number of bright stars which probe different evolutionary phases . of previously considered star formation histories , the only one which is consistent with all of our observations has a star formation rate which is roughly constant for 10 gyr , then increases by a factor of three for the past 2 gyr . contrary to many previous models , this produces a population which is _ not _ dominated by young stars . although the star formation history of the lmc is clearly more complicated , this simple picture should provide a useful guide to understanding the formation of our nearest neighbor . bertelli , g. , bressan , a. , chiosi , c. , fagotto , f. & nasi , e. 1994 , a&as , 106,275 bertelli , g. , mateo , m. , chiosi , c. , & bressan , a. 1992 , apj , 388 , 400 binney , j. & tremaine , s. 1987 , _ galactic dynamics _ , ( princeton university press : princeton , n.j . ) cohen , r. s. _ et al . _ apj 331 , l95 dopita , m. _ et al . _ 1997 , apj , 474 , 188 elson , r. , gilmore , g. & santiago , b. 1997 , mnras , 289 , 157 gallagher , j. s. , _ et al . _ 1996 , apj , 466 , 732 girardi , l. _ et al . _ 1995 , a&a , 298 , 87 iglesias , c. a. , rogers , f. j. , & wilson , b. g. 1992 , apj , 397 , 717 holtzman , j. a. _ et al . _ 1995a , pasp , 107 , 156 holtzman , j. a. _ et al . _ 1995b , pasp , 107 , 1065 holtzman , j. a. _ et al . _ 1997 , aj , 113 , 656 luck , r. e. & lambert , d. l. 1992 , apjs , 79 , 303 madore , b. & freedman , w. 1997 apjl , in press olszewski , e.w . 1993 , in asp conf . 48 , the globular cluster - galaxy connection , ed . g. smith & j. brodie ( san francisco : asp ) , 351 panagia _ et al . _ 1991 , apj , 380 , l23 . et al . _ 1995 , a&a , 304 , 69 . reid , n. 1991 aj , 102 , 1428 . sandage , a. 1961 , _ the hubble atlas of galaxies _ , ( carnegie institution of washington : washington , d.c . ) sarajedini , a. , lee , y. , lee , d. 1995 apj 450 , 712 schwering , p. g. , & israel , f. p. 1991 , a&a , 246 , 231 searle , l. , & sargent , w. l. w. 1972 , apj , 173 , 25 salaris , m. , chieffi , a. , & straniero , o. 1993 , apj , 414 , 580 stappers , b. j. , _ et al . _ 1997 , pasp , 109 , 292 stryker , l. , 1984 , apjs , 55 , 127 suntzeff , n. 1997 , bull . a. a. s. , 190 , 3501 vallenari , a. , chiosi , c. bertelli , g. , & ortolani , s. 1996a , a&a , 309 , 358 vallenari , a. , chiosi , c. bertelli , g. , aparicio , a. , & ortolani , s. 1996b , a&a , 309 , 367 van den bergh , s. 1991 , apj , 369 , 1 westerlund , b. e. , linde , p. & lynga , g. 1995 , a&a , 298 , 39 zaritsky , d. harris , j. , & thompson , i. 1997 , astro - ph/9709055 field 1 & @xmath72 & @xmath73 & @xmath74 + field 2 & @xmath75 & @xmath76 & @xmath77 + field 3 & @xmath78 & @xmath79 & @xmath80 + combined fields & @xmath81 & @xmath82 & @xmath83 + combined + ground & @xmath84 & @xmath85 & @xmath86 + c c c c c c + + & & & & & + observed values & @xmath84 & @xmath85 & @xmath87 & + constant star & @xmath88 & @xmath89 & @xmath90 & @xmath91 + formation & & & & + bertelli _ et al . _ & @xmath92 & @xmath93 & @xmath94 & @xmath95 + ( 1992 ) & & & & + vallenari _ et al . _ & @xmath96 & @xmath97 & @xmath98 & @xmath99 + ( 1996b ) & & & & + holtzman _ et al . _ & @xmath100 & @xmath101 & @xmath102 & 0.14 + ( 1997 ) @xmath59 & & & & + holtzman _ et al . _ & @xmath103 & @xmath104 & @xmath105 & 0.17 + ( 1997 ) @xmath106 & & & & + lmc cluster & @xmath107 & @xmath108 & @xmath109 & @xmath99 + age distribution & & & & +
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we present _ hst _ photometry for three fields in the outer disk of the lmc extending approximately four magnitudes below the faintest main sequence turnoff .
we can not detect any strongly significant differences in the stellar populations of the three fields based on the morphologies of the color - magnitude diagrams , the luminosity functions , and the relative numbers of stars in different evolutionary stages .
our observations therefore suggest similar star formation histories in these regions , although some variations are certainly allowed .
the fields are located in two regions of the lmc : one is in the north - east field and two are located in the north - west . under the assumption of a common star formation history ,
we combine the three fields with ground - based data at the same location as one of the fields to improve statistics for the brightest stars .
we compare this stellar population with those predicted from several simple star formation histories suggested in the literature , using a combination of the r - method of bertelli _ et al .
_ ( 1992 ) and comparisons with the observed luminosity function . the only model which we consider that is not rejected by the observations is one in which the star formation rate is roughly constant for most of the lmc s history and then increases by a factor of three about 2 gyr ago .
such a model has roughly equal numbers of stars older and younger than 4 gyr , and thus is not dominated by young stars .
this star formation history , combined with a closed box chemical evolution model , is consistent with observations that the metallicity of the lmc has doubled in the past 2 gyr .
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this paper presents a unified mathematical framework for probabilistic inference with statistical models , such as graphical models . our approach is summarized as follows : * ( a ) statistical models are algebraic varieties . * * ( b ) every algebraic variety can be tropicalized . * * ( c ) tropicalized statistical models are fundamental for parametric inference . * by a _ statistical model _ we mean a family of joint probability distributions for a collection of discrete random variables @xmath0 . thesis ( a ) states that many families of interest can be characterized by polynomials in the joint probabilities @xmath1 . the emerging field of algebraic statistics @xcite offers algorithms for this polynomial representation . _ tropicalization _ means replacing the arithmetic operations @xmath2 by the operations @xmath3 . this process captures the essence of what happens when the joint probabilities @xmath4 are replaced by their logarithms . the tropicalization of an algebraic variety is a piecewise - linear set which enjoys many features familiar from algebraic geometry @xcite . in particular , the tropicalization of a statistical model is a piecewise - linear set in the space with logarithmic coordinates @xmath5 . thesis ( c ) states that tropical algebraic geometry of statistical models is fundamental in analyzing the behavior of inference algorithms under the variation of model parameters . by _ inference _ we mean the evaluation of one or more coordinates of a single point on the algebraic variety , in either @xmath2 or @xmath3 arithmetic . this is the standard notion of inference used for graphical models in statistical learning theory @xcite , but it differs from other ( more classical ) notions of inference in mathematical statistics . by _ parametric inference _ we mean the analysis of the dependence of inference on parameters . to give a more concrete discussion of parametric inference it is useful to focus on directed graphical models . a _ directed graphical model _ ( or _ bayesian network _ ) is a finite directed acyclic graph @xmath6 with two kinds of vertices , _ observed variables _ @xmath0 and _ hidden variables _ @xmath7 , where each edge is labeled by a transition matrix whose entries are linear forms in some parameters . the rules of discrete probability express the observed probabilities @xmath8 as polynomials of degree @xmath9 in the parameters , where @xmath9 is the number of edges of @xmath6 . the polynomials parametrize the graphical model as an algebraic variety . the two standard types of inference questions for graphical models are : 1 . the calculation of _ marginal probabilities _ : @xmath10 2 . the calculation of _ maximum a posteriori ( map ) _ log probabilities : @xmath11 where the @xmath12 range over all the possible assignments for the hidden random variables @xmath13 . together , these two primitives can be used to effectively solve a range of other inference problems , including the calculation of conditional probabilities and other quantities of interest . the key to inference in graphical models is the _ sum - product algorithm _ @xcite ( also known as the _ generalized distributive law _ this polynomial - time algorithm is used , both in ordinary arithmetic @xmath2 and in tropical arithmetic @xmath3 , to _ efficiently _ solve problems 1 and 2 . for more background on the sum - product algorithm , and for connections to message passing and the junction tree algorithm see @xcite . although the sum - product algorithm provides efficient solutions to the basic inference problems 1 and 2 , it only applies to one coordinate @xmath8 of one distribution at a time . what we are interested in are the _ parametric _ versions of the inference problems . they can be phrased as follows : 1 . find all parameters for a model which result in the same values for all @xmath14 . 2 . given observations @xmath15 and hidden data @xmath16 , identify all parameters such that @xmath17 is the most likely explanation for the observations @xmath18 . as we will see , the following _ modeling _ questions are fundamentally related to problems 3 and 4 : 1 . which ( parameter independent ) relations on the probabilities @xmath8 does the model imply ? 2 . describe the tropicalization of the variety corresponding to a graphical model . problem 5 asks for the ideal of _ polynomial invariants _ of a statistical model @xcite . invariants have been investigated in phylogenetics @xcite where they can help to identify good trees for aligned dna sequences . the primary goal of our study is to give a practical answer to question 4 for graphical models . our main algorithmic result is an efficient procedure for parametric inference that can be viewed as a polytopal analog of the sum - product algorithm . the efficiency is based on the complexity estimates for newton polytopes which we derive in section 4 . the resulting _ polytope propagation algorithm _ is applied to problems in biological sequence analysis in the companion paper @xcite . the mathematics to be developed in sections 3 and 4 is of independent interest . it also furnishes new tools for parametric inference ( problems 3 and 4 ) and parametric modeling ( problems 5 and 6 ) which are applicable to a wide range of statistical problems . we demonstrate this by analyzing the hidden markov model ( hmm ) and the general markov model on a binary tree , in sections 2 and 5 respectively . a graphical model is an algebraic variety which is presented as the image of a highly structured polynomial map @xmath19 . here @xmath20 is the space whose coordinates are the model parameters @xmath21 and @xmath22 is the space whose coordinates @xmath23 are the joint probabilities for the observed random variables . in applications , the integer @xmath24 is much larger than the integer @xmath25 , in fact ; it is so large that one can only look at one coordinate @xmath26 at a time . each coordinate @xmath27 of the map @xmath28 is a polynomial function in @xmath21 . the efficient evaluation of these functions relies on the sum - product algorithm . here we study the ( parametric ) inference and modeling problems in the familiar context of the _ hidden markov model _ ( hmm ) . a discrete hmm has @xmath29 observed states @xmath30 taking on @xmath31 possible values , and @xmath29 hidden states @xmath32 taking on @xmath33 possible values . the hmm can be characterized by the following conditional independence statements for @xmath34 : @xmath35 we consider the homogeneous model with uniform initial distribution , where all transitions @xmath36 are given by the same @xmath37-matrix @xmath38 and all transitions @xmath39 are given by the same @xmath40-matrix @xmath41 . throughout our discussion we disregard for simplicity the usual probabilistic hypothesis that @xmath42 and @xmath43 are non - negative and all row sums are @xmath44 . [ hmmprop ] the hidden markov model is the image of a map @xmath45 , where @xmath46 and each coordinate of @xmath28 is a bi - homogeneous polynomial of degree @xmath47 in @xmath42 and degree @xmath29 in @xmath43 . problem 3 is to compute the fibers of the map @xmath28 . in statistics , this is called _ parameter identification_. we use the term _ coordinate polynomials _ for the polynomials @xmath48 that are coordinates of the map @xmath28 . our running example in this section is the case @xmath49 with binary random variables @xmath50 . the graph of this model is drawn in figure [ fig : hmm ] . the shaded nodes are the observed random variables . here the parameter space is @xmath51 with coordinates @xmath52 , and it maps to @xmath51 with coordinates @xmath53 . the map @xmath54 is given by @xmath55 the hidden markov model ( i.e. the image of @xmath28 ) is the zero set of the quartic polynomial @xmath56 this polynomial was found by a _ grbner basis _ computation . see the discussion on _ implicitization _ in @xcite . in general , the polynomial functions on @xmath57 which vanish on the image of @xmath28 are the called _ invariants of the model_. they form a prime ideal @xmath58 . in our example , @xmath58 is generated by the quartic polynomial above . problem 5 is to compute generators of the ideal @xmath58 . when @xmath59 and @xmath25 are small , this can be done using grbner bases , and in some cases it is possible to characterize @xmath58 based on the structure of the model ( see , for example , conjecture [ luckydeterminants ] ) , but in general problem 5 is hard and the ideal @xmath58 may remain unknown . here is where tropical geometry comes in . the _ tropicalization _ of our map @xmath28 is the map @xmath60 defined by replacing products by sums and sums by minima in the formula for @xmath28 . in our example @xmath61 , the tropicalization is the piecewise - linear map @xmath62 with @xmath63 this minimum is attained by the most likely hidden data @xmath64 , given the observations @xmath65 and given the parameters @xmath66 and @xmath67 . the sequence @xmath64 is known as the _ viterbi sequence _ in the hmm literature @xcite . it solves problem 2 in the introduction . the key observation , which we discuss in more detail in section 4 , is that the set of parameters @xmath68 which select the viterbi sequence @xmath69 is the normal cone at a vertex of the newton polytope of the polynomial @xmath70 . this polytope is @xmath71-dimensional , it has @xmath72 vertices , and its normal fan represents the solution to problem 4 in the introduction when @xmath73 is fixed . we can also consider an extension of problem 4 where @xmath73 ranges over all possible observations . the solution is given by the newton polytope of the map @xmath28 . in our example , this is a @xmath74-dimensional polytope with @xmath75 vertices , @xmath76 edges , @xmath77 two - faces , @xmath78 three - faces and @xmath79 facets , namely , the minkowski sum of eight copies of the earlier @xmath71-dimensional polytope for @xmath80 . for a concrete numerical example , fix the parameters @xmath81 and @xmath82 . we find : @xmath83 the set of all parameters @xmath84 leading to the same conclusions as @xmath85 is the cone defined by @xmath86 our solution to the parametric inference problem with respect to all observations simultaneously consists of @xmath75 such cones . the _ tropical hmm _ is the union of the images of these cones under the piecewise - linear map @xmath87 . this image is a piecewise - linear set of dimension @xmath88 . the cone which contains the chosen parameters @xmath89 mapped to a @xmath88-dimensional cone in the tropical hmm ( it spans the hyperplane @xmath90 ) but most of the other @xmath91 cones are mapped to lower - dimensional cones by the map @xmath92 . the question how the number @xmath75 grows as the length @xmath29 increases will be addressed in corollary 10 . we have seen that a graphical model is the image of a polynomial map @xmath28 from the space of parameters to the space of joint probability distributions on the observed random variables . furthermore , we have seen that the tropicalization of @xmath28 arises naturally in solving problem 4 . in this section we study the geometry of tropicalization in the more general setting where @xmath93 is an arbitrary polynomial map . in statistical applications , it is usually the case that each coordinate @xmath48 of the map @xmath28 is a polynomial with positive coefficients . if this holds then the polynomial map @xmath28 is called _ positive_. we say that @xmath28 is _ surjectively positive _ if , in addition , @xmath28 maps the positive orthant surjectively onto the positive points in the image , in symbols , @xmath94 the set of all polynomial functions which vanish on the image of @xmath28 is a prime ideal @xmath58 in the polynomial ring @xmath95 $ ] . the closure of the image of @xmath28 is the variety of the prime ideal @xmath58 . in tropical geometry , we replace the variety of @xmath58 by a piecewise - linear set as follows . the _ tropical variety _ @xmath96 is the set of all weight vectors @xmath97 such that the initial ideal @xmath98 contains no monomial @xcite . following @xcite , we define the _ positive tropical variety _ @xmath99 as the set of all weight vectors @xmath97 such that the initial ideal @xmath98 contains no polynomial with only positive coefficients . the tropical variety @xmath96 is a _ polyhedral fan _ in @xmath22 , and @xmath99 is a _ polyhedral subcomplex _ of @xmath96 . this means that @xmath96 is a finite union of closed convex polyhedral cones that fit together nicely , and @xmath99 is the union of a subset of these cones . the _ tropicalization _ of the polynomial map @xmath28 is the piecewise - linear map @xmath100 defined by replacing products by sums and sums by minima in the evaluation of @xmath28 . we say that @xmath92 is a _ tropical morphism_. examples of tropical morphisms appear in the displayed formulas ( [ maxformula ] ) , ( [ formula1 ] ) , ( [ formula2 ] ) , ( [ formula3 ] ) , ( [ tropicaltreemap ] ) and ( [ treemapexample3 ] ) . the following theorem describes the geometry of this situation . we define the _ newton polytope _ of a polynomial map @xmath101 as the minkowski sum in @xmath102 of the newton polytopes of its coordinates @xmath103 . for basics on newton polytopes and their normal fans see @xcite . [ hmmmain ] the tropical morphism @xmath92 is linear on each cone in the normal fan of the newton polytope of @xmath28 . its image is a fan contained in @xmath96 . if @xmath28 is positive then @xmath104 is a subset of @xmath99 , but it is generally not a polyhedral subcomplex . if @xmath28 is surjectively positive then @xmath105 . let @xmath106 denote the newton polytope of the polynomial @xmath107 . by definition , @xmath106 is the convex hull in @xmath102 of all non - negative lattice points @xmath108 such that the monomial @xmath109 appears with non - zero coefficient in @xmath110 . the piecewise - linear concave function @xmath111 is the _ support function _ of the polytope @xmath106 . this means that @xmath112 is the minimum value attained on @xmath106 by the linear functional @xmath113 . in particular , the function @xmath114 is linear on each cone in the normal fan of @xmath106 . the newton polytope of the map @xmath28 is the minkowski sum @xmath115 . the normal fan of @xmath116 is the common refinement of the normal fans of @xmath117 . this shows that the function @xmath118 is linear on each cone of the normal fan of the newton polytope of @xmath28 . since @xmath92 is continuous , the image of @xmath92 is a closed polyhedral fan in @xmath22 . consider any vector @xmath119 . we must show that @xmath120 lies in @xmath96 , and if @xmath28 is positive then @xmath120 lies in @xmath99 . let @xmath121 be any polynomial in the ideal @xmath58 . if we substitute @xmath122 into @xmath123 then we get zero . consequently , if we substitute the initial forms @xmath124 into the initial form @xmath125 then the result is zero . see equation ( 11.2 ) on page 100 in @xcite . this implies that @xmath125 is not a monomial . moreover , if @xmath28 is positive then @xmath121 must have two terms whose coefficients have opposite signs . this implies the desired conclusion . the following example shows that @xmath126 need not be a subcomplex of @xmath99 . if @xmath28 is assumed to be surjectively positive , then it follows from ( * ? ? ? * proposition 2.5 ) that @xmath127 . let @xmath128 , @xmath129 and consider the linear map @xmath130 then @xmath131 is the principal ideal generated by the linear form @xmath132 , and @xmath96 is essentially the normal fan of a tetrahedron . we identify @xmath96 with the complete graph @xmath133 . the six edges of @xmath133 are labeled with six monomial - free initial ideals of @xmath58 , namely , @xmath134 the first two of these six initial ideals contain a polynomial with positive coefficients . hence the positive tropical variety @xmath99 is the four - cycle in @xmath133 formed by the remaining four edges . the tropicalization of the linear map @xmath28 is the tropical morphism @xmath135 the image of @xmath92 is the set of all vectors @xmath136 with @xmath137 . each vector @xmath136 with @xmath138 has the initial ideal @xmath139 , so it lies on a particular edge of @xmath133 . but the same edge also accounts for all vectors @xmath136 with @xmath140 , none of which is in the image of @xmath92 . thus @xmath141 is a closed segment which covers only half of the edge of @xmath133 indexed by @xmath142 . here it is easy to replace @xmath28 by a parameterization @xmath143 which is surjectively positive , for instance , @xmath144 @xmath145 we have @xmath146 but now the tropical morphism @xmath147 maps onto the entire four - cycle @xmath99 . in the rest of this section we examine theorem [ hmmmain ] for a small but important graphical model , namely , the _ naive bayes model with two features _ @xcite . there are two observed random variables @xmath148 and @xmath149 dependent on one hidden binary random variable @xmath150 . the two observed variables take @xmath33 and @xmath31 possible values respectively . the parameterization @xmath28 of this model is the map @xmath151 given by @xmath152 thus the model consists of all @xmath40-matrices @xmath153 of the form @xmath154 where @xmath155 is a @xmath156-matrix and @xmath157 is a @xmath158-matrix , i.e. , the model consists of precisely the @xmath40-matrices of rank @xmath159 . [ threebythree ] the parameterization @xmath28 of the naive bayes model with two features is surjectively positive . the ideal @xmath58 is generated by the @xmath160-subdeterminants of the @xmath40-matrix @xmath161 . the map @xmath28 being positive means that if @xmath162 is any positive matrix of rank @xmath163 then @xmath42 and @xmath43 can be chosen to be positive . this is a known result in linear algebra ( see e.g. @xcite ) . the same statement is false for rank @xmath164 , i.e. , the parameterization of the naive bayes model with three or more features is not surjectively positive . a well - known result in commutative algebra states that the @xmath165-minors of a @xmath40-matrix generate a prime ideal . the variety of this ideal is the set of @xmath40-matrices of rank @xmath166 . this our ideal @xmath58 for @xmath167 . the objects of theorem [ hmmmain ] have been studied in @xcite and @xcite . the tropical variety @xmath96 is the set of @xmath40-matrices of _ tropical rank _ @xmath159 , and the tropical variety @xmath168 is the set of @xmath40-matrices of _ barvinok rank _ @xmath159 . develin @xcite determines the combinatorics and topology of these spaces when @xmath169 . he shows that @xmath170 is shellable but @xmath171 can have torsion in its integral homology groups . the newton polytope of the map @xmath28 is an interesting combinatorial object , namely , it is the @xmath172-dimensional zonotope associated with the complete bipartite graph @xmath173 . the newton polytope of each coordinate @xmath174 is a line segment , and the zonotope is their minkowski sum . the normal fan is the hyperplane arrangement @xmath175 . its maximal cones correspond to the acyclic orientations of the complete bipartite graph @xmath173 . west @xcite showed that the number of facets of such a cone can be any integer between @xmath176 and @xmath177 . the total number of cones equals @xmath178 , where @xmath179 is the stirling number of the second kind . here , the tropical morphism @xmath92 is given by @xmath180 the map @xmath181 is piecewise - linear with respect to the hyperplane arrangement . recent work of federico ardilla ( in preparation ) gives a complete classification of all fibers of @xmath92 . -0.3 cm -determinant . ] -1.2 cm let @xmath182 , so the two observed random variables are ternary . the prime ideal is @xmath183 the tropical variety @xmath96 is the fan over a two - dimensional polyhedral complex consisting of six triangles and nine quadrangles . this complex is the @xmath163-skeleton of the product of two triangles , labeled as in figure 2a . this complex is shellable . the positive tropical variety @xmath99 is the subcomplex consisting of the nine quadrangles shown in figure 2b . note that @xmath99 is a torus . the newton polytope of @xmath28 is a five - dimensional zonotope with @xmath184 vertices , one for each acyclic orientation of the complete bipartite graph @xmath185 . the map @xmath92 is linear on each of the @xmath184 cones in the corresponding hyperplane arrangement , but it is rank - deficient on @xmath186 of the cones . the remaining @xmath187 cones are mapped onto the @xmath188 quadrangles of the torus @xmath99 . thus the general fiber of @xmath92 involves @xmath189 cones . of these , eight cones have @xmath74 facets , eight cones have @xmath190 facets , and two cones have @xmath188 facets . consider a graphical model with @xmath9 edges and @xmath29 observed random variables @xmath30 each taking @xmath31 values . such a model is given by a positive polynomial map @xmath191 . each coordinate @xmath192 of @xmath28 is a polynomial of degree @xmath193 in the model parameters @xmath21 . in this section we discuss the statistical meaning and the computational complexity of the mathematical objects introduced in the previous section . we write @xmath194 for the negative logarithms of the model parameters . consider any of the @xmath59 possible observations @xmath18 . the quantity @xmath195 is the probability of making this particular observation , i.e. it is @xmath196 . the quantity @xmath197 is the negative logarithm of the conditional probability @xmath198 where @xmath199 maximizes @xmath200 for the parameters @xmath201 . clearly , the function @xmath202 is piecewise - linear and concave on the logarithmic parameter space . the domains of linearity of the function @xmath203 are the cones in the normal fan of the newton polytope of @xmath192 . each maximal cone @xmath204 is indexed by the hidden data @xmath199 that maximizes @xmath205 for any of the parameters @xmath206 . the hidden data @xmath199 which arise in this manner , for some choice of logarithmic parameters @xmath207 , are called the possible _ explanations _ of the observation @xmath18 . for instance , for the hidden markov model of section 2 , the explanations are the viterbi sequences . let us now vary the observations . each logarithmic parameter vector @xmath208 defines an _ inference function _ @xmath209 from the set of observations to the set of explanations . for the hmm , each inference function @xmath210 takes an observed sequence @xmath211 to the corresponding viterbi sequence @xmath199 . there are @xmath212 such functions , but most of these are * not * inference functions . for instance , consider the binary hmm of length three . there are @xmath213 boolean functions @xmath214 , but , as we have seen at the end of section 2 , only @xmath75 of these are inference functions for the hmm . the inference functions @xmath209 of a graphical model @xmath28 are in bijection with the vertices of the newton polytope of the map @xmath28 . the explanations @xmath215 for a fixed observation @xmath18 in a graphical model are in bijection with the vertices of the newton polytope of the polynomial @xmath192 . in applications of graphical models , the number @xmath25 of parameters and the number @xmath31 of values of the observed random variables is small and fixed , but the number @xmath29 of observed random variables is large . recall that the model is the image of the map @xmath191 . hence the dimension of the model remains fixed but the dimension of its ambient space grows exponentially in @xmath29 . it is therefore algorithmically infeasible to compute the full tropical variety @xmath96 . what we can do efficiently , however , is to compute the newton polytopes of the @xmath192 , or even the newton polytope of @xmath28 . this allows us to glean information about the tropical variety from the domains of linearity of its `` coordinate functions '' @xmath203 . our next goal is to derive an upper bound on the number of vertices of the newton polytopes . [ smallpolytopebound ] consider graphical models @xmath28 whose number of parameters @xmath25 is fixed and whose number @xmath29 of observed random variables and number of edges @xmath9 varies . ( typically , @xmath9 is a linear function of @xmath29 ) . then the number of vertices of the newton polytope @xmath216 of @xmath48 is bounded above by @xmath217 for many important families of graphical models , the number @xmath9 of edges is bounded by a linear function in terms of the number @xmath29 of observed nodes , and in those cases we can replace @xmath9 by @xmath29 . hence , for any given observation @xmath211 , the number of explanations grows polynomially in @xmath29 . for instance , in the hidden markov model of section 2 we have @xmath218 , and a similar relationship holds in the tree model of section 5 . [ hmmcomplexity ] for any fixed observation in the homogeneous hmm , the number of explanations is at most @xmath219 . if all random variables are binary then the upper bound @xmath220 holds . the proof of theorem [ smallpolytopebound ] and corollary [ hmmcomplexity ] are derived from the following classical result on lattice polytopes due to andrews @xcite . the necessary observation is that the newton polytope of @xmath48 is contained in the cube @xmath221^d$ ] , and the volume of this cube equals @xmath222 . ( andrews @xcite ) for every fixed integer @xmath25 there exists a constant @xmath223 such that the number of vertices of any lattice polytope @xmath162 in @xmath102 is bounded above by @xmath224 . the newton polytope of the map @xmath28 was defined as the minkowski sum of the @xmath59 smaller newton polytopes in theorem [ smallpolytopebound ] . from this we infer the following naive bound on its number of vertices . the number of inference functions of a graphical model is at most @xmath225 , hence this number scales at most singly exponentially in the complexity @xmath226 of the graphical model . consider the homogeneous hmm on binary random variables . each inference function is a boolean function @xmath227 , but not conversely . the number of all boolean functions is @xmath228 , which grows doubly exponentially in @xmath29 . however , the number of inference functions is at most @xmath229 . in practical applications of graphical models , it may be infeasible to compute all ( singly - exponentially many ) inference functions . nonetheless , we believe that important insight can be gained by computing and classifying the newton polytopes of graphical models @xmath28 on few random variables . such a study would be the polyhedral analogue to the algebraic classification of @xcite . on the other hand , for a fixed observation @xmath211 , the size of the newton polytope of @xmath192 grows polynomially with the size of the graphical model , and therefore there is hope that the polytopes can be computed efficiently . despite the fact that the newton polytope of @xmath230 has polynomially many vertices in the size of the graphical model , the number of terms in @xmath230 grows exponentially . this is a potential problem because the computation of the newton polytope requires inspecting these terms . the following result states that , in fact , the convex hull computations scales with the running time of the sum - product algorithm , which for many models of interest scales polynomially with the size of the graphical model . [ polyprop ] the newton polytopes of the polynomials @xmath48 can be computed recursively using the decomposition of @xmath48 according to the sum - product algorithm . taken together , theorem 7 and proposition 11 say that * polytope propagation is an efficient algorithm for parametric inference with graphical models*. this statement is thesis ( c ) in our companion paper @xcite . in that paper , the sum - product algorithm and the polytope propagation algorithm are explained and analyzed in more detail . we also demonstrate the practicality of our mathematical theory by explicitly computing ( and statistically interpreting ) various high - dimensional newton polytopes for graphical models that arise in biological sequence analysis . we conclude by illustrating the concepts we have developed in the context of tree markov models . these are directed graphical models where the graph is a directed tree @xmath231 with observed random variables @xmath30 at the leaves . the naive bayes model in section 3 is the special case where @xmath232 . each edge @xmath193 has a different transition matrix @xmath233 $ ] . we consider the general model in allman and rhodes @xcite , which means that the @xmath234 are arbitrary distinct @xmath235-matrices . in most applications , the transition matrices are from a special model family ( e.g. in phylogenetics these may be jukes - cantor model or the hasegawa - kishino - yano model ) . as before , we relax the hypothesis that transition probabilities are non - negative and sum to @xmath44 . hence the @xmath236 are distinct unknowns . for simplicity we shall further assume that the tree @xmath231 is binary . [ treeobservation ] the general markov model for the binary tree @xmath231 is the image of a map @xmath237 , where each coordinate of @xmath28 is a multilinear polynomial in the unknowns @xmath238 , @xmath193 edge of @xmath239 . if we denote an edge between nodes @xmath240 and @xmath241 by @xmath242 and @xmath243 is the tree @xmath231 without the leaves , then the coordinate of the multilinear map @xmath28 indexed by an observed sequence @xmath244 can be written as follows : @xmath245 here @xmath246 ranges over all colorations @xmath247 of the nodes such that @xmath248 for all leaves @xmath241 . our running example in this section is the binary tree in figure [ fig : tree ] with binary random variables @xmath249 . leaves . ] -0.4 cm in this example , the coordinates of the multilinear map @xmath250 are given by the formula @xmath251 the prime ideal @xmath58 of polynomial invariants is generated by the @xmath160-subdeterminants of the matrix @xmath252 thus this particular model is the @xmath253 instance of the determinantal variety in proposition [ threebythree ] . we generalize the determinantal presentation in this example by proposing the following explicit solution to problem 5 for arbitrary binary trees @xmath231 . every edge of @xmath231 induces a _ split _ of the set of leaves @xmath254 , corresponding to the two connected components of the tree obtained by removing that edge . the unrooted tree underlying @xmath231 is uniquely determined by the set of these splits . [ luckydeterminants ] the ideal @xmath58 of phylogenetic invariants of the general markov model for any binary tree @xmath231 on binary random variables is generated by the @xmath160-determinants of all two - dimensional matrices obtained by _ flattening _ the @xmath255-table @xmath256 according to the splits induced by the edges of @xmath231 . we need to explain the meaning of the word `` flattening '' . if @xmath257 is any split of the set @xmath258 then this refers to the @xmath259-matrix whose rows and columns are indexed by functions @xmath260 and @xmath261 respectively , and whose entries are the @xmath262 probabilities @xmath8 . in december 2003 , allman and rhodes announced a proof of the set - theoretic version of our conjecture [ luckydeterminants ] . what this means algebraically is that @xmath58 equals the radical of the ideal generated by the aforementioned @xmath160-determinants . in light of this progress , we wish to offer also the following tropical version of conjecture [ luckydeterminants ] . it would be very nice to show that proposition [ threebythree ] extends to this situation . however , none of the remaining discussion in this section depends on these conjectures . [ luckytropical ] the map @xmath28 is surjectively positive for @xmath263 . the tropical variety ( resp . the positive tropical variety ) of the prime ideal @xmath58 coincides with the set of all @xmath264-tables @xmath265 whose flattenings along the splits of the tree @xmath231 have tropical rank ( resp . barvinok rank ) at most @xmath163 . the sum - product algorithm is used in practice to evaluate the polynomial ( [ treemap ] ) . its running time is linear in @xmath29 , despite the fact that the number @xmath266 of terms in ( [ treemap ] ) grows exponentially . this reduction in complexity is achieved by recursively grouping subsums . for instance , ( [ treemapexample ] ) becomes @xmath267 the rule to remember is this : polynomials are evaluated recursively as sums of products of smaller polynomials . this is the solution to problem 1 . for details on the tree case see @xcite . problem 2 is known in phylogeny as the _ joint ancestral reconstruction _ problem , which asks for the maximum a posteriori ancestral assignments @xmath268 given the observations @xmath269 at the leaves . an efficient method for solving this problem appears in @xcite . this method is nothing but the sum - product algorithm with ordinary arithmetic @xmath270 replaced by tropical arithmetic @xmath271 . the @xmath211-coordinate of the tropicalization @xmath272 of the map ( [ treemap ] ) is @xmath273 this expression can be evaluated efficiently by the same scheme as before . the rule now is this : piecewise - linear concave functions are evaluated recursively as minima of sums of smaller such functions . a simple example illustrating this rule is the tropicalization of ( [ treemapexample2 ] ) : @xmath274 where @xmath275 and similarly for @xmath276 . we saw in section 4 that the number of vertices of the newton polytopes of the coordinate polynomials @xmath48 is critical for efficient parametric inference . that number grows polynomially in @xmath29 if the number of parameters is fixed ( thanks to theorem [ smallpolytopebound ] ) but it may grow exponentially if the number of parameters is not bounded . for the general markov model on a tree @xmath231 , the growth will be exponential unless we restrict the number of parameters . this can be done , for instance , by considering the _ homogeneous tree model _ where the transition matrices along all edges are identical : @xmath277 using theorem [ smallpolytopebound ] , we obtain the following result analogous to corollary [ hmmcomplexity ] . the number of vertices of the newton polytope of any coordinate @xmath48 in the homogeneous tree model is bounded above by @xmath278 times a constant depending only on @xmath31 . for tree models which are used in applications , such as phylogenetics , the number of parameters is likely to be reduced even further . in such cases , the parametric joint ancestral reconstruction problem can be solved efficiently using the polytope propagation algorithm techniques in proposition [ polyprop ] . the algebraic representation for graphical models with hidden variables leads naturally to an interpretation of a parameterized model as a point on an algebraic variety . marginal probabilities are coordinates of points on the variety . varieties can be tropicalized , and the statistical meaning is that the map probabilities ( calculated with logarithms of the parameters ) can be interpreted as coordinates of points on the positive part of the tropical variety . hence , the tropical model is fundamental for understanding map probabilities . although we have not addressed it in this paper , the logarithms of the marginal probabilities are coordinates of points on the _ amoeba _ @xcite of the model . amoebas are likely to be important for understanding the geometry of maximum likelihood estimation . the sum - product algorithm for graphical models is an efficient method for evaluating the coordinate polynomials of a graphical model . this algorithm works in exactly the same way for classical arithmetic @xmath270 and for tropical arithmetic @xmath279 . this means that the same method is used to evaluate coordinates of points on the variety and of points on the tropical variety . an explanation for an observation @xmath18 is a vertex of the newton polytope of @xmath192 . thus , the parametric inference problem is solved by finding the normal fans of the newton polytopes of the coordinate polynomials . for many important applications , the number of vertices of the polytopes is polynomial in the size of the graphical model . the polytope propagation algorithm , which is a geometric analog of the sum - product algorithm , finds the newton polytopes , and is efficient when the sum - product algorithm is fast and the number of vertices on the newton polytopes is small . an inference function for a graphical model is a function from the set of observations to the set of explanations which maximizes the a posteriori probabilities with respect to some choice of parameters . inference functions correspond to vertices of the newton polytope of the map @xmath28 . this polytope is much larger than the newton polytope of a single coordinate @xmath48 , so it can only be computed for small graphical models , but it has the advantage that it encodes the entire piecewise - linear geometry of the model . in a companion paper @xcite , we show that polytope propagation is practical and useful in the important application of biological sequence analysis . in particular , existing parametric alignment methods @xcite can be viewed as special cases of parametric inference for pair hidden markov models . the computation of the newton polytopes is also useful for bayesian computations , where we have priors on the parameters and it is of interest to integrate over the maximal cones in the normal fan of the newton polytope @xcite . lior pachter was supported in part by a grant from the nih ( r01-hg02362 - 02 ) . bernd sturmfels was supported by a hewlett packard visiting research professorship 2003/2004 at msri berkeley and in part by the nsf ( dms-0200729 ) . we are grateful to komei fukuda , michael joswig and kristian ranestad for their help in obtaining the computational results reported in section 2 . 26 g. andrews : a lower bound for the volume of strictly convex bodies with many boundary points , trans . 106 ( 1963 ) 270273 . e. allman and j. rhodes : phylogenetic invariants for the general markov model of sequence mutation , mathematical biosciences , 186 ( 2003 ) 113144 . s. aji and r. j. mceliece : the generalized distributive law , ieee transactions on information theory 46 ( 2000 ) 325343 . p. baldi and s. brunak : bioinformatics . the machine learning approach . a bradford book . the mit press . cambridge , massachusetts 1998 . j. cavender and j. felsenstein : invariants of phylogenies in a simple case with discrete states , journal of classification 4 ( 1987 ) 5771 . j. cohen and u. rothblum : nonnegative ranks , decompositions , and factorizations of nonnegative matrices , linear algebra appl . 190 ( 1993 ) david cox , donal oshea and john little : ideals , varieties and algorithms , springer undergraduate texts in mathematics , 1996 . m. develin , f. santos and b. sturmfels : on the tropical rank of a matrix , preprint , http://front.math.ucdavis.edu/math.co/0312114 . m. develin : the space of @xmath24 points on a tree with @xmath29 leaves , preprint . . r. durbin , s. eddy , a. krogh and g. mitchison : biological sequence analysis ( probabilistic models of proteins and nucleic acids ) , cambridge university press , 1998 . d. fernndez - baca , t. sepplinen and g. slutzki : parametric multiple sequence alignment and phylogeny construction , in combinatorial pattern matching , lecture notes in computer science ( r. giancarlo and d. sankoff eds . ) , vol . 1848 , 2000 , 6882 . garcia , m. stillman and b. sturmfels : algebraic geometry of bayesian networks , journal of symbolic computation , to appear , http://front.math.ucdavis.edu/math.ag/0301255 . d. gusfield , k. balasubramanian , and d. naor : parametric optimization of sequence alignment , algorithmica 12 ( 1994 ) 312326 . f. kschischang , b. frey , and h. a. loeliger : factor graphs and the sum - product algorithm , ieee trans . theory 47 , feb 2001 , 498519 . jordan and y. weiss : graphical models : probabilistic inference , in _ handbook of brain theory and neural networks , 2nd edition _ , m. arbib ( ed . ) , cambridge , ma , mit press , 2002 . t. pupko , i. peer , r. shamir , and d. graur : a fast algorithm for joint reconstruction of ancestral amino acid sequences , molecular biology and evolution 17 ( 2000 ) 890896 . j. richter - gebert , b. sturmfels , and t. theobald : first steps in tropical geometry , in idempotent mathematics and mathematical physics ( eds . litvinov and v.p . maslov ) , american mathematical society , 2004 , preprint posted at http://front.math.ucdavis.edu/math.ag/0306366 . l. pachter and b. sturmfels : parametric inference for biological sequence analysis , companion paper , submitted . g. pistone , e. riccomagno , and h.p . wynn : algebraic statistics : computational commutative algebra in statistics , chapman and hall , boca raton , florida , 2001 . l. r. rabiner : a tutorial in hidden markov models and selected applications in speech recognition , proc . of the ieee , 77 ( 1989 ) 257286 . d. speyer and l. williams : the tropical totally positive grassmannian , preprint , http://front.math.ucdavis.edu/math.co/0312297 . b. sturmfels : grbner bases and convex polytopes , university lecture series , vol . 8 , american mathematical society , 1996 . o. viro : what is an amoeba ? notices of the american math . society 49 ( 2002 ) 916917 . m. waterman , m. eggert and e. lander : parametric sequence comparisons , proc . natl . acad . usa 89 ( 1992 ) 60906093 . d. west : acyclic orientations of complete bipartite graphs , discrete math . 138 ( 1995 ) 393396 .
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this paper presents a unified mathematical framework for inference in graphical models , building on the observation that graphical models are algebraic varieties . from this geometric viewpoint ,
observations generated from a model are coordinates of a point in the variety , and the sum - product algorithm is an efficient tool for evaluating specific coordinates .
the question addressed here is how the solutions to various inference problems depend on the model parameters .
the proposed answer is expressed in terms of tropical algebraic geometry .
a key role is played by the newton polytope of a statistical model .
our results are applied to the hidden markov model and to the general markov model on a binary tree .
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shortly after the big bang , large density fluctuations in the early universe may have resulted in formation of primordial black holes ( pbhs ) @xcite . there is a wide range of allowed masses for pbhs . depending on the epoch and conditions during formation , pbh masses can be anywhere from approximately a gram to a million solar masses . while low mass pbhs would have already evaporated through hawking radiation @xcite , large ones with masses @xmath4 g would still be present today . it is also possible to have prolonged pbh formation during a non - radiation - dominated phase of the universe where pbhs can form with a continuum mass distribution , rather than mostly at one particular mass scale as in the conventional radiation dominated case @xcite . stable pbhs can be cosmologically - significant , and may serve as an ideal dark matter candidate @xcite . depending on the pbh abundance , hawking radiation from pbhs with lifetime longer than the age of the universe may be observable . extragalactic gamma rays strongly constrain pbhs in the mass range @xmath5 g @xcite . pbhs in the mass range @xmath6 g are bounded by femtolensing of gamma - ray bursts @xcite , and pbhs in the mass range @xmath7 m@xmath8 are constrained by gravitational microlensing @xcite . in addition , pbhs with mass @xmath9 m@xmath8 may be constrained by the accretion of matter in the early universe @xcite . black holes at approximately this mass are constrained by x - rays observations @xcite . for recent comprehensive reviews on astrophysical constraints on pbhs see refs . @xcite . in this paper we present a new bound on pbhs in the mass range @xmath5 g using the most recent planck cosmic microwave background ( cmb ) data @xcite . the cmb is sensitive to additional sources of energy injection during the recombination epoch , which leads to damping of the anisotropies . for pbhs , this energy injection is due to hawking radiation . as we show , the planck data now place a stronger bound on pbhs over a larger fraction of this mass range than previous most stringent bounds derived from the extragalactic gamma - ray background ( egb ) . previous authors have used planck data to bound pbhs in the mass regime that we study @xcite ; as discussed below we precisely identify the mass regime over which the cmb and egb bounds are dominant . in addition we note that our analysis is distinct from previous studies that used early - time distortions of the cmb to bound pbhs in the mass range @xmath10 g @xcite . the theoretical formalism that we utilize to constrain pbhs is similar to that used to constrain dark matter annihilation or decay @xcite . from the perspective of the cmb , pbh evaporation is most similar to dark matter decay in that the energy injection only depends on the pbh mass and abundance , and is at a steady rate to the present time . this energy injection can have a significant impact on ionization at low redshift . however unlike the energy from dark matter , which can be injected in the form of heavy standard model particles , pbhs with mass @xmath11 g mostly radiate in electrons and photons , or other ( near ) massless species , but generally not into much heavier particle species . because pbhs with mass @xmath12 g are too cold to emit electrons , their injection into the cmb is unobservable , and the bounds are weak above this mass . this paper is organized as follows . in section [ blackholes ] we briefly discuss the injection of radiation from pbhs . in section [ igm_interactions ] we discuss the impact of this injected energy on the igm . in section [ recomb_hist ] , we show the modifications on the ionization history , and in section [ cmb_alter ] we discuss cmb distortions . in section [ black_hole_constr ] we show the resulting constraints on pbhs , and section [ conclusions ] presents the discussion and conclusions . depending on their mass , pbhs radiate a spectrum of particles , which decay via cascades into photons , electrons , protons , and neutrinos . these particles then deposit energy into the igm . the injection of energy is described by the equation @xmath13 where @xmath14 is the pbh number density , @xmath15 is the critical density of the universe today , @xmath16 is the pbh mass , @xmath17 is the pbh density observed today relative to the critical density . note that the above equations assume that pbhs are comprised of a single mass and the mass does not changes as it radiates . this is satisfied as long as the lifetime is large compared to the age of the universe and is satisfied by the masses considered in this work . apart from some cosmetic differences , equation ( [ equ : bh - energy ] ) is identical to that for decaying dark matter @xcite . in order to evaluate equation ( [ equ : bh - energy ] ) , an expression for @xmath18 is required . to obtain this , we start from the fact that hawking radiation equates the radiation from a black hole to the blackbody radiation of an object with temperature @xmath19 and with an emission spectrum @xmath20 where @xmath21 is the spin of the radiated particle and @xmath22 is the absorption coefficient for the particle . for low @xmath23 the absorption coefficient can deviate greatly from the geometric optic limit @xcite , @xmath24 for standard model particles , the average radiated energy in the massless limit is @xcite @xmath25 for pbhs with masses in the range @xmath26 grams , the fraction of emitted particles of different spins is @xcite @xmath27 it should be noted that these fractions are not normalized to unity , but rather to a @xmath28 g pbh . with this information the energy injection in equation ( [ equ : bh - energy ] ) can be evaluated after using @xcite @xmath29 since only the electrons and photons interact electromagnetically for the pbh masses considered , these are the only fractions that need to be considered for calculating the energy output . the energy deposited into the igm by pbhs is absorbed through multiple channels . following previous studies , here we consider three channels for the igm interaction : hydrogen ionization , lyman - alpha excitations , and heating the igm @xcite . these effects alter the cosmological recombination equations as @xmath30 where @xmath31 is the helium fraction , @xmath32 is the ionization fraction , @xmath33 is the boltzmann constant , and @xmath34 is the hubble parameter . the standard equations without additional energy injection from pbhs are denoted by the subscript orig " and are derived in e.g. ref . the quantities @xmath35 , @xmath36 , and @xmath37 are factors corresponding to the additional energy injection affecting ionization from the ground state , ionization from excited states , and heating the igm . each of these injections are dependent on the injection energy through @xmath38 here @xmath39 is the hydrogen number density , and @xmath40 and @xmath41 are the energies of the ground and the excited hydrogen atom electron levels respectfully . the quantity @xmath42 is related to the probability for an excited hydrogen atom to emit a photon prior to being ionized @xcite . the quantities @xmath43 , @xmath44 , @xmath45 are efficiencies for energy interactions through each channel . commonly referred to as effective efficiencies , they are redshift , energy , and species dependent quantities that equate the total energy injection to the actual amount absorbed through a pathway @xcite . previously , these efficiencies have been approximated by simple @xmath32 dependent equations , with the energy injection taken to be instantaneous , through a technique known as the `` ssck '' method , described in further detail in ref . @xcite . to calculate the effective efficiencies , we follow the approach adopted in ref . these efficiencies have been tabulated for electron and photon particle injection into the igm at various redshifts and particle energies , and divided into five different channels : hydrogen ionization , helium ionization , lyman - alpha excitations , heating , and continuum photons ( energy lost as photons with @xmath46 ) @xcite . the hydrogen , lyman - alpha , and heating efficiencies were used to calculate the various efficiencies in equations ( [ equ : i_xi ] ) , ( [ equ : i_xalpha ] ) , and ( [ equ : k_h ] ) . we do not consider helium ionization because it is subdominant . we additionally note that we do not consider energy deposited into continuum photons . this is because continuum photon energy affects the cmb mostly through spectral distortions rather than through anisotropies , and the anisotropies are the focus of this paper . the impact on spectral distortions is also less significant than on anisotropies @xcite ; a basic demonstration of this is given in the appendix . refs . @xcite tabulate efficiencies specifically for decaying and annihilating dark matter into electron and photon channels . these efficiencies are specifically valid for a small ionization fraction , in which case the efficiency scales linearly with ionization fraction . for dark matter decay , the energy injection is linearly proportional to the density , and for annihilation it is proportional to the density - squared . from equation [ equ : bh - energy ] , the radiation from pbhs is linearly proportional to density , therefore this is most analogous to dark matter decay . for this reason , we utilize the decay efficiencies to produce black hole efficiencies . to obtain the pbh efficiency , we assume that pbh injected particles have the same efficiency as those with the average energy for its species . combining the different species efficiencies through a weighted average based upon their emission fractions gives @xmath47 using the decay efficiencies from refs . @xcite , figure [ fig : effective_efficiency ] shows the effective efficiencies for pbh hawking radiation for the mass range considered in this work . it should be noted that there appears a location where the efficiency drops drastically for all channels used in altering the ionization history , and this location shifts to later times for increasing temperatures . this drop will have a direct impact on the constraints for a given pbh mass . [ cols="^,^ " , ] since there is little variation in the base cosmological parameters , for computational convenience to set upper limits on @xmath17 we take the six principle cosmological parameters to be fixed at their best fit values in the case of no additional energy injection @xcite . figure [ fig : egb_comparison ] shows the result of the 95% confidence limit , where the confidence limit is defined as the cumulative distribution centered around the median , which corresponds closely to the peak of the distributions in figure [ fig : egb_comparison ] . the constraint follows the expected inverse cube relationship to the pbh mass which is predicted by the energy injection formula . at the 95% confidence level ( shaded region ) compared with the same exclusion bound enforced by egb , assuming 100% of the background produced by pbhs ( long dashes ) . also included is an estimation for the bound that is imposed due to femtolensing ( short dashes ) considered in @xcite . ] in addition to the cubic dependence on mass , there is also a highly nonlinear relationship to the effective efficiency . this nonlinearity is most prevalent at a pbh mass around @xmath48 g. comparing effective efficiencies , in figure [ fig : effective_efficiency ] , the trend is correlated with the efficiency values that occur near the time of recombination . as the efficiency value decreases , it is required for a larger amount of total energy to be created in order to produce the same effect . for this reason , as the efficiency experiences a large decrease , the allowable maximum mass fraction increases . we note that since the six base cosmological parameters were fixed , the pbh abundance may be more strongly constrained than in a model in which more parameters are allowed to vary . to check this , we compare to the case in which the base cosmological parameters are allowed to vary for a single pbh mass of @xmath49 g. we find that by freeing all of the cosmological parameters , the constraint on @xmath17 may be weakened by up to a factor of three . however , as stated above for computational convenience we have decided to fix the base cosmological parameters for our main bounds on @xmath50 . as indicated above previous analyses have used the egb to constrain pbhs in the mass regime @xmath0 g @xcite . since pbhs emit a mass - dependent gamma - ray spectrum , there is an upper bound on their density before they would be excluded by egb measurements . following the prescription outlined in ref . @xcite , the number density of photons @xmath51 with energy @xmath52 and their intensity is @xmath53 @xmath54 where @xmath55 is the time when photon creation begins . the quantity @xmath56 is the photon spectrum given by equation ( [ equ : hawking - dist ] ) , which we take at the high energy limit . for pbhs in the mass range studied , peak intensity occurs at @xmath57 mev . constraints were derived by matching the intensity to the upper bound of the comptel egb experimental data @xcite . egb constraints are also shown in figure [ fig : egb_comparison ] as well as those imposed by femtolensing @xcite . we find that planck provides the strongest constraint on the abundance of pbhs for masses @xmath1 g , while the egb dominates for masses @xmath2 g. note that this conclusion differs from that of ref . the planck constraint deviates from a linear relation because of the model for effective efficiencies . pbhs are of great interest in cosmology . they reveal conditions in the early universe and can serve as a dark matter candidate . there are several standard mechanisms that have been proposed to detect pbhs ; these include detection of hawking radiation , detection of radiation produced from accretion disks , and gravitational lensing . each method is capable of targeting different pbh mass ranges . in this paper , we have focused on pbhs with masses in the range @xmath0 g. we have improved and made more precise the constraints in this mass range using the cmb and egb . for our cmb bound , we model the energy absorption not as instantaneous , but rather using redshift dependent efficiency . the energy injection results in an increase in the ionization fraction at late times as well as an increase in the igm temperature , leading to distortions of the cmb anisotropies . larger fractional changes occur at large multipoles because of the increase of the width of the last scattering surface . using planck data , we show that cmb distortions from hawking radiation allow for stringent constraints on the density of @xmath0 g pbhs of @xmath58 . we show that for mass @xmath1 g , the cmb constraints are stronger than the constraints from the @xmath59 mev egb , which imply , @xmath60 . constraints imposed by cmb spectral distortions from hawking radiation producing sub 10.2 ev photons are also much weaker than our constraint . in the future , our theoretical analysis may be improved by including a mass spectrum of pbhs . in addition , even though we have used the egb to bound the contribution of pbhs , it may be interesting to consider the egb as a signal of pbhs . this is an exciting possibility because the origin of this @xmath61 mev gamma - ray background is not yet known @xcite . future missions to measure mev gamma - rays will be especially important for the study of pbhs @xcite . the authors thank daniel meerburg for helpful discussion and modified hyrec codes . we also thank tomohiro harada and tracy slatyer for helpful discussions . bd acknowledges support from doe grant de - fg02 - 13er42020 . les acknowledges support from nsf grant phy-1522717 . yg thanks the mitchell institute for fundamental physics and astronomy and wayne state university for support . sc acknowledges support from nasa astrophysics theory grant nnx12ac71 g . sw is supported in part by nasa astrophysics theory grant nnh12zda001n and doe grant de - fg02 - 85er40237 . 99 s. hawking , mon . not . soc . * 152 * , 75 ( 1971 ) . y. b. zeldovich , i. d. novikov , astron . volume 43 , pages 758 , 1966 ; y. b. zeldovich , i. d. novikov , sov . volume 10 , pages 602 , 1967 . b. j. carr and s. w. hawking , mon . not . * 168 * , 399 ( 1974 ) . s. w. hawking , nature * 248 * , 30 ( 1974 ) . doi:10.1038/248030a0 j. georg , g. sengor , and s. watson , phys . rev . d * 93 * , no . 12 , 123523 ( 2016 ) doi:10.1103/physrevd.93.123523 [ arxiv:1603.00023 [ hep - ph ] ] . t. harada , c. m. yoo , k. kohri , k. i. nakao and s. jhingan , astrophys . j. * 833 * , no . 1 , 61 ( 2016 ) doi:10.3847/1538 - 4357/833/1/61 [ arxiv:1609.01588 [ astro-ph.co ] ] . b. j. carr , k. kohri , y. sendouda and j. yokoyama , phys . rev . d * 81 * , 104019 ( 2010 ) doi:10.1103/physrevd.81.104019 [ arxiv:0912.5297 [ astro-ph.co ] ] . r. j. nemiroff , g. f. marani , j. p. norris and j. t. bonnell , phys . lett . * 86 * , 580 ( 2001 ) doi:10.1103/physrevlett.86.580 [ astro - ph/0101488 ] . k. griest , a. m. cieplak and m. j. lehner , phys . lett . * 111 * , no . 18 , 181302 ( 2013 ) . doi:10.1103/physrevlett.111.181302 m. ricotti , j. p. ostriker and k. j. mack , astrophys . j. * 680 * , 829 ( 2008 ) doi:10.1086/587831 [ arxiv:0709.0524 [ astro - ph ] ] . l. chen , q. g. huang and k. wang , arxiv:1608.02174 [ astro-ph.co ] . y. ali - hamoud and m. kamionkowski , arxiv:1612.05644 [ astro-ph.co ] . d. gaggero , g. bertone , f. calore , r. m. t. connors , m. lovell , s. markoff and e. storm , arxiv:1612.00457 [ astro-ph.he ] . b. carr , f. kuhnel and m. sandstad , phys . rev . d * 94 * , no . 8 , 083504 ( 2016 ) doi:10.1103/physrevd.94.083504 [ arxiv:1607.06077 [ astro-ph.co ] ] . a. m. green , phys . d * 94 * , no . 6 , 063530 ( 2016 ) doi:10.1103/physrevd.94.063530 [ arxiv:1609.01143 [ astro-ph.co ] ] . n. aghanim _ et al . _ [ planck collaboration ] , astron . astrophys . * 594 * , a11 ( 2016 ) doi:10.1051/0004 - 6361/201526926 [ arxiv:1507.02704 [ astro-ph.co ] ] . v. poulin , j. lesgourgues and p. d. serpico , arxiv:1610.10051 [ astro-ph.co ] . h. tashiro and n. sugiyama , phys . d * 78 * , 023004 ( 2008 ) doi:10.1103/physrevd.78.023004 [ arxiv:0801.3172 [ astro - ph ] ] . l. zhang , x. chen , m. kamionkowski , z. g. si and z. zheng , phys . d * 76 * , 061301 ( 2007 ) doi:10.1103/physrevd.76.061301 [ arxiv:0704.2444 [ astro - ph ] ] . m. s. madhavacheril , n. sehgal and t. r. slatyer , phys . d * 89 * , 103508 ( 2014 ) doi:10.1103/physrevd.89.103508 [ arxiv:1310.3815 [ astro-ph.co ] ] . t. r. slatyer , phys . d * 93 * , no . 2 , 023527 ( 2016 ) doi:10.1103/physrevd.93.023527 [ arxiv:1506.03811 [ hep - ph ] ] . t. r. slatyer , phys . d * 93 * , no . 2 , 023521 ( 2016 ) doi:10.1103/physrevd.93.023521 [ arxiv:1506.03812 [ astro-ph.co ] ] . h. liu , t. r. slatyer and j. zavala , phys . d * 94 * , no . 6 , 063507 ( 2016 ) doi:10.1103/physrevd.94.063507 [ arxiv:1604.02457 [ astro-ph.co ] ] . t. r. slatyer and c. l. wu , arxiv:1610.06933 [ astro-ph.co ] . j. h. macgibbon and b. r. webber , phys . rev . d * 41 * , 3052 ( 1990 ) . doi:10.1103/physrevd.41.3052 ; j. h. macgibbon , phys . d * 44 * , 376 ( 1991 ) . doi:10.1103/physrevd.44.376 j. s. bolton , g. d. becker , s. raskutti , j. s. b. wyithe , m. g. haehnelt and w. l. w. sargent , doi:10.1111/j.1365 - 2966.2011.19929.x arxiv:1110.0539 [ astro-ph.co ] . j. s. bolton , g. d. becker , j. s. b. wyithe , m. g. haehnelt and w. l. w. sargent , mon . not . 406 * , 612 ( 2010 ) doi:10.1111/j.1365 - 2966.2010.16701.x [ arxiv:1001.3415 [ astro-ph.co ] ] . k. m. belotsky and a. a. kirillov , jcap * 1501 * , no . 01 , 041 ( 2015 ) doi:10.1088/1475 - 7516/2015/01/041 [ arxiv:1409.8601 [ astro-ph.co ] ] . t. r. slatyer , phys . d * 87 * , no . 12 , 123513 ( 2013 ) doi:10.1103/physrevd.87.123513 [ arxiv:1211.0283 [ astro-ph.co ] ] . y. ali - haimoud and c. m. hirata , phys . d * 83 * , 043513 ( 2011 ) doi:10.1103/physrevd.83.043513 [ arxiv:1011.3758 [ astro-ph.co ] ] . a. lewis , a. challinor and a. lasenby , astrophys . j. * 538 * , 473 ( 2000 ) doi:10.1086/309179 [ astro - ph/9911177 ] . c. howlett , a. lewis , a. hall and a. challinor , jcap * 1204 * , 027 ( 2012 ) doi:10.1088/1475 - 7516/2012/04/027 [ arxiv:1201.3654 [ astro-ph.co ] ] . a. lewis , phys . d * 87 * , no . 10 , 103529 ( 2013 ) doi:10.1103/physrevd.87.103529 [ arxiv:1304.4473 [ astro-ph.co ] ] . a. lewis and s. bridle , phys . d * 66 * , 103511 ( 2002 ) doi:10.1103/physrevd.66.103511 [ astro - ph/0205436 ] . n. padmanabhan and d. p. finkbeiner , phys . d * 72 * , 023508 ( 2005 ) doi:10.1103/physrevd.72.023508 [ astro - ph/0503486 ] . `` planck 2015 results : cosmological parameter tables , '' g. weidenspointner . the origin of the cosmic gamma - ray background in the comptel energy range . phd thesis , technical university of munich , munich , germany ? id=602832 . j. zavala , m. vogelsberger and s. d. m. white , phys . d * 81 * , 083502 ( 2010 ) doi:10.1103/physrevd.81.083502 [ arxiv:0910.5221 [ astro-ph.co ] ] . l. e. strigari , j. f. beacom , t. p. walker and p. zhang , jcap * 0504 * , 017 ( 2005 ) doi:10.1088/1475 - 7516/2005/04/017 [ astro - ph/0502150 ] . p. ruiz - lapuente , l. s. the , d. hartmann , m. ajello , r. canal , f. k. ropke , s. t. ohlmann and w. hillebrandt , astrophys . j. * 820 * , no . 2 , 142 ( 2016 ) doi:10.3847/0004 - 637x/820/2/142 [ arxiv:1502.06116 [ astro-ph.he ] ] . s. horiuchi and j. f. beacom , astrophys . j. * 723 * , 329 ( 2010 ) doi:10.1088/0004 - 637x/723/1/329 [ arxiv:1006.5751 [ astro-ph.co ] ] . a. de angelis _ et al . _ [ e - astrogam collaboration ] , arxiv:1611.02232 [ astro-ph.he ] . continuum photons affect the cmb by creating spectral distortions @xcite . ref . @xcite investigated limits to these spectral distortions modeling the distortions as a bose - einstein distribution with a chemical potentioal @xmath62 , @xmath62-type distortions . in order to get a baseline estimate on the effect of the continuum photons injected by pbh on the cmb a similar approach was taken . assuming the injection will alter the perfect blackbody spectrum with the same @xmath62-type distortions , these distortions will be approximately where @xmath64 and @xmath65 are the energy density of the cmb and the distortion injection rate on the cmb respectfully . @xmath65 becomes the injection rate directly from the black hole energy going into the continuum . the result is current limits on these distortions are @xmath68 at two sigma @xcite . a comparison of this constraint and the cmb value discussed in this work is shown in figure [ fig : spectral_distortion_comparison ] . as can be seen , the constraints produced by this work are much stronger , several orders of magnitude , than limits produced assuming a @xmath62-type or similar distortion . the only limits that approache the cmb result are those that consider alterations at extremely late times . these however can be ignored due to cmb photons at this time being much cooler than the 10.2 ev limit for these injections . thus , the actual alterations at this time is further reduced . confidence level compared with the same exclusion bound produced through spectral distortions . the various curves correspond to different integration limits in equation ( [ equ : spec_dist_mu ] ) . the most constraining uses @xmath69 to infinity , and the second uses @xmath70 to infinity . ]
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we investigate constraints on the abundance of primordial black holes ( pbhs ) in the mass range @xmath0 g using data from the cosmic microwave background ( cmb ) and mev extragalactic gamma - ray background ( egb ) .
hawking radiation from pbhs with lifetime greater than the age of the universe leaves an imprint on the cmb through modification of the ionization history and the damping of cmb anisotropies . using a model for redshift dependent energy injection efficiencies ,
we show that a combination of temperature and polarization data from planck provides the strongest constraint on the abundance of pbhs for masses @xmath1 g , while the egb dominates for masses @xmath2 g. both the cmb and egb now rule out pbhs as the dominant component of dark matter for masses @xmath3 g. planned mev gamma - ray observatories are ideal for further improving constraints on pbhs in this mass range .
mi - th-1636 , wsu - hep-1609
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You are an expert at summarizing long articles. Proceed to summarize the following text:
ls i + 61 303 is a high mass x - ray binary ( hmxb ) that consists of an optical star with spectral type b0 ve and an unknown compact companion in a highly eccentric , 26.5 day orbit @xcite . while the system has a relatively low x - ray luminosity for a hmxb , ls i + 61 303 is the 15th brightest @xmath0-ray source included in the _ fermi _ lat 1-year point source catalogue ( @xcite ) . the be disk interacts with the compact companion , producing emission that has been observed to vary with orbital phase at every wavelength across the electromagnetic spectrum , from radio to tev ( eg . @xcite , @xcite ) . @xcite found periodic radio outbursts that peak near @xmath2 , and they defined the arbitrary reference for zero phase at hjd 2,443,366.775 that remains the conventional definition for ls i + 61 303 . periastron occurs at @xmath3 @xcite . during 2008 october and november , we performed an intense multiwavelength observing campaign on ls i + 61 303 supported by a _ cycle 1 program . we obtained optical h@xmath1 spectra of ls i + 61 303 at the kpno coud feed telescope over 35 consecutive nights to study the evolution of the emission during a complete orbit @xcite , @xcite . the h@xmath1 line profile exhibits a dramatic emission burst near @xmath4 , observed as a redshifted shoulder in the line profile ( see fig . [ gray ] ) as the compact source moves almost directly away from the observer . smaller temporal changes in the red spectra suggest additional h@xmath1 emission variability , so we subtracted the mean emission line profile to investigate the residuals carefully ( see fig . [ diff ] ) . during about half of the orbit , @xmath5 , the difference spectra reveal a partial s - shaped pattern similar to a spiral density wave that is commonly observed in be star disks @xcite . @xcite also observed a strong blue peak near @xmath6 , which supports the development of a spiral density wave near periastron . after this phase , the peculiar red shoulder develops . we measured the equivalent width of h@xmath1 , @xmath7 , for each spectrum by directly integrating over the line profile . ( we use the convention that @xmath7 is negative for an emission line . ) the errors in @xmath7 are typically about 10% due to noise and placement of the continuum . figure [ eqwidth ] shows that during our coud feed run , @xmath7 decreased slightly just before periastron . since @xmath7 is correlated to the radius of a be star s circumstellar disk @xcite , we interpret the decline in emission as a slight decrease in disk radius as gas is stripped away by the compact companion . @xmath7 then rises dramatically with the onset of the red shoulder emission component near @xmath4 . figure [ eqwidth ] also compares our recent @xmath7 with those measured by @xcite . their data were accumulated over six different observing runs over 19982000 , and the long term differences in emission strength are substantial . also during 2008 october and november , g. pooley obtained nearly simultaneous radio flux coverage with the arcminute microkelvin imager ( ami ) array . the 15 ghz ami light curve ( fig . [ radio ] ) reveals emission that peaks at the same time as the h@xmath1 `` red shoulder '' outburst . contemporaneous _ rxte _ light curves from @xciteand _ fermi _ light curves ( @xcite ) also reveal orbitally modulated emission that peaks just before the h@xmath1 red shoulder , although their wide phase bins may mask a true correlation . the h@xmath1 emission clearly traces the high energy emission region in this system . the unusual broadness of the h@xmath1 red shoulder emission is consistent with a balmer - dominated shock ( bds ; @xcite ) . bds are traditionally observed around supernova remnants but are also sometimes produced within pulsar wind nebulae and other evolved stellar systems . they form when high velocity ( 2009000 km s@xmath8 ) shocks collide with the interstellar medium , manifesting themselves as optically emitting filaments . energetic particles and/or photons may be generated in the post - shock region of the collisionless , non - radiative shock . direct collisional excitation of the pre - shock atoms produces a narrow emission line component that reflects thermal conditions within the pre - shock gas . if the energetic particles exceed the shock velocity , the pre - shock hydrogen atoms also exchange electrons with post - shock protons , manifesting themselves as broad neutral hydrogen lines ( widths @xmath9 km s@xmath8 ) . the h@xmath1 line structure in ls i + 61 303 is complicated by the superposition of emission from the circumstellar disk ; however , the broad red shoulder is consistent with such a bds . the temporary nature of the red shoulder , as well as the correlated gev radio emission , suggests that the bds only forms when a high density tidal mass stream interacts with a pulsar wind in ls i + 61 303 . we thank di harmer and the staff at kpno for their hard work to schedule and support the coud feed observations . guy pooley , christina aragona , tabetha boyajian , amber marsh , and rachael roettenbacher helped collect the data presented here and should be cheered for their heroic efforts . this work is supported by nasa dpr numbers nnx08av70 g , nng08e1671 , nnx09at67 g , and an institutional grant from lehigh university .
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the @xmath0-ray binary ls i + 61 303 is one of the brightest fermi sources , with orbitally modulated emission across the electromagnetic spectrum .
here we present h@xmath1 spectra of ls i + 61 303 that exhibit a dramatic emission burst shortly before apastron , observed as a redshifted shoulder in the line profile . a correlated burst in radio , x - ray , and gev emission
is observed at the same orbital phase .
we interpret the source of the emission as a compact pulsar wind nebula that forms when a tidal mass stream from the be circumstellar disk interacts with the relativistic pulsar wind .
the h@xmath1 emission offers an important probe of the high energy emission morphology in this system .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
one of the formidable tasks in heavy - ion physics is to identify a precise understanding of the jet - medium dynamics , the jet - medium interactions , and the jet - energy loss formalism . below , we study the influence of the details of the jet - medium coupling and the medium background on the simultaneous description of the nuclear modification factor ( @xmath1 ) and the high-@xmath0 elliptic flow ( @xmath2 ) measured at rhic and lhc @xcite for a radiative pqcd energy - loss ansatz @xcite . we contrast media determined via the viscous hydrodynamic approach vish2 + 1 @xcite with the parton - cascade bamps @xcite as well as a jet - medium coupling depending on the collision energy with a jet - medium coupling influenced by the energy of the jet , the temperature of the medium and non - equilibrium effects around the phase transition . besides this , we compare the jet - energy loss based on radiative pqcd @xcite with the hybrid ads energy - loss ansatz of ref . we contrast the pion nuclear modification factor obtained via the radiative pqcd - energy loss @xcite and the hybrid ads energy - loss ansatz with a parton - jet nuclear modification factor that can be considered as an idealized lo jet @xmath1 at rhic and lhc energies . the pqcd - based energy loss model studied is parametrized as @xcite @xmath3 with the jet - energy dependence @xmath4 , the path - length dependence @xmath5 , and the energy dependence @xmath6 . in the following , the jet - medium coupling @xmath7 will depend either on the collision energy @xmath8 or the energy of the jet and the temperature of the background medium considered @xmath9 . the jet - energy loss fluctuations are distributed via @xmath10 , allowing for an easy interpolation between non - fluctuating ( @xmath11 ) , uniform dirac distributions and distributions increasingly skewed towards small @xmath12 . the jets are spread according to a transverse initial profile specified by the bulk flow fields given by the vish2 + 1 and bamps backgrounds considered @xcite . on the other hand , the jet - energy loss of the hybrid ads energy - loss ansatz @xcite is based on falling strings @xcite where @xmath13 the initial jet energy is given by @xmath14 and the string stopping distance for quark and gluon jets is determined via @xmath15 with the jet - medium coupling @xmath16 for quarks and @xmath17 for gluons , including the respective casimir operators @xmath18 and @xmath19 . this energy loss ansatz has been integrated into our existing model @xcite . please note that ref . @xcite uses natural units , @xmath20 . for a direct comparison , we quote our results below using a dimensionless coupling . the main differences between the two energy - loss descriptions is the square - root dependence that leads to the formation of a bragg peak with the explosive burst of energy close to the end of the jet s evolution . there have been discussions in literature @xcite on the impact of the bragg peak . in line with previous findings @xcite we will show below that there is a difference between the hybrid ads energy - loss ansatz featuring a bragg peak and the pqcd model without a bragg peak , however , this difference is only marginal . fig . [ fig01 ] shows the pion nuclear modification factor ( @xmath1 ) for central ( left panel ) and mid - central ( middle panel ) collisions at rhic ( black ) and lhc ( red ) as well as the high-@xmath0 elliptic flow ( @xmath2 ) for mid - central events ( right ) . the measured data @xcite is compared to the pqcd - based energy loss of eq . ( [ eq1 ] ) with @xmath21 . jet - energy loss fluctuations ( @xmath22 ) and the transverse expansion of the background flow ( @xmath23 ) are included , as well as a running jet - medium coupling that depends on the energy of the collision , @xmath8 . [ fig01 ] demonstrates that there is a surprising similarity between the results that can not be expected a priori given the fact that the two background media are so different : while the hydrodynamic description of vish2 + 1 @xcite assumes an equilibrated system , the parton cascade bamps @xcite also includes non - equilibrium effects in the bulk medium evolution . in addition , the figure exhibits the so - called high-@xmath0 @xmath2-problem @xcite : the high-@xmath0 elliptic flow below @xmath24 gev is about a factor of two below the measured data @xcite . this effect has been discussed in literature @xcite and recently it has been suggested by cujet3.0 @xcite that a temperature and energy - dependent jet - medium coupling @xmath9 , which includes non - perturbative effects around the phase transition of @xmath25 mev , can overcome this problem . this jet - medium coupling was derived from the dglv gluon number distribution @xcite and is given by the analytic formul @xmath26 it includes a running coupling @xmath27 with @xmath28 , the polyakov - loop suppression of the color - electric scattering @xcite via @xmath29 with pre - factors @xmath30 for quarks and gluons , and the polyakov loop @xmath31 as parametrized from lattice qcd , as well as an enhancement of scattering due to the magnetic monopoles near the critical temperature @xmath32 also derived from lattice qcd @xcite . this temperature and energy - dependent jet - medium coupling shows an effective running as it decreases with temperature . we included the above jet - medium coupling @xmath9 in our jet - energy loss approach @xcite . the result is shown in fig . [ fig02 ] , again for the hydrodynamic background vish2 + 1 ( solid lines ) and a medium determined via the parton cascade bamps ( dashed lines ) . for comparison , we depict the results from cujet3.0 @xcite . as in fig . [ fig01 ] , the ion nuclear modification factor is well described both at rhic and lhc . the high-@xmath0 elliptic flow , however , increases drastically below @xmath24 gev as compared to fig . [ fig01 ] , especially for the bamps background which already includes non - equilibrium effects @xcite . finally , we compare results from the linear pqcd approach of eq . ( [ eq1 ] ) with the highly non - linear hybrid ads holographic model of jet - energy loss , see eq . ( [ eq2 ] ) . we compare the pion nuclear modification factor and an idealized lo jet @xmath1 given by @xmath33 naturally , this lo jet @xmath1 represents a reconstructed jet with vanishing cone radius and is only a lower bound for the nlo jet @xmath1 with jet - cone radii @xmath34 . [ fig03 ] shows this comparison at lhc ( left ) and rhic ( right ) energies for two different jet - medium couplings that are treated as constants : a larger one ( red ) fitted to the pion @xmath1 data ( dashed - dotted lines ) at rhic and a lower one ( blue ) fitted the pion @xmath1 data at lhc . to guide the eye , we include the reconstructed jet @xmath1 from cms @xcite with @xmath35 in fig . [ fig03 ] . the solid blue lines for the lo jet @xmath1 in the left panels of fig . [ fig03 ] lie in the same ballpark as the experimental data . fragmenting this result to pions ( dashed - dotted lines ) leads to an @xmath1 that reproduces the measured pion nuclear modification factor at lhc . a straight extrapolation of this results to rhic energies shows that the lo jet @xmath1 for _ the same _ jet medium couplings lie on top of the measured _ pion _ nuclear modification factor . however , fragmenting this result to pions leads to a @xmath36 that is larger than the measured data at rhic . larger jet - medium couplings ( red lines ) , on the other hand , describe the _ pion _ nuclear modification factor at rhic for the pqcd scenario and the lo jet @xmath1 at lhc is again close to the experimental data . the pion nuclear modification factor at lhc , however , only touches the lower bound of present error bars . in case of the hybrid ads energy loss the results always only touch the lower end of the experimental error bars . [ fig03 ] demonstrates that the results for the pqcd and the hybrid ads energy - loss including a bragg peak are remarkably similar . thus , unfortunately , neither the pion nor a lo jet @xmath1 are sensitive to the difference in the path - length between pqcd and ads models . we compared the measured data on the nuclear modification factor for pions and reconstructed jets as well as on the high-@xmath0 elliptic flow at rhic and lhc energies to results obtained by a linear pqcd and a highly non - linear hybrid ads holographic model of jet - energy loss . we found that the simultaneous description of the @xmath1 and @xmath2 requires a jet - medium coupling that depends on the energy of the jet , the temperature of the medium @xcite , and non - equilibrium effects around the phase transition . we also contrasted a hydrodynamic background ( vish2 + 1 ) @xcite with a medium obtained from the parton cascade bamps @xcite and showed that the influence of the underlying bulk medium considered is suprisingly small . unfortunately , neiter the pion nor the lo jet @xmath1 are sensitive to the difference in the path - length between pqcd and ads models . this work was supported in part through the bundesministerium fr bildung und forschung , the helmholtz international centre for fair within the framework of the loewe program ( landesoffensive zur entwicklung wissenschaftlich - konomischer exzellenz ) launched by the state of hesse , the us - doe nuclear science grant no . de - ac02 - 05ch11231 within the framework of the jet topical collaboration , and the us - doe nuclear science grant no.de-fg02-93er40764 . numerical computations have been performed at the center for scientific computing ( csc ) .
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the measured data on the nuclear modification factor for pions and reconstructed jets as well as on the high-@xmath0 elliptic flow at rhic and lhc energies are compared to results from a linear pqcd and a highly non - linear hybrid ads holographic model of jet - energy loss .
we find that the high-@xmath0 ellitic flow requires to include realistic medium transverse flow fields and a jet - medium coupling including the effects of the energy of the jet , the temperature of the bulk medium , and non - equilibrium effects close to the phase transition .
we extend our jet - energy loss model that is coupled to state - of - the - art hydrodynamic prescriptions to backgrounds generated by the parton cascade bamps .
we demonstrate that the results for the hydrodynamic and the parton - cascade backgrounds show a remarkable similarity .
unfortunately , the results for both the pion and a parton - jet nuclear modification factor are insensitive to the jet - path dependence of the models considered .
jet quenching , viscous hydrodynamics , transport model , jet holography
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You are an expert at summarizing long articles. Proceed to summarize the following text:
frequently it is the case in the study of real - world complex networks that we observe essentially a sample from a larger network . there are many reasons why sampling in networks is often unavoidable and , in some cases , even desirable . sampling , for example , has long been a necessary part of studying internet topology @xcite . similarly , its role has been long - recognized in the context of biological networks , e.g. , protein - protein interaction @xcite , gene regulation @xcite and metabolic networks @xcite . finally , in recent years , there has been intense interest in the use of sampling for monitoring online social media networks . see @xcite , for example , for a representative list of articles in this latter domain . given a sample from a network , a fundamental statistical question is how the sampled network statistics be used to make inferences about the parameters of the underlying global network . parameters of interest in the literature include ( but are by no means limited to ) degree distribution , density , diameter , clustering coefficient , and number of connected components . for seminal work in this direction , see @xcite . in this paper , we propose potential solutions to an estimation problem that appears to have received significantly less attention in the literature to date the estimation of the degrees of individual sampled nodes . degree is one of the most fundamental of network metrics , and is a basic notion of node - centrality . deriving a good estimate of the node degree , in turn , can be helpful in estimating other global parameters , as many such parameters can be viewed as functions that include degree as an argument . while a number of methods are available to estimate the full degree distribution under network sampling ( e.g. , @xcite ) , little work appears to have been done on estimating the individual node degrees . our work addresses this gap . formally , our interest lies in estimation of the degree of a vertex , provided that vertex is selected in a sample of the underlying graph . there are many sampling designs for graphs . see ( * ? ? ? * ch 5 ) for a review of the classical literature , and @xcite for a recent survey . canonical examples include ego - centric sampling@xcite , snowball sampling , induced / incident subgraph sampling , link - tracing and random walk based methods@xcite . under certain sampling designs where one observes the true degree of the sampled node ( e.g. ego - centric and one - wave snowball sampling ) , degree estimation is unnecessary . in this paper , we focus on _ induced subgraph sampling _ , which is structurally representative of a number of other sampling strategies@xcite . formally , in induced subgraph sampling , a set of nodes is selected according to independent bernoulli(@xmath0 ) trials at each node . then , the subgraph induced by the selected nodes , i.e. , the graph generated by selecting edges between selected nodes , is observed . this method of sampling shares stochastic properties with incident subgraph sampling ( wherein the role of nodes and edges is reversed ) and with certain types of random walk sampling @xcite . the problem of estimating degrees of sampled nodes has been given a formal statistical treatment in @xcite , for the specific case of traceroute sampling as a special case of the so - called _ species problem _ @xcite . to the best of our knowledge , a similarly formal treatment has not been applied more generally for other , more canonical sampling strategies . however , a similar problem would be estimating personal network size for a group of people in a survey . some prior works in this direction @xcite consider estimators obtained by scaling up the observed degree in the sampled network , in the spirit of what we term a method of moments estimator below . but no specific graph sampling designs are discussed in these studies . we focus on formulating the problem using the induced subgraph sampling design and exploit network information beyond sampled degree to propose estimators that are better than naive scale - up estimators . key to our formulation is a risk theoretic framework used to derive our estimators of the node degrees , through minimizing frequentist or bayes risks . this contribution is accompanied by a comparative analysis of our proposed estimators and naive scale - up estimators , both theoretical and empirical , in several network regimes . we note that when sampling is coupled with false positive and false negative edges , e.g. , in certain biological networks , our methods are not immediately applicable . sampling designs that result in the selection of a fraction of edges from the underlying global network ( induced and incident subgraph sampling , random walks etc . ) are our primary objects of study . we use induced subgraph sampling as a rudimentary but representative model for this class and aim to simultaneously estimate the true degrees of all the observed nodes with a precision better than that obtained by trivial scale - up estimators with no network information used . let us denote by @xmath1 a true underlying network , where @xmath2 . this network is assumed static and , without loss of generality , undirected . the true degree vector is @xmath3 . the sampled network is denoted by @xmath4 where , again without loss of generality , we assume that @xmath5 . write the sampled degree vector as @xmath6 . throughout the paper , we assume that we have an induced subgraph sample , with ( known ) sampling proportion @xmath0 . it is easy to see from the sampling scheme that @xmath7 . therefore , the method of moments estimator ( mme ) for @xmath8 is @xmath9 . thus , @xmath10 is a natural scale - up estimator of the degree sequence of the sampled nodes . in this section , we propose a class of estimators that minimize the unweighted @xmath11-risk of the sampled degree vector and discuss their theoretical properties . we aim to demonstrate , under several conditions , that the risk minimizers are superior to the regular scale - up estimators , the former taking into account the inherent relationships inside the network . we note that although a maximum likelihood approach to estimation is perhaps intuitively appealing , a closed form derivation of the mle in this setting is probitive . another option is to look at marginal likelihoods . but the mle based on univariate marginal likelihoods are essentially equivalent to the mme for this sampling scheme . we will frequently use the the first and second moments of the sampled degree vector in our estimation methods . the following lemma will be useful . [ lem : meancovariancedegree ] under induced subgraph sampling , the mean and covariance matrix of the observed degree vector are @xmath12 where the diagonals of @xmath13 are @xmath14 and the @xmath15-th off - diagonal is denoted by @xmath16 , which denotes the number of common neighbors of node @xmath17 and node @xmath18 in the network @xmath19 . adopting the standard definition of ( unweighted ) frequentist @xmath11 risk of an estimator @xmath20 of a parameter @xmath21 , i.e. , @xmath22 , the frequentist risks are calculated for a general class of estimators . we also define @xmath23 , a _ restricted risk function _ assuming the sampled graph @xmath24 is restricted to some class @xmath25 . our proposed candidates are the elements in the class of linear functions of the observed degree vector that minimize the risk or the restricted risk w.r.t . some class . it is expected that the optimal estimator will be a function of the parameter and hence another ( naive ) estimator will need to be plugged in . our final estimate will then be a plug - in risk minimizer . here we estimate the node degrees individually , assuming that the estimate for the @xmath26 node is of the form @xmath27 , where @xmath28 is a scalar and @xmath29 is the observed degree in the sample . since @xmath7 , where @xmath8 is the true degree of the @xmath26 node , @xmath30 differentiating w.r.t . @xmath28 and equating to 0 , we get the optimal @xmath31 . plugging in the mme of @xmath32 , we get the plug - in univariate risk minimizer @xmath33 taylor expanding the above formula ( during taylor expansions of functions of @xmath29 , we will assume that @xmath29 is concentrated around its mean , so that the taylor expanded approximation is close ) and taking expectation , we see that @xmath34 = \frac{1}{p } \mathbb{e}\left [ d^*_i\left(1 + \frac{1-p}{d^*_i}\right)^{-1 } \right ] \approx \frac{1}{p } \mathbb{e}\left [ d^*_i\left(1 - \frac{1-p}{d^*_i}\right ) \right ] = d^0_i - \frac{1-p}{p } \enskip . $ ] + the above calculation suggests that an adjustment needs to be made to @xmath35 by bias - correction , so that its risk becomes comparable to that of @xmath36 . in fact , we will show in proposition [ unirisk ] that our bias - corrected plug - in estimator has a lower risk than mme when the true degree is bigger than a lower bound , which can be expressed as a closed form function of the sampling proportion . ultimately , our proposed univariate risk minimizer is given by @xmath37 note that the proposed estimator @xmath38 can be written as @xmath39 this is almost an additive adjustment to the multiplicative scale - up estimate . we also have , @xmath40 . + from the above discussion , it is obvious to see that when @xmath36 overestimates @xmath41 , @xmath38 can not be any better than @xmath36 . we first argue that this happens with small probability . to demonstrate this , we use the erds - rnyi model . + for any node @xmath17 selected in induced subgraph sampling from an erds - rnyi model @xmath42 ( for @xmath43 and @xmath44 denoting the probability of edge formation such that @xmath45 . ) and for any @xmath46 , @xmath47\\ \mathbb{p}\left(\hat{d}^{\rm mme}_i < ( 1-\epsilon)d^0_i\right ) & \leq \exp\left[-np_e\left(1 - e^{-\frac{\epsilon^2p}{2}}\right)\right]\end{aligned}\ ] ] @xmath48 \\ & \hspace{-1.5cm}\leq \mathbb{e}_{d^0_i}\left[\exp\left(- \frac{\epsilon^2 p}{2+\epsilon } d^0_i\right)\right ] \intertext{in the above inequality , we use a version of chernoff inequality . if $ x_1 , x_2 , \cdots , x_n$ are independent with $ 0\leq x_i\leq 1 $ for all $ i$ , $ x = \sum_i x_i$ and $ \mu = \mathbb{e}x$ , then $ $ \mathbb{p}\left[x > ( 1+\epsilon)\mu\right ] \leq \exp\left(-\frac{\epsilon^2}{2+\epsilon}\mu\right)$$ and $ $ \mathbb{p}\left[x < ( 1-\epsilon)\mu\right ] \leq \exp\left(-\frac{\epsilon^2}{2}\mu\right).$$ since $ d^0_i$ approximately follows a poisson$\left(np_e\right)$ , we have } \mathbb{e}_{d^0_i}\left[\exp\left(- \frac{\epsilon^2 p}{2+\epsilon } d^0_i\right)\right ] & = \psi\left(-\frac{\epsilon^2 p}{2+\epsilon}\right ) \\ & \hspace{-1cm}= \exp\left[-np_e\left(1 - e^{-\frac{\epsilon^2p}{2+\epsilon}}\right)\right ] \intertext{where $ \psi$ denotes the moment generating function of poisson distribution . the other inequality follows similarly when the lower tail chernoff bound is used . } \end{aligned}\ ] ] we extend the idea presented in the previous section to the multivariate case , in order to minimize the overall @xmath11 sum over all sampled nodes . the rationale for this extension is to exploit the covariance structure we derived in lemma [ lem : meancovariancedegree ] in estimating the degree vector . accordingly , we consider all estimates of the form @xmath49 , where @xmath50 is an @xmath51 matrix . using lemma [ lem : meancovariancedegree ] , we get the @xmath11 risk + @xmath52 . + the multivariate risk minimizer is defined as + @xmath53 . + differentiating the objective function w.r.t . @xmath50 and equating it to @xmath54 , we get + @xmath55 + plugging in the mme of @xmath56 and @xmath57 , we get the plug - in multivariate risk minimizer @xmath58 where @xmath59 denotes the number of common neighbors of node @xmath17 and node @xmath18 in the sample , and @xmath60 is given by a matrix whose diagonals are @xmath29 and whose off - diagonals are @xmath59 , @xmath61 . in this section , we propose a bayesian solution to our estimation problem , by putting a prior on the degree distribution . the principal motivation behind this approach is the desire to incorporate additional information on global network structure , where the natural candidate in this context is the degree distribution . in case such a subjective prior is not available , an estimate of the degree distribution may be used . we propose and analyze estimators based on both known ( subjective ) and estimated degree distributions below . first , let us assume that we know the degree distribution @xmath62 of the underlying network . under the assumption that the true degree of node @xmath17 follows @xmath62 , and under induced subgraph sampling of @xmath63 , the conditional distribution of @xmath64 is @xmath65 . then it can be easily shown that the bayes estimator under square error loss is @xmath66 if the true degree distribution is not known , then it needs to be estimated , for example using techniques described in or similar to @xcite . let @xmath67 be a reasonable " estimator for @xmath62 . then an empirical bayes estimator is given by @xmath68 generally speaking , if @xmath69 denotes the distribution of @xmath29 given @xmath70 , then this empirical bayes estimate can be expressed as @xmath71 these estimators take the form of a weighted mean , as expected for bayes estimates under quadratic loss . the weights are functionals of both sampling design and the degree distribution . for the latter estimator , only the estimated degree distribution comes into play , and thus the proposed empirical bayes estimator incorporates the sampling and sampled network information . in this section , we present results on the relative performance of our proposed estimators from a risk - theoretic perspective , and we discuss several conditions under which one outperforms the other . all these estimates will be benchmarked against the regular scale - up estimate @xmath72 . proofs may be found in the supplementary materials . in the first part of our risk analysis , we look at the @xmath11 frequentist risk of our proposed univariate and multivariate estimators . our main results in this section will compare the risk incurred by our proposed estimators to the scale up estimator and discuss conditions under which our proposed estimators perform better . [ unirisk ] assuming @xmath73 , we have @xmath74 . in other words , the univariate risk minimizer @xmath75 will outperform the mme when the true degree @xmath8 is sufficiently large . [ multirisk ] let us denote the class of all sampled graphs of size @xmath76 ( where @xmath77 for all @xmath17 , i.e. , there is no isolated node ) as @xmath78 . also assume that there exists an @xmath79 such that @xmath80 are nonempty . then we have @xmath81 over sampled graphs belonging to @xmath82 . scrutiny of the conditions in proposition [ multirisk ] , along with definition of the set @xmath83 , reveals a general characterization of the graphs where the proposed multivariate estimator performs better . it is to be noticed that @xmath84 shrinks @xmath85 by some factor . the term on the right side of the inequality in the definition of @xmath86 provides a lower bound on the shrinkage factor and the term on the left decreases as the cardinality of @xmath87 increases , i.e. , the graph becomes less sparse . hence , the proposed estimator can be expected to work better than the standard scale - up estimator under the assumption of sparsity of the sampled graph . this will also be demonstrated in the simulation section . the eigenvector condition imposes a geometric constraint on the sample degree - degree matrix @xmath60 . what it essentially means is that the angle between the eigenvectors of @xmath88 and @xmath89 should be smaller than @xmath90 . or , in other words , by selecting an @xmath91 sufficiently small but positive , our class of sampled graphs are restricted where the associated matrix @xmath88 has eigenvectors at least @xmath92 angle away from any orthogonal direction to @xmath89 . thus , our estimator performs better for sparse graph satisfying a mild geometric condition . the performance of the bayes estimators is evaluated here under several conditions and network paradigms . note that these estimators are compared to the regular scale - up estimator with respect to their frequentist risk functions . we start with our estimator in its most general form and state conditions on the prior degree distribution that will ensure lower risk . from that , we assess its risk when the prior degree distribution is replaced with an appropriate estimate . we also explicitly derive the bayes estimator for the erds - rnyi class of random graphs and state conditions under which the bayes estimator yields lower risk than the scale - up estimator . [ suffrisk ] let @xmath93 be the true degree of sample node @xmath17 , and @xmath29 , the observed degree . denote by @xmath94 the class of sampled graphs where the following two conditions hold : @xmath95 where @xmath96 . then @xmath97 under induced subgraph sampling . the conditions ( [ suffrisk : cond1 ] ) and ( [ suffrisk : cond2 ] ) essentially constrain the tail behavior of the prior degree disbution . the first condition ensures that the tail decays at a rate such that it is not too `` thick '' and the second condition ensures that it is not too `` thin '' . as @xmath8 becomes bigger , the rhs in condition ( [ suffrisk : cond1 ] ) becomes smaller and that is reminiscent of the sparsity property of the underlying graph , meaning that not a lot of nodes can have very high degree , an observation consistent with sparse graphs . on the other hand , the lhs in the condition ( [ suffrisk : cond2 ] ) can be interpreted as the mean of the tail probabilities weighted by the posterior distribution . this has to be bounded away from zero in order for the bayes estimate to have lower risk than the mme . in real problems , where the true degree distribution is unknown , one either has to choose @xmath98 subjectively or use the data to come up with a reasonable estimate . estimating @xmath98 for a general case is beyond the scope of this paper and will not be discussed here . for our analysis , we will just assume that we have an estimate of the degree distribution at our disposal ( e.g. , @xcite ) , denoted by @xmath99 . using @xmath99 will give us our proposed empirical bayes estimate @xmath100 , the behavior of which can be described as follows . [ pluginapprox ] let @xmath67 be an estimate of @xmath62 such that @xmath101 then under assumption ( [ suffrisk : cond2 ] ) , with @xmath98 replaced by @xmath99 , we have @xmath102 it is easily seen that with the assumption , the upper bound in can be simplified to @xmath103 assuming a large network , the sum in the denominator can be approximated by @xmath104 . then the upper bound is @xmath105 from the above discussion , it is evident that if @xmath106 , @xmath107 for all @xmath17 and hence their risk functions will also be close . thus , using proposition [ suffrisk ] , it is expected that @xmath108 it is well known that the asymptotic degrees in erds - rnyi graph models follow a poisson distribution , under standard conditions . in this section , we study the effects of using a poisson prior degree distribution for large erds - rnyi graphs . the goal is to demonstrate the efficacy of the bayesian approach compared to scale - up estimates as in the last section . however , studying specific models like erds - rnyi will give us more insight about the performance of the proposed bayes estimate . in this scenario , the prior @xmath62 is given by @xmath109 where @xmath110 is the prior mean . for a large erds - rnyi graph with number of nodes @xmath111 and edge probability @xmath44 , @xmath112 . we denote , by @xmath113 , the shifted poisson distribution on @xmath114 whose p.m.f . is given by @xmath115 it is easy to check that with a @xmath116 prior on @xmath70 , the posterior distribution is @xmath117 . hence the bayes estimate with respect to the quadratic loss function is @xmath118 [ erdosbayesrisk ] assuming @xmath119 the quadratic risk of the bayes estimator using a @xmath116 prior is smaller than that of the mme . the above result shows that if the sampled node is such that its true degree belongs to a neighborhood around the mean of the underlying degree distribution , then the bayes estimator is uniformly better than the mme . in case the underlying mean is unknown , it can easily be estimated from the sample . ( e.g. , for known @xmath111 , @xmath120 ) if @xmath121 is a consistent estimator of @xmath110 in the sense that @xmath122 when @xmath123 , @xmath124 and @xmath125 , then the empirical bayes estimator @xmath126 will converge in probability to the bayes estimator in the sense that @xmath127 . hence , the result of prop . ( [ erdosbayesrisk ] ) is expected to hold . this will also be demonstrated in the simulations . for our simulation study , we look at two different regimes of network erds - rnyi random graphs and heavy tailed degree distributions . we compare four methods of estimation - the regular mme , univariate risk minimizer , multivariate risk minimizer and the bayes estimate . as priors in bayes estimation , we use both exponentially decaying ( poisson ) and polynomially decaying degree distribution as priors . table [ ertable ] records the euclidean distance between the true and estimated degree vectors across some combinations of graph size @xmath111 , edge strength @xmath44 and sampling proportion @xmath0 . the errors are averaged over 50 different samples from each given graph @xmath63 . from the output , it is clear that the bayes estimators with true @xmath110 and estimated @xmath110 outperform other estimators by a very wide margin in terms of @xmath11 risk . also , our theoretical prediction in the discussion following proposition [ multirisk ] was that the multivariate risk minimizer ( mrm ) works better than the mme for sparse graphs . this is experimentally verified in this simulation , since we see that the relative risk of mrm compared to mme decreases as the sparsity of the underlying graph increases , i.e. , as @xmath44 decreases . the method with lowest total quadratic loss is shown in red for each condition . we compared four methods of estimation in simulated scale free networks which follow a power law degree distribution . as priors in bayes estimation , we compared the true polynomial prior and quadratic prior . we computed the @xmath128 distances across some combinations of sparsity ( denoted by @xmath129 , given by the ratio of total edges to all possible edges ) , sampling proportion @xmath0 and heaviness of the tail of the degree distribution , controled by @xmath130 . the results are shown in table [ sftable ] . the bayes estimators or the multivariate risk minimizers work better than the other estimators . one important thing to observe here is that for the most sparse graph , the bayes estimator with true prior works the best and as @xmath129 increases , multivariate risk minimizers work better than the rest , but there is hardly any improvement over mme . again , the method with lowest total quadratic loss is shown in red for each condition . [ table : sim1 ] [ table : sim2 ] in february 2015 , the defense advanced research projects agency ( darpa ) , an agency of the u.s . department of defense , announced the _ memex _ program in response to the use of the internet in human trafficking , especially chat forums , advertisements and job services sections . darpa - funded research determined the trafficking industry spent $ 250 m to post more than 60 m advertisements over a two - year time frame@xcite . indexing and cross - referencing the ads with the same contact number , similar address or zip codes help identify and track the illegal trafficking activities . this leads to a massive background network structure where each node represents an advertisement and an edge between two nodes are created if they share certain features . it is not unreasonable to expect that , in surveillance of networks like this , sampling may well arise , either by choice or by circumstance . we mimic this situation by pretending that this underlying network generated by the _ memex _ program is unknown to us and sampling it using induced subgraph sampling . the nodes associated with trafficking activities are flagged in the data . there are 31,248 nodes , of which 12,387 are flagged and there are 10,200,838 edges . our goal was to estimate the true degrees of flagged nodes that we saw in our sample . we compared the @xmath11 distance of regular scale - up estimators , and our proposed univariate , multivariate and bayes estimators . for the bayes estimator , a number of polynomial priors were taken into consideration with varying degree of decay , denoted by @xmath131 . the results are shown in table [ humantraffic ] . almost everything works better than the naive scale - up estimator in terms of total @xmath11 loss , although the relative improvement is more modest than in simulation . [ humantraffic ] in this paper , we addressed the problem of estimation of true degrees of sampled nodes from an unknown graph . we proposed a class of estimators from a risk - theory perspective where the goal was to minimize the overall @xmath11 risk of the degree estimates for the sampled nodes . we considered estimators that minimize both frequentist and bayes risk functions and compared the frequentist @xmath11 risks of our proposed estimator to the naive scale - up estimator . the basic objective of proposing these estimators was to exploit the additional network information inherent in the sampled graph , beyond the observed degrees . our theoretical analyses , simulation studies and real data show clear evidence of superior performance of our estimators compared to mme , especially when the graph is sparse and the sampling ratio is low , mimicking the real - world examples . there are a number of ways our current work could be extended . firstly , a theoretical analysis of the bayes estimators under priors for random graph models beyond erds - rnyi is desirable , although likely more involved . secondly , although induced subgraph sampling serves as a representative structural model for a certain class of adaptive sampling designs , the specific details of the sufficiency conditions discussed in this paper can be expected to vary slightly with the other sampling designs ( e.g. , incident subgraph or random walk designs ) . finally , the success of the bayesian method appears to rely heavily upon appropriate choice of prior distribution , as observed in our theoretical analysis and computational experiments . it would be of interest to explore the performance of the empirical bayes estimate in conjunction with the nonparametric method of degree distribution estimation proposed in @xcite . more generally , the method in @xcite can in principle be extended to estimate individual vertex degrees . but the computational challenge of implementation and the corresponding risk analysis can be expected to be nontrivial . d. killworth , c. mccarty , h. r. bernard , g.a . shelley , and e.c . estimation of seroprevalence , rape , and homelessness in the united states using a social network approach . , 22(2):289308 , 1998 . j. leskovec and c. faloutsos . sampling from large graphs . in _ proceedings of the 12th acm sigkdd international conference on knowledge discovery and data mining _ , kdd 06 , pages 631636 , new york , ny , usa , 2006 . acm . b. ribeiro and d. towsley . estimating and sampling graphs with multidimensional random walks . in _ proceedings of the 10th acm sigcomm conference on internet measurement _ , imc 10 , pages 390403 , new york , ny , usa , 2010 .
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the need to produce accurate estimates of vertex degree in a large network , based on observation of a subnetwork , arises in a number of practical settings .
we study a formalized version of this problem , wherein the goal is , given a randomly sampled subnetwork from a large parent network , to estimate the actual degree of the sampled nodes . depending on the sampling scheme ,
trivial method of moments estimators ( mmes ) can be used . however , the mme is not expected , in general , to use all relevant network information . in this study
, we propose a handful of novel estimators derived from a risk - theoretic perspective , which make more sophisticated use of the information in the sampled network .
theoretical assessment of the new estimators characterizes under what conditions they can offer improvement over the mme , while numerical comparisons show that when such improvement obtains , it can be substantial .
illustration is provided on a human trafficking network .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
diluted magnetic semiconductors @xcite represent an important class of materials and structures where ferromagnetism can be tuned by voltage @xcite . this ability comes from the carrier - mediated character of the ferromagnetic interaction @xcite . the ferromagnetic ordered state in these systems appears due to mobile carriers interacting with stationary spins of magnetic impurities . to date , curie temperatures as high as @xmath0 have been observed in a technologically important class of the mn - doped iii - v semiconductor structures @xcite . when the magnetic semiconductors become combined with the conventional field - effect layered structures , the number of mobile carriers and the ferromagnetic interaction become tunable by the voltage @xcite . this ability to externally control the properties of magnetic crystals with means other than the external magnetic field may have important device applications . a further step from magnetic semiconductor layers would be zero dimensional systems , quantum dots ( qds ) . magnetic quantum dots can be viewed as nano - scale memory elements where information is stored in the form of magnetic polarization . such a system may have important advantages compared to the conventional metal spin - valve memories : ( 1 ) small sizes and relatively small number of carriers and ( 2 ) voltage control of the number of electrons which was already demonstrated in many experiments for non - magnetic qds @xcite . therefore it is important to develop the understanding of magnetic qds with interacting carriers . here we develop a theory of magnetic qds with many carriers where coulomb , ferromagnetic , and single - particle energies contribute to the formation of the equilibrium state . using the quasi - classical description , we show that a qd may be split into three phases with different physical properties . the geometrical sizes of these phases are determined by the coulomb , ferromagnetic , and single - particle contributions to the chemical potential of a qd . for calculations , we employ the mean - field theory and the boltzmann kinetic equation . this approach becomes reduced to the thomas - fermi model at low temperatures . we should note that our approach ignores the discrete structure of single - particle spectrum of qd and is valid when electrons occupy at least several quantum levels . at the same time , this approach has an important advantage : it allows us to describe the coulomb effects in relatively large qds where , as it is shown below , the coulomb interaction is very strong and significantly exceeds the ferromagnetic interaction and the kinetic energy of carriers . the hole - mediated ferromagnetism in quasi - two - dimensional ( 2d ) systems is strongly anisotropic due to the heavy hole - light hole splitting in the valence band . therefore , the magnetic polarization occurs predominantly in the growth direction . then , two magnetic states of a qd with spin polarizations `` up '' and `` down '' can represent a single bit . presently qds and other nano - structures doped with magnetic ( mn ) impurities attract a lot of attention . among other studies , several recent theoretical papers investigate qds and their electron and excitonic states in the presence of a single mn ion @xcite . in particular , it was suggested in ref . @xcite that a qd with a single mn ion can act as a multi - qubit which can be controlled optically . another direction of research describes the magnetic states and polarons in qds with many mn ions and one or several carriers @xcite . among the above publications , the paper @xcite demonstrates that the coulomb - interaction effects in few - electron qds can determine a collective magnetic state of holes and mn spins . ferromagnetism and spin separation in digital layered structures and quantum wells were also studied in several experimental @xcite and theoretical publications @xcite . as a model system , we are going to use a qd `` made out of '' a 2d quantum well . such a zero - dimensional system can be fabricated by etching and lithographical methods . within the lithographical method , a qd can be defined , for example , by using the top metal gates ( fig . [ fig1]a ) . the number of carriers in such a qd is a voltage tunable parameter . . ( b ) calculated 2d spin density as function of the total 2d density of holes at various temperatures ; @xmath1 , @xmath2 , @xmath3 , @xmath4 , @xmath5 . curie temperature for the above parameters is about @xmath6 . ] to describe a state of many carriers confined in a qd , we start from the local properties of the coupled hole - mn system in a 2d quantum well . in our system , a mobile hole and mn spins experience an exchange interaction : @xmath7 , where @xmath8 is the exchange interaction constant , and @xmath9 and @xmath10 are the z - components of the spin operators for a mn spin and hole , respectively ; @xmath11 and @xmath12 are the coordinates of hole and @xmath13-impurity , respectively . the above operator describes the interaction between mn - spins and heavy holes and assumes a sufficiently large energy separation between the heavy - hole and light - hole quantized states in the valence band . the corresponding effective spin - dependent potential of a single hole has a form : @xmath14 where @xmath15}.\end{aligned}\ ] ] here @xmath16 is the net spin 2d density , @xmath17 are the 2d densities of the spin components , @xmath18 is the brillouin function , @xmath19 is the number of cations per unit cell , and @xmath20 . @xmath21 is the total 2d density in a quantum well . for the ground - state wave function in a quantum well , we will use @xmath22 , where @xmath23 is the quantum well width . the chemical potential for a 2d gas depends on @xmath24 and @xmath25 : @xmath26,\end{aligned}\ ] ] where @xmath27 . now we calculate the spin polarization : @xmath28 , \label{nspin}\end{aligned}\ ] ] where @xmath29 . the zener ferromagnetic phase transition occurs when eq . [ nspin ] has a nonzero solution . [ fig1]b shows the data for spin density @xmath30 for a gaas / algaas quantum well with the following parameters : @xmath1 , @xmath31 , @xmath3 , @xmath4 , @xmath5 . the above exchange parameter @xmath8 is comparable to that used in other publications on magnetic semiconductors ( see e.g. @xcite ) . since the exchange interaction is antiferromagnetic ( @xmath32 ) , the case @xmath33 corresponds to the negative average polarization of mn ions , @xmath34 . curie temperature can be analytically calculated in the high - density limit : @xmath35 ( @xmath36 ) . the in - plane potential in a lateral qd near its center can be approximated by the parabolic function : @xmath37 where @xmath38 is the electron charge and @xmath39 is a characteristic frequency of a confining potential . the potential @xmath40 determines a depth of a lateral potential well . in a qd defined by metal gates ( fig . [ fig1]a ) , the parameters @xmath40 and @xmath39 are functions of the gate voltages . in equilibrium , the carrier distribution function , which satisfies the boltzmann equation , has a form : @xmath41 where @xmath42 is the lateral position vector and @xmath43 is the in - plane momentum . the self - consistent scalar potential of a hole is composed of two terms : @xmath44 where @xmath45 is the electrostatic potential induced by a non - uniform spatial distribution of carriers , @xmath46 . in addition , the distribution function ( [ ff ] ) depends on the spin of hole through the exchange interaction which is a function of the local spin density , @xmath47 ( see eqs . [ uspin],[uspin0 ] ) . at the same time , the function @xmath48 itself is determined by the total local density of holes , @xmath49 , and is given by the numerical solution of eq . [ nspin ] ( see the data in fig . [ therefore , it is convenient to regard @xmath50 as a function of @xmath51 , i.e. @xmath52=u_{spin}[j_z , n_{spin}(n_{2d}))]=u_{spin}[j_z , n_{2d}({\bf r})]$ ] . by integrating the function ( [ ff ] ) over momenta we come to two non - local non - linear equations for the densities @xmath53 . then , these equations can be solved for the chemical potential and rewritten in the form resembling the central equation of the thomas - fermi model : @xmath54+k_b t \ln[e^{\frac{2\pi\hbar^2 n_{j_z}}{m_{hh } k_b t}}-1 ] , \hskip 0.5 cm j_z=\pm3/2 , \label{tf}\end{aligned}\ ] ] where @xmath55 and @xmath56 , where @xmath57 is an effective non - local dielectric constant of a system with metal gates @xcite . we should also note that @xmath45 was written as a 2d integral and this it valid if the lateral size of a qd is greater than the quantum - well width @xmath23 . in the system with the top gates closely located to a quantum well , the 2d integral in the equation for @xmath45 is reduced to a linear function of @xmath51 @xcite and is given by a local flat - capacitor formula : @xmath58 where @xmath59 is the distance between the qd plane and the top gate ; the distance to the back metal contact is assumed to be larger , i.e. @xmath60 ; @xmath61 is a dielectric constant of the semiconductor ( @xmath62 ) . the approximation ( [ fc ] ) has been successfully used in the past for description of several experiments on optical and electronic properties of modulated lateral structures @xcite . by using the local approximation for the self - consistent potential ( [ fc ] ) , we reduce eqs . [ tf ] to coupled non - linear local equations which should be solved numerically . the total number of holes in a qd is determined by the chemical potential @xmath63 and the lateral - potential depth @xmath40 . electrostatics of the structure under study ( fig . [ fig1]a ) is similar to that studied in refs . @xcite and we can use here the results of the above publications . if the barrier between the qd and back contact permits efficient tunnelling , the chemical potential in the qd coincides with the potential of the back contact ( i.e. @xmath64 ) . simultaneously , the front barrier ( usually made of algaas ) blocks tunnelling between the qd and the top gate . also , if @xmath65 , the potential @xmath40 becomes close to @xmath66 . [ fig2],[fig3 ] , and [ fig4 ] show numerical calculations for the local spin densities in a circular qd with @xmath67 , @xmath68 , and @xmath64 ; for the qd depth , we take @xmath69 , and @xmath70 . a qd with the minimum free energy is circularly symmetric and can be split into different phases . the total number of holes in a qd is given by an integral @xmath71 . the corresponding @xmath72 for the above values of @xmath40 are estimated as @xmath73 , and @xmath74 . in qds with relatively small @xmath72 ( fig.[fig2 ] ) , the system is split into ferromagnetic ( f ) and paramagnetic ( p ) phases . in fig . [ fig2 ] , the carriers with spin @xmath75 are pushed away from the center of qd , the total spin of holes is negative , and the mn - subsystem has a positive magnetization . this situation corresponds to the antiferromagnetic hole - mn coupling ( @xmath32 ) . with increasing the total number of carriers ( fig . [ fig3 ] ) , one can see the formation of another stripe within the ferromagnetic phase . this stripe is located between the center region of a qd and the paramagnetic phase and the holes in this stripe are almost fully spin - polarized . with further increasing @xmath72 ( fig . [ fig4 ] ) and for relatively low temperatures , the formation of the ferromagnetic stripe ( f2 ) with filly - polarized holes becomes evident . simultaneously , the paramagnetic stripe becomes very narrow . such a magnetic stricture of a qd can be understood by looking at the data in fig . [ fig1]b . at low temperatures , the function @xmath76 becomes very close to the linear function @xmath25 in an extended interval of @xmath25 . for example , at @xmath77 , @xmath78 for @xmath79 . in the above interval of @xmath25 at @xmath77 , the hole subsystem is almost completely spin polarized . it is interesting to estimate different types of energies contributing to the formation of stripes . it is easy to see that the coulomb energy in eqs . [ tf ] dominates the ferromagnetic and single - particle ( kinetic ) energies . the coulomb energy @xmath80 for @xmath81 while the spin energy @xmath82 for @xmath83 at @xmath77 . the single - particle kinetic energy under the similar conditions @xmath84 . for qds with more carriers and higher @xmath51 , the above energies become increased but the condition @xmath85 remains . we can also obtain analytic solutions of eqs . [ tf ] under certain conditions . if both spin subsystems of holes ( @xmath55 ) form a degenerate fermi gas , the last term in eqs . [ tf ] becomes proportional to the fermi energy , @xmath86 . then , we can sum up the equations for @xmath55 . the resulting equation does not contain the spin energy @xmath24 and can be solved analytically : @xmath87.\end{aligned}\ ] ] this formula can be applied , for example , to large qds in the spatial region of the ferromagnetic phase f1 , @xmath88 , where @xmath89 is the radius of the f1 phase ( see fig . [ fig4 ] ) . for this phase , spin densities can also be found analytically , by using the condition @xmath90 , where @xmath91 is a positive constant equal to @xmath30 at high @xmath25 in fig . [ fig1]b ; for @xmath77 , @xmath92 ( see fig . [ the formula ( [ na ] ) also describes the density distribution in the paramagnetic phase in the regions where the hole gas is degenerate ( @xmath93 ) . for many other regimes , the spin densities should be found numerically . since the coulomb energy dominates the magnetic and kinetic terms , the total radius of a qd can be well estimated from eq . [ na ] by putting @xmath94 . the resulting estimate @xmath95}$ ] is valid at low temperatures . experimentally , the stripe structure of a qd can be observed , for example , by spatially - resolved optical spectroscopy @xcite . in optical spectroscopy , a spatial resolution can be as small as @xmath96 @xcite . simultaneously optical emission is sensitive to the spin - polarization of carriers . in an optical experiment , a ferromagnetic qd would be excited with weak nonpolarized illumination ; the resulting local photoluminescence will be circularly polarized and reveal the formation of stripes with different magnetic structures . optical methods can also be used to write a magnetic state of qd ( bit : `` up''-``down '' ) . this may be done with circularly - polarized light . polarized optical pumping can bring a qd into a required collective magnetic state with spins `` up '' or `` down '' . in order to prepare a quantum dot in a required magnetic state , one can also use a magnetic field induced by an electric current driven through a metallic wire on the surface of a sample . another method to observe the magnetic phases in nano - structures is the electrical - capacitance spectroscopy @xcite which was successfully applied to observe , for example , compressible and incompressible stripes in electron quantum wires in the regime of the quantum hall effect @xcite . the capacitance spectroscopy has been also applied to lateral and self - assembled quantum dots ( see e.g. @xcite ) . for the nano - structures with relatively large sizes considered in this paper , the signature of the ferromagnetic phase in the capacitance spectra is expected to be relatively weak because of the inequality @xmath97 . however , the formation of the ferromagnetic phase can be recognized from a critical behavior of the capacitance spectrum as a function of temperature and voltage . as an example , we consider here quantum wires in a structure with the interdigitated metal gate ( see inset in fig . [ fig5 ] ) . in such a system , alternating voltages are applied to the metal strips located on the surface of a quantum well . similar structures were studied experimentally in ref.@xcite . we can calculate the capacitance of a wire as a derivative @xmath98 , where @xmath99 is the linear density of carriers in a quantum wire and @xmath100 is the length of a wire in the in - plane @xmath101-direction . in the local approximation for the coulomb potential ( eq . [ fc ] ) , the problems of quantum dot and wire become similar . fig . [ fig5 ] shows the capacitance of a quantum wire as a function of voltage for two temperatures , just below and above the curie temperature @xmath102 . one can see in fig . [ fig5 ] that at low temperatures ( @xmath103 ) the capacitance becomes increased starting from a critical voltage @xmath104 . this voltage corresponds to the minimum 2d density ( @xmath105 at @xmath77 ; see fig . [ fig1]b ) which is necessary to obtain the ferromagnetic phase stripe in the center of nanowire . starting from this voltage , the central region of a nanowire contains a ferromagnetic stripe . the capacitance of a partially ferromagnetic wire becomes increased since the spin interaction makes a lateral potential well a little deeper and a wire can accommodate more carriers at a given voltage . if temperature increases just by @xmath106 , the ferromagnetic stripe vanishes and capacitance becomes reduced . this peculiar temperature behavior for @xmath107 can be taken as an evidence for the ferromagnetic phase . and @xmath108 . the inset on the right hand side shows a top view of a nanowire with two phases . ] in this paper , we studied quantum dots and wires with many interacting carriers within the quasi - classical approach . the strongest interaction in quantum dots with a relatively weak confinement and a large number of carriers comes from the coulomb forces . however , a weaker ferromagnetic interaction determines the spin structure of a large quantum dot . depending on the parameters , a quantum dot can be split into three phases . the author would like to thank bruce mccombe for motivating discussions . this work was supported by ohio university and the a.v . humboldt foundation . semiconductor spintronics and quantum computation , edited by d. d. awschalom , d. loss , and n. samarth ( springer - verlag , berlin , 2002 ) ; i. zutic , j. fabian , and s. das sarma , rev . mod . 76 , 323 ( 2004 ) . a. d. giddings , m. n. khalid , t. jungwirth , j. wunderlich , s. yasin , r. p. campion , k. w. edmonds , j. sinova , k. ito , k .- y . wang , d. williams , b. l. gallagher , and c. t. foxon , phys . 94 , 127202 ( 2005 ) . y. d. park , a. t. hanbicki , s. c. erwin , c. s. hellberg , j. m. sullivan , j. e. mattson , t. f. ambrose , a. wilson , g. spanos , and b. t. jonker , science 295 , 651 ( 2002 ) ; h. ohno , d. chiba , f. matsukura , t. omiya , e. abe , t. dietl , y. ohno , k. ohtani , nature 408 , 944 ( 2000 ) . for the case of self - assembled quantum dots , see i.e. : h. drexler , d. leonard , w. hansen , j. p. kotthaus , and p. m. petroff , phys . 73 , 2252 ( 1994 ) ; p. schittenhelm , c. engel , f. findeis , g. abstreiter , a. a. darhuber , g. bauer , a. o. kosogov , and p. werner , j. vac . b 16 , 1575 ( 1998 ) . x. chen , m. na , m. cheon , s. wang , h. luo , b. d. mccombe , x. liu , y. sasaki , t. wojtowicz , j. k. furdyna , s. j. potashnik , and p. schiffer , appl . 81 , 511 ( 2002 ) . j. fernandez - rossier and l. j. sham , phys . rev . b 64 , 235323 ( 2001 ) ;
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many - particle electron states in semiconductor quantum dots with carrier - mediated ferromagnetism are studied theoretically within the self - consistent boltzmann equation formalism .
depending on the conditions , a quantum dot may contain there phases : partially spin - polarized ferromagnetic , fully spin - polarized ferromagnetic , and paramagnetic phases .
the physical properties of many - body ferromagnetic confined systems come from the competing carrier - mediated ferromagnetic and coulomb interactions .
the magnetic phases in gated quantum dots with holes can be controlled by the voltage or via optical methods .
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the _ egret _ experiment aboard the _ compton gamma ray observatory _ has detected more than 50 blazars extending out to redshifts greater than 2 ( thompson , _ et al . _ it is expected that @xmath0-rays from blazars with energies above the threshold energy for electron - positron pair production through interactions with low energy intergalactic photons will be annihilated , cutting off the high energy end of blazar spectra . such absorption is strongly dependent on the redshift of the source ( stecker , de jager & salamon 1992 ) . stecker & de jager ( 1997 ) have calculated the absorption of extragalactic @xmath0-rays above 0.3 tev at redshifts up to 0.54 and presented a comparison with the spectral data for the low redshift blazar mrk 421 . the study of blazar spectra at energies below 0.3 tev is a more complex and physically interesting subject . in addition to intergalactic absorption , one must be able to distinguish and to separate out the effects of intrinsic absorption and natural cutoff energies in blazar emission spectra . initial estimates of intergalactic absorption of 10 to 300 gev @xmath0-rays in blazar spectra at higher redshifts have been given by stecker ( 1996 ) , stecker & de jager ( 1996 ) and madau & phinney ( 1996 ) . however , in order to calculate such high - redshift absorption properly , it is necessary to determine the spectral distribution of the intergalactic low energy photon background radiation as a function of redshift as realistically as possible . this calculation , in turn , requires observationally based information on the evolution of the spectral energy distributions ( seds ) of ir through uv starlight from galaxies , particularly at high redshifts . conversely , observations of high - energy cutoffs in the @xmath0-ray spectra of blazars as a function of redshift , which may enable one to separate out intergalactic absorption from redshift - independent cutoff effects , could add to our knowledge of galaxy formation and early galaxy evolution . in this regard , it should be noted that the study of blazar spectra in the 10 to 300 gev range is one of the primary goals of a next generation space - based @xmath0-ray telescope , _ glast ( gamma - ray large area space telescope ) _ ( bloom 1996 ) as well as a number of ground - based @xmath0-ray telescopes currently under construction . our main goal is to calculate the opacity of intergalactic space to high energy @xmath0-rays as a function of redshift . this depends upon the number density of soft target photons ( ir to uv ) as a function of redshift , whose production is dominated by stellar emission . to evaluate the sed of the ir - uv intergalactic radiation field we must integrate the total stellar emissivity over time this requires an estimate of the dependence of stellar emissivity on redshift . previous work has either assumed that all of the background was in place at high redshifts , corresponding to a burst of star formation at the initial redshift ( stecker 1996 ; stecker & de jager 1996 ; macminn and primack 1996 ) or strong evolution ( similar to a burst ) , or no evolution ( madau and phinney 1996 ) . in this paper , we use a more realistic model which is consistent with recent observational data . pei & fall ( 1995 ) have devised a method for calculating stellar emissivity which bypasses the uncertainties associated with estimates of poorly defined luminosity distributions of evolving galaxies . the core idea of their approach is to relate the star formation rate directly to the evolution of the neutral gas density in damped ly@xmath2 systems , and then to use stellar population synthesis models to estimate the mean co - moving stellar emissivity @xmath3 ( erg / s-@xmath4-hz ) of the universe as a function of frequency @xmath5 and redshift @xmath6 ( fall , charlot & pei 1996 ) . our calculation of stellar emissivity closely follows this elegant analysis , with minor modifications as described below . damped ly@xmath2 systems are high - redshift clouds of gas whose neutral hydrogen surface density is large enough ( @xmath7 @xmath8 ) to generate saturated ly@xmath2 absorption lines in the spectra of background quasars that happen to lie along and behind common lines of sight to these clouds . these gas systems are believed to be either precursors to galaxies or young galaxies themselves , since their neutral hydrogen ( hi ) surface densities are comparable to those of spiral galaxies today , and their co - moving number densities are consistent with those of present - day galaxies ( wolfe 1986 ; see also peebles 1993 ) . it is in these systems that initial star formation presumably took place , so there is a relationship between the mass content of stars and of gas in these clouds ; if there is no infall or outflow of gas in these systems , the systems are `` closed '' , so that the formation of stars must be accompanied by a reduction in the neutral gas content . such a variation in the hi surface densities of ly@xmath2 systems with redshift is seen , and is used by pei & fall ( 1995 ) to estimate the mean cosmological rate of star formation back to redshifts as large as @xmath9 . pei & fall ( 1995 ) have estimated the neutral ( hi plus hei ) co - moving gas density @xmath10 in damped ly@xmath2 systems from observations of the redshift evolution of these systems by lanzetta , wolfe , & turnshek ( 1995 ) . ( here @xmath11 is the critical mass density of the universe . the deceleration parameter is assumed throughout to be @xmath12 , with cosmological constant @xmath13 . ) lanzetta , _ et al . _ have observed that while the number density of damped ly@xmath2 systems appears to be relatively constant over redshift , the fraction of higher density absorption systems within this class of objects decreases steadily with decreasing redshift . they attribute this to a reduction in gas density with time , roughly of the form @xmath14 , where @xmath15 is the current gas density in galaxies . pei & fall ( 1995 ) have taken account of self - biasing effects to obtain a corrected value of @xmath16 ; we have reproduced their calculations to obtain @xmath16 under the assumptions that the asymptotic , high redshift value of the neutral gas mass density is @xmath17 , where @xmath18 km / s - mpc ) . in a `` closed galaxy '' model , the change in co - moving stellar mass density @xmath19 , since the gas mass density @xmath10 is being converted into stars . this determines the star formation rate and consequent stellar emissivity ( pei & fall 1995 ) . to determine the mean stellar emissivity from the star formation rate , an initial mass function ( imf ) @xmath20 must be assumed for the distribution of stellar masses @xmath21 in a freshly synthesized stellar population . to further specify the luminosities of these stars as a function of mass @xmath21 and age @xmath22 , fall , charlot , & pei ( 1996 ) use the bruzual - charlot ( bc ) population synthesis models for the spectral evolution of stellar populations ( bruzual & charlot 1993 , charlot & bruzual 1991 ) . in these population synthesis models , the specific luminosity @xmath23 ( erg / s - hz ) , of a star of mass @xmath21 and age @xmath22 is integrated over a specified imf to obtain a total specific luminosity @xmath24 per unit mass ( erg / s - hz - g ) for an entire population , in which all stellar members are produced simultaneously ( @xmath25 ) . following fall , charlot , and pei ( 1996 ) , we have used in our calculations the bc model corresponding to a salpeter imf , @xmath26 , where @xmath27 . the mean co - moving emissivity @xmath28 is then obtained by convolving over time @xmath29 the specific luminosity @xmath30 with the mean co - moving mass rate of star formation , @xmath31 : @xmath32 note that the star mass formation rate @xmath33 that appears in this equation is not the same as @xmath34 , the change in total stellar mass density . this is because @xmath35 is the rate at which mass is _ permanently _ being converted into stars ; since some stellar mass is continuously being returned to the interstellar medium ( ism ) , the _ instantaneous _ mass rate of star formation @xmath36 is larger than @xmath35 , the two being related by @xmath37 where @xmath38 , provided by the bc models , is the fraction of the initial mass of a generation of stars formed at @xmath25 that has been returned to the ism . the bc models specific luminosities @xmath24 are calculated assuming that the metallicity content _ z _ during star formation is fixed at our current solar metallicity value ( @xmath39 ) . however , the metallicity content of the universe is not static , but evolves with redshift as early populations of stars return freshly synthesized metals to the interstellar medium during their various phases of mass loss . for example , in a survey of 1/3 of the known damped lyman - alpha absorbers , pettini _ ( 1994 ) found that the typical metallicity is 0.1 that of the present solar value at a redshift of @xmath40 . since the specific luminosity of a star of a given mass is also a function of its metallicity content ( lower metallicities give bluer spectra ) , the metallicity of a stellar population must be taken into account when integrating the mean emissivity over redshift . the effect of metallicity content in stellar population models has been examined by worthey ( 1994 ) . using the imf @xmath41 with @xmath42 , worthey has calculated the mass - to - light ratios @xmath43 as a function of population age @xmath22 and metallicity @xmath44 , for the color bands @xmath45 through @xmath21 . we have plotted his @xmath43 values for the @xmath45 and @xmath46 bands in figures 1 and 2 respectively . one can see that for a fixed metallicity , the logarithm of the luminosity decreases approximately linearly with the logarithm of population age , and that for a fixed age the @xmath45 and @xmath46 luminosities decrease as the metallicity increases . we have made a linear fit to each fixed - metallicity @xmath47 computed data set , obtaining a metallicity correction factor factor @xmath48 , where @xmath49 designates the color band . > from the parallel linear fits made to the computed data for each @xmath50 value , it is seen that a common correction factor , @xmath51 , applied to each @xmath52 data set will bring these data into rough agreement with the @xmath53 values of @xmath47 . these correction factors are plotted in the inset figures , whose abscissa is @xmath54 and whose ordinate is the correction factor . our fit to worthey s computed data in figure 1 gives a continuous correction factor @xmath55 for @xmath56 @xmath57 m , the center of the @xmath45 band . similar fits to the @xmath46 band data ( figure 2 ) and to the @xmath58 , @xmath59 , and @xmath60 band data ( not shown ) result in the relation @xmath61\approx\left[0.33-\frac{0.30}{\lambda } \right]\log\left(\frac{z}{z_{\odot}}\right ) + \left[0.066-\frac{0.063}{\lambda}\right]\left [ \log\left(\frac{z}{z_{\odot}}\right)\right]^{2},\ ] ] for @xmath62 m @xmath63 m . outside of this wavelength region we take @xmath64 and @xmath65 . note that increased metallicity gives a redder population spectrum ( bertelli _ et al . _ 1994 ) . limitations to this correction factor include the fact that worthey s calculations only apply to stars with ages greater than @xmath66 gyr , and that the upper mass limit of his imf ( @xmath67 ) is much lower than that of the bc model which we employ ( @xmath68 ) . additional uncertainty exists below 0.3 @xmath57 m since worthey s calculations extend only to the @xmath45 band . we have chosen to assume a constant enhancement factor below @xmath69 m . for all of the above reasons , our enhancement factor @xmath51 is really a conservative lower limit to the corrections to the bc models in the ultraviolet . population synthesis models in which varying metallicity is included do exist ( bertelli _ et al . _ 1994 ) , and efforts to reconcile differences in computed spectra generated by these various models have been made ( charlot , worthey , and bressan , 1996 ) . the emissivity @xmath70 given in eq . [ emissivity.eq ] assumes that all stellar emission escapes from the gas system which contains the stars . however , some absorption of stellar radiation occurs both by dust and gas within the larger damped ly@xmath2 systems . above the lyman limit , this absorption is dominated by dust , while below the lyman limit , absorption by neutral hydrogen and singly - ionized helium dominates . defining the mean transmission fractions , averaged over the optical depths of damped ly@xmath2 systems , by @xmath71 and @xmath72 , the final expression for the effective stellar emissivity is @xmath73.\ ] ] the distribution of optical depths @xmath74 of ly@xmath2 clouds due to dust is can be adequately represented by @xmath75 , where @xmath76 , @xmath77 being a characteristic ( redshift dependent ) cloud dust opacity , and @xmath78 ( fall _ et al . _ 1996 ) . under the assumption that both dust and stars are uniformly distributed throughout each ly@xmath2 cloud , the fraction of radiation @xmath79 produced by stars in a given cloud of optical depth @xmath74 that escapes dust absorption is given by @xmath80,\ ] ] where @xmath81 , and @xmath82 is the average albedo of dust , taken to be the same value as in our galaxy ( @xmath83 to 0.6 ; whittet 1992 ) . ( we have calculated eq . [ twostream.eq ] using the 2-stream approximation [ chandrasekhar 1950 ] ) . we note that the dust opacity @xmath74 in eq.[twostream.eq ] is assumed to be proportional to the hi surface column density @xmath84 and metallicity @xmath44 , @xmath85 where @xmath86 is the normalized galactic interstellar dust extinction curve ( savage and mathis 1979 ) . integrating eq.[twostream.eq ] over the ly@xmath2 opacity distribution function @xmath87 of pei & fall ( 1995 ) , we obtain @xmath71 , and find it to have a minor effect on the emissivity , @xmath71 being typically of order unity . below the lyman limit ( @xmath88 m ) , the opacity is dominated by neutral gas absorption : @xmath89 , where @xmath90 and @xmath91 are the hi and hei photoionization cross sections ( osterbrock 1989 ) . with the @xmath92 and @xmath93 distributions of the ly@xmath2 systems being related to the dust opacity distribution @xmath94 through eq.[c ] , the distribution for @xmath95 can be obtained . integrating eq.[twostream.eq ] ( now with @xmath96 ) , weighted with the @xmath95 distribution , gives @xmath72 . figure 3 shows the calculated stellar emissivity as a function of redshift at 0.28 @xmath57 m , 0.44 @xmath57 m , and 1.00 @xmath57 m , both with and without the metallicity correction factor @xmath97 . we have also plotted the observations of the cosmic emissivity by the canada - french redshift survey ( lilly , le fevre , hammer , & crampton 1996 ) at these rest - frame wavelengths for comparison . with a lower mass cutoff of @xmath98 in the imf , we obtain emissivities which are roughly a factor of 2 higher than those obtained by lilly , _ ( 1996 ) . to bring our emissivities down to the observed values requires that we reduce the lower mass limit in the imf to @xmath99 , which puts a fraction ( 0.45 ) of the mass into effectively nonluminous compact objects . we note that a similar reduction was achieved by fall , _ et al . _ ( 1996 ) by modifying the power law index in the imf ; a higher index results in a lower emissivity ( pei 1996 , personal communication ) . overall , our emissivities , both with and without the metallicity corrections , are in reasonable agreement with the data at lower redshifts ( lilly , _ et al . _ although the differences for @xmath70 between the no - metallcity and metallicity cases for @xmath100 are not great , _ they become substantial at larger redshifts for both optical and uv wavelengths_. this has notable effects on the opacity of the radiation background to high energy @xmath0-rays , as will be seen in section 4 . we note that our dotted - line curves in figure 3 ( no metallicity correction ) are essentially a reproduction of the emissivities calculated by fall , _ et al . _ ( 1996 ) . in all cases as shown in figure 3 , the stellar emissivity in the universe peaks at @xmath101 , dropping off at both lower and higher redshifts . indeed , madau , _ et al . _ ( 1996 ) have used observational data from the hubble deep field to show that metal production has a similar redshift distribution , such production being a direct measure of the star formation rate . ( see also the review by madau ( 1996 ) . ) the co - moving radiation energy density @xmath102 ( erg/@xmath4-hz ) is the time integral of the co - moving emissivity @xmath3 , @xmath103 where @xmath104 and @xmath105 is the redshift corresponding to initial galaxy formation . the extinction term @xmath106 accounts for the absorption of ionizing photons by the clumpy intergalactic medium ( igm ) that lies between the source and observer ; although the igm is effectively transparent to non - ionizing photons , the absorption of photons by hi , hei and heii can be considerable ( madau 1995 ) . the presence of damped ly@xmath2 and lyman - limit systems ( lanzetta , _ et al . _ 1995 ) and the lyman - alpha forest , coupled with the absence of a hi gunn - peterson effect ( gunn & peterson 1965 ; steidel & sargent 1987 ) indicates that essentially all of the hi , hei , and heii exists within intergalactic clouds whose measured hi column densities range from approximately @xmath107 to @xmath108 @xmath8 . the effective optical depth @xmath109 between a source at redshift @xmath110 and an observer at redshift @xmath6 owing to poisson - distributed intervening lyman - alpha clouds is given by ( paresce , mckee , & bowyer 1980 ) @xmath111 where @xmath112 $ ] , @xmath113 , and @xmath114 is the distribution function of clouds in redshift @xmath6 and column density @xmath92 . as pointed out by madau & shull ( 1996 ) , when @xmath115 , @xmath109 is just the mean optical depth of the clouds ; when @xmath116 , @xmath109 becomes the number of optically thick clouds between the source and observer , so that the poisson probability of encountering no thick clouds is @xmath106 , as required . for the distribution function of lyman - alpha clouds we use the parameterization of madau ( 1995 ) ( see also miralda - escud & ostriker , 1990 , model a2 ) : @xmath117 using eqs . [ e ] and [ f ] and the stellar emissivity @xmath3 in eq.[b ] , we obtain the background energy density @xmath102 , shown in figures 4 and 5 , calculated with and without the metallicity correction , @xmath51 , respectively . these also include the contribution to the uv background from qsos ( madau 1992 ) , which are believed to dominate the diffuse background radiation below the lyman limit and to be responsible for the early ( @xmath118 ; see schneider , schmidt , & gunn 1991 ) reionization of the igm . although it is possible that uv emission from qsos alone may be able to account for the nearly complete reionization of the igm ( meiksin & madau 1993 ; fall & pei 1993 ; madau & meiksin 1994 ) , it has been argued that additional sources of of ionizing radiation are required ( miralda - escud & ostriker 1990 ) , these perhaps being young galaxies which leak a fraction ( up to @xmath119 ) of their ionizing radiation through hii `` chimneys '' ( dove & shull 1994 ; madau & shull 1996 ) . we have therefore assumed in our calculations that 15% of the stellar emission escapes from the galaxies ( protogalaxies ) through these chimneys , unattentuated by dust or gas . ( we note , however , that recent observations of four starburst galaxies by the hopkins uv telescope ( leitherer _ et al . _ 1995 ) indicate that less than 3% of lyman continuum photons escape from these sources . ) figures 4 and 5 indicate that in our calculation the @xmath120 m background is indeed dominated by qsos , so that the actual value of the escape fraction we choose is not too significant . the intergalactic energy densities given in figures 4 and 5 are quite consistent with the present upper limits in the uv ( martin & bowyer 1989 ; mattila 1990 ; bowyer 1991 ; vogel , weymann , rauch & hamilton 1995 ) . it should be noted that our results as shown in figures 4 and 5 give emissivities _ from starlight only _ and do not include dust emissivities in the mid - infrared and far - infrared . with the co - moving energy density @xmath102 evaluated , the optical depth for @xmath0-rays owing to electron - positron pair production interactions with photons of the stellar radiation background can be determined from the expression ( stecker , _ et al . _ 1992 ) @xmath121\sigma_{\gamma\gamma}[s=2e_{0}h\nu x(1+z)^2],\ ] ] where @xmath122 is the observed @xmath0-ray energy , @xmath123 is the redshift of the @xmath0-ray source , @xmath124 , @xmath125 being the angle between the @xmath0-ray and the soft background photon , @xmath126 is planck s constant , and the pair production cross section @xmath127 is zero for center - of - mass energy @xmath128 , @xmath129 being the electron mass . above this threshold , @xmath130,\ ] ] where @xmath131 . figures 6 and 7 show the opacity @xmath132 for the energy range 10 to 500 gev , calculated with and without the metallicity correction . extinction of @xmath0-rays is negligible below 10 gev . above 500 gev , interactions with photons with wavelengths of tens of @xmath57 m become important , so that one must include interactions from infrared photons generated by dust reradiation ( stecker & de jager 1997 ) , which we have neglected here . for 300 gev @xmath0-rays , at redshifts below 0.5 , our opacities agree with the with the opacities obtained by stecker & de jager ( 1997 ) . our calculated opacity , even with the metallicity correction , is probably somewhat low in the 10 to 30 gev energy range , because we have underestimated the value of @xmath51 in the uv ( see previous discussion ) . note that these calculated opacities are _ independent _ of the value chosen for @xmath133 , as seen in eqs . 4 , 7 , and 10 . the emissivity @xmath70 in eq . 4 scales as @xmath134 , since neither @xmath30 , @xmath51 nor @xmath135 depends on @xmath133 , while @xmath136 scales as @xmath134 . eq . 7 shows then that @xmath137 scales as @xmath133 , and in eq . 10 this @xmath133 factor is cancelled by the integration over time @xmath29 . with the @xmath0-ray opacity @xmath132 calculated out to @xmath138 , the cutoffs in blazar @xmath0-ray spectra caused by extragalactic pair production interactions with stellar photons can be predicted . figure 8 shows the effect of the intergalactic radiation background on a few of the @xmath0-ray blazars ( `` grazars '' ) observed by _ egret _ , _ viz . _ , 1633 + 382 , 3c279 , 3c273 , and mrk 421 . we have assumed that the mean spectral indices obtained for these sources by _ egret _ extrapolate out to higher energies attenuated only by intergalactic absorption . observed cutoffs in grazar spectra may be intrinsic cutoffs in @xmath0-ray production in the source , or may be caused by intrinsic @xmath0-ray absorption within the source itself . whether cutoffs in grazar spectra are primarily caused by intergalactic absorption can be determined by observing whether the grazar cutoff energies have the type of redshift dependence predicted here . figure 8 indicates that the next generation of satellite and ground - based @xmath0-ray detectors , both of which will be designed to explore the energy range between 10 and 300 gev , will be able to reveal information about low - energy radiation produced by galaxies at various redshifts and at different stages in their evolution . our opacity calculations have implications for the determination of the origin of @xmath0-ray bursts , if such bursts are cosmological . as indicated in figure 6 , @xmath0-rays above an energy of @xmath139 15 gev will be attenuated if they at emitted at a redshift of @xmath139 3 . on 17 february 1994 , the _ egret _ telescope observed a @xmath0-ray burst which contained a photon of energy @xmath139 20 gev ( hurley , _ et al._1994 ) . if one adopts the opacity results which include our conservative metallicity correction ( figure 6 ) , this burst would be constrained to have originated at a redshift less than @xmath1392 . ( an estimated redshift constraint of @xmath139 1.5 was given by stecker and de jager ( 1996 ) , based on a simpler model . ) future detectors may be able to place redshift constraints on bursts observed at higher energies . in a previous paper ( stecker & salamon 1996 ) , we presented a model for calculating the extragalactic @xmath0-ray background ( egrb ) due to unresolved grazars . we gave results for @xmath0-ray energies up to 10 gev ( where there is effectively no @xmath0-ray absorption ) which were compared to preliminary _ egret _ data ( kniffen _ et al . _ 1996 ) using the intergalactic @xmath0-ray opacities calculated here , we can now extend the results of this egrb model out to an energy of 0.5 tev . our egrb model assumes that the grazar luminosity function is related to that of flat spectrum radio quasars ( fsrq ) , so that we can use fsrq luminosity and redshift distributions ( dunlop and peacock , 1990 ) to obtain a grazar luminosity function . the effects of grazar flaring states , @xmath0-ray spectral index variation , and redshift dependence are also been included in this model ; see stecker and salamon ( 1996 ) for details . by integrating the grazar luminosity function weighted by our new opacity results , we obtain a grazar background spectrum up to 500 gev which properly includes the effect of @xmath0-ray absorption . figure 9 shows this egrb spectrum compared with the preliminary _ egret _ data . note that the spectrum is concave at energies below 10 gev , reflecting the dominance of hard - spectrum grazars at high energies and softer - spectrum grazars at low energies ; it then steepens above 20 gev , owing to extragalactic absorption by pair - production interactions with radiation from external galaxies , particularly at high redshifts . both the concavity and the steepening are signatures of a blazar dominated @xmath0-ray background spectrum . because the extragalactic @xmath0-ray background in our model is made up of a superposition of _ lower - luminosity , unresolved _ grazars , its intensity is determined by the number of sources in the universe which are below the detection threshhold of a particular telescope . a telescope with a superior point source sensitivity gives a higher source count , thereby reducing the number of unresolved sources which constitute the diffuse @xmath0-ray background . in figure 9 , the upper spectra which are close to the _ egret _ data are obtained using the _ egret _ threshold ; the lower curves correspond to the projected sensitivity of the proposed next generation _ glast _ satellite detector , which is expected to have a detection threshhold of @xmath140 @xmath8s@xmath141 above 0.1 gev . it should also be noted that above 10 gev , blazars may have natural cutoffs in their source spectra ( stecker , de jager & salamon 1996 ) and intrinsic absorption may also be important in some sources ( protheroe & biermann 1996 ) . thus , above 10 gev our calculated background flux from unresolved blazars , shown in figure 9 , may actually be an upper limit . the nature of the dark matter in the universe is one of the most important fundamental problems in astrophysics and cosmology . the non - baryonic mixed dark matter model with a total @xmath142 ( shafi & stecker 1984 ) gave predictions for fluctuations in the cosmic background radiation ( schaefer , shafi & stecker 1989 ; holtzman 1989 ) which were found to be in good agreement with the later cobe measurements . the best agreement appears to be found for @xmath139 20% hot dark matter , of which massive neutrinos are the most likely candidates , and @xmath139 80% cold dark matter ( pogosian & starobinsky 1993 , 1995 ; ma & bertchinger 1994 ; klypin , _ et al._1995 ; primack , _ et al . _ 1995 ; liddle , _ et al . _ 1996 ; babu , schaefer & shafi 1996 ) . the most popular cold dark matter particle candidates are the lightest sypersymmetric particles ( lsps ) , the neutralinos ( hereafter designated as @xmath143 particles ) . cosmologically important @xmath143 particles must annihilate with a weak cross section , @xmath144 10@xmath145 @xmath146s@xmath141 ; calculations show that such cross sections lead to a value for @xmath147 with @xmath148@xmath149@xmath150 . the fact that supersymmetry neutralinos are predicted to have such weak annihilation cross sections is an important reason why they have become such popular dark matter candidates . preliminary lep 2 results give a lower limit on the mass of the @xmath143 of @xmath151 gev ( ellis , falk , olive & schmitt 1996 ) . in the minimal supersymmetry model ( mssm ) , @xmath143 can be generally described as a superposition of two gaugino states and two higgsino states . grand unified models with a universal gaugino mass generally favor states where @xmath143 is almost a pure b - ino ( @xmath152 ) ( _ e.g. _ diehl , _ et al . _ 1995 ) , but other states such as photinos and higgsinos are generally allowed by the theory . kane & wells ( 1996 ) have presented possible accelerator evidence from cdf that @xmath143 may be a higgsino of mass @xmath139 40 gev . dark matter neutralinos will produce @xmath0-rays by mutual pair annihilation . this process is expected to occur because neutralinos are majorana fermions , _ i.e. _ , they are their own antiparticles . indeed , most of this mutual annihilation would have occured in the very early universe , a process which determines the present ( `` freeze out '' ) value of @xmath148 and leads to the relation @xmath153 , where the bracketed quantity is the thermal - averaged annihilation cross section times velocity ( ellis , _ et al . _ 1984 this leads to the relation that the @xmath0-ray flux from neutralino annihilation is inversely proportional to @xmath148 . thus , the annihilation @xmath0-ray flux is limited from below by cosmological constraints on the maximum value of @xmath148 . there are two types of @xmath0-ray spectra produced by @xmath154 annihilations , _ viz . _ , ( 1 ) @xmath0-ray continuum spectra from the decay of secondary particles produced in the annihilation process , and ( 2 ) @xmath0-ray lines , produced primarily from the process @xmath155 ( _ e.g. _ , rudaz 1989 ) . the cosmic @xmath0-ray flux from @xmath154 annihilation is proportional to the line - of - sight integral of the _ square _ of the @xmath143 particle density times @xmath156 . the continuum @xmath0-ray production spectra from @xmath154 annihilation can be calculated for different types of neutralinos by starting with the appropriate branching ratios for annihilation into fermion - antifermion pairs which produce hadronic cascades leading to the subsequent production and decay of neutral pions ( rudaz & stecker 1988 ; stecker 1988 ; stecker & tylka 1989 ) . stecker & tylka ( 1989 ) discuss in detail the various channels involved in continuum @xmath0-ray production via @xmath154 annihilation and give the resulting spectra for some lower mass @xmath143 particles . such continuum fluxes from @xmath154 annihilations would be difficult to observe above the extrapolated cosmic background which we show in figure 9 . however , with good enough sensitivity and energy resolution , it might be possibile to observe a two - photon annihilation line from @xmath154 annihilation . the general considerations for observability of this line were discussed by rudaz & stecker ( 1991 ) . we update this discussion here , using ( 1 ) our new calculation of the @xmath0-ray background flux from blazars shown in figure 9 , ( 2 ) recent accelerator limits on supersymmetric particle masses , and ( 3 ) the proposed sensitivity and energy resolution of a next generation space based @xmath0-ray telescope taken from the _ glast _ proposal ( elliot 1996 ) . the energy of the @xmath155 decay line is @xmath157 . the line width is given by doppler broadening . for galactic halo particles , this width is roughly @xmath158 @xmath159 , much smaller than the energy resolution proposed for any future @xmath0-ray telescope . upper and lower limits on @xmath160 yield lower and upper limits on the @xmath0-ray line flux respectively ( see above ) . other limits can be obtained in flux - energy space ( rudaz & stecker 1991 ) . accelerator determined lower limits on @xmath161 give lower limits on the line energy . lower limits on the mass of the sfermion exchanged in the annihilation process give upper limits on @xmath162 since @xmath162 @xmath163 @xmath164 . in fact , since the particle density @xmath165 and @xmath162 @xmath163 @xmath166 , the predicted annnihilation line flux @xmath167 . further limits are obtained from the inequality @xmath168 , which is the tautology following from the condition that @xmath143 be the lsp . if we assume that annihilations occur mainly through slepton exchange , _ i.e. _ , @xmath169 , we can obtain an upper limit on the 2@xmath0 line flux . this is because lep 1.5 gives a lower limit of @xmath139 70 gev on the slepton mass ( de boer , miquel , pohl & watson 1996 ) , whereas the substantially higher squark mass lower limit of @xmath139 150 gev would imply much lower fluxes , since @xmath170 @xmath171 . the lower limit on the slepton mass implies an upper limit on the line flux from @xmath172 annihilation such that the event rate for a next generation @xmath0-ray telescope with an aperture of 1 m@xmath173sr would be about 5 photons per year for a line in the energy range between 20 and 100 gev . ( if the @xmath143 particles are higgsinos , the event rate would be much lower . ) using all of these constraints , the allowed region for a neutralino annihilation line in flux - energy space is plotted in figure 10 . in constructing figure 10 , we have used the _ glast _ proposed estimate of the point source sensitivity after a one - year full sky survey to estimate the background from unresolved faint blazars ( see figure 9 and the discussion in the previous section ) . we then obtain the background photon number for an appropriate exposure factor of 1 m@xmath173yr - sr and energy resolution of 10% , and plot the square root of this number , which represents the natural background fluctuations above which a line must be observed . of course , a higher exposure factor would reduce the point source background and increase the sensitivity to a line flux , as would a better energy resolution . it should also be noted that the background above 10 gev shown in figures 9 and 10 may be overestimated ( see previous section ) . another possible way in which dark matter may produce @xmath0-rays and neutrinos is if the lsp is allowed to decay to non - supersymmetric , ordinary particles . supersymmetry theories involve a multiplicative quantum number called _ r - parity _ , which is defined so that it is even for ordinary particles and odd for their supersymmetric partners . thus , if r - parity is conserved , as is usually assumed , the lsp is completely stable , making it a potential dark matter candidate . however , such may not be the case . r - parity may be very weakly broken , allowing the lsp to decay with branching ratios involving @xmath0-rays and neutrinos ( _ e.g. _ , berezinsky , masiero & valley 1991 ) . for @xmath143 particles to be the dark matter , their decay time should be considerably longer than the age of the universe . the possible radiative decay @xmath174 will give a @xmath0-ray line with energy @xmath175 . such a line has the potential of being more intense than the annihilation line . whereas the @xmath154 annihilation rate and consequent line flux is cosmologically limited by requiring @xmath148 to be a significant fraction near 1 ( see previous discussion ) , the decay - line flux is limited only by the particular physical supersymmetry model postulated and constraints from related accelerator and astrophysical data . thus , invocation of @xmath143 decay involves a higher order of particle theory model building and speculation . we only wish to mention here that there is a possibility that a decay line may be sufficiently intense to be observable above the background . we have calculated the @xmath0-ray opacity as a function of both energy and redshift for redshifts as high as 3 by taking account of the evolution of both the sed and emissivity of galaxies with redshift . in order to accomplish this , we have adopted the recent analysis of fall , _ et al . _ ( 1996 ) and have also included the effects of metallicity evolution on galactic seds . we have then considered the effects of the @xmath0-ray opacity of the universe on @xmath0-ray bursts , blazar spectra , and on the extragalactic @xmath0-ray background from blazars . in particular , we find that the 17 feb . 1994 _ egret _ burst probably originated at @xmath176 . because the stellar emissivity peaks between a redshift of 1 and 2 , the @xmath0-ray opacity which we derive shows little increase at higher redshifts . this weak dependence indicates that the opacity is not determined by the initial epoch of galaxy formation , contrary to the speculation of macminn and primack ( 1996 ) . the extragalactic @xmath0-ray background , which can be accounted for as a superposition of spectra of unresolved blazars , and which we have predicted to be concave between 0.03 and 10 gev ( stecker & salamon 1996 ) , should steepen significantly above 20 gev owing to our estimates of extragalactic @xmath0-ray absorption at moderate to high redshifts . both the predicted concavity and steepening may be too subtle to detect with present data from _ egret_. however , next generation @xmath0-ray telescopes which are presently being designed , such as _ glast _ , may be able to observe these features and thereby test the blazar background model . we also discuss the possible observability of dark matter lines in the multi - 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[ atten_emiss.eq ] , for three wavelength values , @xmath1880.28 , 0.44 , and 1.0 @xmath57 m , for a hubble constant value of @xmath189 . note that the emissivity scales as @xmath134 ( _ cf . _ eq . 4 and section 3 ) . solid line curves are for the case where the metallicity correction factor ( @xmath51 from eq . [ worthey.eq ] ) is used ; dashed lines give the emissivity when this correction factor is _ not _ included . the data from the canada - french redshift survey ( lilly _ et al . _ 1996 ) are also plotted . figure 4 : the intergalactic radiation energy density from stars and qsos as a function of wavelength for redshifts @xmath6 of 0 , 1 , 2 , and 3 , for a hubble constant value of @xmath189 ( the energy density scales as @xmath133 ; see eq . 7 and section 3 ) these densities are calculated with the metallicity correction factor , @xmath51 , included . figure 6 : the opacity @xmath190 of the universal soft photon background to @xmath0-rays as a function of @xmath0-ray energy and source redshift . these curves are calculated with the metallicity correction factor included in the expression for stellar emissivity . as discussed in the text , these results are independent of the value chosen for @xmath133 . figure 8 : the effect of intergalactic absorption by pair - production on the power - law spectra of four prominent grazars : 1633 + 382 , 3c279 , 3c273 , and mrk 421 . the solid ( dashed ) curves are calculated with ( without ) the metallicity correction factor . figure 9 : the extragalactic @xmath0-ray background energy spectrum from unresolved grazars . the top and bottom sets of curves correspond to point - source sensitivities of @xmath191 and @xmath192 @xmath8s@xmath141 , respectively , for @xmath0-ray energies above 0.1 gev , corresponding to the approximate point - source sensivities of the _ egret _ and _ glast _ detectors respectively . because the fsrq luminosity fuction that we employ scales as @xmath193 ( dunlop and peacock , 1990 ) , our calculated egrb spectrum scales as @xmath134 ( see eq . 10 in stecker and salamon , 1996 ) . figure 10 . the dot - dash polygon shows the allowed region of expected @xmath0-ray photon counts calculated for a bino ( @xmath152 ) annihilation line as a function of @xmath194 . present accelerator and cosmological constraints are indicated by the labels on the sides of the polygon ( see text ) . in the figure labels , the letter `` b '' represents the bino ( @xmath152 ) and the letter `` l '' represents the slepton ( @xmath195 ) . as an illustration of how to read the figure , the arrow within the polygon indicates the line flux upper limit for a bino of mass 100 gev . an exposure factor of 1 m@xmath173sr yr and an energy resolution of 10% are assumed . we also show the background fluctuation count rate appropriate to these parameters for the lower set of flux curves ( _ i.e. _ with and without the metallicity correction ) shown in figure 9 ( see text ) .
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in this paper , we extend previous work on the absorption of high energy @xmath0-rays in intergalactic space by calculating the absorption of 10 to 500 gev @xmath0-rays at high redshifts .
this calculation requires the determination of the high - redshift evolution of the intergalactic starlight photon field , including its spectral energy distribution out to frequencies beyond the lyman limit . to estimate this evolution ,
we have followed a recent analysis of fall , charlot & pei , which reproduces the redshift dependence of the starlight background emissivity obtained by the canada - france redshift survey group .
we also include the uv background from quasars .
we give our results for the @xmath0-ray opacity as a function of redshift out to a redshift of 3 .
we also give predicted @xmath0-ray spectra for selected blazars and extend our calculations of the extragalactic @xmath0-ray background from blazars to an energy of 500 gev with absorption effects included .
our results indicate that the extragalactic @xmath0-ray background spectrum from blazars should steepen significantly above 20 gev , owing to extragalactic absorption .
future observations of a such a steepening would thus provide a test of the blazar origin hypothesis for the @xmath0-ray background radiation .
we also note that our absorption calculations can be used to place limits on the redshifts of @xmath0-ray bursts ; for example , our calculated opacities indicate that the 17 feb .
1994 burst observed by _
egret _ must have originated at @xmath1 .
finally , our estimates of the high - energy @xmath0-ray background spectrum are used to determine the observability of multi - gev @xmath0-ray lines from the annihilation of supersymmetric dark - matter particles in the galactic halo .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the bifurcation of limit cycles by perturbing a planar system which has a continuous family of _ cycles _ , i.e. periodic orbits , has been an intensively studied phenomenon ; see for instance @xcite and references therein . the simplest planar system having a continuous family of cycles is the linear center , and a special family of its perturbations is given by the generalized polynomial linard systems : @xmath0 where @xmath1 , @xmath2 , @xmath3 , @xmath4 and @xmath5 are polynomials for @xmath6 , and @xmath7 is a small parameter . the classical and generalized linard systems appear very often in several branches of science and engineering , as biology , chemistry , mechanics , electronics , etc . see for instance @xcite and references therein . in particular linard systems are frequent specially in physiological processes , see for instance @xcite . in addition , the family of generalized polynomial linard systems is one of the most considered families in the study of limit cycles , see @xcite . we assume that @xmath8 , @xmath9 , @xmath10 , and @xmath11 . for a small enough @xmath7 , let @xmath12 be the maximum number of limit cycles of that bifurcate from cycles of the _ linear center _ @xmath13 , i.e. the maximum number of _ medium amplitude limit cycles _ which can bifurcate from @xmath13 under the perturbation . if @xmath14 , then @xmath15 does not depend on @xmath16 ; hence we only write @xmath17 . the main problem concerning @xmath12 is finding its exact value . we know from @xcite that @xmath18 $ ] , where @xmath19 $ ] denotes the integer part function . moreover , by following ( * ? ? ? * theorem 3.1 ) we can prove that @xmath20 $ ] for @xmath21 ; theorem [ mth1 ] ( below ) is a generalization of this result . also , we know from @xcite that @xmath22 $ ] , @xmath23 , \left[m/2\right]+\left[n/2\right]-1\right\}$ ] , and @xmath24 - 1 $ ] . however , the exact values of @xmath25 , @xmath26 , and @xmath27 were not reported there . in this paper we give the exact value of @xmath12 for two subfamilies of . more precisely , we will give the exact value of @xmath28 and @xmath29 , where @xmath28 is the value of @xmath12 by assuming that @xmath4 is odd for @xmath30 , and @xmath29 is the value of @xmath12 by assuming that @xmath5 is even for @xmath31 , where @xmath32 with @xmath33 is the smallest integer such that @xmath34 . of course , if @xmath35 , then @xmath36 . our main result is the following : [ mth1 ] @xmath37 @xmath38 $ ] . @xmath39 @xmath29 is either @xmath40 $ ] if @xmath41 is odd or @xmath42 + \left[\frac{n}{2}\right]-1 $ ] if @xmath41 is even . the assumptions on @xmath4 and @xmath5 in definitions of @xmath28 and @xmath29 , respectively , are necessary . otherwise , we can construct systems having more medium amplitude limit cycles , see remark 1 in section [ sec3 ] . theorem [ mth1](@xmath43 ) is a generalization of theorem 1.1 in @xcite , where the case @xmath44 was considered . we note that in such a case @xmath45 . hence theorem 1@xmath39 ( ( * ? ? ? * theorem 1.1 ) ) gives the exact value of @xmath46 . the proof of theorem [ mth1 ] is based on computing the maximum number of isolated zeros of the first non - vanishing poincar pontryagin melnikov function of the displacement function of , by taking into account the restrictions : @xmath4 odd for @xmath30 and @xmath5 even for @xmath47 , respectively . the paper is organized as follows . in section [ ppmf ] we recall the definition of the displacement function of , as well as the algorithm to compute the poincar pontryagin melnikov functions . preliminary results that allow us to provide elementary proofs of the main results are given in section [ sec3 ] . finally , in section [ sec4 ] we will prove theorem [ mth1 ] . the linear center @xmath13 is the hamiltonian system associated to the polynomial @xmath48 ; hence its cycles are the circles @xmath49 with @xmath50 . by using @xmath51 as a parameter , the first return map of can be expressed in terms of @xmath7 and @xmath51 : @xmath52 . therefore the corresponding _ displacement function _ @xmath53 is analytic for small enough @xmath7 and can be written as the power series in @xmath7 @xmath54 where @xmath55 with @xmath6 is the _ poincar pontryagin melnikov function _ of order @xmath56 , which is defined for @xmath57 . let @xmath58 with @xmath59 be the first non - vanishing coefficient in . the zeros of @xmath58 are important in the study of medium amplitude limit cycles of because of the _ poincar pontryagin andronov criterion _ : the maximum number of isolated zeros , counting multiplicities , of @xmath58 is an upper bound for @xmath12 . furthermore each simple zero @xmath60 of @xmath58 corresponds to one and only one limit cycle of with @xmath7 small enough bifurcating from the cycle @xmath61 . we know from @xcite that @xmath58 has at most @xmath62 $ ] positive zeros , counting multiplicities . however , this result does not give the value of @xmath12 because the upper bound for @xmath63 depending on @xmath64 , @xmath65 , @xmath41 , and @xmath16 is unknown . now , we will recall the algorithm to compute the functions @xmath55 . system can be written as @xmath66 where @xmath67 , or equivalently as @xmath68 as we know , @xmath69 is given by the classical poincar pontryagin formula @xmath70 the result for computing the higher order poincar pontryagin melnikov functions is the following : [ th3 ] * ( yakovenko franoise iliev algorithm @xcite , @xcite , @xcite)*. if @xmath71 and @xmath72 , then there are polynomials @xmath73 and @xmath74 such that @xmath75 , and @xmath76 where @xmath77 the proof of this result easily follows from the poincar pontryagin formula , and the ilyashenko gavrilov theorem ( @xcite , @xcite ) : if @xmath78 for all @xmath57 , then @xmath79 , where @xmath80 and @xmath81 are polynomials , and by applying an induction argument . for a detailed proof , see for instance @xcite , @xcite . to simplify the computation of the poincar pontryagin melnikov functions , we will give some properties of @xmath82 . for computing @xmath83 for we will use the following two elementary lemmas whose proof is omitted . [ lem1 ] let @xmath84 be a polynomial in the ring @xmath85 $ ] . we define @xmath86 to be the degree of @xmath84 in @xmath87 $ ] . * for @xmath88 there are homogeneous polynomials @xmath89 $ ] with @xmath90 and @xmath91 , such that @xmath92 or @xmath93 . if @xmath94 , then @xmath95 . * for @xmath96 there are homogeneous polynomials @xmath97 $ ] with @xmath98 and @xmath99 , such that @xmath100 . * for @xmath96 we have @xmath101 [ lem2 ] if @xmath102\right\}$ ] and @xmath103\right\}$ ] , then @xmath104 . the next two results are straightforward consequences of these two previous lemmas . [ cor1 ] if @xmath105 , then @xmath106 , @xmath107 with @xmath108 , and @xmath109 . [ cor2 ] if @xmath110 $ ] , then @xmath111 the following two lemmas will be important in the proof of theorem [ mth1 ] . [ lem3 ] suppose theorem [ th3 ] . if @xmath112 for @xmath113 , then @xmath114 for @xmath113 , and @xmath115 we proceed by induction on @xmath63 . if @xmath116 , then @xmath117 . hence @xmath118 , @xmath119 , and @xmath120 by corollary [ cor1 ] . since @xmath121 because of theorem [ th3 ] , @xmath122 . we assume that the lemma is true for @xmath123 , and we will prove it for @xmath63 . by assumption , @xmath112 for @xmath124 . then , by corollary [ cor1 ] , @xmath125 with @xmath126 for all @xmath124 . in addition , by the induction hypothesis , @xmath127 for @xmath128 . thus , @xmath129 with @xmath130 is an element of @xmath131 following lemma [ lem2 ] . since @xmath132 , @xmath133 . hence it is clear that @xmath134 with @xmath130 is an element of @xmath131 , which implies that @xmath135 by corollary [ cor1 ] . finally , from theorem [ th3 ] we have @xmath136 . therefore @xmath137 . before announce next lemma , we note that each polynomial @xmath138 of degree @xmath139 can be written as @xmath140}a_{2r+1}x^{2r } , \quad \mbox{and } \quad \tilde{h}{\left(x^{2}\right)}= \sum_{r=0}^{\left[\frac{m-2}{2}\right]}a_{2r+2}x^{2r}.\ ] ] [ lem4 ] let @xmath141 , where @xmath142 and @xmath143 . * @xmath144}{2(r+1 ) \choose r+1}\frac { a_{2r+1}}{2^{r}(2r+1)}{c^{r}}\right ) . $ ] * if @xmath145 , then @xmath146 with @xmath147 $ ] of degree @xmath148 $ ] , and @xmath149 } \left(\sum_{r=0}^{\left[\frac{m-2}{2}\right]}{2(s+r+1 ) \choose s+r+1 } \frac{{\left(b_{2s}\right ) } ( a_{2r+2})}{2^{s+r}(2s+1)}{c^{s+r } } \right).\ ] ] * @xmath150 if and only if @xmath151 , or @xmath152 . ( @xmath153 ) . by ( @xmath153 ) and ( @xmath43 ) of lemma [ lem1 ] , @xmath154 . finally , the statement follows from corollary [ cor2 ] . ( @xmath43 ) . if @xmath155 , then @xmath156 by statement ( @xmath153 ) . this property implies that @xmath157}a_{2r+2}x^{2r+1}y dx$ ] by . from lemma [ lem1](@xmath43 ) we obtain @xmath158 , thus @xmath159}a_{2r+2 } \overline{q}_{r}\right)\right)+\left(y\left(\sum_{r=0}^{\left[\frac{m-2}{2 } \right]}a_{2r+2}\overline{q}_{r}\right)\right ) dh = dq+\left(y\overline{q}\right ) dh,\ ] ] where @xmath160}$ ] are homogeneous and @xmath161 $ ] . moreover , a simple computation shows that @xmath162 as @xmath163 and @xmath164 , it follows that @xmath165 because of statements @xmath37 and @xmath39 of lemma [ lem1 ] . hence we obtain @xmath166}a_{2r+2}\overline{q}_{r}\right)\left ( \sum_{s=0}^{\left[n/2\right]}b_{2s}x^{2s}\right)y dx.\ ] ] by using expression of @xmath167 , a straightforward computation , and lemma [ lem1](@xmath51 ) we obtain the formula given in the statement . finally , statement ( @xmath51 ) follows from the formula given in statement ( @xmath43 ) . system with @xmath44 , @xmath168 , and @xmath169 does not satisfy the hypothesis in definition of @xmath28 because @xmath170 is not an odd function . here @xmath171 and from theorem [ mth1](@xmath153 ) it follows that @xmath172 ; however , for @xmath7 small enough , this system has one medium amplitude limit cycle . indeed , we need only to prove that the first non - vanishing coefficient of the displacement function , associated to the system , has a simple positive zero . the system can be written in the form as @xmath173 with @xmath174 . by lemma [ lem4]@xmath37 , @xmath175 , and by theorem [ ppmf ] and lemma [ lem4]@xmath39 , @xmath176 . now , system with @xmath177 , @xmath178 , @xmath179 , @xmath180 , and @xmath181 does not satisfy the hypothesis in definition of @xmath29 because @xmath182 is not an even function . in this case @xmath183 and by theorem [ mth1](@xmath43 ) , @xmath184 ; however , for @xmath7 small enough , the resulting system has two medium amplitude limit cycles . indeed , following previous ideas , and using theorem [ ppmf ] and lemma [ lem4 ] it is easy to see that @xmath175 , @xmath185 , and @xmath186 . we can assume , after a linear change of variables if necessary , that @xmath187 for all @xmath188 . suppose that @xmath189 and @xmath190 . thus , @xmath191 and @xmath4 can be written as @xmath192 and @xmath193 , respectively , according to . ( @xmath153 ) . by hypothesis , @xmath4 is odd for @xmath30 , which means that @xmath194 for @xmath30 . let @xmath83 be the first non - vanishing poincar pontryagin melnikov function in . if @xmath195 , then the theorem is true . indeed , we have @xmath196 , and as @xmath197 , and @xmath198 by corollary [ cor1 ] , we obtain @xmath199 . from we have @xmath200 $ ] ; hence @xmath69 has at most @xmath201 $ ] positive zeros because of corollary [ cor2 ] . moreover , we can choose suitable coefficients of @xmath202 in such a way that @xmath69 has exactly @xmath201 $ ] simple positive zeros . therefore , by applying the poincar pontryagin andronov criterion it follows that @xmath203 $ ] . suppose then that @xmath71 and we are therefore in the hypothesis of theorem [ th3 ] . if @xmath112 for @xmath204 , then @xmath205 by lemma [ lem3 ] , and by applying the same idea as in previous paragraph , we obtain @xmath203 $ ] . accordingly , it remains to prove that @xmath112 for @xmath124 . we proceed by induction on @xmath63 . if @xmath116 , then @xmath206 , which implies that @xmath207 . we now assume that the assertion is true for @xmath208 , and we will prove it for @xmath123 . by induction hypothesis , @xmath209 for @xmath210 , which implies that @xmath211 with @xmath212 for @xmath213 by corollary [ cor1 ] . furthermore , by lemma [ lem3 ] , @xmath214 for @xmath215 . hence @xmath129 with @xmath216 is an element of @xmath131 because of lemma [ lem2 ] . since @xmath217 , @xmath218 , whence @xmath219 . therefore @xmath220 , which completes the proof of statement ( @xmath153 ) . for @xmath222 the 1-form @xmath223 is exact , that is , @xmath224 with @xmath225 . hence , by theorem [ th3 ] , @xmath226 and @xmath227 for @xmath222 , and @xmath228 . on the other hand , since @xmath5 is even for @xmath229 , @xmath230 for @xmath31 . thus , @xmath231 for @xmath232 , and as @xmath233 because of lemma [ lem1](@xmath43 ) , we conclude that @xmath234 ; of course @xmath235 for @xmath236 . therefore @xmath237 for all @xmath238 . _ case @xmath41 odd . _ if @xmath41 is odd , then @xmath239 because @xmath5 is an even polynomial for @xmath240 . since @xmath241 has an even degree , @xmath242 . hence , from lemma [ lem4](@xmath153 ) it follows that @xmath243 , and it has at most @xmath201 $ ] positive zeros , counting multiplicities ; moreover , we can choose suitable coefficients of @xmath244 in such a way that @xmath245 has exactly @xmath201 $ ] simple positive zeros . therefore by the poincar pontryagin andronov criterion , @xmath246 $ ] . _ case @xmath41 even . _ let @xmath83 be the first non - vanishing poincar pontryagin melnikov function of . if @xmath247 , then @xmath245 has at most @xmath201 $ ] positive zeros , counting multiplicities , because of lemma [ lem4](@xmath153 ) . since @xmath41 is even , @xmath201\leq \left[m/2\right]+\left[n/2\right]-1 $ ] . hence @xmath245 has at most @xmath248+\left[n/2\right]-1 $ ] positive zeros , counting multiplicities . we claim that if @xmath249 , then @xmath250 , @xmath251 with @xmath212 for @xmath252 , and @xmath253 . by assuming that this assertion is true and by applying lemma [ lem4](@xmath43 ) we conclude that @xmath83 has at most @xmath248+\left[n/2\right]-1 $ ] positive zeros , counting multiplicities ; moreover , we can choose suitable coefficients of @xmath254 and @xmath255 in such a way that @xmath83 has exactly @xmath248+\left[n/2\right]-1 $ ] simple positive zeros . thus , by the poincar pontryagin andronov criterion , @xmath256+\left[n/2\right]-1 $ ] . therefore , to finish the proof of statement ( @xmath43 ) we need only to confirm the assertion , which we prove next by proceeding by induction on @xmath63 . if @xmath257 , then we will prove that @xmath258 with @xmath259 , and that @xmath260 . we know that @xmath261 , and from lemma [ lem4](@xmath43 ) it follows that @xmath262 , where @xmath263 . on the other hand , by theorem [ th3 ] , @xmath264 , where @xmath265 . since @xmath225 for @xmath266 , @xmath267 . moreover , since @xmath268 , @xmath269 . if @xmath270 , then @xmath271 . since @xmath272 , @xmath273 by lemma [ lem4](@xmath51 ) . this implies that @xmath274 , and by corollary [ cor1 ] , @xmath275 with @xmath276 . moreover , we know that @xmath277 with @xmath278 , and @xmath279 . thus , @xmath280 with @xmath281 because of corollary [ cor1 ] . on the other hand , from theorem [ th3 ] we have @xmath282 as @xmath283 and @xmath281 , then we have @xmath284 following lemma [ lem2 ] and @xmath285 by corollary [ cor1 ] . in addition , we know that @xmath225 for @xmath266 and @xmath286 . hence @xmath287 . on the other hand , from the induction hypothesis it follows that @xmath291 , @xmath251 with @xmath212 for @xmath292 , and @xmath293 . since @xmath294 , @xmath295 because of lemma [ lem4](@xmath51 ) , which implies that @xmath296 . therefore , @xmath297 by lemma [ lem2 ] . moreover , we have @xmath298 , and by applying corollary [ cor1 ] we obtain @xmath299 hence @xmath300 by lemma [ lem2 ] . in addition , @xmath301 . thus , we obtain @xmath302 part of the results of this work come from the author s postdoctoral stay at the departament de matemtiques of the universitat autnoma de barcelona . the author would like to thank the centre de recerca matemtica for their support and hospitality during the period in which this paper was written . j. llibre , _ a survey on the limit cycles of the generalization polynomial linard differential equations . _ mathematical models in engineering , biology and medicine , 224233 . , 1124 , amer . melville , ny , 2009 .
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we will consider two special families of polynomial perturbations of the linear center .
for the resulting perturbed systems , which are generalized linard systems , we provide the exact upper bound for the number of limit cycles that bifurcate from the periodic orbits of the linear center .
limit cycle , linard system , periodic orbit
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You are an expert at summarizing long articles. Proceed to summarize the following text:
intermittency has been observed in a variety of real settings as well in a vast number of numerical models . a great deal of effort has therefore gone into understanding these modes of behaviour in the context of deterministic dynamical systems theory . these studies have demonstrated the existence of a number of different types of intermittency ( such as pomeau manneville @xcite , crisis @xcite , on - off @xcite intermittencies ) , each with their own associated signatures and scalings . many of these forms of intermittency have in turn been concretely shown to be present in experiments and numerical studies of dynamical systems in a variety of settings ( see @xcite and references therein ) . an important potential domain of applicability of such behaviour arises in understanding the mechanisms underlying the intermediate time scale variability in the sun @xcite - the occurrence of the so called _ maunder _ or _ grand minima _ - during which solar activity ( as deduced from the sunspot numbers ) was greatly diminished @xcite . this behaviour is also confirmed by evidence coming from the analysis of proxy data @xcite . there is also some evidence for similar types of variability in solar - type stars @xcite . the idea that some type of dynamical intermittency may under pin the grand minima type variability in the sunspot record ( the _ intermittency hypothesis _ @xcite ) goes back at least to the late 1970 s @xcite . this idea has been the subject of intense study over the recent years and has involved the employment of various classes of dynamo models , including ordinary differential equations ( ode ) ( e.g.@xcite as well as partial differential equations ( pde ) models ( e.g. @xcite ) . in addition to the phenomenological evidence for the presence of intermittent - type behaviours in dynamo models @xcite , concrete evidence has recently been found for the presence of particular types of intermittency in both ode dynamo models @xcite as well as a recently discovered generalisation of on - off intermittency , referred to as _ in - out _ intermittency @xcite , in pde models @xcite . here we wish to report concrete evidence for the occurrence of two other types of intermittency , namely the crisis induced and pomeau manneville type - i intermittencies , in pde mean field dynamo models . the organisation of the paper is as follows . in sec . ii we briefly introduce the model studied here . iii summarises our evidence demonstrating the presence of these types of intermittencies in this model and finally in sec . iv we draw our conclusions . ideally one would wish to employ the full 3-dimensional dynamo models with a minimum number of approximations and simplifying assumptions . despite a number of important recent attempts @xcite , the difficulty of dealing with small scale turbulence makes a detailed and extensive self consistent study of such fully turbulent regimes in stars still computationally impractical ( see e.g. @xcite . in view of this an alternative approach in studies of stellar dynamos has been to employ mean field models @xcite . we should mention that there is an ongoing debate regarding the nature and realistic value of such models @xcite . nevertheless , 3-d turbulence simulations do seem to produce magnetic fields whose global properties ( such as field parity and time dependence ) are similar to those expected from corresponding mean field dynamo models @xcite . in this way mean field dynamo models seem to reproduce certain features of the more complicated models and allow the study of certain global properties of magnetic fields in the sun and solar - type stars ( see for example @xcite ) . the standard mean field dynamo equation is given by @xmath0 where @xmath1 and @xmath2 are the mean magnetic field and mean velocity respectively and the turbulent magnetic diffusivity @xmath3 and the coefficient @xmath4 arise from the correlation of small scale turbulent velocities and magnetic fields ( @xmath4 effect ) @xcite . we consider the usual algebraic form of @xmath4quenching namely @xmath5 where @xmath6 and @xmath7 is the co - latitude . we solve eq . [ dynamo ] in an axisymmetric configuration and in the following , as is customary @xcite , we shall discuss the behaviour of the solutions by monitoring the total magnetic energy , @xmath8 , where @xmath9 the induction constant , and the integral is taken over the dynamo region . we split @xmath10 into two parts , @xmath11 , where @xmath12 and @xmath13 are respectively the energies of the antisymmetric and symmetric parts of the field with respect to the equator . the overall parity @xmath14 is given by @xmath15/e$ ] , so @xmath16 denotes an antisymmetric ( dipole - like ) pure parity solution and @xmath17 a symmetric ( quadrupole - like ) pure parity solution . for the numerical results reported in the following section , we used a modified version of the axisymmetric dynamo code of brandenburg _ et al . _ ( 1989 ) @xcite employed recently in @xcite . these models are constructed from a complete sphere of radius @xmath18 by removing an inner concentric sphere of radius @xmath19 and a conical section of semi - angle @xmath20 about the rotation axis , from both the north and south polar regions ( see @xcite for details of the model and the relevant parameters ) . to test the robustness of the code we verified that no qualitative changes were produced by employing a finer grid and different temporal step length ( we used a grid size of @xmath21 mesh points and a step length of @xmath22 in the results presented in this paper ) . for the following results we use @xmath23 , which give the magnitude of the differential rotation and @xmath24 . the magnitude of the @xmath4-effect is given by the dynamo parameter @xmath25 . in the next section we show in turn concrete evidence for the occurrence of crisis induced and pomeau manneville type - i intermittencies . as far as their detailed underlying mechanisms and temporal signatures are concerned , crises come in three varieties @xcite . here we shall be concerned with only one of these types , referred to as `` attractor merging crisis '' , whereby as a system parameter is varied , two or more chaotic attractors merge to form a single attractor . there are both experimental and numerical evidence for this type of intermittency ( see for example @xcite and references therein ) . in particular , this type of behaviour has been discovered in a 6-dimensional truncation of mean field dynamo models @xcite . [ crisis.series ] shows the plots of the energy and parity for the above model as a function of time , calculated with @xmath26 and @xmath27 which show a bimodal behaviour , switching intermittently between two different chaotic states . # 1#20.46#1 to determine the nature of this behaviour more precisely , we have plotted in fig . [ return_map.eps ] the return maps for the pde models ( 1 ) , showing the attractors before and after the merging . as can be seen the resulting merged attractor is , as expected , larger than the superposition of the two pre - existing attractors . these results can be taken as indications for the presence of crisis induced intermittency in this model . to substantiate this further , we recall that another important signature of this type of intermittency is the way @xmath28 , the average time between switches , scales with the system parameter , in this case , @xmath29 . according to grebogi _ @xcite , for a large class of dynamical systems this relation takes the form @xmath30 where the real constant @xmath31 is the critical exponent characteristic of the system under consideration and @xmath32 is the critical value of @xmath29 at which the two chaotic attractors merge . # 1#20.43#1 the model under study here is a pde system which is formally infinite dimensional . such pde models are numerically costly to integrate over long enough intervals of time ( sometimes in excess of 5000 time units ) necessary in order to obtain the scaling of the type ( [ scaling ] ) . furthermore , the demonstration of such scaling requires a precise determination of the critical value @xmath32 which is difficult since as one approaches this value @xmath28 diverges and the integration time becomes prohibitive . despite these difficulties , we have succeeded to obtain strong evidence for the presence of such a scaling as depicted in fig . [ crisis.statistics ] , with the corresponding @xmath33 . @xcite conjecture that there may be a general tendency for @xmath31 to be larger for higher dimensional attractors . we do have a value of @xmath31 higher than the previous one found for a related six dimensional ode dynamo model @xcite but much lower than the value range suggested by grebogi _ therefore , the conjectured range may need modification for large high dimensional systems . # 1#20.46#1 there is also evidence for an enlargement of the final attractor after merging , as shown by the larger amplitudes of variation in the parity , in the sense that the parity gets closer to @xmath34 after the merging , as depicted in fig.[return_map.eps ] . this helped us to numerically arrive at a better estimate for the critical value @xmath32 . these indicators , taken together , amount to strong evidence for the presence of crisis induced intermittency for this model . this type of intermittency , which is brought about through a tangent bifurcation , results in the system switching back and forth between a `` ghost '' periodic orbit and sudden bursts of chaotic behaviour @xcite . there are both experimental and numerical evidence for this type of intermittency ( see for example @xcite and references therein ) . in particular this type of behaviour has been discovered in a 12-dimensional truncation of mean field dynamo model @xcite . to demonstrate the presence of this type of intermittency in the above pde dynamo model , we have plotted in fig . [ typei.series ] the energy and parity as a function of time for the parameter values @xmath35 and @xmath36 , which clearly demonstrates switches between nearly periodic behaviour and sudden bursts . we note that interestingly the energy in this case shows strong modulation which could be of interest in accounting for the occurrence of grand type minima in sunspot activity . # 1#20.46#1 another signature of this type of intermittency is provided by the specific characteristics of its corresponding power spectrum . by employing finite dimensional maps@xcite , it has been shown that the corresponding spectra have a broad - band feature whose shape obeys approximately the inverse - power law @xmath37 for @xmath38 , where @xmath39 is the saturation frequency . below this frequency there is a flat plateau induced by noise that causes arbitrarily long laminar phases to become finite . as further evidence for this type of intermittency in the model ( 1 ) , we have plotted in fig . [ typei.statistics ] the power spectrum at @xmath40 , obtained by averaging over 16 different initial conditions corresponding to different initial parities . as can be seen , the power spectrum shows both the flat plateau and the @xmath37 power law scaling . taken together , these indicators amount to strong evidence for the presence of pomeau manneville type - i intermittency for this model . we have obtained concrete evidence , in terms of phase space signatures , spectra and scalings to demonstrate the presence of crisis induced and the pomeau manneville type - i intermittencies in axisymmetric mean field pde dynamo models . despite the rather idealised nature of these models , this is of potential importance since it shows the occurrence of two more types of intermittency ( in addition to in out intermittency recently discovered @xcite ) in these models which may in turn be taken as an indication that more than one type of intermittency may occur in solar and stellar dynamos . this suggests that any observational programme for identifying the mechanisms underlying grand minima type variability needs to take into account the possibility that multiple intermittency mechanisms may be operative in different stars of the similar type , or even in the same star over different epochs . this would also be of importance in the interpretation of proxy data . in this way a more appropriate hypothesis regarding such variability would be that of _ multiple intermittency hypothesis_. + we would like to thank axel brandenburg for providing us with the original code and andrew tworkowski for helpful discussions . ec is supported by grant bd/5708/95 praxis xxi , jnict . rt benefited from pparc uk grant no . l39094 . 999 y. pomeau , and p. manneville , _ commun . phys . _ * 74 * , 189 ( 1980 ) . c. grebogi , e. ott , and j. a. yorke , _ phys . lett _ * 48 * , 1507 ( 1982 ) ; 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n. o. weiss , f. cattaneo , and c. a. jones , _ geophys . astrophys . fluid dyn . _ * 30 * , 305 ( 1984 ) . e. a. spiegel , in _ chaos in astrophysics _ , edited by j. r. butcher , j. perdang , and e. a. spiegel ( reidel , dordrecht , 1985 ) ; u. feudel , w. jansen , and j. kurths , _ int . j. of bifurcation and chaos _ * 3 * , 131 ( 1993 ) . s. schmalz , and m. stix , _ a&a _ * 245 * , 654 ( 1991 ) . d. schmitt , m. schssler , and a. ferriz - mas , _ a&a _ * 311 * , l1 ( 1996 ) . a. tworkowski , r. tavakol , a. brandenburg , j. m. brooke , d. moss , and i. tuominen , _ mnras _ * 296 * , 287 ( 1998 ) . s. m. tobias , _ a&a _ * 307 * , l21 , ( 1996 ) . j. m. brooke , j. pelt , r. tavakol , and a. tworkowski , _ a&a _ * 332 * , 339 ( 1998 ) . j. m. brooke , _ europhysics letters _ * 37 * , 3 ( 1997 ) . e. covas , and r. tavakol , _ phys . e. _ * 55 * , 6641 ( 1997 ) . e. covas , p. ashwin , and r. tavakol , _ phys . e. _ * 56 * , 6451 ( 1997 ) . p. ashwin , e. covas , and r. tavakol , _ nonlinearity _ * 9 * , 563 ( 1999 ) . e. covas , r. tavakol , p. ashwin , a. tworkowski , and j. m. brooke , submitted to _ phys . a _ ( 1999 ) . preprint available at web address p. a. gilman , _ apj . suppl . _ * 53 * , 243 ( 1983 ) . . nordlund , a. brandenburg , r. l. jennings , m. rieutord , j. ruokolainen , r. f. stein , and i. tuominen , _ apj _ * 392 * , 647 ( 1992 ) . a. brandenburg , r. l. jennings , . nordlund , m. rieutord , r. f. stein , and i. tuominen , _ jfm _ * 306 * , 325 ( 1996 ) . f. cattaneo , d. w. hughes , and n. o. weiss , _ mnras _ * 253 * , 479 ( 1991 ) . d. moss , d. m. barker , a. brandenburg , and i. tuominen , _ a&a _ * 294 * , 155 ( 1995 ) . s. i. vainshtein , f. cattaneo , _ ap . j. _ * 393 * , 165 ( 1992 ) ; 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we find concrete evidence for the presence of crisis induced and pomeau manneville type - i intermittencies in an axisymmetric pde mean field dynamo model .
these findings are of potential importance for two different reasons .
firstly , as far as we are aware , this is the first time detailed evidence has been produced for the occurrence of these types of intermittency for such deterministic pde models . and secondly , despite the rather idealised nature of these models , the concrete evidence for the occurrence of more than one type of intermittency in such models makes it in principle possible that different types of intermittency may occur in different solar - type stars or even in the same star over different epochs . in this way a
_ multiple intermittency framework _ may turn out to be of importance in understanding the mechanisms responsible for grand - minima type behaviour in the sun and solar - type stars and in particular in the interpretation of the corresponding observational and proxy evidence .
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linguistic typology aims to distinguish between logically possible languages and actually observed languages . a fundamental building block for such an understanding is the _ universal implication _ these are short statements that restrict the space of languages in a concrete way ( for instance `` object - verb ordering implies adjective - noun ordering '' ) ; , and provide excellent introductions to linguistic typology . we present a statistical model for automatically discovering such implications from a large typological database @xcite . analyses of universal implications are typically performed by linguists , inspecting an array of @xmath0-@xmath1 languages and a few pairs of features . looking at all pairs of features ( typically several hundred ) is virtually impossible by hand . moreover , it is insufficient to simply look at counts . for instance , results presented in the form `` verb precedes object implies prepositions in 16/19 languages '' are nonconclusive . while compelling , this is not enough evidence to decide if this is a statistically well - founded implication . for one , maybe @xmath2 of languages have prepositions : then the fact that we ve achieved a rate of @xmath3 actually seems really bad . moreover , if the @xmath4 languages are highly related historically or areally ( geographically ) , and the other @xmath5 are not , then we may have only learned something about geography . in this work , we propose a statistical model that deals cleanly with these difficulties . by building a computational model , it is possible to apply it to a very large typological database and search over many thousands of pairs of features . our model hinges on two novel components : a statistical noise model a hierarchical inference over language families . to our knowledge , there is no prior work directly in this area . the closest work is represented by the books _ possible and probable languages _ @xcite and _ language classification by numbers _ @xcite , but the focus of these books is on automatically discovering phylogenetic trees for languages based on indo - european cognate sets @xcite . [ sec : data ] [ fig : world ] [ cols="<,^,^,^,^,^,^ " , ] we have presented a bayesian model for discovering universal linguistic implications from a typological database . our model is able to account for noise in a linguistically plausible manner . our hierarchical models deal with the sampling issue in a unique way , by using prior knowledge about language families to `` group '' related languages . quantitatively , the hierarchical information turns out to be quite useful , regardless of whether it is phylogenetically- or areally - based . qualitatively , our model can recover many well - known implications as well as many more potential implications that can be the object of future linguistic study . we believe that our model is sufficiently general that it could be applied to many different typological databases we attempted not to `` overfit '' it to wals . our hope is that the automatic discovery of such implications not only aid typologically - inclined linguists , but also other groups . for instance , well - attested universal implications have the potential to reduce the amount of data field linguists need to collect . they have also been used computationally to aid in the learning of unsupervised part of speech taggers @xcite . many extensions are possible to this model ; for instance attempting to uncover typologically hierarchies and other higher - order structures . we have made the full output of all models available at http://hal3.name/wals .
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a standard form of analysis for linguistic typology is the universal implication .
these implications state facts about the range of extant languages , such as `` if objects come after verbs , then adjectives come after nouns . ''
such implications are typically discovered by painstaking hand analysis over a small sample of languages .
we propose a computational model for assisting at this process .
our model is able to discover both well - known implications as well as some novel implications that deserve further study .
moreover , through a careful application of hierarchical analysis , we are able to cope with the well - known sampling problem : languages are not independent .
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the insurgence of critical electric fields in the process of gravitational collapse leading to vacuum polarization process @xcite has convinced us of the necessity of critically reexamining the gravitational and electrodynamical properties in neutron stars . in this light we have recently generalized the feynman , metropolis and teller treatment of compressed atoms to the relativistic regimes @xcite . we have so enforced , self - consistently in a relativistic thomas - fermi equation , the condition of @xmath0-equilibrium extending the works of v. s. popov @xcite , ya . b. zeldovich and v. s. popov @xcite , a. b. migdal et al . @xcite , j. ferreirinho et al . @xcite and r. ruffini and l. stella @xcite for heavy nuclei . thanks to the existence of scaling laws ( see @xcite and @xcite ) this treatment has been extrapolated to compressed nuclear matter cores of stellar dimensions with mass numbers @xmath2 or @xmath3 . such configurations fulfill global but not local charge neutrality . they have electric fields on the core surface , increasing for decreasing values of the electron fermi energy @xmath4 reaching values much larger than the critical value @xmath5 , for @xmath6 . the assumption of constant distribution of protons at nuclear densities simulates , in such a treatment , the confinement due to the strong interactions in the case of nuclei and heavy nuclei and due to both the gravitational field and the strong interactions in the case of nuclear matter cores of stellar sizes . in this article we introduce explicitly the effects of gravitation by considering a general relativistic system of degenerate fermions composed of neutrons , protons and electrons in @xmath0-equilibrium : this is the simplest nontrivial system in which new electrodynamical and general relativistic properties of the equilibrium configuration can be clearly and rigorously illustrated . we first prove that the condition of local charge neutrality can never be implemented since it violates necessary conditions of equilibrium at the microphysical scale . we then prove the existence of a solution with global , but not local , charge neutrality by taking into account essential gravito - electrodynamical effects . first we recall the constancy of the general relativistic fermi energy of each specie pioneered by o. klein @xcite . we subsequently introduce the general relativistic thomas - fermi equations for the three fermion species fulfilling relativistic quantum statistics , governed by the einstein - maxwell equations . the solution of this system of equations presents a formidable mathematical challenge in theoretical physics . the traditional difficulties encountered in proving the existence and unicity of the solution of the thomas - fermi equation @xcite are here enhanced by the necessity of solving the general relativistic thomas - fermi equation coupled with the einstein - maxwell system of equations . we present the general solution for the equilibrium configuration , from the center of the star all the way to the border , giving the details of the gravitational field , of the electrodynamical field as well as of the conserved quantities . we illustrate such a solution by selecting a central density @xmath7 , where @xmath8 g @xmath9 is the nuclear density . we point out the existence near the boundary of the core in the equilibrium configuration of three different radii , in decreasing order : @xmath10 corresponding to the vanishing of the fermi momentum of the electron component ; @xmath11 , @xmath12 corresponding to the vanishing of the fermi momentum of the proton component ; @xmath13 and @xmath14 corresponding to the radius at which the fermi momentum of neutrons vanishes : @xmath15 . we then give explicit expressions for the proton versus electron density ratio and the proton versus neutron density ratio for any value of the radial coordinate as well as for the electric potential at the center of the configuration . a novel situation occurs : the description of the pressure and density is not anylonger a local one . their determination needs prior knowledge of the global electrodynamical and gravitational potentials on the entire system as well as of the radii @xmath14 , @xmath12 and @xmath10 . this is a necessary outcome of the self - consistent solution of the eigenfunction within general relativistic thomas - fermi equation in the einstein - maxwell background . as expected from the considerations in @xcite , the electric potential at the center of the configuration fulfills @xmath16 and the gravitational potential @xmath17 . the implementation of the constancy of the general relativistic fermi energy of each particle species and the consequent system of equations illustrated here is the simplest possible example admitting a rigorous nontrivial solution . it will necessarily apply in the case of additional particle species and of the inclusion of nuclear interactions : in this cases however it is not sufficient and the contribution of nuclear fields must be taken into due account . we consider the equilibrium configurations of a degenerate gas of neutrons , protons and electrons with total matter energy density and total matter pressure @xmath18 where @xmath19 is the relativistic single particle energy . in addition , we require the condition of @xmath0-equilibrium between neutrons , protons and electrons @xmath20 where @xmath21 denotes the fermi momentum and @xmath22 is the free - chemical potential of particle - species with number density @xmath23 . we now introduce the extension to general relativity of the thomas - fermi equilibrium condition on the generalized fermi energy @xmath4 of the electron component @xmath24 where @xmath25 is the fundamental charge , @xmath26 is the coulomb potential of the configuration and we have introduced the metric @xmath27 for a spherically symmetric non - rotating neutron star . the metric function @xmath28 is related to the mass @xmath29 and the electric field @xmath30 ( a prime stands for radial derivative ) through @xmath31 thus the equations for the neutron star equilibrium configuration consist of the following einstein - maxwell equations and general relativistic thomas - fermi equation @xmath32 = - 4 \pi \alpha \hbar c \ , { \rm e}^{\nu/2 } { \rm e}^{\lambda } \bigg\ { n_p \nonumber \\ % % & & - \frac{{\rm e}^{-3 \nu/2}}{3 \pi^2}[\hat{v}^2 + 2 m_e c^2 \hat{v } - m^2_e c^4 ( { \rm e}^{\nu}-1)]^{3/2}\bigg\}\ , , \label{eq : grtf}\end{aligned}\ ] ] where @xmath33 denotes the fine structure constant , @xmath34 , @xmath35 and we have used eq . ( [ eq : electroneq ] ) to obtain eq . ( [ eq : grtf ] ) . it can be demonstrated that the assumption of the equilibrium condition ( [ eq : electroneq ] ) together with the @xmath0-equilibrium condition ( [ eq : betaeq ] ) and the hydrostatic equilibrium ( [ eq : tov ] ) is enough to guarantee the constancy of the generalized fermi energy @xmath36 for all particle species separately . here @xmath37 denotes the particle unit charge of the @xmath38-species . indeed , as shown by olson and bailyn @xcite , when the fermion nature of the constituents and their degeneracy is taken into account , in the configuration of minimum energy the generalized fermi energies @xmath39 defined by ( [ eq : olsoneq ] ) must be constant over the entire configuration . these minimum energy conditions generalize the equilibrium conditions of klein @xcite and of kodama and yamada @xcite to the case of degenerate multicomponent fluids with particle species with non - zero unit charge . if one were to assume , as often done in literature , the local charge neutrality condition @xmath40 instead of assuming the equilibrium condition ( [ eq : electroneq ] ) , this would lead to @xmath41 identically ( since there will be no electric fields generated by the neutral matter distribution ) implying via eqs . ( [ eq : betaeq ] ) and ( [ eq : tov ] ) @xmath42 thus the neutron fermi energy would be constant throughout the configuration as well as the sum of the proton and electron fermi energies but not the individual fermi energies of each component . in fig . [ fig:1 ] we show the results of the einstein equations for a selected value of the central density of a system of degenerate neutrons , protons , and electrons in @xmath0-equilibrium under the constraint of local charge neutrality . in particular , we have plotted the fermi energy of the particle species in units of the pion rest - energy . it can be seen that indeed the fermi energies of the protons and electrons are not constant throughout the configuration which would lead to microscopic instability . this proves the impossibility of having a self - consistent configuration fulfilling the condition of local charge neutrality for our system . this result is complementary to the conclusion of eq . ( 4.6 ) of @xcite who found that , at zero temperature , only a dust solution with zero particle kinetic energy can satisfy the condition of local charge neutrality and such a configuration is clearly unacceptable for an equilibrium state of a self - gravitating system . , where @xmath43 g @xmath9 denotes the nuclear density . we turn now to describe the equilibrium configurations fulfilling only global charge neutrality . we solve self - consistently eqs . ( [ eq : gab1 ] ) and ( [ eq : gab2 ] ) for the metric , eq . ( [ eq : tov ] ) for the hydrostatic equilibrium of the three degenerate fermions and , in addition , we impose eq . ( [ eq : betaeq ] ) for the @xmath0-equilibrium . the crucial equation relating the proton and the electron distributions is then given by the general relativistic thomas - fermi equation ( [ eq : grtf ] ) . the boundary conditions are : for eq . ( [ eq : gab1 ] ) the regularity at the origin : @xmath44 , for eq . ( [ eq : tov ] ) a given value of the central density , and for eq . ( [ eq : grtf ] ) the regularity at the origin @xmath45 , and a second condition at infinity which results in an eigenvalue problem determined by imposing the global charge neutrality conditions @xmath46 at the radius @xmath10 of the electron distribution defined by @xmath47 from which follows @xmath48 then the eigenvalue problem consists in determining the gravitational potential and the coulomb potential at the center of the configuration that satisfy the conditions ( [ eq : bound1])([eq : bound3 ] ) at the boundary . the solution for the particle densities , the gravitational potential , the coulomb potential and the electric field are shown in fig . ( [ fig : fig2 ] ) for a configuration with central density @xmath49 . in order to compare our results with those obtained in the case of nuclear matter cores of stellar dimensions @xcite as well as to analyze the gravito - electrodynamical stability of the configuration we have plotted the electric potential in units of the pion rest - energy and the gravitational potential in units of the pion - to - proton mass ratio . one particular interesting new feature is the approach to the boundary of the configuration : three different radii are present corresponding to distinct radii at which the individual particle fermi pressure vanishes . the radius @xmath10 for the electron component corresponding to @xmath50 , the radius @xmath12 for the proton component corresponding to @xmath51 and the radius @xmath14 for the neutron component corresponding to @xmath52 . the smallest radius @xmath14 is due to the threshold energy for @xmath0-decay which occurs at a density @xmath53 g @xmath9 . the radius @xmath12 is larger than @xmath14 because the proton mass is slightly smaller than the neutron mass . instead , @xmath54 due to a combined effect of the difference between the proton and electron masses and the implementation of the global charge neutrality condition through the thomas - fermi equilibrium conditions . for the configuration of fig . [ fig : fig2 ] we found @xmath55 km , @xmath56 km and @xmath57 where @xmath58 denotes the electron compton wavelength . we find that the electron component follows closely the proton component up to the radius @xmath12 and neutralizes the configuration at @xmath10 without having a net charge , contrary to the results e.g in @xcite . bottom panel : proton and electron coulomb potentials in units of the pion rest - energy @xmath59 and @xmath60 respectively and the proton gravitational potential in units of the pion mass @xmath61 where @xmath62.,title="fig : " ] @xmath9 . bottom panel : proton and electron coulomb potentials in units of the pion rest - energy @xmath59 and @xmath60 respectively and the proton gravitational potential in units of the pion mass @xmath61 where @xmath62.,title="fig : " ] normalized to its value at @xmath63 . bottom panel : electric field for @xmath64 normalized to its value at @xmath63 . we have shown also the behavior of the solution of the general relativistic thomas - fermi equation ( [ eq : grtf ] ) for two different eigenvalues close to the one which gives the globally neutral configuration.,title="fig : " ] normalized to its value at @xmath63 . bottom panel : electric field for @xmath64 normalized to its value at @xmath63 . we have shown also the behavior of the solution of the general relativistic thomas - fermi equation ( [ eq : grtf ] ) for two different eigenvalues close to the one which gives the globally neutral configuration.,title="fig : " ] it can be seen from fig . [ fig : fig2 ] that the negative proton gravitational potential energy is indeed always larger than the positive proton electric potential energy . therefore the configuration is stable against coulomb repulsion . this confirms the results in the simplified case analyzed by m. rotondo et al . in @xcite . from eq . ( [ eq : olsoneq ] ) and the relation between fermi momentum and the particle density @xmath65 , we obtain the proton - to - electron and proton - to - neutron ratio for any value of the radial coordinate @xmath66^{3/2}\ , , \\ \frac{n_p ( r)}{n_n(r ) } & = & \left [ \frac{g^2(r ) \mu^2_n(r)-m^2_p c^4}{\mu^2_n(r)-m^2_n c^4 } \right]^{3/2}\ , , \end{aligned}\ ] ] where @xmath67 , @xmath68 and the constant values of the generalized fermi energies are given by @xmath69 a novel situation occurs : the determination of the quantities given in eqs . ( [ eq : nenpratio ] ) and ( [ eq : efi ] ) necessarily require the prior knowledge of the global electrodynamical and gravitational potential from the center of the configuration all the way out to the boundary defined by the radii @xmath10 , @xmath12 and @xmath14 . this necessity is an outcome of the solution for the eigenfunction of the general relativistic thomas - fermi equation ( [ eq : grtf ] ) . from the regularity condition at the center of the star @xmath45 together with eq . ( [ eq : nenpratio ] ) we obtain the coulomb potential at the center of the configuration @xmath70\ , , \end{aligned}\ ] ] which after some algebraic manipulation and defining the central density in units of the nuclear density @xmath71 can be estimated as @xmath72 ^ 2 } \bigg ] \nonumber \\ & \simeq & \frac{1}{2}\bigg[\frac{(3 \pi^2 \eta/2)^{2/3 } m_p}{(3 \pi^2 \eta/2)^{2/3 } m_\pi+ m^2_n / m_\pi}\bigg ] m_\pi c^2\ , , \end{aligned}\ ] ] where we have approximated the gravitational potential at the boundary as @xmath73 . then for configurations with central densities larger than the nuclear density we necessarily have @xmath74 . in particular , for the configuration we have exemplified with @xmath75 in fig . [ fig : fig2 ] , from the above expression ( [ eq : v0app ] ) we obtain @xmath76 . this value of the central potential agrees with the one obtained in the simplified case of nuclear matter cores with constant proton density @xcite . we have proved in the first part of this letter that the treatment generally used for the description of neutron stars adopting the condition of local charge neutrality , is not consistent with the einstein - maxwell equations and microphysical conditions of equilibrium consistent with quantum statistics ( see fig . [ fig:1 ] ) . we have shown how to construct a self - consistent solution for a general relativistic system of degenerate neutrons , protons and electrons in @xmath0-equilibrium fulfilling global but not local charge neutrality . although the mass - radius relation in the simple example considered here in our new treatment , differs slightly from the one of the traditional approaches , the differences in the electrodynamic structure are clearly very large . as is well - known these effects can lead to important astrophysical consequences on the physics of the gravitational collapse of a neutron star to a black hole @xcite . having established in the simplest possible example the new set of einstein - maxwell and general relativistic thomas - fermi equations , we now proceed to extend this approach when strong interactions are present @xcite . the contribution of the strong fields to the energy - momentum tensor , to the four - vector current and consequently to the einstein - maxwell equations have to be taken into account . clearly in this more general case , the conditions introduced in this letter have to be still fulfilled : the @xmath77-independence of the generalized fermi energy of electrons and the fulfillment of the general relativistic thomas - fermi equation @xcite . in addition , the generalized fermi energy of protons and neutrons will depend on the nuclear interaction fields . the fluid of neutrons , protons and electrons in this more general case does not extend all the way to the neutron star surface but is confined to the neutron star core endowed with overcritical electric fields , in precise analogy with the case of the compressed nuclear matter core of stellar dimension described in @xcite .
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we present the self - consistent treatment of the simplest , nontrivial , self - gravitating system of degenerate neutrons , protons and electrons in @xmath0-equilibrium within relativistic quantum statistics and the einstein - maxwell equations .
the impossibility of imposing the condition of local charge neutrality on such systems is proved , consequently overcoming the traditional tolman - oppenheimer - volkoff treatment .
we emphasize the crucial role of imposing the constancy of the generalized fermi energies . a new approach based on the coupled system of the general relativistic thomas - fermi - einstein - maxwell equations
is presented and solved .
we obtain an explicit solution fulfilling global and not local charge neutrality by solving a sophisticated eigenvalue problem of the general relativistic thomas - fermi equation .
the value of the coulomb potential at the center of the configuration is @xmath1 and the system is intrinsically stable against coulomb repulsion in the proton component .
this approach is necessary , but not sufficient , when strong interactions are introduced .
neutron star electrodynamics , general relativistic thomas - fermi treatment .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
since the discovery of increased pinning in heavy - ion irradiated samples , the interaction between flux - lines and columnar defects in high - t@xmath5 superconductors has been the subject of intense experimental and theoretical investigations.@xcite an angle dependent critical current enhancement has been put into evidence both in the moderately anisotropic material yba@xmath0cu@xmath1o@xmath2 @xcite as in highly anisotropic materials such as bi@xmath0sr@xmath0cacu@xmath0o@xmath6.@xcite the influence of correlated disorder on the equilibrium properties of the flux - line lattice has been the subject of fewer experimental investigations . torque and magnetization experiments on bi@xmath0sr@xmath0cacu@xmath0o@xmath6 have revealed a pinning energy contribution to the equilibrium magnetization arising from the presence of the defects , but , in contrast to the irreversible magnetic moment , the equilibrium magnetization did not show any other angle dependent contribution than the one arising from the layering of the material.@xcite here , we investigate the less anisotropic compound yba@xmath0cu@xmath1o@xmath2 and show that there exists a narrow domain in the ( h - t ) diagram where a reversible angular dependent contribution to the torque arises due to the interaction of flux - lines with the linear defects . this constitutes a direct demonstration that vortex lines in the liquid phase distort in order to accomodate to the linear irradiation defects . the experiments were performed on a single crystal of dimensions @xmath7 @xmath8m@xmath9 ; the shortest dimension was along the @xmath10axis . the transition temperature after irradiation was @xmath11 k. a single domain of parallel twin planes was observed , the planes running at 45@xmath12 with respect to the crystal s longest edge . the sample was irradiated with 5.8 gev pb ions to a dose @xmath13 @xmath14 , equivalent to a matching ( dose equivalent ) field @xmath15 kg . the ion beam was directed perpendicular to the longer crystal edge , at an angle of 30 degrees with respect to the @xmath10-axis . the irradiation created continuous linear amorphous defects of radius @xmath16 , oriented along the direction of the ion beam,@xcite with density @xmath17 . after the irradiation , the sample was characterized using the magneto - optic flux visualization technique at 65 , 77 and 82 k ; no evidence of any remaining influence of the twin planes on the flux penetration could be observed . the torque was measured using a piezoresistive microlever from park scientific instruments , as described in ref . . the microlever formed part of a low temperature wheatstone resistive bridge , in which a second lever with no sample was inserted in order to compensate for the background signal originating from the magnetoresistance of the levers . the measuring lever was fed with a current of 300 @xmath8a and thermalized to better than 0.01 k using he@xmath18 exchange gas . the torque setup was calibrated from the meissner slope of the reversible magnetization as a function of field at a fixed angle , as described elsewhere@xcite . in torque experiments with a single rotation axis , the plane in which the applied field @xmath19 is rotated is always at a misorientation angle @xmath20 with respect to the plane enclosing the @xmath10-axis and the irradiation direction ( fig . [ lever ] ) . this angle is not known _ a priori _ : it results from the uncertainty in both the irradiation direction and the sample positioning . we estimate that it is less than a few degrees . the result of the misorientation is that the applied field is never strictly aligned with the irradiation direction ; @xmath20 is therefore the minimum angle between @xmath19 and the ion tracks when the field direction is varied . in a separate experiment , the irreversibility line was measured using squid _ ac_-susceptometry . it was located as the onset of the in phase ( reactive ) component of the _ ac _ susceptibility measured in an oscillatory field of amplitude 0.1 oe and frequency 13 hz , oriented parallel to the dc field . these measurements were performed for two orientations of the static field , applied parallel to the direction of the tracks ( _ i.e . at 30@xmath12 with respect to the @xmath10axis ) , and applied in the symmetric direction with respect to the @xmath10-axis . the irreversibility fields for both orientations were found to be linear with temperature ; the line obtained with the field applied parallel to the tracks clearly lies above the one for the symmetric orientation ( fig . [ irrline ] ) . in contrast to what is observed for bi@xmath0sr@xmath0cacu@xmath0o@xmath6 , and more recently , in heavy - ion irradiated yba@xmath0cu@xmath1o@xmath2 thick films,@xcite there is no change in the behavior at @xmath21 up to our maximum measuring field of @xmath22 koe , and the lines do not merge above the irradiation field . _ typical torque signals are displayed in fig . [ data ] . below the irreversibility line determined by squid _ ac_-susceptometry with the field along the tracks , the torque measurements reveal a hysteretic behavior when the field is aligned with the irradiation direction . above the line , the system is in the so called vortex liquid phase and the torque signal is reversible ; however , in a narrow region typically 1 to 2 k wide , a kink is found , roughly symmetric with respect to the orientation of the columnar defects ( fig . [ data ] ) . this behavior is similar to what is observed for conventional torque on a layered superconductor when the field is rotated across the plane of the layers , and indicates that the vortex lines deform in order to have their direction coincide with that of the linear defects . in other words , the free energy of the vortex liquid phase is lowered by flux line pinning onto the columnar tracks . at low temperature , where thermal fluctuations are not important , theory@xcite predicts that when the external field is applied sufficiently close to the layer / track direction ( _ i.e . the angle between applied field and the tracks @xmath23 where @xmath24 is the lock - in angle ) , the equilibrium configuration of a single flux - line is that in which the whole length of the line is aligned with the defect . at larger angles @xmath25 , one expects a staircase configuration in which line segments aligned with the defects alternate with segments wandering between defects.@xcite for @xmath26 larger than the accomodation angle @xmath27 , the vortices do not readjust to the columnar defects at all . in our experiment , it is unlikely that we achieve the locked configuration , as this would require the alignment of the external field with the track direction to within some angle @xmath28 . in the locked configuration , one should observe a linear variation of the torque signal with angle , with a slope @xmath29 erg @xmath30 rad@xmath31 ( neglecting the anisotropy in the demagnetizing factors ) i.e. @xmath32 erg deg@xmath31 in a 10 koe field in our case . this is three orders of magnitude larger than the highest of the slopes in fig . _ the predicted contribution @xmath33 to the torque signal@xcite is shown in fig . [ theory ] . as the field angle is increased from the irradiation direction , the torque first increases linearly , reaching a maximum at the lock - in angle , and then decreases linearly beyond this . for angles larger than @xmath34 , the torque contribution arising from the interaction between vortices and ion tracks should be zero . in practice , the lock - in angle is quite small , therefore the torque signal should be quasi discontinuous when the field and track direction coincide . in the single vortex regime , the magnitude of the torque signal close to the irradiation direction may be obtained in terms of the lock - in angle,@xcite @xmath35 with @xmath36 the pinning energy per unit length . the accomodation angle @xmath37 , where @xmath38 is the vortex line tension , the energy scale @xmath39 , and @xmath40 is the typical wavevector of the vortex distortion induced by the columns . in optimally doped yba@xmath41cu@xmath42o@xmath2 , the penetration depth @xmath43 , the @xmath44plane coherence length @xmath45 , and the anisotropy parameter @xmath46.@xcite > from eq . ( [ eq : torque ] ) , one sees that one can _ directly obtain an estimate of the lock in angle from the torque jump observed when the field is aligned with the columns and eq . ( [ eq : torque ] ) . a good estimate of the pinning energy @xmath47 is equally obtained from the torque jump : _ @xmath48 ( @xmath49 is the mean separation between vortices ) . this method to obtain the pinning energy@xcite is more direct than estimates based on the angular dependence of the resistivity @xmath50 . those rely on the identification of a shallow maximum of @xmath50 at @xmath27 , or , alternatively , with a `` depinning angle''@xcite determined by the rate at which vortices can liberate themselves from a track ; the relation of the latter with the accomodation angle is not certain . since the pinning energy is predicted to be proportional to @xmath51 , the method based on transport measurements can result in a large uncertainty in @xmath47 . the present approach has the advantage that there is only one assumption , which concerns the precise form of @xmath52 . taking the curve at @xmath53 koe and @xmath54 k ( @xmath55 ) , _ i.e. at the onset of magnetic irreversibility , one has a typical value of the torque jump @xmath56 400 erg @xmath30 ( fig . [ data ] ) ; consequently , @xmath57 deg . the parameter values @xmath58 , @xmath59 , @xmath60 , and @xmath61 , yield the pinning energy per unit length @xmath62 erg cm@xmath31 , and @xmath64 . the obtained value of the accomodation angle seems reasonable : the difference between the extrapolation of the torque from large positive and negative angles to @xmath65 , at which the field and the ion track are nearly aligned , is a good indication that @xmath27 lies beyond the angular range depicted in fig . clearly , @xmath27 greatly exceeds the angular width of the irreversible regime just below the irreversibility line , which is about 8@xmath12 at 88.5 k and @xmath53 koe . the accomodation angle is comparable to the low temperature limit of the `` depinning angle '' measured on an untwinned yba@xmath41cu@xmath42o@xmath2 single crystal irradiated with 1.0 gev u ions to the same nominal dose.@xcite _ returning to the experimental data in fig . [ data ] , one observes that , in spite of the fact that a clear jump in the torque signal can be defined , the discontinuity at the irradiation angle is rather smooth . the smoothness of the curve is possibly due to the non - zero misalignment angle @xmath20 . the effect of misalignment can be quantitatively accounted for using simple trigonometric considerations . projecting the torque as given in ref . on the experimental torque axis * u * ( fig . [ lever ] ) , one obtains the magnitude of the measured torque signal : @xmath66 where @xmath67^{1/2}.\ ] ] @xmath68 is the field rotation angle in the laboratory frame , and @xmath26 , as before , is the real angle between the direction of the magnetic field and that of the ion tracks . the curves plotted in fig . [ theory ] shows that the effect of the misalignment is both to widen the angular interval between the torque maxima ( now @xmath69 from one another ) and to decrease the torque value at the maximum . using @xmath70 , @xmath71 , and @xmath72 we find that the maximum torque is only about 0.6 @xmath73 so that the pinning potential estimated from the apparent torque jump is in this case only about 40@xmath74 of the actual value , which , at @xmath53 koe and @xmath75 k , would amount to @xmath76 erg cm@xmath31 . the absolute value of the pinning energy per unit length is in reasonable agreement with the estimate for core pinning of individual vortices,@xcite @xmath77 ( with @xmath78 erg cm@xmath31 ) in the temperature regime of interest , this mechanism is more relevant than electromagnetic pinning@xcite because @xmath79 greatly exceeds the track radius . using the same parameter values as above , the model yields the theoretical value @xmath80 erg cm@xmath31 ( at @xmath81 ) . recent measurements on heavy ion irradiated bi@xmath41sr@xmath41cacu@xmath41o@xmath6@xcite showed that in that material , the dependence of pinning energy on track diameter and temperature is in agreement with the core pinning model , although the magnitude of the pinning energy exceeded the theoretical expectation ( [ eq : core ] ) by a factor 5 . in the present case , the strong temperature and field dependence of the experimentally obtained pinning energy , displayed in fig . [ fig : upin ] , show that a simple `` zero temperature '' single vortex pinning approach is inadequate . the reasons for this are that ( i ) the fields under consideration are not small with respect to @xmath3 , so that only a fraction of vortices can be expected to be actually trapped on a columnar track , and ( ii ) the proximity to @xmath4 possibly necessitates the inclusion of the effect of strong thermal fluctuations.@xcite a theoretical description of the effect of a field rotation , or even of the total pinning energy , in the case where the vortex density is comparable to the density of a system of strong linear pins has , to our knowledge , not been developped at present . although the decrease of the pinning energy per unit volume as field is increased , and the eventual disappearance of the torque jump at @xmath82 , is the straightforward consequence of the averaging of the pinning energy gain obtained from the restricted number of vortices trapped on an ion track and the ever increasing number of those that are not , there are few predictions about the resulting field dependence of the equilibrium magnetization . extensive numerical calculations of the vortex energy distribution in the presence of columnar pins were carried out by wengel and tuber;@xcite however , they did not make any specific predictions as to the precise temperature or field dependence of the magnetization . the effect of thermal fluctuations must also be considered . in resistivity measurements , such fluctuations are usually accounted for by stating that , at the angle at which depinning occurs , _ i.e. at which the probability to find a pinned vortex segment becomes exponentially small , the thermal energy and the pinning energy of a single trapped vortex segment are equal.@xcite as a consequence , the `` depinning angle '' measured by the angular dependence of the resistivity is smaller than the accomodation angle and is given by:@xcite _ @xmath83 with the parameters values as used above , we obtain @xmath84 15@xmath12 , which is comparable to the angular width of the irreversible regime in fig . [ data ] . the same type of argument leads one to conclude that at the same temperature , thermal fluctuations are much less efficient when the field is aligned with the track direction , because the length and the trapping energy of the pinned line segments are large . nevertheless , the fact that the magnetization is _ reversible and that the resistivity measured under similar condictions is _ linear @xcite implies that although it may be small , the thermal depinning rate is non zero . since the measured pinning energy is proportional to the average vortex length trapped on a columnar defect at any one moment , the rapid decrease of @xmath47 with temperature , which is in agreement with estimates obtained from resistivity data,@xcite does not seem to be an artefact of the method used to analyze torque or resistivity data , but reflects the increasing efficiency of thermal fluctuations in liberating vortices from the tracks . strong vortex wandering above a `` depinning temperature '' @xmath85 , such as proposed in refs . , and would lead to a torque jump that follows an exponential temperature dependence . such a dependence was observed in ref . , where the rapid decrease of the accomodation angle measured in heavy - ion irradiated bi@xmath0sr@xmath0cacu@xmath0o@xmath6 was attributed to the effect of thermal fluctuations . although the reduced range of temperatures over which @xmath47 could be determined in the present experiments makes a direct comparison very difficult , thermal wandering of flux lines could be responsible for the disappearance of the pinning energy and the torque jump at temperatures _ below @xmath86 ( see fig . [ irrline ] ) . we have , from thermodynamic torque measurements , obtained the first evidence for an angle dependent contribution of amorphous columnar defects to the equilibrium magnetization in yba@xmath41cu@xmath42o@xmath2 . the analysis of the torque signal allowed us to directly determine the lock - in angle and the pinning energy of the linear defects . the magnitude of the pinning energy is in qualitative agreement with the core pinning mechanism by columnar defects ; however , the observed strong field dependence means that the interactions between flux lines are not negligible in the range of magnetic fields investigated here . the strong temperature dependence of the torque jump , and the disappearance of the pinning energy below @xmath87 are the consequence of thermal fluctuations in the vortex liquid state , which are increasingly efficient in liberating vortex segments from the tracks as temperature increases . the work of stj is funded by the ec , tmr grant nr . we thank f. holtzberg ( emeritus , i.b.m . thomas j. watson research center , yorktown heights ) for providing the yba@xmath41cu@xmath42o@xmath2 single crystal . this feature is also valid in the presence of weak irreversibility . for a method how to estimate the equilibrium torque in such a situation , see l. fruchter and i.a . campbell , phys . b * 40 * , 5158 ( 1989 )
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we have measured an angle dependent contribution to the equilibrium magnetization of a yba@xmath0cu@xmath1o@xmath2 single crystal with columnar defects created by irradiation with 5.8 gev pb ions .
this contribution manifests itself as a jump in the equilibrium torque signal , when the magnetic field direction crosses that of the defects .
the magnitude of the jump , which is observed in a narrow temperature interval of less than 2 k wide , for fields up to about twice the dose equivalent field @xmath3 , is used to estimate the energy gained by vortex pinning on the defects .
the vanishing of the effective pinning energy at a temperature below @xmath4 is attributed to its renormalization by thermal fluctuations . 2
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You are an expert at summarizing long articles. Proceed to summarize the following text:
believed to be the main origin of the jet quenching phenomena observed @xcite in nucleus nucleus collisions at rhic energy @xmath2@xmath3 , parton energy loss via gluon - radiation is expected to depend on the properties ( gluon density and volume ) of the ` medium ' formed in the collision and on the properties ( color charge and mass ) of the ` probe ' parton @xcite . hard gluons would lose more energy than hard quarks due to the stronger color coupling with the medium . in addition , charm and beauty quarks are qualitatively different probes with respect to light partons , since their energy loss is expected to be reduced , as a consequence of a mass - dependent restriction in the phase - space into which gluon radiation can occur @xcite . we study quenching effects for heavy quarks by supplementing perturbative qcd calculations of the baseline @xmath4 distributions with in - medium energy loss , included via the bdmps quenching weights . the quenching weights , computed for light quarks and gluons in @xcite and for heavy quarks in @xcite , depend on the transport coefficient @xmath5 , a measure of the medium density , and on the in - medium path length . these inputs are evaluated on a parton - by - parton level , using a glauber - model based description of the local @xmath5 profile in the transverse direction @xcite . the @xmath5 value is chosen in order to reproduce the light - flavor particles nuclear modification factor @xmath6 measured in central collisions at @xmath7 ( fig . [ fig : rhic ] , left ) : the range favored by the data for the parton - averaged transport coefficient is @xmath8@xmath9 . [ cols="<,^ " , ] heavy - quark energy loss is presently studied at rhic using measurements of the nuclear modification factor @xmath10 of ` non - photonic ' ( @xmath11-conversion- and @xmath12-dalitz - subtracted ) single electrons . the most recent data by phenix @xcite and star @xcite , reaching out to 5 and 9 gev , respectively , are shown in fig . [ fig : rhic ] ( right ) . since this is an inclusive measurement , with charm decays dominating at low @xmath4 and beauty decays dominating at high @xmath4 , the comparison with mass - dependent energy loss predictions should rely on a solid and data - validated pp baseline . such baseline is still lacking at the moment , as we explain in the following . the state - of - the - art perturbative predictions ( fonll ) , that we use as a baseline , indicate that , in pp collisions , charm decays dominate the electron @xmath4 spectrum up to about 5 gev @xcite . however , there is a large perturbative uncertainty on position in @xmath4 of the @xmath13-decay/@xmath14-decay crossing point : depending on the choice of the factorization and renormalization scales this position can vary from 3 to 9 gev @xcite . in addition , the calculation tends to underpredict the non - photonic electron spectrum measured in pp collisions @xcite . for our electron @xmath10 results ( fig . [ fig : rhic ] , right ) , in addition to the uncertainty on the medium density ( curves for @xmath8 , 10 , @xmath9 ) , we also account for the perturbative uncertainty by varying the values of the scales and of the @xmath13 and @xmath14 quark masses ( shaded band associated to the @xmath15 curve ) @xcite . we find that the nuclear modification factor of single electrons is about 0.2 larger than that of light - flavor hadrons . thus , electrons are in principle sensitive to the mass hierarchy of parton energy loss . the available data neither allow us to support claims of inconsistency between theory and experiment , nor do they support yet the expected mass hierarchy . it is important to note that , in general , the perturbative uncertainty in calculating the partonic baseline spectrum is comparable to the model - intrinsic uncertainty in determining @xmath5 . if future experimental studies at rhic succeeded in disentangling the charm and beauty contributions to single electrons , the sensitivity in the theory - data comparison would be largely improved . ( left ) and @xmath1 ( right ) mesons for the case of realistic heavy - quark masses and for a case study in which the quark mass dependence of parton energy loss is neglected @xcite , scaledwidth=85.0% ] heavy quarks will be produced with large cross sections at lhc energy and the experiments will be equipped with detectors optimized for the separation of charm and beauty decay vertices . thus , it should be possible to carry out a direct comparison of the attenuation of light - flavor hadrons , @xmath0 mesons , and @xmath1 mesons . we calculate the expected nuclear modification factors @xmath10 exploring a conservatively - large range in the medium density for central collisions at @xmath16 : @xmath17 . we use standard nlo perturbative predictions for the @xmath13 and @xmath14 @xmath4-differential cross sections @xcite . figure [ fig : lhc ] ( thick lines ) shows our results for the heavy - to - light ratios of @xmath0 and @xmath1 mesons @xcite , defined as the ratios of the nuclear modification factors of @xmath18 mesons to that of light - flavor hadrons ( @xmath19 ) : @xmath20 . we illustrate the effect of the mass by artificially neglecting the mass dependence of parton energy loss ( thin curves ) . the enhancement above unity that persists in the @xmath21 cases is mainly due to the color - charge dependence of energy loss , since at lhc energy most of the light - flavor hadrons will originate from a gluon parent . our results indicate that , for @xmath0 mesons , the mass effect is small and limited the region @xmath22 , while for @xmath1 mesons a large enhancement can be expected up to @xmath23 . therefore , the comparison of the high-@xmath4 suppression for @xmath0 mesons and for light - flavor hadrons will test the color - charge dependence ( quark parent vs. gluon parent ) of parton energy loss , while the comparison for @xmath1 mesons and for light - flavor hadrons will test its mass dependence @xcite .
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the attenuation of heavy - flavored particles in nucleus
nucleus collisions tests the microscopic dynamics of medium - induced parton energy loss and , in particular , its expected dependence on the identity ( color charge and mass ) of the parent parton .
we discuss the comparison of theoretical calculations with recent single - electron data from rhic experiments .
then , we present predictions for the heavy - to - light ratios of @xmath0 and @xmath1 mesons at lhc energy . address = universit degli studi di padova and infn , padova , italy address = dep . de fsica de partculas and igfae , universidade de santiago de compostela , spain address = lpthe , universit pierre et marie curie ( paris 6 ) , france address = department of physics , cern , theory division , genve , switzerland address = department of physics and astronomy , university of stony brook , ny , usa
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the original parking problem of knuth can be stated as follows . consider a parking lot with @xmath0 spaces , identified with the cyclic group @xmath1 . initially the parking lot is empty , and @xmath2 cars in a queue arrive one by one . car @xmath3 tries to park on a uniformly distributed space @xmath4 among the @xmath0 possible , independently of other cars , but if the space is already occupied , then it tries places labeled @xmath5 until it finally finds a free spot to park . as cars arrive , blocks of consecutive occupied spots are forming . it appears that a _ phase transition _ occurs at the stage where the parking lot is almost full , more precisely when the number of free spots is of order @xmath6 . indeed , while the largest block of occupied spots is of order @xmath7 with high probability as long as @xmath8 , a block of size approximately @xmath0 is present ( while the others are of order at most @xmath9 ) with high probability when @xmath10 . in the meanwhile , precisely when @xmath11 is of order @xmath12 with @xmath13 , a clustering phenomenon occurs as @xmath14 decays . the behavior of this clustering process has been studied precisely by chassaing and louchard @xcite . it turns out that the process of the relative sizes of occupied blocks is related to the so - called _ standard additive coalescent _ @xcite . the model originates from a problem in computer science : spaces in the parking lot should be thought of as elementary memory spaces , each of which can be used to store elementary data ( cars ) . roughly , our aim in this work is to investigate the more general situation where one wants to store larger files , each requiring several elementary memory spaces . in other words , single cars are replaced by _ caravans _ of cars , i.e. several cars may arrive simultaneously at the same spot . in this direction , it will be convenient to consider a continuous version of the problem , that goes as follows . let @xmath15 be a sequence of positive real numbers with sum @xmath16 , and @xmath17 , @xmath18 distinct locations on the unit circle @xmath19 . imagine that @xmath18 drops of paint with masses @xmath20 , fall successively at locations @xmath21 . each time a drop of paint falls , we brush it clockwise in such a way that the resulting painted portion of @xmath22 is covered by a unit density of paint . so at each step the drop of paint is used to cover a new portion of the circle and the total length of the painted part of the circle when @xmath23 drops have fallen is @xmath24 . in this setting , drops of paint play the role of caravans , and the painted portion of the circle corresponds to occupied spots in the parking lot . more precisely , we consider an increasing sequence @xmath25 of open subsets of @xmath26 , starting from @xmath27 and ending at @xmath28 , which can be thought of as the successive painted portions of the circle . given @xmath29 and the location @xmath30 from where the @xmath31-th drop of paint will be brushed , we paint as many space as possible to the right of @xmath30 with the quantity @xmath32 of paint , without covering the already painted parts , i.e. the blocks of @xmath29 . alternatively , we break the @xmath31-th caravan into several pieces , so that to fill as much as possible the holes left by @xmath33 after @xmath30 , when reading in clockwise order . the last car to park arrives at some location @xmath34 , and we let @xmath35 be the union of @xmath29 and the arc between @xmath30 and @xmath34 , see figure [ fig : park ] . more formal definitions will come in sect . [ sec : bridge ] . in particular , @xmath29 is a disjoint union of open intervals and @xmath36 . let @xmath37 be the sequence of the lebesgue measures of the connected components of @xmath38 , ranked by decreasing order . it will be convenient to view @xmath39 as an infinite sequence , by completing with an infinite number of zero terms . now consider the following random problem . let @xmath40 be a random variable with finite expectation @xmath41 $ ] . we say that @xmath42 whenever @xmath43 has a finite second moment @xmath44 $ ] . for @xmath45 , we say that @xmath46 whenever @xmath47 for some @xmath48 . this implies that @xmath43 is in the domain of attraction of a spectrally positive stable random variable with index @xmath49 , and we stress that our results can be extended under this more general hypothesis ; ( [ tail ] ) is only intended to make things easier . we suppose from now on that @xmath46 for some @xmath50 $ ] , and take a random iid sample @xmath51 of variables distributed as @xmath43 , and independently of this sequence , iid uniform random variables on @xmath52 , @xmath53 . for @xmath54 , set @xmath55 so by the elementary renewal theorem , @xmath56 . then introduce the sequence @xmath57 defined by @xmath58 so the terms of @xmath59 sum to @xmath60 . following chassaing and louchard @xcite , we are interested in the formation of macroscopic painted components in the limit when @xmath61 tends to @xmath62 , at times close to @xmath63 , i.e. when the circle is almost entirely painted . specifically , we let @xmath64 for @xmath65 defined as above with the data @xmath66 . observe that @xmath67 $ ] decreases when @xmath68 increases , and therefore , in order to investigate the formation of painted components , we should consider the process @xmath69 backwards in time . this is what we shall do in theorem 1 , using the exponential time change @xmath70 . before describing our main result , let us first recall some features of the additive coalescent . the additive coalescent @xmath71 is a markov process with values in the infinite ordered simplex @xmath72 endowed with the uniform distance , whose evolution is described formally by : given that the current state is @xmath73 , two terms @xmath74 and @xmath75 , @xmath76 , of @xmath73 are chosen and merge into a single term @xmath77 ( which implies some reordering of the resulting sequence ) at a rate equal to @xmath77 . a version @xmath78 of this process defined for times describing the whole real axis is called _ eternal_. we refer to @xcite for background . as shown in @xcite , eternal additive coalescents can be encoded by certain bridges . specifically , let @xmath79 be a cdlg real - valued process with exchangeable increments , such that @xmath80 . suppose further that @xmath81 has infinite variation and no negative jumps a.s . then @xmath81 attains its overall infimum at a unique location @xmath82 ( which is uniformly distributed on @xmath83 $ ] ) , and @xmath81 is continuous at @xmath82 . consider the so - called _ vervaat transform _ which maps the bridge @xmath81 into an excursion @xmath84 defined by @xmath85 where the addition @xmath86 is modulo @xmath16 . finally , we let for @xmath87 @xmath88 and introduce @xmath89 as the random element of @xmath90 defined by the ranked sequence of the lengths of the constancy intervals of the process @xmath91 . here , a constancy interval means a connected component of the complement of the support of the stieltjes measure @xmath92 . finally , if we define @xmath93 , then @xmath94 is an eternal additive coalescent ( see section [ subsec : extreme ] for comments and details ) . in this work , eternal additive coalescent associated to certain remarkable bridges will play a key role . more precisely , we write @xmath95 for the eternal additive coalescent @xmath71 constructed above when @xmath96 is a standard brownian bridge ; so that @xmath97 is the so - called standard additive coalescent ( cf . @xcite ) . next , for @xmath98 , we denote by @xmath99 a standard spectrally positive stable lvy process with index @xmath49 , that is @xmath100 has independent and stationary increments , no negative jumps , and @xmath101 we call _ standard stable loop _ a _ loop _ and not a _ bridge _ to avoid a possible confusion : even though @xmath102 starts from @xmath62 , ends at @xmath62 and has exchangeable increments , it does not have the same law as the stable process @xmath100 conditioned on @xmath103 ! ] of index @xmath49 the process @xmath102 defined by @xmath104 we finally write @xmath105 for the eternal additive coalescent @xmath71 constructed above when the bridge @xmath81 is the standard stable loop of index @xmath49 . we are now able to state our main result . [ convaddcoal ] the process @xmath106 converges as @xmath107 in the sense of weak convergence of finite - dimensional distributions to some process @xmath108 . the exponential time - changed process @xmath109 is an eternal additive coalescent ; more precisely : \(i ) when @xmath110 , @xmath109 is distributed as @xmath111 \(ii ) when @xmath112 , @xmath109 is distributed as @xmath113 it might be interesting to discuss further the role of the parameter @xmath49 and the interpretation in terms of phase transition . as it was already mentioned , the renewal theorem entails than the number of drops of paint needed for the complete covering is @xmath114 , a quantity which is not sensitive to @xmath49 . it is easy to show that for every @xmath115 , there are no macroscopic painted components when only @xmath116 $ ] drops of paint have fallen , so the phase transition ( i.e. the number of drops which is needed for the appearance of macroscopic components ) occurs for numbers close to @xmath117 . more precisely , the regime for the phase transition is of order @xmath118 ; so the phase transition occurs closer to @xmath117 when @xmath49 is larger . we would like also to stress that one - dimensional distributions of the limiting additive coalescent process @xmath119 depend on @xmath49 , but not its semigroup which is the same for all @xmath50 $ ] . a heuristic explanation might be the following : the number of drops needed to complete the covering once the phase transition has occurred is too small ( of order @xmath120 ) to observe significant differences in the dynamics of aggregation of macroscopic painted components . * remark . * our model bears some similarity with another parking problem on the circle , where drops of paints fall uniformly on the circle and then are brushed clockwise , but where overlaps are now allowed ( some points may be covered this way several times ) , call it the `` random covering of an interval '' problem . however , as showed in @xcite , this last model has very different asymptotics from those of the parking problem , as it turns out that the random covering of an interval is related to kingman s coalescent rather than the additive coalescent . a shared feature is that the phase transition of the random covering problem appears also when the circle is almost completely covered , but for example the different fragments are ultimately finite in number rather than infinite . we also mention yet another parking problem , first considered by rnyi ( see @xcite ) . in can be formulated as follows : caravans with size @xmath61 are placed on @xmath26 ( the original work rather considers @xmath121 ) one after another , but the locations @xmath74 where cars park are chosen uniformly among spaces that do not induce overlaps and splitting of caravans , i.e. so that the length of the arc from @xmath74 to @xmath122 is exactly @xmath61 . this is done until no uncovered sub - arc of @xmath26 with size @xmath123 remains . this process does not involve coalescing blocks of cars , and one is rather interested in the properties of the random number of cars that are able to park . the method in @xcite relies on an encoding _ parking _ function which is shown to be asymptotically related to a function of standard brownian bridge , and a representation of the standard additive coalescent due to bertoin @xcite . our approach to theorem [ convaddcoal ] is close in spirit to that of @xcite , and uses the representation of eternal additive coalescent that we presented above ; we briefly sketch it here . first , we encode the process @xmath124 by a bridge with exchangeable increments in sect . [ sec : bridge ] . in sect.[sec : conv ] , we show that this bridge converges to some bridge with exchangeable increments that can be represented in terms of the standard brownian bridge ( for @xmath110 ) or the standard stable loop ( for @xmath112 ) . theorem [ convaddcoal ] then follows readily . the rest of this work is organized as follows . in section [ sec : bridge ] we provide a representation of the painted components in terms of a bridge and its vervaat s transform . the convergence of these bridges when @xmath61 tends to @xmath62 is established in section [ sec : conv ] , and that of the sequence of the sizes of the painted components in section [ convxeps ] . section [ sec : discrete ] is devoted to a brief discussion of the analogous discrete setting ( i.e. knuth s parking for caravans ) , and finally some complements are presented in section [ sec : compl ] . we develop a representation of the parking process with the help of bridges with exchangeable increments , which is crucial to our study . let us first give the proper definition the of sequence @xmath125 of the introduction . we identify the circle @xmath22 with @xmath52 and write @xmath126 for the canonical projection . if @xmath127 is a measurable subset of @xmath26 ( identified with @xmath52 ) , let @xmath128 be its repartition function defined by @xmath129\cap a)$ ] for @xmath130 , where @xmath131 is lebesgue measure . also , extend @xmath128 on the whole real line with the formula @xmath132 . given @xmath29 for some @xmath133 , let @xmath134 notice that the arc @xmath135 oriented clockwise from @xmath30 to @xmath136 has length @xmath137 . then let @xmath35 be the interior of the closure of @xmath138 . the point in taking the closure and then the interior is that we consider that two painted connected components of @xmath22 that are at distance @xmath62 constitute in fact a single painted connected component . define @xmath139 and @xmath140 in @xmath141 , so @xmath142 is a cdlg function ( right - continuous with left - limits ) on @xmath143 . consider it as a function on @xmath26 by letting @xmath144 where @xmath145 is the element of @xmath146 . the quantity @xmath147 can be thought of as the quantity of cars of the @xmath31-th caravan that try to park at @xmath148 . see figure [ fig : park2 ] . we consider the _ profile _ @xmath149 of the parking at step @xmath150 , so @xmath151 is the total quantity of cars that have tried ( successfully or not ) to park at @xmath148 ( with the convention that @xmath152 ) before the @xmath31-th caravan has arrived . [ h ] for @xmath153 , \(i ) the set @xmath29 is the interior of the support of @xmath154 . \(ii ) @xmath155 . \(iii ) @xmath156 jumps at times @xmath157 with respective jump magnitudes @xmath158 , and has a drift with slope @xmath159 on its support.that is , if @xmath160\subseteq\supp(h^{{{\bf p}}}_i)$ ] , @xmath161 properties ( i ) and ( iii ) are easily shown using a recursion on @xmath3 and splitting the behavior of @xmath142 on @xmath162 and @xmath163 . we give some details for ( ii ) . for @xmath164 , notice that by definition @xmath122 can not be a point of increase of @xmath165 , i.e. a point such that @xmath166 for every @xmath54 . therefore , @xmath167 and @xmath168 for @xmath169 . since it follows by continuity of @xmath165 that @xmath170 , ( ii ) is proved . consider the _ bridge function _ : @xmath171 which starts from @xmath172 and ends at @xmath173 . we extend @xmath174 to a function on @xmath175 by setting @xmath176 . for any @xmath177 , it is easily seen using ( iii ) in lemma [ h ] that @xmath178 } ( h^{{{\bf p}}}_i(v-)+b^{{{\bf p}}}_i(u))\right).\ ] ] suppose @xmath179 is such that @xmath180 ( here @xmath181 ) , call such a number a _ last empty spot_. by ( ii ) , lemma [ h ] , the set of last empty spots is not empty since it contains @xmath182 . on the other hand , by ( i ) in the same lemma , the support of @xmath183 is the closure of @xmath184 which has measure @xmath16 , hence it is @xmath26 . by ( iii ) , we conclude by letting @xmath185 that @xmath186 for @xmath187 , so for @xmath188 , @xmath189 necessarily since @xmath183 is non - negative . this implies that the last empty spots are those @xmath179 s such that @xmath190 . we choose one of them by letting @xmath191:b^{{{\bf p}}}_m(x-)=\inf_{u\in[0,1]}b^{{{\bf p}}}_m(u)\},\ ] ] the first location when the infimum of @xmath192 is reached . we have proved [ v ] for any @xmath193 , @xmath194 } b^{{{\bf p}}}_i(u).\ ] ] recall that we are interested in @xmath39 , the ranked sequence of the lengths of the interval components of @xmath38 , where @xmath38 can be viewed as the painted portion of the circle after @xmath3 drops of paint have fallen , or the set of occupied spots after the @xmath3-th caravan has arrived . lemma [ h](i ) enables us to identify @xmath38 as the interior of support of the function @xmath154 , and since the lebesgue measure of the interval components of the interior of the support of @xmath156 is not affected by a cyclic shift , we record the following simple identification [ lamb ] for every @xmath195 , @xmath39 coincides with the ranked lengths of the intervals of constancy of the function @xmath196 } b^{{{\bf p}}}_i(u)\,,\qquad x\in [ 0,1].\ ] ] we now consider a rescaled randomized version of the bridges introduced above . let @xmath197 , where @xmath198 is obtained as above with data @xmath199 , and these quantities are introduced in the introduction . so for @xmath200 @xmath201 because @xmath202 . recall that @xmath203 denotes the standard brownian bridge , and @xmath102 the standard stable loop with index @xmath49 as defined in ( [ pontstable ] ) . [ convbeps ] as @xmath107 , the bridge @xmath204 converges weakly on the space @xmath205 of cdlg paths endowed with skorokhod s topology , to a bridge with exchangeable increments @xmath206 . more precisely : \(i ) if @xmath110 then @xmath81 is distributed as @xmath207 . \(ii ) if @xmath45 , then @xmath81 is distributed as @xmath208 the proof of lemma [ convbeps](ii ) will use the following well - known representation : @xmath209 where @xmath210 is a sequence of i.i.d . uniform@xmath121 r.v.s , @xmath211 is the ranked sequence of the atoms of a poisson measure on @xmath212 with intensity @xmath213 , and these two sequences are independent . more precisely , the series in the right - hand side does not converge absolutely , but is taken in the sense @xmath214 where the limit is uniform in the variable @xmath148 , a.s . this representation follows immediately from the celebrated lvy - it decomposition , specified for the stable process @xmath100 , as the process of the jumps of the latter is a poisson point process on @xmath215 with intensity @xmath216 . see also kallenberg @xcite . following kallenberg @xcite , we represent the jump sizes of the bridge @xmath204 by the random point measure @xmath217 by theorem 2.3 in @xcite , we have to show : @xmath218 and @xmath219 where the convergence is in law with respect to the weak topology on measures on @xmath220 , and in ( [ ii ] ) , @xmath211 is the ranked sequence of the atoms of a poisson measure on @xmath212 with intensity @xmath213 . case ( i ) is easier to treat . indeed , notice that the total mass of @xmath221 is @xmath222 since @xmath223 , the law of large numbers gives @xmath224 . now let @xmath225 so to prove ( [ i ] ) , it suffices to show that @xmath226 in probability . notice that @xmath227 . let @xmath228 and @xmath229 . then @xmath230 the second term converges to @xmath62 since @xmath231 a.s . for the first term , notice that @xmath232 taking logarithms and checking that @xmath233 as @xmath107 ( which holds since @xmath234<\infty$ ] ) , we finally obtain that @xmath235 . this completes the proof of ( [ i ] ) . now we turn our attention to ( [ ii ] ) . it suffices to show that for every function @xmath236 , say of class @xmath237 with bounded derivative @xmath238 see for instance section ii.3 in le gall @xcite . in this direction , recall from the classical formula for poisson random measures that @xmath239 to start with , we observe from the renewal theorem that @xmath240 converges to @xmath62 in probability as @xmath241 , so in ( [ laplace ] ) , we may replace @xmath221 by @xmath242 next , for every @xmath243 , we consider the random measure @xmath244 again , by the ( elementary ) renewal theorem , @xmath245 in probability , so for every @xmath228 , the event @xmath246[encadre ] has a probability which tends to @xmath16 as @xmath241 . now @xmath247 taking logarithms , we have to estimate @xmath248 by ( [ tail ] ) and dominated convergence , we see that the preceding quantity converges as @xmath241 towards @xmath249 taking @xmath250 , using ( [ encadrem ] ) and letting @xmath251 tend to @xmath62 , we see that ( [ laplace ] ) holds , which completes the proof of the statement . in this section , we deduce theorem [ convaddcoal ] from lemmas [ lamb],[convbeps ] . recall the definition of the bridge @xmath174 in section [ sec : bridge ] . for @xmath252 , let @xmath253 be the bridge @xmath254 with data @xmath255 , so @xmath256 . let also @xmath257 be the left - most location of the infimum of @xmath204 , and @xmath258 the vervaat transform of @xmath204 . by lemma [ lamb ] , @xmath259 coincides with the ranked sequence of lengths of constancy intervals of the infimum process of @xmath260 where the constant @xmath261 has no effect and is added for future considerations . [ drift ] for every @xmath87 , the difference @xmath262 converges in probability for the uniform norm to the pure drift @xmath263 as @xmath107 . recall from the renewal theorem that @xmath264 in probability as @xmath265 . therefore , we might start the sum appearing in the statement from @xmath266 . now , the sequences @xmath267 and @xmath268 have the same distribution . up to doing the substitution , lemma [ drift ] for fixed @xmath269 is therefore a simple application of the strong law of large numbers . the conclusion is obtained by standard monotonicity arguments . as a consequence of lemmas [ convbeps ] , [ drift ] , and the fact that @xmath270 is continuous , the process @xmath271 converges in the skorokhod space to @xmath272 where @xmath273 is the vervaat transform of the limiting bridge @xmath81 which appears in lemma [ convbeps ] , @xmath82 being the a.s . unique location of its infimum . now letting @xmath274 be the infimum process of @xmath275 and @xmath89 be the decreasing sequence of lengths of constancy intervals of @xmath274 , we have [ convintervalles ] the process @xmath276 converges to @xmath277 in the sense of weak convergence of finite - dimensional marginals . the technical point is that skorokhod convergence of @xmath278 to @xmath279 , though it does imply convergence of respective infimum processes , does not _ a priori _ imply that of the ranked sequence of lengths of constancy intervals of these processes . however , this convergence does hold because for every @xmath87 , if @xmath280 is such a constancy interval , then @xmath281 for @xmath282 , a.s . see e.g. lemmas 4 and 7 in @xcite . this proposition proves theorem [ convaddcoal ] . indeed , recall from lemma [ convbeps ] that @xmath283 , where @xmath284 and for @xmath98 @xmath285 then plainly , @xmath286 , and hence the limiting process @xmath287 is distributed as @xmath288 . in situations involving parking problems , it may be more natural to consider discrete parking lots , i.e. @xmath1 instead of the unit circle , and caravans with integer sizes , e.g. as in knuth s original parking problem . each caravan chooses a random spot , uniform on @xmath1 , and tries to park at that spot . studying the frequencies of blocks of cars fits with our general framework by taking @xmath43 with integer values , @xmath289 and @xmath290 . rename by @xmath291 the former quantity @xmath63 ( the number of caravans ) . let @xmath292 so @xmath293 would be @xmath294 in the notation above . the analog of lemma [ drift ] is still true when replacing @xmath293 by @xmath295 , without essential change in the proof . thus to obtain the very same conclusions as in the preceding sections , it suffices to check a result similar to lemma [ convbeps ] . namely , we must prove that @xmath296 in the skorokhod space as @xmath297 . now it is easy to check that a.s . , @xmath298 for every @xmath299 $ ] , because no @xmath300 is rational a.s . therefore , it suffices to check that @xmath301 converges to @xmath81 in distribution for the skorokhod topology on @xmath205 . up to using skorokhod s representation theorem , this is done by taking @xmath302 and @xmath303 in the next lemma . [ sko ] let @xmath304 be a sequence of functions converging in @xmath205 to @xmath305 . for @xmath306 let also @xmath307 be a right - continuous non - decreasing function ( not necessarily bijective ) from @xmath83 $ ] to @xmath83 $ ] , such that the sequence @xmath308 converges to the identity function uniformly on @xmath83 $ ] . then @xmath309 in @xmath205 . first consider the case @xmath310 for every @xmath0 . fix @xmath54 . let @xmath311 be the right - continuous inverse of @xmath307 defined by @xmath312:\kappa_n(y)>x\}.\ ] ] it is easy to prove that @xmath313 for every @xmath148 . since @xmath305 is cdlg , one may find @xmath314 such that the oscillation @xmath315 for @xmath316 , where @xmath317 since @xmath307 approaches the identity , for @xmath0 large we may assume @xmath318 for @xmath316 . define a time - change @xmath319 ( i.e. an increasing bijection between @xmath83 $ ] and @xmath83 $ ] ) by interpolating linearly between the points @xmath320 . now let @xmath321 $ ] . suppose @xmath322 for some @xmath316 , and notice that @xmath323 . therefore , @xmath324 belongs to @xmath325 as well as @xmath326 by definition of @xmath327 , and @xmath328 else , one must have @xmath329 or @xmath330 , and the result is similar . finally , doing the same reasoning for @xmath331 converging to @xmath62 slowly enough gives the existence of some time - changes @xmath327 converging to the identity uniformly such that @xmath332}|f(\kappa_{n}(x))-f(\lambda_{n}(x))|\leq 2\eps_{n}$ ] , hence giving convergence of @xmath333 to @xmath305 in the skorokhod space . in the general case , for every @xmath334 let @xmath319 be a time - change such that @xmath319 converges to the identity as @xmath297 and @xmath335 converges to @xmath305 uniformly . take @xmath336 . then @xmath337 uniformly , so it suffices to show that @xmath338 in @xmath205 , which is done by the former discussion . in particular , we recover and extend a certain number of results from @xcite . in this section , we would like to provide some information on the eternal additive coalescents @xmath339 for @xmath340 , which appear in theorem [ convaddcoal ] . to start with , we should like to specify the representation of @xmath339 as a mixture of so - called _ extreme _ eternal additive coalescents ( @xcite , @xcite ) . in this direction , let us first consider a sequence @xmath341 of non - negative numbers satisfying @xmath342 and @xmath343 following kallenberg @xcite we associate to @xmath344 a _ bridge with exchangeable increments _ @xmath345 where @xmath346 denotes a sequence of iid uniform variables and @xmath347 is an independent standard brownian bridge . we write @xmath348 for the eternal additive coalescent associated to the bridge @xmath349 as explained in the introduction and call such @xmath348 extreme . according to ( * theorem 15 ) , every eternal version of the additive coalescent @xmath71 can be obtained as a mixing of shifted versions of extreme eternal additive coalescents @xmath348 , i.e. @xmath71 can be expressed in the form @xmath350 with @xmath351 random . equivalently , @xmath71 can be viewed as the eternal additive coalescent constructed in the introduction from the bridge with exchangeable increments @xmath352 . as observed by aldous and pitman @xcite , the mixing variables @xmath351 can be recovered from the initial behavior of @xmath71 : @xmath353 in the case of the standard stable loop @xmath102 with @xmath112 , recall from the lvy - it decomposition that @xmath354 and @xmath355 is the ranked sequence of the atoms of a poisson random measure on @xmath212 with intensity @xmath356 . in particular , @xmath357 has the law of a ( positive ) stable variable with index @xmath358 and @xmath359 is such that the sequence of squares @xmath360 is distributed according to the poisson - dirichlet law @xmath361 ; see pitman and yor @xcite . we also stress that every coalescent @xmath348 can be obtained as a limit of appropriate caravan parking problems , which are quite natural given the results of @xcite . precisely , suppose that a sequence of probabilities @xmath362 satisfying @xmath363 is given , and satisfies @xmath364 for a sequence @xmath344 as described above , and where @xmath365 when @xmath366 . for every @xmath0 , let @xmath367 be a uniform permutation on @xmath368 . consider the parking problem where the caravans which try to park successively have magnitudes @xmath369 . let @xmath53 be independent uniform@xmath121 random variables independent of @xmath367 , so we may consider the bridge with exchangeable increments @xmath370 kallenberg s theorem shows that under the asymptotic assumptions on @xmath371 , @xmath293 converges in distribution to the bridge @xmath372 defined above . the key to this lemma is to show that @xmath376 in probability as @xmath297 . the result is then obtained via the so - called `` weak law of large numbers for sampling without replacement '' : if @xmath377 is a sequence with sum @xmath68 satisfying @xmath378 as @xmath297 , and if @xmath367 is a uniform permutation on @xmath379 , then for every rational @xmath380 $ ] , @xmath381 in probability ( in fact in @xmath382 ) . the result in probability remains true if @xmath377 is random with sum @xmath68 , and @xmath383 in probability . one concludes that the process @xmath384 converges in probability to @xmath385 for the uniform norm by a monotonicity argument . the lemma is then proved by letting @xmath386 ( note that this last term is @xmath387 , which goes to @xmath62 ) . so let us show ( [ maxzero ] ) . to this end , let @xmath388 , then @xmath389 in probability , since @xmath390\sim\sigma({{\bf p}}^n)^{-1}(1-\rho)$ ] goes to infinity ( notice @xmath391 ) while @xmath392\sim e[x^{\rho}_{n}]^2 $ ] , as a simple computation shows . therefore , @xmath393 in probability . consequently , for any @xmath394 , the quantity @xmath395 goes to @xmath16 , so @xmath396 in probability . but then , for any @xmath54 , if @xmath397 is such that @xmath398 , then @xmath399 for @xmath0 large . up to taking @xmath0 even larger , with probability close to @xmath16 , @xmath400 for @xmath401 and therefore @xmath402 , hence ( [ maxzero ] ) . [ thetacoal ] as @xmath297 , under the asymptotic regime ( [ regime ] ) , the process @xmath405 converges in the sense of weak convergence of finite - dimensional marginals to the time - reversed eternal additive coalescent @xmath406 . it would also be interesting to determine the marginal laws of the fragmentation @xmath407 . the task seems quite difficult if started from the description of @xmath408 in terms of lengths of constancy intervals of vervaat transform of bridges , because excursion theory seems powerless here , unlike in @xcite . in particular , the fact that the fragmentation is based on stable loops and not stable bridges impedes the application of results of miermont @xcite on additive coalescents based on bridges of certain l ' evy processes . another way to start the exploration is to use the representation of fragmentation processes @xmath409 described in the preceding section with the help of inhomogeneous continuum random trees ( icrt ) discussed in @xcite . in particular , it is easy to obtain the first moment of a size - biased pick from a ( random ) positive sequence @xmath410 with sum @xmath411 a.s . is a random variable of the form @xmath412 , where @xmath413 . ] @xmath414 from the sequence @xmath89 for any fixed @xmath68 , as follows . let us recall the basic facts on the icrt@xmath415 construction of @xmath416 . the icrt can be viewed via a _ stick - breaking construction _ as the metric completion of the positive real line @xmath417 endowed with a non standard metric . precisely , suppose we are given the following independent random elements : * a poisson process @xmath418 on the octant @xmath419 , with intensity @xmath420 , so in particular @xmath421 is a poisson process with intensity @xmath422 , * a sequence of independent poisson processes @xmath423 with respective intensities @xmath424 . we distinguish the points @xmath425 as _ joinpoints _ , while @xmath426 are called _ cutpoints_. if @xmath251 is a cutpoint , let @xmath427 be its associated joinpoint , i.e. @xmath428 . by the assumption on @xmath344 , it is a.s . possible to arrange the cutpoints by increasing order @xmath429 . we then construct a family @xmath430 of `` reduced trees '' as follows . cut the set @xmath212 into `` branches '' @xmath431 $ ] , where by convention @xmath432 . let @xmath433 be the segment @xmath434 $ ] , endowed with the usual distance @xmath435 . then given @xmath436 , we obtain @xmath437 by adding the branch @xmath438 $ ] somewhere on @xmath439 , and we plant the left - end @xmath440 on the joinpoint @xmath441 ( since a.s . @xmath442 , the point @xmath443 is indeed an element of @xmath439 ) . precisely , @xmath444 $ ] and @xmath445 if @xmath446 , @xmath447 if @xmath448 $ ] , and @xmath449 if @xmath450,y\in{\cal r}(k)$ ] . as the distances @xmath451 are compatible by definition , this defines a random metric space @xmath452 such that the restriction of @xmath453 to @xmath439 is @xmath451 , we call its metric completion @xmath454 the icrt@xmath415 , its elements are called _ vertices_. the point @xmath455 is distinguished and called the _ root_. one can see that @xmath454 is an @xmath175-tree , i.e. a complete metric space such that for any @xmath456 there is a unique simple path @xmath457 $ ] from @xmath148 to @xmath145 , which is isometric to the segment @xmath458 $ ] , i.e. is a geodesic . moreover , it can be endowed with a natural measure @xmath459 which is the weak limit as @xmath297 of the empirical measures @xmath460 . this measure is non - atomic and supported on _ leaves _ , i.e. vertices @xmath461 such that @xmath462\setminus\{y\}$ ] for any @xmath463 . non - leaf vertices form a set called the _ skeleton_. a second natural measure is the lebesgue measure @xmath14 on @xmath454 , i.e. the unique measure such that @xmath464)=d(x , y)$ ] for any @xmath465 , and this measure is supported on the skeleton . now for each @xmath68 consider a poisson measure on @xmath454 with atoms @xmath466 , with intensity @xmath467 , so the different processes are coupled in the natural way as @xmath68 varies , i.e. @xmath466 increases with @xmath68 . these points disconnect the tree into a forest of disjoint connected tree components , order them as @xmath468 by decreasing order of @xmath459-mass . then the process @xmath469 of these @xmath459-masses has same law as @xmath416 . a size - biased pick from this sequence of masses is then obtained as the @xmath459-mass of the tree component at time @xmath68 that contains an independent @xmath459-sample , conditionally on @xmath470 . therefore , if @xmath471 denotes such a size - biased pick , @xmath472 $ ] is the probability that two independent @xmath459-samples @xmath473 belong to the same tree component of the cut tree , i.e. that no atom of the poisson measure at time @xmath68 falls in the path @xmath474 $ ] , and hence it equals @xmath475 $ ] . it turns out @xcite that @xmath476 has same law as the length @xmath477 of the first branch ( i.e. the length of @xmath433 ) . it is easy to see ( see also @xcite ) that this branch s length has law @xmath478 in our setting , recall that the random sequence @xmath479 is related to that of the atoms @xmath480 of a poisson measure on @xmath212 with intensity @xmath481 by ( [ equt ] ) and ( [ equth ] ) . observe that @xmath482 , and since we must take a poisson process with intensity @xmath483 on the skeleton of the icrt(@xmath479 ) , terms @xmath484 cancel out and @xmath485=e[e^{-t\eta}]$ ] where @xmath486\\ & = & \exp\left(-\int_0^{\infty}\frac{\alpha(\alpha-1)\d x}{\gamma(2-\alpha ) x^{1+\alpha } } ( 1-\exp(-rx+\log(1+rx)))\right)\\ & = & \exp(-(\alpha-1)r^{\alpha}),\end{aligned}\ ] ] which is a weibull distribution . this gives ( at least in principle ) the first moment @xmath487= \int_0^{\infty}\alpha(\alpha-1)r^{\alpha-1}\exp\left(-t r-(\alpha-1 ) r^{\alpha}\right)\d r.\ ] ] in principle , this method could be used for the computation of moments of higher order , where the length @xmath488 of the first branch would simply be replaced by the total length @xmath489 of @xmath439 . unfortunately , the distribution of @xmath489 is complicated for @xmath490 , and seems intractable for higher @xmath491 s ( @xcite ) .
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we consider a generalized version of knuth s parking problem , in which caravans consisting of a random number of cars arrive at random on the unit circle .
then each car turns clockwise until it finds a free space to park . extending a recent work by chassaing and louchard @xcite
, we relate the asymptotics for the sizes of blocks formed by occupied spots with the dynamics of the additive coalescent . according to the behavior of the caravan s size tail distribution ,
several qualitatively different versions of eternal additive coalescent are involved .
* keywords : * parking problem , additive coalescent , bridges with exchangeable increments . *
m.s.c .
code : * 60f17 , 60j25 .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the study of soliton and black hole solutions of the einstein - yang - mills ( eym ) equations has been an active subject for some twenty - five years , triggered by the discovery of regular , static , spherically symmetric , solitons @xcite and `` coloured '' black holes @xcite in @xmath2 eym in four - dimensional asymptotically flat space - time . in these configurations , the nonabelian gauge field is purely magnetic @xcite and described by a single function @xmath4 of the radial coordinate @xmath5 . numerical investigations @xcite show that @xmath4 must have at least one zero , and this is has also been proven analytically @xcite . both the particle - like and black hole solutions arise at discrete points in the phase space of parameters , and can be indexed by @xmath6 , the number of zeros of @xmath4 . the @xmath2 solitons and black holes have no magnetic charge at infinity . the black holes , in particular , are indistinguishable from a schwarzschild black hole at infinity , and therefore present counter - examples to the `` no - hair '' conjecture @xcite . however , both the black hole and soliton solutions are unstable under linear , spherically symmetric , perturbations of the gauge field and metric @xcite . as a consequence of this , bizon postulated a modified `` no - hair '' conjecture @xcite , heuristically stated as `` a stable black hole is uniquely characterized by global charges '' . the `` coloured '' black holes do not contradict this conjecture due to their instability . notwithstanding the instability of the @xmath2 eym solitons and black holes in asymptotically flat space - time , their discovery sparked extensive research into soliton and black hole solutions of more general matter models involving nonabelian gauge fields ( see @xcite for a review ) . in four - dimensional asymptotically flat space - time , enlarging the gauge group to @xmath0 , a richer phase space arises @xcite , but solutions still occur at discrete points in the phase space . furthermore , there is a very general result that all purely magnetic , asymptotically flat , soliton or black hole solutions of the @xmath0 eym equations in four - dimensional space - time are unstable @xcite . the situation is very different if one considers eym with a negative cosmological constant @xmath7 , so that the space - time is asymptotically anti - de sitter ( ads ) rather than asymptotically flat . four - dimensional , purely magnetic , black hole @xcite and soliton @xcite solutions of the @xmath2 eym equations in ads are found in continuous regions of parameter space rather than at discrete points as in the asymptotically flat case . furthermore , if @xmath8 is sufficiently large , there are solutions for which the single magnetic gauge field function @xmath4 has no zeros . these nodeless solutions have no analogue in asymptotically flat space - time and at least some of them are stable under linear , spherically symmetric perturbations @xcite . the proof of stability can be extended to general linear perturbations of the metric and gauge field providing @xmath9 is sufficiently large @xcite . for the larger @xmath0 gauge group , purely magnetic soliton and black hole solutions of eym in ads exist in continuous regions of the parameter space @xcite . in this case the purely magnetic gauge field is described by @xmath1 functions @xmath10 , @xmath11 , of the radial coordinate @xmath5 . there exist both soliton and black hole solutions for which all the @xmath10 have no zeros provided @xmath9 is sufficiently large @xcite . these solutions are of particular interest because it can be proven that at least some of them are stable under linear , spherically symmetric , perturbations of the metric and gauge field @xcite . although these stable black holes are coupled to potentially unlimited amounts of gauge field hair , they can nonetheless be characterized by global charges defined far from the black hole @xcite , in accordance with the modified `` no - hair '' conjecture @xcite ( see also the reviews @xcite ) . returning to @xmath2 eym in ads , the space of solutions revealed another surprise @xcite . in asymptotically flat space - time , all nontrivial ( that is , not corresponding to embedded schwarzschild or abelian reissner - nordstrm configurations ) @xmath2 solitons and black holes must have gauge fields with vanishing electric parts @xcite . this is not the case in asymptotically ads space - time , with dyonic solitons and black holes existing , for which the gauge field has nontrivial electric and magnetic components @xcite . for @xmath2 gauge group , the electric part is described by a single function @xmath12 and the magnetic part by a single function @xmath4 , where @xmath5 is the radial coordinate . the electric gauge field function @xmath12 is monotonic and has no zeros ; there exist solitons and black holes for which the magnetic gauge field function @xmath4 also has no zeros @xcite . although these solutions were found numerically fifteen years ago @xcite , it was only recently that it was shown that at least some of the solutions for which @xmath4 is nodeless are stable under linear , spherically symmetric perturbations @xcite . in this paper we explore the phase space of dyonic eym solitons and black holes in ads when the gauge group is enlarged to @xmath0 . in the purely magnetic case , with the larger gauge group the phase space has a very rich structure @xcite and we anticipate that the same will be true for dyonic solutions . dyonic solutions of @xmath0 eym will be described by @xmath1 electric gauge field functions @xmath13 , @xmath14 , and @xmath1 magnetic gauge field functions @xmath10 , @xmath14 . recently the existence of soliton and black hole solutions ( when @xmath9 is sufficiently large ) for which all the @xmath15 functions have no zeros was proven @xcite . these nodeless solutions are of particular interest because , in the light of results in the purely magnetic sector @xcite and for @xmath2 dyonic solutions @xcite , it is anticipated that at least some of them will be stable under linear , spherically symmetric perturbations . in this paper we present the first numerical dyonic solutions of the @xmath0 eym equations in ads , focusing not just on the nodeless solutions whose existence is proven in @xcite , but also more generally on the rich features of the phase space of solutions . the outline of this paper is as follows . in sec . [ sec : equations ] we introduce the @xmath0 eym model and field equations , and describe our static , spherically symmetric ansatz for the metric and gauge field potential . we also discuss the boundary conditions at the origin ( for regular solitons ) , event horizon ( for black holes ) and at infinity . solutions of the field equations are discussed in detail in sec . [ sec : solutions ] . we describe our numerical method , and explore the phase space of dyonic soliton and black hole solutions with @xmath2 and @xmath3 gauge group . our conclusions are presented in sec . [ sec : conc ] . in this section we describe our gauge field and metric ansatz , the field equations and the boundary conditions for static , spherically symmetric dyon and dyonic black hole solutions of @xmath0 eym in ads . we consider static , spherically symmetric , four - dimensional solitons and black holes with metric @xmath16 where the metric functions @xmath17 and @xmath18 depend only on the radial coordinate @xmath5 . we write the metric function @xmath19 in the form @xmath20 where @xmath7 is the cosmological constant . in ( [ eq : metric ] ) and throughout this paper , the metric has signature @xmath21 and we use units in which @xmath22 . for the static , spherically symmetric @xmath0 gauge field potential @xmath23 , we use the ansatz @xcite @xmath24 d\phi , \label{eq : gaugepot}\end{aligned}\ ] ] where we have set the gauge coupling @xmath25 . in ( [ eq : gaugepot ] ) , @xmath26 , @xmath27 , @xmath28 and @xmath29 are @xmath30 matrices which depend only on the radial coordinate @xmath5 . the matrices @xmath26 and @xmath27 are purely imaginary , diagonal and traceless . the matrix @xmath28 ( with hermitian conjugate @xmath31 ) is upper triangular , with the only nonzero entries immediately above the diagonal , which are written as : @xmath32 where @xmath10 and @xmath33 are real functions . the matrix @xmath29 is constant , diagonal and traceless , and given by @xmath34 the ansatz ( [ eq : gaugepot ] ) has some residual gauge freedom which can be used to set @xmath35 @xcite . it should be emphasized that the choice of ansatz ( [ eq : gaugepot ] ) is not unique @xcite . for purely magnetic solutions @xcite one makes the choice @xmath36 . however , in this paper we are interested in solutions for which the gauge field has nontrivial electric and magnetic parts . therefore @xmath26 will not vanish . we write @xmath26 in terms of matrices @xmath37 , which are the diagonal generators of the cartan subalgebra of @xmath0 . we define the @xmath37 in a similar way to ref . @xcite , but using a slightly different normalization . the nonzero entries of the matrix @xmath37 are : @xmath38 _ { j , k } = - \frac { i}{{\sqrt { 2\ell \left ( \ell + 1 \right ) } } } \left [ \sum _ { p=1}^{\ell } \left ( \delta _ { j , p } \delta _ { k , p } \right ) - \ell \delta _ { j , \ell+1 } \delta _ { k,\ell+1 } \right ] , \label{eq : hdef}\ ] ] where @xmath39 is the usual kronecker delta . for @xmath2 eym , there is a single generator of the cartan subalgebra : @xmath40 while for @xmath3 eym , there are two @xmath41 matrices : @xmath42 for general @xmath43 , there are @xmath1 matrices @xmath37 ( [ eq : hdef ] ) , since @xmath1 is the rank of the @xmath0 lie algebra . in terms of the @xmath37 matrices , the electric part of the gauge potential @xmath23 ( [ eq : gaugepot ] ) takes the form @xmath44 in terms of @xmath1 real scalar functions @xmath45 , depending on the radial coordinate @xmath5 only . our ansatz ( [ eq : adef ] ) for the electric part of the gauge potential at first sight looks rather different from that conventionally used in the literature @xcite , although it is equivalent . usually the @xmath43 diagonal entries of the traceless matrix @xmath26 are denoted by @xmath46 , @xmath47 ( where the @xmath48 are real functions of @xmath5 ) , and written in terms of @xmath1 real quantities @xmath49 as follows @xcite : @xmath50 so that @xmath51 from ( [ eq : adef ] ) , the @xmath48 can be written in terms of the @xmath52 as : @xmath53 and consequently @xmath54 we write the relation ( [ eq : caledef ] ) between the @xmath1 functions @xmath52 and the @xmath1 quantities @xmath55 as @xmath56 where @xmath57 , @xmath58 and @xmath59 is the lower - triangular @xmath60 matrix with entries @xmath61 by inverting @xmath59 ( [ eq : calfdef ] ) , we can write the @xmath52 in terms of the @xmath55 : @xmath62 for the magnetic part of the gauge field potential ( [ eq : gaugepot ] ) , we now assume that all the functions @xmath10 ( [ eq : cj ] ) in the matrix @xmath28 are nonzero . in this case one of the yang - mills equations reduces to @xmath63 for all @xmath64 and @xmath5 @xcite . our ansatz for the gauge potential ( [ eq : gaugepot ] ) then reduces to : @xmath65 d\phi , \label{eq : gaugepotsimp}\end{aligned}\ ] ] and the nonzero entries in the matrix @xmath28 are now simply @xmath66 the above ansatze for the matrices @xmath26 and @xmath28 appearing in the electric and magnetic parts of the gauge field respectively , are related to the expansion of the @xmath0 gauge field in terms of simple roots ( such an expansion is well - known in the construction of the @xmath0 monopole , see , for example , @xcite for a review ) . the matrix @xmath26 ( [ eq : adef ] ) is a linear combination of the generators @xmath67 , where the @xmath68 are the positive roots of @xmath0 , while the matrix @xmath28 with entries given by ( [ eq : cjsimp ] ) is a linear combination of the generators @xmath69 ( the raising operators ) corresponding to the simple roots of @xmath0 . in summary , the gauge field is described by @xmath70 functions of @xmath5 : the @xmath1 electric gauge functions @xmath45 and the @xmath1 magnetic gauge functions @xmath10 . the components of the yang - mills gauge field are given in terms of the gauge potential components ( [ eq : gaugepotsimp ] ) as follows : @xmath71 , \label{eq : gaugefield}\ ] ] when the gauge coupling constant @xmath25 . the yang - mills equations take the form @xmath72 = 0 . \label{eq : yme}\ ] ] the stress - energy tensor of the yang - mills field is @xmath73 , \label{eq : stress}\ ] ] where @xmath74 denotes a lie algebra trace . the stress - energy tensor @xmath75 acts as the source in the einstein equations : @xmath76 since we are using units in which @xmath77 . the yang - mills equations ( [ eq : yme ] ) for the gauge field with potential ( [ eq : gaugepotsimp ] ) take the form ( a prime @xmath78 denotes differentiation with respect to the radial coordinate @xmath5 ) : [ eq : ymefinal ] @xmath79 for @xmath80 . the einstein equations ( [ eq : ee ] ) give two equations for the metric functions @xmath81 and @xmath82 : [ eq : eefinal ] @xmath83 + \sum_{k=1}^{n-1 } \left [ \mu \omega _ { k}'^{2 } + \frac { k(k+1)}{4r^{2 } } \left ( 1 - \frac { \omega_{k}^{2}}{k } + \frac { \omega _ { k+1}^{2}}{k+1 } \right ) ^{2 } \right ] , \label{eq : mprime } \\ \sigma ' & = & \sum_{k=1}^{n-1 } \left [ \frac { 2\sigma \omega _ { k}'^{2}}{r } + \frac { 2\omega _ { k}^{2}}{\sigma \mu ^{2 } r } \left ( h_{k } { \sqrt { \frac { k+1}{2k } } } - h_{k-1 } { \sqrt { \frac { k-1}{2k } } } \right ) ^{2 } \right ] . \label{eq : sigmaprime}\end{aligned}\ ] ] setting @xmath84 for all @xmath85 , the field equations ( [ eq : ymefinal][eq : eefinal ] ) reduce to the purely magnetic field equations studied in @xcite . the field equations ( [ eq : ymefinal][eq : eefinal ] ) are singular at the origin @xmath86 ( relevant for soliton solutions ) , the black hole event horizon @xmath87 ( if there is one ) and infinity @xmath88 . we therefore need to specify appropriate boundary conditions at each of these points . as with the purely magnetic solutions @xcite the boundary conditions at the origin are the most complex . local existence of solutions of the field equations near the singular points @xmath86 , @xmath87 and @xmath88 is proven in @xcite , using a different representation of the electric part of the gauge field potential . in this section we cast the results of @xcite into our notation for completeness . we assume that the space - time is asymptotically ads and that the field variables have regular taylor series expansions about @xmath88 : @xmath89 substituting the expressions ( [ eq : infinity ] ) into the field equations ( [ eq : ymefinal][eq : eefinal ] ) , we find that @xmath90 , @xmath91 , @xmath92 and @xmath93 are free parameters , @xmath94 and that @xmath95 and @xmath96 are given by ( cf . @xcite ) @xmath97 , \nonumber \\ \sigma _ { 4 } & = & - \frac { 1}{2 } \sum_{k=1}^{n-1 } \left [ l^{4 } \omega _ { k,\infty } ^{2 } \left ( h_{k,\infty } { \sqrt { \frac { k+1}{2k } } } - h_{k-1,\infty } { \sqrt { \frac { k-1}{2k } } } \right ) ^{2 } + c_{k,1}^{2 } \right ] , \label{eq : m1sigma4infinity}\end{aligned}\ ] ] where @xmath98 is the ads radius of curvature . we assume that there is a regular , nonextremal black hole event horizon at @xmath87 . at @xmath87 , the metric function @xmath19 ( [ eq : mu ] ) has a single zero , so that @xmath99 to avoid a singularity at the event horizon , it must be the case that the electric gauge functions @xmath100 vanish at @xmath87 . we therefore assume the following taylor series expansions near the event horizon : @xmath101 from the field equations , we find that @xmath102 , @xmath103 and @xmath104 are free parameters and that @xmath105 , @xmath106 and @xmath107 are given in terms of them as follows ( cf . @xcite ) : @xmath108 , \nonumber \\ \sigma ' ( r_{h } ) & = & 2\sum _ { k=1}^{n-1 } \left [ \frac { \omega _ { k}(r_{h})^{2}}{\sigma ( r_{h } ) \mu ' ( r_{h})^{2 } r_{h } } \left ( h_{k}'(r_{h } ) { \sqrt { \frac { k+1}{2k } } } - h_{k-1}'(r_{h } ) { \sqrt { \frac { k-1}{2k } } } \right ) ^{2 } \right ] + 2\sum_{k=1}^{n-1 } \frac { \sigma ( r_{h } ) \omega _ { k}'(r_{h})^{2}}{r_{h } } , \nonumber \\ \omega _ { k}'(r_{h } ) & = & \frac { \omega _ { k}(r_{h})}{\mu ' ( r_{h } ) r_{h}^{2 } } \left ( \omega _ { k}(r_{h})^{2 } - 1 - \frac{1}{2 } \left [ \omega _ { k-1}(r_{h})^{2 } + \omega _ { k+1}(r_{h})^{2 } \right ] \right ) . \label{eq : horexp}\end{aligned}\ ] ] the value of @xmath102 is fixed in practice by the requirement that the metric function @xmath82 approaches unity as @xmath88 . this leaves the @xmath109 free parameters @xmath103 and @xmath104 for @xmath80 , whose values are restricted by the requirement @xmath110 which is needed for a regular nonextremal horizon at @xmath87 . in the purely magnetic case @xcite , the boundary conditions for the @xmath0 gauge potential near the origin are rather complicated , with a power series expansion up to @xmath111 necessary in order to completely specify the gauge field functions . it is no surprise that the addition of a nontrivial electric part to the gauge field potential only complicates matters further @xcite . we begin by assuming regular taylor series expansions for all field variables in a neighbourhood of the origin @xmath86 : @xmath112 the constant @xmath113 must be nonzero in order for the metric ( [ eq : metric ] ) to be regular at the origin . it is otherwise arbitrary as far as the expansions near the origin are concerned , and will be fixed in practice by the requirement that @xmath114 as @xmath88 . regularity of the field equations ( [ eq : ymefinal][eq : eefinal ] ) , metric ( [ eq : metric ] ) and curvature at @xmath86 gives @xcite @xmath115 from the equation for @xmath116 ( [ eq : mprime ] ) , we also have @xmath117 ^{2 } = 0 , \label{eq : originconds2}\ ] ] which is solved by @xmath118 the field equations ( [ eq : ymefinal][eq : eefinal ] ) are unchanged by the transformation @xmath119 , for each @xmath85 separately . therefore we take the positive sign in ( [ eq : omegaorigin ] ) without loss of generality . we consider next the magnetic gauge field functions @xmath120 . the coupling between these and the electric gauge field functions @xmath100 in the yang - mills equation for @xmath120 ( [ eq : omegakeqn ] ) does not affect the first two terms in the expansion of this equation near @xmath86 , because @xmath121 as @xmath122 . following @xcite , we define two vectors @xmath123 and @xmath124 . the first two terms in the expansion about @xmath86 of the yang - mills equation ( [ eq : omegakeqn ] ) then give the following equations for @xmath125 and @xmath126 : @xmath127 where @xmath128 is the @xmath129 matrix @xcite @xmath130 as discussed in @xcite , the eigenvalues of the matrix @xmath128 are @xmath131 and the eigenvectors have a complicated form involving hahn polynomials @xcite . writing a basis of normalized eigenvectors of @xmath128 as @xmath132 ( where @xmath133 is the eigenvector having eigenvalue @xmath134 ( [ eq : eigenvalues ] ) ) , we have , from ( [ eq : omega23 ] ) , @xmath135 where @xmath136 and @xmath137 are arbitrary constants . we proceed in a similar way for the electric gauge field functions @xmath100 , defining two vectors @xmath138 and @xmath139 . as with the magnetic gauge field functions , the coupling between the @xmath100 and @xmath120 in the yang - mills equation ( [ eq : hkeqn ] ) does not affect the first two terms in the expansion of this equation about @xmath86 , which then give the following two equations : @xmath140 where @xmath141 is the @xmath129 matrix @xmath142 in @xcite , an expansion similar to ( [ eq : origin ] ) is performed for the alternative electric gauge field functions @xmath143 , defined in terms of the @xmath144 functions that we use here by ( [ eq : caledef ] , [ eq : caleh ] ) : @xmath145 again we have @xmath146 for all @xmath85 , and the vectors @xmath147 and @xmath148 satisfy the equations @xcite @xmath149 using the relationship between @xmath150 and @xmath151 in ( [ eq : caleh ] ) , it is clear that the matrices @xmath128 ( [ eq : calm ] ) and @xmath141 ( [ eq : caln ] ) are related by @xmath152 and therefore the matrices @xmath128 and @xmath141 have the same eigenvalues @xmath134 ( [ eq : eigenvalues ] ) . in particular , we have , from ( [ eq : h23 ] ) , @xmath153 where @xmath154 and @xmath155 are arbitrary constants and the vectors @xmath156 , @xmath157 , are eigenvectors of @xmath141 , given by @xmath158 where an overall multiplicative constant is required so that the @xmath156 are normalized . once we have found the eigenvectors @xmath125 , @xmath126 , @xmath159 and @xmath160 , the first two terms in the einstein equations ( [ eq : mprime ] , [ eq : sigmaprime ] ) give the values of @xmath161 and @xmath162 ( which also depend on the cosmological constant @xmath163 ) @xcite . from the above analysis , the expansion of the @xmath100 to order @xmath5 and @xmath120 functions to order @xmath164 depends on just two arbitrary constants , @xmath136 ( [ eq : omega23sol ] ) and @xmath154 ( [ eq : h23sol ] ) , while the expansion to next order in @xmath5 ( that is , to order @xmath164 for the electric gauge field functions @xmath100 and order @xmath165 for the magnetic gauge field functions @xmath120 ) adds a further two arbitrary constants , @xmath137 and @xmath155 . it is shown in proposition 8 in @xcite , in analogy with the purely magnetic case @xcite , that a total of @xmath166 arbitrary constants are required to completely specify the gauge field functions in a neighbourhood of the origin . each additional power of @xmath5 in the expansion of the @xmath100 and @xmath120 depends on just two further arbitrary constants , one for the @xmath100 and one for the @xmath120 . to see this , define the vectors @xmath167 examination of the appropriate term in the expansion of the relevant yang - mills equation ( [ eq : ymefinal ] ) shows that the vectors @xmath168 , @xmath169 satisfy equations of the form @xmath170 { { \mbox{{\boldmath{{$\omega $ } } } } } } _ { j+1 } & = & { { \mbox{{\boldmath{{$p$}}}}}}_{j+1 } , \nonumber \\ \left [ { \mathcal { n}}_{n-1 } - j \left ( j + 1 \right ) \right ] { { \mbox{{\boldmath{{$h$}}}}}}_{j } & = & { { \mbox{{\boldmath{{$q$}}}}}}_{j+1 } , \label{eq : higherorigin}\end{aligned}\ ] ] where @xmath171 and @xmath172 are complicated vectors depending on @xmath173 , @xmath174 , @xmath175 , @xmath176 , whose detailed form can be found in @xcite . the solutions of ( [ eq : higherorigin ] ) are @xcite : @xmath177 where @xmath178 and @xmath179 are arbitrary constants . in ( [ eq : highersol ] ) , the vectors @xmath180 and @xmath181 are particular solutions of ( [ eq : higherorigin ] ) . it is shown in @xcite that these particular solutions can be chosen such that @xmath180 and @xmath181 are linear combinations of @xmath182 . we can therefore write the electric and magnetic gauge field functions in the following vectorial form , where @xmath183 and @xmath184 : @xmath185 where @xmath186 and the @xmath187 and @xmath188 functions have the following behaviour near the origin : @xmath189 as is explained in more detail in the next section , in our numerical procedure we integrate the field equations for the @xmath187 and @xmath188 functions rather than @xmath100 and @xmath120 , to improve numerical accuracy . in this paper we present dyonic solutions for the @xmath190 and @xmath191 cases only , so it is sufficient for our purposes to find @xmath160 , @xmath192 , @xmath125 and @xmath126 . we do not need to consider the complicated vectors @xmath171 , @xmath172 ( [ eq : higherorigin ] ) or @xmath193 , @xmath194 ( [ eq : highersol ] ) . we now present our new soliton and black hole solutions of the field equations ( [ eq : ymefinal][eq : eefinal ] ) . these field equations have a number of trivial solutions , which we describe first in sec . [ sec : trivial ] , before discussing our numerical method in sec . [ sec : numerics ] and the nontrivial solutions for @xmath2 and @xmath3 gauge groups in secs . [ sec : su2 ] and [ sec : su3 ] respectively . the first trivial solution arises on setting the electric gauge functions @xmath195 for @xmath80 , and the magnetic gauge functions @xmath120 to be the following constants : @xmath196 the metric functions @xmath81 and @xmath82 are then constants . setting @xmath197 without loss of generality gives the schwarzschild - ads metric with @xmath198 . the second trivial solution is reissner - nordstrm - ads . the metric function @xmath197 and @xmath19 takes the form @xmath199 where the mass @xmath200 and charge @xmath201 are constants . this solution of the field equations arises on setting the magnetic gauge functions @xmath202 for all @xmath80 , in which case the electric gauge functions are exactly @xmath203 where the @xmath90 and @xmath91 are constants . from the einstein equation ( [ eq : mprime ] ) , the charge @xmath201 is given by @xmath204 the charge @xmath201 ( [ eq : rncharge ] ) is an effective charge , with @xmath205 having two components . the first , @xmath206 , is a magnetic charge , and the second , @xmath207 , is an electric charge . the reissner - nordstrm - ads solution is therefore dyonic in this case . setting all the @xmath208 yields the purely magnetically charged reissner - nordstrm - ads solution . note that purely electrically charged reissner - nordstrm - ads is _ not _ a solution of the field equations ( [ eq : ymefinal][eq : eefinal ] ) due to the coupling between the electric gauge field functions @xmath100 and the magnetic gauge field functions @xmath120 . the third class of trivial solutions is @xmath2 embedded solutions , obtained in @xcite with an alternative parametrization of the electric part of the gauge field potential . we start by writing the @xmath1 magnetic gauge functions @xmath120 in terms of a single function @xmath209 , and the @xmath1 electric gauge functions @xmath100 in terms of a single function @xmath210 , as follows : @xmath211 where @xmath212 and @xmath213 are constants . the yang - mills equations ( [ eq : ymefinal ] ) reduce to those for the @xmath2 case with gauge functions @xmath210 and @xmath209 if the following conditions hold : @xmath214 substituting ( [ eq : su2gauge ] ) into the einstein equations ( [ eq : eefinal ] ) , we obtain the @xmath2 equations if @xmath215 and @xmath216 the conditions ( [ eq : ymsu2conds][eq : esu2conds2 ] ) are solved by taking @xmath217 the values of @xmath212 are the same as those used to embed purely magnetic @xmath2 solutions into @xmath0 eym @xcite . substituting ( [ eq : su2gauge ] , [ eq : absu2 ] ) into the field equations ( [ eq : ymefinal][eq : eefinal ] ) and defining new rescaled variables as follows @xcite : @xmath218 where @xmath219 gives the @xmath2 field equations @xmath220 \right ) , \end{aligned}\ ] ] where the magnetic gauge field function @xmath209 and the metric functions @xmath19 and @xmath82 are not scaled . setting @xmath221 gives , as expected , the purely magnetic embedded @xmath2 equations . the field equations ( [ eq : ymefinal][eq : eefinal ] ) form a set of @xmath222 ordinary differential equations . note that , unlike the purely magnetic case @xcite , here the equation for the metric function @xmath18 ( [ eq : sigmaprime ] ) does not decouple from the other equations . this does not complicate the numerical method significantly . we employ standard `` shooting '' techniques @xcite , using a bulirsch - stoer algorithm in c++ to integrate the ordinary differential equations . the field equations are singular at the origin or a black hole event horizon . for black hole solutions , we start our integration just outside the event horizon , at @xmath223 , using the expansions ( [ eq : horizon ] ) as initial conditions . we then integrate outwards with @xmath5 increasing until the solution either becomes singular or the field variables @xmath224 and @xmath82 have converged to constant asymptotic values to within a suitable numerical tolerance . for soliton solutions , we start our integration close to the origin . the need to include higher order terms in the expansions of the gauge field functions @xmath100 and @xmath120 means that these functions are not suitable for numerical integration . with limited numerical precision , we can not keep adding powers of @xmath5 in our initial conditions ( the expansions ( [ eq : origin ] ) ) without losing accuracy . for each @xmath43 we therefore first make a change of variables , writing the electric gauge functions @xmath100 in terms of new variables @xmath33 , @xmath14 , and the magnetic gauge functions @xmath120 in terms of new variables @xmath225 , @xmath14 ( [ eq : betagamma ] ) , where the @xmath225 and @xmath33 are chosen so that their expansions near the origin have the form ( [ eq : betagammaorigin ] ) . in secs . [ sec : su2dyons ] and [ sec : su3dyons ] the details of this change of variables will be presented for the @xmath190 and @xmath191 cases respectively . in the following sections , we present examples of numerical solutions of the field equations ( [ eq : ymefinal][eq : eefinal ] ) representing both dyons and dyonic black holes , for @xmath190 and @xmath191 . following @xcite , we also study the structure of the space of solutions by examining the phase space of parameters characterizing the solutions near the event horizon or origin , as applicable . in the @xmath2 case , it is straightforward to show that the single electric gauge field function @xmath12 has no zeros , as follows . the equation for @xmath12 takes the form ( [ eq : hkeqn ] ) @xmath226 if the function @xmath12 has a turning point at @xmath227 , then @xmath228 and ( [ eq : h1su2 ] ) gives @xmath229 since the metric function @xmath19 is strictly positive for all @xmath230 for soliton solutions and for all @xmath231 for black hole solutions , we conclude from ( [ eq : h1ppsu2r0 ] ) that the turning point is a minimum if @xmath232 and a maximum if @xmath233 . from the expansions ( [ eq : origin ] ) , noting that @xmath234 , we see that @xmath235 has the same sign as @xmath12 very close to the origin . similarly , near a black hole event horizon , @xmath235 and @xmath12 have the same sign from ( [ eq : horizon ] ) . we deduce that it is not possible for @xmath12 to have a turning point , and therefore it is monotonic and has no zeros . for @xmath0 , it is proven in @xcite that the electric gauge field functions @xmath55 , defined in terms of the @xmath100 by ( [ eq : caledef ] ) , are monotonic and have no zeros . while it is not necessarily the case that our alternative electric gauge field functions @xmath100 have no zeros , since all the @xmath55 are nodeless any zeros of @xmath100 are a quirk of our parametrization of the electric part of the gauge potential , rather than revealing any underlying structure of the space of solutions . we therefore divide our numerical solutions into classes depending on the numbers of zeros @xmath236 of the magnetic gauge field functions @xmath120 respectively . we use coloured plots to show this phase space structure , but hope that the key features will still be apparent to readers using black and white . dyons and dyonic black holes in @xmath2 eym in ads were first found by bjoraker and hosotani @xcite . in this section we study the phase space of @xmath2 solutions , checking that we reproduce the results of @xcite and exploring in more detail those key features which will extend to the larger @xmath43 case . four parameters are required to describe the @xmath2 black hole solutions : @xmath237 , @xmath163 , @xmath238 and @xmath239 ( [ eq : horizon ] , [ eq : horexp ] ) . we fix @xmath240 . eym in ads . the parameters are : @xmath240 , @xmath241 , @xmath242 and @xmath243 . the electric gauge field function @xmath12 is monotonic and nodeless ; the magnetic gauge field function @xmath4 has a single zero.,width=321 ] a typical black hole solution with @xmath241 is shown in fig . as anticipated , the electric gauge field function @xmath12 is monotonic and has no zeros . in fig . [ fig1 ] , we have chosen the initial values @xmath242 and @xmath244 . for these initial values we see that the magnetic gauge field function @xmath4 has a single zero . we now study the phase space by fixing @xmath163 and varying @xmath238 and @xmath239 . setting @xmath245 gives purely magnetic solutions . the field equations ( [ eq : ymefinal][eq : eefinal ] ) are invariant under the separate transformations @xmath246 and @xmath247 . therefore it suffices to consider @xmath248 and @xmath249 . the values of @xmath238 and @xmath239 are not completely free : the constraint ( [ eq : murh ] ) must be satisfied . as in the purely magnetic case @xcite , for each value of @xmath163 studied we find a region in the @xmath250-plane for which ( [ eq : murh ] ) is satisfied , but we are unable to find regular black hole solutions which converge as @xmath88 . eym , with @xmath240 and @xmath241 . all shaded points in the plot correspond to black hole solutions . the solutions are indexed by @xmath251 , the number of zeros of the magnetic gauge field function @xmath4 . the different values of @xmath251 are indicated by colour - coding the regions - in black and white the different colours are different shades of grey . solutions with the largest values of @xmath251 are found towards the right - hand - side of the coloured region.,width=321 ] solutions from fig . all shaded points in the plot correspond to black hole solutions . the @xmath252 region ( the small red or dark grey region near the point ( 0,1 ) ) is in agreement with ref . @xcite.,width=321 ] in fig . [ fig2 ] we show the phase space for black holes with @xmath240 and @xmath241 , part of which has previously been shown in @xcite . all points in the plot in fig . [ fig2 ] represent black hole solutions with particular values of @xmath238 and @xmath239 . in fig . [ fig2 ] we find a richly structured parameter space , with solutions for which @xmath4 has up to 17 nodes . the number of zeros of @xmath4 increases as @xmath238 increases for each fixed value of @xmath239 . the corresponding plot in @xcite focused on the small region near @xmath245 , @xmath253 for which there are nodeless @xmath252 solutions , so for comparison with @xcite , in fig . [ fig3 ] we show a close - up of the parameter space near the @xmath252 region , which is in agreement with @xcite . eym , with @xmath240 and @xmath254 . all black hole solutions that we find have @xmath252 . the region labeled `` no solution '' ( the red / darker grey region containing larger values of @xmath238 and @xmath239 ) is the region where the constraint ( [ eq : murh ] ) is satisfied , giving a regular event horizon , but we have not found any regular black hole solutions.,width=321 ] for purely magnetic solutions , increasing the magnitude of the negative cosmological constant @xmath9 increased the size of the region of phase space where solutions were found @xcite . the @xmath252 region of nodeless solutions also expanded as a proportion of the total solution space @xcite . we find the same effects for dyonic black hole solutions . to illustrate this , in fig . [ fig4 ] we show the phase space of solutions for @xmath254 and @xmath240 . in this case the only solutions we find are such that the magnetic gauge field function @xmath4 has no zeros . in fig . [ fig4 ] , we have also shown the region of the @xmath250-plane where the constraint ( [ eq : murh ] ) is satisfied but we have not been able to find regular solutions . this region is marked `` no solution '' in fig . [ fig4 ] . eym with @xmath240 and @xmath255 . the solutions are indexed by @xmath251 , the number of zeros of the magnetic gauge function @xmath4 . regions with different colours ( or shades of grey ) correspond to different values of @xmath251 . we find solutions with very large values of @xmath251 as @xmath9 decreases . the blue / darkest region ( second from the right ) corresponds to @xmath256 black hole solutions , and extends to @xmath257 , with @xmath258 in this limit.,width=321 ] eym with @xmath240 and @xmath259 . the solutions are indexed by @xmath251 , the number of zeros of the magnetic gauge function @xmath4 . regions with different colours ( or shades of grey ) correspond to different values of @xmath251 . we find solutions with very large values of @xmath251 as @xmath9 decreases . however , the phase space of solutions does not extend to @xmath257 as there is no solution in this limit for this value of @xmath239.,width=321 ] bjoraker and hosotani @xcite found a rich , `` fractal''-like , structure in the phase space of solutions as @xmath257 ( see also @xcite for similar behaviour for solitons in the purely magnetic case ) . in asymptotically flat space , there are no dyonic black hole solutions of @xmath2 eym @xcite . however , there are purely magnetic solutions found by setting the electric part of the gauge potential equal to zero @xcite . in figs . [ fig5 ] and [ fig6 ] we investigate the structure of the phase space as @xmath257 . we fix @xmath239 and vary @xmath238 and @xmath163 . in fig . [ fig5 ] we set @xmath260 , which is the value for the first `` coloured '' black hole solution which exists in the limit @xmath257 @xcite . in fig . [ fig6 ] we set @xmath259 , which does not correspond to a regular black hole solution in the limit @xmath257 . in both figs . [ fig5 ] and [ fig6 ] , we have marked `` no solution '' the region where the constraint ( [ eq : murh ] ) for a regular event horizon is satisfied , but our numerical solution becomes singular before @xmath88 . for @xmath9 sufficiently large , for both values of @xmath239 the solutions are such that @xmath4 is nodeless . as @xmath9 decreases , the number of zeros of @xmath4 increases rapidly . we find solutions for which @xmath4 has more than fifty zeros . the phase spaces shown in figs . [ fig5 ] and [ fig6 ] are subtly different when @xmath9 is very small . when @xmath260 ( fig . [ fig5 ] ) , the @xmath256 part of the phase space ( the second region from the right ) extends to @xmath257 as we have the first `` coloured '' black hole solution in this limit @xcite . however , for @xmath259 , there is no solution in the limit @xmath257 and so the phase space ends at a small but nonzero value of @xmath9 . we are unable to find solutions for @xmath259 and @xmath9 smaller than about @xmath261 . as well as the cosmological constant @xmath163 , dyonic solitons in @xmath2 eym are parameterized by the quantities @xmath262 and @xmath263 , so that the expansions ( [ eq : origin ] ) of the gauge field functions take the form @xmath264 unlike the black hole case , where the constraint ( [ eq : murh ] ) for a regular nonextremal event horizon restricts the values of the parameters describing the solutions , for solitons there are no _ a priori _ constraints on the values that @xmath262 and @xmath263 can take . in particular , @xmath262 can take either positive or negative values . however , since the field equations are invariant under the transformation @xmath265 , we can restrict attention to @xmath266 without loss of generality . to avoid numerical errors in terms in the field equations ( [ eq : ymefinal][eq : eefinal ] ) , we define a new variable @xmath267 by @xmath268 which satisfies the first order differential equation @xmath269 and add this differential equation to those ( [ eq : ymefinal][eq : eefinal ] ) to be integrated numerically . near the origin , @xmath270 in the @xmath2 case , the relation ( [ eq : betagamma ] ) for the gauge field functions takes the form @xmath271 therefore our new variable @xmath267 ( [ eq : psidef ] ) is related to @xmath272 by @xmath273 and , using ( [ eq : betagammaorigin ] ) , @xmath274 eym in ads . the parameters are : @xmath241 , @xmath275 , @xmath276 . the electric gauge field function @xmath12 is monotonic and nodeless ; the magnetic gauge field function @xmath4 has a single zero.,width=321 ] a typical soliton solution with @xmath241 is shown in fig . [ fig7 ] , where the parameters are @xmath277 and @xmath276 . for these initial values we see that the magnetic gauge field function @xmath4 has a single zero . as expected , the electric gauge field function @xmath12 is monotonic and has no zeros . eym , with @xmath241 . all shaded points in the plot correspond to soliton solutions . the solutions are indexed by @xmath251 , the number of zeros of the magnetic gauge field function @xmath4 . the different values of @xmath251 are indicated by colour - coding the regions - in black and white the different colours are different shades of grey . solutions with the largest values of @xmath251 are found towards the right - hand - side of the coloured region.,width=321 ] solutions from fig . the @xmath252 region ( the green or light grey region containing the point ( 0,0 ) ) is in agreement with ref . the region labelled `` no solution '' ( the red / darker grey region containing larger values of @xmath262 for smaller values of @xmath263 ) is the region where we have not found any regular soliton solutions.,width=321 ] the entire phase space of soliton solutions for @xmath241 is shown in fig . all points in the plot in fig . [ fig8 ] represent dyonic soliton solutions with particular values of @xmath263 and @xmath262 . in fig . [ fig8 ] , as in the black hole case ( fig . [ fig2 ] ) , the phase space is very complicated , and we find solutions for which @xmath4 has up to 17 nodes . the corresponding plot in @xcite focused on the small region near @xmath278 , @xmath279 for which there are nodeless @xmath252 solutions . for comparison , in fig . [ fig9 ] we show a close - up of the parameter space near the @xmath252 region , which is in agreement with the corresponding plot in @xcite . eym , with @xmath254 . the different values of @xmath251 are indicated by colour - coding the regions - in black and white the different colours are different shades of grey . the region labelled `` no solution '' ( the red / mid grey region containing larger values of @xmath262 and @xmath263 ) is the region where we have not found any regular soliton solutions . there are solutions for which the number of nodes @xmath251 of the magnetic gauge field function @xmath4 is either zero or one.,width=321 ] as @xmath9 increases , the phase space of dyonic soliton solutions simplifies , in the same way as we observed for the black hole solutions . this can be seen in fig . [ fig10 ] , where we plot the phase space of solutions for @xmath254 . we find solutions where the magnetic gauge field function @xmath4 has either zero nodes or one node . having discussed the phase space of @xmath2 dyonic black holes and solitons in some detail , we now present new @xmath3 dyonic black holes and solitons , and explore the phase space of solutions . dyonic black hole solutions of @xmath3 eym are described by the following six parameters : @xmath237 , @xmath163 , @xmath238 , @xmath280 , @xmath239 and @xmath281 . we fix the event horizon radius @xmath240 . the field equations ( [ eq : ymefinal][eq : eefinal ] ) are symmetric under the transformations @xmath282 , @xmath283 , for each function separately , and so we can consider @xmath284 , @xmath285 , for @xmath286 without loss of generality . eym in ads . the parameters are : @xmath240 , @xmath241 , @xmath242 , @xmath287 and @xmath288 . both electric gauge field functions @xmath12 and @xmath289 are monotonic and nodeless . the magnetic gauge field function @xmath4 has two zeros and @xmath290 has three.,width=321 ] fig . [ fig11 ] shows a typical @xmath3 dyonic black hole solution , with @xmath241 and the initial values @xmath291 , @xmath292 , @xmath293 . for this solution the two electric gauge field functions are monotonic and have no zeros ; @xmath12 is monotonically decreasing and @xmath289 monotonically increasing . the two magnetic gauge field functions @xmath4 and @xmath290 have zeros ; @xmath4 has two zeros and @xmath290 has three . in order to explore the phase space using two - dimensional plots , it is necessary to fix two of the four parameters @xmath238 , @xmath280 , @xmath239 and @xmath281 as well as the event horizon radius @xmath237 and cosmological constant @xmath163 . overall , the structure of the phase space of solutions is extremely complicated ; we give a flavour of some of the key features in figs . [ fig12 ] and [ fig13 ] . as explained in sec . [ sec : numerics ] , we can classify the solutions according to the numbers of zeros @xmath294 and @xmath295 of the magnetic gauge field functions @xmath4 and @xmath290 respectively . in both figs . [ fig12 ] and [ fig13 ] , we have fixed the values of @xmath239 and @xmath281 , scanning the values of @xmath238 and @xmath280 such that the constraint ( [ eq : murh ] ) is satisfied . as in the @xmath2 case , there are values of the parameters such that ( [ eq : murh ] ) is satisfied but for which we do not find regular black hole solutions . eym with @xmath240 , @xmath241 , and @xmath296 . all shaded points in the plot correspond to black hole solutions . the solutions are indexed by @xmath297 , the number of zeros of the magnetic gauge field functions @xmath4 , @xmath290 respectively . the different combinations of values of @xmath298 are indicated by colour - coding the regions - in black and white the different colours are different shades of grey . in general the number of zeros of the magnetic gauge field functions increases as we move towards the edges of the phase space . for these values of the parameters , there are no nodeless solutions . the number of zeros of the magnetic gauge field functions is @xmath299 in the green / lighter grey region containing the origin.,width=321 ] eym with @xmath240 , @xmath254 , @xmath300 and @xmath301 . the solutions are indexed by @xmath297 , the number of zeros of the magnetic gauge field functions @xmath4 , @xmath290 respectively . the different combinations of values of @xmath298 are indicated by colour - coding the regions - in black and white the different colours are different shades of grey . for these values of the parameters , all nontrivial solutions are nodeless . we have also shown in red / darker grey ( marked `` no solution '' ) those values of the parameters @xmath302 for which the constraint ( [ eq : murh ] ) for a regular event horizon holds , but we have not found regular solutions.,width=321 ] when the magnitude of the cosmological constant is small , we typically find a rich phase space structure with solutions with many different numbers of nodes . this is illustrated in fig . [ fig12 ] , where we set @xmath241 . we have also fixed the values of the magnetic gauge field functions on the event horizon to be @xmath296 , and scanned the phase space by varying @xmath238 , @xmath280 . with these values of the parameters , for all the solutions both magnetic gauge field functions @xmath4 and @xmath290 have at least two zeros . as we have set the magnetic gauge field functions to be equal on the event horizon , from ( [ eq : su2gauge ] , [ eq : absu2 ] ) there are embedded @xmath2 solutions along the line @xmath303 . in @xcite the existence of dyonic black hole solutions of @xmath0 eym in ads in a neighbourhood of embedded @xmath2 dyonic black holes is proven . in fig . [ fig12 ] we see the neighbourhood of the embedded @xmath2 solutions for which there are nontrivial @xmath3 dyonic black holes . also from fig . [ fig12 ] , the number of nodes of the magnetic gauge field functions increases as @xmath238 and @xmath280 increase . as the magnitude of the cosmological constant increases , for @xmath3 black hole solutions we find ( as for the @xmath2 case ) that the phase space simplifies considerably . this is illustrated in fig . [ fig13 ] , where we have fixed the cosmological constant to be @xmath254 and the magnetic gauge field functions on the horizon have the values @xmath300 , @xmath301 . for these values of the parameters , all nontrivial solutions are such that the two magnetic gauge field functions @xmath4 and @xmath290 have no zeros . since @xmath304 in fig . [ fig13 ] , there are no embedded @xmath2 solutions in this part of the phase space . for @xmath3 gauge group , the electric and magnetic gauge field functions need to be expanded to @xmath305 in a neighbourhood of the origin , as follows ( see the discussion in sec . [ sec : origin ] ) : @xmath306 for @xmath286 , where we have assumed without loss of generality that @xmath307 . the vectors @xmath308 and @xmath309 are eigenvectors of the matrix @xmath310 ( [ eq : calm ] ) with eigenvalues @xmath311 and @xmath312 respectively . for @xmath191 , the matrix @xmath310 simplifies to @xmath313 and the relevant normalized eigenvectors are @xcite @xmath314 in terms of which @xmath125 and @xmath125 are given by ( [ eq : omega23sol ] ) . similarly , the vectors @xmath315 and @xmath316 are eigenvectors of the matrix @xmath317 ( [ eq : caln ] ) with eigenvalues @xmath311 and @xmath312 respectively . for @xmath191 , the matrix @xmath317 is @xmath318 with the relevant normalized eigenvectors ( in terms of which @xmath160 and @xmath192 are given by ( [ eq : h23sol ] ) ) being @xmath319 it is straightforward to check that the eigenvectors @xmath133 ( [ eq : u1u2 ] ) , @xmath320 ( [ eq : v1v2 ] ) , @xmath321 , are related by @xmath322 where , for @xmath191 , the matrix @xmath323 ( [ eq : calfdef ] ) takes the form @xmath324 in order to numerically integrate the field equations ( [ eq : ymefinal ] , [ eq : eefinal ] ) with the initial conditions ( [ eq : su3origin ] ) , we seek new variables @xmath225 , @xmath33 , @xmath321 , with the behaviour ( [ eq : betagammaorigin ] ) near the origin . the @xmath225 depend only on the magnetic gauge field functions and the @xmath33 depend only on the electric gauge field functions . using the relations ( [ eq : betagamma ] ) and the eigenvectors ( [ eq : u1u2 ] , [ eq : v1v2 ] ) , we define the @xmath225 functions so that the magnetic gauge field functions @xmath10 take the form @xmath325 , \nonumber \\ \omega_{2}(r ) & = & \sqrt{2 } + \frac{1}{\sqrt{2}}\left[\beta_{1}(r ) - \beta_{2}(r ) \right ] , \label{eq : omegasu3sol}\end{aligned}\ ] ] and the @xmath33 functions so that the electric gauge field functions @xmath13 take the form @xmath326 the new variables @xmath272 , @xmath327 , @xmath328 and @xmath329 satisfy the following equations , which are derived from ( [ eq : ymefinal ] ) : [ eq : su3soleqns ] @xmath330 , \\ \beta_2 '' & = & -\left(\frac{\sigma'}{\sigma } + \frac{\mu'}{\mu}\right)\beta_2 ' + \frac{1}{4\mu r^2}(7\beta_1 ^ 2 + 28\beta_2 + \beta_2 ^ 2 + 24)\beta_2 \nonumber\\ & \ , & - \frac{1}{\sqrt{2}\sigma^2\mu^2}\left [ \sqrt{6}\gamma_1 \gamma_2 + \frac{\sqrt{3}\beta_2\gamma_1 \gamma_2}{\sqrt{2 } } - \frac{\beta_1}{\sqrt{2}}\left(\frac{9\gamma_1 ^ 2}{16 } + \frac{3\gamma_2 ^ 2}{2}\right)\right ] , \\ \gamma_1 '' & = & \left ( \frac{\sigma'}{\sigma } - \frac{2}{r}\right)\gamma_1 ' + \frac{2\gamma_1}{\mu r^2 } + \frac{1}{\mu r^2}\left[\frac{1}{2}(\beta_1 ^ 2 + \beta_2 ^ 2 ) + 2(\beta_1 - \beta_2 ) - \beta_1\beta_2\right]\left(\frac{1}{2}\gamma_1 + \frac{\sqrt{3}}{2}\gamma_2\right ) \nonumber \\ & \ , & + \frac{1}{\mu r^2}\left[\frac{1}{2}(\beta_1 ^ 2 + \beta_2 ^ 2 ) + 2(\beta_1 + \beta_2 ) + \beta_1\beta_2\right]\left(\frac{1}{2}\gamma_1 - \frac{\sqrt{3}}{2}\gamma_2\right ) , \\ \gamma_2 '' & = & \left ( \frac{\sigma'}{\sigma } - \frac{2}{r}\right)\gamma_2 ' + \frac{6\gamma_2}{\mu r^2 } + \frac{1}{\mu r^2}\left[\frac{1}{2}(\beta_1 ^ 2 + \beta_2 ^ 2 ) + 2(\beta_1 - \beta_2 ) - \beta_1\beta_2\right]\left(\frac{\sqrt{3}}{2}\gamma_1 + \frac{3}{2}\gamma_2\right ) \nonumber\\ & \ , & + \frac{1}{\mu r^2}\left[\frac{1}{2}(\beta_1 ^ 2 + \beta_2 ^ 2 ) + 2(\beta_1 + \beta_2 ) + \beta_1\beta_2\right]\left(\frac{\sqrt{3}}{2}\gamma_1 - \frac{3}{2}\gamma_2\right).\end{aligned}\ ] ] we also substitute for @xmath13 and @xmath10 from ( [ eq : omegasu3sol ] , [ eq : hsu3sol ] ) into the einstein equations ( [ eq : eefinal ] ) , and then numerically integrate the resulting equations , together with ( [ eq : su3soleqns ] ) , using the initial conditions ( [ eq : betagammaorigin ] ) . the solutions are parametrized by the cosmological constant @xmath163 , and the four parameters @xmath136 , @xmath137 , @xmath154 and @xmath155 . as in the @xmath2 case , there are no _ a priori _ constraints on the values of these four parameters . in general @xmath136 and @xmath137 can take both positive and negative values . in our phase space plots , we have restricted our attention to @xmath331 , @xmath332 since this reveals the key features of the phase space . eym in ads . the parameters are : @xmath241 , @xmath333 , @xmath334 , @xmath335 , @xmath336 . both electric gauge field functions @xmath12 and @xmath289 are monotonic and nodeless . the magnetic gauge field functions @xmath4 and @xmath290 both have a single zero.,width=321 ] a typical dyonic soliton solution of @xmath3 eym is shown in fig . [ fig14 ] . the cosmological constant is @xmath241 and the other parameters are @xmath333 , @xmath334 , @xmath335 , @xmath336 . for these values of the parameters , the two electric gauge field functions @xmath12 and @xmath289 are monotonic and have no zeros ; @xmath12 is monotonically decreasing and @xmath289 is monotonically increasing . the two magnetic gauge field functions @xmath4 and @xmath290 are monotonically decreasing , and both have a single zero . eym with @xmath241 , @xmath333 , @xmath334 . all shaded points correspond to soliton solutions . the solutions are indexed by @xmath297 , the number of zeros of the magnetic gauge field functions @xmath4 , @xmath290 respectively . the different combinations of values of @xmath298 are indicated by colour - coding the regions - in black and white the different colours are different shades of grey . we find solutions where the magnetic gauge field functions have a wide variety of numbers of nodes . there is a small region close to the origin where both @xmath4 and @xmath290 have no zeros.,width=321 ] eym with @xmath254 , @xmath337 , @xmath338 . all shaded points correspond to soliton solutions . the solutions are indexed by @xmath297 , the number of zeros of the magnetic gauge field functions @xmath4 , @xmath290 respectively . the different combinations of values of @xmath298 are indicated by colour - coding the regions - in black and white the different colours are different shades of grey . the phase space has a much simpler structure for this larger value of @xmath8 . for most of the solution space , both magnetic gauge field functions @xmath4 and @xmath290 have no zeros . there are small regions at the edges of the space of solutions where either @xmath4 or @xmath290 has a single zero.,width=321 ] once again we find a very rich space of solutions , and illustrate some features in figs . [ fig15 ] and [ fig16 ] . [ fig15 ] shows the phase space for @xmath241 . the parameters @xmath333 and @xmath334 which govern the behaviour of the magnetic field functions near the origin are fixed ; we have scanned over positive values of the parameters @xmath154 and @xmath155 describing the electric gauge field functions near the origin . as we have seen previously for both @xmath2 solutions and @xmath3 dyonic black holes , for @xmath3 dyonic solitons with @xmath9 comparatively small , the phase space is very complicated . there are many different regions in which the two magnetic gauge field functions @xmath4 and @xmath290 have different numbers of zeros . for these values of the parameters , we find that @xmath4 and @xmath290 can have up to four zeros . we find a small region close to @xmath339 for which both magnetic gauge field functions have no zeros . embedded @xmath2 solutions correspond to @xmath340 , and therefore there are no embedded solutions in fig . [ fig15 ] . as the magnitude of the cosmological constant increases , the phase space simplifies considerably . this can be seen in fig . [ fig16 ] , where @xmath254 . the size of the space of nontrivial solutions also expands as @xmath9 increases . in fig . [ fig16 ] , we have fixed @xmath337 and @xmath338 , and varied @xmath331 and @xmath332 . with these values of the parameters , most of the nontrivial dyonic soliton solutions are such that @xmath341 and both magnetic gauge field functions have no zeros . there are small regions close to the edge of the solution space where one of @xmath342 ( but not both ) has a single zero . in this paper we have presented new dyonic soliton and black hole solutions of the @xmath0 einstein - yang - mills ( eym ) field equations in asymptotically anti - de sitter ( ads ) space - time with a negative cosmological constant @xmath7 . the metric is static and spherically symmetric . the gauge field has nontrivial electric and magnetic components , and is described by @xmath343 independent gauge field functions , with equal numbers of electric and magnetic gauge field functions . we have explored the phase space of soliton and black hole solutions for @xmath2 and @xmath3 gauge groups . the solutions can be categorized by the numbers of zeros , @xmath344 , of the magnetic gauge field functions @xmath10 . in general the phase space is very rich , with many different combinations of @xmath344 possible . however , we find the following general features , many of which are in common with the phase space of purely magnetic @xmath0 solutions @xcite : * for small @xmath9 , we find solutions in which the magnetic gauge field functions have large numbers of zeros , particularly for the @xmath2 case ; * for small @xmath9 , the phase space is particularly complicated , with many different combinations of values of @xmath344 ; * as @xmath9 increases , the phase space expands in parameter space and the number of different combinations of values of @xmath344 decreases ; * for large @xmath9 , there are solutions for which all the magnetic gauge field functions have no zeros . * in a neighbourbood of the embedded trivial solution , either pure ads ( for solitons ) or schwarzschild - ads ( for black holes ) ; * in a neighbourhood of embedded nontrivial @xmath2 dyonic soliton and black hole solutions ; * in a neighbourhood of nontrivial purely magnetic @xmath0 solutions ( whose existence is proven in @xcite ) . of particular interest are the solutions in the intersection of the last items in the two lists above , namely nontrivial nodeless solutions in the intersection of a neighbourbood of embedded @xmath2 solutions and a neighbourhood of embedded purely magnetic @xmath0 solutions . we conjecture that it may be possible to prove that such solutions are stable under linear , spherically symmetric perturbations . recently the existence of stable dyonic soliton and black hole solutions of @xmath2 eym in ads has been proven @xcite , and it would be interesting to attempt to extend that analysis to the case of a larger gauge group . in the @xmath2 case , the perturbation equations for dyonic solutions are much more complicated than the corresponding equations for purely magnetic background solutions , and the same will be true for @xmath0 gauge group . we therefore leave this question open for future work . the dyonic soliton and black hole solutions studied in this paper are spherically symmetric , with the event horizon being a surface of constant positive curvature . the existence proof in @xcite is more general , and applies also to topological black holes for which the event horizon has either zero or constant negative curvature . a natural question would be to investigate the phase space of dyonic topological black hole solutions of the @xmath0 eym equations , extending the recent study of the phase space of purely magnetic topological black hole solutions @xcite . black holes with a flat event horizon in particular have attracted a great deal of attention in the literature as models of holographic superconductors ( see @xcite for a recent review ) . planar black holes with a @xmath2 gauge field have been used to model @xmath345-wave superconductors ( see , for example , @xcite for a selection of papers ) , and enlarging the gauge group in these models would also be of interest . we plan to return to this topic in a future publication . finally , we anticipate that the thermodynamics of the dyonic black holes presented here would be very interesting . the thermodynamics of purely magnetic @xmath2 black holes in ads has recently been studied @xcite . in @xcite , a complex picture emerges : it is found that purely magnetic , spherically symmetric @xmath2 black holes with unit magnetic charge at infinity are globally thermodynamically unstable ; 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we present new spherically symmetric , dyonic soliton and black hole solutions of the @xmath0 einstein - yang - mills equations in four - dimensional asymptotically anti - de sitter space - time .
the gauge field has nontrivial electric and magnetic components and is described by @xmath1 magnetic gauge field functions and @xmath1 electric gauge field functions .
we explore the phase space of solutions in detail for @xmath2 and @xmath3 gauge groups .
combinations of the electric gauge field functions are monotonic and have no zeros ; in general the magnetic gauge field functions may have zeros .
the phase space of solutions is extremely rich , and we find solutions in which the magnetic gauge field functions have more than fifty zeros . of particular interest
are solutions for which the magnetic gauge field functions have no zeros , which exist when the negative cosmological constant has sufficiently large magnitude .
we conjecture that at least some of these nodeless solutions may be stable under linear , spherically symmetric , perturbations .
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weakly interacting massive particles ( wimps ) are a promising dark matter candidate as they are generically produced in the early universe with roughly the right density . furthermore supersymmetry ( susy ) provides a well - motivated concrete wimp candidate in the form of the lightest neutralino ( for reviews see e.g. refs . ) . wimps can be directly detected in the lab via their elastic scattering off target nuclei @xcite . numerous direct detection experiments are underway and they are probing the regions of wimp mass - cross - section parameter space populated by susy models ( see e.g. ref . ) . the realisation that uncertainties in the velocity distribution , @xmath0 , will affect the direct detection signals dates back to some of the earliest direct detection papers written in the 1980s ( e.g. ref . ) . we first discuss the standard halo model and other approaches to modelling the milky way halo ( sec . [ modelling ] ) . we then discuss what observations and simulations tell us about the dark matter distribution ( secs . [ astro1 ] and [ astro2 ] respectively ) . we then examine the direct detection signals ( sec . [ signals ] ) and how the astrophysical uncertainties affect these signals ( sec . [ affect ] ) . we conclude in sec . [ strategies ] by discussing strategies for handling the uncertainties , including both ` astrophysics independent ' approaches and parameterizing the wimp speed distribution . the steady - state phase space distribution function , @xmath1 , of a collection of collisionless particles is given by the solution of the collisionless boltzmann equation : @xmath2 in cartesian coordinates this becomes @xmath3 where @xmath4 is the potential . the standard halo model ( shm ) is an isotropic , isothermal sphere with density profile @xmath5 . in this case the solution of the collisionless boltzmann equation is a so - called maxwellian velocity distribution , given by @xmath6 where @xmath7 is a normalisation constant . the isothermal sphere has a flat rotation curve at large radii and the velocity dispersion is related to the asymptotic value of the circular speed ( the speed with which objects on circular orbits orbit the galactic centre ) @xmath8 . it is usually assumed that the rotation curve has already reached its asymptotic value at the solar radius , @xmath9 , so that @xmath10 where @xmath11 is the local circular speed . in the shm the peak speed @xmath12 and the circular speed are identical , @xmath13 , and these parameters are often used interchangeably . however this is not the case in general , for instance for a nfw @xcite density profile @xmath14 at @xmath15 ( where the scale radius , @xmath16 , is the radius at which the logarithmic slope of the density profile is equal to -2 ) @xcite . the density distribution of the shm is formally infinite and hence the velocity distribution extends to infinity too . in reality the milky way halo is finite , and particles with speeds greater than the escape speed , @xmath17 , will not be gravitationally bound to the mw . this is often addressed by simply truncating the velocity distribution at the measured local escape speed , @xmath18 , so that @xmath0 is given by eq . ( [ max ] ) for @xmath19 and @xmath20 for @xmath21 . this sharp truncation is clearly unphysical . an alternative , but still ad hoc , approach is to make the truncation smooth : @xmath22 \ , , & 0 \ , , & |{\bf v}| \geq v_{\rm esc } \ , . \end{cases}\ ] ] another approach , used in the king model / lowered isothermal sphere , is to modify the shm distribution function so that it becomes zero for large energies @xcite . the standard parameter values used for the shm are a local density @xmath23 , a local circular speed @xmath24 @xcite , and a local escape speed @xmath25 @xcite . we will discuss the determination of these parameters and the associated uncertainties in sec . [ astro1 ] . for spherical , isotropic systems there is a unique relationship between the density profile and the distribution function known as the eddington equation ( see e.g. ref . ): @xmath26\,,\ ] ] where @xmath27 and @xmath28 , where @xmath29 is the kinetic energy . the eddington equation has been used to calculate the speed distribution for a range of spherically symmetric density profiles @xcite . ref . found , using a bayesian analysis incorporating various dynamical constraints , speed distributions that are not too dissimilar to the standard maxwellian ( when the same value of the circular speed is used ) . ref . used the eddington equation to motivate a phenomenological form for @xmath30 which matches the high speed tail of the speed distributions found in simulated halos ( see sec . [ astro2 ] ) . the osipkov - merritt models @xcite assume that the distribution function depends on the energy , @xmath31 , and angular momentum , @xmath32 , only via a single parameter , @xmath33 , @xmath34 where the constant @xmath35 is the anisotropy radius . in this model the velocity anisotropy @xmath36 where @xmath37 are the velocity dispersions in the @xmath38 and @xmath39 directions , has a particular , radially dependent , form : @xmath40 and the distribution function can be calculated from a modified version of the eddington equation ( see e.g. refs . ) . in these models the peak of the speed distribution becomes narrower and there is an excess of high speed particles ( e.g. ref . ) . in general there is no unique relationship between the density profile and the velocity distribution for triaxial and/or anisotropic systems . in these cases a common approach is to use the jeans equations to calculate the lower order moments of the distribution function . multiplying the collisionless boltzmann equation , eq . ( [ cme ] ) , by one of the velocity components and integrating gives the jeans equations , which in cartesian coordinates take the form @xmath41 we now have three equations ( corresponding to @xmath42 ) for six unknowns ( @xmath43 ) . to make further progress it is therefore necessary to make assumptions about the alignment of the velocity ellipsoid , i.e. to choose coordinates such that @xmath44 for @xmath45 . the velocity distribution is then approximated by a multivariate gaussian in these coordinates : @xmath46 where @xmath47 . evans , carollo and de zeeuw @xcite studied the logarithmic ellipsoidal model , the simplest triaxial generalisation of the isothermal sphere . they argue that aligning the velocity ellipsoid with conical coordinates is physically well motivated . in the planes of the halo , conical coordinates are locally equivalent to cylindrical polar coordinates . hence , in this model , the velocity distribution can be approximated by a multi - variate gaussian in cylindrical polar coordinates , with velocity dispersions depending on the shape of the halo , the velocity anisotropy and also location within the halo . both the width of the speed distribution and the peak speed can change ( e.g. refs . ) . analytic models inevitably make assumptions ( regarding the shape and anisotropy of the halo , and their radial dependence ) which are almost certainly not completely valid . the relationship between the observed properties of dark matter halos and the velocity distribution is non - trivial ; models with the same bulk properties ( e.g. shape and local velocity anisotropy ) can have velocity distributions with very different forms . furthermore all analytic approaches to halo modelling rely on the assumption that the phase space distribution function has reached a steady state . to some extent analytic halo models have been superseded by results from numerical simulations . it should be emphasized ( see sec . [ micro ] and [ baryon ] ) that velocity distributions from numerical simulations also involve approximations and extrapolations . the event rate is directly proportional to the local density , @xmath48 . the standard value used is @xmath49 . as discussed in ref . , the origin of the use of this particular value , rather than say @xmath50 , is unclear . the local density is calculated via mass modelling of the milky way ( mw ) . this involves constructing a model of the mw ( including its luminous components ) and then finding the range of values of the local density that are consistent with all observational data ( including , for instance , rotation curve measurements , velocity dispersions of halo stars , local surface mass density , total mass ) @xcite . as emphasised in ref . the shape of the mw halo has a crucial effect on the local density extracted . for a fixed circular speed , in a flattened halo the local density , in the galactic plane , is higher . this work has recently been updated to include new data sets , models for the mw halo motivated by numerical simulations , and , in some cases , bayesian statistical techniques . ref . , assuming a spherical halo with a cuspy density profile ( @xmath51 as @xmath52 ) found @xmath53 , while ref . , assuming spherically symmetric nfw @xcite and einasto @xcite profiles for the mw halo found , @xmath54 . while these determinations have @xmath55 errors , they differ from each other by significantly more than this . this suggests that the systematic errors are bigger than the statistical errors . . finds , considering a range of density profiles including cored profiles ( @xmath56 as @xmath52 ) , values in the range @xmath57 . other recent work has investigated ` model independent ' techniques , which do nt involve global mass - modelling of the galaxy , and hence have smaller hidden systematic errors . ref . proposed using the equation of centrifugal equilibrium and subtracting the contribution of the stellar component of the mw . the resulting determination of the local density is @xmath58 , where the uncertainties come from the uncertainty in the slope of the circular speed at the solar radius and the ratio of the solar radius to the length scale of the thin stellar disk . ref . introduced a method which involves solving the jeans and poisson equations with minimal assumptions . using hipparcos data they find @xmath59 , at @xmath60 confidence , assuming the stellar tracer populations are isothermal and @xmath61 if they have a non - isothermal profile . an accurate determination of the vertical dispersion profile of the tracer population is therefore required to make an unbiased estimate of the local density using this method . ref . used a high - resolution simulation of a milky way like galaxy , including baryons @xcite , to investigate the effect of halo shape on the local density . specifically they examined how the local density varies from the density averaged in a spherical shell , as determined by observations . they find that the density within the stellar disk , at a distance @xmath62 from the centre , is a factor of between 1.01 and 1.41 , with an average of 1.21 , larger than the value averaged in a spherical shell . the local circular speed , @xmath63 , is an important quantity to determine . it appears in the galilean transformation of the velocity distribution into the lab frame ( see sec . [ signals ] ) . it is also related to the radial component of the velocity dispersion , @xmath64 , by one of the jeans equations ( e.g. ref . ): @xmath65 where the anisotropy parameter @xmath66 is defined in eq . ( [ beta ] ) . the shm has @xmath5 and the the velocity distribution is isotropic ( @xmath67 so that @xmath68 ) and independent of radius , so that @xmath69 . the standard value of @xmath70 of @xmath71 dates back to a 1980s review of the galactic constants @xcite and was found by taking an average of the values found from a wide range of different analyses . note that the ratio @xmath72 is better determined than either @xmath70 or @xmath73 individually ( e.g. refs . ) . a recent analysis using measurements of the motions of galactic masers , found a significantly higher value , @xmath74 , @xcite . reanalyzed the data using a more general model for the maser velocity distribution ( including allowing a non - zero velocity dispersion tensor ) and found that the maser data places only a relatively weak constraint on @xmath70 . when combined with other measurements ( from the proper motion of sgr @xmath75 , and the orbit of the gd-1 stellar stream ) , they found @xmath76 , assuming a flat rotation curve . meanwhile ref . found the value of @xmath70 determined from the maser data depends strongly on the mw model used . using a range of models for the rotation curve , including a power - law with free slope , they found values in the range @xmath77 to @xmath78 . this illustrates that , as in the case of the local density , systematic , modelling errors are important . the escape speed is the speed required to escape the local gravitational field of the mw , @xmath79 the local escape speed , @xmath18 , is estimated from the speeds of high velocity stars . to do this it is necessary to parameterise the shape of the high speed tail of the velocity distribution . assuming that the velocities are isotropic and the jeans theorem applies , the asymptotic form of the velocity distribution can be written as @xcite : @xmath80^{k } \ , , & |{\bf v}| < v_{\rm esc } \ , , \\ 0 \ , , & |{\bf v}| \geq v_{\rm esc } \ , . \end{cases}\ ] ] traditionally a value @xmath81 , corresponding to the upper @xmath60 confidence limit from ref . , has been used . ref . has updated these measurements , using additional data from the rave survey and using a prior on @xmath82 , @xmath83 $ ] ( motivated by analysis of the speed distributions of stellar particles in simulated halos ) . they find that the escape speed lies in the range @xmath84 at @xmath60 confidence , with a median likelihood of @xmath85 . a number of high resolution , dark matter only , simulations of the formation of milky way - like halos in a cosmological context have been carried out ( e.g. aquarius @xcite , ghalo @xcite and via lactea @xcite ) . the velocity distributions of these halos deviate systematically from a multivariate gaussian @xcite . there are more low speed particles and the peak in the distribution is lower and flatter in shape ( i.e. the distribution is platykurtic ) . several fitting functions have been considered ; a tsallis distribution ( which arises from non - extensive statistical mechanics ) @xcite and a modified maxwellian @xcite @xmath86 } \ f(v_{\rm t } ) & = & \frac{v_{\rm t}}{n_{\rm t } } \exp { \left [ - \left ( \frac{v_{\rm t}^2}{\bar{v}_{\rm t}^2 } \right)^{\alpha_{\rm t } } \right ] } \,,\end{aligned}\ ] ] where @xmath87 and @xmath88 are the radial and tangential components of the velocity , @xmath89 are normalisation factors and @xmath90 are free - parameters which parameterise the deviations from a maxwellian distribution . the most likely speed deviates from the circular speed : for vl2 and ghalo @xmath91 and @xmath92 respectively @xcite . the velocity distributions also have stochastic features at high speeds . there are broad bumps which vary from halo to halo , but are independent of position within a given halo and are thought to reflect the formation history of the halo @xcite . kuhlen et al . @xcite also find narrow spikes in some locations , corresponding to tidal streams . ref . presented an ansatz for the velocity distribution @xmath93^{k } \theta ( v_{\rm esc } - component of the speed distributions found in simulations . the shape of the high speed tail of the distribution is determined by the slope of the density profile at large radii . using the eddington equation the parameter @xmath82 can be related to the outer slope of the density profile , @xmath94 , ( @xmath95 for large @xmath96 ) , by @xmath97 for @xmath98 @xcite . as @xmath99 ( the value corresponding to the nfw profile @xcite ) the calculation breaks down , and numerical fits to eq . ( [ k ] ) find @xmath100 . more generally for outer slopes in the range @xmath101 , @xmath82 lies in the range @xmath102 $ ] . ( [ k ] ) with @xmath82 in this range provides a good fit to the high speed tails of the speed distributions found in simulation and has less high speed particles than the tail of the standard maxwellian distribution with a sharp cut - off . note , however , that eq . ( [ k ] ) does not match the low and moderate speed behaviour of the simulation speed distributions , possibly due to the assumption of isotropy @xcite . the simulations discussed above contain dark matter only , while baryons dominate in the inner regions of the milky way . simulating baryonic physics is extremely difficult , and producing galaxies whose detailed properties match those of real galaxies is a major challenge . some recent simulations have found that late merging sub - halos are preferentially dragged towards the disc by dynamical friction , where they are destroyed leading to the formation of a co - rotating dark disc ( dd ) @xcite . ref . modelled the dd velocity distribution as a gaussian with isotropic dispersion , @xmath103 and lag @xmath104 , matching ( roughly ) the kinematics of the milky way s stellar thick disc and considered dd densities in the range @xmath105 , where @xmath106 is the local halo density . the properties ( and even existence ) of the dark disc are highly uncertain . ref . argues that to be consistent with the observed morphological and kinematic properties of the milky way s thick disc , the milky way s merger history must be quiescent compared with typical @xmath107cdm merger histories , and hence the dd density must be relatively small , @xmath108 . the total ( halo + disc ) local density can be probed by the kinematics of stars ( e.g. ref . ) . there is no evidence for a dark matter disc ( i.e. the data are well fit without a dark disc ) and a thick , dense , @xmath109 , dark disc is excluded @xcite . refs . and also argue that the dd velocity dispersion is likely to be substantially larger than that of the stellar thick disc . baryonic physics will also affect the halo speed distribution . for instance gas cooling makes halos more spherical ( e.g. ref . ) . the local speed distribution found in ref . is well fit by a tsallis distribution , but appears to deviate less from the standard maxwellian than the distributions found in dark matter only simulations . a further caution is that the scales resolved by simulations are many orders of magnitude larger than those relevant for direct detection experiments . the earth moves at @xmath110 , therefore direct detection experiments probe the dark matter on sub mpc scales , while the numerical simulations discussed above have gravitational softening of order @xmath111 . vogelsberger and white have developed a new technique to follow the fine - grained phase space distribution in simulations as it stretches and folds under the action of gravity @xcite . they find that the median density of the resulting streams is of order @xmath112 the local halo density . schneider et al . @xcite have studied the evolution of the first , and smallest , roughly earth mass @xcite , microhalos to form in the universe . they find that tidal disruption and encounters with stars produce tidal streams with average density @xmath113 the local halo density . these results ( see also ref . ) suggest that the ultra - local dark matter density and velocity distribution should not be drastically different to those on the scales resolved by simulations . the ultra - local velocity distribution may , however , contain some features or fine - grained structure @xcite . the differential event rate for elastic scattering , assuming spin - independent coupling with identical couplings to the proton and neutron , is given by ( e.g. refs . ): @xmath114 where @xmath48 is the local wimp density , @xmath115 the wimp velocity distribution in the lab frame , @xmath116 the wimp scattering cross section on the proton , @xmath117 the wimp - proton reduced mass , @xmath118 and @xmath119 the mass number and form factor of the target nuclei respectively and @xmath120 is the recoil energy . the lower limit of the integral , @xmath121 , is the minimum wimp speed that can cause a recoil of energy @xmath120 : @xmath122 where @xmath123 is the atomic mass of the detector nuclei and @xmath124 the wimp - nucleon reduced mass . the event rate for inelastic scattering of wimps @xcite also depends on the wimp density and velocity distribution . however , due to the altered kinematics the relationship between @xmath125 and @xmath120 is changed , and hence the effects of changes in the velocity distribution are different @xcite . the wimp speed distribution must be transformed from the galactic rest frame to the lab frame . this is done by carrying out , a time dependent , galilean transformation : @xmath126 . the earth s motion relative to the galactic rest frame , @xmath127 , is made up of three components : the motion of the local standard of rest ( lsr ) , @xmath128 , the sun s peculiar motion with respect to the lsr , @xmath129 , and the earth s orbit about the sun , @xmath130 , . the motion of the lsr is defined as @xmath131 , where @xmath70 is the local circular speed ( see sec . [ vc ] for a discussion of recent determinations ) . the most recent determination of the sun s motion with respect to the lsr , taking into account the effects of the metallicity gradient in the disc , finds @xmath132 @xcite in galactic co - ordinates ( where @xmath133 points towards the galactic center , @xmath134 is the direction of galactic rotation and @xmath135 towards the north galactic pole ) . note that the value of @xmath136 is significantly ( @xmath137 ) larger than previously found @xcite . the new larger value is also supported by the analysis of the motions of masers @xcite . accurate expressions for the earth s orbit can be found in ref . . simpler expressions , which are acceptable for most practical purposes , can be found in refs . . the differential event rate , eq . ( [ drde ] ) , depends on the target nuclei mass , the ( a priori unknown ) wimp mass and the integral of the velocity distribution . it is therefore useful to rewrite eq . ( [ drde ] ) as @xmath138 where @xmath139 and the prefactor @xmath140 contains the wimp and target dependent terms and is independent of the astrophysical wimp distribution . we will now discuss the energy ( sec . [ er ] ) , time ( sec . [ am ] ) and direction ( sec . [ dd ] ) dependence of the differential event rate . in order to make concrete statements we will for now assume the standard maxwellian velocity distribution , eq . ( [ max ] ) . in sec . [ affect ] we will discuss how uncertainties in the wimp distribution affect these signals . the shape of the energy spectrum depends on both the wimp mass and the mass of the target nuclei . this can be seen , following lewin and smith @xcite , by assuming a standard maxwellian velocity distribution , eq . ( [ max ] ) , and , initially , neglecting the earth s velocity and the galactic escape speed . this allows the time averaged energy spectrum to be written as @xmath141 where @xmath142 is the event rate in the @xmath143 limit , and @xmath144 , the characteristic energy scale , is given by @xmath145 when the earth s velocity and the galactic escape speed are taken into account eq . ( [ drde0 ] ) is still a reasonable approximation to the event rate if @xmath146 and @xmath144 are both multiplied by constants of order unity . if @xmath147 , @xmath148 , while if @xmath149 , @xmath144 is independent of the wimp mass . this indicates that the wimp mass can be determined from the energy spectrum , provided it is not significantly larger than the mass of the target nuclei @xcite . furthermore measuring consistent spectra for two different target nuclei could in principle confirm the wimp origin of these spectra ( e.g. ref . ) . this is sometimes referred to as the ` materials signal ' . due to the earth s orbit about the sun , the net velocity of the lab with respect to the galactic rest frame varies annually . the net speed is largest in the summer and hence there are more high speed wimps , and less low speed wimps , in the lab frame . this produces an annual modulation of the event rate @xcite . the differential event rate peaks in winter for small recoil energies and in summer for larger recoil energies @xcite . the energy at which the annual modulation changes phase is often referred to as the ` crossing energy ' . its value depends on the wimp and target masses , and hence could be used to determine the wimp mass @xcite . since the earth s orbital speed is significantly smaller than the sun s circular speed the amplitude of the modulation is small and , to a first approximation , the differential event rate can , for the standard halo model , be written approximately as a taylor series : @xmath150 \,,\ ] ] where @xmath151 , @xmath152 year and @xmath153 days . for the standard halo model the fractional amplitude of the modulation is approximately given by @xcite @xmath154 where @xmath155 , @xmath156 is the value of @xmath157 at which the sign of the modulation reverses and @xmath158 . for @xmath159 , in the extreme tail of the speed distribution , the shape of the modulation is non - sinusoidal . for very small energies @xmath160 is negative ( i.e. the maximum occurs in december rather than june ) . as @xmath120 is increased , @xmath161 initially decreases to zero at which point the phase of the annual modulation changes so that the maximum occurs in summer . as @xmath120 is increased further the fractional amplitude continues increasing . this is potentially misleading however , as the mean event rate , and hence the raw amplitude , becomes very small at large @xmath120 . after the phase change on increasing @xmath120 further the raw amplitude increases to a local maximum before decreasing again and tending to zero . for the shm , for measurable energies , the amplitude of the modulation lies in the range @xmath162 . our motion with respect to the galactic rest frame also produces a direction dependence of the event rate . the wimp flux in the lab frame is peaked in the direction of motion of the sun ( towards the constellation cygnus ) , and hence the recoil spectrum is , for most energies , peaked in the direction opposite to this @xcite . this directional signal is far larger than the annual modulation ; the event rate in the backward direction is several times larger than that in the forward direction @xcite . a detector which can measure the recoil directions is required to detect this signal ( see ref . for an overview of the status of directional detection experiments ) . the full direction dependence of the event rate is most compactly written in terms of the radon transform of the wimp velocity distribution @xcite @xmath163 where @xmath164 , @xmath165 is the recoil direction and @xmath166 is the 3-dimensional radon transform of the wimp velocity distribution @xmath167 geometrically the radon transform , @xmath168 , is the integral of the function @xmath0 on a plane orthogonal to the direction @xmath165 at a distance @xmath125 from the origin . see ref . for an alternative , but equivalent , expression . while the directional recoil rate depends on both of the angles which specify a given direction , the strongest signal is the differential of the event rate with respect to the angle between the recoil and the direction of solar motion , @xmath94 , @xcite @xmath169 for the standard maxwellian velocity distribution @xcite @xmath170 } \,,\ ] ] where @xmath171 is the component of the earth s velocity parallel to the direction of solar motion . an ideal detector capable of measuring the nuclear recoil vectors ( including their senses , @xmath172 versus @xmath173 ) in 3-dimensions , with good angular resolution , could reject isotropy of wimp recoils with only of order 10 events @xcite . most , but not all , backgrounds would produce an isotropic galactic recoil distribution ( due to the complicated motion of the earth with respect to the galactic rest frame ) . an anisotropic galactic recoil distribution would therefore provide strong , but not conclusive , evidence for a galactic origin of the recoils . roughly 30 events would be required for an ideal detector to confirm that the peak recoil direction coincides with the inverse of the direction of solar motion , hence confirming the galactic origin of the recoil events @xcite . in this section we will discuss how the uncertainties described in secs . [ astro1 ] and [ astro2 ] affect the signals expected in direct detection experiments . since the normalisation of the event rate is directly proportional to the product of the cross - section , @xmath174 , and the local density , @xmath48 , the uncertainty in @xmath48 leads directly to an uncertainty in measurements of , or constraints on , @xmath175 . since the density is not expected to vary on mpc scales , the uncertainty in @xmath174 is the same for all direct detection experiments , and does not affect comparisons of e.g. the exclusion limits from different experiments . it does however affect comparisons with collider and other constraints on @xmath174 . furthermore it can significantly bias the determination of the wimp mass @xcite . the characteristic energy scale , @xmath144 , given by eq . ( [ ereq ] ) , depends on the local circular speed . therefore the uncertainty in the local circular speed , @xmath70 , leads to a @xmath176 uncertainty in the differential event rate , and hence exclusion limits @xcite . the nature of the change in the exclusion limits is similar , but not identical , for different experiments @xcite . it will also lead to a bias in determinations of the wimp mass @xcite . this can be seen by differentiating the expression for the characteristic energy , eq . ( [ ereq ] ) , @xmath177 \frac{\delta v_{\rm c}}{v_{\rm c } } \,.\ ] ] since the differential event rate is given by an integral over the velocity distribution the differential event rate depends only weakly on the detailed form of the velocity distribution @xcite . consequently the resulting uncertainty in exclusion limits @xcite and determinations of the wimp mass @xcite is usually fairly small . there are , however , some exceptions to this statement . firstly if the wimp is light and/or the experimental energy threshold is high compared with the characteristic energy , then only the high speed tail of the speed distribution is probed . in this case the uncertainties in its shape can have a significant effect on the expected energy spectrum and hence exclusions limits or allowed regions @xcite . secondly if there is a dark disc there will be an additional population of low speed wimps . if the dark disc density is sufficiently high and its velocity dispersion sufficiently small this will significantly change the energy spectrum and hence exclusion limits and mass determinations @xcite . the size of these changes depends on the properties of the dark disc , which are currently uncertain @xcite . a wimp stream , with sufficiently high density , would add a sloping step to the differential event rate @xcite . the annual modulation arises from the , relatively small , shift in the speed distribution in the lab frame over the course of the year . it is therefore far more sensitive to the speed distribution than the time averaged energy spectrum . both the amplitude and phase of the modulation , and hence the regions of parameter space compatible with an observed signal , can vary significantly ( e.g. refs . ) . the uncertainty in the value of @xmath178 can change the amplitude by a factor of order unity , while the uncertainty in the shape of the halo velocity distribution changes the amplitude by @xmath179 @xcite . a high density dark disc could change the annual modulation signal significantly , producing a 2nd , low energy , maximum in the amplitude of the modulation and changing the phase significantly @xcite . for a wimp stream the position and height of the step in the energy spectrum produced would vary annually ( e.g. ref . ) . the detailed direction dependence of the event rate is sensitive to the velocity distribution , however the anisotropy is robust @xcite . the number of events required to detect anisotropy or to demonstrate that the median inverse recoil direction coincides with the direction of solar motion only vary by of order @xmath180 @xcite . the features in the high speed tail can cause the median inverse direction of high energy recoils to deviate from the direction of solar motion @xcite . a stream of wimps would produce recoils which are strongly peaked in the opposite direction @xcite . in the long term , studying the median direction of high energy recoils could allow high speed features to be detected and the formation history of the milky way probed . it is possible to carry out an ` astrophysics independent ' comparison of data from multiple experiments . this effectively involves making comparisons of the energy spectra , so that the integral over the velocity distribution , @xmath181 defined in eq . ( [ gvmin ] ) , cancels . for instance , with data from two different experiments using two different targets , the wimp mass can in principle be determined , without any assumptions about @xmath0 , by taking moments of the energy spectra @xcite . however this method leads to a systematic underestimate of the wimp mass if it is comparable to or larger than the mass of the target nuclei . ref . , see also ref . , showed how the differential event rates in two experiments , ` 1 ' and ` 2 ' with target nuclei with mass numbers @xmath182 and @xmath183 are related . if the first experiment is sensitive to recoil energies in the range @xmath184 $ ] it probes @xmath181 for @xmath185 , where @xmath125 is related to the recoil energy , wimp mass and target nuclei mass by eq . ( [ vmin ] ) . in experiment 2 these values of @xmath125 correspond to recoils in the range @xmath186 = \frac{\mu_{a2 \chi}^2 m_{a1}}{\mu_{a1 \chi}^2 m_{a2 } } [ e_{1}^{\rm low } , e_{1}^{\rm high } ] \,.\ ] ] for recoil energies in this range the differential event rates in the two experiments are then related by @xmath187 this relation depends on the ( unknown ) wimp mass , therefore the comparison has to be made separately for a range of @xmath188 values . this method has recently been extended to include annual modulation data @xcite . this approach has proved particularly useful @xcite in comparing the event rate excesses and annual modulations found in the cogent @xcite , cresst @xcite and dama @xcite experiments with exclusions limits from cdms @xcite and xenon @xcite . ref . found that the dama , cogent and cresst - ii data are compatible which each other , but not cdms and xenon , if the velocity distribution is very anisotropic ( which leads to a large modulation fraction ) . the astrophysics independent methods are invaluable for assessing the compatibility of data from different experiments . however the wimp mass is not a priori known , and the goal of wimp direct detection experiments is not just to detect wimps , but to also measure their mass and cross - section . while there are uncertainties in the velocity distribution it is not completely unknown , and therefore does not need to be completely removed from the analysis . strigari and trotta @xcite have shown how the wimp properties can be constrained by a single future direct detection experiment when astronomical data ( such as the kinematics of halo stars ) are used to jointly constrain the wimp properties and the parameters of a model for the milky way halo . they assumed an isotropic maxwellian speed distribution characterised by the peak speed , @xmath12 , and the escape speed , @xmath189 . peter @xcite has examined how data sets from multiple direct detection experiments could be used to jointly constrain a maxwellian parametrisation of the wimp speed distribution and the wimp parameters ( mass and cross - section ) . pato et al . @xcite have taken a similar approach using the parameterization in eq . ( [ k ] ) . peter has recently extended this work by allowing the peak speed and circular speed to differ ( as expected if the density profile is not @xmath5 ) and also considering an empirical speed distribution consisting of a five or ten bin step function @xcite . combining data from multiple experiments with different targets significantly increases the accuracy with which the wimp parameters can be measured @xcite . however fixing the form of the speed distribution leads to biases in the wimp properties , if the true speed distribution differs significantly from the parameterization assumed @xcite . with a suitable parameterization o the wimp parameters and speed distribution can be jointly probed @xcite . the form of the optimal parameterization of the speed distribution is an open question . direct detection event rate calculations often assume the standard halo model , with an isotropic maxwellian velocity distribution with dispersion @xmath190 and a local wimp density @xmath191 . however it has long been realised @xcite that uncertainties in the speed distribution will affect the signals expected in experiments . we have discussed the standard halo model and other approaches to halo modelling , and the assumptions behind them . we then reviewed observational determinations of quantities that are relevant to direct detection experiments , namely the local dark matter density , the local circular speed ( which is related to the velocity dispersion by the jeans equations ) and the local escape speed . while the statistical errors on these quantities are often small the systematic errors , from uncertainties in the modelling of the milky way , can be significantly larger . next we turned our attention to high resolution , dark matter only simulations of the formation of milky way - like halos in a cosmological context . they find velocity distributions that deviate significantly from the standard maxwellian , and have features at high speeds . the effect of baryonic physics on the dark matter distribution is not yet well understood . some recent simulations have found that a dark disc may be formed , however its properties ( and even existence ) are highly uncertain . simulations resolve the dark matter distribution on scales many orders of magnitude larger than those probed by direct detection experiments . the latest results suggest that the ultra - local dark matter distribution is largely smooth , but some features may exist . we then reviewed the resulting uncertainties in the energy , time and direction dependence of the energy spectrum , and hence constraints on , or measurements of , the wimp parameters . the uncertainty in the local circular speed has a significant effect on the event rate and hence exclusion limits and determinations of the wimp mass . the uncertainty in the local density leads directly to an uncertainty in measurements of , or constraints on , the cross - section . only the high energy tail of the energy spectrum is particularly sensitive to the exact shape of the speed distribution . the annual modulation is far more sensitive to the shape of the velocity distribution ; its amplitude and phase can change significantly . the anisotropy of the recoil rate is robust to changes in the velocity distribution , however high speed features can change the peak direction of high energy recoils , and hence provide a way of probing the formation of the milky way . finally we discussed techniques for handling the astrophysical uncertainties when analysing direct detection data . astrophysics independent comparisons between different experiments can be made using the integral of the velocity distribution , @xmath181 . this approach is extremely useful for assessing the compatibility of various experiments , but requires the wimp mass as input . parameterising the wimp speed distribution , and using data from multiple experiments to jointly constrain the wimp mass and cross - section and speed distribution is a promising approach , but the optimal form for the parameterisation is not yet known . the author is supported by stfc . she is grateful to the organisers of , and participants in , the ` dark matter underground and in the heavens ' workshop ( dmuh11 ) at cern , where she had various useful discussions about the topics covered in this review . she is also grateful to bradley kavanagh for useful comments on a draft of this article .
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direct detection experiments are poised to detect dark matter in the form of weakly interacting massive particles ( wimps ) .
the signals expected in these experiments depend on the ultra - local wimp density and velocity distribution .
firstly we review methods for modelling the dark matter distribution .
we then discuss observational determinations of the local dark matter density , circular speed and escape speed and the results of numerical simulations of milky way - like dark matter halos . in each case
we highlight the uncertainties and assumptions made .
we then overview the resulting uncertainties in the signals expected in direct detection experiments , specifically the energy , time and direction dependence of the event rate .
finally we conclude by discussing techniques for handling the astrophysical uncertainties when interpreting data from direct detection experiments .
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infrared spectroscopy provides a crucial means in the identification of interstellar dust compositions . recent infrared satellite observations by the _ infrared telescope in space _ ( _ irts _ ; * ? ? ? * ) and the _ infrared space observatory _ ( _ iso _ ; * ? ? ? * ) have revealed several new dust features in the diffuse emission , indicating the presence of new dust components in the interstellar medium @xcite . observations by _ iso _ also clearly show the presence of crystalline silicates around young and evolved stars for the first time @xcite , while it is not yet certain whether crystalline silicates exist commonly in interstellar space . in the present paper we report the observations of active star - forming regions , the carina nebula and the sharpless 171 ( s171 ) region with the long - wavelength spectrometer ( lws ; * ? ? ? * ) on board _ iso _ and the detection of far - infrared features around 65@xmath0 m and 100@xmath0 m in the diffuse emission . the carina nebula is one of the most active regions on the galactic plane and known to contain a number of early - type stars @xcite . the s171 region is a typical region and molecular cloud complex @xcite . both regions are supposed to represent the characteristics of active regions in the galaxy . possible carriers of the two detected features are discussed and we investigate the possibility that carbon onion grains of curved graphitic shells are the carrier for the broad interstellar 100@xmath0 m feature . the central @xmath1 portion of the carina nebula was observed by two - dimensional raster scans with the lws full grating scan mode and the far - infrared spectra for 43197@xmath0 m were obtained for 132 positions ( for details of the observations , see * ? ? ? a one - dimensional scan was made for 24 positions on a line from the heating source to the molecular cloud region in s171 with the same lws observing mode @xcite . the observed area of the both objects includes ionized regions and molecular clouds and the spectra sample the diffuse emission from the interstellar matter rather than the emission from point - like sources . the off - line processing data of version 10.1 ( olp 10.1 ) provided by the iso archival data center were used in the present study . the spectra were defringed , converted into the surface brightness , and the extended source correction was applied by the iso spectral analysis package ( isap ) software . the beam size and the correction factors were taken from the latest lws handbook @xcite . there are gaps in the spectra between the detector channels , which can be ascribed to the uncertainties either in the responsivity , in the dark current , or in the spatial brightness distribution in the beam . figure [ fig1 ] shows examples of the obtained original spectra of the two regions , while figure [ fig2 ] indicates their stitched spectra to correct the gaps . both spectra were taken at the interface regions between the molecular cloud and ionized gas , where the far - infrared intensity is sufficiently large to investigate dust features . the stitched spectra are made by scaling each detector signal because the observed regions are bright enough that the uncertainty in the dark current should be less significant than those in the responsivity or in the spatial brightness distribution . as can be seen in figure [ fig1 ] , the amount of the gaps is small ( @xmath2 5% ) except for the three longest channels ( @xmath3@xmath0 m ) , where 1020% scaling is necessary to correct the gaps . the presence of a relatively narrow band feature at 65@xmath0 m is seen even in the unstitched spectra , particularly in s171 . in the spectrum of the carina nebula , the appearance of this feature is slightly disturbed by the higher levels of the adjacent channel spectra ( sw2 and sw4 ) relative to the level of the sw3 channel , but it can still be seen in the individual spectrum of the sw3 channel . a broad feature centered around 100@xmath0 m is also noticeable in the unstitched spectrum of s171 . the slope of the continuum starts to become flatter around the boundary between the sw4 and sw5 channels , indicating a feature starting around 80@xmath0 m . in the unstitched spectrum of the carina nebula , the gap between the sw4 and sw5 channels makes the feature less obvious , but the change in the slope in the sw5 channel can still be seen . the stitched spectrum clearly indicates the presence of the feature . however there is no appreciable abrupt change in the slope at longer wavelengths and the longer wavelength end of the feature is difficult to estimate from these spectra . neither spurious features have been reported nor the relative spectral response functions have the corresponding features in these spectral ranges @xcite . we will discuss possible underlying continua to confirm the presence of the feature and estimate the feature profile in next section . similar features are seen at about a half of the observed positions both in the carina and s171 regions . since these features are seen in a wide area of the interstellar medium , the band carriers must be ubiquitous species in interstellar space . @xcite reported the presence of 65@xmath0 m and 90@xmath0 m features in the spectra of evolved stars . figure [ fig3]a shows a spectrum of ngc6302 taken from the iso archival data center for comparison ( cf . * ? ? ? the continuum emission indicates a much higher temperature than those in figure [ fig2 ] and the features are weakly seen on the steep continuum . to see the features more clearly , the flux is multiplied by the square of the wavelength ( @xmath4 ) and plotted in figure [ fig3]b . the interstellar 65@xmath0 m feature seems very similar to that detected in evolved stars . the peak of the 65@xmath0 m feature is located obviously longer than [ ] 63@xmath0 m line ( fig . [ fig4 ] ; see also fig . [ fig6 ] ) and thus is not compatible with the crystalline ice band at 62@xmath0 m @xcite . @xcite have proposed a ca - bearing crystalline silicate , diopside ( camgsi@xmath5o@xmath6 ) , as a possible carrier of the 65@xmath0 m feature in evolved stars . cryogenic measurements of the optical properties of diopside support the identification @xcite . figure [ fig4 ] shows a comparison of the observed spectra with the laboratory data . the best fit continua described in next subsection are also plotted . the laboratory spectrum shows a narrower profile than those observed and other species , such as water ice and dolomite ( camg(co@xmath7)@xmath5 ) , have been suggested to contribute also to the 65@xmath0 m band emission @xcite . diopside has a weak feature also at 44.5@xmath0 m . the lws detector in this spectral range ( sw1 ) is less sensitive and known to have strong hysteresis . in the present spectra a band feature is seen around 45@xmath0 m both in the upward and downward scans of both spectra , suggesting the presence of the 45@xmath0 m feature . however large noises in this spectral range preclude the firm detection and further observations are needed to confirm the feature . band features of other crystalline minerals , such as the 69@xmath0 m band of forsterite seen in ngc6302 , are not seen in the present lws spectra . diopside also has strong features in 3040@xmath0 m . the carina nebula was observed by short - wavelength spectrometer ( sws ; * ? ? ? * ) and the spectra of 2.345@xmath0 m have been obtained . however the sws spectra were not taken at the same positions as the lws spectrum and thus the direct examination is difficult . the sws spectra are dominated by strong continuum and do not clearly show any solid bands except for the broad 22@xmath0 m feature @xcite . if the identification of the interstellar 65@xmath0 m with diopside is correct , this is the first detection of a crystalline silicate in the diffuse interstellar medium . efficient destruction of dust grains by interstellar shocks suggests that a large fraction of interstellar dust must be formed in interstellar space in addition to those supplied from stars @xcite . diopside is a high - temperature condensate and may survive harsh conditions . calcium is a less abundant element than magnesium or silicon , but it is highly depleted in the gas phase of the interstellar medium @xcite . therefore the presence of ca - containing dust should not be surprising in interstellar space . based on the measured band strength of diopside @xcite and the observed strength relative to the continuum , we roughly estimate that 510% of solar abundance calcium in diopside grains is sufficient to account for the observed band emission if we take the commonly used mass absorption coefficient of 50 @xmath8g@xmath9 for the continuum emission at 100@xmath0 m @xcite . @xcite have suggested that the crystallinity of the silicates is less than 12% in the interstellar medium based on the observations of protostars . since the feature seen around 100@xmath0 m is quite broad and weak , we investigate several cases for the underlying continuum to examine the presence of the broad 100@xmath0 m feature in detail and to make a rough estimate of the 100@xmath0 m feature profile . in the estimate of the continuum we assume the baseline positions to be at 5560@xmath0 m , 7080@xmath0 m , and 140190@xmath0 m or 120190@xmath0 m ( see below ) and we try to fit the observed spectra in these ranges with the model continuum as much as possible . because the shortest spectral range ( 5560@xmath0 m ) has higher noises , less weight is put on this range in the fitting . we first adopt the dust model with the power - law emissivity ( @xmath10 , where @xmath11 is a constant ) for the continuum emission . we found that the single - temperature graybody model can not fit the entire baseline positions satisfactorily . particularly the model always gives a higher flux at long wavelengths than the observed spectra . this discrepancy can not be solved by increasing @xmath11 because then the model would provide unnecessarily large fluxes at shorter wavelengths . introducing a second component with a low temperature improves the fit drastically . the model of @xmath12 both for warm and cold grains gives reasonable fits , but still has slightly larger fluxes at longest wavelengths ( @xmath13@xmath0 m ) than the observed spectra . increase of @xmath11 from 2 to unrealistically large 3 for the cold grains does not improve the fit appreciably . the power - law emissivity model has a spectral dependence of @xmath14 in the rayleigh - jeans regime . the discrepancy in the fit at longest wavelengths comes from the fact that the observed spectra have a gradually changing power - law index . the brightness distribution within the lws beam affects the global shape of the spectrum . as shown in figure [ fig1 ] the unstitched spectra have relatively large gaps in longer wavelengths ( @xmath15@xmath0 m ) , suggesting an uncertainty associated with the slope in this spectral range . it also suggests a difficulty in defining the assumed baseline in the longer wavelengths . in the following we present two cases for the baseline ; one with 140190@xmath0 m ( case a ) and 120190@xmath0 m ( case b ) to examine the effect of the assumed baseline and as a more realistic model we examine the astronomical silicate and graphite grain model @xcite the silicate and graphite grains both have approximately a power - law emissivity of @xmath16 in the far - infrared and this model provides slightly better fits than the power - law emissivity model of @xmath17 . we present the results of the silicate - graphite model in the following . we assume different single temperatures for each of the astronomical silicate and graphite grains and search for the best fit temperatures . the observed regions may contain various temperature components of various dust grains and thus these fits are a simple approximation for the underlying continuum . in figure [ fig2 ] the best fit results are plotted together with the observed spectra . the dotted lines indicate the results for the case a baseline , which fit the observed spectra reasonably well even in the longest wavelengths . they overlap mostly with the observed spectra for @xmath18@xmath0 m in the plot . the dashed lines show those for the case b baseline , which have obviously higher fluxes at longest wavelengths ( @xmath13@xmath0 m ) than the observed spectra . both cases clearly indicate the presence of an excess feature starting around 80@xmath0 m . the slope change around 80@xmath0 m is steep and can not be accounted for by extra graybodies . the similarity of the excess profile in two different regions supports the presence of the feature and suggests the common origin . @xcite have attributed the 90@xmath0 m feature in evolved stars to calcite ( caco@xmath7 ) , a carbonate mineral . in figure [ fig3]b we also plot a single - temperature graybody as a simple reference continuum . comparison with figure [ fig2 ] indicates that the 90@xmath0 m feature in ngc6302 is narrower than the interstellar 100@xmath0 m feature . the spectrum of ngc6302 shows a clear slope change around 100@xmath0 m , which indicates the longer wavelength edge of the feature . in contrast , the interstellar spectra do not show the clear change in the slope and suggest that the feature is extended to longer wavelengths than the 90@xmath0 m feature . the longer wavelength edge of the interstellar feature can not be well determined . although the exact peak position and width of the feature depend on the assumed continuum and the location of the baseline , the interstellar 100@xmath0 m feature seems to be shifted to longer wavelengths and have a wider width than the 90@xmath0 m feature seen in evolved stars . while carbonate grains are a likely candidate for the 90@xmath0 m emission around evolved stars and may partly contribute to the interstellar 100@xmath0 m feature , we examine the possibility of alternative species which has a broad feature around 100@xmath0 m for the band carrier in the diffuse emission . in the following we investigate whether carbon onion grains consisting of concentric curved graphitic sheets @xcite can account for the observed broad 100@xmath0 m feature or not . graphite is an anisotropic material and has different optical properties in the directions parallel and perpendicular to the c - axis ( the c - axis is perpendicular to the graphitic plane ) . it has an interband transition around 80@xmath0 m in the direction perpendicular to the c - axis @xcite . the emission efficiency of graphite spheres can be calculated by the so - called approximation @xcite , in which the efficiencies in the perpendicular and parallel to the c - axis are averaged with the weight of and , respectively . this approximation is valid in the small particle limit if the sphere consists of layered graphitic sheets and the optical properties in both directions are independent . in the graphite sphere , the emission efficiency in the direction parallel to the c - axis is much larger than that in the direction perpendicular to the axis in the far - infrared region . therefore the interband transition feature mentioned above is not visible in the averaged efficiency @xcite . in carbon onions , on the other hand , the graphitic layer is curved and approximately constitutes closed shells . thus the optical properties in the both directions should be mutually coupled and the interband feature can become visible in the emission efficiency of carbon onion grains . figure [ fig5 ] shows the emission efficiency factors divided by the grain radius for a graphite sphere and a carbon onion grain under the assumption that the grain radius is much smaller than the wavelengths in question . here the dielectric constants of graphite in the directions parallel and perpendicular to the c - axis at room temperature measurements are adopted in the calculations ( * ? ? ? * ; * ? ? ? * see below for discussion ) . the efficiency for the carbon onion is calculated by the formulation by henrard et al . ( 1993 ) and is assumed to have a central cavity of 0.7 in radius relative to the particle size . the appearance of the feature is insensitive to the size of the cavity . a broad feature around 100@xmath0 m is seen in the emission efficiency of the graphite sphere in the direction perpendicular to the c - axis , but it is hardly seen in the averaged efficiency . on the other hand , the far - infrared feature is evident in the emission efficiency of carbon onion particles . figure [ fig6 ] shows a comparison of the observed feature with that of carbon onion grains . to make the comparison easy the observed spectra are divided by the assumed continuum , while the efficiency of the carbon onion is divided by @xmath19 . two lines in the upper two panels indicate the effect of the assumed continuum . carbon onion grains show a similar broad feature to that observed in the diffuse interstellar emission , but details of the profile do not match perfectly . taking account of the uncertainties in the shape of the underlying continuum and the optical properties of carbon onions ( see below ) , the similarity of the band feature suggests that carbon onions are a possible carrier of the interstellar 100@xmath0 m feature . the 100@xmath0 m feature of carbon onions results from the surface resonance of small particles @xcite and appears near the wavelength where the real part of the dielectric constants in the perpendicular direction just becomes below zero . the exact position and profile of the feature thus depend on the adopted dielectric constants . while the electronic structure of carbon onions has been suggested to not differ significantly from that of graphite @xcite , the contribution of free electrons , which dominates in the far - infrared regions , may be different . in fact , measurements of electron spin resonance and electron energy - loss spectroscopy suggest that @xmath20 electrons in carbon onions are mostly localized in small domains @xcite . the localization of @xmath20 electron will decrease the contribution of free electrons , shifting the surface mode to wavelengths longer than 100@xmath0 m . we surmise that the shift can be more than 10@xmath0 m . but it is difficult to estimate the possible range of the shift because the behavior of free electrons depends also on the temperature and the strength of the interband transition in carbon onions could also be affected by the localization . the @xmath20 electron localization , the temperature dependence , and the possible change in the interband transition strength should affect the optical properties of carbon onions in the far - infrared . the match seen in figure [ fig6 ] may be just a coincidence in this sense . the present calculation suggests that the observed band feature can be accounted for if carbon onion grains contribute to 2030% of the far - infrared emission . carbon is an abundant element , but the exact form of carbon dust in the interstellar medium is not yet clear ( e.g. * ? ? ? carbon onions are a likely form other than graphite or amorphous carbon in addition to small aromatic particles or large molecules whose presence has been confirmed by the infrared emission bands in the diffuse interstellar radiation @xcite . carbon onions have recently attracted attentions as a new form of carbon material following the discovery of fullerens and their family . in astronomy , they are suggested to be formed in interstellar processes @xcite and the harsh conditions accompanying interstellar dust formation are favorable for the formation of onions @xcite . they have been proposed as a likely candidate for the interstellar 220 nm extinction hump @xcite . the quenched carbonaceous composite ( qcc ) , which shows a feature similar to the interstellar 220 nm hump @xcite , has also been shown to contain graphitic shell structures @xcite . it is not unexpected that the band features of carbon onions , if exist , also appear in the infrared region . in the present paper we simply propose the possibility that the interband feature of graphite in the far - infrared could appear in the emissivity spectrum of particles consisting of curved graphitic sheets and the observed broad interstellar feature around 100@xmath0 m may be accounted for by carbon onion grains . experimental work is definitely needed for further investigations . in the present paper we reported the detection of two far - infrared features at 65@xmath0 m and 100@xmath0 m in the diffuse infrared emission . the 65@xmath0 m band can reasonably be attributed to the ca - rich silicate , diopside . if this identification is correct , this is the first detection of a crystalline silicate feature in the interstellar diffuse emission . the interstellar 100@xmath0 m feature seems to be broader and peaked at longer wavelengths than the calcite feature seen in evolved stars although the precise estimate of the band profile is difficult . as a possible band carrier we investigate the possibility that the feature originates from carbon onion grains . while the observed feature may be accounted for by carbon onion grains if the assumed optical properties are adequate , the appearance of the feature is sensitive to the electronic structure of carbon onions . the origin of the interstellar 100@xmath0 m feature must be investigated in further experimental studies . the authors thank k. kawara , y. satoh , t. tanab , h. okuda , t. tsuji , h. shibai , and other members of the japanese iso group for their continuous help and support . we also thank s. tomita and s. hayashi for stimulating discussions on the optical properties of carbon onions and h. chihara and c. koike for providing us the far - infrared data of diopside and calcite . this work was supported in part by grant - in - aids for scientific research from the japan society of promotion of science ( jsps ) . 1982 , absorption and scattering of light by small particles ( new york : wiley ) chan , k .- w . , & onaka , t. 2000 , , 533 , l33 chihara , h. , koike , c. , & tsuchiyama , a. 2001 , , 53 , 243 clegg , p. , et al . 1996 , , 315 , l38 cohen , m. , barlow , m. j. , sylvester , r. j. , liu , x .- w . , cox , p. , lim , t. , schmitt , b. , & speck , a. k. 1999 , , 513 , l135 de graauw , t. , et al . 1996 , , 315 , l49 demyk , k. , jones , a. p. , dartois , e. , cox , p. , & dhendecourt , l. 1999 , , 349 , 267 draine , b. t. , & lee , h .- m . 1984 , 285 , 89 draine , b. t. , & malhotra , s. 1993 , , 414 , 632 feinstein , a. 1995 , rev . astrofis . , 2 , 57 gry , c. , et al . 2002 , iso handbook ; volume iv : lws the long - wavelength spectrometer , version 2.0 ( esa ) henrard , l. , lucas , a. a. , & lambin , ph . 1993 , , 406 , 92 henrard , l. , lambin , ph . , & lucas , a. a. 1997 , , 487 , 719 hildebrand , r. h. 1983 , , 24 , 267 jones , a. p. , tielens , a. g. g. m. , & hollenbach , d. j. 1996 , , 469 , 740 kemper , f. , jger , c. , waters , l. b. f. m. , henning , th . , molster , f. j. , barlow , m. j. , lim , t. , & de koter , a. 2002 , , 415 , 295 kessler , m. f. , et al . 1996 , , 315 , l27 koike , c. , et al . 2000 , , 363 , 1115 mattila , k , lemke , d. , haikala , l. k. , laureijs , r. j. , leger , a. , lehtinen , k. , leinert , c. , & mezger , p. g. 1996 , , 315 , l353 mizutani , m. , onaka , t. , & shibai , h. 2002 , , 382 , 610 molinari , s. , ceccarelli , c. , white , g. j. , saraceno , p. , nisini , b. , giannini , t. , & caux , e. 1999 , , 521 , l71 molster , f. j. , lim , t. l. , sylvester , r. j. , waters , l. b. f. m. , barlow , m. j. , beintema , d. a. , cohen , m. , cox , p. , & schmitt , b. 2001 , , 372 , 165 molster , f. j. , waters , l. b. f. m. , & tielens , a. g. g. m. 2002 , , 382 , 222 murakami , h. , et al . 1996 , , 48 , l41 nuth , j. 1985 , , 318 , 166 okada , y. , onaka , t. , shibai , h. & doi , y. 2002 , , submitted onaka , t. , yamamura , i. , tanab , t. , roellig , t. l. , & yuen , l. 1996 , , 48 , l59 philipp , h. r. 1977 , , 16 , 2896 pichler , t. , knupfer , m. , golden , m. s. , fink , j. , & cabioch , t. 2001 , , 63 , 155415 sakata , a. , wada , s. , okutsu , y. , shintani , h. , & nakada , y. 1983 , , 301 , 493 savage , b. d. , & sembach , k. r. 1996 , , 34 , 279 smith , r. g. , robinson , g. , hyland , a. r. , & carpenter , g. l. 1994 , , 271 , 481 tomita , s. , sakurai , m. , ohta , h. , & hayashi , s. 2001 , , 114 , 7477 ugarte , d. 1992 , , 359 , 707 ugarte , d. 1995 , , 443 , l85 venghaus , h. 1977 , phys . solidi ( b ) , 81 , 221 wada , s. , kaito , c. , kitamura , s. , ono , h. , & tokunaga , t. a. 1999 , , 345 , 259 waelkens , c. , et al . 1996 , , 315 , l245 walborn , n. r. 1995 , rev . , 2 , 51 waters , l. b. f. m. , et al . 1996 , , 315 , l361 yang , j. , & fukui , y. 1992 , , 386 , 618
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we report the detection of a feature at 65@xmath0 m and a broad feature around 100@xmath0 m in the far - infrared spectra of the diffuse emission from two active star - forming regions , and .
the features are seen in the spectra over a wide area of the observed regions , indicating that the carriers are fairly ubiquitous species in the interstellar medium .
a similar 65@xmath0 m feature has been detected in evolved stars and attributed to diopside , a ca - bearing crystalline silicate .
the present observations indicate the first detection of a crystalline silicate in the interstellar medium if this identification holds true also for the interstellar feature . a similar broad feature around 90@xmath0 m reported in the spectra of evolved stars has been attributed to calcite , a ca - bearing carbonate mineral .
the interstellar feature seems to be shifted to longer wavelengths and have a broader width although the precise estimate of the feature profile is difficult . as a carrier for the interstellar 100@xmath0 m feature ,
we investigate the possibility that the feature originates from carbon onions , grains consisting of curved graphitic shells .
because of the curved graphitic sheet structure , the optical properties in the direction parallel to the graphitic plane interacts with those in the vertical direction in carbon onion grains .
this effect enhances the interband transition feature in the direction parallel to the graphitic plane in carbon onions , which is suppressed in graphite particles .
simple calculations suggest that carbon onion grains are a likely candidate for the observed 100@xmath0 m feature carrier , but the appearance of the feature is sensitive to the assumed optical properties .
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the equation of state ( eos ) in the vicinity of saturation density is pretty well understood . but beyond saturation , the theories of dense matter present uncertainties . neutron stars provide a fantastic astrophysical environment for testing theories of cold and dense matter . in the core of neutron stars , densities could reach values of several times @xmath2 g @xmath3 . at such high densities , the fermi energies of the constituent particles could exceed the rest masses of heavier particles , and hence favour the appearance of such particles in the core . further , as the timescales associated with neutron stars are much greater than those associated with weak interactions , violation of strangeness conservation due to weak reactions in the core would result in the appearance of strangeness - containing particles such as hyperons . by producing new degrees of freedom , the appearance of strange particles is expected to result in a softer eos of dense matter in the neutron star interior . the highest neutron star mass that can be supported depends crucially on the eos . observations of pulsars in neutron star binaries provide a precise measurement of neutron star masses through general relativistic effects . the best determined pulsar mass ( @xmath4 ) is the hulse - taylor pulsar , and masses of most other pulsars are found to be clustered around this canonical value . recently , several neutron stars with larger masses have been discovered . radio timing observations of three post keplerian parameters led to the most precise measurement of the mass of a millisecond pulsar of @xmath5 @xcite . shapiro delay measurements from radio timing observations of the binary millisecond pulsar psr j1614 - 2230 indicated a mass of [email protected]@xmath7 of the neutron star @xcite . this is the largest rather precisely observed pulsar mass so far and thus poses the tightest reliable lower bound on the maximum mass of neutron stars . any theory of ultradense matter requires that the eos produce a maximum mass at least as high as this measured value , i.e. , models with @xmath8 @xmath9 would be ruled out . + according to existing models of dense matter , the presence of hyperons leads to a considerable softening of the eos , resulting in a reduction of the maximum mass of the neutron star . with hyperons , including only the hyperon - nucleon interaction , brueckner - hartree - fock ( bhf ) calculations obtain maximum masses of the order 1.47@xmath7 @xcite . the inclusion of the hyperon - hyperon interaction leads to a further softening of the eos and reduces the obtained masses to 1.34@xmath7 @xcite . on inclusion of three - body interactions , the maximum mass achieved in the bhf framework is 1.26@xmath7 @xcite . by employing a recently constructed hyperon - nucleon potential , the maximum masses of neutron stars with hyperons is computed to be well below 1.4@xmath7 @xcite . another approach is to adopt a relativistic mean field ( rmf ) model @xcite , which we employ for the following investigation . several attempts have been made using the rmf model to explain neutron star masses higher than 2@xmath7 , by artificially increasing the hyperon vector coupling away from their su(6 ) values @xcite . dexheimer et al . @xcite obtained large neutron star masses using a chirally motivated model including fourth - order self - interaction terms of the vector mesons @xmath10 and @xmath11 . recently , bednarek et al . @xcite achieved a stiffening of the eos by invoking quartic vector - meson terms proportional to @xmath12 and cross terms like @xmath13 and @xmath14 . when the stiffening on inclusion of hyperons is not sufficient , a transition to the quark phase can be considered to produce large maximum masses @xcite . bonanno and sedrakian @xcite also succeeded in obtaining a large neutron star mass with a hyperon and quark core using a stiff eos and vector repulsion among quarks . + as the parameters in the rmf model are fitted to the saturation properties of infinite nuclear matter , extrapolation to higher densities and asymmetry involve uncertainties . three of these properties - the saturation density , the binding energy and the asymmetry energy are more precisely known than the remaining ones - the effective nucleon mass and the compression modulus of nuclear matter . the uncertainty in the dense matter eos is basically related to the uncertainty in these two saturation properties . in this paper , we determine which of the two influences most the high density part of the eos that determines the highest attained neutron star mass . the parameters associated with attractive interaction among hyperons and nucleons are fitted to the potential depths of hyperons in nuclear matter , known from hypernuclear experiments . we investigate in this paper how the uncertainty in hyperon potential depths influences the stiffness of the eos and hence the maximum mass . we want to address the question : under which conditions can hyperons be present in massive neutron stars ? we achieve this through a controlled parameter study within the rmf model . in our model , we succeed in generating stiff eos by taking small values of effective nucleon mass @xmath15 , in the range of those obtained from fits to nuclei rather than bulk nuclear matter , and by including the strange vector meson @xmath11 . in a subsequent paper @xcite , we will focus on the assumption of the underlying symmetries that govern the repulsive interactions among hyperons and nucleons . + this paper is organized in the following way : in sec . 2 , we describe the model to calculate the eos . the parameters of the model are listed in sec . the results of our calculations are discussed in sec . 4 , and the summary and conclusions are given in sec . one of the possible approaches to describe neutron star matter is to adopt a rmf model subject to chemical equilibrium and charge neutrality . for our investigation of nucleons and hyperons in neutron star matter we will choose the full standard ( @xmath16 ) baryon octet as well as electrons and muons . in this model , baryon - baryon interaction is mediated by the exchange of scalar ( @xmath17 ) , vector ( @xmath18 ) and isovector ( @xmath19 ) mesons . the lagrangian density is given by @xcite @xmath20 the isospin multiplets for baryons b are represented by the dirac spinor @xmath21 with vacuum baryon mass @xmath22 , isospin operator @xmath23 , and @xmath24 and @xmath25 are field strength tensors . to reproduce the saturation properties of nuclear matter , the scalar self - interaction term @xmath26 is introduced @xcite . we also included an additional self - interaction term @xmath27 for the vector field as proposed by bodmer @xcite . due to inclusion of this term , the vector field increases proportional to @xmath28 for high densities , where @xmath19 is the baryon density , instead of the linear dependence in absence of this term . this results in a good agreement with brueckner - hartree - fock calculations . we will denote the above model with the three exchange mesons as `` model @xmath29 '' . the hyperon - hyperon interaction is usually incorporated through the exchange of additional strange scalar ( @xmath30 ) and strange vector ( @xmath11 ) mesons , again the scalar meson being responsible for attractive and the vector meson for repulsive interactions respectively : @xmath31 since it is our goal to obtain the stiffest possible eos within the model , we will only make use of the @xmath11 but omit the @xmath30 . this choice is also in accordance with @xmath32-hypernuclear data , where now it is clear that the @xmath32 interaction is only weakly attractive ( see ref @xcite for a discussion ) . the model obtained with this additional lagrangian we will call `` model @xmath33 '' . the calculation is performed using the mean field approximation @xcite . the effective baryon mass is given by @xmath34 while the chemical potential of baryon @xmath35 is @xmath36 , with @xmath37 denoting the corresponding fermi momentum . charge neutrality is described by the condition @xmath38 where @xmath39 is the number density of baryon b , @xmath40 is the electric charge and @xmath41 and @xmath42 are charge densities of electrons and muons respectively . we consider the hadronic matter to behave like an ideal fluid . the energy - momentum tensor for such a fluid yields the eos , defined by the relationship between the total energy density @xmath43 and the pressure @xmath44 there are five input parameters for the rmf models : the nuclear saturation density @xmath45 ( or equivalently , the number density @xmath46 ) as well as the binding energy per baryon number b / a , the effective mass of the nucleon @xmath47 , the nuclear compression modulus k and the asymmetry coefficient @xmath48 , all taken at saturation density . the parameters @xmath49 , @xmath50 , @xmath51 , @xmath52 and @xmath53 of `` model @xmath54 '' are determined from the saturation properties of nuclear matter in an analytic way ( see @xcite for explicit formulae ) . the binding energy , the asymmetry coefficient and the saturation density are well determined . we set them to @xmath55 , @xmath56 mev , @xmath57 mev @xcite . the effective nucleon mass and the compression modulus are less well known . in principle , the parameters of the model can be fitted to properties of nuclei , which result in low effective masses @xcite . however , here we want to explore the full parameter range of the model , as model parameters fitted to bulk nuclear matter and usually adopted in the community have large effective mass parameters . hence for this study , we use the parameter sets such as gm1 ( @xmath58=300 mev , @xmath59 ) and gm3 ( @xmath58=240 mev , @xmath60 ) which are fitted to the properties of bulk nuclear matter and have rather large effective nucleon masses in the range @xmath61 @xcite , and also the parameter sets fitted to the properties of nuclei , e.g. the whole nl parameter family and tm1 which yield very low values of @xmath62 @xcite . the value of compression modulus for symmetric nuclear matter can be extracted from the energies of the isoscalar giant monopole resonances ( isgmr ) in nuclei ( see e.g. @xcite and references therein ) . systematic studies of isgmr energies in various nuclei suggest a value of @xmath63 mev for the incompressibility of symmetric nuclear matter @xcite . however , there are strong surface effects , as well as large uncertainties in the value of the compression modulus for higher densities and asymmetric nuclear matter . another semiempirical parameter of nuclear matter crucial for determining the stiffness of the eos is the density dependence of symmetry energy , denoted by @xmath64 . we computed the values of the density dependence of symmetry energy @xmath64 for the sets discussed above , and they were found to be fairly comparable with the values existing in the literature @xcite . these however differ from the values of @xmath64 extracted from isospin diffusion data ( @xmath65 mev ) @xcite and isoscaling data ( @xmath66 mev ) @xcite . we note here that when the effective mass @xmath67 is varied between 0.55 and 0.8 , @xmath64 changes from 108 to 94 mev ( @xmath6 1 mev depending on the compressibility ) . the hyperon coupling constants @xmath68 , @xmath69 and @xmath70 are determined by using su(6 ) symmetry ( the quark model ) @xcite : @xmath71 the scalar meson - hyperon coupling constants @xmath72 are adjusted to the potential depths @xmath73 felt by a hyperon y in a bath of nucleons at saturation @xcite following the relation @xmath74 where the hypernuclear potential depths in nuclear matter @xmath75 are fixed in accordance with the available hypernuclear data . the best known hyperon potential is that of the @xmath76 , having a value of about @xmath77 mev @xcite . in case of the @xmath78s and the @xmath79s the potential depths are not as firmly known as for the @xmath76 . the following values @xmath80 mev , @xmath81 mev are generally adopted from the hypernuclear experimental data ( see @xcite for a discussion ) . we begin our investigations with `` model @xmath54 '' , namely with a @xmath82 model including non - linear scalar and vector self - interactions as well as @xmath19 mesons and the full baryon octet . in the previous section , we already mentioned that there are uncertainties in the values for the potential depths of hyperons in nuclear matter , obtained from hypernuclear experiments . we vary the potentials @xmath83 and @xmath84 systematically to study how they influence the stiffness of the hadronic eos and the limiting neutron star mass . we start with a fixed @xmath79 potential depth of @xmath85 mev @xcite . the potential depth of @xmath78 is varied in the range @xmath86 mev @xmath87 mev . we plot in fig . [ figuresigmafreeeos ] the eos for different values of @xmath83 ( lower branch ) for the gm1 parameter set and also for the same model with additional @xmath11 meson ( upper branch ) . for the sake of clarity only the cases @xmath88 mev are displayed . in mev , indicated by the corresponding numbers in the key . the lower branch is obtained for `` model @xmath54 '' for the gm1 parameter set while the upper branch is obtained including additionally the @xmath11 meson.,width=12 ] it can be inferred from the figure that for less deep potentials @xmath83 the eos becomes stiffer for each model . however , this is only true for negative potentials , since the values @xmath89 mev and @xmath90 mev basically result in the same eos . for negative values of the potential @xmath83 , the @xmath78s are bound in nuclear matter and the deeper the potential is , the more attractive must be the effective mesonic interaction and thus the softer the eos . for @xmath91 mev , the @xmath78s are no longer bound to nuclear matter and the effective mesonic interaction becomes more and more repulsive for increasing @xmath83 . this would in principle stiffen the eos . however , up to neutron star densities , i.e. about @xmath92 , the @xmath78s are not present in hadronic matter if the potential is repulsive and thus the eos up to these densities becomes insensitive to the actual value of @xmath83 . the composition of hadronic matter for positive and negative values of @xmath78 potential depth has already been investigated in @xcite . + . the variation of @xmath93 in model @xmath54 can not account for the observed neutron star mass limit ( lower branch ) , unless the @xmath11 meson is included in the model ( upper branch).,width=453 ] the repulsive potentials @xmath94 give the highest maximum star masses for each parameter set ( also for others which we did not plot , as e.g. tm1 and nl3 ) . in fig . [ figuresigmafreemr ] we plot the parts of the mass - radius relations around the maximum mass star configurations for the eos in fig . [ figuresigmafreeeos ] . the maximum mass for the lower branch of curves in fig . [ figuresigmafreemr ] corresponding to `` model @xmath54 '' is @xmath95 . we note , that the variation of @xmath83 in `` model @xmath54 '' for gm1 does not help to increase the maximum neutron star mass above the limiting value of [email protected]@xmath96 . investigating other parameter sets like tm1 would not solve the problem since the maximum masses are comparable to those of gm1 . the various nl - models would be good candidates , since they are amongst the stiffest available eos @xcite . however , the nl - models have ( like all models which are fitted to finite nuclei ) very small effective nucleon masses already at saturation density ( @xmath97 ) . at around five times nuclear saturation density the @xmath98fields are so large that the effective nucleon mass becomes zero or in principle even negative . this causes a well - known instability which makes it impossible to find a physical solution to the rmf equations @xcite . however , the critical density for the onset of the instability is higher than the maximum central density in the neutron star interior , and hence this is not a relevant problem for the present investigation . + the `` model @xmath99 '' gives us the upper bundle of eos in fig . [ figuresigmafreeeos ] . in fig . [ figuresigmafreemr ] the upper branch of the mass - radius relations corresponds to these eos from `` model @xmath99 '' . while the variation of @xmath78 potential only varies the maximum mass in the range @xmath100 0.03 @xmath96 , the impact of @xmath11 on the maximum mass is an order of magnitude larger : the maximum mass is increased by nearly 0.21 @xmath101 to @xmath102=1.96 @xmath101 . we should also mention that for purely nucleonic matter eos the obtained maximum masses are even higher ( about 2.4@xmath96 for gm1 and nearly 2.8@xmath101 for nl3 ) . we have seen that the potential depth @xmath83 has but a little influence on the maximum mass of a hadronic neutron star . as it does not play an important role in our investigations , we shall fix it to the value @xmath83=+30 mev @xcite which means that the @xmath78s are not present and the eos is as stiff as possible . we then vary the @xmath79 potential @xmath84 in the range @xmath103 + 40mev . as before , we compute the eos for the gm1 parameter set for `` model @xmath54 '' and `` model @xmath99 '' for different @xmath79 potentials . we plot in fig . [ figurexifreeeos ] the results for the values of @xmath84 from -40 mev to + 40 mev in steps of 20 mev . . the lower bundle of curves corresponds to `` model @xmath54 '' and the upper to `` model @xmath99 '' . all eos are obtained for gm1 parameter set.,width=453 ] '' for potential depths @xmath84 = + 28 mev ( a ) and -28 mev ( b ) are displayed . similarly , particle fractions for `` model @xmath99 '' are shown for @xmath79 potentials + 28 mev ( c ) and -28 mev ( d ) . the threshold for appearance of the @xmath79 hyperons is pushed to higher densities with increasing @xmath79 potential.,width=453 ] we see that for increasing @xmath79 potential , the eos stiffens . a difference to the influence of the @xmath78 potential depth is found for positive values of @xmath84 : here the eos remains sensitive to the value of the potential due to the contributions from @xmath104 at neutron star densities while for even higher densities both @xmath79s contribute . in fig . [ figureuxifrac ] , we display how the composition of the neutron star core varies for various values of @xmath79 potential in models `` @xmath54 '' and `` @xmath99 '' . it is clear from the figure that for positive @xmath79 potential , @xmath76 hyperons dominate particle fractions at high densities , while for negative @xmath105 the particle fractions from all hyperons contribute . as @xmath79 potential increases from negative to positive values , the threshold density for appearance of @xmath104 and @xmath106 hyperons is pushed to higher densities , and this effect further increases on inclusion of repulsion from the @xmath11 mesons . + . as before , the variation of @xmath107 in model @xmath54 can not explain the observed neutron star mass limit ( lower branch ) , unless the @xmath11 meson is included ( upper branch).,width=453 ] in fig . [ figurexifreemr ] we plot the mass - radius relations of the neutron stars obtained from the various eos in fig . [ figurexifreeeos ] . within `` model @xmath54 '' the maximum masses range from 1.69 @xmath96 for @xmath108 mev to 1.92 @xmath96 for @xmath84=+40 mev , i.e. over an interval of @xmath1090.22 @xmath96 . the maximum mass for the case @xmath84=+40 mev including @xmath11 is 2.04@xmath96 . although the value of the @xmath79 potential depth helps to stiffen the eos far better than the @xmath83 it is still not enough to be consistent with the mass of psr j1614 - 2230 for `` model @xmath54 '' . however , the mass constraint can be fulfilled within `` model @xmath110 '' for the gm1 parameter set . we combine the variations of @xmath83 and @xmath84 and plot in fig . [ figurepotentialcontour ] lines of constant maximum mass in the @xmath111 plane within the range @xmath86 to @xmath112 mev for both potentials . as before , we compare the results from `` model @xmath54 '' with those of `` model @xmath33 '' . ) in the @xmath111 plane . upper plot for `` model @xmath54 '' in gm1 parameterization , lower one for the same model including also @xmath11 mesons.,title="fig:",width=453,height=453 ] - 3 cm ) in the @xmath111 plane . upper plot for `` model @xmath54 '' in gm1 parameterization , lower one for the same model including also @xmath11 mesons.,title="fig:",width=453,height=453 ] - 1 cm in the light of our previous discussions we can quickly explain the shape of the lines in the plot : on each side of the @xmath113 line the maximum masses are dominated by that potential which is smaller and are rather insensitive to the value of the larger potential . this means , that in the region where @xmath114 the lines of constant maximum mass are parallel to the @xmath115axis . in the other part of the plot , where @xmath116 , the behaviour is mirrored . since the influence on the maximum masses is larger for smaller potentials , the mass lines ( which are plotted for equidistant mass steps of 0.01@xmath96 ) are densest towards the lower and the left parts of the plots but thin out with increasing potentials . we conclude this section by noting that for the whole range @xmath86 mev @xmath117,@xmath118 mev the maximum masses of `` model @xmath54 '' for the gm1 parameter set are not compatible with the observed pulsar mass @xmath1190.04@xmath96 . inclusion of the @xmath11 meson helps to sufficiently stiffen the eos and the whole range of @xmath120,@xmath121 + 40mev is now allowed . however , in this case the new mass limit does not help to restrict the parameter range of the hyperon potentials . . the upper curves refer to purely nucleonic stars while for the lower curves hyperons are included.,width=491 ] in the previous section , we found that the variation of hyperon potentials were not sufficient to explain the observed large neutron star mass . we then proceed to find out how the maximum mass depends on the nucleon coupling constants . we vary the least well known of the saturation parameters , namely the effective nucleon mass @xmath47 and the compression modulus k. we first consider purely nucleonic stars , and then repeat the exercise including hyperons in the `` model @xmath29 '' . in fig . [ figuremefffree ] we plot the maximum masses as a function of the effective nucleon mass at saturation @xmath122 for several values of k. we see that the maximum masses for the nucleonic case ( upper bunch of curves ) show a strong dependence on the effective nucleon mass : they drop from 2.88@xmath96 at @xmath123 to 1.87 - 2.01@xmath96 at @xmath124 . from fig . [ figuremefffree ] it is clear that for moderate effective masses , @xmath125 , the compression modulus gains a little influence on the maximum mass for pure nucleonic stars , and causes a mass difference of less than 0.05@xmath96 . above this value of the effective mass , the compression modulus causes a larger splitting of the maximum masses until at @xmath124 the mass difference is about 0.1@xmath96 . for the hyperonic case ( lower bunch of curves ) , the maximum masses range from 2.0 - 2.1@xmath96 at @xmath123 to about 1.39 - 1.63@xmath96 at @xmath124 . the mass splitting at @xmath126 for different k is about 0.2@xmath96 , increases to nearly 0.35@xmath96 at @xmath127 , and then again decreases to about 0.25@xmath96 at @xmath128 . for comparison we also mark in fig . [ figuremefffree ] the maximum masses and the effective masses of several other rmf sets fitted to properties of nuclei , like tm1 , nl3 or nl - sh @xcite . for the `` model @xmath33 '' , the maximum masses range from 2.4 @xmath7 at @xmath129 to 1.6 @xmath7 at @xmath128 . we also note that the results do not change substantially from the results obtained using `` model @xmath29 '' when we include the @xmath30 meson ( in `` model @xmath130 '' ) . contrary to the statement made in @xcite , that k as well as @xmath47 strongly influence the high - density behaviour of the equation of state of symmetric nuclear matter this being an effect [ which ] shows up in neutron rich and neutron star matter , we find that the compression modulus has nearly no influence on the high - density behaviour of the nucleonic eos whereas the effective mass has a stark impact . this finding is well known for the rmf model @xcite . the effect of the compressibility on hyperonic stars is however remarkable , although this dependence also varies considerably with the effective nucleon mass . in @xcite , the investigated range of the effective nucleon mass was 0.7 - 0.8 for the hyperonic case , in which k has a greater influence on the maximum mass than @xmath122 , which we also confirm in our study . however , we also extend our investigation for a broader range of @xmath122 , and find that this conclusion no longer holds for low effective nucleon masses . also , at large @xmath122 where k is influential the maximum masses of hyperonic stars do not comply with the limiting pulsar mass of [email protected]@xmath96 . in `` model @xmath29 '' with hyperons this mass is only reached for low values of the effective nucleon mass @xmath131 . within the `` model @xmath132 '' the mass constraint can only be achieved for @xmath133 . in the case of nucleonic stars , for @xmath134 the pulsar mass constraint can not be reached if the compression modulus is below 225 mev . + plane.,width=453,height=453 ] to explore the whole range of k continuously , we plot lines of constant maximum mass of nucleonic stars in the @xmath135 plane in fig . [ figuremeffkcontour ] . the results support our previous observation : the lines of constant maximum mass are nearly parallel to the x - axis , which means that the compression modulus does not influence the stiffness of the eos at neutron star densities . only for effective masses above @xmath136 do the lines become slightly slanted and show that a higher compression modulus then slightly stiffens the eos . we note , that in the investigated case of purely nucleonic neutron stars the observed mass of psr j1614 - 2230 requires an effective nucleon mass at saturation density of maximally @xmath137 at @xmath138 mev down to @xmath139 at @xmath140 mev . the possibility of existence of hyperons in massive neutron stars has been investigated in this paper . within hadronic models , we varied the coupling strengths of the scalar @xmath17 meson to the @xmath78 and @xmath79 hyperons by changing the potential depths of these hyperons in nuclear matter . meanwhile , we assumed su(6 ) relations for the vector couplings and kept the @xmath76 potential fixed . we found that for deeper potential depths the eos becomes softer and the maximum masses correspondingly smaller . which of the two potentials has a larger impact on the stiffness of the eos depends mainly on their relative values : if @xmath141 the @xmath79s will appear earlier ( i.e. at lower densities ) than the @xmath78s and therefore have the greater influence on the eos . however , this can not exactly be mirrored , since for @xmath142 the @xmath143 might appear earliest of all @xmath78s and @xmath79s , but as soon as the @xmath104 appears , the number density of the @xmath104 overtakes that of the @xmath143 by far . the overall effect of varying the two potential depths is rather small ( @xmath1090.2@xmath96 over the probed range ) . + since the impact of the potential depths on the eos is not enough to raise the maximum masses of neutron stars for `` model @xmath54 '' for gm1 above the limit of @xmath1190.04@xmath96 , we decided to increase the repulsive interaction amongst hyperons by including the vector @xmath11 meson in the rmf model . the additional repulsion between hyperons results in a sufficiently stiff eos accompanied by an increase in the maximum masses of about @xmath1090.2@xmath96 . for the gm1 parameter set , the investigated area of the @xmath144 plane complies now to almost full extent with the mass of psr j1614 - 2230 . + in the case of purely nucleonic stars we varied the effective mass of the nucleon as well as the compression modulus at nuclear saturation density which are input parameters for the rmf model . we found that the compression modulus has very little influence on the maximum mass of pure nucleonic neutron stars : in the range @xmath145 mev its influence is negligible for low effective masses @xmath146 , while its most prominent effect is for large effective masses @xmath147 where it causes a mass difference of just @xmath148 0.1@xmath96 . the effective mass of the nucleon at saturation proved to be a very good parameter for obtaining large maximum masses . over the investigated range @xmath149 the maximum masses change about @xmath100 1.1@xmath96 . the most massive stars ( up to @xmath1502.9@xmath96 ) are obtained for low effective nucleon masses independently of the compression modulus , while for @xmath151 we can not reach the mass limit of [email protected]@xmath96 if the compression modulus is too low . for the hyperonic case , maximum masses varied from 2.0 - 2.1@xmath96 at @xmath123 to about 1.39 - 1.63@xmath96 at @xmath124 in the `` model @xmath29 '' , whereas in the `` model @xmath33 '' the variation is from 2.4 to 1.6 @xmath7 respectively . hyperonic neutron stars in the `` model @xmath29 '' are only compatible with the new pulsar mass constraint for low values of the effective nucleon mass @xmath131 mev , while within the `` model @xmath132 '' the mass constraint can only be achieved for @xmath133 mev . we conclude that the nuclear compression modulus at saturation might be a good indicator for the stiffness of the nuclear matter eos at low densities ( around saturation ) , but the maximum mass is much more sensitive to the effective nucleon mass at saturation . this result is in accordance with the fact that the high density behaviour of the eos in the rmf model is solely controlled by the effective nucleon mass at saturation density by virtue of the hugenholtz - van hove theorem @xcite . + as we have seen , a 2@xmath96 star like psr j1614 - 2230 helps to constrain the nuclear matter eos . then a possible 2.4@xmath96 star - as has been recently suggested to exist @xcite - would pose even tighter constraints on the neutron star composition . should this measurement be confirmed , we could rule out many hadronic models ( e.g. tm1 or gm3 ) even in the limit of purely nucleonic stars . furthermore , in most such models the appearance of hyperons together with the use of su(6 ) relations for the vector coupling constants would then become impossible . in an associated paper @xcite , we further investigate the possibility of existence of hyperons in massive neutron stars , by questioning the assumption of su(6 ) symmetry for the determination of vector meson - hyperon couplings . + @xmath152 + _ acknowledgements : _ b . is supported by the dfg through the heidelberg graduate school of fundamental physics . d.c . acknowledges the support from the alexander von humboldt foundation . is supported by the state of baden - wrttemberg through a lgfg stipend . this work is supported by bmbf under grant fkz 06hd9127 , by the helmholtz alliance ha216/emmi and by compstar , a research networking program of the european science foundation . [ 2]#2 p. c. c. freire , c. g. bassa , n. wex , i. h. stairs , d. j. champion , s. m. ransom , p. lazarus , v. m. kaspi , j. w. t. hessels , m. kramer , j. m. cordes , j. p. w. verbiest , p. podsiadlowski , d. j. nice , j. s. deneva , d. r. lorimer , b. w. stappers , m. a. mclaughlin , and f. camilo , http://dx.doi.org/10.1111/j.1365-2966.2010.18109.x [ _ mon . not . r. astron . * 412 * ( 2011 ) 2763 ] .
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the constituents of cold dense matter are still far from being understood .
however , neutron star observations such as the recently observed pulsar psr j1614 - 2230 with a mass of @xmath0 m@xmath1 help to considerably constrain the hadronic equation of state ( eos ) .
we systematically investigate the influence of the hyperon potentials on the stiffness of the eos .
we find that they have but little influence on the maximum mass compared to the inclusion of an additional vector - meson mediating repulsive interaction amongst hyperons .
the new mass limit can only be reached with this additional meson regardless of the hyperon potentials .
further , we investigate the impact of the nuclear compression modulus and the effective mass of the nucleon at saturation density on the high density regime of the eos .
we show that the maximum mass of purely nucleonic stars is very sensitive to the effective nucleon mass but only very little to the compression modulus .
neutron stars , equation of state , hypernuclei , hadronic matter
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in recent years , the theoretical and experimental use of quantum systems to store , transmit and process information has spurred the study of how much of classical information theory can be extended to the new territory of quantum information and , vice versa , how much novel strategies and concepts are needed that have no classical counterpart . we shall compare the relations between the rate at which entropy is produced by classical , respectively quantum , ergodic sources , and the complexity of the emitted strings of bits , respectively qubits . according to kolmogorov @xcite , the complexity of a bit string is the minimal length of a program for a turing machine ( _ tm _ ) that produces the string . more in detail , the algorithmic complexity @xmath0 of a string @xmath1 is the length ( counted in the number of bits ) of the shortest program @xmath2 that fed into a universal _ tm _ ( _ utm _ ) @xmath3 yields the string as output , i.e. @xmath4 . for infinite sequences @xmath5 , in analogy with the entropy rate , one defines the _ complexity rate _ as @xmath6 , where @xmath1 is the string consisting of the first @xmath7 bits of @xmath5 , @xcite . the universality of @xmath3 implies that changing the _ utm _ , the difference in the complexity of a given string is bounded by a constant independent of the string ; it follows that the complexity rate @xmath8 is _ utm_-independent . different ways to quantify the complexity of qubit strings have been put forward ; in this paper , we shall be concerned with some which directly generalize the classical definition by relating the complexity of qubit strings with their algorithmic description by means of quantum turing machines ( _ qtm _ ) . for classical ergodic sources , an important theorem , proved by brudno @xcite and conjectured before by zvonkin and levin @xcite , establishes that the entropy rate equals the algorithmic complexity per symbol of almost all emitted bit strings . we shall show that this essentially also holds in quantum information theory . for stationary classical information sources , the most important parameter is the _ entropy rate _ @xmath9 , where @xmath10 is the shannon entropy of the ensembles of strings of length @xmath7 that are emitted according to the probability distribution @xmath11 . according to the shannon - mcmillan - breiman theorem @xcite , @xmath12 represents the optimal compression rate at which the information provided by classical ergodic sources can be compressed and then retrieved with negligible probability of error ( in the limit of longer and longer strings ) . essentially , @xmath13 is the number of bits that are needed for reliable compression of bit strings of length @xmath7 . intuitively , the less amount of patterns the emitted strings contain , the harder will be their compression , which is based on the presence of regularities and on the elimination of redundancies . from this point of view , the entropy rate measures the randomness of a classical source by means of its compressibility on the average , but does not address the randomness of single strings in the first instance . this latter problem was approached by kolmogorov @xcite , ( and independently and almost at the same time by chaitin @xcite , and solomonoff @xcite ) , in terms of the difficulty of their description by means of algorithms executed by universal turing machines ( _ utm _ ) , see also @xcite . on the whole , structureless strings offer no catch for writing down short programs that fed into a computer produce the given strings as outputs . the intuitive notion of random strings is thus mathematically characterized by kolmogorov by the fact that , for large @xmath7 , the shortest programs that reproduce them can not do better than literal transcription @xcite . intuitively , one expects a connection between the randomness of single strings and the average randomness of ensembles of strings . in the classical case , this is exactly the content of a theorem of brudno @xcite which states that for ergodic sources , the complexity rate of @xmath14-almost all infinite sequences @xmath5 coincides with the entropy rate , i.e. @xmath15 . quantum sources can be thought as black boxes emitting strings of qubits . the ensembles of emitted strings of length @xmath7 are described by a density operator @xmath16 on the hilbert spaces @xmath17 , which replaces the probability distribution @xmath11 from the classical case . the simplest quantum sources are of bernoulli type : they amount to infinite quantum spin chains described by shift - invariant states characterized by local density matrices @xmath16 over @xmath7 sites with a tensor product structure @xmath18 , where @xmath19 is a density operator on @xmath20 . however , typical ergodic states of quantum spin - chains have richer structures that could be used as quantum sources : the local states @xmath16 , not anymore tensor products , would describe emitted @xmath21qubit strings which are correlated density matrices . similarly to classical information sources , quantum stationary sources ( shift - invariant chains ) are characterized by their entropy rate @xmath22 , where @xmath23 denotes the von neumann entropy of the density matrix @xmath16 . the quantum extension of the shannon - mcmillan theorem was first obtained in @xcite for bernoulli sources , then a partial assertion was obtained for the restricted class of completely ergodic sources in @xcite , and finally in @xcite , a complete quantum extension was shown for general ergodic sources . the latter result is based on the construction of subspaces of dimension close to @xmath24 , being typical for the source , in the sense that for sufficiently large block length @xmath7 , their corresponding orthogonal projectors have an expectation value arbitrarily close to @xmath25 with respect to the state of the quantum source . these typical subspaces have subsequently been used to construct compression protocols @xcite . the concept of a universal quantum turing machine ( _ uqtm _ ) as a precise mathematical model for quantum computation was first proposed by deutsch @xcite . the detailed construction of _ uqtms _ can be found in @xcite : these machines work analogously to classical _ tm_s , that is they consist of a read / write head , a set of internal control states and input / output tapes . however , the local transition functions among the machine s configurations ( the programs or quantum algorithms ) are given in terms of probability amplitudes , implying the possibility of linear superpositions of the machine s configurations . the quantum algorithms work reversibly . they correspond to unitary actions of the _ uqtm _ as a whole . an element of irreversibility appears only when the output tape information is extracted by tracing away the other degrees of freedom of the _ uqtm_. this provides linear superpositions as well as mixtures of the output tape configurations consisting of the local states @xmath26 and blanks @xmath27 , which are elements of the so - called _ computational basis_. the reversibility of the _ uqtm _ s time evolution is to be contrasted with recent models of quantum computation that are based on measurements on large entangled states , that is on irreversible processes , subsequently performed in accordance to the outcomes of the previous ones @xcite . in this paper we shall be concerned with bernstein - vazirani - type _ uqtms _ whose inputs and outputs may be bit or qubit strings @xcite . given the theoretical possibility of universal computing machines working in agreement with the quantum rules , it was a natural step to extend the problem of algorithmic descriptions as a complexity measure to the quantum case . contrary to the classical case , where different formulations are equivalent , several inequivalent possibilities are available in the quantum setting . in the following , we shall use the definitions in @xcite which , roughly speaking , say that the algorithmic complexity of a qubit string @xmath19 is the logarithm in base @xmath28 of the dimension of the smallest hilbert space ( spanned by computational basis vectors ) containing a quantum state that , once fed into a _ uqtm _ , makes the _ uqtm _ compute the output @xmath19 and halt . in general , quantum states can not be perfectly distinguished . thus , it makes sense to allow some tolerance in the accuracy of the machine s output . as explained below , there are two natural ways to deal with this , leading to two ( closely related ) different complexity notions @xmath29 and @xmath30 , which correspond to asymptotically vanishing , respectively small but fixed tolerance . both quantum algorithmic complexities @xmath29 and @xmath30 are thus measured in terms of the length of _ quantum _ descriptions of qubit strings , in contrast to another definition @xcite which defines the complexity of a qubit string as the length of its shortest _ classical _ description . a third definition @xcite is instead based on an extension of the classical notion of universal probability to that of universal density matrices . the study of the relations among these proposals is still in a very preliminary stage . for an approach to quantum complexity based on the amount of resources ( quantum gates ) needed to implement a quantum circuit reproducing a given qubit string see @xcite . the main result of this work is the proof of a weaker form of brudno s theorem , connecting the quantum entropy rate @xmath31 and the quantum algorithmic complexities @xmath29 and @xmath30 of pure states emitted by quantum ergodic sources . it will be proved that there are sequences of typical subspaces of @xmath17 , such that the complexity rates @xmath32 and @xmath33 of any of their pure - state projectors @xmath34 can be made as close to the entropy rate @xmath31 as one wants by choosing @xmath7 large enough , and there are no such sequences with a smaller expected complexity rate . the paper is divided as follows . in section 2 , a short review of the @xmath35-algebraic approach to quantum sources is given , while section 3 states as our main result a quantum version of brudno s theorem . in section 4 , a detailed survey of _ qtm_s and of the notion of _ quantum kolmogorov complexity _ is presented . in section 5 , based on a quantum counting argument , a lower bound is given for the quantum kolmogorov complexity per qubit , while an upper bound is obtained in section 5 by explicit construction of a short quantum algorithm able to reproduce any pure state projector @xmath34 belonging to a particular sequence of high probability subspaces . in order to formulate our main result rigorously , we start with a brief introduction to the relevant concepts of the formalism of quasi - local @xmath35-algebras which is the most suited one for dealing with quantum spin chains . at the same time , we shall fix the notations . we shall consider the lattice @xmath36 and assign to each site @xmath37 a @xmath35-algebra @xmath38 being a copy of a fixed finite - dimensional algebra @xmath39 , in the sense that there exists a @xmath40-isomorphism @xmath41 . to simplify notations , we write @xmath42 for @xmath43 and @xmath44 . the algebra of observables associated to a finite @xmath45 is defined by @xmath46 . observe that for @xmath47 we have @xmath48 and there is a canonical embedding of @xmath49 into @xmath50 given by @xmath51 , where @xmath52 and @xmath53 denotes the identity of @xmath54 . the infinite - dimensional quasi - local @xmath35-algebra @xmath55 is the norm completion of the normed algebra @xmath56 , where the union is taken over all finite subsets @xmath57 . in the present paper , we mainly deal with qubits , which are the quantum counterpart of classical bits . thus , in the following , we restrict our considerations to the case where @xmath39 is the algebra of observables of a qubit , i.e. the algebra @xmath58 of @xmath59 matrices acting on @xmath60 . since every finite - dimensional unital @xmath35-algebra @xmath39 is @xmath40-isomorphic to a subalgebra of @xmath61 for some @xmath62 , our results contain the general case of arbitrary @xmath39 . moreover , the case of classical bits is covered by @xmath39 being the subalgebra of @xmath58 consisting of diagonal matrices only . similarly , we think of @xmath49 as the algebra of observables of qubit strings of length @xmath63 , namely the algebra @xmath64 of @xmath65 matrices acting on the hilbert space @xmath66 . the quasi - local algebra @xmath55 corresponds to the doubly - infinite qubit strings . the ( right ) shift @xmath67 is a @xmath40-automorphism on @xmath68 uniquely defined by its action on local observables @xmath69}\mapsto a\in\mathcal{a}_{[m+1,n+1]}\end{aligned}\ ] ] where @xmath70 \subset \mathbb{z}$ ] is an integer interval . a state @xmath71 on @xmath55 is a normalized positive linear functional on @xmath68 . each local state @xmath72 , @xmath73 finite , corresponds to a density operator @xmath74 by the relation @xmath75 , for all @xmath76 , where @xmath77 is the trace on @xmath78 . the density operator @xmath79 is a positive matrix acting on the hilbert space @xmath80 associated with @xmath81 satisfying the normalization condition @xmath82 . the simplest @xmath79 correspond to one - dimensional projectors @xmath83 onto vectors @xmath84 and are called pure states , while general density operators are linear convex combinations of one - dimensional projectors : @xmath85 , @xmath86 , @xmath87 . we denote by @xmath88 the convex set of density operators acting on a ( possibly infinite - dimensional ) hilbert space @xmath89 , whence @xmath90 . a state @xmath91 on @xmath55 corresponds one - to - one to a family of density operators @xmath74 , @xmath92 finite , fulfilling the consistency condition @xmath93 for @xmath94 , where @xmath95 denotes the partial trace over the local algebra @xmath49 which is computed with respect to any orthonormal basis in the associated hilbert space @xmath80 . notice that a state @xmath71 with @xmath96 , i.e. a shift - invariant state , is uniquely determined by a consistent sequence of density operators @xmath97 in @xmath98 corresponding to the local states @xmath99 , where @xmath100 denotes the integer interval @xmath101\subset \mathbb{z}$ ] , for each @xmath102 . as motivated in the introduction , in the information - theoretical context , we interpret the tuple @xmath103 describing the quantum spin chain as a stationary quantum source . the von neumann entropy of a density matrix @xmath19 is @xmath104 . by the subadditivity of @xmath105 for a shift - invariant state @xmath71 on @xmath55 , the following limit , the quantum entropy rate , exists @xmath106 the set of shift - invariant states on @xmath55 is convex and compact in the weak@xmath40-topology . the extremal points of this set are called ergodic states : they are those states which can not be decomposed into linear convex combinations of other shift - invariant states . notice that in particular the shift - invariant product states defined by a sequence of density matrices @xmath107 , @xmath108 , where @xmath109 is a fixed @xmath110 density matrix , are ergodic . they are the quantum counterparts of bernoulli ( i.i.d . ) most of the results in quantum information theory concern such sources , but , as mentioned in the introduction , more general ergodic quantum sources allowing correlations can be considered . more concretely , the typical quantum source that has first been considered was a finite - dimensional quantum system emitting vector states @xmath111 with probabilities @xmath112 . the state of such a source is the density matrix @xmath113 being an element of the full matrix algebra @xmath114 ; furthermore , the most natural source of qubit strings of length @xmath7 is the one that emits vectors @xmath115 independently one after the other at each stroke of time . is a vector in a hilbert space and a ket @xmath116 is its dual vector . ] the corresponding state after @xmath7 emissions is thus the tensor product @xmath117 in the following , we shall deal with the more general case of _ ergodic _ sources defined above , which naturally appear e.g. in statistical mechanics ( compare 1d spin chains with finite - range interaction ) . when restricted to act only on @xmath7 successive chain sites , namely on the local algebra @xmath118 , these states correspond to density matrices @xmath16 acting on @xmath17 which are not simply tensor products , but may contain classical correlations and entanglement . the qubit strings of length @xmath7 emitted by these sources are generic density matrices @xmath119 acting on @xmath17 , which are compatible with the state of the source @xmath71 in the sense that @xmath120 , where @xmath121 denotes the support projector of the operator @xmath119 , that is the orthogonal projection onto the subspace where @xmath119 can not vanish . more concretely , @xmath16 can be decomposed in uncountably many different ways into convex decompositions @xmath122 in terms of other density matrices @xmath123 on the local algebra @xmath124 each one of which describes a possible qubit string of length @xmath7 emitted by the source . it turns out that the rates of the complexities @xmath29 ( approximation - scheme complexity ) and @xmath30 ( finite - accuracy complexity ) of the typical pure states of qubit strings generated by an ergodic quantum source @xmath125 are asymptotically equal to the entropy rate @xmath126 of the source . a precise formulation of this result is the content of the following theorem . it can be seen as a quantum extension of brudno s theorem as a convergence in probability statement , while the original formulation of brudno s result is an almost sure statement . we remark that a proper introduction to the concept of quantum kolmogorov complexity needs some further considerations . we postpone this task to the next section . + in the remainder of this paper , we call a sequence of projectors @xmath127 , @xmath108 , satisfying @xmath128 a _ sequence of @xmath71-typical projectors_. [ theqbrudno ] ' '' '' let @xmath129 be an ergodic quantum source with entropy rate @xmath31 . for every @xmath130 , there exists a sequence of @xmath71-typical projectors @xmath131 , @xmath108 , i.e. @xmath132 , such that for @xmath7 large enough every one - dimensional projector @xmath133 satisfies @xmath134 moreover , @xmath31 is the optimal expected asymptotic complexity rate , in the sense that every sequence of projectors @xmath135 , @xmath108 , that for large @xmath7 may be represented as a sum of mutually orthogonal one - dimensional projectors that all violate the lower bounds in ( [ eq1 ] ) and ( [ eq2 ] ) for some @xmath130 , has an asymptotically vanishing expectation value with respect to @xmath71 . algorithmic complexity measures the degree of randomness of a single object . it is defined as the minimal description length of the object , relative to a certain machine ( classically a _ utm _ ) . in order to properly introduce a quantum counterpart of kolmogorov complexity , we thus have to specify what kind of objects we want to describe ( outputs ) , what the descriptions ( inputs ) are made of , and what kind of machines run the algorithms . in accordance to the introduction , we stipulate that inputs and outputs are so - called ( pure or mixed ) _ variable - length qubit strings _ , while the reference machines will be _ qtm_s as defined by bernstein and vazirani @xcite , in particular universal _ qtm_s . let @xmath136 be the hilbert space of @xmath137 qubits ( @xmath138 ) . we write @xmath139 for @xmath20 to indicate that we fix two orthonormal _ computational basis vectors _ @xmath140 and @xmath141 . since we want to allow superpositions of different lengths @xmath137 , we consider the hilbert space @xmath142 defined as @xmath143 the classical finite binary strings @xmath144 are identified with the computational basis vectors in @xmath145 , i.e. @xmath146 , where @xmath147 denotes the empty string . we also use the notation @xmath148 and treat it as a subspace of @xmath145 . a ( variable - length ) _ qubit string _ @xmath149 is a density operator on @xmath145 . we define the _ length _ @xmath150 of a qubit string @xmath149 as @xmath151 or as @xmath152 if this set is empty ( this will never occur in the following ) . there are two reasons for considering variable - length and also mixed qubit strings . first , we want our result to be as general as possible . second , a _ qtm _ will naturally produce superpositions of qubit strings of different lengths ; mixed outputs appear naturally while tracing out the other parts of the _ qtm _ ( input tape , control , head ) after halting . in contrast to the classical situation , there are uncountably many qubit strings that can not be perfectly distinguished by means of any quantum measurement . if @xmath153 are two qubit strings with finite length , then we can quantify their distance in terms of the trace distance @xmath154 where the @xmath155 are the eigenvalues of the hermitian operator @xmath156 . in subsection [ subqac ] , we will define quantum kolmogorov complexity @xmath157 for qubit strings @xmath19 . due to the considerations above , it can not be expected that the qubit strings @xmath19 are reproduced exactly , but it rather makes sense to demand the strings to be generated within some trace distance @xmath158 . another possibility is to consider approximation schemes , i.e. to have some parameter @xmath159 , and to demand the machine to approximate the desired state better and better the larger @xmath137 gets . we will pursue both approaches , corresponding to equations ( [ berthdelta ] ) and ( [ berth ] ) below . note that we can identify every density operator @xmath160 on the local @xmath7-block algebra with its corresponding qubit string @xmath161 such that @xmath162 . similarly , we identify qubit strings @xmath149 of finite length @xmath163 with the state of the input or output tape of a _ qtm _ ( see subsection [ basicdefqtmsec ] ) containing the state in the cell interval @xmath164 $ ] and vice versa . due to the equivalence of various models for quantum computation , the definition of quantum kolmogorov complexity should be rather insensitive to the details of the underlying machine . nevertheless , there are some details which are relevant for our theorem . thus , we have to give a thorough definition of what we mean by a _ qtm_. bernstein and vazirani ( @xcite , def . 3.2.2 ) define a quantum turing machine @xmath165 as a triplet @xmath166 , where @xmath167 is a finite alphabet with an identified blank symbol @xmath27 , and @xmath168 is a finite set of states with an identified initial state @xmath169 and final state @xmath170 . the function @xmath171 is called the _ quantum transition function_. the symbol @xmath172 denotes the set of complex numbers @xmath173 such that there is a deterministic algorithm that computes the real and imaginary parts of @xmath174 to within @xmath175 in time polynomial in @xmath7 . one can think of a _ qtm _ as consisting of a two - way infinite tape @xmath176 of cells indexed by @xmath177 , a control @xmath178 , and a single read / write head @xmath179 that moves along the tape . a ( classical ) configuration is a triplet @xmath180 such that only a finite number of tape cell contents @xmath181 are non - blank ( @xmath34 and @xmath137 are the state of the control and the position of the head respectively ) . let @xmath182 be the set of all configurations , and define the hilbert space @xmath183 , which can be written as @xmath184 . the transition function @xmath158 generates a linear operator @xmath185 on @xmath186 describing the time evolution of the _ qtm_. we identify @xmath149 with the initial state of @xmath165 on input @xmath119 , which is according to the definition in @xcite a state on @xmath186 where @xmath119 is written on the input track over the cell interval @xmath187 $ ] , the empty state @xmath27 is written on the remaining cells of the input track and on the whole output track , the control is in the initial state @xmath169 and the head is in position @xmath188 . then , the state @xmath189 of @xmath165 on input @xmath119 at time @xmath190 is given by @xmath191 . the state of the control at time @xmath192 is thus given by partial trace over all the other parts of the machine , that is @xmath193 . in accordance with @xcite , def . 3.5.1 , we say that the _ qtm _ @xmath165 _ halts at time @xmath190 on input @xmath149 _ , if and only if @xmath194 where @xmath195 is the special state of the control ( specified in the definition of @xmath165 ) signalling the halting of the computation . denote by @xmath196 the set of vector inputs with equal halting time @xmath192 . observe that the above definition implies that @xmath197 is equal to the linear span of @xmath198 , i.e. @xmath199 is a linear subspace of @xmath145 . moreover for @xmath200 the corresponding subspaces @xmath199 and @xmath201 are mutually orthogonal , because otherwise one could perfectly distinguish non - orthogonal vectors by means of the halting time . it follows that the subset of @xmath202 on which a _ qtm m _ halts is a union @xmath203 . for our purpose , it is useful to consider a special class of _ qtms _ with the property that their tape @xmath176 consists of two different tracks , an _ input track _ @xmath204 and an _ output track _ @xmath205 . this can be achieved by having an alphabet which is a cartesian product of two alphabets , in our case @xmath206 . then , the tape hilbert space @xmath207 can be written as @xmath208 . ' '' '' a partial map @xmath209 will be called a _ qtm _ , if there is a bernstein - vazirani two - track qtm @xmath210 ( see @xcite , def . 3.5.5 ) with the following properties : * @xmath206 , * the corresponding time evolution operator @xmath211 is unitary , * if @xmath212 halts on input @xmath119 with a variable - length qubit string @xmath213 on the output track starting in cell @xmath188 such that the @xmath214-th cell is empty for every @xmath215 $ ] , then @xmath216 ; otherwise , @xmath217 is undefined . in general , different inputs @xmath119 have different halting times @xmath192 and the corresponding outputs are essentially results of different unitary transformations given by @xmath218 . however , as the subset of @xmath219 on which @xmath165 is defined is of the form @xmath220 , the action of the partial map @xmath165 on this subset may be extended to a valid quantum operation on the system hilbert space @xmath89 , see @xcite . ] on @xmath221 : [ lemmaqtmsareoperations ] ' '' '' for every _ qtm _ @xmath165 there is a quantum operation @xmath222 , such that for every @xmath223 @xmath224 * proof . * let @xmath225 and @xmath226 be an orthonormal basis of @xmath199 , @xmath227 , and the orthogonal complement of @xmath228 within @xmath145 , respectively . we add an ancilla hilbert space @xmath229 to the _ qtm _ , and define a linear operator @xmath230 by specifying its action on the orthonormal basis vectors @xmath231 : @xmath232 since the right hand side of ( [ eqonb ] ) is a set of orthonormal vectors in @xmath233 , the map @xmath234 is a partial isometry . thus , the map @xmath235 is trace - preserving , completely positive ( @xcite ) . its composition with the partial trace , given by @xmath236 , is a quantum operation . the typical case we want to study is the ( approximate ) reproduction of a density matrix @xmath237 by a qtm @xmath165 . this means that there is a quantum program @xmath238 , such that @xmath239 in a sense explained below . we are particularly interested in the case that the program @xmath119 is shorter than @xmath19 itself , i.e. that @xmath240 . on the whole , the minimum possible length @xmath241 for @xmath19 will be defined as the _ quantum algorithmic complexity _ of @xmath19 . as already mentioned , there are at least two natural possible definitions . the first one is to demand only approximate reproduction of @xmath19 within some trace distance @xmath158 . the second one is based on the notion of an approximation scheme . to define the latter , we have to specify what we mean by supplying a _ qtm _ with _ two _ inputs , the qubit string and a parameter : [ defencoding ] ' '' '' let @xmath159 and @xmath242 . we define an encoding @xmath243 of a pair @xmath244 into a single string @xmath245 by @xmath246 where @xmath247 denotes the ( classical ) string consisting of @xmath248 @xmath25 s , followed by one @xmath188 , followed by the @xmath249 binary digits of @xmath137 , and @xmath250 is the corresponding projector in the computational basis and @xmath251 . ] . for every _ qtm _ @xmath165 , we set @xmath252 note that @xmath253 the _ qtm _ @xmath165 has to be constructed in such a way that it is able to decode both @xmath137 and @xmath119 from @xmath245 , which is an easy classical task . [ defqk ] ' '' '' let @xmath165 be a _ qtm _ and @xmath254 a qubit string . for every @xmath255 , we define the _ finite - accuracy quantum complexity _ @xmath256 as the minimal length @xmath241 of any quantum program @xmath257 such that the corresponding output @xmath217 has trace distance from @xmath19 smaller than @xmath158 , @xmath258 similarly , we define an _ approximation - scheme quantum complexity _ @xmath259 by the minimal length @xmath241 of any density operator @xmath257 , such that when given @xmath165 as input together with any integer @xmath137 , the output @xmath260 has trace distance from @xmath19 smaller than @xmath261 : @xmath262 some points are worth stressing in connection with the previous definition : * this definition is essentially equivalent to the definition given by berthiaume et . al . in @xcite . the only technical difference is that we found it convenient to use the trace distance rather than the fidelity . * the _ same _ qubit program @xmath119 is accompanied by a classical specification of an integer @xmath137 , which tells the program to what accuracy the computation of the output state must be accomplished . * if @xmath165 does not have too restricted functionality ( for example , if @xmath165 is universal , which is discussed below ) , a noiseless transmission channel ( implementing the identity transformation ) between the input and output tracks can always be realized : this corresponds to classical literal transcription , so that automatically @xmath263 for some constant @xmath264 . of course , the key point in classical as well as quantum algorithmic complexity is that there are sometimes much shorter qubit programs than literal transcription . * the exact choice of the accuracy specification @xmath265 is not important ; we can choose any computable function that tends to zero for @xmath266 , and we will always get an equivalent definition ( in the sense of being equal up to some constant ) . + the same is true for the choice of the encoding @xmath267 : as long as @xmath137 and @xmath119 can both be computably decoded from @xmath245 and as long as there is no way to extract additional information on the desired output @xmath19 from the @xmath137-description part of @xmath245 , the results will be equivalent up to some constant . both quantum algorithmic complexities @xmath30 and @xmath29 are related to each other in a useful way : [ lemrelation ] for every _ qtm _ @xmath165 and every @xmath159 , we have the relation @xmath268 * proof . * suppose that @xmath269 , so there is a density matrix @xmath270 with @xmath271 , such that @xmath272 for every @xmath159 . then @xmath273 , where @xmath267 is given in definition [ defencoding ] , is an input for @xmath165 such that @xmath274 . thus @xmath275 , where the second inequality is by ( [ encoding_length ] ) . the term @xmath276 in ( [ eqrelation ] ) depends on our encoding @xmath267 given in definition [ defencoding ] , but if @xmath165 is assumed to be universal ( which will be discussed below ) , then ( [ eqrelation ] ) will hold for _ every _ encoding , if we replace the term @xmath277 by @xmath278 , where @xmath279 denotes the classical ( self - delimiting ) algorithmic complexity of the integer @xmath137 , and @xmath264 is some constant depending only on @xmath165 . for more details we refer the reader to @xcite . in @xcite , it is proved that there is a universal _ qtm _ ( _ uqtm _ ) @xmath3 that can simulate with arbitrary accuracy every other machine @xmath165 in the sense that for every such @xmath165 there is a classical bit string @xmath280 such that @xmath281 where @xmath282 . as it is implicit in this definition of universality , we will demand that @xmath283 is able to perfectly simulate every classical computation , and that it can apply a given unitary transformation within any desired accuracy ( it is shown in @xcite that such machines exist ) . we choose an arbitrary _ uqtm _ @xmath283 which is constructed such that it decodes our encoding @xmath245 given in definition [ defencoding ] into @xmath137 and @xmath119 at the beginning of the computation . like in the classical case , we fix @xmath283 for the rest of the paper and simplify notation by @xmath284 as already mentioned at the beginning of section [ secergodicquantumsources ] , without loss of generality , we give the proofs for the case that @xmath39 is the algebra of the observables of a qubit , i.e. the complex @xmath59-matrices . for classical _ tm_s , there are no more than @xmath285 different programs of length @xmath286 . this can be used as a counting argument for proving the lower bound of brudno s theorem in the classical case ( @xcite ) . we are now going to prove a similar statement for _ qtm_s . our first step is to elaborate on an argument due to @xcite which states that there can not be more than @xmath287 mutually orthogonal one - dimensional projectors @xmath2 with quantum complexity @xmath288 . the argument is based on holevo s @xmath289-quantity associated to any ensemble @xmath290 consisting of weights @xmath291 , @xmath292 , and of density matrices @xmath293 acting on a hilbert space @xmath89 . setting @xmath294 , the @xmath289-quantity is defined as follows @xmath295 where , in the second line , the relative entropy appears @xmath296 if @xmath297 is finite , ( [ eqholevologdim1 ] ) is bounded by the maximal von neumann entropy : @xmath298 in the following , @xmath299 denotes an arbitrary ( possibly infinite - dimensional ) hilbert space , while the rest of the notation is adopted from subsection [ basicdefqtmsec ] . [ countingargument ] let @xmath300 , @xmath301 such that @xmath302 , @xmath303 an orthogonal projector onto a linear subspace of an arbitrary hilbert space @xmath299 , and @xmath304 a quantum operation . let @xmath305 be a subset of one - dimensional mutually orthogonal projections from the set @xmath306 that is , the set of all pure quantum states which are reproduced within @xmath158 by the operation @xmath307 on some input of length @xmath308 . then it holds that @xmath309 * proof . * let @xmath310 , @xmath311 , be a set of mutually orthogonal projectors and @xmath312 . by the definition of @xmath313 , for every @xmath314 , there are density matrices @xmath315 with @xmath316 consider the equidistributed ensemble @xmath317 , where @xmath318 also acts on @xmath319 . using that @xmath320 , inequality ( [ eqholevologdim3 ] ) yields @xmath321 we define a quantum operation @xmath322 on @xmath323 by @xmath324 . applying twice the monotonicity of the relative entropy under quantum operations , we obtain @xmath325 moreover , for every @xmath326 , the density operator @xmath327 is close to the corresponding one - dimensional projector @xmath328 . indeed , by the contractivity of the trace distance under quantum operations ( compare thm . 9.2 in @xcite ) and by assumption ( [ delta_distance ] ) , it holds @xmath329 let @xmath330 . the trace - distance is convex ( @xcite , ( 9.51 ) ) , thus @xmath331 whence , since @xmath332 , fannes inequality ( compare thm . 11.6 in @xcite ) gives @xmath333 where @xmath334 . combining the two estimates above with ( [ dim_estim ] ) and ( [ chi_estimate ] ) , we obtain @xmath335 assume now that @xmath336 . then it follows ( [ constr ] ) that @xmath337 . so if @xmath338 is larger than this expression , the maximum number @xmath339 of mutually orthogonal projectors in @xmath340 must be bounded by @xmath341 . the second step uses the previous lemma together with the following theorem ( * ? ? ? it is closely related to the quantum shannon - mcmillan theorem and concerns the minimal dimension of the @xmath342typical subspaces . [ qaep ] let @xmath129 be an ergodic quantum source with entropy rate @xmath31 . then , for every @xmath343 , @xmath344 where @xmath345 . notice that the limit ( [ eqboltzmann ] ) is valid for all @xmath346 . by means of this property , we will first prove the lower bound for the finite - accuracy complexity @xmath30 , and then use lemma [ lemrelation ] to extend it to @xmath29 . [ cor1 ] ' '' '' let @xmath129 be an ergodic quantum source with entropy rate @xmath31 . moreover , let @xmath300 , and let @xmath347 be a sequence of @xmath71-typical projectors . then , there is another sequence of @xmath71-typical projectors @xmath348 , such that for @xmath7 large enough @xmath349 is true for every one - dimensional projector @xmath133 . * proof . * the case @xmath350 is trivial , so let @xmath351 . fix @xmath108 and some @xmath300 , and consider the set @xmath352 from the definition of @xmath353 , to all @xmath2 s there exist associated density matrices @xmath354 with @xmath355 such that @xmath356 , where @xmath357 denotes the quantum operation @xmath358 of the corresponding _ uqtm _ @xmath3 , as explained in lemma [ lemmaqtmsareoperations ] . using the notation of lemma [ countingargument ] , it thus follows that @xmath359 let @xmath360 be a sum of a maximal number of mutually orthogonal projectors from @xmath361 . if @xmath7 was chosen large enough such that @xmath362 is satisfied , lemma [ countingargument ] implies that @xmath363 and there are no one - dimensional projectors @xmath364 such that @xmath365 , namely , one - dimensional projectors @xmath366 must satisfy @xmath367 . since inequality ( [ eqcountargused ] ) is valid for every @xmath368 large enough , we conclude @xmath369 using theorem [ qaep ] , we obtain that @xmath370 . finally , set @xmath371 . the claim follows . [ cor2 ] ' '' '' let @xmath129 be an ergodic quantum source with entropy rate @xmath31 . let @xmath347 with @xmath372 be an arbitrary sequence of @xmath71-typical projectors . then , for every @xmath373 , there is a sequence of @xmath71-typical projectors @xmath348 such that for @xmath7 large enough @xmath374 is satisfied for every one - dimensional projector @xmath133 . * according to corollary [ cor1 ] , for every @xmath375 , there exists a sequence of @xmath71-typical projectors @xmath376 with @xmath377 for every one - dimensional projector @xmath378 if @xmath7 is large enough . we have @xmath379 where the first estimate is by lemma [ lemrelation ] , and the second one is true for one - dimensional projectors @xmath380 and @xmath368 large enough . fix a large @xmath137 satisfying @xmath381 . the result follows by setting @xmath382 . in the previous section , we have shown that with high probability and for large @xmath383 , the finite - accuracy complexity rate @xmath384 is bounded from below by @xmath385 , and the approximation - scheme quantum complexity rate @xmath386 by @xmath387 . we are now going to establish the upper bounds . [ propupperbound ] ' '' '' let @xmath129 be an ergodic quantum source with entropy rate @xmath31 . then , for every @xmath300 , there is a sequence of @xmath71-typical projectors @xmath388 such that for every one - dimensional projector @xmath389 and @xmath383 large enough @xmath390 we prove the above proposition by explicitly providing a quantum algorithm ( with program length increasing like @xmath391 ) that computes @xmath34 within arbitrary accuracy . this will be done by means of quantum universal typical subspaces constructed by kaltchenko and yang in @xcite . [ kaltc ] ' '' '' let @xmath392 and @xmath393 . there exists a sequence of projectors @xmath394 , @xmath368 , such that for @xmath7 large enough @xmath395 and for every ergodic quantum state @xmath396 with entropy rate @xmath397 it holds that @xmath398 we call the orthogonal projectors @xmath399 in the above theorem universal typical projectors at level @xmath31 . suited for designing an appropriate quantum algorithm , we slightly modify the proof given by kaltchenko and yang in @xcite . * let @xmath400 and @xmath401 . we consider an abelian quasi - local subalgebra @xmath402 constructed from a maximal abelian @xmath403block subalgebra @xmath404 . the results in @xcite imply that there exists a universal sequence of projectors @xmath405 with @xmath406 such that @xmath407 for any ergodic state @xmath14 on the abelian algebra @xmath408 with entropy rate @xmath409 . notice that ergodicity and entropy rate of @xmath14 are defined with respect to the shift on @xmath410 , which corresponds to the @xmath411-shift on @xmath55 . the first step in @xcite is to apply unitary operators of the form @xmath412 , @xmath413 unitary , to the @xmath414 and to introduce the projectors @xmath415 let @xmath416 be a spectral decomposition of @xmath417 ( with @xmath418 some index set ) , and let @xmath419 denote the orthogonal projector onto a given subspace @xmath420 . then , @xmath421 can also be written as @xmath422 it will be more convenient for the construction of our algorithm in [ subsubconstr ] to consider the projector @xmath423 it holds that @xmath424 . for integers @xmath425 with @xmath102 and @xmath426 we introduce the projectors in @xmath427 @xmath428 we now use an argument of @xcite to estimate the trace of @xmath429 . the dimension of the symmetric subspace @xmath430 is upper bounded by @xmath431 , thus @xmath432 now we consider a stationary ergodic state @xmath71 on the quasi - local algebra @xmath68 with entropy rate @xmath433 . let @xmath434 . if @xmath411 is chosen large enough then the projectors @xmath435 , where @xmath436 , are @xmath437typical for @xmath71 , i.e. @xmath438 , for @xmath439 sufficiently large . this can be seen as follows . due to the result in ( * ? ? ? 3.1 ) the ergodic state @xmath71 convexly decomposes into @xmath440 states @xmath441 each @xmath442 being ergodic with respect to the @xmath403shift on @xmath68 and having an entropy rate ( with respect to the @xmath403shift ) equal to @xmath443 . we define for @xmath444 the set of integers @xmath445 then , according to a density lemma proven in ( * ? ? ? * lemma 3.1 ) it holds @xmath446 let @xmath447 be the maximal abelian subalgebra of @xmath448 generated by the one - dimensional eigenprojectors of @xmath449 . the restriction of a component @xmath442 to the abelian quasi - local algebra @xmath450 is again an ergodic state . it holds in general @xmath451 for @xmath452 , where we set @xmath453 , we additionally have the upper bound @xmath454 . let @xmath455 be a unitary operator such that @xmath456 . for every @xmath452 , it holds that @xmath457 we fix an @xmath458 large enough to fulfill @xmath459 and use the ergodic decomposition ( [ erg_decomp ] ) to obtain the lower bound @xmath460 from ( [ erg_comp ] ) we conclude that for @xmath7 large enough @xmath461 we proceed by following the lines of @xcite by introducing the sequence @xmath462 , @xmath439 , where each @xmath462 is a power of @xmath28 fulfilling the inequality @xmath463 let the integer sequence @xmath464 and the real - valued sequence @xmath465 be defined by @xmath466 and @xmath467 . then we set @xmath468 observe that @xmath469 where the second inequality is by estimate ( [ dim_estimate ] ) and the last one by the bounds on @xmath464 @xmath470 thus , for large @xmath383 , it holds @xmath471 by the special choice ( [ l_m ] ) of @xmath462 it is ensured that the sequence of projectors @xmath472 is indeed typical for any quantum state @xmath71 with entropy rate @xmath397 , compare @xcite . this means that @xmath473 is a sequence of universal typical projectors at level @xmath31 . we proceed by applying the latter result to universal typical subspaces for our proof of the upper bound . let @xmath474 be an arbitrary real number such that @xmath475 is rational , and let @xmath476 be the universal projector sequence of theorem [ kaltc ] . recall that the projector sequence @xmath477 is _ independent _ of the choice of the ergodic state @xmath71 , as long as @xmath478 . because of ( [ eqtraceissmall ] ) , for @xmath383 large enough , there exists some unitary transformation @xmath479 that transforms the projector @xmath477 into a projector belonging to @xmath480 , thus transforming every one - dimensional projector @xmath481 into a qubit string @xmath482 of length @xmath483 . as shown in @xcite , a _ uqtm _ can implement every classical algorithm , and it can apply every unitary transformation @xmath484 ( when given an algorithm for the computation of @xmath484 ) on its tapes within any desired accuracy . we can thus feed @xmath485 ( plus some classical instructions including a subprogram for the computation of @xmath484 ) as input into the _ uqtm _ @xmath283 . this _ uqtm _ starts by computing a classical description of the transformation @xmath484 , and subsequently applies @xmath484 to @xmath485 , recovering the original projector @xmath486 on the output tape . since @xmath487 depends on @xmath71 only through its entropy rate @xmath126 , the subprogram that computes @xmath484 does not have to be supplied with additional information on @xmath71 and will thus have fixed length . we give a precise definition of a quantum decompression algorithm @xmath488 , which is , formally , a mapping ( @xmath489 is rational ) @xmath490 we require that @xmath488 is a short algorithm in the sense of short in description , _ not _ short ( fast ) in running time or resource consumption . indeed , the algorithm @xmath488 is very slow and memory consuming , but this does not matter , since kolmogorov complexity only cares about the description length of the program . the instructions defining the quantum algorithm @xmath488 are : * read the value of @xmath383 , and find a solution @xmath491 for the inequality @xmath492 such that @xmath411 is a power of two . ( there is only one such @xmath411 . ) * compute @xmath493 . * read the value of @xmath489 . compute @xmath494 . * compute a list of codewords @xmath495 , belonging to a classical universal block code sequence of rate @xmath496 . ( for the construction of an appropriate algorithm , see ( * ? ? ? * thm . 2 and 1 ) . ) since @xmath497 @xmath498 can be stored as a list of binary strings . every string has length @xmath499 . ( note that the exact value of the cardinality @xmath500 depends on the choice of @xmath495 . ) during the following steps , the quantum algorithm @xmath488 will have to deal with * rational numbers , * square roots of rational numbers , * binary - digit - approximations ( up to some specified accuracy ) of real numbers , * ( large ) vectors and matrices containing such numbers . a classical _ tm _ can of course deal with all such objects ( and so can _ qtm _ ) : for example , rational numbers can be stored as a list of two integers ( containing numerator and denominator ) , square roots can be stored as such a list and an additional bit denoting the square root , and binary - digit - approximations can be stored as binary strings . vectors and matrices are arrays containing those objects . they are always assumed to be given in the computational basis . operations on those objects , like addition or multiplication , are easily implemented . the quantum algorithm @xmath488 continues as follows : * compute a basis @xmath501 of the symmetric subspace @xmath502 this can be done as follows : for every @xmath7-tuple @xmath503 , where @xmath504 , there is one basis element @xmath505 , given by the formula @xmath506 where the summation runs over all @xmath7-permutations @xmath119 , and @xmath507 with @xmath508 a system of matrix units in @xmath448 . + there is a number of @xmath509 different matrices @xmath510 which we can label by @xmath511 . it follows from ( [ eqwillberational ] ) that these matrices have integer entries . + they are stored as a list of @xmath512-tables of integers . thus , this step of the computation is exact , that is without approximations . * for every @xmath513 and @xmath514 , let @xmath515 where @xmath516 denotes the computational basis vector which is a tensor product of @xmath140 s and @xmath141 s according to the bits of the string @xmath517 . compute the vectors @xmath518 one after the other . for every vector that has been computed , check if it can be written as a linear combination of already computed vectors . ( the corresponding system of linear equations can be solved exactly , since every vector is given as an array of integers . ) if yes , then discard the new vector @xmath518 , otherwise store it and give it a number . + this way , a set of vectors @xmath519 is computed . these vectors linearly span the support of the projector @xmath520 given in ( [ eqjoin2 ] ) . * denote by @xmath521 the computational basis vectors of @xmath522 . if @xmath523 , then let @xmath524 , and let @xmath525 . otherwise , compute @xmath526 for every @xmath527 and @xmath528 . the resulting set of vectors @xmath529 has cardinality @xmath530 . + in both cases , the resulting vectors @xmath531 will span the support of the projector @xmath532 . * the set @xmath529 is completed to linearly span the whole space @xmath533 . this will be accomplished as follows : + consider the sequence of vectors @xmath534 where @xmath535 denotes the computational basis vectors of @xmath533 . find the smallest @xmath214 such that @xmath536 can be written as a linear combination of @xmath537 , and discard it ( this can still be decided exactly , since all the vectors are given as tables of integers ) . repeat this step @xmath538 times until there remain only @xmath539 linearly independent vectors , namely all the @xmath540 and @xmath541 of the @xmath542 . * apply the gram - schmidt orthonormalization procedure to the resulting vectors , to get an orthonormal basis @xmath543 of @xmath533 , such that the first @xmath544 vectors are a basis for the support of @xmath532 . + since every vector @xmath540 and @xmath542 has only integer entries , all the resulting vectors @xmath545 will have only entries that are ( plus or minus ) the square root of some rational number . up to this point , every calculation was _ exact _ without any numerical error , comparable to the way that well - known computer algebra systems work . the goal of the next steps is to compute an approximate description of the desired unitary decompression map @xmath484 and subsequently apply it to the quantum state @xmath485 . according to section 6 in @xcite , a _ uqtm _ is able to apply a unitary transformation @xmath484 on some segment of its tape within an accuracy of @xmath158 , if it is supplied with a complex matrix @xmath546 as input which is within operator norm distance @xmath547 of @xmath484 ( here , @xmath548 denotes the size of the matrix ) . thus , the next task is to compute the number of digits @xmath549 that are necessary to guarantee that the output will be within trace distance @xmath550 of @xmath34 . * read the value of @xmath137 ( which denotes an approximation parameter ; the larger @xmath137 , the more accurate the output of the algorithm will be ) . due to the considerations above and the calculations below , the necessary number of digits @xmath549 turns out to be @xmath551 . compute this number . + afterwards , compute the components of all the vectors @xmath543 up to @xmath549 binary digits of accuracy . ( this involves only calculation of the square root of rational numbers , which can easily be done to any desired accuracy . ) + call the resulting numerically approximated vectors @xmath552 . write them as columns into an array ( a matrix ) @xmath553 . + let @xmath554 denote the unitary matrix with the exact vectors @xmath545 as columns . since @xmath549 binary digits give an accuracy of @xmath555 , it follows that @xmath556 if two @xmath557-matrices @xmath484 and @xmath546 are @xmath558-close in their entries , they also must be @xmath559-close in norm , so we get @xmath560 so far , every step was purely classical and could have been done on a classical computer . now , the quantum part begins : @xmath485 will be touched for the first time . * compute @xmath561 , which gives the length @xmath562 . afterwards , move @xmath485 to some free space on the input tape , and append zeroes , i.e. create the state @xmath563 on some segment of @xmath383 cells on the input tape . * approximately apply the unitary transformation @xmath484 on the tape segment that contains the state @xmath564 . + the machine can not apply @xmath484 exactly ( since it only knows an approximation @xmath546 ) , and it also can not apply @xmath546 directly ( since @xmath546 is only approximately unitary , and the machine can only do unitary transformations ) . instead , it will effectively apply another unitary transformation @xmath420 which is close to @xmath546 and thus close to @xmath484 , such that @xmath565 + let @xmath566 be the output that we want to have , and let @xmath567 be the approximation that is really computed by the machine . then , @xmath568 a simple calculation proves that the trace distance must then also be small : @xmath569 * move @xmath570 to the output tape and halt . we have to give a precise definition how the parameters @xmath571 are encoded into a single qubit string @xmath119 . ( according to the definition of @xmath29 , the parameter @xmath137 is not a part of @xmath119 , but is given as a second parameter . see definitions [ defencoding ] and [ defqk ] for details . ) we choose to encode @xmath383 by giving @xmath572 1 s , followed by one 0 , followed by the @xmath573 binary digits of @xmath383 . let @xmath574 denote the corresponding projector in the computational basis . the parameter @xmath489 can be encoded in any way , since it does not depend on @xmath383 . the only constraint is that the description must be self - delimiting , i.e. it must be clear and decidable at what position the description for @xmath489 starts and ends . the descriptions will also be given by a computational basis vector ( or rather the corresponding projector ) @xmath575 . the descriptions are then stuck together , and the input @xmath576 is given by @xmath577 if @xmath383 is large enough such that ( [ eqlogtrqm ] ) is fulfilled , it follows that @xmath578 , where @xmath301 is some constant which depends on @xmath489 , but not on @xmath383 . it is clear that this qubit string can be fed into the reference _ uqtm _ @xmath3 together with a description of the algorithm @xmath488 of fixed length @xmath579 which depends on @xmath489 , but not on @xmath383 . this will give a qubit string @xmath580 of length @xmath581 where @xmath582 is again a constant which depends on @xmath489 , but not on @xmath383 . recall the matrix @xmath484 constructed in step 11 of our algorithm @xmath583 , which rotates ( decompresses ) a compressed ( short ) qubit string @xmath485 back into the typical subspace . conversely , for every one - dimensional projector @xmath481 , where @xmath584 was defined in ( [ eqtypicalprojector ] ) , let @xmath585 be the projector given by @xmath586 . then , since @xmath583 has been constructed such that @xmath587 it follows from ( [ eqlength ] ) that @xmath588 if @xmath383 is large enough , equation ( [ upperzero ] ) follows . now we continue by proving equation ( [ upperdelta ] ) . let @xmath589 . then , we have for every one - dimensional projector @xmath481 and @xmath383 large enough @xmath590 where the first inequality follows from the obvious monotonicity property @xmath591 , the second one is by lemma [ lemrelation ] , and the third estimate is due to equation ( [ upperzero ] ) . _ proof of the main theorem [ theqbrudno ] . _ let @xmath592 be the @xmath71-typical projector sequence given in proposition [ propupperbound ] , i.e. the complexities @xmath386 and @xmath593 of every one - dimensional projector @xmath594 are upper bounded by @xmath595 . due to corollary [ cor1 ] , there exists another sequence of @xmath71-typical projectors @xmath596 such that additionally , @xmath597 is satisfied for @xmath598 . from corollary [ cor2 ] , we can further deduce that there is another sequence of @xmath71-typical projectors @xmath599 such that also @xmath600 holds . finally , the optimality assertion is a direct consequence of the quantum counting argument , lemma [ countingargument ] , combined with theorem [ qaep ] . classical algorithmic complexity theory as initiated by kolmogorov , chaitin and solomonoff aimed at giving firm mathematical ground to the intuitive notion of randomness . the idea is that random objects can not have short descriptions . such an approach is on the one hand equivalent to martin - lf s which is based on the notion of _ typicalness _ @xcite , and is on the other hand intimately connected with the notion of entropy . the latter relation is best exemplified in the case of longer and longer strings : by taking the ratio of the complexity with respect to the number of bits , one gets a _ complexity per symbol _ which a theorem of brudno shows to be equal to the _ entropy per symbol _ of almost all sequences emitted by ergodic sources . the fast development of quantum information and computation , with the formalization of the concept of _ uqtms _ , quite naturally brought with itself the need of extending the notion of algorithmic complexity to the quantum setting . within such a broader context , the ultimate goal is again a mathematical theory of the randomness of quantum objects . there are two possible algorithmic descriptions of qubit strings : either by means of bit - programs or of qubit - programs . in this work , we have considered a qubit - based _ quantum algorithmic complexity _ , namely constructed in terms of quantum descriptions of quantum objects . the main result of this paper is an extension of brudno s theorem to the quantum setting , though in a slightly weaker form which is due to the absence of a natural concatenation of qubits . the quantum brudno s relation proved in this paper is not a pointwise relation as in the classical case , rather a kind of convergence in probability which connects the _ quantum complexity per qubit _ with the von neumann entropy rate of quantum ergodic sources . possible strengthening of this relation following the strategy which permits the formulation of a quantum breiman theorem starting from the quantum shannon - mcmillan noiseless coding theorem @xcite will be the matter of future investigations . in order to assert that this choice of quantum complexity as a formalization of quantum randomness " is as good as its classical counterpart in relation to classical randomness , one ought to compare it with the other proposals that have been put forward : not only with the quantum complexity based on classical descriptions of quantum objects @xcite , but also with the one based on the notion of _ universal density matrices _ @xcite . in relation to vitanyi s approach , the comparison essentially boils down to understanding whether a classical description of qubit strings requires more classical bits than @xmath31 qubits per hilbert space dimension . an indication that this is likely to be the case may be related to the existence of entangled states . in relation to gacs approach , the clue is provided by the possible formulation of quantum martin - lf tests in terms of measurement processes projecting onto low - probability subspaces , the quantum counterparts of classical untypical sets . one can not however expect classical - like equivalences among the various definitions . it is indeed a likely consequence of the very structure of quantum theory that a same classical notion may be extended in different inequivalent ways , all of them reflecting a specific aspect of that structure . this fact is most clearly seen in the case of quantum dynamical entropies ( compare for instance @xcite ) where one definition can capture dynamical features which are precluded to another . therefore , it is possible that there may exist different , equally suitable notions of `` quantum randomness '' , each one of them reflecting a different facet of it . this work was supported by the dfg via the project `` entropie , geometrie und kodierung groer quanten - informationssysteme '' and the dfg - forschergruppe `` stochastische analysis und groe abweichungen '' at the university of bielefeld . a. k. zvonkin , l. a. levin , `` the complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms '' , _ russian mathematical surveys _ * 25 no . 6 * 83 - 124 ( 1970 )
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in classical information theory , entropy rate and algorithmic complexity per symbol are related by a theorem of brudno . in this paper , we prove a quantum version of this theorem , connecting the von neumann entropy rate and two notions of quantum kolmogorov complexity , both based on the shortest qubit descriptions of qubit strings that , run by a universal quantum turing machine , reproduce them as outputs .
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let @xmath0 be a reiamannian 2-sphere . denote the area of @xmath0 by @xmath6 . in this paper we consider the problem of slicing @xmath0 by short curves . we start with the following isoperimetric problem : when is it possible to subdivide @xmath0 into two regions of relatively large area by a short simple closed curve ? papasoglu @xcite used besicovitch inequality to show that there exists a simple closed curve of length @xmath7 subdividing @xmath0 into two regions of area @xmath8 . a similar result was independently proved by balacheff and sabourau @xcite using a variation of gromov s filling argument . on the other hand , consider the 3-legged starfish example on figure [ starfish ] . as it has been observed in @xcite , for any @xmath9 , if the tentacles are sufficiently thin and long , the length of the shortest simple closed curve subdividing @xmath0 into two regions of area @xmath10 can be arbitrarily large . a. nabutovsky asked ( @xcite ) the following question : what is the maximal value of @xmath11 $ ] such that for some @xmath12 each riemannian 2-sphere of area @xmath13 can be subdivided into two discs of area @xmath14 by a simple closed curve of length @xmath15 ? our first result provides an answer for this question . [ 1/3 area ] there exists a simple closed curve @xmath16 of length @xmath17 subdividing @xmath0 into two subdiscs of area @xmath18 . to prove theorem [ 1/3 area ] we obtain the following result of independent interest . [ tree ] there exists a map @xmath19 from @xmath0 into a trivalent tree @xmath20 , such that fibers of @xmath19 have length @xmath17 and controlled topology : preimage of every interior point is a simple closed curve , preimage of every terminal vertex is a point and preimage of every vertex of degree @xmath21 is homeomorphic to the greek letter @xmath22 . theorem [ tree ] follows from a more general theorem [ tree1 ] in section 3 for spheres with @xmath1 holes . using different methods guth @xcite proved existence of a map from @xmath0 into a trivalent tree with lengths of fibers bounded in terms of hypersphericity of @xmath0 . it follows from theorem 0.3 in @xcite that there exists such a map with fibers of length @xmath23 . the second main result of this paper is about slicing @xmath0 by @xmath24cycles . if instead of simple closed curves we allow subdividison by 1-cycles , we show that @xmath0 can be subdivided into two regions with arbitrary prescribed ratio of areas by a 1-cycle of length @xmath25 . in fact , we prove the following [ morse ] there exists a morse function @xmath2 with fibers of length @xmath25 . this improves the result of balacheff and sabourau @xcite that there exists a sweep - out of @xmath0 by @xmath24cycles of length @xmath26 . theorem [ morse ] follows from a more general result in section 4 , where we prove existence of a morse function on a sphere with @xmath27 holes , such that the function is constant on each connected component of the boundary and the length of fibers are bounded in terms of area and boundary length . alvarez paiva , balacheff and tzanev @xcite show that existence of a morse function on a riemannian 2-sphere with bounded fibers yields a length - area inequality for the shortest periodic geodesic on a finsler 2-sphere ( for both reversible and non - reversible metrics ) . note that arguments used by croke @xcite to prove the length - area bound for the shortest closed geodesic on a riemannian 2-sphere ( see also @xcite for the best known constant ) can not be directly generalized to the finsler case because co - area inequality fails for non - reversible finsler metrics . hence , theorem [ morse ] yields a better constant for theorem vi in @xcite . [ finsler ] let @xmath0 be a finsler two - sphere with holmes - thompson area @xmath28 . then @xmath0 carries a closed geodesic of length @xmath29 . the reason for constants @xmath30 and @xmath31 in our theorems is the following . we obtain the desired slicing of the sphere by repeatedly using the result of papasoglu @xcite to subdivide the sphere into smaller regions by a curve of length at most @xmath32 times the square root of the area of the region . at each step the area of the region reduces at least by a factor of @xmath33 . we then assemble these subdividing curves into one foliation with lengths bounded by the geometric progression @xmath34 in the proof of theorem [ morse ] some subdividng curves are used twice so an additional factor of @xmath35 appears . after the first subdivision happens our regions are no longer spheres , but rather spheres with a finite number of holes . it may not be possible to find a short simple closed curve subdividing it into two parts of area @xmath36 of its area . instead we may have to use a collection of arcs with endpoints on the boundary subdividng the region into many pieces , each of small area . the issue is then how to assemble all of these subdividing curves into one foliation . the main technical result of this paper ( proved in the next section , see proposition [ subdivision ] ) is that we can always choose these subdividing arcs in such a way that they belong to a single connected component of the boundary of a certain subregion @xmath37 with area of @xmath37 between @xmath38 and @xmath33 of the area of the region . this result make assembling curves into one foliation a straightforward procedure . after the first version of this article appeared on the web , f. balacheff in @xcite improved constant @xmath30 in theorems 1 and 2 to @xmath39 , and constant @xmath40 in theorem 4 to @xmath41 , but not constant @xmath31 in theorem 3 . * acknowledgements . * the author is grateful to alexander nabutovsky for reading the first draft of this paper and to juan carlos alvarez paiva , florent balacheff , gregory chambers and regina rotman for valuable discussions . the author would like to thank the referee for helpful comments . the author acknowledges the support by natural sciences and engineering research council ( nserc ) cgs graduate scholarship , and by ontario graduate scholarship . let @xmath0 be a riemannian 2-sphere . for @xmath42 let @xmath43 denote the set of simple closed curves on @xmath0 that divide it into subdiscs of area @xmath44 . define @xmath45 and @xmath46 , where the supremum is taken over all metrics on @xmath47 of area @xmath13 . by definition @xmath12 is increasing and by proposition [ semicontinuity ] below it is lower semicontinuous . for any @xmath9 it follows from the example in the introduction ( see figure [ starfish ] ) that @xmath48 . for @xmath49 we have the following result of papasoglu . [ papasoglu ] ( papasoglu , @xcite ) @xmath50 we will need to generalize theorem [ papasoglu ] to spheres with finitely many holes and allow a larger class of subdividing curves than just simple closed curves . when the surface has boundary we will allow the subdividing curve @xmath16 to consist of several arcs with endpoints on the boundary . in this case we will define a distinguished connected component @xmath37 of @xmath51 and require that @xmath16 is contained in a connected component of @xmath52 . this is a technical condition that will make it easer to repeatedly cut the surface into smaller pieces and concatenate the subdivding curves to obtain a slicing of @xmath0 . let @xmath53 be a complete riemannian 2-surface with boundary homeomorphic to a sphere with @xmath27 holes . let @xmath16 be a simple closed curve in the interior of @xmath53 or a union of finitely many arcs @xmath54 , where @xmath55 are arcs with endpoints on @xmath56 that do not pairwise intersect and have no self - intersections . let @xmath57 be the set of connected components of @xmath58 . let @xmath59 denote the set of all such @xmath16 on @xmath53 that in addition satisfy 1 . @xmath60 2 . @xmath16 is contained in a connected component of @xmath52 in particular , ( 1 ) implies that the area of every connected component of @xmath58 is bounded from above by @xmath61 . define @xmath62 and @xmath63 , where the supremum is taken over all metrics on a sphere with @xmath27 holes that have area @xmath13 . we have the following useful fact . [ semicontinuous ] @xmath64 and @xmath12 are lower semi - continuous for @xmath65 $ ] . we prove the result for @xmath64 and for @xmath12 it will follow as a special case from the argument below . let @xmath66 be an increasing sequence converging to @xmath67 for some @xmath68 . fix @xmath69 . let @xmath53 be a complete riemannian surface of area @xmath13 diffeomorphic to a sphere with @xmath27 holes . we would like to show that for some @xmath70 there exists @xmath71 with @xmath72 . we can find @xmath73 small enough so that it satisfies the following requirements : 1 . the area of the @xmath74-tubular neighbourhood of @xmath75 satisfies @xmath76 . any ball @xmath77 around @xmath78 is bilipschitz diffeomorphic to the euclidean disc of radius @xmath74 with lipschitz constant between @xmath79 and @xmath80 . in particular , @xmath81 and @xmath82 . 3 . @xmath83 for such a @xmath74 choose @xmath70 so that @xmath84 . let @xmath85 be the subdividing arcs of length @xmath86 . in the following argument it will be more convenient to consider @xmath87 , the connected component of @xmath52 that contains @xmath16 ( recall that by definition of @xmath64 @xmath16 lies in a single boundary component of @xmath52 ) . if @xmath16 is a closed curve then @xmath88 . if @xmath16 is a union of arcs then @xmath87 is a closed curve made out of arcs of @xmath16 and arcs of @xmath75 . let @xmath89 denote the element of @xmath90 of smaller area and @xmath91 denote the element of larger area . we can assume that @xmath92 for otherwise we are done . therefore , we have @xmath93 and @xmath94 . let @xmath95 denote @xmath96 . by our choice of @xmath74 we have that the area of a ball @xmath97 for @xmath98 and @xmath99 . by fubini s theorem we obtain @xmath100 . hence , for some @xmath101 we have @xmath102 . since @xmath103 is non - empty we can always find such a ball so that @xmath104 is non - empty . we will now construct a new curve @xmath105 that coincides with @xmath87 outside of @xmath106 and divides @xmath53 into regions @xmath107 and @xmath108 so that one of them is connected and each of them has area @xmath14 . moreover , @xmath109 . this implies the desired inequality @xmath110 . we construct @xmath105 by cutting @xmath87 at the points of intersection with @xmath111 and attaching arcs of @xmath111 . we do it in such a way that @xmath77 is now entirely contained in the smaller of two regions . this increases the area of the smaller region by at least @xmath112 and increases the length of the subdividing curve by at most @xmath113 . the procedure is illustrated in figure [ semicontinuity ] . let @xmath114 be connected components of @xmath115 and @xmath116 denote connected components of @xmath117 . first we erase all arcs of @xmath87 that are in @xmath77 . for each @xmath118 we erase the arc @xmath119 from @xmath111 . for each @xmath120 , @xmath121 , we add a copy of @xmath120 and perturb it so that the new curve @xmath105 does not intersect @xmath77 in the neighbourhood of @xmath120 . this does not increase the number of connected components of either region . the only example that i know where upper semicontinuity of @xmath12 fails is the three - legged starfish on figure [ starfish ] ( showing @xmath48 for @xmath9 , while @xmath122 by theorem [ 1/3 area ] ) . the following question seems natural : * question * is @xmath12 continuous for @xmath123 ? it can be easily shown from the definition that @xmath124 . under the additional assumption @xmath125 we are able to prove that they are equal . [ subdivision ] suppose @xmath125 , then @xmath126 . in the proof of proposition [ subdivision ] we will use the following topological fact . [ 2-gon ] let @xmath0 be a submanifold ( with boundary ) of @xmath47 and let @xmath16 be a simple closed curve in @xmath47 . suppose the intersection of @xmath16 and @xmath75 is non - empty and transversal . if @xmath28 denotes a connected component of @xmath127 then there exists an arc @xmath128 and an arc @xmath129 , such that @xmath130 bounds a disc @xmath131 and @xmath132 is connected . ] the proof of this lemma is the main technical part of the paper . the proof is illustrated on figure [ 2gon_fig ] . the reader can draw several pictures like on figure [ 2gon_fig ] and convince him or herself that the statement is correct . the strategy of the proof is to find some disc @xmath133 , with possibly a large number of components of @xmath134 . we then show that there exists a subdisc of @xmath133 whose intersection with @xmath0 has a smaller number of connected components . let @xmath135 and @xmath136 be the connected component of @xmath75 that contains @xmath137 . define an interval @xmath138 to be a connected component of @xmath139 that has @xmath137 as an enpoint ( note that it is unique ) . we denote the other endpoint of @xmath140 by @xmath141 . let @xmath142 be a tangent vector to @xmath16 at @xmath137 pointing inside @xmath0 . let @xmath143 be an arc of @xmath16 that starts at @xmath137 in the direction @xmath142 and ends at @xmath141 . observe that the curve @xmath144 is a simple closed curve enclosing a disc @xmath145 that contains a non - empty subset of @xmath146 . if @xmath147 has one connected component then we are done . assume it has more than one . let @xmath37 be a component of @xmath148 with @xmath149 and let @xmath150 be a different component . define a point @xmath151 by @xmath152 it follows from the definition that @xmath153 . we can find a point @xmath154 and an arc @xmath155 from @xmath156 to @xmath157 . the interior of @xmath158 is contained in the interior of @xmath159 . it follows that @xmath160 . denote the arc of @xmath143 between @xmath156 and @xmath157 by @xmath161 . we have that @xmath162 separates @xmath47 into a disc @xmath163 that contains @xmath150 and its complement that contains @xmath37 . set @xmath164 . it is non - empty and has fewer connected components than @xmath148 . we iterate this procedure until we are left with just one connected component . we now prove proposition [ subdivision ] . the direction @xmath124 is simple . given a riemannian 2-sphere we can make @xmath27 holes in it of small area and small boundary length . we subdivide the resulting sphere with holes and use the fact that the connected component of @xmath52 that contains @xmath16 will have length close to @xmath64 . here we did not use that @xmath165 . to prove the other direction we proceed as follows . let @xmath166 be a small positive constant . given a sphere with @xmath27 holes @xmath53 of area @xmath13 we attach @xmath27 discs of total area @xmath166 to the boundary of @xmath53 . we obtain a riemannian 2-sphere @xmath0 . let @xmath167 be a simple closed curve of length @xmath168 subdividing @xmath0 into two subdiscs @xmath28 and @xmath169 of area between @xmath170 and @xmath171 . suppose @xmath28 is a subdisc of area @xmath172 . if @xmath167 is disjoint from @xmath56 we conclude that @xmath173 . suppose @xmath167 intersects @xmath56 . by lemma [ 2-gon ] there exists an arc @xmath143 of @xmath167 and an arc @xmath140 of @xmath56 , such that @xmath174 is connected . we consider two possibilities . first , suppose @xmath175 . let @xmath176 be the intersection of @xmath143 with the interior of @xmath53 . @xmath37 is a connected component of @xmath177 of area between @xmath178 and @xmath179 . the rest of connected components of @xmath177 have area less than @xmath179 . so @xmath180 . alternatively , suppose @xmath181 . in this case we can define a new curve @xmath176 , such that the number of connected components of @xmath177 is smaller than the number of connected components of @xmath182 . we do this by replacing the arc @xmath143 of @xmath167 by @xmath183 . note that we can slightly perturb the part of the new curve that coincides with @xmath140 so that it is entirely in @xmath184 , in particular , the intersection of @xmath176 with @xmath56 is transversal and the length of the intersection of @xmath176 with the interior of @xmath53 is smaller than that of @xmath167 . as a result we transferred the area of @xmath37 from @xmath28 to @xmath169 . since @xmath185 , @xmath125 and @xmath186 we obtain that the area of each of the two discs @xmath187 is at least @xmath170 . in this way we can continue reducing the number of connected components of @xmath188 until one of the subdiscs of @xmath189 contains only one connected component of @xmath53 or until we encounter the first possibility above . by proposition [ semicontinuous ] we conclude that @xmath190 . a map @xmath19 from @xmath53 to a trivalent tree @xmath20 is called a @xmath20-map if the topology of fibers of @xmath19 is controlled in the following way : the preimage of any point in an edge of @xmath20 is a circle , there exist @xmath27 terminal vertices @xmath191 , such that @xmath192 is a connected component of @xmath56 , the preimage of any other terminal point of @xmath20 is a point , and the preimage of a trivalent vertex of @xmath20 is homeomorphic to the greek letter @xmath22 . [ tree1 ] for @xmath193 $ ] and any @xmath69 there exists a @xmath194map @xmath19 from @xmath195 , @xmath196 , so that each fiber of the map has length less than @xmath197 . theorem [ tree ] follows by taking @xmath198 and applying theorem [ papasoglu ] . in the proof we will repeatedly use the following simple fact , so it is convenient to state it as a separate lemma . [ concatenation ] let @xmath37 and @xmath150 be two closed smooth submanifolds ( with boundary ) of @xmath195 , such that @xmath199 is a connected arc . let @xmath200 denote the connected component of @xmath201 that contains @xmath202 . suppose @xmath203 and that each @xmath204 admits a @xmath194map with fibers of length @xmath205 , then @xmath206 admits a @xmath194map with fibers of length @xmath207 . ] from the assumption that @xmath204 admits a @xmath194map it follows that there exists an embedded cylinder @xmath208 and a map @xmath209 $ ] with fiber @xmath210 and the length of all fibers @xmath205 . the boundary @xmath211 with @xmath212 contained in the interior of @xmath204 . let @xmath143 and @xmath161 be the endpoints of @xmath202 . for a sufficiently small @xmath73 we perform a surgery on the the closed curves in the foliation @xmath213 . the surgery happens in the @xmath214neighbourhood of @xmath215 and is depicted on figure [ concat_fig ] . as a result of the surgery we obtain three families of curves . the outer " family converging to @xmath216 , and two inner " families each converging to @xmath217 or @xmath218 . these three families are separated by a @xmath22 graph ( drawn in red on figure [ concat_fig ] ) . this surgery defines the desired @xmath194map . we will need first a version of theorem [ tree1 ] for very small balls . [ small length ] for any @xmath219 there exists @xmath220 , such that for every disc @xmath221 with @xmath222 there exists a diffeomorphism @xmath19 from @xmath223 to the standard closed disc @xmath224 so that the preimage of each concentric circle @xmath225 , has length @xmath226 . for @xmath73 sufficiently small every ball @xmath227 of radius @xmath74 is @xmath228bilipschitz diffeomorphic to a disc in the closed upper half - plane @xmath229 . let @xmath230 denote the image of @xmath223 under such a diffeomorphism . ( here we are assuming that @xmath231 and @xmath223 is contained within a ball of radius @xmath74 ) . after a small perturbation we may assume that the projection @xmath232 of @xmath233 onto @xmath234 coordinate is a morse function . define a 1-dimensional simplicial complex @xmath235 as follows . let @xmath236 be a regular value of the projection function @xmath237 restricted to the boundary of @xmath230 . @xmath238 is a finite union of disjoint closed intervals @xmath239 . we set @xmath240 to be the midpoints of @xmath241s . if @xmath236 and @xmath242 are two consecutive critical values of @xmath232 , it follows that @xmath243 is a collection of disjoint simple arcs as on figure [ fig_small length ] . at a critical point @xmath233 locally looks like the graph of a function @xmath244 . we connect the endpoints of the intervals of @xmath20 by a horizontal arc tangent to the critical point and contained in @xmath230 . note that @xmath230 retracts onto @xmath20 , so in particular @xmath20 must be connected and simply connected , hence a tree . we contract @xmath223 along the edges of @xmath20 in the obvious way . as a result we obtain a contraction of @xmath245 inside @xmath223 to a point through curves of length @xmath226 . after a small perturbation we can assume that this homotopy realizes the desired diffeomorphism . for details we refer the reader to @xcite , where it is shown that if there exists a homotopy of the boundary of @xmath223 to a point through curves of length @xmath246 , then there exists a diffeomorphism from @xmath223 to @xmath247 so that preimages of concentric circles have length @xmath246 . in @xmath230 ] [ small area_s^2 ] for any @xmath219 there exists @xmath249 , such that for every disc @xmath221 with @xmath250 there exists a @xmath194map @xmath19 from @xmath223 with fibers of length less than @xmath251 . the proof is similar to that of lemma 2.2 in @xcite . choose @xmath252 , where @xmath253 is as in lemma [ small length ] . the proof is by induction on @xmath254 . for @xmath255 the result follows by lemma [ small length ] . assume the lemma to be true for all subdiscs with @xmath256 and consider the case when this quantity equals @xmath257 . subdivide @xmath245 into 4 arcs of equal length . by besicovitch lemma we can find an arc @xmath202 of length @xmath258 connecting two opposite arcs . @xmath202 subdivides @xmath223 into two subdiscs @xmath259 and @xmath260 of area @xmath261 and boundary length @xmath262 . hence , by inductive assumption @xmath259 and @xmath260 admit @xmath194maps with fibers of length @xmath263 . by inductive assumption each disc admits a @xmath194map with fibers of length @xmath264 . we also have @xmath265 so the result follows by lemma [ concatenation ] . we need to generalize this result about small discs to other small submanifolds of @xmath195 . [ small area_mk ] for any @xmath219 there exists @xmath249 , such that for every submanifold with boundary @xmath266 , with @xmath267 there exists a @xmath194map @xmath19 from @xmath53 with fibers of length @xmath268 . let @xmath53 be a closed submanifold ( with boundary ) of @xmath195 and let @xmath269 be a connected component of @xmath56 . if @xmath270 then there exists an open ball @xmath271 of radius @xmath272 whose interior does not intersect the boundary of @xmath53 ( and , in particular , it does not intersect the boundary of @xmath195 ) . as @xmath273 the area of @xmath274 approaches the area of a euclidean disc of the same diameter . since @xmath195 is compact , this happens uniformly for all balls of radius @xmath272 disjoint from the boundary . for @xmath275 sufficiently small we may conclude that @xmath276 . hence , for a sufficiently small @xmath28 , if @xmath277 then the distance @xmath278 . we attach @xmath279 arcs @xmath280 to the boundary of @xmath56 of total length less than @xmath281 and so that @xmath282 is connected and its complement in @xmath53 is homeomorphic to a disc . denote this disc by @xmath223 . consider the normal @xmath214neighbourhood @xmath283 of @xmath245 in @xmath53 ( see figure [ fig_small area ] ) for some small @xmath74 . by lemma [ small area_s^2 ] the complement of @xmath283 in @xmath53 admits a @xmath20 map with fibers of length @xmath268 . let @xmath202 be a short closed curve in @xmath283 which separates one connected component of @xmath56 from other connected components as on figure [ fig_small area ] . curve @xmath202 separates @xmath283 into two regions . let @xmath169 denote the region that contains only one connected component of @xmath56 . by lemma [ concatenation ] we can extend the @xmath194map to @xmath284 . by chopping off connected components of @xmath56 and applying lemma [ concatenation ] repeatedly we obtain the desired @xmath194map . ] the proof of theorem [ tree1 ] proceeds inductively by cutting @xmath0 into smaller pieces until their size is small enough so that lemma [ small area_mk ] can be applied . we assemble @xmath194maps on these smaller regions to obtain one map from @xmath0 with the desired bound on lengths of fibers . fix @xmath285 and let @xmath28 be as in lemma [ small area_mk ] . let @xmath286 . we claim that for every @xmath266 with @xmath287 there exists a @xmath194map with fibers of length @xmath288 when @xmath289 the inequality ( [ inductive statement ] ) is true by lemma [ small area_mk ] . we assume it to be true for @xmath290 and prove it for @xmath291 . by proposition [ subdivision ] there exists @xmath292 of length @xmath293 . we have two possibilities . * * @xmath16 is a simple closed curve . in this case @xmath16 separates @xmath53 into two regions @xmath294 and @xmath295 of area @xmath296 . the number of connected components of @xmath297 is at most @xmath298 . by inductive assumption @xmath294 and @xmath295 admit @xmath194maps into trees @xmath299 and @xmath300 respectively . we construct @xmath20 by identifying the terminal vertex @xmath301 of @xmath299 with the terminal vertex @xmath302 of @xmath300 . @xmath194map @xmath19 is defined by setting it equal to @xmath303 when restricted to @xmath304 . a simple calculation shows that lengths of fibers of @xmath19 satisfy the desired bound . * case 2 . * @xmath16 is a collection of arcs with endpoints on @xmath56 . let @xmath305 be connected components of @xmath56 that intersect arcs of @xmath16 . for a sufficiently small @xmath74 the normal neighbourhood @xmath306 is foliated by closed curves of length very close to @xmath307 , which are transverse to the arcs of @xmath16 . in particular , each @xmath306 admits a @xmath194map with fibers of length @xmath308 . let @xmath37 be the connected component of @xmath51 with area satisfying @xmath309 ( recall that it exists by definition of @xmath64 ) . let @xmath169 denote @xmath310 and @xmath311 denote @xmath312 . ( see figure [ fig_horseshoe ] ) . ] let @xmath313 be the connected component of @xmath314 that contains @xmath315 . we say that an arc @xmath202 of @xmath16 is a horseshoe if @xmath202 is in @xmath311 and the endpoints of @xmath202 lie on the same connected component of @xmath316 . we will use the following simple observation . [ horseshoe ] if @xmath16 contains no horseshoes , then there exists a connected component @xmath317 of @xmath316 , such that @xmath318 is connected . let @xmath317 be a connected component of @xmath316 and suppose @xmath318 contains more than one interval . let @xmath319 be an interval of @xmath320 and let @xmath236 and @xmath242 denote the endpoints of @xmath319 . @xmath321 consists of two arcs , call them @xmath136 and @xmath322 . let @xmath323 be a connected component of @xmath316 that intersects @xmath136 . we claim that @xmath322 does not intersect @xmath324 . for suppose it does , consider then a subarc @xmath325 of @xmath136 from @xmath236 until the fist point of intersection with @xmath324 and denote this point by @xmath143 . similarly , denote by @xmath326 the subarc of @xmath322 from @xmath236 until the fist point of intersection with @xmath324 and call it @xmath161 . let @xmath327 be an arc of @xmath324 from @xmath143 to @xmath161 , then @xmath328 is a closed curve separating @xmath311 into two connected components . moreover , the point @xmath242 and points of @xmath329 belong to different connected components of @xmath330 . this is a contradiction since @xmath169 is connected . suppose the intersection @xmath331 is not connected . we can then find a subarc @xmath332 that intersects @xmath324 only at the endpoints . the number of connected components of @xmath316 that intersect @xmath333 is strictly smaller than the number of components that intersect @xmath136 . proceeding in this way we can find an arc of @xmath313 that has endpoints on @xmath334 , and its interior intersects only one connected component @xmath335 of @xmath316 . it follows that @xmath336 is connected . we can now construct the desired @xmath194map . by inductive assumption there exists a @xmath194map on @xmath169 with fibers @xmath337 . note that @xmath338 and the number of connected components @xmath339 of @xmath314 satisfies @xmath340 . suppose first that @xmath341 is a horseshoe . it separates @xmath311 into two connected components . let @xmath91 denote the connected component that does not contain @xmath169 . since @xmath342 , it admits a @xmath194map with the desired bound on length of fibiers . by lemma [ concatenation ] we can extend the @xmath194map to @xmath343 . inductively we extend the @xmath194map to a subset @xmath344 , such that @xmath345 and @xmath346 does not contain any horseshoes . by lemma [ horseshoe ] @xmath347 is connected for some @xmath348 . by lemma [ concatenation ] we can extend the @xmath194map to @xmath349 . we iterate this procedure until we extended the @xmath194map to a set @xmath350 , which contains @xmath306 for all @xmath348 . the last step is to extend the @xmath194map to the whole of @xmath53 exactly as we did it in case 1 . to finish the proof of theorem [ tree1 ] recall that @xmath351 as @xmath352 . since @xmath28 can be chosen arbitrarily small theorem [ tree1 ] follows from ( [ inductive statement ] ) . now we can prove theorem [ 1/3 area ] . let @xmath0 be a riemannian 2-sphere and @xmath353 a @xmath194map with fibers of length @xmath17 . if there are no trivalent vertices in @xmath20 we can find a short closed curve that separates @xmath0 into two halves of equal area . for each trivalent vertex @xmath354 let @xmath355 . denote the closed curve @xmath356 by @xmath357 . assume that for every @xmath358 ( @xmath359 ) @xmath357 separates @xmath0 into two discs , s.t . the area of the smaller disc is strictly smaller than @xmath360 . let @xmath358 be such that the area of the smaller disc is maximized among all such curves . let @xmath361 denote the smaller and @xmath362 denote the larger of the subdiscs of @xmath363 . let @xmath364 denote the edge of @xmath20 adjacent to @xmath365 , such that @xmath366 . we observe that the other two edges adjacent to @xmath365 are contained in @xmath367 , for otherwise it would follow that the area of the smaller disc is not maximal for @xmath357 . we conclude that for some @xmath368 @xmath369 subdivides @xmath0 into two discs of area @xmath18 for otherwise it would again contradict our choice of @xmath357 . in this section we prove the existence of a morse function @xmath19 from @xmath0 to @xmath370 with short preimages . [ m - function def ] let @xmath0 be a manifold with boundary . a function @xmath2 is called an @xmath371function if it is morse on the interior of @xmath0 , constant on each boundary component and maps boundary components to distinct points disjoint from the critical values of @xmath19 . the proof of theorem [ morse1 ] proceeds along the same lines as the proof of theorem [ tree1 ] . there are two main differences . the fist difference is that we would like the function to be smooth ( in particular , curves in the foliation @xmath369 are not allowed to have corners ) and have singularities of morse type . this is accomplished by a simple surgery described in lemma [ theta ] . the second difference is that we would like to bound the length of the whole level set , not just of the individual connected components . we will need one technical lemma similar to concatenation lemma [ concatenation ] . let @xmath374 denote a union of three non - intersecting simple arcs @xmath375 with common endpoints . @xmath22 subdivides @xmath195 into three regions @xmath376 , @xmath377 , @xmath378 , choosing the indices so that @xmath379 . let @xmath380 denote a curve obtained by pushing @xmath381 inside @xmath382 by a small perturbation , so that it is contained in a small normal neighbourhood of @xmath22 , smooth and has length @xmath383 . let @xmath384 denote the neighbourhood of @xmath22 bounded by curves @xmath380 ( see figure [ fig_theta ] ) . let @xmath236 and @xmath242 denote the endpoints of @xmath375 . since @xmath0 is orientable , the normal tubular neighbourhood of @xmath381 is homeomorphic to the cylinder . let @xmath380 be the boundary component of the tubular neighbourhood that does not intersect @xmath399 , @xmath400 . after a small perturbation in the neighbourhood of @xmath236 and @xmath242 we can assume that @xmath380 is smooth and there is a diffeomorphism @xmath401 from @xmath402 \times s^1 $ ] onto the region between @xmath381 and @xmath380 with @xmath403 . let @xmath269 be the midpoint of @xmath404 . we perform a straightforward surgery to the curves @xmath405 depicted on figure [ fig_theta2 ] . in the neighbourhood of @xmath269 we can choose a coordinate chart so that curves in the new foliation are given by @xmath406 . [ m - small area disc ] for any @xmath219 there exists @xmath249 , such that for every disc @xmath407 with @xmath250 there exists an @xmath371map @xmath19 from @xmath223 with fibers of length less than @xmath251 . as in the proof of lemma [ small area_s^2 ] , we proceed by induction on @xmath254 . however , the inductive assumption is now different . we would like to show that for every @xmath69 a subdisc @xmath407 admits an @xmath371map with fibers of length @xmath408 assume the lemma to be true for all subdiscs with @xmath409 . we take small tubular neigbourhood of @xmath245 in @xmath223 and foliate it by closed curves of length @xmath410 . we subdivide the innermost curve @xmath16 into 4 arcs of equal length . by besicovitch lemma we can find an arc @xmath202 of length @xmath411 connecting two opposite subarcs of @xmath16 . let @xmath412 be the neighbourhood of @xmath413 as in lemma [ theta ] . note that @xmath414 has 3 connected components : one of them is @xmath245 and denote the other two by @xmath136 and @xmath322 , bounding subdiscs @xmath259 and @xmath260 respectively . assume that @xmath259 is a disc of smaller area , hence @xmath415 . since @xmath420 by inductive assumption @xmath259 admits an @xmath371map @xmath421 with fibers of length @xmath422 . after appropriately scaling @xmath421 on a small neigbourhood of @xmath136 and multiplying by @xmath423 if necessary we can assume that @xmath424 is the minimum point of @xmath421 . furthermore , we scale and shift @xmath421 so that @xmath425 and @xmath426 is contained between @xmath13 and @xmath427 we now extend the @xmath371map to @xmath428 with fibers of length @xmath429 @xmath430 by inductive assumption @xmath260 admits an @xmath371function @xmath431 with fibers of length @xmath432 . we modify @xmath431 so that it takes on its minimum at @xmath433 . we can now extend @xmath19 to the whole of @xmath223 . [ m - small area mk ] for any @xmath219 there exists @xmath249 , such that for every submanifold with boundary @xmath266 , with @xmath267 there exists an @xmath371map @xmath19 from @xmath53 with fibers of length @xmath436 . as in the proof of lemma [ small area_mk ] we connect all components of @xmath56 with @xmath437 closed curves @xmath55 of total length @xmath438 and denote the union of @xmath55s and @xmath56 by @xmath313 . denote the normal @xmath214neighbourhood of @xmath313 in @xmath53 by @xmath412 . after a small perturbation we can assume that the boundary of @xmath412 is smooth . the complement @xmath223 of @xmath412 in @xmath53 is a disc and so by lemma [ m - small area disc ] it admits an @xmath371map with fibers of length @xmath268 . next we proceed as in the proof of lemma [ small area_mk ] extending the domain of the @xmath371function over connected components of @xmath56 one by one . let @xmath439 be a collection of nested simple closed curves in @xmath412 and @xmath304 denote the subset of @xmath412 between @xmath440 and @xmath441 . we require that @xmath442 , @xmath443 and for each @xmath348 @xmath304 contains exactly one connected component of @xmath56 . we can then apply lemma [ theta ] to extend the @xmath371function from @xmath223 to @xmath294 and inductively to the whole of @xmath53 . let @xmath294 and @xmath295 denote the two components of @xmath448 . in this case there is an @xmath371function @xmath421 from @xmath294 onto @xmath449 $ ] and an @xmath371function @xmath431 onto @xmath450 $ ] . hence , we obtain an @xmath371function @xmath451 $ ] satisfying the same bound on the length of the fibers . using the inductive assumption we calculate that this length bound is exactly what we want . as in the proof theorem [ tree1 ] we add a small collar around boundary components of @xmath195 that intersect @xmath16 . we apply the inductive assumption to @xmath227 and extend this map to regions separated from @xmath169 by horseshoes " . then we extend the map to collars whose intersection with @xmath314 is a connected arc . we iterate this procedure until we extended the map to all of @xmath195 . by appropriately scaling the @xmath371functions obtained for each region we ensure the correct bound on the lengths of fibers .
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we show that for every complete riemannian surface @xmath0 diffeomorphic to a sphere with @xmath1 holes there exists a morse function @xmath2 , which is constant on each connected component of the boundary of @xmath0 and has fibers of length no more than @xmath3 .
we also show that on every 2-sphere there exists a simple closed curve of length @xmath4 subdividing the sphere into two discs of area @xmath5 .
[ theorem]definition [ theorem]lemma [ theorem]proposition [ theorem]corollary
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You are an expert at summarizing long articles. Proceed to summarize the following text:
paraxial propagation of linearly polarized laser beams through transparent optical media with intensity dependent refractive index is mathematically equivalent to the free evolution of the wavefunction order parameter used in the mean - field description of a two - dimensional gas of @xmath0 interacting atoms in a bose - einstein condensate ( bec ) at temperature @xmath1 @xcite . both systems can be modeled by identical nonlinear schrdinger equations@xcite . for photons in the laser beam , the @xmath2 component of the nonlinear optical susceptibility plays the same role as n - body interactions between atoms in the cloud and the propagation constant can be identified with a chemical potential for the light distribution . as all the photons in a coherent wave are equal , the laser beam can be treated on equal foot as any system of @xmath0 identical interacting bosons at zero temperature@xcite . the previous point of view , which takes into account the equivalence between laser beams and becs of ultracold atoms , has led to an interesting suggestion made by chiao@xcite , who recently proposed to verify the superfluidity of coherent light , in analogy with degenerate quantum atomic gases . more recently , similar concepts have been successfully used to analyze condensation phenomena of nonlinear waves@xcite and quantum phase transitions of photons in periodic lattices@xcite . in chiao s model the key point of the analysis is _ `` same equations , same predictions '' _ and therefore photons from a monochromatic laser source are considered as an ideal bosonic gas at zero temperature in which continuous phase transitions ( cpt ) can take place due to long range quantum fluctuations around the ground state@xcite . these critical phenomena are thus called _ quantum _ phase transitions ( qpt)@xcite to distinguish them from the standard phase changes which are well - known in classical thermodynamics . as we will show in this work , cpt may occur in any _ classical _ system at zero temperature without long range quantum correlations involved , if opposite nonlinear interactions are present . in the case of chiao s `` superfluid light '' the phase transition is produced by effect of a defocusing intensity - dependent refractive index@xcite and thus a waveguide is used to avoid spreading of the beam ( in the same way as magneto - optical traps are employed to hold atomic becs ) . however , as superfluidity may occur both in gases and liquids , it can not be considered as a trace of the presence of a liquid state@xcite ; it is also required the appearance of surface tension effects@xcite . moreover , in chiao s model , the interactions between particles are repulsive and they can not drive a gas - liquid transition . thus , in this work we will follow the same lines of thought to suggest the possibility of obtaining a gas - to - liquid phase transition in a classical gas at @xmath1 described in a mean field theory by the so - called _ cubic - quintic _ ( cq ) model with competing nonlinearities . several pioneering works have highlighted the interesting properties of this cq - model@xcite . cavitation , superfluidity and coalescence have been investigated@xcite in the context of liquid he , where the model is a simple approach if nonlocal interactions are not taken into account . stable optical vortex solitons and the existence of top - flat states have been also reported in optical materials with cq optical susceptibility@xcite . the surface tension properties that appear in this system@xcite have been considered as a trace of a `` liquid state of light''@xcite . on the other hand , recent experiments about filamentation of high - power laser pulses in @xmath3 have shown that the cq nonlinearity is achievable in this material@xcite . it has been also suggested that atomic coherence may be used to induce a giant cq - like refractive index of a rb gas@xcite . thus , the practical realization of the first `` liquid of light '' state as an example of non - quantum liquefaction at zero temperature is close . in bec systems , a cq - model can be used in the mean field description of an ultracold gas at zero temperature in the presence of efimov states with tunable two- and three - body interactions , which have been recently proposed@xcite . the cubic - quintic model describes a coherent bosonic system of @xmath0 particles with two- and three - body interactions . the mathematical formulation of the mean field theory yields a generalised non - linear schrdinger ( nls , also called gross - pitaevskii ) equation of the form : @xmath4 if the system modeled by the previous equation is a photon gas , the above nls describes paraxial propagation of a continuous linearly - polarized laser beam of wavelength @xmath5 in a nonlinear medium with a refractive index depending on the intensity @xmath6 in the form @xmath7 and the adimensional variables are : @xmath8 the propagation distance multiplied by @xmath9 , @xmath10 the beam irradiance multiplied by the kerr coefficient @xmath11 , @xmath12 an adimensional parameter indicating the strength of the quintic nonlinear optical susceptibility , and @xmath13 the transverse laplacian operator , where @xmath14 and @xmath15 are the transverse spatial dimensions multiplied by @xmath16 . in the case of a two - dimensional atomic bec tightly trapped along one axis by a parabolic potential of frequency @xmath17 and thickness @xmath18 , the adimensional variables correspond to : @xmath8 the time in units of @xmath19 , @xmath10 the atomic density multiplied by two - body coefficient @xmath20 , @xmath21 an adimensional parameter indicating the strength of the three - body interactions , and @xmath13 the transverse laplacian operator , where @xmath14 and @xmath15 are the transverse spatial dimensions divided by @xmath22 . recent experiments for beam propagation in @xmath3@xcite at @xmath23 yield the following values for the above parameters in the case of laser beams : @xmath24 , @xmath25 and @xmath26 . other materials like air@xcite or chalcogenide glasses@xcite seem to display the c - q behavior usually accompanied by ionization and non - linear losses . it has been also pointed the possibility of engineering this type of optical response by quantum techniques which allow to access this nonlinear regime with miliwatt ultrastabilized lasers@xcite . for atomic bec systems it has been proposed that a combination of two - body ( attractive ) and three - body ( repulsive ) elastic interactions can yield liquid _ this behaviour can be explained in terms of the efimov states@xcite . however , three - body scattering in becs has inelastic contributions and yields highly nonlinear losses . this means that in the most general case the coefficient @xmath21 in equation may be complex both for laser systems as well as for atomic gases . the above cq - nlse admits soliton - like solutions of finite size@xcite of the form @xmath27 , being @xmath28 the propagation constant in the case of light and the chemical potential for atomic becs . these solitons can only be calculated numerically and coexist with plane waves solutions of constant amplitude @xmath29 , which lead by substitution in eq . to @xmath30 . in fig.[fig1 ] we have plotted the maximum value of @xmath31 vs. @xmath28 in units of @xmath32 , for different kinds of stationary solutions of eq . . the continuous and dotted lines correspond respectively to stable and modulationally unstable plane waves@xcite , whereas the dashed line stands for numerically calculated localized eigenstates@xcite . it is known@xcite that the shape of the solitons ( see dashed profiles in fig.[fig2 ] ) vary from quasi - gaussian shapes for low powers to almost square profiles for beam amplitudes close to a certain critical value of @xmath33@xcite . at this point the size of the solutions tends to infinity whereas the amplitude of the beam stabilizes at @xmath34 . we will now clarify the reasons of this behavior by using a thermodynamic model . the appropriate tool to study the equilibrium condition as a function of the number of particles @xmath35 is landau s grand potential @xmath36,\end{aligned}\ ] ] where the lagrange multiplier @xmath28 is the chemical potential for a bec or the propagation constant in the case of an optical system . for our two - dimensional model the area is @xmath37 and we have that @xmath38 is the pressure . the equilibrium configurations are obtained when @xmath39 is minimized . in particular @xmath40 implies that the pressure has to be zero , which for the plane waves of fig.[fig1 ] yields @xmath41 . there are two possibilities : the trivial case @xmath42 and a uniform phase , with @xmath43 and @xmath44 . in the language of field theory , these solutions are the two possible vacuum states of the system . as it can be seen in fig . [ fig1 ] the two vacua can be `` connected '' by the soliton solutions of the dashed line in fig . [ fig1 ] , which constitute the _ instantons _ of the theory@xcite . it is interesting to notice that the non - zero vacuum solution implies a spontaneously symmetry breaking of the global phase symmetry , which is preserved in the case with @xmath45 . this result about the pressure has already been discussed by authors in ref @xcite within the framework of the madelung transformation ( mt ) , which allows one to establish a formal analogy between nonlinear optics and classical fluid dynamics@xcite . with the aid of the mt , an analytical expression for the effective pressure has been derived@xcite . very remarkably , this effective pressure , which is an approximation since it does not take into account the _ quantum - mechanical pressure term_@xcite , calculated for the plane - wave solutions of the cq - nlse has the same mathematical expression as the one shown before . unlike the previous works related in the references , we have calculated the effective pressure by means of the potential given by eq . which contains all the relevant information about the nonlinear system which is currently being studied . while the analysis of the infinite plane - wave solutions has already been considered in the literature @xcite , our discussion of the pressure also applies to the localised soliton solutions , being a central contribution of the present work . we also find a significant difference between the low - power quasi - gaussian solutions and the high - power top - flat solutions of eq .. the expression for plane - waves and top - flat eigenstates can be approximated by the same `` reduced '' expression for @xmath46 given by both the mt and the omega density calculation . in fact , we have shown in fig.[fig1 ] that the two branches of solutions , i.e. the localised solitons and the stable plane waves , merge as @xmath28 approaches @xmath47 , having a radius increasing to infinity , thus in this case it is justified to neglect @xmath48 in eq .. on the other hand , for the quasi - gaussian eigenstates the previous gradient term is not negligible . in this case , we should calculate @xmath46 numerically in the lack of analytical solutions . the analysis of the pressure can be refined calculating numerically its distribution for different eigenstates as it is plotted in fig.[fig2 ] . the pressure of the plane wave branches in fig.[fig1 ] is plotted as a function of @xmath49 . very remarkable is the fact that the pressure of the stable branch is higher than its counterpart of the unstable branch . this implies that the stable plane waves free energy density is smaller than the free energy of the modulationally unstable solutions branch . in fig.[fig2 ] it is also plotted the curve of the eigenstates central pressure . as it can be seen in the graph , the curve corresponding to the eigenstates is bounded by the two curves of the plane - waves branches , so that the existence domain of the filaments phase is limited by them . in the insets of fig.[fig2 ] , we show the shape profiles of the eigenstates ( dashed line ) superposed with their effective pressure profiles ( solid lines ) which have been conveniently rescaled to fit in the graph . as it can be appreciated in inset a ) , solitons with @xmath50 have smooth pressure distributions with a central maximum located at the centroid of the soliton and two negative - valued minima . as the value of @xmath51 increases , the soliton profiles and their corresponding shapes of @xmath46 narrow , reaching a minimum width at @xmath52 , which corresponds to a filament soliton solution with the same peak amplitude as the `` critical '' plane wave with @xmath53 . this plane - wave marks the border between stable and modulationally unstable plane waves@xcite . when the chemical potential reaches the value @xmath54 ( see inset b ) , the absolute maximum of @xmath46 for eigenstates is obtained , i.e , this filament has the minimum of @xmath55 at its centroid . we also consider that close to @xmath56 , the liquid top - flat eigenstates begin to exist@xcite . in fact , the curve of stable localised solutions in fig.[fig1 ] has a maximum in that region and both localised solution and stable plane - waves branches seem to merge there . for @xmath57 , the pressure maximum is located in a flat region ( see inset c ) , tending to zero as @xmath51 approaches the critical value @xmath58 , point d ) in fig.[fig1 ] , over which no localized solutions exist@xcite . in this section , we will provide a set of numerical simulations showing the condensation process , i.e. , the phase transition from the gaseous phase to an homogeneus coherent `` liquid '' plane wave solution corresponding to the upper branch in fig.[fig1 ] . in order to achieve this result , we will perturb an unstable plane wave , corresponding to the lower branch in fig [ fig1 ] , with a randomly varying noise . this will produce a filamented phase , made of coherent structures whose shape remains qualitatively unchanged , up to smaller scale fluctuations@xcite . in order to achieve the non - quantum liquefaction we have included both a linear incoherent pumping mechanism and nonlinear three - body losses@xcite . in other words , we have considered @xmath59 and we have introduced a linear gain term @xmath60 in eq .. this corresponds to a continuous load of particles in the system . note that in this way , we are describing a more realistic non - conservative version of eq . , which models an experimentally achievable scenario in the framework of current bec experiments . in fact , in condensed matter systems it is possible to control the load of particles and two and three - body recombination within the coherent atomic cloud@xcite . on the other hand , in nonlinear optics , this kind of nonlinear models are well - known in the frame of complex ginzburg - landau ( g - l ) equations used to describe wide - aperture laser cavities@xcite . although it is always possible to control the linear gain introduced in the system , three - body losses are often imposed by the nonlinear response of the material , so it is not possible to manage the dissipation terms of the system . however , by means of electromagnetic induced transparency techniques , it is possible to customize the nonlinear optical response of cold atomic ensembles like rb@xcite so that the nonlinear refractive index corresponds to the one given by the modified eq . analyzed in the current section of the paper . therefore our model , and its predicted phenomenon of liquefaction that we will demostrate below , can correspond both to realistic bec and nonlinear optical systems . in fig.[fig3 ] , the initial state consists of an incoherent set of filaments with a randomly varying phase distribution . within this apparent disorder , some coherent uncorrelated structures ( filaments ) exist and can be observed in fig.[fig3 ] a ) . in this simulation , we have considered a nonlinear parameter regime where the linear gain was not enough to compensate the three - body dissipative term . as a consequence , certain degree of coherence is lost since the filaments disappear and only the noisy background is observed , as shown by snapshot c ) in fig.[fig3 ] ) . starting from the same initial condition but increasing the linear gain term over a certain threshold , we have performed the simulation shown in fig.[fig4 ] . in this case , we see that the system evolves towards an homogeneus plane wave by the combined effect of adding particles to the initial random state and dissipation due to many - body inelastic processes . this relaxation process is well - known in the context of the complex g - l equations and is attributted to the non - conservative nature of the model@xcite . nevertheless , as our theory predicts , the system will tend to form a particular plane wave of the stable `` liquid '' upper branch of fig.[fig1 ] , the one with zero pressure . this illustrates the tendency of the system to reach the non - zero vacuum state with @xmath61 . very remarkably , during the process pairs of vortex - antivortex with topological charges @xmath62 are formed ( see bottom phase maps in fig.[fig4 ] ) , so that the constant phase of the emerging coherent wave remains hidden by the overlapping of the different vortex rotating phase - distributions . vortices are very robust topological structures@xcite and in our simulations they remain stable as far as we could follow the numerical simulations . finally , we have considered the effect of replacing the linear gain term by a nonlinear gain term of the form @xmath63 . in this situation , the numerical results are qualitatively the same as in fig.[fig4 ] , although it can be observed that in the last stage of the evolution the vortices are annihilated . in snapshot f ) of fig.[fig4 ] , it can be seen how the underlying plane wave , which emerges after the dynamical process as described above , has a homogeneus phase distribution . we think that this result is a very interesting example of a self - organization process , where coherence is produced from disorder , with evident practical applications . we have shown in the present work that a system of @xmath0 equal bosons in a system with competing nonlinearities can undergo a phase transition from a gas to a liquid state . the process takes place at zero temperature without any quantum effect and it is only ruled by nonlinear interactions . finally , we have shown that a cubic - quintic medium with complex susceptibilities exhibiting linear gain and nonlinear losses , will tend to produce a homogeneous phase liquid distribution starting from a collection of non - correlated filaments . this opens the door for experiments in the field of bec systems in ultracold gases . h. m. thanks a. j. legget and a. ferrando for useful discussions and y. castin , j. ho and g. v. shlyapnikov for warm hospitality at inst . henri poincar . this work was supported by mec , spain ( projects fis2006 - 04190 and fis2007 - 62560 ) xunta de galicia ( project pgidit04tic383001pr and d.n . grant from consellera de educacin , xunta de galicia ) . e. l. bolda , r. y. chiao and w. h. zurek , , 416419 ( 2001 ) ; r. y. chiao _ et al . _ , j. phys . b : at . mol . phys * 37 * , s81s89 ( 2004 ) ; r. y. chiao _ et al . _ , , 063816 ( 2004 ) ; a. tanzini and s.p . sorella a * 263 * , 4347 ( 1999 ) . h. michinel , j. campo - tboas , m. l. quiroga - teixeiro , j. r. salgueiro , and r. garca fernndez , j. opt . b : quantum semiclass . opt . * 3 * , 314 ( 2001 ) ; m. j. paz - alonso and h. michinel , phys . lett . * 94 * , 093901 ( 2005 ) . crasovan , ba . malomed and d. mihalache , , 016605 ( 2001 ) ; p. grelu , j. soto - crespo and n. akhmediev , opt . express * 13 * , 23 , pp : 9352 - 9360 ( 2005 ) s. mancas , sr . choudhury , chaos , solitons and fractals , * 27 * , 5 , pp : 1256 - 1271 ( 2006 )
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we show that a gas - to - liquid phase transition at zero temperature may occur in a coherent gas of bosons in the presence of competing nonlinear effects .
this situation can take place both in atomic systems like bose - einstein condensates in alkalii gases with two and three - body interactions of opposite signs , as well as in laser beams which propagate through optical media with kerr ( focusing ) and higher order ( defocusing ) nonlinear responses .
the liquefaction process takes place in absence of any quantum effect and can be formulated in the frame of a mean field theory , in terms of the minimization of a thermodynamic potential .
we also show numerically that the effect of linear gain and three - body recombination also provides a rich dynamics with the emergence of self - organization behaviour .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the most interesting objects in theoretical physics are arguably black holes . to understand their dynamics we need to put together two widely accepted theories of nature : general relativity ( einstein s classical theory of gravity ) and quantum mechanics . black hole thermodynamics is the crossroad between the classical and the quantum pictures . discovery of hawking radiations lead to the identification of black holes as thermodynamic objects with physical temperature and entropy @xcite . this discovery paved the way for progress in the understanding of spacetime , quantum mechanically @xcite . variation in the mass , @xmath1 , of a rotating black hole having angular momentum , @xmath2 , and electric charge @xmath3 , obeys the formalism @xmath4known as first law of thermodynamics . here @xmath5 is the angular velocity of the horizon , @xmath6 is the electric potential on the horizon and @xmath7 is the entropy of the event horizon @xmath8 . the important results of the black hole thermodynamics are the association of temperature ( @xmath9 ) and entropy @xmath10 with surface gravity @xmath11 and area @xmath12 of the black hole event horizon respectively . the phenomena of phase transition in black hole thermodynamics was first observed long ago @xcite . schwarzschild black hole has negative specific heat and evaporates via hawking radiation . the ads schwarzschild black holes have a different behavior towards temperature and heat capacity . there are two types of black holes in ads spacetime : the smaller one , like the usual schwarzschild black hole , with negative specific heat ( unstable ) , and the big black holes having positive specific heat ( locally stable ) . axisymmetric , stationary , and electrically charged black holes in einstein - maxwell theory with arbitrary surrounding matter , always have regular inner horizon ( cauchy horizon ) ( @xmath13 ) and an outer horizon ( event horizon ) ( @xmath8 ) , if the angular momentum and charge of the black hole do not vanish at the same time @xcite . in recent years products of thermodynamics parameters , specially area and entropy , at both horizons of black holes has gained attention in general relativity and string theory @xcite . it is observed that the area product of the outer and inner horizons is independent of black hole mass @xmath1 . for a regular axisymmetric and stationary spacetime in einstein maxwell gravity these products are @xcite : @xmath14 and @xmath15 note that both the above given products are mass independent so these are universal quantities . this universal sense of area and entropy products holds for all known five dimensional asymptotically flat black rings , and for black strings @xcite . the microscopic degrees of freedom of the black hole are described in terms of those of a conformal field theory ( cft ) . the area product of the inner and outer horizons of a black hole in three dimensions is @xmath16 where @xmath17 and @xmath18 are the number of right and left moving excitations of the two - dimensional cft @xcite , i.e. @xmath19 in other words , the products of the areas of the killing horizons is independent of the mass of any asymptotically flat black hole in d - dimensional spacetime , therefore depends on the quantized charges @xcite . hence mass independence of area products , is necessary condition for holographic cft description . in @xcite the thermal products for rotating black holes are studied . in @xcite area products for stationary black hole horizons are calculated . it has been shown that the area products are independent of adm ( arnowitt - deser - misner ) mass parameter and depends on the quantized charge and quantized angular momentum parameter for all known five dimensional asymptotically flat black rings and black strings . it may sometimes also fail , e.g. in @xcite authors show that entropy products are not mass independent in general myers - perry black holes when spacetime dimension @xmath20 , and kerr - ads black holes with @xmath21 . the kerr / cft correspondence from the thermodynamics of both outer and inner horizons was investigated recently @xcite . authors prove that the first law of thermodynamics of the outer horizon guaranty that of the inner horizon , under some assumption , and mass independence of the entropy product @xmath22 is equivalent to the condition @xmath23 . furthermore , using the thermodynamics method , information of the dual cft could be obtained easily , because thermodynamics of the outer and inner horizons give the thermodynamics in the left and right moving sectors of the dual cft @xcite . so the central charges and the temperatures in all possible pictures can be obtained in a simple way . newman et . al . , obtained the solution of the einstein - maxwell equation in kerr space - time , as a rotating ring of mass and electric charge @xcite . applying the ernst s formulation ( for axisymmetric stationary fields ) , tomimatsu and sato discovered the series of solutions for the gravitational field of a rotating mass @xcite . yamazaki obtained the charged kerr - tomimatsu - sato family of solutions with some distortion parameter @xmath24 ( integer ) in the gravitational fields of rotating masses @xcite . static spherically symmetric julia - zee dyon solution in curved spacetime were obtained by kasuya et . later , an exact stationary rotating dyon solution in tomimatsu - sato- yamazaki space - time , was proposed @xcite . furthermore , both the `` schwinger '' and the `` julia - zee '' dyon exact solutions in kerr - newman space - time , i.e. for @xmath25 , were studied . this solution is known as kerr - newman - kasuya ( knk ) black hole in literature @xcite . this solution is featured by four physical parameters ( mass @xmath1 , angular momentum @xmath2 , electric charge @xmath3 , and magnetic charge @xmath26 ) . in this work we consider the knk black hole and discuss its thermodynamic at horizons and generalize some already existing results for cauchy horizon @xmath13 . we also find the knk / cft correspondence using the thermodynamics method . plan of the work is as follows : in section 2 , we discuss the knk black hole metric and the products of surface gravity , temperature and komar energy , angular velocities and electromagnetic potentials calculated on the inner and outer horizons . products of area and entropy at @xmath27 are also calculated , and these products are independent of mass of the black hole . in subsection 2.1 , the smarr formula for the black hole is derived and first law of thermodynamics is verified . in subsection 2.2 , we have discussed the irreducible mass and obtained the expression relating , the rest mass and irreducible mass . in subsection 2.3 , heat capacity of black hole is calculated , and phase transition is discussed . in 2.4 , we find the cft relation of knk black hole and re - obtained the expressions of temperature and entropy . in 2.5 section we concluded the work . we use units in which @xmath28 . the spherically asymmetric and rotating solutions for einstein - maxwell field equations , in the presence of electric and magnetic field , was proposed by kasuya @xcite . metric in the boyer- lindquist coordinates is : @xmath29 where @xmath30the electromagnetic potential is : @xmath31 + \epsilon p d\phi,\ ] ] here @xmath1 is mass of the black hole , @xmath3 is the electric charge , @xmath26 is the magnetic charge , @xmath2 is the angular momentum , and @xmath32 . the horizons of the black hole are obtained by solving @xmath33 , given as @xmath34 where @xmath35 is the outer horizon named as event horizon @xmath8 and @xmath36 is the inner horizon known as cauchy horizon @xmath37 . both cauchy horizon and event horizon are null surfaces with infinite blue and red - shifts respectively . cauchy horizon is a null surface of infinite blue - shift , while the event horizon is an infinite red - shift surface @xcite . product of the two horizons : @xmath38 does not directly depend on mass of the black hole but on charge ( electric and magnetic ) and spin parameter of the black hole . area and bakenstein - hawking entropy at the horizons @xmath39 of the black hole are @xcite : @xmath40 and @xmath41 surface gravity is the acceleration due to gravity at the horizon of a black hole . it is defined as the force required to an observer at infinity , for holding a particle ( of unit mass ) in place at the event horizon , given as @xcite @xmath42 @xmath43 hawking temperature of the horizons is : @xmath44 the komar energy is given by @xcite : @xmath45 electromagnetic potential due to electric and magnetic charges is @xmath46 and @xmath47 respectively . angular velocity at @xmath39 is @xmath48 product of surface gravities , surface temperatures and komar energies at @xmath39 are @xmath49},\label{k11}\ ] ] @xmath50},\ ] ] and @xmath51 respectively . products of electromagnetic potentials @xmath52 at @xmath53 , ( due to electric and magnetic charges ) are @xmath54 and products of angular velocities @xmath5 at @xmath39 is @xmath55it is clear that all these products ( except product of horizons @xmath56 ) are depending on mass of the black hole directly , so these quantities are not universal . we also calculate the products of areas and entropies at @xmath39 , @xmath57,\label{k area}\ ] ] and @xmath58.\ ] ] note that both products are independent of mass , so these are universal quantities . the physical parameters defined in previous section for knk black hole obey the symmetries @xmath59 solving together @xmath60 , the black hole parameters @xmath61 and @xmath62 can be written in terms of horizons ( @xmath63 ) , @xmath3 and @xmath26 as @xmath64 @xmath65},\nonumber\\ & & { \cal a}_{\pm}= 4\pi [ r_{\pm}(r_++r_-)-q^2-p^2],\nonumber\\ & & \phi_{\pm}^q= \frac{q r_{\pm}}{r_+(r_++r_-)-q^2-p^2},\nonumber\\ & & \phi_{\pm}^p= \frac{p r_{\pm}}{r_+(r_++r_-)-q^2-p^2},\nonumber\\ & & \omega_{\pm}=\frac{\sqrt{r_+r_--q^2-p^2}}{r_{\pm}(r_++r_-)-q^2-p^2}.\end{aligned}\ ] ] using the above relations the first law of thermodynamics @xmath66 can be verified easily . the first law at the inner horizon can be written by exchanging @xmath67 in all the quantities given in eq . ( [ k2.5 ] ) , because @xmath63 are on equal footing . this symmetry of @xmath63 gives @xcite @xmath68 further @xmath69 these products are independent of @xmath1 . in @xcite authors prove that mass independence of entropy products together with first laws at both the horizons imply that @xmath70 for several black holes . the expression for area of the black hole can be rewritten using the idea proposed by smarr @xcite as : @xmath72.\ ] ] the area of both horizons must be constant given by : @xmath73.\ ] ] using eq . ( [ k15 ] ) mass of the black hole or adm mass is expressed in terms of the areas of the horizons @xmath74 since the first law of thermodynamics states that change in mass of a black hole is related to change in its area , angular momentum and electric charge , in other words @xmath75 where @xmath76\nonumber\\ \phi_{\pm } ^q&= & \text{electromagnetic potential at horizons due to electric charge $ q$}\nonumber\\ & = & \frac{q r_{\pm}}{2{\cal m}r_{\pm}-q^2-p^2}\nonumber\\ \phi_{\pm } ^p&= & \text{electromagnetic potential at horizons due to magnetic charge $ p$}\nonumber\\ & = & \frac{p r_{\pm}}{2{\cal m}r_{\pm}-q^2-p^2}\nonumber\\ \omega_{\pm}&= & \text{angular velocity at horizons}\nonumber\\ & = & \frac{a}{r_{\pm}^2+a^2}.\end{aligned}\ ] ] we can rewrite effective surface tension as : @xmath77 \nonumber\\ & & = \frac{1}{32 \pi { \cal m}}\big[1-\frac{16\pi { \cal m}^2_{\pm}-{\cal a}_{\pm}-8 \pi ( p^2+q^2)}{{\cal a}}\big]\nonumber\\ & & = \frac{1}{32 \pi { \cal m}}\big[1-\frac{(8\pi { \cal m}^2 - 4 \pi(p^2+q^2))}{4\pi(r_{\pm}^2+a^2 ) } \big ] \nonumber\\ & & = \frac{1}{8 \pi}\big[\frac{r_{\pm}^2-{\cal m}}{r_{\pm}^2+a^2}\big]\nonumber\\ & & = \frac{\kappa_{\pm}}{8\pi},\end{aligned}\ ] ] hence first law of thermodynamics is derived from smarr formula . the mass of a black hole could be increased or decreased but there is no way to decrease the irreducible mass @xmath78 of a black hole . since second law of black hole thermodynamics states that the surface area @xmath79 of both the horizons @xmath80 never decreases @xcite i.e. @xmath81 so there is a one to one connection between the surface area and irreducible mass of a black hole . christodoulou showed that the irreducible mass of a black hole is proportional to the square root of its surface area @xcite . the irreducible mass of knk black hole expressed in terms of its area is:@xmath82 the irreducible mass defined on inner and outer horizons is @xmath83 and @xmath84 respectively . the product of the irreducible mass at the horizons @xmath85 is : @xmath86 this product is universal because it does not depends directly on mass of the black hole . the expression for the rest mass of the rotating charged black hole given by christodoulou and ruffini in terms of its irreducible mass , angular momentum and charge is @xcite : @xmath87 for knk black hole , expression of mass in terms of irreducible mass is : @xmath88 or @xmath89 where @xmath90 . the nature ( positivity or negativity ) of heat capacity reflects the change in the stability properties of the thermal system ( black hole ) . a black hole with negative heat capacity is in unstable equilibrium state i.e. by emitting hawking radiations it may decay to a hot flat space or by absorbing a radiation it may grow without limit @xcite . heat capacity of a black hole is given by @xmath92 where mass @xmath1 in terms of @xmath63 is : @xmath93 the partial derivatives of mass @xmath1 and temperature @xmath94 with respect to @xmath35 are : @xmath95 and from eq . ( [ k9 ] ) we have @xmath96 and @xmath97 while positive for @xmath98 , so it undergoes phase transition from instable to stable region . heat capacity diverges at @xmath99 . here we chose @xmath100 , @xmath101 , @xmath102 , and divergence occurs at @xmath103 . , width=340 ] the expression for heat capacity @xmath104 for knk black hole becomes : @xmath105 note that heat capacity will be positive if + @xmath106 , and @xmath107 are positive or negative together , and heat capacity is negative if these factors are of opposite signs . the region where heat capacity is negative , corresponds to an instable region around black hole , whereas a region in which the heat capacity is positive , represents a stability region . behavior of heat capacity given in eq . ( [ k21 ] ) is shown in fig . ( [ 1k ] ) . in this section we calculate the central charge , temperature and entropy of knk . there are three parameters @xmath2 , @xmath3 and @xmath26 of knk black hole besides @xmath108 . these three parameters correspond to three @xmath109 symmetries ( rotational and gauge symmetries ) of the back ground . since for each @xmath109 symmetry there is a cft dual , we call these j - pictures , q - picture and p - picture . the first law of thermodynamics is satisfied for both , the left and right moving sectors independently @xcite . temperatures and entropies of the left and right moving sectors can be defined as @xmath110 where @xmath111 the temperature of these two sectors can be calculated by considering some results as mentioned in @xcite , i.e. mass independence of the area products guarantees the existence of cft description , further the central charges of the two sectors of dual cft would be equal , since in knk black hole , product of entropies given in eq . ( [ k entropy ] ) is mass independent , so these results hold for knk black hole . to find the temperatures directly from the thermodynamical relations we use the same procedure as discussed in @xcite . as in @xcite authors started discussion with btz black hole , ( then extended to other black holes , in 4d and 5d ) , that the first law of thermodynamics for left and right sectors can be written as @xmath112where @xmath113 are the left and right moving temperatures in dual cft , also known as j - picture cft temperatures . note that both sides of eq . ( [ k5.17 ] ) are dimensionally correct . the cardy formula for entropies of the left and right - moving sectors becomes @xmath114 whereas @xmath115 , as mentioned earlier . once we have exact values of the central charges , the temperatures can be calculated using cardy formula . these temperatures are proportional to the microscopic temperatures obtained by hidden conformal symmetries in low frequency scattering @xcite . the hidden conformal symmetries of knk black hole have been discussed in @xcite , these are exactly same as that of kerr newman black hole , but replacing @xmath116 by @xmath117 . in this work we find the temperatures and central charges of cft duals , by applying thermodynamics method . we use eqs . ( [ k5.2 ] ) and ( [ k5.3 ] ) with eq . ( [ k2.5 ] ) to find the temperatures and entropies of the left and right moving sectors for knk black hole , @xmath118}\nonumber\\ t_l&=&\frac{1}{4\pi(r_++r_- ) } \nonumber\\ { \cal s}_r&=&\frac{\pi}{2}(r_+^2-r_-^2)\nonumber\\ { \cal s}_l&=&\frac{\pi}{2}[(r_++r_-)^2 - 2q^2 - 2p^2]\nonumber\\ \omega_r&=&\frac{\sqrt{r_+r_--q^2-p^2}}{(r_++r_-)^2 - 2q^2 - 2p^2}\nonumber\\ \omega_l&=&0\nonumber\\ \phi^q_r&=&\frac{q(r_++r_-)}{2[(r_++r_-)^2 - 2q^2 - 2p^2]}\nonumber\\ \phi^q_l&=&\frac{q}{2(r_++r_-)}\nonumber\\ \phi^p_r&=&\frac{p(r_++r_-)}{2[(r_++r_-)^2 - 2q^2 - 2p^2]}\nonumber\\ \phi^p_l&=&\frac{p}{2(r_++r_-)},\end{aligned}\]]where @xmath119 and @xmath120 are the electromagnetic potentials of right and left moving sectors , due to the electric and magnetic charges , @xmath3 and @xmath26 , defined as @xmath121the first law of thermodynamics for the right and left moving sectors is @xmath122 for @xmath123 , eq . ( [ k5.19 ] ) reduces to eq . ( [ k5.16 ] ) , so we calculate the microscopic temperatures for knk in j - picture , defined in eq . ( [ k5.17 ] ) , as @xmath124 using eq . ( [ k5.4 ] ) with eq . ( [ k5.22 ] ) we get the central charges in so called j - picture as @xmath125 if we set @xmath126 we get @xmath127 to balance eq . ( [ k5.27 ] ) dimensionally we use quantized charges , and define the electric and magnetic charges @xcite@xmath128 where @xmath129 are integers . using these charges one can define @xmath130 and @xmath131 for both horizons @xcite @xmath132 the corresponding electromagnetic potentials for left and right sectors are @xcite : @xmath133 using eqs . ( [ k9 ] ) , ( [ 4.13rn ] ) and ( [ 4.14rn ] ) , we have @xmath134,\end{aligned}\ ] ] @xmath135}{2(r_+r_-)(r_+^2+r_-^2 + 2a^2)},\end{aligned}\ ] ] and @xmath136,\end{aligned}\ ] ] @xmath137}{2(r_+r_-)(r_+^2-r_-^2)},\end{aligned}\ ] ] the angular velocities at left and right sectors are : @xmath138 and we obtain @xmath139and the first law of thermodynamics in terms of the quantized charges becomes @xmath140 or we can write it as @xmath141 for knk black hole , three cft pictures are there , named as the q , p and j - picture . setting @xmath142 , we get cft dual in q - picture @xmath143 with temperature as @xmath144 where @xmath145 is the scale factor , we have @xmath146 @xmath147 from the cardy formula given in eq . ( [ k5.4 ] ) , we can find central charges for q - picture using @xmath148 given as @xmath149 and @xmath150 setting @xmath151 , we get cft dual in p - picture @xmath152 with temperature as @xmath153 where @xmath154 , and are given by @xmath155},\ ] ] and @xmath156}.\ ] ] the central charges for p - picture are @xmath157for knk black hole it becomes @xmath158},\ ] ] and @xmath159}.\ ] ] the q - picture and p - pictures are electronic and magnetic pictures . for @xmath160 , we are left with cft dual in j - picture and the change in @xmath2 is linked with temperature and entropy as @xmath161 where @xmath162 where @xmath163 and we get @xmath164and @xmath165 using @xmath166 we obtain the central charges for j - picture as @xmath167 if there are two dual pictures , there could be a class of dual pictures related by @xmath168 transformations with each other @xcite , as explained for @xmath169d kerr - newman and @xmath170d myers - perry black holes in @xcite . similarly , one can generate more dual pictures from @xmath168 for knk black hole . some important thermodynamical parameters for kerr - newman - kasuya black hole with reference to their event and cauchy horizons are studied . we find the products of these parameters calculated at both horizons of the black hole , and observe that the product of surface gravities , surface temperature product , product of komar energies , products of electromagnetic potentials and angular velocities at the horizons are not universal quantities , being mass dependant , while products of areas and entropies at both the horizons are independent of mass of the black hole . we derive the expression for heat capacity at the horizons . using the heat capacity expression , stability regions of the black hole can be estimated , and it is observed graphically for knk black hole . the first law of thermodynamics is also obtained from the smarr formula approach . the relations that are independent of the black hole mass are of particular interest because these may turn out to be `` universal '' and hold for more general solutions , with nontrivial surroundings . the relation @xmath171 holds for knk black hole , so it has a cft dual in the einstein gravity and knk / cft correspondence is achievable . we redefine the thermodynamics by applying the quantization approach on the first laws . the temperatures of the q , p , and j - pictures of cft duals and the central charges are calculated , with the use of cardy formula . it is observed that the central charges of the two sectors are identical , which is in agreement with the work done in @xcite , i.e. the mass independence of the area product of the horizons implies the similarity of the central charges of the two sectors . furthermore , we obtained the temperature of the black hole horizon by thermodynamics approach and by microscopic cft , both temperatures differ by a scale factor . we would like to thank prof . bin chen , and jia - ju zhang for valuable discussions on cft correspondence of knk black hole . 99 d. christodoulou , _ phys . lett . _ * 25 * , 1596 ( 1970 ) . d. christodoulou and r. ruffini , _ phys . rev . _ * d 4 * , 3552 ( 1971 ) . r. penrose and r. m. floyd , _ nature _ * 229 * , 177 ( 1971 ) ; s. hawking , _ phys . * 26 * , 1344 ( 1971 ) . j. d. bekenstein , _ phys . * d 7 * 2333 ( 1973 ) ; j. d. bekenstein , _ phys . _ * d 9 * 3292 ( 1974 ) . s. w. hawking , _ nature _ * 248 * 30 ( 1974 ) . s. w. hawking , " _ commun . phys . _ * 43 * , 199 ( 1975 ) . f. larsen , _ phys . rev . _ * d56 * , 1005 ( 1997 ) . 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* abstract : * we investigate the thermodynamics of kerr - newman - kasuya black hole on the inner and outer horizons .
products of surface gravities , surface temperatures , komar energies , electromagnetic potentials , angular velocities , areas , entropies , horizon radii and the irreducible masses at the cauchy and the event horizons are calculated .
it is observed that the product of surface gravities , surface temperature product and product of komar energies , electromagnetic potentials and angular velocities at the horizons are not universal quantities for kerr - newman - kasuya black hole .
products of areas and entropies at both the horizons are independent of mass of the black hole . heat capacity is calculated and phase transition is observed , under certain conditions on @xmath0 . using the thermodynamics method with quantized charges ( known as refined thermodynamics ) , the central charges and the holographic pictures ( j - picture , q - picture , and p - picture ) of the dual cft for kerr - newman - kasuya black hole are determined .
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in this letter we address dynamical processes in highly ordered complex plasmas associated with _ spontaneous symmetry breaking_. spontaneous symmetry breaking ( ssb ) plays a crucial role in elementary particle physics but is also very common in classical physics @xcite . it happens whenever the system goes from a state which has a certain symmetry , e.g. rotational symmetry , into an ordered state , which does not have this symmetry anymore . in general , this state not necessarily has to be the ground ( vacuum ) state and the transition to the new state may or may not be associated with a phase transition . for example , in the case of magnetization the spins point all in one direction ( ordered state ) whereas above the curie temperature there is no preferred direction . another example from a mechanical system without phase transition is a vertical stick which bends under a sufficiently high force from above to one side breaking the rotational symmetry of the system without the force . different symmetries coexisting in the same phase , and symmetry transformations escorting phase transitions are widely spread in nature . for instance , the mechanisms of symmetry breaking are thought to be inherent in the molecular basis of life @xcite . ssb is also an important feature of elementary particle physics @xcite . the universe itself is believed to have experienced a cascade of symmetry - breaking phase transitions which broke the symmetry of the originally unified interaction giving rise to all known fundamental forces @xcite . symmetry effects are crucial either in 3d and 2d systems . chiral ( mirror - isomeric ) clusters @xcite , magic clusters of a new symmetry frozen - in by a solid surface @xcite , or dynamical symmetry breaking by the surface stress anisotropy of a two - phase monolayer on an elastic substrate @xcite are examples of the importance of 2d or quasi-2d systems in many applications . low pressure , low temperature plasmas are called _ complex plasmas _ if they contain microparticles as an additional thermodynamically active component . in the size domain of 1 - 10@xmath0 m ( normally used in experiments with complex plasmas ) these particles can be visualized individually , providing hence an atomistic ( kinetic ) level of investigations @xcite . the interparticle spacing can be of the order of 0.1 - 1 mm and characteristic time - scales are of the order of 0.01 - 0.1 s. these unique characteristics allow to investigate the microscopic mechanism of ssb and phase transitions at the kinetic level . common wisdom dictates that symmetry breaking is an inherent attribute of systems in an active state . hence these effects are naturally important in complex plasmas where the _ particle cloud - plasma _ feedback mechanisms underlying many dynamical processes are easy to vitalize . also in complex plasmas where different kind of phase transitions exist , e.g. in the elelectrorheological plasmas @xcite , one can find examples for classical ssb . another option , interesting in many applications , is the clustering of a new phase which is dissymmetric with regard to a background symmetry ( as an example of fluid phase separation in binary complex plasmas see @xcite ) . it is important to mention that the microparticles , collecting electrons and ions from the plasma background , become charged ( most often negatively @xcite ) and hence should be confined by external electric fields . the configuration of the confining forces might deeply affect the geometry and actual structure of the microparticle cloud . in rf discharge complex plasmas the particles are self - trapped inside the plasma because of a favorable configuration of the electric fields @xcite . one of the interesting things is the possibility to levitate a monolayer of particles under gravity conditions . in this case the particle suspension has a flat practically two dimensional structure . this is , of course , a very attractive simplification ( from a theoretical point of view ) , significantly lowering the description difficulties . below we concentrate mostly on 2d complex plasmas . depending on the discharge conditions , the monolayer can have crystalline or liquid order . 2d configurations of dust particles either in crystalline or liquid state were successfully used to study phase transitions , dynamics of waves and many transport phenomena in complex plasmas @xcite . a symmetry disordering escorting a crystalline - liquid phase transition has been investigated experimentally in @xcite . dislocation nucleation ( a shear instability ) has been reported in @xcite , albeit the importance of ssb for this phenomenon has not been explained . the results of these recent experimental observations can not be properly addressed without a deep understanding of this important issue . we would like to highlight this in the paper and report on the physics of spontaneous disordering of a cold plasma crystal , simulated melting and crystallization process , including associated defect clusters nucleation , dissociation , and symmetry alternation . these options are realizable in experimental complex plasmas , and can be mimicked in simulations , as we demonstrate below . it is well known that two broken symmetries distinguish the crystalline state from the liquid : the broken translational order and the broken orientational order . in two dimensions for ordinary crystals it is also well known that even at low temperatures the translational order is broken by spontaneous disordering mediated by thermal fluctuations @xcite . as a result , the fluctuation deflections ( disordering ) grow with distance and translational correlations decay ( algebraically , see @xcite ) . 2d plasma crystals also obey this common rule . the character of disordering may be deeply affected by the confinement forces , though . usually such an in - plane confinement is due to the bowl - shaped potential well self - maintained inside the discharge chamber , which to first order is approximately parabolic ( see , e.g. @xcite ) , that is @xmath1 , where @xmath2 is the distance , @xmath3 is the particle mass , and @xmath4 is the _ confinement parameter _ @xcite . ( the out - of - plane confining forces , controlling the position of the entire lattice , are normally much stronger ; below we consider the pure 2d - case , assuming , hence , an absolutely stiff out - of - plane confinement . ) the fluctuation spectra can be calculated in the following manner . the long - range phonon contribution to the free energy of a 2d system of particles interacting via the yukawa potential and confined by a shallow isotropic parabolic well can be conveniently represented as @xcite : @xmath5 @xmath6 where @xmath7 are the transverse ( shear wave ) and the longitudinal ( compressional wave ) sound speed , and @xmath8 are the fourier components of the vorticity @xmath9 and the divergency @xmath10 of the particle displacements @xmath11 , and @xmath12 is the wave vector ( @xmath13 ) . the unperturbed crystal is supposed to be hexagonal . the relationship ( [ eq.a ] ) provides ( see , e.g. @xcite ) the probability of the fluctuation @xmath14 . next , using it , we can calculate the averaged fluctuation spectral intensity per unit mass as @xmath15 where @xmath16 is the particle thermal velocity . it is known ( see @xcite ) that a lattice layer of a finite size @xmath17 is stably confined if roughly : @xmath18 here the parameter @xmath19 stands for an effective number of the nearest neighbors of any edge particle . note that according to ( [ eq.4 ] ) formally @xmath20 at @xmath21 . without confinement ( @xmath22 ) the fluctuation spectrum ( [ eq.3 ] ) apparently diverges @xmath23 at @xmath24 , and , as a consequence , in agreement with @xcite , the crystal ordering decays algebraically with the distance @xmath2 , i.e. the density - density correlation behaves as @xmath25 here @xmath26 , @xmath27 is the vector of the reciprocal lattice , and @xmath28 . it is assumed that @xmath29 is large compared to the interparticle separation @xmath30 . in the experiments @xmath31 is always finite ( though noticeably small , one or two orders of magnitude less than the frequency of the local caged oscillations of the individual particles @xcite ) . from ( [ eq.3 ] ) at non - vanishing @xmath4 it immediately follows that the fluctuations remain finite even at @xmath32 . this absence of a singularity alters the character of disordering from algebraic ( [ eq.5 ] ) to exponential at a scale depending on the confinement parameter : @xmath33 it is essential that both asymptotes algebraic and exponential must be treated as near - field ( @xmath34 ) and far - field ( @xmath35 ) approximations . hence it would be logical to assume that the ordering decay alternates with distance from algebraic to exponential . this is indeed in qualitative agreement with observations @xcite . remarkably ( [ eq.a])-([eq.3 ] ) are formally similar to the equations describing director fluctuations in nematic crystals in the presence of a magnetic field @xcite . the action of the magnetic field is known as suppressing the large - scale director fluctuations in liquid crystals . the length scale @xmath36 seems to be of a fundamental importance . the particles , experiencing a horizontal confinement , are distributed non - uniformly . the steady - state displacements of the particles @xmath37 in the plasma crystal from their ideal locations in a uniform 2d crystal represent a growing function with distance @xmath38 @xcite . the lattice breaks up when @xmath39 where @xmath40 is the lindemann parameter . since @xmath40=0.16 - 0.18 ( see , e.g. @xcite ) , it follows that the first row of defects most probably appears at @xmath41 . making use of ( [ eq.4 ] ) , ( [ eq.7 ] ) one can estimate the size of the domains ( or equivalent correlation length ) as : @xmath42 the correlation length ( [ eq.9 ] ) does not depend explicitly on the temperature . in other words , for purely topological reasons the big crystal spontaneously splits , assembling an array of sub - domains , even at zero temperature . the estimated values of @xmath43 agree well with those obtained in the simulation see fig . [ fig : cluster ] , and in experiments . for instance , it has been observed in @xcite that the crystal orientational order had a power law decay at distances @xmath44 in fairly good agreement with @xmath45 following from ( [ eq.9 ] ) . the one - plus correlation length ( [ eq.9 ] ) , unavoidably introducing a network of sub - domains to a lattice layer , is of crucial importance , e.g. , for observations of the so called _ hexatic state _ in the plasma crystals that is still an outstanding and controversial issue in complex plasma studies @xcite . one of the possible scenarios for melting ( recrystallization ) in a 2d complex plasma is a precipitous increase ( decrease ) in the density of the dislocations and the dislocation aggregates ( such as defect clusters , grain boundaries etc . ) @xcite . to realize this scenario in simulations , it is desirable to avoid any aforementioned complications associated with the lattice layer sectioning from the very beginning. a promising tool in that sense , allowing to create a defect - free initial lattice layer , is a hexagonal confinement cell proposed in @xcite . we performed a series of simulations that revealed several peculiarities in symmetry that are worth to mention . first , the order parameter of the paired defects dislocations , was systematically lower for 7-fold cells . this is not surprising actually from a purely geometric point of view because the 5-fold cell in a pair is more compact . second , simulations manifested that not only isolated pairs dislocations ( @xmath46@xmath47 ) , but also compact triplets like ( @xmath46@xmath47@xmath46 ) , quadruplets ( @xmath46@xmath47@xmath48@xmath49 ) etc . , or even elongated defect chains were quite frequent . actually they dominantly defined the symmetry of the entire particle suspension . it would certainly be promising to connect the cluster formation in ordered complex plasmas @xcite with the general percolation process known in many similar applications ( see , e.g. , @xcite ) . third , in such melted clusters , in agreement with recent experimental observations @xcite , the defect density permanently decreased upon cooling . at higher temperatures in the beginning of the recrystallization process , while the mutual interparticle collisions were still frequent , the defect density dropped exponentially . then , at lower temperatures , the decay rate significantly slowed down ( see fig.[fig : defect ] ) . a sharp drop in the defect numbers followed by a quasi - saturation resembles the well - known situation @xcite in which both thermal activation and tunneling events occur . hence , by analogy , the fact that in our case the system of defects behaves in a similar way could be naturally explained by an annihilation scenario which is presumably of the _ dissipative tunneling _ type @xcite at lower mean kinetic energies . nucleation of dislocations is another important example of spontaneous symmetry breaking on a scale of elementary cells . whatever the melting scenario would be true , still there would remain a question what mechanism explains nucleation of the primary dislocation clusters . recently this issue has been studied experimentally : spontaneous nucleation of the edge - dislocation pairs ( followed by their dissociation ) has been successfully observed at the kinetic level in the experiments with plasma crystals @xcite . since the burgers vector of the entire lattice is kept constant ( e.g. zero ) spontaneously created dislocations must be paired forming defect quadruplets of the type ( @xmath46@xmath47@xmath48@xmath49 ) . ( the burgers vector characterizes the magnitude and direction of the crystalline lattice distortion by a dislocation @xcite . ) these dislocation clusters were created in the lattice locations where the internal shear stress exceeded a threshold . it has also been shown that even an elementary act of nucleation is in fact a multi - scale process consisting of the latent pre - phase , prompt nucleation of a defect cluster , and dissociation of the cluster followed by the escape of free dislocations @xcite . in the experiments @xcite it was suggested that the stress that finally caused nucleation was affected by the differential crystal rotation . the exact reason of nucleation , however , was difficult to determine consistently . in simulations the nucleation conditions are certainly easier to identify . to demonstrate nucleation in simulations a deformable hexagonal cell is used ( fig.[fig : nucleation ] ) . it confines a 2d cloud of equally charged particles interacting pairwise via the yukawa ( the screened coulomb ) force : @xmath50 where @xmath51 is the relative coordinate and @xmath52 is the distance between the particles @xmath53 with the coordinates @xmath54 ; @xmath55 is the particle charge and @xmath56 is the screening length . the cell design is similar to that applied in @xcite to simulate melting and recrystallization process of the plasma crystal . the hexagonal simulation cell has the evident advantage of flexible shape , compared to , e.g. , a parabolic cell confinement . deforming the boundary of the cell , it is simple to manipulate the particles in a tractable way . an additional option of variable geometry enables an opportunity to separate or consolidate pure shear and simple shear deformation @xcite if desirable . the strain rate is controllable during deformation as well . [ fig : nucleation ] shows a simple - sheared particle lattice layer . at a properly chosen loading rate deformation affects the _ shear instability _ that ends up with nucleation of defect clusters in the bulk of the lattice layer . after a while , when deformation becomes stronger , the components of the clusters decoupled and the newly born free dislocations glided away in a similar manner as the dislocations observed in experiments . symmetry alternation is of primary importance for understanding nucleation of dislocation clusters . the compact cluster design is _ magic _ in the sense that the _ hexagonal _ symmetry of the particle system neatly turns into a nearly _ tetratic _ symmetry of the cluster core ( like lead turns into gold when touched by the philosophers stone ) , see fig . [ fig : topology ] . despite an apparent simplicity of the cluster interior only four nearest neighbor particles ( marked abcd in fig . [ fig : topology ] ) , the centers of the 5- and 7-fold cells , are in the core , to discover the cluster topology was certainly a challenge @xcite . in our case the interparticle interaction potential is of the screened yukawa type , hence more compact in contrast to the @xmath57 interaction potential in case of magnetically interacting super - paramagnetic colloid particles considered in @xcite . thus there is a unique opportunity to verify whether the core topology is universal . let us start with a simple model treating the cluster as constituted of two point - like dislocations , which are set apart at a distance @xmath2 and allowed to glide only along two fixed crystallographic planes separated by one lattice period @xmath30 , so that @xmath58 , where @xmath59 is the angle of mutual orientation of the cluster components with respect to the gliding plane . the interaction energy of the point dislocations having the counter - directed burgers vectors is @xcite : @xmath60+const.\ ] ] it has a minimum , a _ stable ground state _ , at @xmath61 . it corresponds to @xmath62 , hence tetragonal symmetry of the cluster core might be considered as preferred . this prediction agrees noticeably well with the results of simulations of finite clusters : on average in fig . [ fig : topology ] the edge - to - diagonal angle in the cluster core is @xmath63 . it is worth noting that the cluster core is nearly cyclic . a measure of it immediately follows from the famous ptolemy s inequality valid for any quadrilateral : @xmath64 where @xmath65 denote the ( ordered ) sides , and @xmath66 are the diagonals of the quadrilateral . over 80 % of the recognized clusters have @xmath67 for the simulation results shown in fig . [ fig : topology ] . for comparison a hexagonal four - side cell corresponds to @xmath68 . note also that a stable defect cluster could not be obtained only by shifting positions of four central particles from a hexagonal configuration to tetragonal one . such deformation would be reversible , hence unstable . a weakly deformed environment , impeding relaxation of the core particles back to the stable hexagonal configuration , is indeed a necessary lock making the deformation plastic , i.e. irreversible . at sufficiently strong external stress even a stable cluster dissociates . whatever is the orientation of the cluster as a whole , escaping dislocations can glide only along two crystallographic directions ( along burgers vectors , see fig . [ fig : topology ] ( a , b ) ) . this naturally explains the asymmetry of the escape directions and chirality of the defect configurations revealed by the newly nucleated dislocations in experiments @xcite . spontaneous symmetry breaking is a common and inherent feature of many systems in physics as well as other fields , it plays an important role , for example , from classical one - component plasmas to modern string representations @xcite , from the evolution of the early universe @xcite to the dynamics of a wide variety of small - scale systems @xcite . therefore it is not surprising that ssb is present also in the physics of plasma crystals . as example we have considered dislocations in plasma crystals which exhibit a spontaneous disordering , involved in the process of melting , and form clusters that are caused by a shear instability and show an interesting topological symmetry . the authors appreciate valuable discussions with dr . ivlev and dr . nosenko . k. r. stterlin , a. wysocki , a. v. ivlev , c. rth , h. m. thomas , m. rubin - zuzic , w. j. goedheer , v. e. fortov , a. m. lipaev , v. i. molotkov , o. f. petrov , g. e. morfill , and h. lwen , phys . lett . * 102 * , 085003 ( 2009 ) .
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spontaneous symmetry breaking is an essential feature of modern science .
we demonstrate that it also plays an important role in the physics of complex plasmas .
complex plasmas can serve as a powerful tool for observing and studying discrete types of symmetry and disordering at the kinetic level that numerous many - body systems exhibit .
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a mono - layer of graphite sheet , called graphene , has attracted growing interests recently.@xcite graphene exhibits a dirac cone with a linear dispersion at the corner of the first brillouin zone , resulting in a variety of novel transport phenomena of electrons . they stimulate theoretical and experimental studies taking account of analogy to physics of relativistic electron , such as klein tunneling@xcite and zitterbewegung@xcite . moreover , semi - infinite graphene and finite stripe of graphene with zigzag edges support a novel edge state with nearly flat dispersion.@xcite on the contrary , armchair edge does not support such an edge state . the flat dispersion implies that the density of state ( dos ) diverges at the flat band energy , in a striking contrast to the zero dos in bulk . so far , theoretical investigation of graphene heavily relies on the tight - binding model . it is not clear to what extent the edge states change in other models on the honeycomb lattice . here , we study a photonic analog of graphene model,@xcite namely , two - dimensional photonic crystal ( phc ) composed of the honeycomb lattice of dielectric cylinders embedded in a background substance . the honeycomb lattice consists of two inter - penetrating triangular lattices ( called a and b sub - lattices ) with the same lattice constant . in phc it is not rare to have the dirac cone in the dispersion diagram . triangular and honeycomb lattices of identical circular rods support multiple dirac cones at the corner of the first brillouin zone . it should be noted that they have the same point group of six - fold symmetry . doubly degenerate modes at the corner of the first brillouin zone exhibit the dirac cone owing to the point group symmetry.@xcite this fact suggests that , the symmetry is crucial and the dirac cone is not limited in the tight - binding model of electrons on the honeycomb lattice . some perturbation breaks the symmetry of the original honeycomb lattice and causes a crucial influence on the linear dispersion . in electronic systems the energy difference between a- and b - site atomic orbitals,@xcite periodic magnetic flux of zero average,@xcite and rashba spin - orbit interaction@xcite are such examples of the symmetry breaking . they break at least either of time - reversal symmetry ( trs ) or space - inversion symmetry ( sis ) or parities in plane . therefore , the point group of the original honeycomb lattice is reduced into a smaller group . as a result , the two - dimensional irreducible representations are prohibited , and the doubly - degenerate modes are lifted . the gap between the lifted modes is correlated with the magnitude of the symmetry breaking . the effective theory around a nearly degenerate point is described by the massive dirac hamiltonian , where the mass gap can be controlled via the degree of the symmetry breaking . in the honeycomb lattice phcs the trs is efficiently broken by applying a magnetic field parallel to the cylindrical axis . nonzero static magnetic field induces imaginary off - diagonal elements in the permittivity or permeability tensors , through the magneto - optical effect . the sis is broken if the a - site rods are different from the b - site rods.@xcite therefore , we can continuously tune the degree of the symmetry breaking in the honeycomb lattice phcs . this tunability is a great advantage of the photonic analog of graphene model and its extension.@xcite from a theoretical point of view , the tight - binding approximation , which is commonly used in modeling of graphene , is not widely applicable for photonic band calculation . accordingly , the non - bonding orbital in the nearest neighbor tight - binding approximation , which is responsible for flatness of the dispersion curve of the zigzag edge state , is completely absent in phcs . for example , even in the original honeycomb phc with both the trs and the sis , the dispersion of the zigzag edge states is not flat because of the absence of the non - bonding orbital . regarding the system with boundary , photonic system is quite distinct from electronic system . in the latter system the electrons near fermi level are prohibited to escape to the outer region via the work function , _ i.e. _ , a confining potential , and the wave functions of the electrons are evanescent in the outer region . therefore , to sustain an edge state , formation of the band gap in bulk is the minimum requirement . on the other hand , in the former system confining potentials for photon are absent at the boundary . energy of photon is always positive as in free space , and no energy barrier exists between the phc and free space . the simplest way to confine photonic edge states in the phc is to utilize the light cone . this restriction of the confinement makes photonic systems quit nontrivial in various aspects . in particular , the topological relation between bulk and edge@xcite is of high interest in photonic systems without trs . in quantum hall system nontrivial topology of bulk states leads to the emergence of chiral edge states , which are robust against localization effect . the edge states play a crucial role in this system.@xcite recently , haldane and raghu proposed one - way light waveguide realized in phcs without trs.@xcite explicit construction of such waveguides is demonstrated by several authors.@xcite this paper also shades light to this topic , by using simpler structure than those demonstrated so far . this paper is organized as follows . section ii is devoted to present bulk properties of the phc with and/or without trs and sis . a numerical method to deal with edge states is given in sec . properties of zigzag and armchair edge states are investigated in detail in secs . iv and v , respectively . a one - way light transport along the edge of a rectangular - shaped phc is demonstrated in sec . finally , summary and discussions are given in sec . let us consider two - dimensional phcs composed of the honeycomb array of circular cylinders embedded in air . the photonic band structure of the phcs with and without trs is shown in fig . [ band ] for the transverse magnetic ( tm ) polarization . for comparison , the photonic band structure of the transverse electric ( te ) polarization is also shown for the phc with trs . the sis holds in all the cases . ( color online ) the photonic band structure of the honeycomb lattice phcs of dielectric cylinders embedded in air . solid ( dashed ) line stands for the tm band structure of the phc with ( without ) trs . the sis holds in both the cases . the dielectric constant and radius of the cylinders are taken to be 12 and @xmath0 , respectively , where @xmath1 is the lattice constant . the magnetic permeability of the cylinders is taken to be 1 for the phc with trs and is given by eq . ( [ rodmu ] ) for the phc without trs . for comparison , the te band structure of the phc with trs is shown by doted line . , scaledwidth=45.0% ] here , the dielectric constants @xmath2 and radius @xmath3 of the a(b)-cylinders are taken to be 12 and @xmath0 , respectively . the magnetic permeability of the cylinders is taken to be 1 for the phc with trs , and has the tensor form given by @xmath4 for the phc without trs . the first , second , and third rows ( columns ) stand for @xmath5 and @xmath6 cartesian components , respectively . the cylindrical axis is taken to be parallel to the @xmath6 axis . the imaginary off - diagonal components of @xmath7 are responsible for the magneto - optical effect and break the trs . thus , parameter @xmath8 represents the degree of the trs breaking . as mentioned in introduction , for the phc with trs the dirac cone is found at the k point . in particular , the first ( lowest ) and second tm bands are in contact with each other at the k point . they are also in contact with the k point because of the spatial symmetry . this property is quite similar to the tight - binding electron in graphene . as for the dirac point at @xmath9 of the tm polarization , the fourth band is in contact with the fifth band at k ( and k ) , whereas the former and the latter are also in contact with the third and sixth bands , respectively at the @xmath10 point . concerning the te polarization , the dirac cones are not clearly visible , but are indeed formed between the second and third and between the fourth and fifth . on the other hand , in the phc without trs , all the degenerate modes at @xmath10 and k are lifted . the point group of this phc becomes @xmath11 and the point group of @xmath12 at the k point is @xmath13 . they are abelian groups , allowing solely one - dimensional representations . therefore , the degeneracy is forbidden . the energy gap between the lifted modes is proportional to @xmath8 if it is small enough . the sis breaking , @xmath14 , lifts the double degeneracy at k , but not at @xmath10 when the trs is preserved . the energy gap between the lifted modes is proportional to @xmath15.@xcite let us focus on the gap between the first and second tm bands of the phc as a function of the sis and trs breaking parameters . the phase diagram of the phc concerning the gap is shown in fig . [ phase ] . phase diagram of the honeycomb lattice phcs for the tm polarization . phase space is spanned by two parameters , @xmath16 and @xmath8 , which represent the sis and trs breaking , respectively . the average dielectric constant and the radius of the cylinders are kept fixed to @xmath17 and @xmath18 , respectively . , scaledwidth=45.0% ] at generic values of the parameters the gap opens . however , if we change the parameters along a certain curve in the parameter space , the gap remains to close as shown in fig . [ phase ] . this property implies that at finite @xmath8 the gap closes only at a certain value of @xmath15 . although the gap opens in both the regions above and below the curve , the two regions are topologically different , and are characterized by the chern numbers of the first and second photonic bands . the chern number is a topological integer defined by @xmath19 for each non - degenerate band . here , bz , uc , and @xmath20 stand for brillouin zone , unit cell , and the area of unit cell , respectively . the envelop function @xmath21 of the @xmath22-th bloch state at @xmath12 is of @xmath23 ( _ i.e. _ , the @xmath6 component of the electric field ) . in the upper region of fig . [ phase ] , @xmath24 and @xmath25 , whereas in the lower region @xmath26 . at the gap closing point , the chern number transfers between the upper and lower band under the topological number conservation law.@xcite we will see that the phase diagram correlates with a property of edge states in corresponding phc stripes . this correlation is a guiding principle to design a one - way light transport near phc edges.@xcite figure [ phase ] shows solely the phase diagram in the first quadrant of real @xmath15 and @xmath8 . the mirror reflection with respect to the @xmath15 axis gives the phase diagram in the fourth quadrant , where @xmath27 and @xmath28 are interchanged due to the inversion of @xmath8 . the phase diagram in the second and third quadrants is obtained by the mirror reflection with respect to the @xmath8 axis . the resulting phase diagram is similar to that obtained in a triangular lattice phc with anisotropic rods.@xcite so far , we have concentrated on properties of the phcs of infinite extent in plane . if the system has edges , there can emerge edge states which are localized near the edges and are evanescent both inside and outside the phc . in this section we introduce a phc stripe with two parallel boundaries . the boundaries are supposed to have infinite extent , so that the translational invariance along the boundary still holds . the edge states are characterized by bloch wave vector parallel to the boundary . optical properties of the phc stripe are described by the s - matrix . it relates the incident plane wave of parallel momentum @xmath29 to the outgoing plane wave of parallel momentum @xmath30 , where @xmath31 and @xmath32 are the reciprocal lattice vectors relevant to the periodicity parallel to the stripe.@xcite both the waves can be evanescent . to be precise , the s - matrix is defined by @xmath33 where @xmath34 is the plane - wave - expansion components of upward ( + ) and downward ( - ) incoming ( outgoing ) waves of parallel momentum @xmath30 , respectively . in our phcs under consideration the s - matrix can be calculated via the photonic layer - korringa - kohn - rostoker method@xcite as a function of parallel momentum @xmath35 and frequency @xmath36 . if the s - matrix is numerically available , the dispersion relation of the edge states is obtained according to the following secular equation : @xmath37.\end{aligned}\ ] ] strictly speaking , this equation also includes solutions of bulk states below the light line . if we search for the solutions inside pseudo gaps ( _ i.e. _ @xmath35-dependent gaps ) , solely the dispersion relations of the edge states are obtained . in actual calculation , however , the magnitude of @xmath38 $ ] becomes extremely small with increasing size of the matrix . the matrix size is given by the number of reciprocal lattice vectors taken into account in numerical calculation . in order to obtain numerical accuracy , we have to deal with larger matrix . therefore , this procedure to determine the edge states is generally unstable . instead , we employ the following scheme . suppose that the s - matrix is divided into two parts @xmath39 and @xmath40 that correspond to the division of the phc stripe into the upper and lower parts . this division is arbitrary , unless the upper or lower part is not empty . the following secular equation also determines the dispersion relation of the edge states : @xmath41.\end{aligned}\ ] ] this scheme is much stable for larger matrix . as far as true edge states are concerned , the secular equation has the zeros in real axis of frequency for a given real @xmath35 . here we should also mention leaky edge states ( _ i.e. _ , resonances near the edges ) , which are not evanescent outside the phc but are evanescent inside the phc . such an edge state is still meaningful , because the dos exhibits a peak there . the peak frequency as a function of parallel momentum @xmath35 follows a certain curve that is connected to the dispersion curve of the true edge states . to evaluate the leaky edge states , the method developed by ohtaka _ et al_@xcite is employed . in this method , the dos at fixed @xmath35 and @xmath36 is calculated with the truncated s - matrix of open diffraction channels . the unitarity of the truncated s - matrix enables us to determine the dos via eigen - phase - shifts of the s - matrix . a peak of the dos inside the pseudo gap corresponds to a leaky edge state . first , let us consider the zigzag edge . figure [ pband ] shows four sets of the projected band diagram of the honeycomb phc and the dispersion relation of the edge states localized near the zigzag edges . in fig . [ pband ] the shaded regions represent bulk states , whereas the blank regions correspond to the pseudo gap . inside the pseudo gap edge states can emerge . in the evaluation of the edge states , we assumed the phc stripe of @xmath42 , being @xmath43 the number of the layers along the direction perpendicular to the zigzag edges . the projected band diagrams at point @xmath44 in the phase diagram ( fig . [ phase ] ) and the dispersion curves of the edge states . the zigzag edge is assumed . the shaded regions represent bulk states . the edge states are of the phc stripes with 16 layers . thin solid line stands for the light line . the surface brillouin zone is taken to be @xmath45 in order to see the connectivity of the edge - state dispersion curves . , scaledwidth=45.0% ] here , we close up the first and second bands . higher bands are well separated from the lowest two bands . each set refers to either of four points indicated in the phase diagram of fig . [ phase ] . in accordance with the dirac cone in fig . [ band ] , the projected band structure of point @xmath46 also exhibits a point contact at @xmath47 and @xmath48 . the first and second bands are separated for @xmath49 and @xmath50 , but are nearly in contact at @xmath47 for @xmath51 . this is because @xmath51 is close to the phase boundary . except for the lower right panel , in which the trs and the sis are broken , the projected band diagrams and the edge - state dispersion curves are symmetric with respect to @xmath52 . this symmetry is preserved if either the trs or the sis holds . the time - reversal transformation implies @xmath53 where @xmath54 is the eigen - frequency of the @xmath22-th bloch state at given parameters of @xmath15 and @xmath8 , and @xmath55 is the momentum perpendicular to the edge . since @xmath8 is inverted , eq . ( [ t - transform ] ) is not a symmetry of the phc , but is just a transformation law . in the case of @xmath56 , after the projection concerning @xmath55 , the symmetry with respect to @xmath57 is obtained . this symmetry combined with the translational invariance under @xmath58 results in the symmetry with respect to @xmath52 . similarly , the space inversion results in @xmath59 the symmetry with respect to @xmath60 and @xmath61 is obtained at @xmath62 . when edge states are well defined in phcs with enough number of layers , their dispersion relation satisfies @xmath63 owing to the time - reversal and space - inversion transformations , respectively . here , @xmath64 and @xmath65 denote the dispersion relation of opposite edges of the phc stripe . at @xmath56 , both @xmath64 and @xmath65 are symmetric under the inversion of @xmath35 . in contrast , at @xmath62 they are interchanged . the resulting band diagram is symmetric with respect to @xmath60 and @xmath61 as in fig . [ pband ] . the upper left panel of fig . [ pband ] shows two almost - degenerate curves that are lifted a bit near the dirac point . this lifting comes from the hybridization between edge states of the opposite boundary , owing to finite width of the stripe . the lifting becomes smaller with increasing @xmath43 , and eventually two curves merge with each other . since @xmath46 corresponds to @xmath66 , we obtain @xmath67 owing to eqs . ( [ t - transform_edge ] ) and ( [ p - transform_edge ] ) , irrespective of @xmath35 . as is the same with in graphene , our edge states appear only in the region @xmath68 . however , the edge - state curves are not flat , in a striking contrast to the zigzag edge state in the nearest - neighbor tight - binding model of graphene . in the upper right panel two edge - state curves are separated in frequency and each curve terminates in the same bulk band . on the contrary , in the lower two panels the dispersion curves of the two edge states intersect one another at a particular point and each curve terminates at different bulk bands . for instance , in the lower left panel , the curve including @xmath69 terminates at the upper band near @xmath47 and at the lower band near @xmath70 . at other points in the parameter space , we found that the two edge - state curves are separated if the system is in the phase of zero chern number . otherwise , if the system is in the phase of non - zero chern number , the two curves intersect one another . the wave function of the edge state at marked points @xmath69 and @xmath71 is plotted in fig . [ edge ] . ( color online ) the electric field intensity @xmath72 of the true edge state at @xmath69 ( left panel ) and @xmath71 ( right panel ) in fig . [ pband ] . the intensity maxima is normalized as 1 . in the enlarged panels the poynting vector flow is also shown . , scaledwidth=50.0% ] we can easily see that the edge states at @xmath69 and @xmath71 are localized near different edges . this property is consistent with the fact that at @xmath62 , @xmath64 and @xmath65 are interchanged under the inversion of @xmath35 . the field configuration at @xmath69 is identical to that at @xmath71 after the space - inversion operation ( @xmath73 rotation ) . since the sis is preserved in this case , they are the sis partners . it is also remarkable that the electric field intensity is confined almost in the rods forming one particular sub - lattice . this field pattern is reminiscent of the non - bonding orbital of the zigzag edge state in graphene . the edge state at @xmath74 has the negative ( positive ) group velocity . moreover , no other bulk and edge states exist at the frequency . therefore , solely the propagation from left to right is allowed near the upper edge , while the propagation from left to right is allowed in the lower edge . in this way a one - way light transport is realized near a given edge . the one - way transport is robust against quenched disorder with long correlation length , because the edge states are out of the light line and the bulk states at the same frequency is completely absent.@xcite this is also the case in the lower right panel of fig . [ pband ] , although the frequency range of the one - way transport is very narrow . it should be noted that the non - correlated disorder would cause the scattering into the states above the light line , where the energy leakage takes place . detailed investigation of disorder effects is beyond the scope of the present paper . the results obtained in this section strongly support the bulk - edge correspondence , which was originally proven in the context of quantum hall systems@xcite and was discussed in the context of photonic systems recently.@xcite namely , the number of one - way edge states in a given two - dimensional omni - directional gap ( _ i.e. _ @xmath35-independent gap ) is equal to the sum of the chern numbers of the bulk bands below the gap . in our case the chern number of the lower ( upper ) band is equal to -1 ( 1 ) . a negative sign of the sum corresponds to the inverted direction of the edge propagation . accordingly , there is only one ( one - way ) state per edge in the gap between the first and second bands . moreover , no edge state is found between the second and third bands . this behavior is consistent with the chern numbers of the first and second bands , according to the bulk - edge correspondence . finally , let us briefly comment on the edge states in @xmath46 and @xmath49 . in @xmath46 the two edge states are completely degenerate at @xmath75 . for the system with narrow width , there appear the bonding and anti - bonding orbitals , each of which has an equal weight of the field intensity in both the zigzag edges . as for the edge states in @xmath49 , the upper ( lower ) edge states are localized near the upper ( lower ) zigzag edge . next , let us consider the armchair edge . the projected band diagram and the dispersion curves of the edge states are shown in fig . [ pband_armchair ] . we assumed the phc stripe with @xmath76 . the projected band diagrams at point @xmath44 in the phase diagram ( fig . [ phase ] ) and the dispersion curves of the edge states . the armchair edge is assumed . the shaded regions correspond to bulk states . the edge states are of the phc stripes with 64 layers . thin solid line stands for the light line . , scaledwidth=45.0% ] it should be noted that they are symmetric with respect to @xmath57 regardless of sis and trs . this property is understood by the combination of a parity transformation and eq . ( [ t - transform ] ) . under the parity transformation with respect to the mirror plane parallel to the armchair edges , @xmath77 by combining eqs . ( [ t - transform ] ) and ( [ parity ] ) , we obtain the symmetric projected band diagram with respect to @xmath57 . concerning the edge states , the parity transformation results in @xmath78 therefore , by combining eqs . ( [ t - transform_edge ] ) and ( [ parity_edge ] ) , we can derive that @xmath64 and @xmath65 are interchanged by the inversion of @xmath35 , regardless of sis and trs : @xmath79 equation ( [ interchange ] ) results in the degeneracy between @xmath64 and @xmath65 at the boundary of the surface brillouin zone . moreover , it is obvious from eq . ( [ parity_edge ] ) that the two edge states are completely degenerate at @xmath56 in the entire surface brillouin zone . in the armchair projection the k and k points in the first brillouin zone are mapped on the same point @xmath57 in the surface brillouin zone , being above the light line . therefore , possible edge states relevant to the dirac cone are leaky , unless the region outside the phc is screened . accordingly , the dos of an armchair edge state at fixed @xmath35 shows up as a lorentzian peak , in a striking contrast to that of a zigzag edge state being a delta - function peak . the dispersion relation of the leaky edge states depends strongly on the number of layers @xmath43 . however , if @xmath43 is large enough , the @xmath43-dependence disappears . we found that at large enough @xmath43 , the leaky edge states correlate with the chern number fairly well . in the case as @xmath50 where the chern numbers of the upper and lower bands are nonzero , we found a segment of the dispersion curve of the leaky edge state whose bottom is at the lower band edge , as shown in the lower left panel of fig . [ pband_armchair ] . there also appear another segment of the dispersion curve which crosses the light line . across the phase boundary , the upper band touches to and separates from the lower band . after the separation as the case @xmath80 , the bottom of the former segment moves from the lower band edge to the upper band edge as shown in the lower right panel of fig . [ pband_armchair ] . by increasing @xmath15 , this segment hides among the upper bulk band ( not shown ) . we should note that the dispersion curve of the leaky edge states is obtained by tracing the peak frequencies of the dos as a function of @xmath35 . if a peak becomes a shoulder , we stopped tracing the curve and indicated shoulder frequencies by dotted curve . this is the case for @xmath50 and @xmath80 . for @xmath50 , the dos changes its shape from peak to shoulder at @xmath81 . this is why the segment including @xmath82 and @xmath83 seems to terminate around there . however , we can distinguish this shoulder in the region @xmath84 and crosses the light line . in the dos spectrum of @xmath80 , we can find two shoulders just below the peaks of bulk states in the region @xmath85 . again , they merge each other and become an asymmetric peak for @xmath86 . such an asymmetric peak consists of two peaks with different heights and widths , which come from the lifting of the degenerate edge states in the limit of @xmath56 . actually , for @xmath46 and @xmath49 in which the edge states are doubly - degenerate , we can see a nearly - symmetric single peak for the leaky edge states in each case . as in the case of zigzag edge , the leaky edge states in the two - dimensional omni - directional gap exhibit a one - way light transport if the relevant chern number is nonzero . here , we consider the structure with two horizontal armchair edges . the incident wave with positive @xmath35 coming from the bottom can not excite the leaky edge state just above the lower band edge , _ e.g. _ , state @xmath83 in fig . [ pband_armchair ] . however , the incident wave with negative @xmath35 coming from the bottom can excite the leaky edge state , _ e.g. _ , at @xmath82 . in the latter case , the leaky edge state has negative group velocity , traveling from right to left . this relation becomes inverted for the plane wave coming from the top . the incident plane wave with positive ( negative ) @xmath35 can ( can not ) excite the leaky edge state localized near the upper armchair edge . this edge state has positive group velocity , traveling from left to right . in this way , one - way light transport is realized as in the zigzag edge case . under quenched disorder the one - way transport is protected against the mixing with bulk states , because no bulk state exists in the omni - directional gap . however , in contrast to the zigzag edge case , even the disorder with long correlation length could enhance the energy leakage to the outer region . figure [ gkedgeconf ] shows the electric field intensity @xmath72 induced by the incident plane wave whose @xmath36 and @xmath35 are at the marked points ( @xmath82 and @xmath83 ) in fig . [ pband_armchair ] . the intensity of the incident plane wave is taken to be 1 and the field configuration above @xmath87 is omitted . ( color online ) the electric field intensity @xmath72 induced by the incident plane wave having @xmath88 at @xmath82 ( left panel ) and @xmath83 ( right panel ) in fig . [ pband_armchair ] . the incident plane wave of unit intensity comes from the bottom of the structure . in the enlarged panels the poynting vector flow is also shown . , scaledwidth=50.0% ] although , the dispersion curve is symmetric with respect to @xmath57 , the field configuration is quite asymmetric . of particular importance is the near - field pattern around the lower edge . in the left panel the strongest field intensity of order 40 is found in the boundary armchair layer , whereas in the right panel it is found outside the phc with much smaller intensity . in both the cases , the transmittances are the same and nearly equal to zero . accordingly , no field enhancement is observed near the upper edge ( not shown ) . the remarkable contrast of the field profiles indicates that the leaky edge state with horizontal energy flow is excited in the left panel , but is not in the right panel . if the plane wave is incident from the top , the field pattern exhibits an opposite behavior . that is , the plane wave with @xmath36 and @xmath35 at @xmath83 from the top excites the leaky edge state localized near the upper edge , but at @xmath82 it can not excite the leaky edge state . the property of each edge state is also understood as follows . when we scan @xmath35 from negative to positive along the dispersion curve of the leaky edge state , the localized center of the edge state transfers from one edge to the other . the critical point is at the bottom of the dispersion curve , where the edge state merges to the bulk state of the lower band . it is extended inside the phc , making a bridge from one edge to the other . the entire picture is consistent with the interchange of @xmath64 and @xmath89 under the inversion of @xmath35 . finally , let us comment on the field configuration of other edge states . for @xmath46 and @xmath49 , the edge states are degenerate between the upper and lower edges . accordingly , the incident plane wave coming from the bottom ( top ) of the structure excites the leaky edge states localized around the bottom ( top ) edge . it is regardless of the sign of @xmath35 . for @xmath50 and @xmath80 , the edge - state curve that crosses the light line corresponds to an asymmetric peak in the dos , which is actually the sum of two peaks . it is difficult to separate the two peaks , because they are overlapped in frequency . thus , the edge states can be excited by the incident wave coming from both top and bottom of the phc . concerning the quadratic edge state around @xmath57 of @xmath80 , a similar contrast in the field configuration between positive and negative @xmath35 is obtained as in @xmath82 and @xmath83 . however , under quenched disorder this edge state readily mixes with bulk states that exist at the same frequency . the direction of the one - way transport in the zigzag edge is consistent with that in the armchair edge . let us consider a rectangular - shaped phc whose four edges are zigzag , armchair , zigzag , and armchair in a clockwise order . the one - way transport found in figs . [ pband ] and [ pband_armchair ] must be clockwise in this geometry . to verify it certainly happens , we performed a numerical simulation of the light transport in the rectangular - shaped phc . the multiple - scattering method is employed along with a gaussian beam incidence.@xcite we assume @xmath90 for the zigzag edges and @xmath76 for the armchair edges . the incident gaussian beam is focused at the midpoint of the front armchair edge . the electric field intensity @xmath72 at the focused point is normalized as 1 and the beam waist is @xmath91 . the frequency and the incident angle of the beam are taken to be @xmath92 and @xmath93 , which corresponds to the leaky edge state very close to the @xmath82 point . the beam waist size is chosen to avoid possible diffraction at the corner of the phc and not to excite the states near the @xmath83 point at the same time . resulting electric field intensity @xmath72 is plotted in fig . [ onewayconf ] . ( color online ) the electric field intensity @xmath72 induced by the time - harmonic gaussian beam coming from the left of the rectangular - shaped phc . the four edges are either zigzag ( top and bottom ) or armchair ( left and right ) . the beam is focused at the mid point of the left armchair edge with the unit electric field intensity and beam waist of @xmath91 . , scaledwidth=45.0% ] the incident beam is almost reflected at the left ( armchair ) edge , forming the interference pattern in the left side of the phc . however , as in the left panel of fig . [ gkedgeconf ] , the leaky edge state is certainly excited there . this edge state propagates upward , and is diffracted at the upper left corner . a certain portion of the energy turns into the zigzag edge state localized near the upper edge . this edge state propagates from left to right . the energy leakage at the upper edge is very small compared to that in the left and right edges . this zigzag edge state is more or less diffracted at the upper right corner . however , the down - going armchair edge state is certainly excited in the right edge . obviously , the field intensity of the right edge reduces with reducing @xmath94 coordinate . this behavior is consistent with the energy leakage of the armchair edge state . finally , the field intensity almost vanished at the lower right corner . in this way , the clockwise one - way light transport is realized in the rectangular - shaped phc . we also confirmed that the incident beam with the same parameters but inverted incident angle ( @xmath95 ) does not excite the counterclockwise one - way transport along the edges . the incident beam is just reflected without exciting the relevant leaky edge state in accordance with the right panel of fig . [ gkedgeconf ] . in summary , we have presented a numerical analysis on the bulk and edge states in honeycomb lattice phcs as a photonic analog of graphene model and its extension . in the tm polarization the dirac cone emerges between the first and second bands . the mass gap in the dirac cone is controllable by the parameters of the sis or trs breaking . on a certain curve in the parameter space , the band touching takes place . this curve divides the parameter space into two topologically - distinct regions . one is characterized by zero chern number of the upper and lower bands , and the other is characterized by chern number of @xmath96 . of particular importance is the correlation between the chern number in bulk and light transport near edge . non - zero chern number in bulk photonic bands results in one - way light transport near the edge . it is quite similar to the bulk - edge correspondence found in quantum hall systems . in this paper we focus on the tm polarization in rod - in - air type phcs . this is mainly because the band touching takes place between the lowest two bands and they are well separated from higher bands by the wide band gap , provided that the refractive index of the rods are high enough . in rod - in - air type phcs the te polarization results in the band touching between the second and third bands . however , the dirac cone is not clearly visible , although it is certainly formed . as for hole - in - dielectric type phcs , an opposite tendency is found . namely , the band touching between the lowest two bands takes place only in the te polarization . in this case the distance between the boundary column of air holes and the phc edge affect edge states . therefore , we must take account of this parameter to determine the dispersion curves of the edge states . concerning the trs breaking , we have introduced imaginary off - diagonal components in the permeability tensor . this is the most efficient way to break the trs for the tm polarization . such a permeability tensor is normally not available in visible frequency range.@xcite however , in ghz range it is possible to obtain @xmath8 of order 10 . such a large @xmath8 is necessary to obtain a robust one - way transport against thermal fluctuations , etc . in the numerical setup we assume an intermediate frequency range with smaller @xmath8 . on the other hand , in the te polarization , the trs can be efficiently broken by imaginary off - diagonal components in the permittivity tensor . in this case the phc without the trs can operate in visible frequency range . however , strong magnetic field is necessary in order to induce large imaginary off - diagonal components of the permittivity tensor . thus , it is strongly desired to explorer low - loss optical media with large magneto - optical effect , in order to have robust one - way transport . recently , another photonic analog of graphene , namely , honeycomb array of metallic nano - particles , is proposed and analyzed theoretically.@xcite particle plasmon resonances in the nano - particles act as if localized orbitals in carbon atom . the tight - binding picture is thus reasonably adapted to this system , and nearly flat bands are found in the zigzag edge . vectorial nature of photon plays a crucial role there , giving rise to a remarkable feature in the dispersion curves of the edge states in the quasi - static approximation . in contrast , vectorial nature of photon is minimally introduced in our model , but a full analysis including possible retardation effects and symmetry - breaking effects has been made . effects of the te - tm mixing in off - axis propagation are an important issue in our system . in particular , it is interesting to study to what extent the bulk - edge correspondence is modified . we hope this paper stimulates further investigation based on the analogy between electronic and photonic systems on honeycomb lattices .
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this paper presents a theoretical analysis on bulk and edge states in honeycomb lattice photonic crystals with and without time - reversal and/or space - inversion symmetries .
multiple dirac cones are found in the photonic band structure and the mass gaps are controllable via symmetry breaking .
the zigzag and armchair edges of the photonic crystals can support novel edge states that reflect the symmetries of the photonic crystals .
the dispersion relation and the field configuration of the edge states are analyzed in detail in comparison to electronic edge states .
leakage of the edge states to free space is inherent in photonic systems and is fully taken into account in the analysis .
a topological relation between bulk and edge , which is analogous to that found in quantum hall systems , is also verified .
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we study degrees of freedom , or the `` effective number of parameters , '' in @xmath2-penalized linear regression problems . in particular , for a response vector @xmath3 , predictor matrix @xmath4 and tuning parameter @xmath5 , we consider the lasso problem [ @xcite , @xcite ] @xmath6 the above notation emphasizes the fact that the solution @xmath7 may not be unique [ such nonuniqueness can occur if @xmath8 . throughout the paper , when a function @xmath9 may have a nonunique minimizer over its domain @xmath1 , we write @xmath10 to denote the set of minimizing @xmath11 values , that is , @xmath12 . a fundamental result on the degrees of freedom of the lasso fit was shown by @xcite . the authors show that if @xmath13 follows a normal distribution with spherical covariance , @xmath14 , and @xmath15 are considered fixed with @xmath16 , then @xmath17 where @xmath18 denotes the active set of the unique lasso solution at @xmath13 , and @xmath19 is its cardinality . this is quite a well - known result , and is sometimes used to informally justify an application of the lasso procedure , as it says that number of parameters used by the lasso fit is simply equal to the ( average ) number of selected variables . however , we note that the assumption @xmath16 implies that @xmath20 ; in other words , the degrees of freedom result ( [ eq : lassodffull ] ) does not cover the important `` high - dimensional '' case @xmath21 . in this case , the lasso solution is not necessarily unique , which raises the questions : * can we still express degrees of freedom in terms of the active set of a lasso solution ? * if so , which active set ( solution ) would we refer to ? in section [ sec : lasso ] , we provide answers to these questions , by proving a stronger result when @xmath0 is a general predictor matrix . we show that the subspace spanned by the columns of @xmath0 in @xmath22 is almost surely unique , where `` almost surely '' means for almost every @xmath3 . furthermore , the degrees of freedom of the lasso fit is simply the expected dimension of this column space . we also consider the generalized lasso problem , @xmath23 where @xmath24 is a penalty matrix , and again the notation emphasizes the fact that @xmath7 need not be unique [ when @xmath8 . this of course reduces to the usual lasso problem ( [ eq : lasso ] ) when @xmath25 , and @xcite demonstrate that the formulation ( [ eq : genlasso ] ) encapsulates several other important problems including the fused lasso on any graph and trend filtering of any order by varying the penalty matrix @xmath1 . the same paper shows that if @xmath13 is normally distributed as above , and @xmath26 are fixed with @xmath16 , then the generalized lasso fit has degrees of freedom @xmath27.\ ] ] here @xmath28 denotes the boundary set of an optimal subgradient to the generalized lasso problem at @xmath13 ( equivalently , the boundary set of a dual solution at @xmath13 ) , @xmath29 denotes the matrix @xmath1 after having removed the rows that are indexed by @xmath30 , and @xmath31 , the dimension of the null space of @xmath29 . it turns out that examining ( [ eq : genlassodffull ] ) for specific choices of @xmath1 produces a number of interpretable corollaries , as discussed in @xcite . for example , this result implies that the degrees of freedom of the fused lasso fit is equal to the expected number of fused groups , and that the degrees of freedom of the trend filtering fit is equal to the expected number of knots @xmath32 @xmath33 , where @xmath34 is the order of the polynomial . the result ( [ eq : genlassodffull ] ) assumes that @xmath16 and does not cover the case @xmath21 ; in section [ sec : genlasso ] , we derive the degrees of freedom of the generalized lasso fit for a general @xmath0 ( and still a general @xmath1 ) . as in the lasso case , we prove that there exists a linear subspace @xmath35 that is almost surely unique , meaning that it will be the same under different boundary sets @xmath30 corresponding to different solutions of ( [ eq : genlasso ] ) . the generalized lasso degrees of freedom is then the expected dimension of this subspace . our assumptions throughout the paper are minimal . as was already mentioned , we place no assumptions whatsoever on the predictor matrix @xmath36 or on the penalty matrix @xmath24 , considering them fixed and nonrandom . we also consider @xmath37 fixed . for theorems [ thm : lassodfequi ] , [ thm : lassodfact ] and [ thm : genlassodf ] we assume that @xmath13 is normally distributed , @xmath38 for some ( unknown ) mean vector @xmath39 and marginal variance @xmath40 . this assumption is only needed in order to apply stein s formula for degrees of freedom , and none of the other lasso and generalized lasso results in the paper , namely lemmas [ lem : lassoproj ] through [ lem : invbound ] , make any assumption about the distribution of @xmath13 . this paper is organized as follows . the rest of the introduction contains an overview of related work , and an explanation of our notation . section [ sec : prelim ] covers some relevant background material on degrees of freedom and convex polyhedra . though the connection may not be immediately obvious , the geometry of polyhedra plays a large role in understanding problems ( [ eq : lasso ] ) and ( [ eq : genlasso ] ) , and section [ sec : poly ] gives a high - level view of this geometry before the technical arguments that follow in sections [ sec : lasso ] and [ sec : genlasso ] . in section [ sec : lasso ] , we derive two representations for the degrees of freedom of the lasso fit , given in theorems [ thm : lassodfequi ] and [ thm : lassodfact ] . in section [ sec : genlasso ] , we derive the analogous results for the generalized lasso problem , and these are given in theorem [ thm : genlassodf ] . as the lasso problem is a special case of the generalized lasso problem ( corresponding to @xmath25 ) , theorems [ thm : lassodfequi ] and [ thm : lassodfact ] can actually be viewed as corollaries of theorem [ thm : genlassodf ] . the reader may then ask : why is there a separate section dedicated to the lasso problem ? we give two reasons : first , the lasso arguments are simpler and easier to follow than their generalized lasso counterparts ; second , we cover some intermediate results for the lasso problem that are interesting in their own right and that do not carry over to the generalized lasso perspective . section [ sec : disc ] contains some final discussion . all of the degrees of freedom results discussed here assume that the response vector has distribution @xmath14 , and that the predictor matrix @xmath0 is fixed . to the best of our knowledge , @xcite were the first to prove a result on the degrees of freedom of the lasso fit , using the lasso solution path with @xmath41 moving from @xmath42 to @xmath43 . the authors showed that when the active set reaches size @xmath34 along this path , the lasso fit has degrees of freedom exactly @xmath34 . this result assumes that @xmath0 has full column rank and further satisfies a restrictive condition called the `` positive cone condition , '' which ensures that as @xmath41 decreases , variables can only enter , and not leave , the active set . subsequent results on the lasso degrees of freedom ( including those presented in this paper ) differ from this original result in that they derive degrees of freedom for a fixed value of the tuning parameter @xmath41 , and not a fixed number of steps @xmath34 taken along the solution path . as mentioned previously , @xcite established the basic lasso degrees of freedom result ( for fixed @xmath41 ) stated in ( [ eq : lassodffull ] ) . this is analogous to the path result of @xcite ; here degrees of freedom is equal to the expected size of the active set ( rather than simply the size ) because for a fixed @xmath41 the active set is a random quantity , and can hence achieve a random size . the proof of ( [ eq : lassodffull ] ) appearing in @xcite relies heavily on properties of the lasso solution path . as also mentioned previously , @xcite derived an extension of ( [ eq : lassodffull ] ) to the generalized lasso problem , which is stated in ( [ eq : genlassodffull ] ) for an arbitrary penalty matrix @xmath1 . their arguments are not based on properties of the solution path , but instead come from a geometric perspective much like the one developed in this paper . both of the results ( [ eq : lassodffull ] ) and ( [ eq : genlassodffull ] ) assume that @xmath16 ; the current work extends these to the case of an arbitrary matrix @xmath0 , in theorems [ thm : lassodfequi ] , [ thm : lassodfact ] ( the lasso ) and [ thm : genlassodf ] ( the generalized lasso ) . in terms of our intermediate results , a version of lemmas [ lem : lcequi ] , [ lem : lcact ] corresponding to @xmath16 appears in @xcite , and a version of lemma [ lem : lcbound ] corresponding to @xmath16 appears in @xcite [ furthermore , @xcite only consider the boundary set representation and not the active set representation ] . lemmas [ lem : nonexp ] , [ lem : locaff ] and the conclusions thereafter , on the degrees of freedom of the projection map onto a convex polyhedron , are essentially given in @xcite , though these authors state and prove the results in a different manner . in preparing a draft of this manuscript , it was brought to our attention that other authors have independently and concurrently worked to extend results ( [ eq : lassodffull ] ) and ( [ eq : genlassodffull ] ) to the general @xmath0 case . namely , @xcite prove a result on the lasso degrees of freedom , and @xcite prove a result on the generalized lasso degrees of freedom , both for an arbitrary @xmath0 . these authors results express degrees of freedom in terms of the active sets of special ( lasso or generalized lasso ) solutions . theorems [ thm : lassodfact ] and [ thm : genlassodf ] express degrees of freedom in terms of the active sets of any solutions , and hence the appropriate application of these theorems provides an alternative verification of these formulas . we discuss this in detail in the form of remarks following the theorems . in this paper , we use @xmath44 , @xmath45 and @xmath46 to denote the column space , row space and null space of a matrix @xmath47 , respectively ; we use @xmath48 and @xmath49 to denote the dimensions of @xmath44 [ equivalently , @xmath45 ] and @xmath46 , respectively . we write @xmath51 for the the moore penrose pseudoinverse of @xmath47 ; for a rectangular matrix @xmath47 , recall that @xmath52 . we write @xmath53 to denote the projection matrix onto a linear subspace @xmath54 , and more generally , @xmath55 to denote the projection of a point @xmath11 onto a closed convex set @xmath56 . for readability , we sometimes write @xmath57 ( instead of @xmath58 ) to denote the inner product between vectors @xmath59 and @xmath60 . for a set of indices @xmath61 satisfying @xmath62 , and a vector @xmath63 , we use @xmath64 to denote the subvector @xmath65 . we denote the complementary subvector by @xmath66 . the notation is similar for matrices . given another subset of indices @xmath67 with @xmath68 , and a matrix @xmath69 , we use @xmath70 to denote the submatrix @xmath71 \in{\mathbb{r}}^{k \times\ell}.\ ] ] in words , rows are indexed by @xmath72 , and columns are indexed by @xmath73 . when combining this notation with the transpose operation , we assume that the indexing happens first , so that @xmath74 . as above , negative signs are used to denote the complementary set of rows or columns ; for example , @xmath75 . to extract only rows or only columns , we abbreviate the other dimension by a dot , so that @xmath76 and @xmath77 ; to extract a single row or column , we use @xmath78 or @xmath79 . finally , and most importantly , we introduce the following shorthand notation : * for the predictor matrix @xmath80 , we let @xmath81 . * for the penalty matrix @xmath24 , we let @xmath82 . in other words , the default for @xmath0 is to index its columns , and the default for @xmath1 is to index its rows . this convention greatly simplifies the notation in expressions that involve multiple instances of @xmath83 or @xmath84 ; however , its use could also cause a great deal of confusion , if not properly interpreted by the reader ! the following two sections describe some background material needed to follow the results in sections [ sec : lasso ] and [ sec : genlasso ] . if the data vector @xmath3 is distributed according to the homoskedastic model @xmath85 , meaning that the components of @xmath13 are uncorrelated , with @xmath86 having mean @xmath87 and variance @xmath88 for @xmath89 , then the degrees of freedom of a function @xmath90 with @xmath91 , is defined as @xmath92 this definition is often attributed to @xcite or @xcite , and is interpreted as the `` effective number of parameters '' used by the fitting procedure @xmath93 . note that for the linear regression fit of @xmath3 onto a fixed and full column rank predictor matrix @xmath36 , we have @xmath94 , and @xmath95 , which is the number of fitted coefficients ( one for each predictor variable ) . furthermore , we can decompose the risk of @xmath96 , denoted by @xmath97 , as @xmath98 a well - known identity that leads to the derivation of the @xmath99 statistic [ @xcite ] . for a general fitting procedure @xmath93 , the motivation for the definition ( [ eq : df ] ) comes from the analogous decomposition of the quantity @xmath100 , @xmath101 therefore a large difference between risk and expected training error implies a large degrees of freedom . why is the concept of degrees of freedom important ? one simple answer is that it provides a way to put different fitting procedures on equal footing . for example , it would not seem fair to compare a procedure that uses an effective number of parameters equal to 100 with another that uses only 10 . however , assuming that these procedures can be tuned to varying levels of adaptivity ( as is the case with the lasso and generalized lasso , where the adaptivity is controlled by @xmath41 ) , one could first tune the procedures to have the same degrees of freedom , and then compare their performances . doing this over several common values for degrees of freedom may reveal , in an informal sense , that one procedure is particularly efficient when it comes to its parameter usage versus another . a more detailed answer to the above question is based the risk decomposition ( [ eq : riskd ] ) . the decomposition suggests that an estimate @xmath102 of degrees of freedom can be used to form an estimate of the risk , @xmath103 furthermore , it is straightforward to check that an unbiased estimate of degrees of freedom leads to an unbiased estimate of risk ; that is , @xmath104 $ ] implies @xmath105 $ ] . hence , the risk estimate ( [ eq : riskhat ] ) can be used to choose between fitting procedures , assuming that unbiased estimates of degrees of freedom are available . [ it is worth mentioning that bootstrap or monte carlo methods can be helpful in estimating degrees of freedom ( [ eq : df ] ) when an analytic form is difficult to obtain . ] the natural extension of this idea is to use the risk estimate ( [ eq : riskhat ] ) for tuning parameter selection . if we suppose that @xmath93 depends on a tuning parameter @xmath106 , denoted @xmath107 , then in principle one could minimize the estimated risk over @xmath41 to select an appropriate value for the tuning parameter , @xmath108 this is a computationally efficient alternative to selecting the tuning parameter by cross - validation , and it is commonly used ( along with similar methods that replace the factor of @xmath109 above with a function of @xmath110 or @xmath111 ) in penalized regression problems . even though such an estimate ( [ eq : tunsel ] ) is commonly used in the high - dimensional setting ( @xmath21 ) , its asymptotic properties are largely unknown in this case , such as risk consistency , or relatively efficiency compared to the cross - validation estimate . @xcite proposed the risk estimate ( [ eq : riskhat ] ) using a particular unbiased estimate of degrees of freedom , now commonly referred to as _ stein s unbiased risk estimate _ ( sure ) . stein s framework requires that we strengthen our distributional assumption on @xmath13 and assume normality , as stated in ( [ eq : normal ] ) . we also assume that the function @xmath93 is continuous and almost differentiable . ( the precise definition of almost differentiability is not important here , but the interested reader may take it to mean that each coordinate function @xmath112 is absolutely continuous on almost every line segment parallel to one of the coordinate axes . ) given these assumptions , stein s main result is an alternate expression for degrees of freedom , @xmath113,\ ] ] where the function @xmath114 is called the divergence of @xmath93 . immediately following is the unbiased estimate of degrees of freedom , @xmath115 we pause for a moment to reflect on the importance of this result . from its definition ( [ eq : df ] ) , we can see that the two most obvious candidates for unbiased estimates of degrees of freedom are @xmath116 \bigr ) y_i.\ ] ] to use the first estimate above , we need to know @xmath117 ( remember , this is ultimately what we are trying to estimate ! ) . using the second requires knowing @xmath118 $ ] , which is equally impractical because this invariably depends on @xmath117 . on the other hand , stein s unbiased estimate ( [ eq : steindfhat ] ) does not have an explicit dependence on @xmath117 ; moreover , it can be analytically computed for many fitting procedures @xmath93 . for example , theorem [ thm : lassodfact ] in section [ sec : lasso ] shows that , except for @xmath13 in a set of measure zero , the divergence of the lasso fit is equal to @xmath119 with @xmath18 being the active set of a lasso solution at @xmath13 . hence , stein s formula allows for the unbiased estimation of degrees of freedom ( and subsequently , risk ) for a broad class of fitting procedures @xmath93something that may have not seemed possible when working from the definition directly . a set @xmath120 is called a _ convex polyhedron _ , or simply a _ polyhedron _ , if @xmath56 is the intersection of finitely many half - spaces , @xmath121 where @xmath122 and @xmath123 . ( note that we do not require boundedness here ; a bounded polyhedron is sometimes called a polytope . ) see figure [ fig : poly ] for an example . there is a rich theory on polyhedra ; the definitive reference is @xcite , and another good reference is @xcite . as this is a paper on statistics and not geometry , we do not attempt to give an extensive treatment of the properties of polyhedra . we do , however , give two properties ( in the form of two lemmas ) that are especially important with respect to our statistical problem ; our discussion will also make it clear why polyhedra are relevant in the first place . . _ _ ] from its definition ( [ eq : poly ] ) , it follows that a polyhedron is a closed convex set . the first property that we discuss does not actually rely on the special structure of polyhedra , but only on convexity . for any closed convex set @xmath120 and any point @xmath124 , there is a unique point @xmath125 minimizing @xmath126 . to see this , note that if @xmath127 is another minimizer , @xmath128 , then by convexity @xmath129 , and @xmath130 , a contradiction . therefore , the projection map onto @xmath56 is indeed well defined , and we write this as @xmath131 , @xmath132 for the usual linear regression problem , where @xmath3 is regressed onto @xmath36 , the fit @xmath133 can be written in terms of the projection map onto the polyhedron @xmath134 , as in @xmath135 . furthermore , for both the lasso and generalized lasso problems , ( [ eq : lasso ] ) and ( [ eq : genlasso ] ) , it turns out that we can express the fit as the residual from projecting onto a suitable polyhedron @xmath136 , that is , @xmath137 this is proved in lemma [ lem : lassoproj ] for the lasso and in lemma [ lem : genlassoproj ] for the generalized lasso ( the polyhedron @xmath56 depends on @xmath15 for the lasso case , and on @xmath26 for the generalized lasso case ) . our first lemma establishes that both the projection map onto a closed convex set and the residual map are nonexpansive , hence continuous and almost differentiable everywhere . these are the conditions needed to apply stein s formula . [ lem : nonexp ] for any closed convex set @xmath120 , both the projection map @xmath138 and the residual projection map @xmath139 are nonexpansive . that is , they satisfy @xmath140 therefore , @xmath141 and @xmath142 are both continuous and almost differentiable . the proof can be found in appendix [ app : nonexp ] . lemma [ lem : nonexp ] will be quite useful later in the paper , as it will allow us to use stein s formula to compute the degrees of freedom of the lasso and generalized lasso fits , after showing that these fits are indeed the residuals from projecting onto closed convex sets . the second property that we discuss uses the structure of polyhedra . unlike lemma [ lem : nonexp ] , this property will not be used directly in the following sections of the paper ; instead , we present it here to give some intuition with respect to the degrees of freedom calculations to come . the property can be best explained by looking back at figure [ fig : poly ] . loosely speaking , the picture suggests that we can move the point @xmath11 around a bit and it will still project to the same face of @xmath56 . another way of saying this is that there is a neighborhood of @xmath11 on which @xmath141 is simply the projection onto an affine subspace . this would not be true if @xmath11 is in some exceptional set , which is made up of rays that emanate from the corners of @xmath56 , like the two drawn in the bottom right corner of figure . however , the union of such rays has measure zero , so the map @xmath141 is locally an affine projection , almost everywhere . this idea can be stated formally as follows . [ lem : locaff ] let @xmath120 be a polyhedron . for almost every @xmath143 , there is an associated neighborhood @xmath144 of @xmath11 , linear subspace @xmath145 and point @xmath146 , such that the projection map restricted to @xmath144 , @xmath147 , is @xmath148 which is simply the projection onto the affine subspace @xmath149 . the proof is given in appendix [ app : locaff ] . these last two properties can be used to derive a general expression for the degrees of freedom of the fitting procedure @xmath150 , when @xmath120 is a polyhedron . [ a similar formula holds for @xmath151 . ] lemma [ lem : nonexp ] tells us that @xmath142 is continuous and almost differentiable , so we can use stein s formula ( [ eq : steindf ] ) to compute its degrees of freedom . lemma [ lem : locaff ] tells us that for almost every @xmath152 , there is a neighborhood @xmath144 of @xmath13 , linear subspace @xmath153 , and point @xmath146 , such that @xmath154 therefore , @xmath155 and an expectation over @xmath13 gives @xmath156.\ ] ] it should be made clear that the random quantity in the above expectation is the linear subspace @xmath157 , which depends on @xmath13 . in a sense , the remainder of this paper is focused on describing @xmath158the dimension of the face of @xmath56 onto which the point @xmath13 projects in a meaningful way for the lasso and generalized lasso problems . section [ sec : lasso ] considers the lasso problem , and we show that @xmath54 can be written in terms of the equicorrelation set of the fit at @xmath13 . we also show that @xmath54 can be described in terms of the active set of a solution at @xmath13 . in section [ sec : genlasso ] we show the analogous results for the generalized lasso problem , namely , that @xmath54 can be written in terms of either the boundary set of an optimal subgradient at @xmath13 ( the analogy of the equicorrelation set for the lasso ) or the active set of a solution at @xmath13 . in this section we derive the degrees of freedom of the lasso fit , for a general predictor matrix @xmath0 . all of our arguments stem from the karush tucker ( kkt ) optimality conditions , and we present these first . we note that many of the results in this section can be alternatively derived using the lasso dual problem . appendix [ app : dual ] explains this connection more precisely . for the current work , we avoid the dual perspective simply to keep the presentation more self - contained . finally , we remind the reader that @xmath83 is used to extract columns of @xmath0 corresponding to an index set @xmath73.=1 the kkt conditions for the lasso problem ( [ eq : lasso ] ) can be expressed as @xmath159 & \quad if $ { { \hat{\beta}}}_i = 0$. } & \ ] ] here @xmath160 is a subgradient of the function @xmath161 evaluated at @xmath162 . hence @xmath7 is a minimizer in ( [ eq : lasso ] ) if and only if @xmath7 satisfies ( [ eq : lassokkt ] ) and ( [ eq : lassosg ] ) for some @xmath163 . directly from the kkt conditions , we can show that @xmath133 is the residual from projecting @xmath13 onto a polyhedron . [ lem : lassoproj ] for any @xmath0 and @xmath5 , the lasso fit @xmath133 can be written as @xmath164 , where @xmath165 is the polyhedron @xmath166 given a point @xmath3 , its projection @xmath167 onto a closed convex set @xmath120 can be characterized as the unique point satisfying @xmath168 hence defining @xmath169 , and @xmath56 as in the lemma , we want to show that ( [ eq : opt ] ) holds for all @xmath125 . well , @xmath170 \\[-8pt ] & = & \langle x{{\hat{\beta } } } , y - x{{\hat{\beta}}}\rangle - \langle x{^t}u , { { \hat{\beta}}}\rangle.\nonumber\end{aligned}\ ] ] consider the first term above . taking an inner product with @xmath7 on both sides of ( [ eq : lassokkt ] ) gives @xmath171 . furthermore , the @xmath2 norm can be characterized in terms of its dual norm , the @xmath172 norm , as in @xmath173 therefore , continuing from ( [ eq : innerprod ] ) , we have @xmath174 which is @xmath175 for all @xmath125 , and we have hence proved that @xmath176 . to show that @xmath56 is indeed a polyhedron , note that it can be written as @xmath177 which is a finite intersection of half - spaces . showing that the lasso fit is the residual from projecting @xmath13 onto a polyhedron is important , because it means that @xmath178 is nonexpansive as a function of @xmath13 , and hence continuous and almost differentiable , by lemma [ lem : nonexp ] . this establishes the conditions that are needed to apply stein s formula for degrees of freedom . in the next section , we define the equicorrelation set @xmath179 , and show that the lasso fit and solutions both have an explicit form in terms of @xmath179 . following this , we derive an expression for the lasso degrees of freedom as a function of the equicorrelation set . according to lemma [ lem : lassoproj ] , the lasso fit @xmath133 is always unique ( because projection onto a closed convex set is unique ) . therefore , even though the solution @xmath7 is not necessarily unique , the optimal subgradient @xmath163 is unique , because it can be written entirely in terms of @xmath133 , as shown by ( [ eq : lassokkt ] ) . we define the unique _ equicorrelation set _ @xmath179 as @xmath180 an alternative definition for the equicorrelation set is @xmath181 which explains its name , as @xmath179 can be thought of as the set of variables that have equal and maximal absolute inner product ( or correlation for standardized variables ) with the residual . the set @xmath179 is a natural quantity to work with because we can express the lasso fit and the set of lasso solutions in terms of @xmath179 , by working directly from equation ( [ eq : lassokkt ] ) . first we let @xmath182 the signs of the inner products of the equicorrelation variables with the residual . since @xmath183 by definition of the subgradient , the @xmath179 block of the kkt conditions can be rewritten as @xmath184 because @xmath185 , we can write @xmath186 , so rearranging ( [ eq : lassokkt2 ] ) we get @xmath187 therefore , the lasso fit @xmath188 is @xmath189 and any lasso solution must be of the form @xmath190 where @xmath191 . in the case that @xmath192for example , this holds if @xmath16the lasso solution is unique and is given by ( [ eq : lassosol ] ) with @xmath193 . but in general , when @xmath194 , it is important to note that not every @xmath191 necessarily leads to a lasso solution in ( [ eq : lassosol ] ) ; the vector @xmath60 must also preserve the signs of the nonzero coefficients ; that is , it must also satisfy @xmath195_i + b_i \bigr ) = s_i\nonumber \\[-8pt ] \\[-8pt ] \eqntext{\mbox{for each $ i$ such that } \bigl[(x_{\mathcal{e}})^+ \bigl(y - ( x_{\mathcal{e}}{^t})^+ \lambda s \bigr ) \bigr]_i + b_i \not= 0.}\end{aligned}\ ] ] otherwise , @xmath163 would not be a proper subgradient of @xmath196 . using relatively simple arguments , we can derive a result on the lasso degrees of freedom in terms of the equicorrelation set . our arguments build on the following key lemma . [ lem : lassosol0 ] for any @xmath197 and @xmath5 , a lasso solution is given by @xmath198 where @xmath179 and @xmath199 are the equicorrelation set and signs , as defined in ( [ eq : equiset ] ) and ( [ eq : equisigns ] ) . in other words , lemma [ lem : lassosol0 ] says that the sign condition ( [ eq : lassosign ] ) is always satisfied by taking @xmath193 , regardless of the rank of @xmath0 . this result is inspired by the lars work of @xcite , though it is not proved in the lars paper ; see appendix b of @xcite for a proof . next we show that , almost everywhere in @xmath13 , the equicorrelation set and signs are locally constant functions of @xmath13 . to emphasize their functional dependence on @xmath13 , we write them as @xmath200 and @xmath201 . [ lem : lcequi ] for almost every @xmath3 , there exists a neighborhood @xmath144 of @xmath13 such that @xmath202 and @xmath203 for all @xmath204 . define @xmath205_{(i,\cdot ) } \bigl(z - ( x_{\mathcal{e}}{^t})^+ \lambda s \bigr ) = 0 \bigr\},\ ] ] where the first union above is taken over all subsets @xmath206 and sign vectors @xmath207 , but we exclude sets @xmath179 for which a row of @xmath208 is entirely zero . the set @xmath209 is a finite union of affine subspaces of dimension @xmath210 , and therefore has measure zero . let @xmath211 , and abbreviate the equicorrelation set and signs as @xmath212 and @xmath213 . we may assume no row of @xmath208 is entirely zero . ( otherwise , this implies that @xmath214 has a zero column , which implies that @xmath215 , a trivial case for this lemma . ) therefore , as @xmath211 , this means that the lasso solution given in ( [ eq : lassosol0 ] ) satisfies @xmath216 for every @xmath217 . now , for a new point @xmath218 , consider defining @xmath219 we need to verify that this is indeed a solution at @xmath218 , and that the corresponding fit has equicorrelation set @xmath179 and signs @xmath199 . first notice that , after a straightforward calculation , @xmath220 also , by the continuity of the function @xmath221 , @xmath222 there exists a neighborhood @xmath223 of @xmath13 such that @xmath224 for all @xmath225 . hence @xmath226 has equicorrelation set @xmath227 and signs . to check that @xmath228 is a lasso solution at @xmath218 , we consider the function @xmath229 , @xmath230 the continuity of @xmath93 implies that there exists a neighborhood @xmath231 of @xmath13 such that @xmath232_i \not= 0 \qquad\mbox{for } i \in{\mathcal{e } } , \quad\mbox{and } \\ { \operatorname{sign}}({{\hat{\beta}}}_{\mathcal{e}}(y ' ) ) & = & { \operatorname{sign}}\bigl((x_{\mathcal{e}})^+ \bigl(y ' - ( x_{\mathcal{e}}{^t})^+ \lambda s \bigr ) \bigr)\end{aligned}\ ] ] for each @xmath233 . defining @xmath234 completes the proof . this immediately implies the following theorem . [ thm : lassodfequi ] assume that @xmath13 follows a normal distribution ( [ eq : normal ] ) . for any @xmath0 and @xmath5 , the lasso fit @xmath133 has degrees of freedom @xmath235,\ ] ] where @xmath212 is the equicorrelation set of the lasso fit at @xmath13 . by lemmas [ lem : nonexp ] and [ lem : lassoproj ] we know that @xmath178 is continuous and almost differentiable , so we can use stein s formula ( [ eq : steindf ] ) for degrees of freedom . by lemma [ lem : lcequi ] , we know that @xmath212 and @xmath213 are locally constant for all @xmath211 . therefore , taking the divergence of the fit in ( [ eq : lassofit ] ) , we get @xmath236 taking an expectation over @xmath13 ( and recalling that @xmath209 has measure zero ) gives the result . next , we shift our focus to a different subset of variables : the active set @xmath22 . unlike the equicorrelation set , the active set is not unique , as it depends on a particular choice of lasso solution . though it may seem that such nonuniqueness could present complications , it turns out that all of the active sets share a special property ; namely , the linear subspace @xmath237 is the same for any choice of active set @xmath22 , almost everywhere in @xmath13 . this invariance allows us to express the degrees of freedom of lasso fit in terms of the active set ( or , more precisely , any active set ) . given a particular solution @xmath7 , we define the _ active set _ @xmath22 as @xmath238 this is also called the support of @xmath7 and written @xmath239 . from ( [ eq : lassosol ] ) , we can see that we always have @xmath240 , and different active sets @xmath22 can be formed by choosing @xmath191 to satisfy the sign condition ( [ eq : lassosign ] ) and also @xmath241_i + b_i = 0 \qquad\mbox{for } i \notin{{\mathcal{a}}}.\ ] ] if @xmath16 , then @xmath193 , so there is a unique active set , and furthermore @xmath242 for almost every @xmath3 ( in particular , this last statement holds for , where @xmath209 is the set of measure zero set defined in the proof of lemma [ lem : lcequi ] ) . for the signs of the coefficients of active variables , we write @xmath243 and we note that @xmath244 . by similar arguments as those used to derive expression ( [ eq : lassofit ] ) for the fit in section [ sec : lassoequi ] , the lasso fit can also be written as @xmath245 for the active set @xmath22 and signs @xmath246 of any lasso solution @xmath7 . if we could take the divergence of the fit in the expression above , and simply ignore the dependence of @xmath22 and @xmath246 on @xmath13 ( treat them as constants ) , then this would give @xmath247 . in the next section , we show that treating @xmath22 and @xmath246 as constants in ( [ eq : lassofit2 ] ) is indeed correct , for almost every @xmath13 . this property then implies that the linear subspace @xmath237 is invariant under any choice of active set @xmath22 , almost everywhere in @xmath13 ; moreover , it implies that we can write the lasso degrees of freedom in terms of any active set . we first establish a result on the local stability of @xmath248 and @xmath249 [ written in this way to emphasize their dependence on @xmath13 , through a solution @xmath250 . [ lem : lcact ] there is a set @xmath251 , of measure zero , with the following property : for @xmath252 , and for any lasso solution @xmath253 with active set @xmath248 and signs @xmath249 , there is a neighborhood @xmath144 of @xmath13 such that every point @xmath204 yields a lasso solution @xmath228 with the same active set @xmath254 and the same active signs @xmath255 . the proof is similar to that of lemma [ lem : lcequi ] , except that it is longer and somewhat more complicated , so it is delayed until appendix [ app : lcact ] . combined with expression ( [ eq : lassofit2 ] ) for the lasso fit , lemma [ lem : lcact ] now implies an invariance of the subspace spanned by the active variables . [ lem : invact ] for the same set @xmath251 as in lemma [ lem : lcact ] , and for any @xmath252 , the linear subspace @xmath237 is invariant under all sets @xmath18 defined in terms of a lasso solution @xmath253 at @xmath13 . let @xmath252 , and let @xmath253 be a solution with active set @xmath18 and signs @xmath256 . let @xmath144 be the neighborhood of @xmath13 as constructed in the proof of lemma [ lem : lcact ] ; on this neighborhood , solutions exist with active set @xmath22 and signs @xmath246 . hence , recalling ( [ eq : lassofit2 ] ) , we know that for every @xmath257 , @xmath258 now suppose that @xmath259 and @xmath260 are the active set and signs of another lasso solution at @xmath13 . then , by the same arguments , there is a neighborhood @xmath261 of @xmath13 such that @xmath262 for all @xmath263 . by the uniqueness of the fit , we have that for each @xmath264 , @xmath265 since @xmath266 is open , for any @xmath267 , there is an @xmath268 such that @xmath269 . plugging @xmath270 into the above equation implies that @xmath271 , so @xmath272 . a similar argument gives @xmath273 , completing the proof . again , this immediately leads to the following theorem . [ thm : lassodfact ] assume that @xmath13 follows a normal distribution ( [ eq : normal ] ) . for any @xmath0 and @xmath5 , the lasso fit @xmath133 has degrees of freedom @xmath274,\ ] ] where @xmath18 is the active set corresponding to any lasso solution @xmath253 at @xmath13 . note : by lemma [ lem : invact ] , @xmath119 is an invariant quantity , not depending on the choice of active set ( coming from a lasso solution ) , for almost every @xmath13 . this makes the above result well defined . proof of theorem [ thm : lassodfact ] we can apply stein s formula ( [ eq : steindf ] ) for degrees of freedom , because @xmath178 is continuous and almost differentiable by lemmas [ lem : nonexp ] and [ lem : lassoproj ] . let @xmath18 and @xmath256 be the active set and active signs of a lasso solution at @xmath252 , with @xmath275 as in lemma [ lem : invact ] . by this same lemma , there exists a lasso solution with active set @xmath22 and signs @xmath246 at every point @xmath218 in some neighborhood @xmath144 of @xmath13 , and therefore , taking the divergence of the fit ( [ eq : lassofit2 ] ) , we get @xmath276 taking an expectation over @xmath13 completes the proof . the proof of lemma [ lem : lcact ] showed that , for almost every @xmath13 , the equicorrelation set @xmath179 is actually the active set @xmath22 of the particular lasso solution defined in ( [ eq : lassosol0 ] ) . hence theorem [ thm : lassodfequi ] can be viewed as a corollary of theorem [ thm : lassodfact ] . when @xmath16 , the lasso solution is unique , and there is only one active set @xmath22 . and as the columns of @xmath0 are linearly independent , we have @xmath277 , so the result of theorem [ thm : lassodfact ] reduces to @xmath278 as shown in @xcite . an interesting result on the lasso degrees of freedom was recently and independently obtained by @xcite . their result states that , for a general @xmath0 , @xmath279 where @xmath280 is the smallest cardinality among all active sets of lasso solutions . this actually follows from theorem [ thm : lassodfact ] , by noting that for any @xmath13 there exists a lasso solution whose active set @xmath259 corresponds to linear independent predictors @xmath281 , so @xmath282 [ e.g. , see theorem 3 in appendix b of @xcite ] , and furthermore , for almost every @xmath13 no active set can have a cardinality smaller than @xmath280 , as this would contradict lemma [ lem : invact ] . consider the elastic net problem [ @xcite ] , @xmath283 where we now have two tuning parameters @xmath284 . note that our notation above emphasizes the fact that there is always a unique solution to the elastic net criterion , regardless of the rank of @xmath0 . this property ( among others , such as stability and predictive ability ) is considered an advantage of the elastic net over the lasso . we can rewrite the elastic net problem ( [ eq : enet ] ) as a ( full column rank ) lasso problem , @xmath285 \beta \right\|_{2}^2 + \lambda_1 \|\beta\|_{1},\ ] ] and hence it can be shown ( although we omit the details ) that the degrees of freedom of the elastic net fit is @xmath286,\ ] ] where @xmath18 is the active set of the elastic net solution at @xmath13 . it is often more appropriate to include an ( unpenalized ) intercept coefficient in the lasso model , yielding the problem @xmath287 where @xmath288 is the vector of all @xmath289s . defining @xmath290 , we note that the fit of problem ( [ eq : lassoint ] ) can be written as @xmath291 , and that @xmath7 solves the usual lasso problem @xmath292 now it follows ( again we omit the details ) that the fit of the lasso problem with intercept ( [ eq : lassoint ] ) has degrees of freedom @xmath293,\ ] ] where @xmath18 is the active set of a solution @xmath253 at @xmath13 ( these are the nonintercept coefficients ) . in other words , the degrees of freedom is one plus the expected dimension of the subspace spanned by the active variables , once we have centered these variables . a similar result holds for an arbitrary set of unpenalized coefficients , by replacing @xmath294 above with the projection onto the orthogonal complement of the column space of the unpenalized variables , and @xmath289 above with the dimension of the column space of the unpenalized variables . as mentioned in the , a nice feature of the full column rank result ( [ eq : lassodffull ] ) is its interpretability and its explicit nature . the general result is also explicit in the sense that an unbiased estimate of degrees of freedom can be achieved by computing the rank of a given matrix . in terms of interpretability , when @xmath16 , the degrees of freedom of the lasso fit is @xmath295this says that , on average , the lasso `` spends '' the same number of parameters as does linear regression on @xmath19 linearly independent predictor variables . fortunately , a similar interpretation is possible in the general case : we showed in theorem [ thm : lassodfact ] that for a general predictor matrix @xmath0 , the degrees of freedom of the lasso fit is @xmath296 $ ] , the expected dimension of the linear subspace spanned by the active variables . meanwhile , for the linear regression problem @xmath297 where we consider @xmath22 fixed , the degrees of freedom of the fit is @xmath298 . in other words , the lasso adaptively selects a subset @xmath22 of the variables to use for a linear model of @xmath13 , but on average it only `` spends '' the same number of parameters as would linear regression on the variables in @xmath22 , if @xmath22 was pre - specified . how is this possible ? broadly speaking , the answer lies in the shrinkage due to the @xmath2 penalty . although the active set is chosen adaptively , the lasso does not estimate the active coefficients as aggressively as does the corresponding linear regression problem ( [ eq : lsa ] ) ; instead , they are shrunken toward zero , and this adjusts for the adaptive selection . differing views have been presented in the literature with respect to this feature of lasso shrinkage . on the one hand , for example , @xcite point out that lasso estimates suffer from bias due to the shrinkage of large coefficients , and motivate the nonconvex _ scad _ penalty as an attempt to overcome this bias . on the other hand , for example , loubes and massart ( @xcite ) discuss the merits of such shrunken estimates in model selection criteria , such as ( [ eq : tunsel ] ) . in the current context , the shrinkage due to the @xmath2 penalty is helpful in that it provides control over degrees of freedom . a more precise study of this idea is the topic of future work . in this section we extend our degrees of freedom results to the generalized lasso problem , with an arbitrary predictor matrix @xmath0 and penalty matrix @xmath1 . as before , the kkt conditions play a central role , and we present these first . also , many results that follow have equivalent derivations from the perspective of the generalized lasso dual problem ; see appendix [ app : dual ] . we remind the reader that @xmath84 is used to extract to extract rows of @xmath1 corresponding to an index set @xmath72 . the kkt conditions for the generalized lasso problem ( [ eq : genlasso ] ) are @xmath299 & \quad if $ ( d{{\hat{\beta}}})_i = 0$. } & \ ] ] now @xmath300 is a subgradient of the function @xmath301 evaluated at @xmath302 . similar to what we showed for the lasso , it follows from the kkt conditions that the generalized lasso fit is the residual from projecting @xmath13 onto a polyhedron . [ lem : genlassoproj ] for any @xmath0 and @xmath5 , the generalized lasso fit can be written as @xmath164 , where @xmath165 is the polyhedron @xmath303 the proof is quite similar to that of lemma [ lem : lassoproj ] . as in ( [ eq : innerprod ] ) , we want to show that @xmath304 for all @xmath125 , where @xmath56 is as in the lemma . for the first term above , we can take an inner product with @xmath7 on both sides of ( [ eq : genlassokkt ] ) to get @xmath305 , and furthermore , @xmath306 therefore ( [ eq : innerprod2 ] ) holds if @xmath307 for some @xmath308 , in other words , if @xmath309 . to show that @xmath56 is a polyhedron , note that we can write it as @xmath310 where @xmath311 is taken to mean the inverse image under the linear map @xmath312 , and @xmath313 , a hypercube in @xmath314 . clearly @xmath315 is a polyhedron , and the image or inverse image of a polyhedron under a linear map is still a polyhedron . as with the lasso , this lemma implies that the generalized lasso fit @xmath178 is nonexpansive , and therefore continuous and almost differentiable as a function of @xmath13 , by lemma [ lem : nonexp ] . this is important because it allows us to use stein s formula when computing degrees of freedom . in the next section we define the boundary set @xmath30 , and derive expressions for the generalized lasso fit and solutions in terms of @xmath30 . the following section defines the active set @xmath22 in the generalized lasso context , and again gives expressions for the fit and solutions in terms of @xmath22 . though neither @xmath30 nor @xmath22 are necessarily unique for the generalized lasso problem , any choice of @xmath30 or @xmath22 generates a special invariant subspace ( similar to the case for the active sets in the lasso problem ) . we are subsequently able to express the degrees of freedom of the generalized lasso fit in terms of any boundary set @xmath30 , or any active set @xmath22 . like the lasso , the generalized lasso fit @xmath133 is always unique ( following from lemma [ lem : genlassoproj ] , and the fact that projection onto a closed convex set is unique ) . however , unlike the lasso , the optimal subgradient @xmath163 in the generalized lasso problem is not necessarily unique . in particular , if @xmath316 , then the optimal subgradient @xmath163 is not uniquely determined by conditions ( [ eq : genlassokkt ] ) and ( [ eq : genlassosg ] ) . given a subgradient @xmath163 satisfying ( [ eq : genlassokkt ] ) and ( [ eq : genlassosg ] ) for some @xmath7 , we define the _ boundary set _ @xmath30 as @xmath317 this generalizes the notion of the equicorrelation set @xmath179 in the lasso problem [ though , as just noted , the set @xmath30 is not necessarily unique unless ] . we also define @xmath318 now we focus on writing the generalized lasso fit and solutions in terms of @xmath30 and @xmath199 . abbreviating @xmath319 , note that we can expand @xmath320 . therefore , multiplying both sides of ( [ eq : genlassokkt ] ) by @xmath321 yields @xmath322 since @xmath323 , we can write @xmath324 . also , we have @xmath325 by definition of @xmath30 , so @xmath326 . these two facts allow us to rewrite ( [ eq : genlassokktb ] ) as @xmath327 and hence the fit @xmath328 is @xmath329 where we have un - abbreviated @xmath319 . further , any generalized lasso solution is of the form @xmath330 where @xmath331 . multiplying the above equation by @xmath29 , and recalling that @xmath325 , reveals that @xmath332 ; hence @xmath333 . in the case that @xmath334 , the generalized lasso solution is unique and is given by ( [ eq : genlassosol ] ) with @xmath193 . this occurs when @xmath16 , for example . otherwise , any @xmath335 gives a generalized lasso solution in ( [ eq : genlassosol ] ) as long as it also satisfies the sign condition @xmath336 necessary to ensure that @xmath163 is a proper subgradient of @xmath337 . we define the _ active set _ of a particular solution @xmath7 as @xmath338 which can be alternatively expressed as @xmath339 . if @xmath7 corresponds to a subgradient with boundary set @xmath30 and signs @xmath199 , then @xmath340 ; in particular , given @xmath30 and @xmath199 , different active sets @xmath22 can be generated by taking @xmath341 such that ( [ eq : genlassosign ] ) is satisfied , and also @xmath342 if @xmath16 , then @xmath193 , and there is only one active set @xmath22 ; however , in this case , @xmath22 can still be a strict subset of @xmath30 . this is quite different from the lasso problem , wherein @xmath242 for almost every @xmath13 whenever @xmath16 . [ note that in the generalized lasso problem , @xmath16 implies that @xmath22 is unique but implies nothing about the uniqueness of @xmath30this is determined by the rank of @xmath1 . the boundary set @xmath30 is not necessarily unique if @xmath316 , and in this case we may have @xmath343 for some @xmath344 , which certainly implies that @xmath345 for any @xmath3 . hence some boundary sets may not correspond to active sets at any @xmath13 . ] we denote the signs of the active entries in @xmath346 by @xmath347 and we note that @xmath244 . following the same arguments as those leading up to the expression for the fit ( [ eq : genlassofit ] ) in section [ sec : genlassobound ] , we can alternatively express the generalized lasso fit as @xmath348 where @xmath22 and @xmath246 are the active set and signs of any solution . computing the divergence of the fit in ( [ eq : genlassofit2 ] ) , and pretending that @xmath22 and @xmath246 are constants ( not depending on @xmath13 ) , gives @xmath349 . the same logic applied to ( [ eq : genlassofit ] ) gives @xmath350 . the next section shows that , for almost every @xmath13 , the quantities @xmath351 or @xmath352 can indeed be treated as locally constant in expressions ( [ eq : genlassofit2 ] ) or ( [ eq : genlassofit ] ) , respectively . we then prove that linear subspaces @xmath353 are invariant under all choices of boundary sets @xmath30 , respectively active sets @xmath22 , and that the two subspaces are in fact equal , for almost every @xmath13 . furthermore , we express the generalized lasso degrees of freedom in terms of any boundary set or any active set . we call @xmath354 an _ optimal pair _ provided that @xmath355 and @xmath253 jointly satisfy the kkt conditions , ( [ eq : genlassokkt ] ) and ( [ eq : genlassosg ] ) , at @xmath13 . for such a pair , we consider its boundary set @xmath356 , boundary signs @xmath201 , active set @xmath248 , active signs @xmath249 , and show that these sets and sign vectors possess a kind of local stability . [ lem : lcbound ] there exists a set @xmath357 , of measure zero , with the following property : for @xmath211 , and for any optimal pair @xmath354 with boundary set @xmath356 , boundary signs @xmath201 , active set @xmath248 , and active signs @xmath249 , there is a neighborhood @xmath144 of @xmath13 such that each point @xmath257 yields an optimal pair @xmath358 with the same boundary set @xmath359 , boundary signs @xmath360 , active set @xmath254 and active signs @xmath255 . the proof is delayed to appendix [ app : lcbound ] , mainly because of its length . now lemma [ lem : lcbound ] , used together with expressions ( [ eq : genlassofit ] ) and ( [ eq : genlassofit2 ] ) for the generalized lasso fit , implies an invariance in representing a ( particularly important ) linear subspace . [ lem : invbound ] for the same set @xmath357 as in lemma [ lem : lcbound ] , and for any @xmath211 , the linear subspace @xmath361 is invariant under all boundary sets @xmath28 defined in terms of an optimal subgradient at @xmath355 at @xmath13 . the linear subspace @xmath362 is also invariant under all choices of active sets @xmath18 defined in terms of a generalized lasso solution @xmath253 at @xmath13 . finally , the two subspaces are equal , @xmath363 . let @xmath211 , and let @xmath355 be an optimal subgradient with boundary set @xmath28 and signs @xmath213 . let @xmath144 be the neighborhood of @xmath13 over which optimal subgradients exist with boundary set @xmath30 and signs @xmath199 , as given by lemma [ lem : lcbound ] . recalling the expression for the fit ( [ eq : genlassofit ] ) , we have that for every @xmath204 @xmath364 if @xmath253 is a solution with active set @xmath18 and signs @xmath256 , then again by lemma [ lem : lcbound ] there is a neighborhood @xmath365 of @xmath13 such that each point @xmath366 yields a solution with active set @xmath22 and signs @xmath246 . [ note that @xmath365 and @xmath144 are not necessarily equal unless @xmath355 and @xmath253 jointly satisfy the kkt conditions at @xmath13 . ] therefore , recalling ( [ eq : genlassofit ] ) , we have @xmath367 for each @xmath366 . the uniqueness of the generalized lasso fit now implies that @xmath368 for all @xmath369 . as @xmath370 is open , for any @xmath371 , there exists an @xmath268 such that @xmath372 . plugging @xmath270 into the equation above reveals that @xmath373 , hence @xmath374 . the reverse inclusion follows similarly , and therefore@xmath375 . finally , the same strategy can be used to show that these linear subspaces are unchanged for any choice of boundary set @xmath28 , coming from an optimal subgradient at @xmath13 and for any choice of active set @xmath18 coming from a solution at @xmath13 . noticing that @xmath376 for matrices @xmath377 gives the result as stated in the lemma . this local stability result implies the following theorem . [ thm : genlassodf ] assume that @xmath13 follows a normal distribution ( [ eq : normal ] ) . for any @xmath378 and @xmath5 , the degrees of freedom of the generalized lasso fit can be expressed as @xmath379,\ ] ] where @xmath28 is the boundary set corresponding to any optimal subgradient @xmath355 of the generalized lasso problem at @xmath13 . we can alternatively express degrees of freedom as @xmath380,\ ] ] with @xmath18 being the active set corresponding to any generalized lasso solution @xmath253 at @xmath13 . note : lemma [ lem : invbound ] implies that for almost every @xmath152 , for any @xmath30 defined in terms of an optimal subgradient , and for any @xmath22 defined in terms of a generalized lasso solution , @xmath381 . this makes the above expressions for degrees of freedom well defined . proof of theorem [ thm : genlassodf ] first , the continuity and almost differentiability of @xmath178 follow from lemmas [ lem : nonexp ] and [ lem : genlassoproj ] , so we can use stein s formula ( [ eq : steindf ] ) for degrees of freedom . let @xmath211 , where @xmath209 is the set of measure zero as in lemma [ lem : lcact ] . if @xmath28 and @xmath213 are the boundary set and signs of an optimal subgradient at @xmath13 , then by lemma [ lem : invbound ] there is a neighborhood @xmath144 of @xmath13 such that each point @xmath204 yields an optimal subgradient with boundary set @xmath30 and signs @xmath199 . therefore , taking the divergence of the fit in ( [ eq : genlassofit ] ) , @xmath382 and taking an expectation over @xmath13 gives the first expression in the theorem . similarly , if @xmath18 and @xmath256 are the active set and signs of a generalized lasso solution at @xmath13 , then by lemma [ lem : invbound ] there exists a solution with active set @xmath22 and signs @xmath246 at each point @xmath218 in some neighborhood @xmath365 of @xmath13 . the divergence of the fit in ( [ eq : genlassofit2 ] ) is hence @xmath383 and taking an expectation over @xmath13 gives the second expression . if @xmath16 , then @xmath384 for any linear subspace @xmath54 , so the results of theorem [ thm : genlassodf ] reduce to @xmath385 = { \mathrm{e}}[{\operatorname{nullity}}(d_{-{{\mathcal{a}}}})].\ ] ] the first equality above was shown in @xcite . analyzing the null space of @xmath29 ( equivalently , @xmath386 ) for specific choices of @xmath1 then gives interpretable results on the degrees of freedom of the fused lasso and trend filtering fits as mentioned in the introduction . it is important to note that , as @xmath16 , the active set @xmath22 is unique , but not necessarily equal to the boundary set @xmath30 [ since @xmath30 can be nonunique if @xmath316 ] . if @xmath25 , then @xmath388 for any subset @xmath389 . therefore the results of theorem [ thm : genlassodf ] become @xmath390 = { \mathrm{e}}[{\operatorname{rank}}(x_{{\mathcal{a}}})],\ ] ] which match the results of theorems [ thm : lassodfequi ] and [ thm : lassodfact ] ( recall that for the lasso the boundary set @xmath30 is exactly the same as equicorrelation set @xmath179 ) . recent and independent work of @xcite shows that , for arbitrary @xmath378 and for any @xmath13 , there exists a generalized lasso solution whose active set @xmath259 satisfies @xmath391 ( calling @xmath259 the `` smallest '' active set is somewhat of an abuse of terminology , but it is the smallest in terms of the above intersection . ) the authors then prove that , for any @xmath378 , the generalized lasso fit has degrees of freedom @xmath392,\ ] ] with @xmath259 the special active set as above . this matches the active set result of theorem [ thm : genlassodf ] applied to @xmath259 , since @xmath393 for this special active set . we conclude this section by comparing the active set result of theorem [ thm : genlassodf ] to degrees of freedom in a particularly relevant equality constrained linear regression problem ( this comparison is similar to that made in lasso case , given at the end of section [ sec : lasso ] ) . the result states that the generalized lasso fit has degrees of freedom @xmath394 $ ] , where @xmath18 is the active set of a generalized lasso solution at @xmath13 . in other words , the complement of @xmath22 gives the rows of @xmath1 that are orthogonal to some generalized lasso solution . now , consider the equality constrained linear regression problem @xmath395 in which the set @xmath22 is fixed . it is straightforward to verify that the fit of this problem is the projection map onto @xmath396 , and hence has degrees of freedom @xmath397 . this means that the generalized lasso fits a linear model of @xmath13 , and simultaneously makes the coefficients orthogonal to an adaptive subset @xmath22 of the rows of @xmath1 , but on average it only uses the same number of parameters as does the corresponding equality constrained linear regression problem ( [ eq : lsb ] ) , in which @xmath22 is pre - specified . this seemingly paradoxical statement can be explained by the shrinkage due to the @xmath2 penalty . even though the active set @xmath22 is chosen adaptively based on @xmath13 , the generalized lasso does not estimate the coefficients as aggressively as does the equality constrained linear regression problem ( [ eq : lsb ] ) , but rather , it shrinks them toward zero . roughly speaking , his shrinkage can be viewed as a `` deficit '' in degrees of freedom , which makes up for the `` surplus '' attributed to the adaptive selection . we study this idea more precisely in a future paper . we showed that the degrees of freedom of the lasso fit , for an arbitrary predictor matrix @xmath0 , is equal to @xmath296 $ ] . here @xmath18 is the active set of any lasso solution at @xmath13 , that is , @xmath398 . this result is well defined , since we proved that any active set @xmath22 generates the same linear subspace @xmath237 , almost everywhere in @xmath13 . in fact , we showed that for almost every @xmath13 , and for any active set @xmath22 of a solution at @xmath13 , the lasso fit can be written as @xmath399 for all @xmath218 in a neighborhood of @xmath13 , where @xmath400 is a constant ( it does not depend on @xmath218 ) . this draws an interesting connection to linear regression , as it shows that locally the lasso fit is just a translation of the linear regression fit of on @xmath401 . the same results ( on degrees of freedom and local representations of the fit ) hold when the active set @xmath22 is replaced by the equicorrelation set @xmath179 . our results also extend to the generalized lasso problem , with an arbitrary predictor matrix @xmath0 and arbitrary penalty matrix @xmath1 . we showed that degrees of freedom of the generalized lasso fit is @xmath394 $ ] , with @xmath18 being the active set of any generalized lasso solution at @xmath13 , that is , @xmath402 . as before , this result is well defined because any choice of active set @xmath22 generates the same linear subspace @xmath403 , almost everywhere in @xmath13 . furthermore , for almost every @xmath13 , and for any active set of a solution at @xmath13 , the generalized lasso fit satisfies @xmath404 for all @xmath218 in a neighborhood of @xmath13 , where @xmath400 is a constant ( not depending on @xmath13 ) . this again reveals an interesting connection to linear regression , since it says that locally the generalized lasso fit is a translation of the linear regression fit on @xmath0 , with the coefficients @xmath405 subject to @xmath406 . the same statements hold with the active set @xmath22 replaced by the boundary set @xmath30 of an optimal subgradient . we note that our results provide practically useful estimates of degrees of freedom . for the lasso problem , we can use @xmath119 as an unbiased estimate of degrees of freedom , with @xmath22 being the active set of a lasso solution . to emphasize what has already been said , here we can actually choose any active set ( i.e. , any solution ) , because all active sets give rise to the same @xmath119 , except for @xmath13 in a set of measure zero . this is important , since different algorithms for the lasso can produce different solutions with different active sets . for the generalized lasso problem , an unbiased estimate for degrees of freedom is given by @xmath407 , where @xmath22 is the active set of a generalized lasso solution . this estimate is the same , regardless of the choice of active set ( i.e. , choice of solution ) , for almost every @xmath13 . hence any algorithm can be used to compute a solution . first , we prove the statement for the projection map . note that @xmath409 where the first inequality follows from ( [ eq : pfact ] ) , and the second is by cauchy dividing both sides by @xmath410 gives the result . we have shown that @xmath141 and @xmath142 are lipschitz ( with constant @xmath289 ) ; they are therefore continuous , and almost differentiability follows from the standard proof of the fact that a lipschitz function is differentiable almost everywhere . we write @xmath413 to denote the set of faces of @xmath56 . to each face @xmath414 , there is an associated normal cone @xmath415 , defined as @xmath416 the normal cone of @xmath417 satisfies @xmath418 for any @xmath419 . [ we use @xmath420 to denote the relative interior of a set @xmath47 , and @xmath421 to denote its relative boundary . ] now let @xmath427 . we have @xmath428 for some @xmath424 , and by construction @xmath429 . furthermore , we claim that projecting @xmath428 onto @xmath56 is the same as projecting @xmath11 onto the affine hull of @xmath417 , that is , @xmath430 . otherwise there is some @xmath431 with @xmath432 , and as @xmath433 , this means that @xmath434 . by definition of @xmath435 , there is some @xmath436 such that @xmath437 . but @xmath438 , which is a contradiction . this proves the claim , and writing @xmath439 , we have @xmath440 as desired . in the first type of points above , vertices are excluded because @xmath448 when @xmath417 is a vertex . in the second type , @xmath56 is excluded because @xmath449 . the lattice structure of @xmath413 tells us that for any face @xmath424 , we can write @xmath450 . this , and the fact that the normal cones have the opposite partial ordering as the faces , imply that points of the first type above can be written as @xmath451 with @xmath452 and @xmath453 for some @xmath454 . note that actually we must have @xmath455 because otherwise we would have @xmath456 . therefore it suffices to consider points of the second type alone , and @xmath442 can be written as @xmath457 as @xmath56 is a polyhedron , the set @xmath413 of its faces is finite , and @xmath458 for each @xmath459 . therefore @xmath442 is a finite union of sets of dimension @xmath460 , and hence has measure zero . now let @xmath465^\perp } [ ( x_{\mathcal{e}})^+]_{(-{{\mathcal{a}}},\cdot ) } \bigl(z - ( x_{\mathcal{e}}{^t})^+\lambda s \bigr ) = 0 \bigr\}.\ ] ] the first union is taken over all possible subsets @xmath466 and all sign vectors @xmath467 ; as for the second union , we define for a fixed subset @xmath179 @xmath468^\perp } [ ( x_{\mathcal{e}})^+]_{(-{{\mathcal{a}}},\cdot ) } \not= 0 \bigr\}.\ ] ] notice that @xmath275 is a finite union of affine subspace of dimension @xmath460 , and hence has measure zero . let @xmath252 , and let @xmath253 be a lasso solution , abbreviating @xmath18 and @xmath256 for the active set and active signs . also write @xmath212 and @xmath213 for the equicorrelation set and equicorrelation signs of the fit . we know from ( [ eq : lassosol ] ) that we can write @xmath469 where @xmath191 is such that @xmath470_{(-{{\mathcal{a}}},\cdot ) } \bigl(y - ( x_{\mathcal{e}}{^t})^+ \lambda s \bigr ) + b_{-{{\mathcal{a } } } } = 0.\ ] ] in other words , @xmath471_{(-{{\mathcal{a}}},\cdot ) } \bigl(y - ( x_{\mathcal{e}}{^t})^+ \lambda s \bigr ) = -b_{-{{\mathcal{a } } } } \in \pi_{-{{\mathcal{a } } } } ( { \operatorname{null}}(x_{\mathcal{e}})),\ ] ] so projecting onto the orthogonal complement of the linear subspace@xmath472 gives zero , @xmath473^\perp } [ ( x_{\mathcal{e}})^+]_{(-{{\mathcal{a}}},\cdot ) } \bigl(y - ( x_{\mathcal{e}}{^t})^+\lambda s \bigr ) = 0.\ ] ] since @xmath252 , we know that @xmath473^\perp } [ ( x_{\mathcal{e}})^+]_{(-{{\mathcal{a}}},\cdot ) } = 0,\ ] ] and finally , this can be rewritten as @xmath474_{(-{{\mathcal{a}}},\cdot ) } \bigr ) \subseteq\pi_{-{{\mathcal{a}}}}({\operatorname{null}}(x_{\mathcal{e}})).\ ] ] consider defining , for a new point @xmath218 , @xmath475 where @xmath476 , and is yet to be determined . exactly as in the proof of lemma [ lem : lcequi ] , we know that @xmath477 , and for all @xmath225 , a neighborhood of @xmath13 . now we want to choose @xmath478 so that @xmath228 has the correct active set and active signs . for simplicity of notation , first define the function @xmath479 , @xmath480 equation ( [ eq : colsp ] ) implies that there is a @xmath476 such that @xmath481 , hence @xmath482 . however , we must choose @xmath478 so that additionally @xmath483 for @xmath484 and @xmath485 . write @xmath486 by the continuity of @xmath487 , there exits a neighborhood of @xmath231 of @xmath13 such that @xmath488 for @xmath484 and @xmath489 , for all @xmath233 . therefore we only need to choose a vector @xmath476 , with @xmath481 , such that @xmath490 sufficiently small . this can be achieved by applying the bounded inverse theorem , which says that the bijective linear map @xmath491 has a bounded inverse ( when considered a function from its row space to its column space ) . therefore there exists some @xmath492 such that for any @xmath218 , there is a vector @xmath476 , @xmath481 , with @xmath493 finally , the continuity of @xmath494 implies that @xmath495 can be made sufficiently small by restricting @xmath496 , another neighborhood of @xmath13 . define the set @xmath501^\perp}\cdot d_{{\mathcal{b}}\setminus{{\mathcal{a } } } } \bigl(x p_{{\operatorname{null}}(d_{-{\mathcal{b}}})}\bigr)^+ \\ & & \hspace*{165pt}{}\times \bigl(z - \bigl(p_{{\operatorname{null}}(d_{-{\mathcal{b } } } ) } x{^t}\bigr)^+ d_{\mathcal{b}}{^t}\lambda s \bigr ) = 0 \bigr\}.\end{aligned}\ ] ] the first union above is taken over all subsets @xmath502 and all sign vectors @xmath503 . the second union is taken over subsets @xmath504 , where @xmath505^\perp } d_{{\mathcal{b}}\setminus{{\mathcal{a } } } } \bigl(x p_{{\operatorname{null}}(d_{-{\mathcal{b}}})}\bigr)^+ \not= 0 \bigr\}.\ ] ] since @xmath209 is a finite union of affine subspaces of dimension @xmath506 , it has measure zero . now fix @xmath211 , and let @xmath354 be an optimal pair , with boundary set @xmath28 , boundary signs @xmath213 , active set @xmath18 , and active signs @xmath256 . starting from ( [ eq : genlassokktb ] ) , and plugging in for the fit in terms of @xmath352 , as in ( [ eq : genlassofit ] ) we can show that @xmath507 where @xmath508 . by ( [ eq : genlassosol ] ) , we know that @xmath509 where @xmath510 . furthermore , @xmath511 or equivalently , @xmath512 projecting onto the orthogonal complement of the linear subspace@xmath513 therefore gives zero , @xmath514^\perp } d_{{\mathcal{b}}\setminus{{\mathcal{a } } } } \bigl(x p_{{\operatorname{null}}(d_{-{\mathcal{b}}})}\bigr)^+ \bigl(y - \bigl(p_{{\operatorname{null}}(d_{-{\mathcal{b } } } ) } x{^t}\bigr)^+ d_{\mathcal{b}}{^t}\lambda s \bigr ) = 0,\ ] ] and because @xmath211 , we know that in fact @xmath514^\perp } d_{{\mathcal{b}}\setminus{{\mathcal{a } } } } \bigl(x p_{{\operatorname{null}}(d_{-{\mathcal{b}}})}\bigr)^+ = 0.\ ] ] this can be rewritten as @xmath515 at a new point @xmath218 , consider defining @xmath516 , @xmath517 and @xmath518 where @xmath519 is yet to be determined . by construction , @xmath520 and @xmath228 satisfy the stationarity condition ( [ eq : genlassokkt ] ) at @xmath218 . hence it remains to show two parts : first , we must show that this pair satisfies the subgradient condition ( [ eq : genlassosg ] ) at @xmath218 ; second , we must show this pair has boundary set , boundary signs @xmath521 , active set @xmath499 and active signs @xmath522 . actually , it suffices to show the second part alone , because the first part is then implied by the fact that @xmath355 and @xmath253 satisfy the subgradient condition at @xmath13 . well , by the continuity of the function @xmath523 , @xmath524 we have @xmath525 provided that @xmath225 , a neighborhood of @xmath13 . this ensures that @xmath520 has boundary set @xmath526 and signs @xmath521 . as for the active set and signs of @xmath228 , note first that @xmath527 , following directly from the definition . next , define the function @xmath528 , @xmath529 so @xmath530 . equation ( [ eq : colsp2 ] ) implies that there is a vector @xmath519 such that @xmath531 , which makes @xmath532 . however , we still need to choose @xmath478 such that @xmath533 for all @xmath484 and @xmath534 . to this end , write @xmath535 the continuity of @xmath536 implies that there is a neighborhood @xmath231 of @xmath13 such that @xmath537 for all @xmath484 and @xmath538 , for @xmath233 . since @xmath539 where @xmath540 is the operator norm of the @xmath541 , we only need to choose @xmath519 such that @xmath531 , and such that @xmath490 is sufficiently small . this is possible by the bounded inverse theorem applied to the linear map @xmath542 : when considered a function from its row space to its column space , @xmath542 is bijective and hence has a bounded inverse . therefore there is some @xmath492 such that for any @xmath218 , there is a @xmath519 with @xmath531 and @xmath543 the continuity of @xmath544 implies that the right - hand side above can be made sufficiently small by restricting @xmath496 , a neighborhood of @xmath13 . the dual of the lasso problem ( [ eq : lasso ] ) has appeared in many papers in the literature ; as far as we can tell , it was first considered by @xcite . we start by rewriting problem ( [ eq : lasso ] ) as @xmath545 then we write the lagrangian @xmath546 and we minimize @xmath547 over @xmath548 to obtain the dual problem @xmath549 taking the gradient of @xmath547 with respect to to @xmath548 , and setting this equal to zero gives @xmath550 where @xmath160 is a subgradient of the function @xmath551 evaluated at @xmath162 . from ( [ eq : lassodual1 ] ) , we can immediately see that the dual solution @xmath552 is the projection of @xmath13 onto the polyhedron @xmath56 as in lemma [ lem : lassoproj ] , and then ( [ eq : lassopd1 ] ) shows that @xmath553 is the residual from projecting @xmath13 onto @xmath56 . further , from ( [ eq : lassopd2 ] ) , we can define the equicorrelation set @xmath179 as @xmath554 noting that together ( [ eq : lassopd1 ] ) , ( [ eq : lassopd2 ] ) are exactly the same as the kkt conditions ( [ eq : lassokkt ] ) , ( [ eq : lassosg ] ) , and all of the arguments in section [ sec : lasso ] involving the equicorrelation set @xmath179 can be translated to this dual perspective . there is a slightly different way to derive the lasso dual , resulting in a different ( but of course , equivalent ) formulation . we first rewrite problem ( [ eq : lasso ] ) as @xmath555 and by following similar steps to those above , we arrive at the dual problem @xmath556 each dual solution @xmath552 ( now no longer unique ) satisfies @xmath557 the dual problem ( [ eq : lassodual2 ] ) and its relationship ( [ eq : lassopd3 ] ) , ( [ eq : lassopd4 ] ) to the primal problem offer yet another viewpoint to understand some of the results in section [ sec : lasso ] . for the generalized lasso problem , one might imagine that there are three different dual problems , corresponding to the three different ways of introducing an auxiliary variable @xmath558 into the generalized lasso criterion : @xmath559 { { \hat{\beta}}},\hat{z } & \in&\mathop{{\operatorname{argmin}}}_{\beta\in{\mathbb{r}}^p , z \in{\mathbb{r}}^p } { \frac{1}{2}}\|y - x\beta\|_{2}^2 + \lambda\|dz\|_{1}\qquad\mbox{subject to } z=\beta ; \\[2pt ] { { \hat{\beta}}},\hat{z } & \in&\mathop{{\operatorname{argmin}}}_{\beta\in{\mathbb{r}}^p , z \in{\mathbb{r}}^m } { \frac{1}{2}}\|y - x\beta\|_{2}^2 + \lambda\|z\|_{1}\qquad\mbox{subject to } z = d\beta.\end{aligned}\ ] ] however , the first two approaches above lead to lagrangian functions that can not be minimized analytically over @xmath548 . only the third approach yields a dual problem in closed - form , as given by @xcite , @xmath560 \\[-7pt ] \eqntext{\mbox{subject to } \|v\|_{\infty}\leq\lambda , d{^t}v \in{\operatorname{row}}(x).}\end{aligned}\ ] ] the relationship between primal and dual solutions is @xmath561 \label{eq : genlassopd2 } { \hat{v}}&= & \lambda\gamma,\end{aligned}\ ] ] where @xmath300 is a subgradient of @xmath301 evaluated at @xmath302 . directly from ( [ eq : genlassodual ] ) we can see that @xmath562 is the projection of the point @xmath563 onto the polyhedron @xmath564 by ( [ eq : genlassopd1 ] ) , the primal fit is @xmath565 , which can be rewritten as @xmath566 where @xmath56 is the polyhedron from lemma [ lem : genlassoproj ] , and finally @xmath567 because @xmath142 is zero on @xmath568 . by ( [ eq : genlassopd2 ] ) , we can define the boundary set @xmath30 corresponding to a particular dual solution @xmath552 as @xmath569 ( this explains its name , as @xmath30 gives the coordinates of @xmath552 that are on the boundary of the box @xmath570 . ) as ( [ eq : genlassopd1 ] ) , ( [ eq : genlassopd2 ] ) are equivalent to the kkt conditions ( [ eq : genlassokkt ] ) , ( [ eq : genlassosg ] ) [ following from rewriting ( [ eq : genlassopd1 ] ) using @xmath571 , the results in section [ sec : genlasso ] on the boundary set @xmath30 can all be derived from this dual setting .
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we derive the degrees of freedom of the lasso fit , placing no assumptions on the predictor matrix @xmath0 .
like the well - known result of zou , hastie and tibshirani [ _ ann .
statist . _
* 35 * ( 2007 ) 21732192 ] , which gives the degrees of freedom of the lasso fit when @xmath0 has full column rank , we express our result in terms of the active set of a lasso solution .
we extend this result to cover the degrees of freedom of the generalized lasso fit for an arbitrary predictor matrix @xmath0 ( and an arbitrary penalty matrix @xmath1 ) . though our focus is degrees of freedom , we establish some intermediate results on the lasso and generalized lasso that may be interesting on their own . .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the exact renormalization group ( rg ) equation @xcite ( erge ) also called non - perturbative or functional rg equation can not be concretely used without recourse to approximation ( for modern reviews or introductory lectures see , e.g. , @xcite ) . the best known approximation framework for the erge is the derivative expansion @xcite . the leading order of that expansion , @xmath3-order , also named the local potential approximation ( lpa ) @xcite , completely discards any momentum dependence from the study . in principle the lpa amounts to projecting the rg flow of the complete action @xmath4 $ ] ( a functional of the field @xmath5 ) onto the space of simple functions @xmath6 of a uniform field @xmath7 by assuming that:@xmath8 = \omega _ { d}\,u\left ( \phi \right ) \label{1}\]]where @xmath9 is the volume of the @xmath10-dimensional space . due to its simplicity and because it is thought that it qualitatively involves most of the properties of the complete erge in the large distance regime ( e.g. , stability properties and number of fixed points ) , the lpa is currently utilized in many studies . numerically , the lpa is considered as a reasonable approximation because the estimations of the critical properties would only be vitiated by the obligatory zero value of the critical exponent @xmath0 ( characterizing the large distance behavior of the two - point correlation function at the critical point ) which , in many circumstances , is actually a small parameter . in the early studies , the condition @xmath2 in the lpa has been justified as a consequence of the neglect of the detailed momentum dependence in the rg @xcite ( the same kind of justification of @xmath2 may be found in @xcite ) . though this argumentation by default is sometimes reused @xcite , it is not very strong . it has been argued that , in the lpa , _ it is not possible to consistently determine _ @xmath0 , or @xmath0 _ is set to zero as there is no mechanism to determine _ its value @xcite ( see also @xcite ) . the alert reader could express some surprise and argue that the vanishing of @xmath0 in the lpa has been clearly demonstrated a long time ago by hasenfratz and hasenfratz @xcite as often put forward in current studies ( see , e.g , @xcite ) . unfortunately , the arguments are not unassailable because they rely , at least allusively ( see section [ invalidity ] ) , on the following truncation of @xmath11 $ ] : @xmath8 = \omega _ { d}\,u\left ( \phi \right ) + \frac{\bar{z}}{2}% \int d^{d}x\left ( \partial _ { x}\phi \left ( x\right ) \right ) ^{2 } \label{2}\]]in which the coefficient @xmath12 of the kinetic term would be maintained unaltered ( equal to unity ) along a rg flow of @xmath13 . a condition which would imply @xmath2 @xcite . pending to show that the argument is actually artificial ( see section [ invalidity ] ) , we may already notice that truncation ( [ 2 ] ) differs in nature from the pure lpa ( [ 1 ] ) since it refers partly to the @xmath14-order of the derivative expansion . consequently , assuming it was correct , this currently accepted argument , basis of what is referred to in the following as the conventional lpa , spoils the logic of the expansion based on a systematic projection of the complete erge onto the space of actions successively truncated according to the number of the derivatives of the field @xmath5 . normally the lpa should correspond to ( [ 1 ] ) and not to ( [ 2 ] ) so that the supposedly proof of @xcite , even true , would be inappropriate . hence , only remains the poor default argument . it is then legitimate to wonder whether the condition @xmath2 is actually obligatory in the pure lpa . it is a matter of fact that the conventional value @xmath2 in the lpa not accidentally but associated with a systematic absence of field renormalization raises some questions : 1 . the absence of any field renormalization precludes in the lpa the eventual setting up of a non - classical power law behavior of correlation functions at criticality @xcite other than that purely induced by a diverging correlation length . according to the theory of critical phenomena , two critical exponents are necessary to determine all the other critical indices of a second order critical point . with the erge considered around a wilson - fisher - like ( wf ) of the correlation length . ] fixed point @xcite , these two exponents are @xmath0 and @xmath15 . the two exponents arise differently in the erge . the index @xmath15 ( which characterizes the divergence of the correlation length @xmath16 when the temperature @xmath17 approaches its critical value @xmath18 ) occurs as a positive eigenvalue of the rg equation linearized around a fixed point ( the number of such positive values determines the order of the transition ) . the role of the index @xmath0 is more subtle . it is associated with the field renormalization allowing for a non - classical power law behavior of the correlation function at criticality . usually one introduces @xmath0 in order to reproduce the critical behavior of the correlation function at large distances ( momenta going to zero ) and @xmath19 ; this manner of doing tightly links the field renormalization to the momentum dependence and suggests that no field renormalization is required when the momentum dependence is neglected @xcite implying @xmath20 . however , the fluctuation - dissipation theorem relates the correlation function to the susceptibility . this allows another introduction of @xmath0 via the critical exponent @xmath21 characterizing the critical behavior of the two - point correlation function at zero momenta and @xmath22 . no reference to an explicit momentum is required in that case but the field should be renormalized nonetheless ( to take this eventual non - trivial power law with @xmath23 into account ) . in the lpa ( conventional or not ) @xmath15 takes on a non - classical value 6137 when the fixed point is a non - trivial one . then , there is a priori no reason why @xmath24 would take on a classical value at this fixed point ( a priori no reason for keeping the field unrenormalized ) . 2 . [ invariance]although broken by the derivative expansion , the reparametrization invariance of the complete erge is expected to be progressively restored as the order of the expansion grows . when it is satisfied , this invariance specifies , in particular , that a change of normalization of the field by a pure constant [ like the parameter @xmath12 in ( [ 2 ] ) ] generates a line of equivalent fixed points characterized by a unique set of critical exponents with the joint existence of a zero eigenvalue mode in the solutions of the erge linearized around those fixed points . the breaking of that invariance by the derivative expansion has been concretely observed at next - to - leading order [ @xmath14-order ] @xcite ; it is such that , for a given smooth cutoff function , a line of fixed points is well generated by the change of normalization of @xmath7 but those fixed points are not equivalent . nonetheless , in agreement with a remark of bell and wilson @xcite in such a situation , one observes the existence at @xmath25-order of a vestige of the invariance via an extremum by a constant factor @xmath26 . ] of @xmath0 @xcite accompanied by the presence of a zero mode . this gives a preferred estimate for @xmath0 ( one sometimes also refers to a principle of minimal sensitivity ) . ] , see , e.g , 2267,5469,5805 ) . because the conventional lpa offers no opportunities to look at the state of the invariance being arbitrarily fixed to zero there is no line of fixed points to be observed . ] , then no signs of progressive restoration of the invariance may be observed by going from the @xmath27-order to the @xmath14-order of the derivative expansion . thus , considered as the leading order of that expansion , the conventional treatment creates confusion about the convergence property of the derivative expansion . this is a pity because the issue is of some importance . 3 . [ etadet]having defined the rg - time @xmath28 ( with @xmath29 the running cutoff scale and @xmath30 an arbitrary fixed momentum scale ) the critical exponent @xmath31 is actually defined in the rg as the limit of a function @xmath32 on approaching a given fixed point ( when @xmath33 ) . according to wilson s prescriptions @xcite , the function @xmath32 is determined by keeping fixed the coefficient of one particular term in the action @xmath34 $ ] with the initial condition @xmath35 . it is customary to keep constant the coefficient @xmath12 of the kinetic term because , due to a symmetry of the action linked to the reparametrization invariance , the flow of such a term is non - essential ( redundant ) so that it may be constrained without altering the model integrity . when the kinetic term is not part of the approximation , as in the pure lpa , the redundancy still exists at least formally and it seems logical to wonder what the state of the reparametrization process is in the pure lpa . to this end one should introduce a function @xmath36 that would maintain fixed a particular monomial of @xmath37 . the line of fixed points mentioned in point [ invariance ] would presumably be generated and this would give back the status of genuine leading order [ @xmath3-order ] of the derivative expansion to the lpa ( see section [ revisedlpa ] ) . in section [ revisedlpa ] we present and discuss a version of the lpa with @xmath38 . we show that it satisfies all the required conditions for being a genuine @xmath3-order of the derivative expansion . in particular we study , in section [ fixedpoints ] , the structure of the fixed points for any value of the dimension @xmath10 and show explicitly , for the first time in the lpa , how the reparametrization invariance is broken . we also introduce , in section [ lt ] , a legendre transformation of the potential adapted to the case studied ( @xmath38 ) . this allows us to utilize easy quasi - analytic methods ( section [ tsm ] ) of integration of an ordinary differential equation ( ode ) well adapted to the obtention of the eigenvalues of the rg flow linearized around a fixed point . it is then shown that the principle of minimal sensitivity ( pms ) does not necessarily indicate preferred values of the critical exponents ( section pms ) . in section [ versuspseudolpa ] , we first show that the conventional argument which is generally put forward to justify @xmath2 in the lpa , is actually artificial ( section [ invalidity ] ) . we then discuss briefly , according to the rg rules , how the reparametrization invariance could be studied with the partial truncation ( [ 2 ] ) ( section pseudolpa ) . in appendix [ movable ] , we illustrate some reason why the quasi - analytic methods of integration of ode of section [ tsm ] do not work in the case of the potential of the action whereas they work after a legendre transformation is performed . we conclude in section [ conclusion ] . let us consider the rg flow equation of the polchinski erge extended to include the parameter @xmath38 in the lpa @xcite , it reads@xmath39where @xmath40 stands for a simple function of @xmath7 and @xmath41 , @xmath42 , @xmath43 , @xmath44 , @xmath10 is the spatial dimension , and @xmath36 the field renormalization parameter which , at a fixed point , takes on the value @xmath45 . in principle and with the complete erge , @xmath45 should coincide with the critical index @xmath0 . the flow equation ( [ lpa0 ] ) has already been studied by kubyshin et al @xcite for the derivative @xmath46 . but they have considered @xmath47 in the lpa for technical reasons exclusively @xcite . hence , in accordance with the conventional lpa , they have left @xmath36 undefined and focused their interest on @xmath45 considered as an arbitrarily adjustable parameter while emphasizing that physically @xmath45 should be zero at this order of the derivative expansion . with a view to study the fixed point equation ( @xmath48 ) for any @xmath10 at one time , we perform the following change of normalization of @xmath7:@xmath49then eq . ( [ lpa0 ] ) transforms into @xcite:@xmath50 \label{lpa } \\ \mu \left ( t\right ) & = & \frac{d-2+\eta \left ( t\right ) } { 2d } \label{mu}\end{aligned}\]]for a given @xmath10 , @xmath51 plays the role of @xmath32 and for any @xmath10 , the fixed point equation involves only one parameter instead of two in the preceding case of ( [ lpa0 ] ) . considering exclusively the issue of finding a non - singular solution @xmath52 to the fixed point equation corresponding to ( [ lpa ] ) [ or ( [ lpa0 ] ) ] , gives no possibility for determining a value of @xmath53 . then @xmath54 may rightly be considered as an extra parameter . indeed @xmath52 is a solution of the following two - point boundary value problem of a second order non - linear ode:@xmath55now the whole of the two integration constants are fixed by the property of parity ( [ cond1 ] ) and by the adjustment of @xmath56 so as to get a non - singular @xmath57 in the whole range @xmath58 -\infty , + \infty \right [ $ ] as prescribed by condition ( [ cond2 ] ) ; then there is no room for determining @xmath59 ( @xmath45 at fixed @xmath10 ) without a supplementary condition . in the conventional lpa , the supplementary condition is merely @xmath60 which would be obtained ( see section [ invalidity ] however ) by an explicit reference to a larger space of truncation functions [ see eq . ( [ 2 ] ) ] . however , even correct , this procedure would not be justified because the rg theory gives precise rules to determine both the function @xmath36 and its fixed point value @xmath45 . as recalled in point [ etadet ] of the introduction , the function @xmath61 is determined by keeping one particular term of the action fixed along the rg flows @xcite ; then @xmath45 is the value reached by @xmath36 in approaching a given fixed point . this procedure is a direct consequence of the reparametrization invariance of the complete action which induces the redundancy of the flow of one term of the action . in absence of any kinetic term , as in the pure lpa , it is logical , and coherent with the rg theory , to define @xmath36 by keeping constant the coefficient of the quadratic term @xmath62 let us examine , on a general ground , the approach of the wf fixed point @xmath63 with the flow equation ( [ lpa ] ) starting at @xmath64 with the following simple potential:@xmath65where the coefficient of the quadratic term has been intentionally noted @xmath66 . because @xmath63 has only one relevant direction , to make the flow approaching @xmath63 only one coefficient of ( [ vdephi0 ] ) , say @xmath67 , must be fine - tuned ( in terms of the other three coefficients ) @xcite . this adjustment is necessary to place the initial potential on the critical surface ( within the domain of attraction ) of @xmath63 @xcite . in order to follow a rg flow , the function @xmath36 must be defined . we do it such that @xmath68 all along the rg flow , with the initial condition @xmath35 and @xmath69 a constant independent of @xmath70 . at the fixed point ( reached at infinite rg - time provided the initial potential lies in the domain of attraction of the fixed point ) , @xmath71 takes on the value of @xmath72 and this defines a line of fixed points ( parametrized by @xmath26 ) . if it was satisfied , the reparametrization invariance would imply that @xmath73 be independent of @xmath26 and equal to @xmath0 . of course , in the pure lpa one rather expects to observe the breaking of that marvelous property and the true question is : to which extent is that invariance broken in the lpa ? to look at this question , suffices to express the variation of @xmath74 in terms of @xmath75 . this is precisely what kubyshin et al have done in @xcite : they studied eq . ( lpa0 ) for @xmath76 and @xmath77 ( for @xmath78 in @xcite ) . the purpose of kubyshin et al was not the status of the reparametrization invariance in the lpa however . in fact , having considered the flow equation for the derivative @xmath79 , the connection parameter of their fixed point equation was not @xmath80 but instead @xmath75 that they have noted @xmath81 . then they have naturally drawn the variation of @xmath82 on changing the value of @xmath45 ( or the reverse ) without relating this variation to the reparametrization process . it is however clear that , with our prescription of keeping @xmath83 fixed along a rg flow , the fixed point is reached with @xmath84 where obviously @xmath26 may be considered as the normalization of the field . consequently , the evolutions of @xmath85 drawn by kubyshin et al are nothing but illustrations of the breaking of the reparametrization invariance in the lpa . let us redo the study of kubyshin et al using our own conventions . using a standard numerical shooting method , we have looked for regular solutions ( the values of @xmath86 ) of the two - point boundary value problem ( [ fp]-[cond2 ] ) . clearly , those solutions are parametrized by @xmath87 . since the coefficient of the quadratic term @xmath88 is linked to @xmath87 via the differential equation as @xmath89 , one easily gets functions @xmath90 corresponding to the functions ) the value of @xmath91 in the study of kubyshin et al is related to our @xmath26 as @xmath92 . ] @xmath93 of kubyshin et al 5255 . the four first solutions are displayed in fig ( [ figpf1 ] ) as continuous curved lines ; this figure involves simultaneously the two graphs of fig . ( 1 ) of kubyshin et al @xcite and displays the same features . let us discuss them . to @xmath94 obtained as regular solutions of eqs . ( [ fp]-[cond2 ] ) . the full circles represent the thresholds of instability of the gaussian fixed point . the horizontal lines indicate three values of @xmath10 where such instabilities occur when @xmath95 see text for more details.,width=453 ] each curved line drawn on fig . ( [ figpf1 ] ) corresponds to a line of fixed points of a particular nature . it appears as bifurcating from the gaussian fixed point ( full circles ) on varying @xmath59 each time @xmath59 falls below the thresholds @xmath96 $ ] , @xmath97 [ horizontal lines on fig . ( [ figpf1 ] ) , corresponding to the usual dimensional thresholds @xmath98 ( for @xmath99 ) ] . in particular fig . ( [ figpf1 ] ) shows the well - known fact that the gaussian fixed point is stable for @xmath100 and @xmath101 ( @xmath102 where there is no regular solution to eq . ( [ fp ] ) . a new fixed point bifurcates each time the gaussian fixed point acquires a new direction of instability ; then the fixed points belonging to a line @xmath90 have @xmath103 directions of instability each . fig . ( [ figpf1 ] ) shows the four first lines of fixed points @xmath104 to @xmath105 ) having respectively one , two , three and four directions of instability . all the lines accumulate at the horizontal line @xmath106 as @xmath107 along which @xmath26 reaches @xmath108 or stop at @xmath109 in agreement with the analytical solution found by kubyshin et al @xcite for @xmath110 . let us focus our attention on the line @xmath111 which is a line of wf fixed points . because , for a given @xmath10 , @xmath112 varies along the line , it is obvious that the reparametrization invariance is broken . of course , this was expected in the lpa but surprisingly had never been explicitly emphasized before the present study . because the line is smooth and monotonous , there is no vestige of the invariance , the breaking is perfect except at the limiting value @xmath113 where , for @xmath78 , @xmath45 takes on the values @xmath114 . notice that nothing particular distinguishes the value @xmath101 from the other values lies at the intersection of @xmath111 and the horizontal line @xmath115 , @xmath101 with @xmath116 ( i.e. @xmath117 ) . ] except at the limiting cases @xmath106 , @xmath76 and @xmath118 , @xmath119 . one observes also that @xmath120 on the whole line of fixed points @xmath121 ( except the gaussian fixed point ) ; this means that the basin of attraction of a non - trivial wf fixed point implies the condition @xmath122 on the initial potential @xmath123 ( otherwise the rg flow goes away from the critical surface towards the trivial high - temperature fixed point ) . notice that the perfect breaking of the reparametrization invariance does not completely spoil the universal character of the critical behavior since the infinite number of initial potentials with a given @xmath122 lying on the critical surface are characterized by the same critical behavior governed by a unique value of @xmath45 and , subsidiarily , of the other critical exponents . from the line @xmath111 , it is also interesting to notice that non - trivial fixed points may be formally considered as existing for @xmath119 provided that @xmath45 be strictly negative . usually such fixed points are rejected but in the present study there is no reason to reject them a priori since it is a consequence of the breaking of the reparametrization invariance to generate also negative values of @xmath124 . it would be puzzling however if a vestige of that invariance led us to choose such a negative value of @xmath45 . fortunately we observe no sign of such a preferred value of @xmath45 along the line @xmath125 except the limit case @xmath106 . at this level we conclude that the lpa does not allow one to determine any estimate of @xmath45 but only ranges of possible values . for example , if one excludes negative values of @xmath126 then for @xmath78 this range would be @xmath127 for the only line @xmath128 , the other lines being excluded . from this example taken alone , one could be inclined to conclude that the conventional lpa , by imposing @xmath101 , would be merely a reasonable choice since one knows that @xmath45 is most often small . however , fig . ( [ figpf1 ] ) shows that from @xmath129 down to @xmath76 , emerges a rich structure of various fixed points with possible different positive and growing values of @xmath45 for which the conventional lpa would impose , increasingly poorly as @xmath10 decreases , the same zero value ( one knows that at @xmath76 , @xmath130 what is not small ) . notice that , for @xmath131 we have excluded the limit case @xmath132 on the line @xmath133 . ] from the range of possible values of @xmath45 though it corresponds to the only point on @xmath111 where a zero eigenvalue exists ( see section [ pms ] ) . indeed , since @xmath134 at this point , this zero mode is not a vestige of the reparametrization invariance ; instead it indicates that the nature of the gaussian fixed point is going to change by losing one direction of instability . hence , if a direct derivative of the fixed point equation with respect to @xmath26 shows that an extremum of @xmath135 implies the appearance of a zero eigenvalue , the reverse is not true . in order to better illustrate the role of the zero mode in the process of restoration of the reparametrization invariance , let us look at the critical exponents in the lpa and at their variations on changing the normalization of the field . to this end , we perform a legendre transformation of the potential which will allow us to make use of user - friendly quasi - analytic methods of integration of ode . the interest of using some quasi - analytic methods to solve the rg flow equation in the lpa is the extremely easy access that they offer to estimate : 1 . the fixed point value @xmath136 of the connection parameter 2 . a set of critical ( and subcritical ) exponents at one time . on the contrary , the purely numerical shooting method necessitates a skillful adjustment of an initial guess of the final value of @xmath136 or , independently , of each critical exponent sought . the use of quasi - analytic methods based on taylor series , in solving a two - point boundary value problem like ( [ fp]-[cond2 ] ) , has been recently reviewed and illustrated in @xcite . among such methods is an extremely simple procedure @xcite ( named the simplistic method in @xcite ) that merely consists of imposing the vanishing of the last term @xmath137 of the maclaurin series of the truncated solution : @xmath138the coefficients @xmath139 being determined such that the edo considered be satisfied order by order in powers of @xmath140 . the auxiliary condition @xmath141 gives a condition from which one tries to extract an estimate of the connection parameter @xmath142 corresponding to the only regular solution of ( [ fp]-[cond2 ] ) . of course , because it is too simple , this simplistic method is not always ( most often never ) efficient . firstly the finite character of the radius of convergence of the series limits the accuracy of the method @xcite . secondly the method may simply not work at all ( in the sense that even a rough estimate of @xmath136 may not be approachable ) . indeed , in trying to solve ( [ fp]-[cond2 ] ) , the issue we are faced with amounts to pushing a movable singularity to infinity . the efficiency of the simplistic method then depends on whether or not that singularity lies within the circle of convergence of the maclaurin series or not ( see appendix [ movable ] ) . a variant of the simplistic method , referred to below as the taylor method , is frequently used which is based on a taylor expansion around the minimum of the potential , as proposed in @xcite ( see also @xcite ) . the solution of the ode is thus expressed as : @xmath143 where @xmath144 is the expansion point chosen to coincide with the minimum of the potential arbitrarily and the two unknowns would have been @xmath67 and @xmath145 . this would offer the possibility of improving the apparent convergence of the taylor method by varying @xmath146 , see @xcite . ] since @xmath147 , and @xmath148 , whereas the original connection parameter is , by definition , given by:@xmath149there are two unknowns ( @xmath67 and @xmath146 ) to be determined . the taylor method consists of imposing the vanishing of the two last terms of the taylor series to get two auxiliary conditions on @xmath67 and @xmath146 . this method may improve considerably the simplistic method ( it has provided excellent estimates of the critical exponents in the lpa @xcite ) . the reason is due to the fact that , other things being equal compared to the simplistic method , one starts closer to the movable singularity . but the taylor method requires that the expansion point ( the minimum of the potential ) lies within the circle of convergence of the maclaurin series ( otherwise one could not get a reliable estimate of @xmath136 by summing the series back to the origin ) . also , the accuracy of the method is naturally limited by the finite range of convergence of the taylor expansion . it is a matter of fact that , in the conventional lpa with @xmath60 , the two quasi - analytic methods presented just above do not work when they are applied to the polchinski rg flow equation of @xmath13 but they work if one first performs a legendre transformation ( @xmath150 ) as that defined in @xcite . with a view to make use of these user - friendly quasi - analytic methods that allow anyone to easily verify the content of the present paper , let us introduce a legendre transformation appropriate to the case @xmath38 . to begin with , we consider the legendre transformation originally introduced for @xmath152 in @xcite and we apply it to the flow equation of @xmath153 extended to include @xmath38 , namely:@xmath154 according to @xcite , the legendre transformation reads @xmath155from which we have:@xmath156 thus applied to ( [ lpamorris0 ] ) we get the following flow equation for @xmath13 : @xmath157which differs from the usual polchinski equation ( [ lpa0 ] ) when @xmath158 . the appearance of this coefficient may be seen as the consequence of a non - linear introduction of @xmath61 in the erge @xcite instead of the linear introduction of @xcite that corresponds to ( [ lpa0 ] ) . though , near a fixed point , the coefficient @xmath159 may be removed from ( lpa0bis ) through the change @xmath160 to get the same equation as ( [ lpa0 ] ) , we have numerically studied ) corresponding to applying the legendre transformation ( [ tl1]-[tl3 ] ) on ( [ lpa0 ] ) , but this would have made the quasi - analytic methods heavier and thus less attractive . ] ( [ lpa0bis ] ) explicitly for @xmath78 ( using a shooting method ) . we have , this way , verified explicitly ( in the case of wf fixed points ) both that we get the same kind of line of fixed points as previously ( monotonous function @xmath161 ) and that the simplistic and taylor methods applied to ( lpamorris0 ) work well also for @xmath38 [ at least for the values of @xmath162 shown in fig ( [ fignudez ] ) ] . let us focus our interest on the eigenvalue problem corresponding to eq . ( [ lpamorris0 ] ) linearized around a fixed point @xmath163 [ solutions of ( [ lpamorris0 ] ) such as @xmath164 . we get the following second order linear ode ( once @xmath165 is known):@xmath166where @xmath167 is the eigenfunction and @xmath168 the eigenvalue parameter . for a given set of initial conditions , such as @xmath169 , @xmath170 in the even case , one expects to obtain an infinite set of discrete couples @xmath171 ordered according to the magnitude of @xmath172 . the number of positive values depends on the fixed point considered . except the trivial eigenvalue @xmath173 , the wf fixed point is characterized by the existence of only one positive value @xmath174 corresponding to the critical exponent @xmath175 , the next eigenvalue @xmath176 is negative and corresponds to the leading subcritical exponent @xmath177 characterizing the leading correction - to - scaling with @xmath178 . these two exponents have been estimated , for @xmath78 , with a very high accuracy in the conventional lpa ( with @xmath20 ) to get @xcite:@xmath179of course , due to the legendre transformation , the same set of critical exponents is obtained in both cases of the flow equations of @xmath153 and @xmath13 @xcite . it is clear that this is also the case in the present study with @xmath38 , provided the methods used converge . we have determined the evolution in terms of @xmath180 of the two first exponents @xmath15 and @xmath181 using both the simplistic and taylor methods and obtained the curves shown in figures ( [ fignudez ] , [ figomegadez ] ) . , of the critical exponent @xmath182 as function of the field - normalization @xmath26 . the full line corresponds to calculations done using the taylor method , open circles correspond to results obtained with the simplistic method . the point located at @xmath183 corresponds to the gaussian fixed point with @xmath184.,width=377 ] , of the subcritical exponent @xmath185 as function of the field - normalization @xmath26 ( full line ) obtained using the taylor method . the point located at @xmath186 @xmath187 is the only possibility of having a zero mode ; it is located far from the value corresponding to @xmath188 for which @xmath189 undergoes a minimum [ see fig . ( [ fignudez])].,width=377 ] fig . ( [ fignudez ] ) shows that @xmath190 undergoes a minimum at @xmath191 corresponding to @xmath192 whereas , at this point we get @xmath193 [ see fig . ( [ figomegadez ] ) ] . according to a principle of minimal sensitivity ( pms ) @xcite sometimes used in calculations at higher orders of the derivative expansion of the erge ( see , e.g , @xcite) one could be inclined to propose those values as being the preferred estimates of the critical exponents in the lpa for @xmath78 . however , one may observe that those values are not designated as the consequence of a vestige of the reparametrization invariance which is the only reason that fundamentally led us to vary @xmath26 . indeed no zero eigenvalue is obtained at this point as shown in table [ table1 ] . ( [ figomegadez ] ) clearly shows that the only point where a zero mode occurs corresponds to the gaussian fixed point which is losing one direction of instability ( @xmath194 , @xmath195 , @xmath132 , for @xmath78 ; or @xmath194 , @xmath196 , @xmath101 , for @xmath119 ) but at this point @xmath197 does not vanish [ see fig . ( [ figpf1 ] ) ] . notice that this is only a confirmation of the absence of any extrema in the function @xmath73 . indeed , if one performs a derivation with respect to @xmath26 of the fixed point equation corresponding to ( [ lpamorris0 ] ) , assuming @xmath198 , then one gets ( [ vplitim ] ) with @xmath199 . the reverse is not true however : the presence of a zero mode may reveal instead the change of the stability properties of the fixed point . .comparison of the six first eigenvalues of eq . ( [ vplitim ] ) obtained for @xmath200 , @xmath78 and with the two quasi - analytic methods considered in the study . the first line of numbers corresponds to @xmath201 ( 1/@xmath202 ) , the second to @xmath203 ( @xmath204 ) , etc . no zero eigenvalue is present . [ cols="<,<",options="header " , ] we may thus conclude that , because it has no link with the reparametrization invariance , the observed minimum of @xmath15 occurs accidentally and that the pms can not be utilized in the circumstances as a tool to determine a preferred set of values of the critical indices . in this section , we first show that the argument of hasenfratz - hasenfratz @xcite , by which the rg flow projected on ( [ 2 ] ) would imply @xmath205 if @xmath12 is kept unaltered by the flow of @xmath13 , is artificial and reduces to a triviality that poorly justifies the default argument . then we briefly illustrate that , correctly treated , the projection of the erge on ( [ 2 ] ) gets an intermediate order [ between the @xmath3 and @xmath14 ] of the derivative expansion that we name pseudo - lpa . this partial @xmath25-order differs from the approximation introduced in @xcite in that we try to account for the reparametrization invariance . to get the rg flow equations of the wilson - polchinski erge correctly projected on ( [ 2 ] ) , suffices to consider the complete @xmath25-order equations available in the literature as , e.g. , in 3491,3836 , and to impose within them that the kinetic term is a pure number that remains constant along a rg flow of the potential . for example , let us consider eqs . ( 12 ) of @xcite for the derivative @xmath207 and a function @xmath208 reduced to @xmath209 , it comes : @xmath210 \,f^{\prime } \left ( 0,t\right ) \label{plpaz}\end{aligned}\]]where @xmath211 is a constant parameter depending on the choice of cutoff function . up to inessential changes , eq . ( [ plpav ] ) is the same flow equation as ( [ lpa ] ) of the pure potential @xmath13 discussed previously in section revisedlpa . ( [ plpaz ] ) shows that the flow of @xmath13 induces a flow of @xmath12 so that keeping it constant , i.e. imposing @xmath212 , yields:@xmath213 \,f^{\prime } \left ( 0,t\right ) = 0 \label{p1}\]]this condition considered at a fixed point of ( [ plpav ] ) may be rewritten as:@xmath214 \,\gamma = 0 \label{p0}\]]where , as seen in section [ revisedlpa ] ( footnote [ gamma ] ) , @xmath215 is a function of @xmath45 as that given implicitly by the lines of fixed points @xmath216 drawn in fig . ( [ figpf1 ] ) where @xmath217 was playing the role of @xmath12 . for @xmath218 and @xmath219 ( the conventional values ) , eq . ( [ p0 ] ) implies @xmath220 . ( [ figpf1 ] ) shows that this is not possible along the line of wf fixed points @xmath128 except trivially at @xmath119 where the fixed point is gaussian@xmath221 the conventional argument of @xcite actually relies upon the arbitrary requirement that no contribution of @xmath13 must alter the flow of @xmath12 so that the right - hand - side of ( [ plpaz ] ) would be reduced to the first term exclusively , thus implying @xmath152 for a constant @xmath222 . clearly , this is only an illustration of an obvious fact : the non - necessity of renormalizing the field ( here forced by the obligatory absence of contribution coming from @xmath13 ) , induces @xmath223 ( what is true by definition ) . we add that , not only truncation ( [ 2 ] ) is incompatible with a pure @xmath3-order , it is also in contradiction with the default argument by which @xmath61 is absent because there is _ no momenta _ in the lpa . actually , the conventional argument is merely artificial . as shown in section [ revisedlpa ] , it is the basis of a conventional lpa which is misleading concerning the concept of reparametrization invariance and in contradiction with the logic of the derivative expansion . considered as an actual truncation of the action @xmath11 $ ] , eq . ( [ 2 ] ) gives access to an intermediate approximate order of the erge [ between @xmath3 and @xmath14 ] . by allowing the coefficient of the kinetic term to flow ( whereas it remains independent of @xmath7 ) one obtains the partial truncation first used by tetradis and wetterich @xcite in order to easily have @xmath224 in an improved(conventional ) lpa . but the way of determining @xmath45 , in the original proposal , is limited to the erge for the effective average action @xmath225 $ ] ( for a review see @xcite ) . this is because one utilizes the available momentum dependence of the exact propagator @xmath226 to determine the function @xmath227 yielding a value for @xmath45 . that way of doing is not convenient to the polchinski erge , with which an easy access to @xmath226 is not possible . moreover , the tetradis and wetterich approach does not give a clear account of the reparametrization invariance ( being in the spirit of the conventional lpa criticized in the present paper ) . let us look at truncation ( [ 2 ] ) for the polchinski erge by strictly applying the basic rules of the rg theory . the rg rules prescribe a field renormalization in order to maintain constant one term of the action . with truncation ( [ 2 ] ) , we choose it to be the kinetic term - order we could have pursued the process of maintaining constant the quadratic term of the action ( instead of the kinetic term ) . in the present pseudo - lpa this procedure would give nothing new compared to the pure lpa . this underlines the particular character of that truncation.[pseudofoot ] ] ( the only momentum - dependent monomial of the approximation ) . hence we get eq . ( [ p1 ] ) and , at a fixed point , eq . ( [ p0 ] ) . considering @xmath12 as a free parameter at hand , @xmath45 and @xmath82 appear to be functions of @xmath222 . we thus get lines of fixed points parametrized by @xmath12 . the relation between @xmath45 and @xmath82 being unchanged compared to the lpa [ eq . ( [ plpav ] ) is the derivative with respect to @xmath7 of ( lpa0 ) ] , we have:@xmath228\]]where @xmath229 $ ] has been determined at leading order in section [ fixedpoints ] [ implicitly through the lines of fixed points displayed by fig . ( [ figpf1 ] ) ] . it is then easy to seek for a vestige of the reparametrization invariance eventually displayed by the new lines of fixed points ( parametrized by @xmath230 ) . suffices to look at possible values of @xmath45 where @xmath231 . differentiating ( [ p0 ] ) with respect to @xmath12 , we get:@xmath232 \,\gamma \,_{\mathrm{lpa}}^{\prime } \right\ } + \eta ^{\ast } + 4\,\gamma _ { \mathrm{lpa}}=0\]]where @xmath233 . finally , imposing the required condition gives a preferred value of @xmath45 defined by:@xmath234 notice that this condition is independent of the choice of the cutoff function , contrary to what is observed with the complete @xmath25-order @xcite ( for another difference with the complete order , see footnote [ pseudofoot ] ) . in terms of the quantity @xmath235 defined by ( [ mu ] ) and taking into account the change of field variable ( [ changephi ] ) , this condition writes , for @xmath78 and @xmath236 ( see footnote [ gamma])@xmath237 from the calculations done in section [ fixedpoints ] , we obtain:@xmath238which are not excellent values . this result may however be considered as an improvement compared to the lpa for which no vestige of the reparametrization invariance was observed . we have justified the presence of a non - vanishing value of @xmath0 in the lpa as a strict consequence of the general principles of the rg theory . without field renormalization , as usually prescribed in the conventional lpa ( with @xmath2 ) , the approximation could not be considered as the genuine @xmath27-order of the derivative expansion . if no particular estimate of @xmath0 can actually be proposed at this order , @xmath31 is not an arbitrary parameter , it varies monotonously ( within some limits ) on changing the normalization of the field as a consequence of a perfect breaking of the reparametrization invariance . the situation is coherent with the idea that , if the derivative expansion converges then , at least , a progressive and smooth restoration of the invariance must be observed from the few first terms of the expansion ( this was not possible with the conventional view ) . we have done explicit calculations of the lines of fixed points generated by the change of the normalization of the field by a constant @xmath26 using both purely numerical and quasi - analytic methods in order to offer the possibility to anyone to easily redo the calculations . we also emphasize that , despites the minimum observed ( for @xmath78 ) in the evolution of the critical exponent @xmath15 on varying @xmath26 , the principle of minimal sensitivity can not be applied being not compatible with a possible vestige of the reparametrization invariance in the lpa . we have shown ( in section [ invalidity ] ) that it is purely artificial the conventional argument stating that @xmath0 should vanish if one keeps the kinetic term unchanged along a rg flow of the potential . we have illustrated ( in appendix movable ) the respective roles of the movable and fixed singularities of the fixed point solutions in the convergence property of the quasi - analytic methods of integration utilized in the study . in this appendix we illustrate the role of the movable singularity of the solution of the fixed point equation of the lpa in the convergence and efficiency of the simplistic method ( see section [ tsm ] ) . let us consider the fixed point equations of respectively the rg flow ( lpa0 ) of @xmath13 and the rg flow ( [ lpamorris0 ] ) of the legendre transformed potential @xmath153 [ see ( [ tl1]-[tl3 ] ) ] . for @xmath78 and @xmath239 the common value of the connection parameter @xmath240 , corresponding to the respective regular fixed point solutions , is known with a huge number of digits @xcite to be:@xmath241 that value corresponds precisely to the only solution of the two - point boundary problem ( [ fp]-[cond2 ] ) with @xmath242 . if one forgets about the condition at infinity , then it exists a solution involving a singularity for each value , is named movable singularity . ] of @xmath243 different from @xmath244 . hence getting the value ( [ kstar ] ) may be viewed as the consequence of pushing that movable singularity to infinity . in addition to a movable singularity , a solution of ( [ fp]-[cond1 ] ) displays also fixed singularities . ] . potentially , those singularities control the convergence properties of the maclaurin series of the ultimate @xmath245 . we have performed pad approximants on the maclaurin series of solutions for various @xmath246 of the fixed point equations in the two cases of @xmath13 and @xmath153 . the complex zeros of the denominators of the corresponding rational fractions give an approximate image of the location of the singularities in the complex plane of the variables @xmath247 and @xmath248 of respectively @xmath13 and @xmath153 . , of the singularities of the solutions @xmath249 of the fixed point equation , for @xmath250 and @xmath78 , of the rg flow eq . ( [ lpamorris0 ] ) in the process of determining @xmath251 @xmath252 the value of which is given by eq . ( [ kstar ] ) . two steps are shown : @xmath253 ( top ) and @xmath254 ( bottom ) . as one approaches @xmath136 , the movable singularity is pushed on the right ( ideally up to infinity ) . the efficiency of the simplistic method ceases when the movable singularity is about to leave the disc of convergence of the maclaurin series ( determined by the location of the fixed singularity the closest to the origin ) . the method yields a rather high accuracy on the estimate of @xmath255 with 9 accurate figures . the radius of convergence of the maclaurin series for @xmath256 is @xmath257.,width=377 ] figure ( [ figsinlitim ] ) shows the singularity structure of @xmath153 for two values of @xmath246 on approaching @xmath136 . one clearly sees that the movable singularity still lies within the circle of convergence of the maclaurin series though @xmath246 is already close to @xmath136 . on approaching closer to @xmath258 the movable singularity is pushed to the right and the simplistic method ceases to converge when the singularity comes out of the disc of convergence of the series . concretely the method provides an estimate of @xmath258 with 9 accurate figures . , of the singularities of the solution of eq . ( [ fp ] ) , for @xmath250 and @xmath78 , in the process of determining @xmath251 @xmath259 given by eq . ( [ kstar ] ) . same presentation as in fig . ( [ figsinlitim ] ) . the step corresponds to @xmath260 which is relatively far from @xmath136 whereas the movable singularity is already outside the disc of convergence of the maclaurin series . the simplistic method does not work in that case . the radius of convergence of the maclaurin series for @xmath261 is @xmath262,width=377 ] on the contrary , fig . ( [ figsingpol ] ) shows that the movable singularity is already well outside the circle of convergence of the maclaurin series of @xmath13 when @xmath246 is still far from @xmath136 . in that case the simplistic method can not even give a poor estimate of @xmath136 . k. i. aoki , int . b * * 14 * * ( 2000 ) 1249 ; j. polonyi , cent . * 1 * ( 2003 ) 1 . [ arxiv : hep - th/0110026 ] ; j. m. pawlowski , ann . ( n.y . ) * 322 * ( 2007 ) 2831 . [ arxiv : hep - th/0512261 ] ; b. delamotte , in _ order , disorder and criticality . advanced problems of phase transition theory , vol 2 _ , p. 1 , _ ed . by _ yu . holovatch ( world scientific , publ . co. , singapore , 2007 ) ; also in lecture notes in physics vol . * 852 * ( 2012 ) 49 . [ arxiv : cond - mat/0702365 ] ; o. j. rosten , phys . rep . * 511 * ( 2012 ) 177 . [ arxiv:1003.1366 ] ; a. wipf , lecture notes in physics * 864 * ( 2013 ) 257 . c. wetterich , z. phys . c * 57 * ( 1993 ) 451 ; t. r. morris , nucl . b ( proc . suppl . ) * 42 * ( 1995 ) 811 . [ arxiv : hep - lat/9411053 ] ; int . j. mod b * 12 * ( 1998 ) 1343 . [ arxiv : hep - th/9610012 ] ; phys . lett . b * * 329 * * ( 1994 ) 241 . [ arxiv : hep - ph/9403340 ] t. r. morris , nucl . b * 458 * ( 1996 ) 477 . [ arxiv : hep - th/9508017 ] ; j. comellas and a. travesset , nucl . b * 498 * ( 1997 ) 539 . [ arxiv : hep - th/9701028 ] ; t. r. morris and m. d. turner , nucl . phys . b * * 509 * * ( 1998 ) 637 . [ arxiv : hep - th/9704202 ] y. kubyshin , r. neves and r. potting , in _ the exact renormalization group _ , p. 159 , _ ed . by _ a. krasnitz , y. a. kubyshin , r. potting and p. s ( world scientific , publ . co. , singapore , 1999 ) . [ arxiv : hep - th/9811151 ]
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the conventional absence of field renormalization in the local potential approximation ( lpa ) implying a zero value of the critical exponent @xmath0 is shown to be incompatible with the logic of the derivative expansion of the exact renormalization group ( rg ) equation .
we present a lpa with @xmath1 that strictly does not make reference to any momentum dependence .
emphasis is made on the perfect breaking of the reparametrization invariance in that pure lpa ( absence of any vestige of invariance ) which is compatible with the observation of a progressive smooth restoration of that invariance on implementing the two first orders of the derivative expansion whereas the conventional requirement ( @xmath2 in the lpa ) precluded that observation .
local potential approximation , derivative expansion , exact renormalization group equation , reparametrization invariance , anomalous dimension 05.10.cc , 11.10.gh , 11.10.hi , 64.60.ae , 02.60.lj
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You are an expert at summarizing long articles. Proceed to summarize the following text:
we begin this section by fixing the notation . let @xmath3 be monic polynomials of degree @xmath4 in @xmath5 and orthogonal , with respect to a weight , @xmath6 $ ] , @xmath7 where @xmath8 is the square of the weighted @xmath9 norm of @xmath10 also , @xmath11 for convenience we set @xmath12 . the recurrence relation follows from the orthogonality condition : @xmath13 where @xmath14 , the @xmath15 are real and @xmath16 are strictly positive . in this paper we describe a formalism which will facilitate the determination of the recurrence coefficients for polynomials with singular weights . two points of view lead to this problem : on one hand the x - ray problem @xcite of condensed matter theory , on the other hand related problems in random matrix theory which involve the asymptotics of the fredholm determinant of finite convolution operators with discontinuous symbols @xcite . this paper is the first in a series that systematically study orthogonal polynomial where the otherwise smooth weights have been singularly deformed . the ultimate aim is the computation for large @xmath4 of the determinant @xmath17 of the @xmath18 moments or hankel matrix @xmath19 with moments @xmath20 where @xmath21 , thereby doing what has been done for the determinants of @xmath18 toeplitz matrices with singular generating functions @xcite . the deformed weight with one jump is @xmath22 where @xmath23 is the position of the jump , @xmath24 is the heaviside step function and the real @xmath25 parametrises the height of the jump . more generally , we take @xmath26 to be the canonical jump function @xmath27 and @xmath28 . the actions of the ladder operators on @xmath29 and @xmath30 are @xmath31 @xmath32 where @xmath33 . if @xmath34 and @xmath35 are non - vanishing one must add @xmath36 to and respectively . now @xmath0 and @xmath1 , the coefficient functions in the ladder operators , satisfy identities analogous to those found for smooth weights @xcite : @xmath37 the derivation of - will be published in a forthcoming paper where the weight has several jumps and @xmath38 $ ] is the jacobi weight . multiplying the recurrence relation evaluated at @xmath39 by @xmath40 and noting as well as we arrive at the universal equality @xmath41 similarly , squaring @xmath42 we find a second universal equation @xmath43 note that in the expressions for @xmath0 and @xmath1 only @xmath44 , the `` potential '' associated with the smooth reference weight , appears . the discontinuities give rise to @xmath45 and @xmath46 it is clear from and that if @xmath47 is rational , then @xmath0 and @xmath1 are also rational . this is particularly useful for our purpose which is the determination of the recurrence coefficients , for in this situation by comparing residues on both sides of and we should find the required difference equations @xcite . in the following section the above approach is exemplified by the hermite weight , @xmath48 and @xmath49 given by . it turns out that in this situation @xmath50 and @xmath51 are related to @xmath52 and @xmath53 in a very simple way . now , @xmath48 , so that @xmath54 , and @xmath55 as in . also , @xmath56 which are independent of the particular choice of @xmath2 and @xmath57 particular to @xmath58 note that @xmath59 is the value of @xmath60 at @xmath61 instead of proceeding with the full machinery of and we take advantage of the fact that @xmath54 . from orthogonality and the recurrence relation , we have @xmath62 by integration by parts . the string equation , @xmath63 is an immediate consequence of the orthogonality condition . again , an integration by parts and noting that @xmath64 produces @xmath65 it should be pointed out here that in general neither the string equation nor will provide the complete set of difference equations for the recurrence coefficients which can be seen if @xmath2 were the jacobi weight . in such a situation the compatibility conditions and must be used . now and become @xmath66 and @xmath67 equations and , supplemented by the initial conditions @xmath68^{-1 } \quad \textrm{and } \quad r_0({{\tilde x } } ) = 0 \ , \ ] ] can be iterated to determine the recurrence coefficients numerically . also , explicit solutions to and can be produced for small @xmath4 . if and are combined with the evolution equations to be derived in this section , the painlev iv mentioned in the abstract is found . we begin with the @xmath9 norm @xmath70 , , which entails @xmath71 and thus @xmath72 since @xmath73 . with , @xmath74 which is the first toda equation . taking the derivative with respect to @xmath69 of at @xmath75 and using the definition of the monic polynomials then gives @xmath76 since @xmath77 is an immediate consequence of the recurrence relation . therefore @xmath78 the second toda equation . eliminating @xmath79 from and the second toda equation , gives @xmath53 in terms of @xmath50 and @xmath80 : @xmath81 using the first toda equation to express @xmath82 in terms of @xmath50 and @xmath83 and substituting into produces a particular painlev iv @xcite , @xmath84 which can be brought into the canonical form with the replacements @xmath85 and @xmath86 . is supplemented by the boundary conditions @xmath87 . in a recent paper @xcite , a painlev iv was derived for the discontinuous hermite weight using an entirely different method . based on and the derivative of the logarithm of the hankel determinant @xmath88 can be computed as @xmath89 where has been used in the first line , which can be summed by the christoffel - darboux formula , @xmath90 in the limit @xmath91 we find , in general , @xmath92 using the the ladder operators and . with this entails @xmath93 the apparent pole at @xmath94 can be shown to have vanishing residue by considering @xmath95 : @xmath96 where the last equality is due to . a further regular term can be found as a contribution from the taylor series of @xmath97 about @xmath94 namely , @xmath98 where @xmath99 . using the fact that @xmath64 and , we see that the first two terms of combined into @xmath100 cancel the third . we are therefore left with the regular term : @xmath101 using it follows from @xmath102 that reproduces the second derivative , @xmath103 . let @xmath104 be the free energy . expressing in terms of @xmath105 and finally in terms of @xmath106 , where @xmath107 , the free energy reads @xmath108 where @xmath109 is the free energy corresponding to @xmath110 . note that @xmath111 , which gives rise to the sum rule @xmath112 with a minor change of variables becomes the toda molecule equation . first we note that @xmath113 defining @xmath114 it then follows @xmath115 we may express @xmath105 in terms of the derivatives of the free energy , by noting that @xmath116 and @xmath117 . one finds with @xmath118 where @xmath119 . for @xmath120 we find the asymptotic expansion @xmath121 \nonumber \\ \fl r_n(0 ) & = & -b ( -1)^n \cos ( 2 b \ln n + b ) - \frac{b^2}{2n } \big ( 1+\sin^2(2b\ln n+b ) \big ) + \mathcal{o}(n^{-2 } ) { \label{eq : asymptote_rn}}\end{aligned}\ ] ] guided by the numerics on the difference equations and . the constant @xmath122 is given by @xmath123 and @xmath124 is a phase independent of @xmath4 . unfortunately , the formalism developed in this paper does not seem to shed any light on its delicate dependence on @xmath25 . it can be verified by a direct calculation that and satisfy and to order @xmath125 . the top panel of shows a comparison between the numerical results and the above asymptotes for suitably rescaled @xmath50 . in principle , @xmath50 and @xmath53 can be determined analytically to any order in @xmath4 because an approximation of @xmath50 to order @xmath126 gives rise to a difference equation for @xmath53 to order @xmath127 via . in turn , @xmath53 to order @xmath127 produces an equation for @xmath50 to order @xmath127 and so forth . this scheme breaks down for @xmath128 . for fixed @xmath128 , the numerics does not suggest an ansatz for the asymptotes . most remarkably , the effect of @xmath128 persists for very large @xmath4 , even for very small @xmath128 , as illustrated in . also shown in this figure as dashed ( dotted ) lines are the approximations of @xmath105 from the first three ( two ) terms of a taylor - series in @xmath69 around @xmath120 based on the iterative results for @xmath129 , and . in principle , the painlev iv , , provides a way to express @xmath130 in terms of lower order derivatives @xmath131 with @xmath132 , yet the results in suggest that for sufficiently large @xmath4 a finite taylor series eventually deviates wildly from the correct @xmath105 . note that both @xmath133 and @xmath134 are bounded in @xmath4 for large @xmath4 . shows the rescaled @xmath50 for fixed @xmath4 and varying @xmath69 . it resembles a hermite polynomial because of its direct relation to @xmath135 , and . for the same reason @xmath105 vanishes for @xmath136 . xxx e. l. basor and y. chen , `` the x - ray problem revisited '' , j. phys . a. : math . gen . * 36 * ( 2003 ) l175-l180 ; `` a note on the wiener - hopf determinants and the borodin - okunkov identity '' , integr . oper . theory , * 45 * ( 2003 ) 301 - 308 . h. widom , `` toeplitz determinants with singular generating functions '' , amer . , * 95 * ( 1973 ) 333 - 383 ; e. l. basor , `` asymptotic formulas for toeplitz determinants '' , trans . * 239 * ( 1978 ) 33 - 65 . a. p. bassom , p. a. clarkson , a. c. hicks , and j. b. mcleod , `` integral equations and exact solutions of the fourth painlev equation '' , proc . london ser . a * 437 * ( 1992 ) 1 - 24 . y. chen and m. e. h. ismail , `` jacobi polynomials from compatibility conditions '' , proc . amer . math . * 133 * ( 2005 ) 465 - 472 , 225 - 237 . y. chen and m. e. h. ismail , `` ladder operators and differential equations for orthogonal polynomials '' , j. phys . a. : math . gen . * 30 * ( 1997 ) 7817 - 7829 . p. j. forrester and n. s. witte , `` discrete painlev equations and random matrix averages '' , nonlinearity , * 16 * ( 2003 ) 1919 - 1944 . m. e. h. ismail and j. wimp , `` on differential equations for orthogonal polynomials '' , methods appl . anal . * 5 * ( 1998 ) 439 - 452 . m. jimbo , t. miwa , y. mri and m. sato , `` density matrix of an impenetrable bose gas and the fifth painlev transcendent '' , physica d * 1 * ( 1980 ) 80158 .
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in this paper we present a brief description of a ladder operator formalism applied to orthogonal polynomials with discontinuous weights .
the two coefficient functions , @xmath0 and @xmath1 , appearing in the ladder operators satisfy the two fundamental compatibility conditions previously derived for smooth weights .
if the weight is a product of an absolutely continuous reference weight @xmath2 and a standard jump function , then @xmath0 and @xmath1 have apparent simple poles at these jumps .
we exemplify the approach by taking @xmath2 to be the hermite weight . for this simpler case
we derive , without using the compatibility conditions , a pair of difference equations satisfied by the diagonal and off - diagonal recurrence coefficients for a fixed location of the jump .
we also derive a pair of toda evolution equations for the recurrence coefficients which , when combined with the difference equations , yields a particular painlev iv .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
unlike the equilibrium thermodynamics and statistical mechanics , which are well developed after the pioneering works of boltzmann and gibbs , our understanding of non - equilibrium thermodynamics is restricted to some special models and cases . stochastic lattice gases have provided a fertile testing ground for studying non - equilibrium stationary states in driven systems @xcite . these models exhibit a variety of phase transition arising from a diffusive ( collisional ) relaxation . for some of these models local equilibrium and hydrodynamic equations have been derived rigorously @xcite . there are , however , other physical systems for which the approach to final stationary state is through a process of collisionless relaxation @xcite . gravitational systems and confined one component plasmas are just two such examples . for these systems the collision duration time diverges and the relaxation is governed by the collisionless boltzmann ( vlasov ) equation @xcite . in the thermodynamic limit , the collisionless relaxation process leads to non - maxwell - boltzmann velocity distributions , even for stationary states without macroscopic currents . unlike normal thermodynamic equilibrium , the stationary state which follows the collisionless relaxation depends explicitly on the initial distribution of particle positions and velocities . in spite of this complication , it was recently shown that it is possible to construct a statistical theory that quantitatively describes these states @xcite . beams of electrons driven by accelerating vacuum devices , like the thermionic valves , diodes , and magnetrons , also do not relax to the maxwell - boltzmann distribution @xcite . unlike the driven stochastic lattice gases , these systems , however , are intrinsically collisionless . an important practical question concerns the kinetic temperature distribution in thermionic devices in which the directed velocity produced by the electric field is comparable to the thermal velocity @xcite . this is particularly the case for the transitional region between child - langmuir and no - cutoff regimes in magnetrons , where the electric potential becomes comparable to the thermal energy @xcite . even when the final directed velocity is larger than the thermal velocity , there is a region near the emitting cathode where thermal effects are important . it is of great practical interest to determine the extent of these regions @xcite . furthermore , since in these systems the collision duration time diverges , there is no local equilibrium , and one can not _ a priori _ postulate an equation of state relating the beam density and the beam temperature , as for adiabatic or isothermal processes . instead , given the properties of thermionic filaments such as say the velocity distribution of the emitted electrons one should solve the boundary value problem posed by the vlasov equation . the purpose of this letter is to develop a theoretical framework which will allow us to study the relaxation dynamics and the stationary states of collisionless driven systems . as a prototype of a collisionless driven system , we consider a beam of electrons , accelerated by an external electric field , traveling from an emitting ( planar ) cathode to a collecting ( planar ) anode across the device gap . the cathode , located at position @xmath0 , is kept at electrostatic potential @xmath1 and is heated to temperature @xmath2 , resulting in the emission of electrons . after traversing the device gap , these electrons are collected at the cold anode ( @xmath3 ) located at @xmath4 and kept at potential @xmath5 . during the steady state operation , the region between the cathode and anode contains a total of @xmath6 electrons , resulting in a current density @xmath7 . our goal is to relate @xmath7 , to the potential difference @xmath8 , the number of electrons @xmath6 , the device width @xmath9 , and the cathode temperature @xmath2 . for planar electrodes , particle distribution transverse to the @xmath10 axis can be taken to be uniform . furthermore , the one particle distribution function for a collisionless system in a steady state must satisfy the stationary vlasov equation , @xmath11 where @xmath12 is the elementary charge , @xmath13 is the electron mass , and @xmath14 is the static distribution function . in the thermodynamic limit , vlasov equation becomes exact for particles interacting by long range potentials @xcite . it can be readily seen that the distribution functions of the form @xmath15 $ ] , where @xmath16 is the mean particle energy , @xmath17 , satisfy eq . ( [ equa1 ] ) . therefore , if @xmath18 is specified at @xmath0 , @xmath19 is then also determined for any other position , provided that the electrostatic potential @xmath20 is known . this potential can , in turn , be calculated self - consistently from the solution of the poisson equation @xmath21 where the particle density @xmath22 is given by @xmath23 , the total particle number is @xmath24 , and the transverse cross sectional area of the essentially 1d device is @xmath25 . to represent both the thermal distribution near the cathode , and the fact that only particles with positive velocities actually move into the device gap , we choose at @xmath0 a unidirectional maxwellian distribution of the form @xmath26 where @xmath27 is the boltzmann constant , @xmath2 is the cathode temperature , and @xmath28 is the beam density at the cathode after the stationary state is achieved . the value of @xmath28 can only be obtained once the full problem has been resolved . the distribution function over the length of the whole diode is then integrating the distribution function @xmath31 $ ] over the possible values of velocity , we arrive at a nonlinear integro - differential equation for the electrostatic potential , @xmath32 it is important to note the difference between this equation and the poisson - boltzmann equation obtained for usual collisional plasmas and electrolytes in the mean - field limit @xcite . equation ( [ equa4 ] ) can be solved numerically , to yield the electrostatic potential and the distribution function for the electron beam in the stationary state . for systems with long range interactions , vlasov equation should become exact in the thermodynamic limit . to confirm this for our system , we have performed molecular dynamics simulation of an equivalent one dimensional model . the simulated system consists of @xmath33 mutually interacting charged sheets of area @xmath25 each containing @xmath34 electrons of the same velocity moving along the @xmath10 axis , under the action of the external electric field produced by the grounded cathode @xmath35 and an anode kept at a fixed potential @xmath36 . the interaction potential between the two sheets @xmath37 is the green s function @xcite of the laplace equation , @xmath38 with the boundary conditions @xmath39 . solving this equation we obtain @xmath40 , where @xmath41 and @xmath42 are the smaller and the larger of the two particle coordinates @xmath43 and @xmath44 . the effective hamiltonian for the sheet dynamics is then @xmath45 where @xmath46 and @xmath47 are the charge and mass of each sheet respectively . the acceleration of each simulated sheet then follows from the canonical equations of motion , where @xmath49}$ ] is the number of sheets to the left(right ) of the one considered , and @xmath50 denotes the positional average @xmath51 . since @xmath52 , from eq . ( [ equa5 ] ) one sees that the electron acceleration at the device entrance where @xmath53 } \rightarrow 0 $ ] and @xmath54 } \rightarrow n_s$ ] satisfies @xmath55 , which reveals that in _ space - charge dominated _ devices where @xmath56 , acceleration at beam entrance may be zero or even negative @xcite . when the acceleration vanishes , the associated current is denoted as the _ limiting _ one . since we wish to describe a hot cathode and a cycling current inside the device , we adopt the following strategy . we advance the simulation in small time steps , always obeying eq . ( [ equa5 ] ) . whenever a particle crosses the anode and exits the system , it is re - injected at the cathode position . at this point all the particles in a small region @xmath57 around the cathode are re - thermalized , so as to ensure that the distribution there keeps its original form of a truncated maxwellian . the width @xmath57 must be sufficiently small , @xmath58 , but apart from this condition its precise value is arbitrary . the simulations were performed with @xmath59 and @xmath60 . in all cases we start with a uniform distribution of sheets and compute the observables only after the system reaches its final stationary state . to compare the predictions of the theory with the results of the simulations , we consider the density and the temperature distributions inside the diode . the kinetic temperature is defined as @xmath61 where the over - bar denotes the velocity average at a given position @xmath10 . the theoretical averages are calculated using the distribution function @xmath31 $ ] , while in the simulations , the averages are performed over the particle velocities within narrow bins along the @xmath10 axis . note that because of the asymmetry of the velocity distribution at @xmath0 , @xmath62 . it is convenient to scale space and time with the diode length @xmath9 and the plasma frequency @xmath63 , respectively . dimensionless coordinate and velocity can then be defined as @xmath64 and @xmath65 . in addition , eqs . ( [ equa5 ] ) and ( [ equa3 ] ) show that adimensional voltage and adimensional temperature can be defined as @xmath66 and @xmath67 , respectively , and serve as the control parameters for the system . in fig . [ fig1.eps](a ) the scaled temperature @xmath68 is plotted against the scaled coordinate @xmath69 . we consider @xmath70 and also consider a device operating at its limiting current , @xmath71 . a striking feature of this plot is that the temperature drops rapidly as one moves away from cathode towards anode . we next study the dependence of scaled density @xmath72 along the length of the diode . the density is very high near the cathode , where the average velocity is small . it then drops rapidly towards the anode , where particles are accelerated up to high speeds , see fig . [ fig1.eps](b ) . agreement between the simulations and the theory for both the kinetic temperature and density is excellent . we now study the current - voltage phase diagram of the device . in general , current is a function of the voltage drop , the temperature , the gap length , and the total charge of the device . however , by measuring the time in units of one over the plasma frequency @xmath73 , and the length in units of the gap length @xmath9 , we can scale away two of these variables . the current density can be calculated using @xmath74 since in the steady state the current does not depend on either time or coordinate , integration along the @xmath10 axis and over the cross sectional area @xmath25 yields @xmath75 furthermore , since the current density is measured in units @xmath76 \sim [ ne v / a l]$ ] , rescaling it in terms of the gap length and the plasma frequency , we can write the reduced current density as @xmath77 , which then satisfies @xmath78 where @xmath79 is the reduced velocity averaged over _ all _ the particles . the reduced average velocity , in turn , must be a function of the two previously introduced control parameters : the reduced voltage and temperature . eq.([sim ] ) is in fact a similarity transformation relating systems with different charge , length , temperature , and potential difference . in fig . [ fig2.eps ] we plot @xmath80 vs. @xmath81 for various @xmath68 . the phase diagram provides all the information about the current - voltage characteristics for all possible planar diodes . the first feature to note is that all the different curves emanate from the limiting current backbone , which traces a temperature dependent path in the @xmath82 plane . to the left of the limiting current border , indicated by the solid line in fig . [ fig2.eps ] , the distribution function can no longer be described by a unidirectional maxwellian , such as the expression ( [ equa3 ] ) . the transition resembles bose - einstein condensation ( bec ) . in the case of bec below the critical temperature a macroscopically populated ground state appears , and only a fraction of particles remains in the excited states . similarly , in the case of our diode , to the left of the limiting curve , part of the charge must be expelled from the system before a stationary state can be achieved . as @xmath83 , the voltage effects become negligible compared to the thermal ones , and the beam density becomes uniform across the gap . in this limit , it is possible to show that the dimensionless backbone curve asymptotes to a vertical line , @xmath84 , fig . [ fig2.eps ] . vs. @xmath85 . the thick solid line represents the limiting current and the thick dashed line , the zero temperature limit . to the left of the solid curve , charge must be expelled from the system before a stationary state can be achieved . dotted lines represent the theoretical results for the indicated temperatures . the circles are the results of the simulations at the same temperatures.[fig2.eps],width=302,height=264 ] to conclude , we have studied the dynamics of collisionless driven systems . unlike the stochastic lattice gasses which are significantly abstracted from reality , the models studied in this paper are very similar to real electronic devises , such as the thermionic valves , diodes , and magnetrons . furthermore , differently from the lattice gases whose dynamics is diffusive , the distribution function of collisionless systems satisfies the vlasov equation . for the class of driven systems introduced in this letter , the stationary state vlasov equation can be solved exactly . the theory developed in this paper should , therefore , be relevant to the design and operation of real electronic devises . it is important to stress that in the absence of collisions , a charged beam does not relax to an equilibrium with a known equation of state . in fact , the thermodynamic temperature is defined only in the vicinity of the hot emitting cathode . away from the cathode , dynamics is controlled by the collisionless vlasov equation , which has to be solved as a boundary value problem . once the solution is obtained , all the macroscopic quantities can be determined via appropriate averages . the kinetic temperature is found to vary strongly across the device gap , precluding the use of conventional isothermal or adiabatic assumptions and of the hydrodynamic formalisms .
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a statistical theory is presented which allows to calculate the stationary state achieved by a driven system after a process of collisionless relaxation .
the theory is applied to study an electron beam driven by an external electric field .
the vlasov equation with appropriate boundary conditions is solved analytically and compared with the molecular dynamics simulation .
a perfect agreement is found between the theory and the simulations . the full current - voltage phase diagram is constructed .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
one of the first striking applications of gromov s theory of pseudoholomorphic curves @xcite was that a closed exact lagrangian immersion @xmath4 inside a liouville manifold must have a double - point , given the assumption that it is hamiltonian displaceable . gromov s result has the following contact - geometric reformulation , which will turn out to be useful . consider the so - called _ contactisation _ @xmath5 of the liouville manifold @xmath6 , which is a contact manifold with the choice of a contact form . recall that a ( generic ) exact lagrangian immersion @xmath4 lifts to a legendrian ( embedding ) @xmath7 . one says that @xmath1 is _ horizontally displaceable _ given that @xmath8 is hamiltonian displaceable . the above result thus translates into the fact that a horizontally displaceable legendrian submanifold @xmath1 must have a _ reeb chord _ for the above standard contact form i.e. a non - trivial integral curve of @xmath9 having endpoints on @xmath1 . a similar result holds for legendrian submanifolds of boundaries of subcritical weinstein manifolds , as proven in @xcite by mohnke . in the spirit of arnold @xcite , the following conjectural refinement of the above result was later made : the number of reeb chords on a chord - generic legendrian submanifold @xmath7 whose lagrangian projection is hamiltonian displaceable is at least @xmath10 . however , as was shown by sauvaget in @xcite by the explicit counter - examples inside the standard contact vector space @xmath11 , @xmath12 , the above inequality is not true without additional assumptions on the the legendrian submanifold ; also , see the more recent examples constructed in @xcite by ekholm - eliashberg - murphy - smith . the latter result is based upon the h - principle proven in @xcite by eliashberg - murphy for lagrangian cobordisms having loose negative ends in the sense of murphy @xcite . on the positive side , the above arnold - type bound has been proven using the legendrian contact homology of the legendrian submanifold , under the additional assumption that the legendrian contact homology algebra is sufficiently well - behaved . legendrian contact homology is a legendrian isotopy invariant independently constructed by chekanov @xcite and eliashberg - givental - hofer @xcite , and later developed by ekholm - etnyre - sullivan @xcite . this invariant is defined by encoding pseudoholomorphic disc counts in the legendrian contact homology differential graded algebra ( dga for short ) which usually is called the _ chekanov - eliashberg algebra _ of the legendrian submanifold . in the case when the chekanov - eliashberg algebra of a legendrian admits an augmentation ( this should be seen as a form of non - obstructedness for its floer theory ) , the above arnold - type bound was proven by ekholm - etnyre - sullivan in @xcite and by ekholm - etnyre - sabloff in @xcite . in @xcite , the authors generalised this proof to the case when the chekanov - eliashberg algebra admits a finite - dimensional matrix representation , in which case the same lower bound also is satisfied . the above arnold - type bound is also related to the one regarding the number of hamiltonian chords between the zero - section in @xmath13 ( or , more generally , any exact closed lagrangian submanifold of a liouville manifold ) and its image under a generic hamiltonian diffeomorphism . namely , such hamiltonian chords correspond to reeb chords on a legendrian lift of the union of the lagrangian submanifold and its image under the hamiltonian diffeomorphism . in fact , as shown by laudenbach - sikorav in @xcite , the number of such chords is bounded from below by the stable morse number of the zero - section ( and hence , in particular , it is bounded from below by half of the betti numbers of the disjoint union of _ two _ copies of the zero - section ) . arnold originally asked whether this bound can be improved , and if in fact the _ morse number _ of the zero - section is a lower bound . however , this question seems to be out of reach of current technology . on the other hand , we note that the stable morse number is equal to the morse number in a number of cases ; see @xcite as well as section [ sec : gendefs ] below for more details . finally , we mention the remarkable result by ekholm - smith in @xcite , which shows that the smooth structure itself can predict the existence of more double points than the original bound given in terms of the homology . namely , a @xmath14-dimensional manifold @xmath15 for @xmath16 that admits a legendrian embedding having precisely one transverse reeb chord in the standard contact space must be _ diffeomorphic _ to the standard sphere unless @xmath17 . also see @xcite for similar results in other dimensions . in this paper , we will explore a priori lower bounds for the number of reeb chords on a legendrian submanifold @xmath7 , given that it admits an exact lagrangian filling @xmath18 inside the symplectisation . recall that the condition of admitting an exact lagrangian filling is invariant under legendrian isotopy ; see e.g. @xcite . the bound will be given in terms of the simple homotopy type of @xmath19 . first , we recall that such a legendrian submanifold automatically has a well - behaved chekanov - eliashberg algebra ; namely , an exact lagrangian filling induces an augmentation by @xcite . in the case when the projection of @xmath1 to @xmath20 is displaceable , the aforementioned result can thus be applied , giving the above arnold - type bound . however , in this case , there are even stronger bounds that can be obtained from the topology of the exact lagrangian filling @xmath19 ( and without the assumption of horizontal displaceability ) . see section [ sec : wrapped ] below for previous such results as well as an outline of the proof , which is based upon seidel s isomorphism in wrapped floer homology . this is also the starting point of the argument that we will use in order to prove our results here . in the following we assume that a legendrian submanifold @xmath21 is chord - generic and has an exact lagrangian filling @xmath22 . here @xmath23 denotes the coordinate on the first @xmath24-factor . in particular , the set of reeb chords @xmath25 of @xmath1 is finite . further , the set of reeb chords @xmath26 in degree @xmath27 will be denoted by @xmath28 , where the grading is induced by the conley - zehnder index modulo the maslov number @xmath29 of @xmath19 as defined in @xcite . observe that @xmath30 in particular implies that the first chern class of @xmath6 vanishes on @xmath31 . for a group @xmath32 being the epimorphic image of @xmath33 , consider the morse homology complex @xmath34),\partial_f)$ ] of @xmath19 with coefficients in the group ring @xmath35 $ ] twisted by the fundamental group , where @xmath36 is a unital commutative ring and @xmath37 is a morse function satisfying @xmath38 outside of a compact set . ( the generators of this complex are graded by the morse index , and the differential counts negative gradient flow lines . ) [ mainthmsmplehomeqnts ] let @xmath22 be an exact lagrangian filling of an @xmath39-dimensional closed legendrian submanifold @xmath40 with fundamental group @xmath41 and maslov number @xmath29 . * in the case when the filling is spin and when @xmath30 , the morse homology complex @xmath42),\partial_f)$ ] is simple homotopy equivalent to a @xmath43$]-equivariant complex @xmath44\langle \mathcal{q}_{n-\bullet}(\lambda ) \rangle,\partial)$ ] ; * in the general case , it follows that the complex @xmath45),\partial_f)$ ] is homotopy equivalent in the category of @xmath32-equivariant complexes to a complex @xmath46\langle \mathcal{q}_{n-\bullet}(\lambda ) \rangle,\partial)$ ] with grading in @xmath47 . here we can always take @xmath48 , while we are free to choose an arbitrary unital commutative ring in the case when @xmath19 is spin . we prove theorem [ mainthmsmplehomeqnts ] in section [ maintheoremanditsconsequences ] . now let @xmath49 denote the stable morse number of a manifold @xmath50 with possibly non - empty boundary , see definition [ defn : stablemorse ] . using theorem [ mainthmsmplehomeqnts ] and the adaptation of ( * ? ? ? * theorem 2.2 ) to the case of manifolds with boundary ( see proposition [ prop : morse ] ) , the following result is immediate : [ maininequalitystablemorsenumberofafilling ] suppose that @xmath51 is a chord - generic closed legendrian submanifold admitting an exact lagrangian filling @xmath19 which is spin and has vanishing maslov number . it follows that the bound latexmath:[\ ] ] where @xmath604 is a subgroup of @xmath605 $ ] generated by @xmath606-[\phi(h)]$ ] . the proof can , for example , be deduced from ( * ? ? ? * proposition 29 ) . then we describe a series of finite solvable groups @xmath607 with the property that @xmath608 , @xmath609)=1 $ ] . let @xmath610 be a finite field with @xmath611 elements , where @xmath612 is a prime number . we define a group @xmath613 , where @xmath614 acts on the additive group @xmath615 by @xmath616 , @xmath617 , @xmath618 . it is easy to see that @xmath607 is a solvable group . this follows from the existence of the following subnormal series @xmath619 where @xmath620 and @xmath621 . then we observe that from formula [ absemiddirprreltohsh ] it follows that @xmath622\simeq { \mathbb{f}}^{\ast}_q$ ] is a non - trivial cyclic group . therefore , @xmath609)=1 $ ] . finally we prove that @xmath608 . since @xmath623 and @xmath624 , we get that @xmath625 . we first show that @xmath626 . assume that @xmath607 has a generating set @xmath627 with @xmath628 , @xmath629 and @xmath630 . then , note that every element in the group generated by @xmath631 has a form @xmath632 , where @xmath633 , @xmath634 , and @xmath635 . this leads to the contradiction with the fact that @xmath623 . then we take a set of generators @xmath631 and @xmath636 with the property that @xmath330 is a generator of @xmath637 ( @xmath637 is a cyclic group ) . such an element definitely exists since if all the elements of @xmath631 are of the form @xmath638 , where @xmath496 is not a generator of @xmath637 , then @xmath631 is not a generating set of @xmath607 . again , @xmath637 is a cyclic group of order @xmath639 , and the order of @xmath330 that we denote by @xmath640 is coprime to @xmath641 . this implies that @xmath642 is coprime to @xmath641 , and hence none of the primes which divide @xmath643 will divide @xmath644=|{\mathbb{f}}^m_q|$ ] . let @xmath282 be the set of primes which divide @xmath640 . then @xmath645 and @xmath646 are two hall @xmath282-subgroups , and hence by theorem [ conjpihallsubgr ] they are conjugate by some element @xmath647 . this implies that @xmath631 , after conjugation , contains an element @xmath648 , where @xmath649 is a generator of @xmath637 . we also would like to mention that it is possible to find @xmath173 explicitly without relying on the theory of hall @xmath282-subgroups . then , already knowing that @xmath626 , we can apply the previous argument and see that , in fact , @xmath650 . together with the fact that @xmath625 we get that @xmath608 . using proposition [ constrrealizoffundgroup ] , we construct an exact lagrangian filling @xmath651 of a legendrian @xmath652 inside the standard contact vector space . then theorem [ intrineqabssi ] tells us that @xmath653 is satisfied for all representatives . on the other hand , the bound given by seidel s isomorphism is @xmath654 finally , note that the difference between the previous two bounds gets arbitrarily large as @xmath655 . we would like to thank franois charette and jarek kdra for very helpful conversations and interest in our work . in addition , we are grateful to the referee of an earlier version of this paper for many valuable comments and suggestions . b. chantraine , g. dimitroglou rizell , p. ghiggini , and r. golovko , _ floer homology and lagrangian concordance _ , proceedings of 21st gkova geometry - topology conference 2014 , 76113 , gkova geometry / topology conference ( ggt ) , gkova , 2015 . t. ekholm , _ rational sft , linearized legendrian contact homology , and lagrangian floer cohomology _ , perspectives in analysis , geometry , and topology . on the occasion of the 60th birthday of oleg viro , volume 296 , pages 109145 . springer , 2012 . k. fukaya , p. seidel and i. smith , _ the symplectic geometry of cotangent bundles from a categorical viewpoint _ , homological mirror symmetry , lecture notes in phys . 757 , springer , berlin , 2009 , 126 .
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assume that we are given a closed chord - generic legendrian submanifold @xmath0 of the contactisation of a liouville manifold , where @xmath1 moreover admits an exact lagrangian filling @xmath2 inside the symplectisation . under the further assumptions that this filling is spin and has vanishing maslov class , we prove that the number of reeb chords on @xmath1 is bounded from below by the stable morse number of @xmath3 . given a general exact lagrangian filling @xmath3
, we show that the number of reeb chords is bounded from below by a quantity depending on the homotopy type of @xmath3 , following ono - pajitnov s implementation in floer homology of invariants due to sharko .
this improves previously known bounds in terms of the betti numbers of either @xmath1 or @xmath3 .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
since the recent rediscovery of low - density parity - check ( ldpc ) codes , a great deal of effort has been devoted to constructing ldpc codes that can work well with the belief - propagation iterative decoder . the studies of long block - length ldpc codes are very much established . the recent works of @xcite , @xcite have shown that , for long block - lengths , the best performing ldpc codes are irregular codes and these codes can outperform turbo codes of the same block - length and code - rate . these long ldpc codes have degree distributions which are derived from differential evolution @xcite or gaussian approximation @xcite . it can be shown that , using the concentration theorem @xcite , the performance of infinitely long ldpc codes of a given degree distribution can be characterised by the average performance of the ensemble based on cycle - free assumption . this assumption , however , does not work for short and moderate block - length ldpc codes due to the inevitable existence of cycles in the underlying tanner graphs . consequently , for a given degree distribution , the performance of short block - length ldpc codes varies considerably from the ensemble performance . various methods exist for the construction of finite block - length irregular codes @xcite,@xcite,@xcite . in addition to irregular ldpc codes , algebraic constructions exist and the resulting codes are regular and usually cyclic in nature . some examples of algebraic ldpc codes are the euclidean and projective geometry codes @xcite . it has been noticed by the authors that , in general , there is a performance association between the code minimum distance ( @xmath2 ) and decoding convergence . the irregular ldpc codes converge very well with iterative decoding , but their @xmath2 are reasonably low . on the other hand , the algebraically constructed ldpc codes , which have high @xmath2 , tend not to converge well with the iterative decoder . it is not surprising that algebraically constructed codes may outperform the irregular codes . the latter have error - floor which is caused by the @xmath2 error - events . on the encoding side , the existence of algebraic structure in the codes is of benefit . rather than depending on the parity - check or generator matrices for encoding , as in the case of irregular codes , a low - complexity encoder can be built for the algebraic ldpc codes . one such example is the linear shift - register encoder for cyclic ldpc codes . assuming that @xmath3 and @xmath4 denote the codeword and information length respectively , algebraic codes that are cyclic offer another decoding advantage . the iterative decoder has @xmath3 parity - check equations to iterate with instead of @xmath5 equations , as in the case of non - cyclic ldpc codes , and this leads to improved performance . it has been shown that the performance of ldpc codes can be improved by going beyond the binary field @xcite , @xcite . _ showed that , under iterative decoding , the non binary ldpc codes have better convergence properties than the binary codes @xcite . they also demonstrated that a coding gain of @xmath6db is achieved by moving from @xmath7 to @xmath8 . non binary ldpc codes in which each symbol takes values from gf(@xmath0 ) offer an attractive scheme for higher - order modulation . the complexity of the symbol - based iterative decoder can be simplified as the extrinsic information from the component codes can be evaluated using the frequency domain dual codes decoder based on the fast - walsh - hadamard transform . based on the pioneering works of macwilliams @xcite,@xcite on the idempotents and the mattson - solomon polynomials , we present a generalised construction method for algebraic gf(@xmath0 ) codes that are applicable as ldpc codes . the construction for binary codes using idempotents has been investigated by shibuya and sakaniwa @xcite , however , their investigation was mainly focused on half - rate codes . in this paper , we construct some higher code - rate non binary ldpc codes with good convergence properties . we focus on the design of short block - length ldpc codes in view of the benefits for thin data - storage , wireless , command / control data reporting and watermarking applications . one of the desirable features in any code construction technique is an effective method of determining the @xmath2 and this feature is not present in irregular code construction methods . with our idempotent - based method , the @xmath2 of a constructed code can be easily lower - bounded using the well - known bch bound . the rest of the paper is organised as follows . in section [ sec : cyclotomic_coset ] , we briefly review the theory of the cyclotomic cosets , idempotents and mattson - solomon polynomials . based on the theory , we devise a generalised construction algorithm and present an example in section [ sec : construction ] . we also outline an efficient and systematic algorithm to search for algebraic ldpc codes in section [ sec : construction ] . in section [ sec : performance ] , we demonstrate the performance of the constructed codes by means of simulation and section [ sec : conclusion ] concludes this paper . we briefly review the theory of cyclotomic cosets , idempotents and mattson - solomon polynomials to make this paper relatively self - contained . let us first introduce some notations that will be used throughout this paper . let @xmath9 and @xmath10 be positive integers with @xmath11 , so that @xmath12 is a subfield of @xmath13 . let @xmath3 be a positive odd integer and @xmath13 be the splitting field for @xmath14 over @xmath12 , so that @xmath15 . let @xmath16 , @xmath17 , @xmath18 be a generator for @xmath13 and @xmath19 be a generator for @xmath12 , where @xmath20 . let @xmath21 be the set of polynomials of degree at most @xmath22 with coefficients in @xmath23 . [ def : ms ] if @xmath24 , then the finite - field transform of @xmath25 is : @xmath26 where @xmath27 . this transform is widely known as the mattson - solomon polynomial . the inverse transform is : @xmath28 [ def : idempotent ] consider @xmath29 , @xmath30 is an idempotent if the property of @xmath31 is satisfied . in the case of @xmath32 , the property of @xmath33 is also satisfied . an @xmath34 cyclic code @xmath35 can be described by the generator polynomial @xmath36 of degree @xmath5 and the parity - check polynomials @xmath37 of degree @xmath4 such that @xmath38 . it is widely known that idempotents can be used to generate @xmath35 . any @xmath12 cyclic code can also be described by a unique idempotent @xmath39 which consists of a sum of primitive idempotents . this unique idempotent is known as the generating idempotent and , as the name implies , @xmath40 is a divisor of this idempotent , i.e. @xmath41 , where @xmath42 contains the repeated factors or non - factors of @xmath14 . [ lemma : ms_idempotent ] if @xmath29 is an idempotent , @xmath43 . * ch 8) ) since @xmath31 , from equation [ eqn : mattson_solomon ] , it follows that @xmath44 , @xmath45 for some integers @xmath46 and @xmath47 . clearly , @xmath48 implying that @xmath49 is a binary polynomial . [ def : cyclotomic_cosets ] if @xmath50 is a positive integer , the binary cyclotomic coset of @xmath51 is : @xmath52 where we shall always assume that the subscript , @xmath50 , is the smallest element in the set @xmath53 , and @xmath54 is the smallest positive integer with the property that @xmath55 . if @xmath56 is the set consisting of the smallest elements of all possible cyclotomic cosets then @xmath57 [ lemma : nonbinary_idempotent ] let @xmath58 and let @xmath59 represents the @xmath60th element of @xmath53 . let the polynomial @xmath61 be given by @xmath62 where @xmath63 is the number of elements in @xmath53 and @xmath64 is defined below . ) if @xmath32 , @xmath65 , if @xmath66 , @xmath64 is defined recursively as follows : @xmath67 the polynomial so defined , @xmath68 , is an idempotent . we term @xmath68 a _ cyclotomic _ idempotent . [ def : parity_check_idempotent ] let @xmath69 and let @xmath70 be @xmath71 then ( refer to lemma [ lemma : nonbinary_idempotent ] ) @xmath72 is an idempotent and we call @xmath72 a _ parity - check _ idempotent . the parity - check idempotent @xmath72 can be used to describe the code @xmath35 , the parity check matrix being made up of the @xmath3 cyclic shifts of the polynomial @xmath73 . if @xmath74 denotes the greatest common divisor of @xmath75 and @xmath76 then , in general , @xmath77 is much lower than @xmath78 denotes the weight of polynomial @xmath79 . ] . based on this observation and the fact that @xmath72 contains all the roots of @xmath80 , we can construct cyclic codes that have a low - density parity - check matrix . [ def : difference_enumerator ] let the polynomial @xmath81 . the difference enumerator of @xmath79 , denoted as @xmath82 , is defined as follows : @xmath83 where we assume that @xmath82 is a modulo @xmath84 polynomial with real coefficients . [ lemma : orthogonal ] let @xmath32 and let @xmath85 for @xmath86 denote the coefficients of @xmath87 . if @xmath88 , @xmath89 , the parity - check polynomial derived from @xmath72 is orthogonal on each position in the @xmath3-tuple . consequently ( i ) the @xmath2 of the resulting @xmath35 is @xmath90 and ( ii ) the underlying tanner graph has girth of at least @xmath91 . \(i ) ( cf . * theorem 10.1 ) ) let a codeword @xmath92 and @xmath93 . for each non zero bit position @xmath94 of @xmath95 where @xmath96 , there are @xmath77 parity - check equations orthogonal to position @xmath94 . each of the parity - check equation must check another non zero bit @xmath97 @xmath98 so that the equation is satisfied . clearly , @xmath99 must equal to @xmath90 and this is the minimum weight of all codewords.(ii ) the direct consequence of having orthogonal parity - check equation is the absence of cycles of length @xmath100 in the tanner graphs . it can be shown that there exists three integers @xmath75 , @xmath101 and @xmath102 , such that @xmath103 for @xmath104 . if these three integers are associated to the variable nodes in the tanner graphs , a cycle of length @xmath91 can be formed between these variable nodes and some check nodes . from lemma [ lemma : orthogonal ] we can deduce that @xmath72 is the parity - check polynomial for one - step majority - logic decodable codes if @xmath88 , @xmath105 or the parity - check polynomial for difference - set cyclic codes if @xmath106 , @xmath105 . [ lemma : nonbinary_dmin_bound ] for the non binary gf(@xmath0 ) cyclic codes , the @xmath2 is bounded by : @xmath107 where @xmath108 denotes the maximum run of consecutive ones in @xmath109 taken cyclically modulo @xmath3 . the lower - bound of the @xmath2 of a cyclic code , bch bound is determined from the number of consecutive roots of @xmath110 and from lemma [ lemma : ms_idempotent ] , it is equivalent to the run of consecutive ones in @xmath109 . based on the mathematical theories outlined above , we devise an algorithm to construct gf(@xmath0 ) @xmath35 which are applicable for iterative decoding . the construction algorithm can be described in the following procedures : 1 . given the integers @xmath9 and @xmath3 , find the splitting field ( @xmath13 ) of @xmath14 over @xmath12 . we can only construct @xmath12 cyclic codes of length @xmath3 if and only if the condition of @xmath11 is satisfied . 2 . generate the cyclotomic cosets modulo @xmath111 and denote it @xmath112 . 3 . derive a polynomial @xmath113 from @xmath112 . let @xmath114 be the smallest positive integer such that @xmath115 . the polynomial @xmath113 is the minimal polynomial of @xmath116 : @xmath117 construct all elements of gf(@xmath0 ) using @xmath113 as the primitive polynomial . 4 . let @xmath118 be the cyclotomic cosets modulo @xmath3 and @xmath56 be a set containing the smallest number in each coset of @xmath118 . assume that there exists a non empty set @xmath119 and following definition [ def : parity_check_idempotent ] , construct the parity - check idempotent @xmath72 . the coefficients of @xmath72 can be assigned following lemma [ lemma : nonbinary_idempotent ] . generate the parity - check matrix of @xmath35 using the @xmath3 cyclic shifts of @xmath73 . compute @xmath46 and @xmath47 , then take the mattson - solomon polynomial of @xmath72 to produce @xmath109 . obtain the code dimension and the lower - bound of the @xmath2 from @xmath109 . note that care should be taken to ensure that there is no common factor between @xmath3 and all of the exponents of @xmath72 , apart from unity , in order to avoid a degenerate code . [ cols="^,<,^,^,^,^",options="header " , ] ^@xmath120^the code minimum distance in binary level . + ^@xmath121^distance to the sphere - packing - bound constrained for binary transmission . [ ex : code - construction ] let us assume that we want to construct a @xmath122 @xmath123 cyclic idempotent code . the splitting field for @xmath124 over @xmath122 is @xmath122 and this implies that @xmath125 , @xmath126 and @xmath127 . let @xmath118 and @xmath112 denote the cyclotomic cosets modulo @xmath3 and @xmath111 respectively . @xmath128 and therefore the primitive polynomial @xmath113 has roots of @xmath129 , @xmath130 , i.e. @xmath131 . by letting @xmath132 , all of the elements of @xmath122 can be defined . if we let @xmath72 be the parity - check idempotent generated by the sum of the cyclotomic idempotents defined by @xmath53 where @xmath133 and @xmath134 , @xmath135 be @xmath136 , @xmath137 and @xmath137 respectively , @xmath138 and its mattson - solomon polynomial @xmath109 tells us that it is @xmath139 cyclic code with @xmath140 . a systematic algorithm has been developed to sum up all combinations of the cyclotomic idempotents to search for all possible @xmath12 cyclic codes of a given length . the search algorithm is targeted on the following key parameters : 1 . sparseness of the resulting parity - check matrix . since the parity - check matrix of @xmath35 is directly derived from @xmath72 which consists of the sum of the cyclotomic idempotents , we are only interested in low - weight cyclotomic idempotents . let us define @xmath141 as the maximum @xmath77 then the search algorithm will only choose the cyclotomic idempotents whose sum has total weight less than or equal to @xmath141 . high code - rate . the number of roots of @xmath72 which are also roots of unity define the dimension of @xmath35 and let us define @xmath142 as the minimum information length of @xmath35 . we are only interested in the sum of the cyclotomic idempotents whose mattson - solomon polynomial has at least @xmath142 zeros . high @xmath2 . let us define @xmath143 as the minimum value of the @xmath2 of @xmath35 . the sum of the cyclotomic idempotents should have at least @xmath144 consecutive powers of @xmath19 which are roots of unity but not roots of @xmath72 . the search algorithm can be relaxed to allow the existence of cycles of length 4 in the resulting parity - check matrix of @xmath35 . the condition of cycles - of - length-4 is not crucial as we will show later that there are codes that have good convergence properties when decoded using iterative decoder . clearly , by eliminating the cycles - of - length-4 constraint , we can construct more codes . following definitions [ def : ms ] and [ def : parity_check_idempotent ] : @xmath145 and hence it is possible to maximise the run of the consecutive ones in @xmath109 if the coefficients of @xmath68 are aligned appropriately . it is therefore important that all possible non zero values of @xmath134 , @xmath146 are included in the search in order to guarantee that we can obtain codes with the highest possible @xmath2 or at least to obtain a better estimate of the @xmath2 . as an example of the performance attainable from an iterative decoder , computer simulations have been carried out for several @xmath12 cyclic ldpc codes . we assume bpsk signalling and the iterative decoder used is the modified belief - propagation decoder which approximates the performance of a maximum - likelihood decoder @xcite,@xcite . the frame - error - rate ( fer ) performance of the @xmath147 cyclic ldpc code is shown in fig . [ fig : fer - gf64 - 21 - 15 ] and is compared with the sphere - packing - bound @xcite,@xcite for binary codes of length @xmath148 bits offset by the binary transmission loss . we can see that the performance of the code is within @xmath149db away from this bound at @xmath150 fer . the binary level minimum - distance of this @xmath139 cyclic ldpc code is @xmath151 . frame error performance of the @xmath147 cyclic ldpc code , width=216 ] frame error performance of the @xmath152 cyclic ldpc code , width=216 ] frame error performance of the @xmath153 cyclic ldpc code , width=216 ] fig . [ fig : fer - gf4 - 255 - 175 ] shows the fer curve of the @xmath152 cyclic ldpc code which is equivalent to @xmath154 binary code . at @xmath150 fer , the performance of this code is approximately @xmath155db away from the sphere - packing - bound of length @xmath156 bits . while both of the codes mentioned above are free from cycles of length 4 , good convergence codes exist even if they have cycles of length 4 in the underlying tanner graph . one such example is the @xmath153 cyclic code whose fer performance is shown in fig . [ fig : fer - gf8 - 91 - 63 ] . at @xmath150 fer , the code performs around @xmath157db away from the sphere - packing - bound of length @xmath158 bits . the parameters of the codes in fig . [ fig : fer - gf4 - 255 - 175 ] and [ fig : fer - gf8 - 91 - 63 ] are available in table [ tbl : code - parameter ] . some other examples of the non binary @xmath12 cyclic ldpc codes with their parameters and distance from the sphere - packing - bound are also shown in table [ tbl : code - parameter ] . an algebraic construction technique for gf(@xmath0 ) ( @xmath1 ) ldpc codes based on summing the cyclotomic idempotents to define the parity - check polynomial is able to produce a large number of cyclic codes . the fact that we consider step - by - step summation of the cyclotomic idempotents , we are able to control the sparseness of the resulting parity - check matrix . the lower - bound of the @xmath2 and the dimension of the codes can be easily determined from the mattson - solomon polynomial of the resulting idempotent . for gf(@xmath159 ) case where the parity - check polynomials are orthogonal on each bit position , we can even determine the true @xmath2 of the codes regardless of the code length . in fact , this special class of binary cyclic codes are the difference - set cyclic and the one - step majority - logic decodable codes which can be easily constructed using our method . for non - binary cases , if the constructed code has low @xmath2 , we can concatenate this code with an inner binary code to trade improvement in @xmath2 with loss in code - rate . simulation results have shown that these codes can converge well under iterative decoding and their performance is very close to the sphere - packing - bound of binary codes for the same code length and rate . the excellent performance of these codes coupled with their low - complexity encoder offers an attractive coding scheme for applications that required short block - lengths such as thin data - storage , wireless , command / control data reporting and watermarking . this research is partially funded by the uk overseas research students award scheme . s. y. chung , g. d. forney , jr . , t. j. richardson , and r. l. urbanke , `` on the design of low - density parity check codes within 0.0045 db of the shannon limit , '' _ ieee comm . letters _ , vol . 3 , pp . 5860 , feb . s. y. chung , t. j. richardson , and r. l. urbanke , `` analysis of sum - product decoding of low - density parity - check codes using a gaussian approximation , '' _ ieee trans . inform . theory _ 47 , pp . 657670 , feb . 2001 . c. j. tjhai , e. papagiannis , m. tomlinson , m. a. ambroze , and m. z. ahmed , `` improved iterative decoder for ldpc codes with performance approximating to a maximum likelihood decoder . '' uk patent application 0409306.8 , apr . e. papagiannis , m. ambroze , and m. tomlinson , `` improved decoding of low - density parity - check codes with low , linearly increased added complexity . '' submitted to 4^th^ iasted international conference on communication systems and networks , 2005 .
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based on the ideas of cyclotomic cosets , idempotents and mattson - solomon polynomials , we present a new method to construct gf(@xmath0 ) , where @xmath1 cyclic low - density parity - check codes .
the construction method produces the dual code idempotent which is used to define the parity - check matrix of the low - density parity - check code . an interesting feature of this construction method is the ability to increment the code dimension by adding more idempotents and so steadily decrease the sparseness of the parity - check matrix .
we show that the constructed codes can achieve performance very close to the sphere - packing - bound constrained for binary transmission .
coding , idempotent , non binary ldpc , mattson - solomon polynomial
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the aim of this work is to study the dynamics of the gravitational instability , _ i.e. _ , the evolution of scalar density perturbations , in presence of viscous effects . viscosity can be divided in two different kinds : _ bulk viscosity _ ( bv ) @xmath0 and _ shear viscosity _ @xmath4 . in this paper , we will concentrate on the analysis of isotropic ( or almost quasi - isotropic ) cosmological models . in this respect , we can safely neglect shear viscosity since there is no displacement of matter layers wrt each other ( in the zeroth - order motion ) and this kind of viscosity represents the energy dissipation due to this feature . indeed , in presence of small inhomogeneities , such effects should be taken into account but we focus on the study of scalar density perturbations and volume changes of a given mass scale are essentially involved . we expect that the non - equilibrium dynamics of matter compression and rarefaction is more relevant than frictions , thus , we concentrate on bv only since it outcomes from the difficulty for a thermodynamical system to follow the equilibrium configuration . in this paper , we assume bv as function of the state parameters of the fluid following the line of the fundamental analysis due to the _ landau school _ @xcite . in particular , we implement the so - called _ hydrodynamical description _ of the fluid , _ i.e. _ , an arbitrary state is consistently characterized by the particle - flow vector and the energy momentum tensor alone and viscosity is fixed by the macroscopic fluid parameters . in the homogeneous models @xmath0 depends only on time and we assume it as a power - law of the density of the fluid @xmath1 , _ i.e. _ , @xmath5 with @xmath3 , @xmath6 the study of the density - perturbation dynamics during the matter - dominated era can be consistently described using the newtonian picture , as soon as sub - horizon - sized scales are treated . the fundamental result is the _ jeans mass _ @xcite ( @xmath7 ) , which is the threshold value for the fluctuation masses to condense , generating a real structure . if masses greater than @xmath7 are addressed , the density contrast ( @xmath8 ) diverges ( in time ) giving rise to the gravitational collapse . in the following , we brief discuss how bv affects such a scheme . the starting point is the eulerian set of motion equations on which a perturbative theory is developed ( the background is assumed static and uniform ) . if we introduce bv effects in the first - order analysis , the density - contrast growth results to be dumped by viscosity , suppressing the structure formation _ without changing the threshold value of @xmath7_. in particular , the grater @xmath3 is , the slower @xmath8 diverges in time and , furthermore , viscosity generates a decreasing exponential regime and a damped oscillatory one in place of the standard pure oscillatory behavior @xcite . in this respect , we now analyze the _ top - down mechanism _ of sub - structure formation , _ i.e. _ , the comparison of one collapsing agglomerate with @xmath9 and an internal non - collapsing sub - structure with @xmath10 . the sub - structure mass must be compared with a decreasing @xmath7 since the latter is inversely proportional to the collapsing agglomerate background mass . as soon as such value reaches the sub - structure mass , this begins to condense . in the standard jeans model , such mechanism is always allowed since the amplitude for perturbations characterized by @xmath10 remains constant in time but , the presence of decreasing @xmath8 in the viscous model , requires a discussion on the effective damping and on the efficacy of the top - down scheme . as shown in figure 1 , in correspondence of a very small viscosity coefficient ( case a ) , we can show how the sub - structure survives in the oscillatory regime during the background collapse and the fragmentation occurs . on the other hand , if bv is strong enough ( case b ) , the damping becomes very strong and the sub - structure vanishes during the agglomerate evolution . thus , the top - down mechanism results to be deeply unfavored . a b moreover , if the static and uniform background solution is corrected for the expansion of the universe @xcite , a jeans - like relation and a considerable damping of the density contrast growth can be found . in 1963 @xcite , e.m . lifshitz and i.m . khalatnikov firstly proposed the quasi - isotropic solution ( qis ) based on the idea that , as a function of time , the _ 3_-metric @xmath11 ( where @xmath12 , @xmath13=1 , 2 , 3 ) is expandable in powers of @xmath14 , _ i.e. _ , a taylor expansion is addressed : @xmath15 where @xmath16 ( space contraction ) and @xmath17 ( consistence of the perturbation scheme ) . we now focus on the relevance of dealing with bv properties of the cosmological fluid approaching the big - bang singularity @xcite . as far as we characterize the bv coefficient like @xmath18 , it is easy to realize that the choice @xmath19 prevents dominating viscous effects . on the other hand , simple considerations indicate that the case @xmath20 leads to negligible contributions of bv in the asymptotic regime towards the big - bang . as a consequence , in studying the singularity physics , the most appropriate form of the power - law is @xmath21 and our aim is to determine the conditions on the parameter @xmath3 ( _ i.e. _ , on the viscosity intensity ) which allows for the existence of a qis for the radiation dominated universe . to this purpose , we separate zeroth- and first - order terms into the 3-metric tensor and the whole analysis of einstein equations follows this scheme of approximation . in the search for a self - consistent solution , we make use of the hydrodynamical equations in view of fixing the form of the energy density . of course , in our solution the power - law for the leading _ 3_-metric term is sensitive to @xmath3 . as a result , by guaranteeing the consistence of the model , we find that the qis exists if and only if @xmath3 remains smaller than a certain critical value , _ i.e. _ , @xmath22 ( here @xmath23 ) . in fact , for values of the viscous parameter greater than @xmath3 , the perturbative expansion towards the singularity can not be addressed , since fluctuations would grow more rapidly than zeroth - order terms . it is worth noting that the study of the perturbation dynamics in a pure isotropic picture yields a very similar asymptotic behavior when viscous effects are taken into account @xcite . the friedmann scheme is preserved only if we deal with limited values of the viscosity parameter , obtaining the condition @xmath24 . this smaller constraint is physically motivated if we consider , _ as it is _ , the friedmann model a particular case of the qis . finally , analyzing @xmath8 , we confirm and generalize the result obtained in @xcite about the damping of density perturbations by the viscous correction .
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* abstract : * this paper focuses on the analysis of the gravitational instability in presence of bulk viscosity both in newtonian regime and in the fully - relativistic approach . the standard jeans mechanism and the quasi - isotropic solution are treated expressing the bulk - viscosity coefficient @xmath0 as a power - law of the fluid energy density @xmath1 , _
i.e. _ , @xmath2 . in the newtonian regime ,
the perturbation evolution is founded to be damped by viscosity and the top - down mechanism of structure fragmentation is suppressed .
the value of the jeans mass remains unchanged also in presence of viscosity . in the relativistic approach
, we get a power - law solution existing only in correspondence to a restricted domain of @xmath3 . ' '' '' ' '' ''
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You are an expert at summarizing long articles. Proceed to summarize the following text:
among the great variety of the works devoted to random motions at finite speed in the euclidean spaces @xmath8 ( see @xcite , @xcite , @xcite , @xcite for the markovian case and @xcite , @xcite for different non - markovian cases ) , the markov random flight in the three - dimensional euclidean space @xmath1 is , undoubtedly , the most difficult and hard to study . while in the low even - dimensional spaces @xmath9 and @xmath10 one managed to obtain the distributions of the motions in an explicit form ( see @xcite , @xcite and @xcite , respectively ) , in the important three - dimensional case only a few results are known . the absolutely continuous part of the transition density of the symmetric markov random flight with unit speed in the euclidean space @xmath1 was presented in ( * ? ? ? * formulas ( 1.3 ) and ( 4.21 ) therein ) . it has an extremely complicated form of an integral with variable limits whose integrand involves inverse hyperbolic tangent function . this formula has so complicated form that can not even be evaluated by means of standard computer environments . moreover , the lack of the speed parameter in this formula impoverishes somewhat the model because it does not allow to study the limiting behaviour of the motion under various scaling conditions ( under kac s condition , for example ) . the presence of both parameters ( i.e. the speed and the intensity of switchings ) in any process of markov random flight makes it , undoubtedly , the most adequate and realistic model for describing the finite - velocity diffusion in the euclidean spaces . these parameters can not be considered as independent because they are connected with each other through the time ( namely , the speed is the distance passed _ per unit of time _ and the intensity is the mean number of switchings _ per unit of time _ ) . another question concerning the density presented in @xcite is the infinite discontinuity at the origin @xmath11 . while the infinite discontinuity of the transition density on the border of the diffusion area is a quite natural property in some euclidean spaces of low dimensions ( see @xcite for the euclidean plane @xmath12 and ( * ? ? ? * the second term of formulas ( 1.3 ) and ( 4.21 ) ) , @xcite , ( * ? ? ? * formula ( 3.12 ) ) in the space @xmath1 ) , the discontinuity at the origin looks somewhat strange and hard to explain . the difficulty of analysing the three - dimensional markov random flight and , on the other hand , the great theoretical and applied importance of the problem of describing the finite - velocity diffusion in the space @xmath1 suggest to look for other methods of studying this model . that is why various asymptotic theorems yielding a good approximation would be a fairly desirable aim of the research . such asymptotic results could be obtained by using the characteristic functions technique . in the case of the three - dimensional symmetric markov random flight some important results for its characteristic functions were obtained . in particular , the closed - form expression for the laplace transform of the characteristic function was obtained by different methods in ( * ? ? ? * formulas ( 1.6 ) and ( 5.8 ) ) ( for unit speed ) and in ( * ? ? ? * formula ( 45 ) ) , @xcite ( for arbitrary speed ) . a general relation for the conditional characteristic functions of the three - dimensional symmetric markov random flight conditioned by the number of changes of direction , was given in ( * ? ? ? * formula ( 3.8 ) ) . the key point in these formulas is the possibility of evaluating the inverse laplace transforms of the powers of the inverse tangent functions in the complex right half - plane . this is the basic idea of deriving the series representations of the conditional characteristic functions corresponding to two and three changes of direction given in section 3 . based on these representations , an asymptotic formula , as time @xmath5 , for the unconditional characteristic function is obtained in section 4 and the error in this formula has the order @xmath6 . the inverse fourier transformation of the unconditional characteristic function yields an asymptotic formula for the transition density of the process which is presented in section 5 . this formula shows that the density is discontinuous on the border , but it is continuous at the origin @xmath11 , as it must be . the unexpected and interesting peculiarity is that the conditional density corresponding to two changes of direction contains a term having an infinite discontinuity on the border of the diffusion area . from this fact it follows that such conditional density is discontinuous itself on the border and this differs the 3d - model from its 2d - counterpart where only the conditional density of the single change of direction has an infinite discontinuity on the border . the error in the obtained asymptotic formula has the order @xmath6 . in section 6 we estimate the accuracy of the asymptotic formula and show that it gives a good approximation on small time intervals whose lengths depend on the intensity of switchings . finally , in appendices we prove a series of auxiliary lemmas that have been used in our analysis . consider the stochastic motion of a particle that , at the initial time instant @xmath13 , starts from the origin @xmath14 of the euclidean space @xmath1 and moves with some constant speed @xmath15 ( note that @xmath15 is treated as the constant norm of the velocity ) . the initial direction is a random three - dimensional vector with uniform distribution on the unit sphere @xmath16 the motion is controlled by a homogeneous poisson process @xmath17 of rate @xmath3 as follows . at each poissonian instant , the particle instantaneously takes on a new random direction distributed uniformly on @xmath18 independently of its previous motion and keeps moving with the same speed @xmath15 until the next poisson event occurs , then it takes on a new random direction again and so on . let @xmath19 be the particle s position at time @xmath20 which is referred to as the three - dimensional symmetric markov random flight . at arbitrary time instant @xmath20 the particle , with probability 1 , is located in the closed three - dimensional ball of radius @xmath21 centred at the origin @xmath22 : @xmath23 consider the probability distribution function @xmath24 of the process @xmath4 , where @xmath25 is the infinitesimal element in the space @xmath1 . for arbitrary fixed @xmath20 , the distribution @xmath26 consists of two components . the singular component corresponds to the case when no poisson events occur on the time interval @xmath27 and it is concentrated on the sphere @xmath28 in this case , at time instant @xmath29 , the particle is located on the sphere @xmath30 and the probability of this event is @xmath31 if at least one poisson event occurs on the time interval @xmath32 , then the particle is located strictly inside the ball @xmath33 and the probability of this event is @xmath34 the part of the distribution @xmath26 corresponding to this case is concentrated in the interior @xmath35 of the ball @xmath33 and forms its absolutely continuous component . let @xmath36 be the density of distribution @xmath37 . it has the form @xmath38 where @xmath39 is the density ( in the sense of generalized functions ) of the singular component of @xmath26 concentrated on the sphere @xmath30 and @xmath40 is the density of the absolutely continuous component of @xmath26 concentrated in @xmath41 . the singular part of density ( [ struc2 ] ) is given by the formula : @xmath42 where @xmath43 is the dirac delta - function . the absolutely continuous part of density ( [ struc2 ] ) has the form : @xmath44 where @xmath45 is some positive function absolutely continuous in @xmath41 and @xmath46 is the heaviside unit - step function given by @xmath47 asymptotic behaviour of the transition density ( [ struc2 ] ) on small time intervals is the main subject of this research . since its singular part is explicitly given by ( [ denss ] ) , then our efforts are mostly concentrated on deriving the respective asymptotic formulas for the absolutely continuous component ( [ densac ] ) of the density . our main tool is the characteristic functions technique because , as it was mentioned above , some closed - form expressions for the characteristic functions ( both conditional and unconditional ones ) of the three - dimensional symmetric markov random flight @xmath4 are known . in this section we obtain the series representations of the conditional characteristic functions corresponding to two and three changes of direction . these formulas are the basis for our further analysis leading to asymptotic relations for the unconditional characteristic function and the transition density of the three - dimensional symmetric markov random flight @xmath4 on small time intervals . the main result of this section is given by the following theorem . * theorem 1 . * _ the conditional characteristic functions @xmath48 and @xmath49 corresponding to two and three changes of direction are given , respectively , by the formulas : _ @xmath50 @xmath51 @xmath52 _ where @xmath53 is bessel function , @xmath54 is the generalized hypergeometric function given by _ ( [ hypergeom54 ] ) _ ( see below ) and the coefficients @xmath55 are given by the formula _ @xmath56 0.2 cm _ proof . _ it was proved in ( * ? ? ? * formula ( 3.8 ) ) that , for arbitrary @xmath20 , the characteristic function @xmath57 ( that is , fourier transform @xmath58 with respect to spatial variable @xmath59 ) of the conditional density @xmath60 of the three - dimensional markov random flight @xmath4 corresponding to @xmath61 changes of directions is given by the formula @xmath62(\boldsymbol\alpha ) = \frac{n!}{t^n } ( c\vert\boldsymbol\alpha\vert)^{-(n+1 ) } \mathcal l_s^{-1 } \left [ \left ( \text{arctg } \frac{c\vert\boldsymbol\alpha\vert}{s } \right)^{n+1 } \right](t ) , \ ] ] @xmath63 where @xmath64 is the inverse laplace transformation with respect to complex variable @xmath65 and @xmath66 is the right half - plane of the complex plane @xmath67 . in particular , in the case of two changes of directions @xmath68 , formula ( [ eq1 ] ) yields : @xmath69(\boldsymbol\alpha ) = \frac{2!}{t^2 } ( c\vert\boldsymbol\alpha\vert)^{-3 } \mathcal l_s^{-1 } \left [ \left ( \text{arctg } \frac{c\vert\boldsymbol\alpha\vert}{s } \right)^3 \right](t ) , \qquad \boldsymbol\alpha\in\bbb r^3 , \quad s\in\bbb c^+ .\ ] ] applying lemma b3 of the appendix b to the power of inverse tangent function in ( [ eq2 ] ) , we obtain : @xmath70(t ) \\ & = \frac{2}{\sqrt{\pi } \ ; t^2 } \ ; \sum_{k=0}^{\infty } \frac{\gamma\left ( k+\frac{1}{2 } \right)}{k ! \ ; ( 2k+1 ) } \ ; ( c\vert\boldsymbol\alpha\vert)^{2k } \\ & \qquad \times \ ; _ 5f_4\left ( 1,1,1,-k ,- k-\frac{1}{2 } ; \ ; -k+\frac{1}{2 } , -k+\frac{1}{2 } , \frac{3}{2 } , 2 ; \ ; 1 \right ) \mathcal l_s^{-1 } \biggl [ \frac{1}{\left ( s^2 + ( c\vert\boldsymbol\alpha\vert)^2 \right)^{k+3/2 } } \biggr](t ) . \endaligned\ ] ] note that evaluating the inverse laplace transformation of each term of the series separately is justified because it converges uniformly in @xmath65 everywhere in @xmath71 and the complex functions @xmath72 are holomorphic and do not have any singular points in this half - plane . moreover , each of these functions contains the inversion complex variable @xmath73 in a negative power and behaves like @xmath74 , as @xmath75 , and , therefore , all these complex functions rapidly tend to zero at infinity . according to ( * table 8.4 - 1 , formula 57 ) , we have @xmath76(t ) = \frac{\sqrt{\pi}}{\gamma\left ( k+\frac{3}{2 } \right ) } \left ( \frac{t}{2c\vert\boldsymbol\alpha\vert } \right)^{k+1 } j_{k+1}(ct\vert\boldsymbol\alpha\vert ) .\ ] ] substituting this into ( [ eq3 ] ) , after some simple calculations we obtain ( [ char2 ] ) . for @xmath77 , formula ( [ eq1 ] ) yields : @xmath78(\boldsymbol\alpha ) = \frac{3!}{t^3 } ( c\vert\boldsymbol\alpha\vert)^{-4 } \mathcal l_s^{-1 } \left [ \left ( \text{arctg } \frac{c\vert\boldsymbol\alpha\vert}{s } \right)^4 \right](t ) , \qquad \boldsymbol\alpha\in\bbb r^3 , \quad s\in\bbb c^+ .\ ] ] applying lemma b4 of the appendix b to the power of inverse tangent function in ( [ eq4 ] ) and taking into account that @xmath79(t ) = \frac{\sqrt{\pi}}{(k+1 ) ! } \left ( \frac{t}{2c\vert\boldsymbol\alpha\vert } \right)^{k+3/2 } j_{k+3/2}(ct\vert\boldsymbol\alpha\vert ) , \ ] ] we obtain : @xmath80(t ) \\ & = 3\pi^{3/2 } \ ; \sum_{k=0}^{\infty } \frac{\gamma_k \ ; ( ct\vert\boldsymbol\alpha\vert)^{k-3/2}}{2^{k+3/2 } \ ; ( k+1 ) ! } \ ; j_{k+3/2}(ct\vert\boldsymbol\alpha\vert ) , \endaligned\ ] ] where the coefficients @xmath55 are given by ( [ coef1 ] ) . the theorem is proved . @xmath81 _ _ the series in formulas ( [ char2 ] ) and ( [ char3 ] ) are convergent for any fixed @xmath20 , however this convergence is not uniform in @xmath82 . therefore , we can not invert each term of these series separately . moreover , one can see that the inverse fourier transform of each term does not exist for @xmath83 . thus , while there exist the inverse fourier transforms of the whole series ( [ char2 ] ) and ( [ char3 ] ) , it is impossible to invert their terms separately and , therefore , we can not obtain closed - form expressions for the respective conditional densities . these formulas can , nevertheless , be used for obtaining the important asymptotic relations and this is the main subject of the next sections . using the results of the previous section , we can now present an asymptotic relation on small time intervals for the characteristic function @xmath84 of the three - dimensional symmetric markov random flight , where @xmath85 are the conditional characteristic functions corresponding to @xmath86 changes of direction . this result is given by the following theorem . * theorem 2 . * _ for the characterictic function @xmath87 of the three - dimensional markov random flight @xmath4 the following asymptotic formula holds : _ @xmath88 \\ & \qquad\qquad\qquad + \frac{\lambda^2 t}{c\vert\boldsymbol\alpha\vert } j_1(ct\vert\boldsymbol\alpha\vert ) + \frac{\lambda^3 \ ; \sqrt{\pi } \ ; t^{3/2}}{(2 c\vert\boldsymbol\alpha\vert)^{3/2 } } \ ; j_{3/2}(ct\vert\boldsymbol\alpha\vert)\biggr\ } + o(t^3 ) , \endaligned\ ] ] @xmath52 _ where _ @xmath89 _ and _ @xmath90 _ are the incomplete integral sine and cosine , respectively , given by the formulas : _ @xmath91 0.2 cm _ proof . _ we have : @xmath92 .\ ] ] since all the conditional characteristic functions are uniformly bounded in both variables , that is , @xmath93 then @xmath94 and , therefore , @xmath95.\ ] ] in view of ( [ char2 ] ) , we have : @xmath96 . \endaligned\ ] ] from the asymptotic formula @xmath97 we get @xmath98 and , therefore , @xmath99 thus , we obtain the following asymptotic relation : @xmath100 similarly , according to ( [ char3 ] ) , we have : @xmath101 . \endaligned\ ] ] in view of ( [ asbes ] ) , we have @xmath102 and , therefore , @xmath103 thus , taking into account that @xmath104 ( see ( [ coef1 ] ) ) , we arrive at the formula : @xmath105 since ( see ( * ? ? ? * formula ( 3.11 ) ) ) @xmath106\ ] ] and @xmath107 ( that is , characteristic function of the uniform distribution on the surface of the three - dimensional sphere of radius @xmath21 ) , then by substituting these formulas , as well as ( [ eq8 ] ) and ( [ eq9 ] ) into ( [ eq7 ] ) , we finally obtain asymptotic relation ( [ eq6 ] ) . the theorem is completely proved . asymptotic formula ( [ eq6 ] ) for the unconditional characteristic function enables us to obtain the respective asymptotic relation for the transition density of the process @xmath4 . this result is given by the following theorem . * theorem 3 . * _ for the transition density @xmath108 of the three - dimensional markov random flight @xmath4 the following asymptotic relation holds : _ @xmath109 \theta(ct-\vert\bold x\vert ) + o(t^3 ) , \endaligned\ ] ] @xmath110 0.2 cm _ proof . _ applying the inverse fourier transformation @xmath111 to both sides of ( [ eq6 ] ) , we have : @xmath112(\bold x ) \\ & \qquad\quad + \mathcal f_{\boldsymbol\alpha}^{-1 } \biggl [ \frac{\lambda}{c^2 t \vert\boldsymbol\alpha\vert^2 } \biggl ( \sin{(ct\vert\boldsymbol\alpha\vert ) } \text{si}(2ct\vert\boldsymbol\alpha\vert ) + \cos{(ct\vert\boldsymbol\alpha\vert ) } \text{ci}(2ct\vert\boldsymbol\alpha\vert ) \biggr ) \biggr](\bold x ) \\ & \qquad\quad + \mathcal f_{\boldsymbol\alpha}^{-1 } \biggl [ \frac{\lambda^2 t}{c\vert\boldsymbol\alpha\vert } j_1(ct\vert\boldsymbol\alpha\vert ) \biggr](\bold x ) \\ & \qquad\quad + \mathcal f_{\boldsymbol\alpha}^{-1 } \biggl [ \frac{\lambda^3 t}{2(c\vert\boldsymbol\alpha\vert)^2 } \left ( \frac{\sin{(ct\vert\boldsymbol\alpha\vert)}}{ct\vert\boldsymbol\alpha\vert } - \cos{(ct\vert\boldsymbol\alpha\vert ) } \right ) \biggr](\bold x ) \biggr\ } + o(t^3 ) . \endaligned\ ] ] note that here we have used the fact that , due to the continuity of the inverse fourier transformation , the asymptotic formula @xmath113(\bold x ) = o(t^3)$ ] holds . let us evaluate separately the inverse fourier transforms on the right - hand side of ( [ dens2 ] ) . the first one is well known ( see @xcite ) : @xmath114(\bold x ) = \frac{1}{4\pi ( ct)^2 } \ ; \delta(c^2t^2-\vert\bold x\vert^2)\ ] ] that is the uniform density concentrated on the surface of the sphere @xmath115 of radius @xmath21 centred at the origin @xmath11 . the second fourier transform on the right - hand side of ( [ dens2 ] ) is also well known ( see ( * ? ? ? * , the theorem ) or ( * ? ? ? * formulas ( 3.11 ) and ( 3.12 ) ) ) : @xmath116(\bold x ) \\ & \hskip 4 cm = \frac{\lambda}{4\pi c^2 t \vert\bold x\vert } \ln\left ( \frac{ct+\vert\bold x\vert}{ct-\vert\bold x\vert } \right ) \ ; \theta(ct-\vert\bold x\vert ) . \endaligned\ ] ] applying the hankel inversion formula , we have for the third fourier transform on the right - hand side of ( [ dens2 ] ) : @xmath117(\bold x ) = \frac{\lambda^2 t}{c } \ ; ( 2\pi)^{-3/2 } \vert\bold x\vert^{-1/2 } \int_0^{\infty } j_{1/2}(\vert\bold x\vert \xi ) \ ; \xi^{3/2 } \ ; \xi^{-1 } j_1(ct\xi ) \ ; d\xi .\ ] ] taking into account that @xmath118 and applying ( * ? ? ? * formula 2.12.15(2 ) ) , we have : @xmath119(\bold x ) & = \frac{\lambda^2 t}{2\pi^2 c \vert\bold x\vert } \int_0^{\infty } \sin{(\vert\bold x\vert \xi ) } \ ; j_1(ct\xi ) \ ; d\xi \\ & = \frac{\lambda^2 t}{2\pi^2 c \vert\bold x\vert } \ ; ( c^2t^2-\vert\bold x\vert^2)^{-1/2 } \ ; \left ( \frac{\vert\bold x\vert}{ct } \right ) \ ; \theta(ct-\vert\bold x\vert)\\ & = \frac{\lambda^2}{2\pi^2 c^2 \ ; \sqrt{c^2t^2-\vert\bold x\vert^2 } } \ ; \theta(ct-\vert\bold x\vert ) . \endaligned\ ] ] this is a fairly unexpected result showing that the conditional density @xmath120 corresponding to two changes of direction has an infinite discontinuity on the border of the three - dimensional ball @xmath33 . this property is similar to that of the conditional density @xmath121 corresponding to the single change of direction ( for the respective joint density see ( [ dens4 ] ) ) . applying the hankel inversion formula and taking into account ( [ bessin ] ) , we have for the fourth term on the right - hand side of ( [ dens2 ] ) : @xmath122(\bold x ) \\ & = \frac{\lambda^3 \ ; \sqrt{\pi } \ ; t^{3/2}}{(2 c)^{3/2 } } \ ; ( 2\pi)^{-3/2 } \vert\bold x\vert^{-1/2 } \int_0^{\infty } j_{1/2}(\vert\bold x\vert \xi ) \ ; \xi^{3/2 } \ ; \xi^{-3/2 } j_{3/2}(ct\xi ) \ ; d\xi \\ & = \frac{\lambda^3 \ ; \sqrt{2 } \ ; t^{3/2}}{8c^{3/2 } \ ; \pi\sqrt{\pi } \ ; \vert\bold x\vert } \int_0^{\infty } \xi^{-1/2 } \ ; \sin{(\vert\bold x\vert \xi ) } \ ; j_{3/2}(ct\xi ) \ ; d\xi . \endaligned\ ] ] using ( * ? ? ? * formula 6.699(1 ) ) , we obtain : @xmath123(\bold x ) \\ & = \frac{\lambda^3 \ ; \sqrt{2 } \ ; t^{3/2}}{8c^{3/2 } \ ; \pi\sqrt{\pi } \ ; \vert\bold x\vert } \ ; \frac{2^{-1/2 } \ ; \sqrt{\pi } \ ; \vert\bold x\vert \ ; ( ct)^{-3/2}}{\gamma(1 ) } \ ; \theta(ct-\vert\bold x\vert ) \\ & = \frac{\lambda^3}{8\pi c^3 } \ ; \theta(ct-\vert\bold x\vert ) . \endaligned\ ] ] substituting now ( [ dens3 ] ) , ( [ dens4 ] ) , ( [ dens5 ] ) and ( [ dens6 ] ) into ( [ dens2 ] ) we arrive at ( [ dens1 ] ) . the theorem is proved . @xmath81 ) at instant @xmath124 @xmath125 ( for @xmath126 ) on the interval @xmath127_,width=377,height=302 ] -1 cm the shape of the absolutely continuous part of density ( [ dens1 ] ) at time instant @xmath124 ( for @xmath128 ) on the interval @xmath127 is plotted in fig . the error in these calculations does not exceed 0.001 . we see that the density increases slowly as the distance @xmath129 from the origin @xmath11 grows , while near the border this growth becomes explosive . from this fact it follows that , for small time @xmath29 , the greater part of the density is concentrated outside the neighbourhood of the origin @xmath11 and this feature of the three - dimensional markov random flight is quite similar to that of its two - dimensional counterpart . the infinite discontinuity of the density on the border @xmath130 is also similar to the analogous property of the two - dimensional markov random flight ( see , for comparison , ( * ? ? ? * formula ( 20 ) and figure 2 therein ) ) . note that density ( [ dens1 ] ) is continuous at the origin , as it must be . _ remark 2 . _ using ( [ dens1 ] ) , we can derive an asymptotic formula , as @xmath5 , for the probability of being in a subball @xmath131 of some radius @xmath132 centred at the origin @xmath11 . applying ( * ? ? ? * formula 4.642 ) and ( * ? ? ? * formula 1.513(1 ) ) , we have : @xmath133 this series can be expressed through the special lerch @xmath134-function . applying again ( * ? ? ? * formula 4.642 ) , we get : @xmath135 where we have used the easily checked equality : @xmath136 then , by integrating the absolutely continuous part of ( [ dens1 ] ) over the ball @xmath137 and taking into account ( [ dens7 ] ) and ( [ dens8 ] ) . we have ( for arbitrary @xmath132 ) : @xmath138 \\ & = e^{-\lambda t } \biggl [ \frac{\lambda}{4\pi c^2 t } \ ; 8\pi r ct \sum_{k=1}^{\infty } \frac{1}{4k^2 - 1 } \ ; \left ( \frac{r^2}{c^2t^2 } \right)^k \\ & \qquad\qquad + \frac{\lambda^2}{2\pi^2 c^2 } \biggl ( 2\pi ( ct)^2 \arcsin\left ( \frac{r}{ct } \right ) - 2\pi r \sqrt{c^2t^2-r^2 } \biggr ) + \frac{\lambda^3}{8\pi c^3 } \ ; \frac{4}{3 } \pi r^3 \biggr ] , \endaligned\ ] ] and after some simple computations we finally arrive at the following asymptotic formula ( for @xmath132 ) : @xmath139 , \qquad t\to 0 . \endaligned\ ] ] the error in asymptotic formula ( [ dens1 ] ) has the order @xmath6 . this means that , for small @xmath29 , this formula yields a fairly good accuracy . to estimate it , let us integrate the function in square brackets of ( [ dens1 ] ) over the ball @xmath33 . for the first term in square brackets of ( [ dens1 ] ) we have : @xmath140 because the second integrand is the conditional density corresponding to the single change of direction ( see ( * ? ? ? * the theorem ) or ( * ? ? ? * formula ( 3.12 ) ) ) and , therefore , the second integral is equal to 1 . applying ( * ? ? ? * formula 4.642 ) , we have for the second term in square brackets of ( [ dens1 ] ) : @xmath141 for the third term in square brackets of ( [ dens1 ] ) we get : @xmath142 hence , in view of ( [ est1 ] ) , ( [ est2 ] ) and ( [ est3 ] ) , the integral of the absolutely continuous part in asymptotic formula ( [ dens1 ] ) is : @xmath143 dx_1 dx_2 dx_3 \\ & = e^{-\lambda t } \left ( \lambda t + \frac{\lambda^2t^2}{2 } + \frac{\lambda^3 t^3}{6 } \right ) . \endaligned\ ] ] note that ( [ est4 ] ) can also be obtained by passing to the limit , as @xmath144 , in asymptotic formula ( [ dens9 ] ) . on the other hand , according to ( [ struc1 ] ) and ( [ densac ] ) , the integral of the absolutely continuous part of the transition density of the three - dimensional markov random flight @xmath4 is @xmath145 the difference between the approximating function @xmath146 and the exact function @xmath147 given by ( [ est4 ] ) and ( [ est5 ] ) enables us to estimate the value of the probability generated by all the terms of the density aggregated in the term @xmath6 of asymptotic relation ( [ dens1 ] ) . the shapes of functions @xmath147 and @xmath146 on the time interval @xmath148 for the values of the intensity of switchings @xmath149 are presented in figures 2 and 3 . + + + + we see that , for @xmath150 , the function @xmath146 yields a very good coincidence with function @xmath147 on the subinterval @xmath151 ( fig . 2 ( left ) ) , while for @xmath152 ( fig . 2 ( right ) ) such coincidence is good only on the subinterval @xmath153 . the same phenomenon is also clearly seen in figure 3 where , for @xmath154 , the function @xmath146 yields a very good coincidence with function @xmath147 on the subinterval @xmath155 ( fig . 3 ( left ) ) , while for @xmath156 such good coincidence takes place only on the subinterval @xmath157 ( fig . 3 ( right ) ) . thus , we can conclude that the greater is the intensity of switchings @xmath7 , the shorter is the subinterval of coincidence . this fact can easily be explained . really , the greater is the intensity of switchings @xmath7 , the shorter is the time interval , on which no more than three changes of directions can occur with big probability . this means that , for increasing @xmath7 , the asymptotic formula ( [ dens1 ] ) yields a good accuracy on more and more small time intervals . however , for arbitrary fixed @xmath7 , there exists some @xmath158 such that formula ( [ dens1 ] ) yields a good accuracy on the time interval @xmath159 and the error of this approximation does not exceed @xmath160 . this is the essence of the asymptotic formula ( [ dens1 ] ) . * appendices * in the following appendices we establish some lemmas that have been used in the proofs of the above theorems . note that some of them are of a separate mathematical interest because no similar results can be found in the mathematical handbooks . * lemma a1 . * _ for arbitrary integer @xmath161 and for arbitrary real @xmath162 , the following formula holds : _ @xmath163 @xmath164 0.2 cm _ proof . _ using the well - known relations for pochhammer symbol @xmath165 and the formula for euler gamma - function @xmath166 we can easily check that the sum on the left - hand side of ( [ appa1 ] ) is @xmath167 where @xmath168 is the generalized hypergeometric function . according to ( * item 7.4.4 , page 539 , formula 88 ) @xmath169 substituting this into ( [ appa3 ] ) , we obtain ( [ appa1 ] ) . the lemma is proved . in this appendix we derive series representations for some powers of the inverse tangent function that have been used in the proofs of the above theorems . moreover , these results are of a more general mathematical interest because , to the best of the author s knowledge , there are no series representations , similar to ( [ appb2 ] ) , ( [ appb4 ] ) and ( [ appb6 ] ) ( see below ) , in mathematical handbooks , including @xcite , @xcite , @xcite . @xmath184 substituting these coefficients into ( [ appb3 ] ) we obtain ( [ appb2 ] ) . the uniform convergence of the series in formula ( [ appb2 ] ) can be established similarly to that of lemma b1 . this completes the proof of the lemma . @xmath81 * lemma b3 . * _ for arbitrary @xmath185 , the following series representation holds : _ @xmath186 _ where _ @xmath187 _ is the generalized hypergeometric function . the series in _ ( [ appb4 ] ) _ is convergent uniformly in @xmath172 . _ 0.2 cm _ proof . _ from ( [ appb1 ] ) and ( [ appb2 ] ) it follows that @xmath188 where the coefficients @xmath55 are given by @xmath189 applying ( [ apa3 ] ) , ( [ appa2 ] ) and the formula @xmath190 after some simple computations , we arrive at the relation @xmath191 substituting these coefficients into ( [ appb5 ] ) we obtain ( [ appb4 ] ) . the lemma is proved . @xmath81 * lemma b4 . * _ for arbitrary @xmath177 , the following series representation holds : _ @xmath192 _ where the coefficients @xmath55 are given by the formula _ @xmath193 _ the series in _ ( [ appb6 ] ) _ is convergent uniformly in @xmath172 . _ _ proof . _ according to lemma b2 , we have : @xmath194 where the coefficients @xmath195 are : @xmath196\\ & = \frac{2}{k+2 } \sum_{l=0}^k \frac{l ! \ ; ( k - l)!}{(l+1 ) \ ; \gamma\left ( l+\frac{3}{2 } \right ) \ ; \gamma\left ( k - l+\frac{3}{2 } \right ) } . \endaligned\ ] ] substituting this into ( [ appb7 ] ) , we get the statement of the lemma . @xmath81
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we consider the markov random flight @xmath0 in the three - dimensional euclidean space @xmath1 with constant finite speed @xmath2 and the uniform choice of the initial and each new direction at random time instants that form a homogeneous poisson flow of rate @xmath3 .
series representations for the conditional characteristic functions of @xmath4 corresponding to two and three changes of direction , are obtained .
based on these results , an asymptotic formula , as @xmath5 , for the unconditional characteristic function of @xmath4 is derived . by inverting it
, we obtain an asymptotic relation for the transition density of the process .
we show that the error in this formula has the order @xmath6 and , therefore , it gives a good approximation on small time intervals whose lengths depend on @xmath7 .
estimate of the accuracy of the approximation is analysed . * asymptotic relation for the transition density + of the three - dimensional markov random flight + on small time intervals * alexander d. kolesnik + institute of mathematics and computer science + academy street 5 , kishinev 2028 , moldova + e - mail : [email protected] 0.2 cm 0.1 cm _ keywords : _ markov random flight , persistent random walk , conditional density , fourier transform , characteristic function , asymptotic relation , transition density , small time intervals 0.2 cm _ ams 2010 subject classification : _
60k35 , 60k99 , 60j60 , 60j65 , 82c41 , 82c70
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You are an expert at summarizing long articles. Proceed to summarize the following text:
it has recently become possible to collect large samples of high resolution cloud particle images in real time , opening up the modelling of cloud dynamics to detailed comparison with nature . figure 1 shows ice crystal aggregates from a cirrus cloud over the usa , captured by non - contact aircraft - based imaging . such aggregates can be seen to be comprised of varied rosette ice crystal types , and detailed statistics have recently been published on both the cluster aspect ratios [ _ korolev and isaac _ 2003 ] and size distributions [ _ field and heymsfield _ 2003 ] in cirrus clouds . such aggregation is a key feature of cloud development in the troposphere and can be quite crucial to the development of precipitation , whether it reaches the ground as snow or melts first to arrive as rain . the openness of the aggregates significantly accelerates their growth . two clusters ( labelled by @xmath0 ) pass approach with their centres closer than the sum of their radii @xmath1 at a rate proportional to@xmath2 where for each cluster the sedimentation speed @xmath3 varies inversely with its radius @xmath4 and mass @xmath5 as @xmath6 here @xmath7 and @xmath8 are the viscosity and density of the air , @xmath9 is the acceleration due to gravity , and we have assumed that only one geometrical radius is relevant : _ mitchell _ [ 1996 ] discusses an elaboration . given the above , the rates of aggregation per unit time for fixed cluster masses vary linearly overall with the cluster radii , and openness of aggregate structure enhances aggregation rates despite lowering fall speed . for real aggregates this is a significant factor : using data from _ heymsfield et al . _ , [ 2002 ] , one finds that rosette aggregates 2 mm across ( which yield 0.5 mm droplets ) aggregate four times faster than when melted . for cloud particles it is also relevant to consider the rates of aggregation per unit of distance fallen ( rather than per unit time ) , which at fixed mass is proportional to the square of radius , leading to 16 times enhancement for ice over water in the example cited . we have made computer simulations of ice aggregation based on equations ( [ eq : kernel ] ) and ( [ eq : fallspeed ] ) , tracing trajectories through possible collisions to obtain accurate collision geometries . we assumed that all collisions led to rigid irreversible joining of clusters , as the openness of the experimentally observed clusters suggests little large scale consolidation of structure upon aggregation , and that cluster orientations were randomised in between collisions but did not change significantly during them . we took the sedimentation speeds to be governed by inertial flow , for which the mass dependence function in equation ( [ eq : fallspeed ] ) is given by @xmath10 . details of implementation are given in a longer paper [ _ westbrook et al . _ , 2004 ] . some representative computer aggregates are shown in figure 1 alongside the experimental ones . our simulations used three dimensional cross shapes for the initial particles as a crude representation of the experimental bullet rosettes . figure 2 shows a quantitative comparison of aggregate geometry , in terms of the ratio of cluster spans perpendicular to and along the direction of maximal span , as measured from projections in an arbitrary plane of observation . we find that different initial particle geometries ( rosettes , rods ) approach a common asymptotic average cluster value . the aspect ratio of cpi images have been similarly calculated [ _ korolev and isaac _ 2003 ] , the results of which have been overlayed onto figure 2 , and these appear to approach the same value . this universality of aspect ratios provides direct support for our hypothesis of rigid cluster joining upon contact . a deeper indicator of universality is provided by the fractal scaling of ice crystal aggregates , where one tests the relation @xmath11 between aggregate mass @xmath5 and linear span @xmath12 . our simulations and experimental observations [ _ heymsfield et al . _ , 2002 ] , rather accurately agree on the fractal dimension @xmath13 and in _ westbrook et al . _ [ 2004 ] we discuss theoretical arguments leading to @xmath14 . our simulations conform well to dynamical scaling of the cluster size distribution . this means that number of clusters per unit mass varies with mass and time together as a single scaling function,@xmath15 where @xmath16 is the weight average cluster mass . this relationship is confirmed in figure 3 , where we rescale the mass distribution from different times in the simulations onto a universal curve . the scaling function which we observe in figure 3 exhibits power law behaviour with @xmath17 for @xmath18 with @xmath19 . this is not intrinsically surprising ( and indeed it matches theoretical expectations [ _ van dongen and ernst _ , 1985 ] ) but it has forced us to abandon the way experimentally observed distributions of cluster linear size have hitherto been plotted . the problem is that given equation ( [ eq : dyn.scalg ] ) and its observed power law form , we must expect that the distribution of clusters by linear span @xmath20 should at small @xmath20 take the form @xmath21 which diverges as @xmath22 using our observed exponents . for small enough crystal sizes this behaviour will be modified by the role of growth processes other than aggregation , but that lies outside the scaling regime . because of the divergence one has to take great care in constructing a characteristic linear size @xmath23 , where the natural choices are @xmath24 and the lowest whole number @xmath25 for which the denominator is not dominated by the smallest clusters is @xmath26 . the simplest natural scaling ansatz for the cluster span distribution is then found [ _ westbrook et al . _ , 2004 ] to be@xmath27 where @xmath28 . figure 4 shows that this scaling ansatz works acceptably for our simulation data and well for the experimental observations . the latter are rich data because cluster span is one of the simplest automated measurements to take . the rescaled distributions from simulation and experiment agree fairly well but not perfectly , as shown in figure 4 . one experimental reservation is the fall - off of experimental observation efficiency at small sizes , where clusters can be missed . however our scaling procedure itself is in effect expressly designed to avoid sensitivity to this , and the superposition of the experimental data down to small reduced sizes looks good . indeed it looks so good that the transient flattening around @xmath29 , which is absent from the simulations , appears to be significant . one suggestion for the flattening around the middle of the rescaled distribution is that it might be associated with the peculiar feature of the collision rate being zero between clusters of equal sedimentation speed . our simulations include this feature but for low relative approach speeds , where each cluster has more time to adjust its momentum in response to the other perturbing the local airflow , our approximation of ignoring hydrodynamic interactions ( and hence phenomena such as wake capture ) is less accurate . in summary , we have a fairly complete understanding of the geometry of the atmospheric ice crystal aggregates , dominated by sticking upon encounter . further details of the sticking mechanism ( which we did not include ) appear not to be important for the cluster geometry , and the excellent scaling superposition of the experimental cluster size distributions suggests sticking efficiency does not favour aggregation at particular sizes . the simplest interpretation of these observations is that although the sticking probability might be low for a single microscopic event , many such contacts will be attempted during a cluster - cluster encounter so that eventual escape is unlikely . the actual sticking mechanism between ice crystals remains an intriguing open question , particularly for the low temperatures of figure 1 . the fact that the same evolution is seen for differing initial monomer populations ( rods and rosettes ) suggests that a single set of geometric relationships for ice aggregates can successfully be applied in a wide range of cloud conditions . this would lead to greater accuracy in retrieving cloud properties such as precipitation rate and predicting the radiative affect of ice crystal aggregates upon the climate system . c to @xmath30c ( @xmath319 km altitude ) , using a cloud particle imager ( cpi , spec inc . the pictures shown are aggregates of rosette ice crystal types . ( b ) aggregates as simulated by our computer model which assumed rigid joining when clusters collide under differential sedimentation . , width=317 ] divided into the span perpendicular to the longest @xmath32 . grey lines show cloud data of _ korolev and isaac _ [ 2003 ] plotted against longest span in microns for a range of temperatures between @xmath33c and @xmath34c . black lines show simulation data plotted against longest span in arbitrarily scaled units , where the initial particles were three dimensional crosses ( solid line ) and simple rods ( dashed ) . , width=317 ] c ) to 6.6 km ( @xmath37c ) in the cirrus cloud of _ field and heymsfield _ [ 2003 ] obtained during an arm ( atmospheric radiation measurement program ) flight ( 9th march 2000 ) . each experimental size distribution represents an in - cloud average over 15 km . black lines show simulation data . , width=317 ]
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aggregation of ice crystals is a key process governing precipitation .
individual ice crystals exhibit considerable diversity of shape , and a wide range of physical processes could influence their aggregation ; despite this we show that a simple computer model captures key features of aggregate shape and size distribution reported recently from cirrus clouds .
the results prompt a new way to plot the experimental size distributions leading to remarkably good dynamical scaling .
that scaling independently confirms that there is a single dominant aggregation mechanism at play , albeit our model ( based on undeflected trajectories to contact ) does not capture its form exactly .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
surface plasmon resonance ( spr ) is a well accepted direct detection technique for monitoring biological processes @xcite . while ellipsometry is another well known method for analyzing thin film properties @xcite , its use in liquid medium for monitoring biochemical reactions is made difficult by the varying environment through which the probing light beam has to propagate . in the kretschmann configuration , the laser generating the spr evanescent wave is only propagating through the substrate , leading to a better control over the influence of the various buffer solutions used during a protein adsorption experiment . the use of various acoustic wave devices for monitoring bound mass changes in liquid media is well known @xcite . love mode devices , based on the propagation of a guided shear acoustic wave , present sensitivity improvements over the more usual quartz crystal microbalance @xcite as well as a compatibility with measurements in liquids . we take advantage of the unique geometrical setup of the surface acoustic wave ( saw ) device which leaves the backside of the quartz wafer free of electrodes to inject a laser in order to generate an evanescent surface plasmon on the gold coated sensing area . such a setup enables simultaneous estimates of the bound mass and dielectric surface properties changes during electrochemical and biochemical reactions occurring on the sensing electrode @xcite . estimates of water content in protein layers is an important topic in the development of biosensors , since the detection sensitivity is directly related to the number of active sites on the surface of the sensor to which the receptor molecules are bound . when physical measurements of this bound layer are performed , such as via spr or acoustic measurements , a large water content will lead to overestimates of the potential detection limit of the biosensor due to an overestimate of number of molecules bound to the surface . from a more fundamental point of view , determining water level content should provide more accurate physical parameters of the protein layer itself such as density and optical index by allowing the subtraction of the influence of water once it has been identified . we use a modified commercial spr instrument ( ibis ii , ibis technologies bv , the netherlands ) to detect the surface plasmon resonance angle after replacing the gold coated glass slide by a love mode saw device ( fig . [ fig1 ] ) . the excitation laser in this instrument has a wavelength of 670 nm . all reflected intensity _ vs. _ angle curves ( 6 @xmath3 angle span recorded on a 200 pixels ccd array ) were recorded and later fitted by a polynomial to extract with high accuracy the position of the dip . the acoustic wave device is made of a 500 @xmath4 m thick st - cut quartz wafer on which 200 nm thick sputtered @xmath5 interdigitated electrodes are patterned . the surface is coated by a 1.13 @xmath4 m pecvd silicon dioxide layer acting as a guiding layer , and the [email protected] mm@xmath7 sensing area is coated with 10 nm @xmath8 and 50 nm @xmath9 . this area acts both as a working electrode for electrochemistry or a grounded electrode during biochemical experiments , as well as a supporting layer for the surface plasmon resonance generation . the influence of the saw device substrate over the detection of the spr is limited to interference patterns due to the birefringence of the quartz , and the optical index mismatch between the quartz and the deposited silicon dioxide layer . the former effect is reduced by orienting the optical axis of quartz so that it is in the plane defined by the normal to the sensing surface and the wavevector of the laser , thus minimizing the optical index difference between the ordinary and the extraordinary axis . the remaining interference patterns lead to fringes with a low enough contrast that the surface plasmon peak is easily identified and tracked during the experiments . the phase of the saw delay line is monitored at a fixed frequency using a network analyzer hp4396a at 123.200 mhz . the phase shift is converted to a frequency shift thanks to the linear phase to frequency relationship recorded in the bode plot . the observed phase shift leads to a frequency shift ( as would be observed in a phase locked loop configuration ) which in turn can be converted to an adsorbed mass change through the sensitivity of the device . we first electrodeposited copper on the surface in order to calibrate the mass sensitivity of the acoustic wave device in liquid medium ( fig . [ fig2 ] ) @xcite . at the same time , the spr displayed resonance angle shifts due to the varying potential @xcite . when the voltage applied by the potentiostat is above 0.2 v with respect to the @xmath10 pseudo reference electrode , the spr angle slightly shifts due to the electroreflectance effect described by ktz _ _ @xcite . below 0.2 v , under potential deposition starts depositing a mono layer of @xmath10 atoms on the @xmath9 surface as visible both in the phase shift of the saw device and as a reversal of the trend of the angle shift of the spr . below 0 v , a rough @xmath10 layer is deposited , leading to a loss of the spr peak and a large phase shift of the saw device due to the large added mass . the rough , discontinuous film of copper clusters a few tens to hundreds of nanometers high was observed by in - situ afm imaging ( data not shown @xcite ) . simulating the influence of sub - wavelength metallic structures on the surface plasmon resonance is a difficult task @xcite which we have not attempted to solve . simultaneously , the saw device displays a frequency drop @xmath11 due to the added mass @xmath12 to the sensing electrode , leading with the combination of the current measurement from the potentiostat to a direct estimate of the mass sensitivity @xmath13 where @xmath14 cm@xmath7 is the area of the sensing working electrode and @xmath16 mhz the base frequency of the saw device . the electrodeposited mass is deduced from the current measurement @xmath17 with a sampling period @xmath18 : @xmath19 where @xmath20 c is the faraday constant , @xmath21 the molar weight of the electrodeposited metal and @xmath22 the number of electrons exchanged for each metallic atom reduced . the resulting sensitivity for the love mode device under consideration here is 150@xmath215 cm@xmath7/g . once the mass sensitivity of the saw is calibrated , we adsorbed on the same device a crystalline monolayer of s - layer proteins ( 100 @xmath4g / ml of the protein sbpa of _ bacillus sphaericus _ ccm 2177 in 0.5 mm tris , 10 mm @xmath23 , ph=9 buffer ) @xcite while monitoring simultaneously with the saw device the mass variation and with spr the surface dielectric change . the adsorption and desorption steps , the latter being possible with the use of 2 % @xmath24 , was repeated several times ( fig . [ fig3 ] ) . a property of this protein which makes it suitable for calibration of a new instrument is that it only forms a monolayer and will not stack to multiple layers even at high concentrations . the parameters required for analyzing the saw data are the layer thickness @xmath25 and the layer density @xmath26 . the parameters required for analyzing the spr data are the common layer thickness @xmath25 and the optical index @xmath27 of the adsorbed layer . we thus have three free parameters and only two measurements , namely the adsorbed mass per unit area and an spr resonance angle shift which is a function of both @xmath27 and @xmath25 . the two options are to add one more measurement parameter , such as an spr dip position as a function of wavelength which would lead to a unique identification of @xmath25 and @xmath27 @xcite , or to reduce the number of variables by adding the assumption that the layer is made of a homogeneous mixture of a proportion @xmath28 of protein and @xmath29 of water . the layer density @xmath26 is then @xmath30 where @xmath31 is assumed to be equal in the @xmath32 g/@xmath33 to 1.4 g/@xmath33 range @xcite and @xmath34 g/@xmath33 is the density of water . similarly , we use then the optical index of the layer @xmath35 with the optical index of the protein layer assumed to be in the @xmath36 to @xmath37 range @xcite and that of water @xmath38 . the remaining unknown parameters are then the layer thickness @xmath25 and the water content proportion @xmath29 . by simulating a stack of planar multilayers ( glass @xmath39 , 1200 nm silicon dioxide @xmath40 , 2 nm titanium @xmath41 @xcite , 50 nm gold @xmath42 @xcite , proteins , water ) for the angle shift as a function of water content and layer thickness , one obtains a set of pairs of values for these two parameters compatible with the observed angle shift ( fig . [ fig4 ] , bottom ) . then , by calculating the mass per unit area @xmath43 , a comparison with the experimentally observed adsorbed mass as seen from the phase shift of the saw device leads to a unique set of parameters both compatible with the optical index change and the adsorbed mass ( fig . [ fig4 ] , top ) . in the case under consideration here , an observed angle shift of 380 to 400 m@xmath3 and an adsorbed mass of 540 to 580 ng / cm@xmath7 is only compatible with @xmath44 and @xmath45 nm . the thickness result is compatible with atomic force microscope ( afm ) measurements in liquid @xcite while the mass per unit area is compatible with morphological data obtained by electronic microscopy @xcite . while the main source of uncertainty is due to the wide possible values of the density and optical index reported in the literature , the experimental results display good reproducibility . the water content is lower than that observed by other authors for different kinds of proteins @xcite but compatible with the 30 to 78% value cited in ref . the dense packing of protein is explained by the regular arrangement of identical s - layer subunits in the p4 lattice @xcite . we have shown here how the combination of love mode saw device with spr provides advantageous combinations of information on the bound mass as well as on the dielectric changes when monitoring protein adsorption . the resulting protein layer thickness and protein content percentage , respectively @xmath45 nm and @xmath46 % , is in agreement with independent afm estimates . the saw transducers were fabricated within the framework of a belgian phd scholarship program ( fonds pour la formation la recherche dans lindustrie et dans lagriculture fria ) . we wish to thank r. giust ( lopmd , besanon , france ) for kindly providing spr simulation routines . b. liedberg , c. nylander , and i. lundstrm , sensors and actuators * 4 * , 299 ( 1983 ) b. ivarsson , m. malmqvist , _ surface plasmon resonance : development and use of biacore instruments for biomolecular interaction analysis _ , in biomolecular sensors , ed . e. gizeli and c.r . lowe ( taylor & francis , london ) 2002 f. hk , b. kasemo , t. nylander , c. fant , k. sott , and h. elwing , anal . chem . * 73 * 5796 ( 2001 ) e. gizeli , _ acoustic transducers _ , in biomolecular sensors , ed . e. gizeli and c.r . lowe ( taylor & francis , london ) 2002 l.e . bailey , d. kambhampati , k.k . kanazawa , w. knoll , and c.w . franck , langmuir * 18 * 479 ( 2002 ) a. laschitsch , b. menges , and d. johansmann , appl . phys . lett . * 77 * 2252 ( 2000 ) j .- m . friedt , l. francis , k .- h . choi , and a. campitelli , j. vac . a ( to be published ) r. ktz , d.m . kolb , and j.k . sass , surf . sci . * 64 * 96 ( 1977 ) t.w . ebbesen , h.j . lezec , h.f . ghaemi , t. thio , and p.a . wolff , nature * 391 * 667 ( 1998 ) a.j . haes , and r.p . van duyne , j. am . 124 * 10596 ( 2002 ) m. sra , and u.b . sleytr , micron * 27 * 141 - 156 ( 1996 ) e.s . gyrvary , a. oriordan , a.j . quinn , g. redmon , d. pum , and u.b . sleytr , nanoletters * 3 * 315 ( 2003 ) b.p . nelson , a.g . frutos , j.m . brockman , and r.m . corn , anal . chem . * 71 * 3928 ( 1999 ) f. caruso , d.n . furlong , k. ariga , i. ichinose , and t. kunitake , langmuir * 14 * 4559 ( 1998 ) f. hk , j. vrs , m. rodahl , r. kurrat , p. bni , j.j . ramsden , m. textor , n.d . spencer , p. tengvall , j. gold , and b. kasemo , colloids and surfaces b * 24 * 155 ( 2002 ) r.j . marsch , r.a.l . jones , and m. sferrazza , _ adsorption and displacement of a globular protein on hydrophilic and hydrophobic surfaces _ colloids and surfaces b * 23 * 31 ( 2002 ) e. palik , _ handbook of optical constants of solids _ , academic press ( 1997 ) e. stenberg , b. persson , h. roos , and c. urbaniczky , journal of colloid and interface science * 143 * 513 ( 1991 ) m. weygand , b. wetzer , d. pum , u.b . sleytr , n. cuvillier , k. kjaer , p.b . howes and m. lsche , j. * 76 * 458 ( 1999 ) m. weygand , m. schalke , p.b . howes , k. kjaer , j. friedmann , b. wetzer , d. pum , u.b . sleytr , and m. lsche , j. mater . chem . * 10 * 141 ( 2000 )
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we present results from an instrument combining surface acoustic wave ( saw ) propagation and surface plasmon resonance ( spr ) measurements .
the objective is to use two independent methods , the former based on adsorbed mass change measurements and the latter on surface dielectric properties variations , to identify physical properties of protein layers , and more specifically their water content .
we display mass sensitivity calibration curves using electrodeposition of copper leading to a sensitivity in liquid of 150@xmath0 @xmath1 for the love mode device used here , and the application to monitoring biological processes .
the extraction of protein layer thickness and protein to water content ratio is also presented for s - layer proteins under investigation .
we obtain respectively [email protected] nm and 75@xmath215% .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
gaussian process classifiers are a very effective family of non - parametric methods for supervised classification @xcite . in the binary case , the class label @xmath0 associated to each data instance @xmath1 is assumed to depend on the sign of a function @xmath2 which is modeled using a gaussian process prior . given some data @xmath3 , learning is performed by computing a posterior distribution for @xmath2 . nevertheless , the computation of such a posterior distribution is intractable and it must be approximated using methods for approximate inference @xcite . a practical disadvantage is that the cost of most of these methods scales like @xmath4 , where @xmath5 is the number of training instances . this limits the applicability of gaussian process classifiers to small datasets with a few data instances at most . recent advances on gaussian process classification have led to sparse methods of approximate inference that reduce the training cost of these classifiers . sparse methods introduce @xmath6 inducing points or pseudoinputs , whose location is determined during the training process , leading to a training cost that is @xmath7 @xcite . a notable approach combines in @xcite the sparse approximation suggested in @xcite with stochastic variational inference @xcite . this allows to learn the posterior for @xmath2 and the hyper - parameters ( inducing points , length - scales , amplitudes and noise ) using stochastic gradient ascent . the consequence is that the training cost is @xmath8 , which does not depend on the number of instances @xmath5 . similarly , in a recent work , expectation propagation ( ep ) @xcite is considered as an alternative to stochastic variational inference for training these classifiers @xcite . that work shows ( i ) that stochastic gradients can also be used to learn the hyper - parameters in ep , and ( ii ) that ep performs similarly to the variational approach , but does not require one - dimensional quadratures . a disadvantage of the approach described in @xcite is that the memory requirements scale like @xmath9 since ep stores in memory @xmath10 parameters for each data instance . this is a severe limitation when dealing with very large datasets with millions of instances and complex models with many inducing points . to reduce the memory cost , we investigate in this extended abstract , as an alternative to ep , the use of stochastic propagation ( sep ) @xcite . unlike ep , sep only stores a single global approximate factor for the complete likelihood of the model , leading to a memory cost that scales like @xmath11 . we now explain the method for gaussian process classification described in @xcite . consider @xmath12 the observed labels . let @xmath13 be a matrix with the observed data . the assumed labeling rule is @xmath14 , where @xmath15 is a non - linear function following a zero mean gaussian process with covariance function @xmath16 , and @xmath17 is standard normal noise that accounts for mislabeled data . let @xmath18 be the matrix of inducing points ( _ i.e. _ , virtual data that specify how @xmath2 varies ) . let @xmath19 and @xmath20 be the vectors of @xmath2 values associated to @xmath21 and @xmath22 , respectively . the posterior of @xmath23 is approximated as @xmath24 , with @xmath25 a gaussian that approximates @xmath26 , _ i.e. _ , the posterior of the values associated to @xmath22 . to get @xmath25 , first the full independent training conditional approximation ( fitc ) @xcite of @xmath27 is employed to approximate @xmath26 and to reduce the training cost from @xmath4 to @xmath7 : @xmath28 where @xmath29 , @xmath30 and @xmath31 , with @xmath32 , @xmath33 , @xmath34 and @xmath35 is the marginal likelihood . furthermore , @xmath36 is a matrix with the prior covariances among the entries in @xmath37 , @xmath38 is a row vector with the prior covariances between @xmath39 and @xmath37 and @xmath40 is the prior variance of @xmath39 . finally , @xmath41 denotes the p.d.f of a gaussian distribution with mean vector equal to @xmath42 and covariance matrix equal to @xmath43 . next , the r.h.s . of ( [ eq : posterior ] ) is approximated in @xcite via expectation propagation ( ep ) to obtain @xmath25 . for this , each non - gaussian factor @xmath44 is replaced by a corresponding un - normalized gaussian approximate factor @xmath45 . that is , @xmath46 , where @xmath47 is a @xmath48 dimensional vector , and @xmath49 , @xmath50 and @xmath51 are parameters estimated by ep so that @xmath44 is similar to @xmath45 in regions of high posterior probability as estimated by @xmath52 . namely , @xmath53 , where @xmath54 is the kullback leibler divergence . we note that each @xmath45 has a one - rank precision matrix and hence only @xmath10 parameters need to be stored per each @xmath45 . the posterior approximation @xmath25 is obtained by replacing in the r.h.s . of ( [ eq : posterior ] ) each exact factor @xmath44 with the corresponding @xmath45 . namely , @xmath55 , where @xmath56 is a constant that approximates @xmath35 , which can be maximized for finding good hyper - parameters via type - ii maximum likelihood @xcite . finally , since all factors in @xmath25 are gaussian , @xmath25 is a multivariate gaussian . in order for gaussian process classification to work well , hyper - parameters and inducing points must be learned from the data . previously , this was infeasible on big datasets using ep . in @xcite the gradient of @xmath57 w.r.t @xmath58 ( _ i.e. _ , a parameter of the covariance function @xmath59 or a component of @xmath22 ) is : @xmath60 where @xmath61 and @xmath62 are the expected sufficient statistics under @xmath25 and @xmath63 , respectively , @xmath64 are the natural parameters of @xmath63 , and @xmath65 is the normalization constant of @xmath66 . we note that ( [ eq : gradient ] ) has a sum across the data . this enables using stochastic gradient ascent for hyper - parameter learning . a batch iteration of ep updates in parallel each @xmath45 . after this , @xmath25 is recomputed and the gradients of @xmath57 with respect to each hyper - parameter are used to update the model hyper - parameters . the ep algorithm in @xcite can also process data using minibatches of size @xmath67 . in this case , the update of the hyper - parameters and the reconstruction of @xmath25 is done after processing each minibatch . the update of each @xmath45 corresponding to the data contained in the minibatch is also done in parallel . when computing the gradient of the hyper - parameters , the sum in the r.h.s . of ( [ eq : gradient ] ) is replaced by a stochastic approximation , _ i.e. _ , @xmath68 , with @xmath69 the set of indices of the instances of the current minibatch . when using minibatches and stochastic gradients the training cost is @xmath8 the method described in the previous section has the disadvantage that it requires to store in memory @xmath70 parameters for each approximate factor @xmath45 . this leads to a memory cost that scales like @xmath9 . thus , in very big datasets where @xmath5 is of the order of several millions , and in complex models where the number of inducing points @xmath48 may be in the hundreds , this cost can lead to memory problems . to alleviate this , we consider training via stochastic expectation propagation ( sep ) as an alternative to expectation propagation @xcite . sep reduces the memory requirements by a factor of @xmath5 . r0.5 ll + & for each approximate factor @xmath45 to update : + 1.1 : & + 1.2 : & @xmath71 + 2 : & reconstruct @xmath25 : @xmath72 + ll + & set the new global factor @xmath73 to be uniform . + 2 : & for each exact factor @xmath44 to incorporate : + 2.1 : & + 2.2 : & @xmath71 + 2.3 : & @xmath74 + 3 : & reconstruct @xmath25 : @xmath75 + ll + & set @xmath25 to the prior . for each @xmath44 to process : + 1.1 : & + 1.2 : & @xmath71 + 2 : & update @xmath25 : @xmath76 + in sep the likelihood of the model is approximated by a single global gaussian factor @xmath77 , instead of a product of @xmath5 gaussian factors @xmath45 . the idea is that the natural parameters @xmath78 of @xmath77 approximate the sum of the natural parameters @xmath79 of the ep approximate factors @xmath45 . this approximation reduces by a factor of @xmath5 the memory requirements because only the natural parameters @xmath78 of @xmath77 need to be stored in memory , and the size of @xmath78 is dominated by the precision matrix of @xmath77 , which scales like @xmath11 . when sep is used instead of ep for finding @xmath25 some things change . in particular , the computation of the cavity distribution @xmath80 is now replaced by @xmath81 , @xmath82 . furthermore , in the case of the batch learning method described in the previous section , the corresponding approximate factor @xmath45 for each instance is computed as @xmath53 to then set @xmath83 . this is equivalent to adding natural parameters , _ i.e. _ , @xmath84 . in the case of minibatch training with minibatches of size @xmath85 the update is slightly different to account for the fact that we have only processed a small amount of the total data . in this case , @xmath86 , where @xmath69 is a set with the indices of the instances contained in the current minibatch . finally , in sep the computation of the gradients for updating the hyper - parameters is done exactly as in ep . figure [ fig : fig_ep_vs_sep ] compares among ep , sep and adf @xcite when used to update @xmath25 . in the figure training is done in batch mode and the update of the hyper - parameters has been omitted since it is exactly the same in either ep , sep or adf . in adf the cavity distribution @xmath87 is simply the posterior approximation @xmath25 , and when @xmath25 is recomputed , the natural parameters of the approximate factors are simply added to the natural parameters of @xmath25 . adf is a simple baseline in which each data point is _ seen _ by the model several times and hence it underestimates variance @xcite . we evaluate the performance of the model described before when trained using ep , sep and adf . * performance on datasets from the uci repository : * first , we consider 7 datasets from the uci repository . the experimental protocol followed is the same as the one described in @xcite . in these experiments we consider a different number of inducing points @xmath48 . namely , @xmath88 , @xmath89 and @xmath90 of the total training instances and the training of all methods is done in batch mode for 250 iterations . table [ tab : ll_uci ] shows the average negative test log likelihood of each method ( the lower the better ) on the test set . the best method has been highlighted in boldface . we note that sep obtains similar and sometimes even better results than ep . by contrast , adf performs worse , probably because it underestimating the posterior variance . in terms of the average training time all methods are equal . .average negative test log likelihood for each method and average training time in seconds . [ cols="<,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^ " , ] * performance on big datasets : * we carry out experiments when the model is trained using minibatches . we follow @xcite and consider the mnist dataset , which has 70,000 instances , and the airline delays dataset , which has 2,127,068 data instances ( see @xcite for more details ) . in both cases the test set has 10,000 instances . training is done using minibatches of size 200 , which is equal to the number of inducing points @xmath48 . in the case of the mnist dataset we also report results for batch training ( in the airline dataset batch training is infeasible ) . figure [ fig : stochastic ] shows the avg . negative log likelihood obtained on the test set as a function of training time . in the mnist dataset training using minibatches is much more efficient . furthermore , in both datasets sep performs very similar to ep . however , in these experiments adf provides equivalent results to both sep and ep . furthermore , in the airline dataset both sep and adf provide better results than ep at the early iterations , and improve a simple linear model after just a few seconds . the reason is that , unlike ep , sep and adf do not initialize the approximate factors to be uniform , which has a significant cost in this dataset . r0.5 @xmath91 the results obtained in the large datasets contradict the results obtained in the uci datasets in the sense that adf performs similar to ep . we believe the reason for this is that adf may perform similar to ep only when the model is simple ( small @xmath48 ) and/or when the number of training instances is very large ( large @xmath5 ) . to check that this is the case , we repeat the experiments with the mnist dataset with an increasing number of training instances @xmath5 ( from @xmath92 to @xmath93 ) and with an increasing number of inducing points @xmath48 ( from @xmath94 to @xmath95 ) . the results obtained are shown in figure [ fig : n_vsm ] , which confirms that adf only performs similar to ep in the scenario described . by contrast , sep seems to always perform similar to ep . finally , increasing the model complexity ( @xmath48 ) seems beneficial . stochastic expectation propagation ( sep ) @xcite can reduce the memory cost of the method recently proposed in @xcite to address large scale gaussian process classification . such a method uses expectation propagation ( ep ) for training , which stores @xmath9 parameters in memory , where @xmath6 is some small constant and @xmath5 is the training set size . this cost may be too expensive in the case of very large datasets or complex models . sep reduces the storage resources needed by a factor of @xmath5 , leading to a memory cost that is @xmath11 . furthermore , several experiments show that sep provides similar performance results to those of ep . a simple baseline known as adf may also provide similar results to sep , but only when the number of instances is very large and/or the chosen model is very simple . finally , we note that applying bayesian learning methods at scale makes most sense with large models , and this is precisely the aim of the method described in this extended abstract . * acknowledgments : * yl thanks the schlumberger foundation for her faculty for the future phd fellowship . jmhl acknowledges support from the rafael del pino foundation . ret thanks epsrc grant # s ep / g050821/1 and ep / l000776/1 . tb thanks google for funding his european doctoral fellowship . dhl and jmhl acknowledge support from plan nacional i+d+i , grant tin2013 - 42351-p , and from comunidad autnoma de madrid , grant s2013/ice-2845 casi - cam - cm . dhl is grateful for using the computational resources of _ centro de computacin cientfica _ at universidad autnoma de madrid . j. hensman , a. matthews , and z. ghahramani . scalable variational gaussian process classification . in _ proceedings of the eighteenth international conference on artificial intelligence and statistics _ , 2015 .
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a method for large scale gaussian process classification has been recently proposed based on expectation propagation ( ep ) .
such a method allows gaussian process classifiers to be trained on very large datasets that were out of the reach of previous deployments of ep and has been shown to be competitive with related techniques based on stochastic variational inference . nevertheless , the memory resources required scale linearly with the dataset size , unlike in variational methods .
this is a severe limitation when the number of instances is very large .
here we show that this problem is avoided when stochastic ep is used to train the model .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the analysis and interpretation of ongoing and future neutrino oscillation experiments strongly rely on the nuclear modeling for describing the interaction of neutrinos and anti - neutrinos with the detector . moreover , neutrino - nucleus scattering has recently become a matter of debate in connection with the possibility of extracting information on the nucleon axial mass . specifically , the data on muon neutrino charged - current quasielastic ( ccqe ) cross sections obtained by the miniboone collaboration @xcite are substantially underestimated by the relativistic fermi gas ( rfg ) prediction . this has been ascribed either to effects in the elementary neutrino - nucleon interaction , or to nuclear effects . the most poorly known ingredient of the single nucleon cross section is the cutoff parameter @xmath0 employed in the dipole prescription for the axial form factor of the nucleon , which can be extracted from @xmath1 and @xmath2 scattering off hydrogen and deuterium and from charged pion electroproduction . if @xmath0 is kept as a free parameter in the rfg calculation , a best fit of the miniboone data yields a value of the order of 1.35 gev / c@xmath3 , much larger than the average value @xmath4 gev / c@xmath3 extracted from the ( anti)neutrino world data @xcite . this should be taken more as an indication of incompleteness of the theoretical description of the data based upon the rfg , rather than as a true indication for a larger axial mass . indeed it is well - known from comparisons with electron scattering data that the rfg model is too crude to account for the nuclear dynamics . hence it is crucial to explore more sophisticated nuclear models before drawing conclusions on the value of @xmath0 . several calculations have been recently performed and applied to neutrino reactions . these include , besides the approach that will be presented here , models based on nuclear spectral functions @xcite , relativistic independent particle models @xcite , relativistic green function approaches @xcite , models including nn correlations @xcite , coupled - channel transport models @xcite , rpa calculations @xcite and models including multinucleon knock - out @xcite . the difference between the predictions of the above models can be large due to the different treatment of both initial and final state interactions . as a general trend , the models based on impulse approximation , where the neutrino is supposed to scatter off a single nucleon inside the nucleus , tend to underestimate the miniboone data , while a sizable increase of the cross section is obtained when two - particle - two - hole ( 2p-2h ) mechanisms are included in the calculations . furthermore , a recent calculation performed within the relativistic green function ( rgf ) framework has shown that at this kinematics the results strongly depend on the phenomenological optical potential used to describe the final state interaction between the ejected nucleon and the residual nucleus @xcite . with an appropriate choice of the optical potential the rgf model can reproduce the miniboone data without the need of modifying the axial mass ( see giusti s contribution to this volume @xcite ) . the kinematics of the miniboone experiment , where the neutrino flux spans a wide range of energies reaching values as high as 3 gev , demands relativity as an essential ingredient . this is illustrated in fig . 1 , where the relativistic and non - relativistic fermi gas results for the ccqe double differential cross section of 1 gev muon neutrinos on @xmath5 are shown as a function of the outgoing muon momentum and for two values of the muon scattering angle . the relativistic effects , which affect both the kinematics and the dynamics of the problem , have been shown to be relevant even at moderate momentum and energy transfers @xcite . @xmath6ccqe double differential cross sections on @xmath5 displayed versus the outgoing muon momentum for non - relativistic ( nrfg ) and relativistic ( rfg ) fermi gas.,title="fig : " ] @xmath6ccqe double differential cross sections on @xmath5 displayed versus the outgoing muon momentum for non - relativistic ( nrfg ) and relativistic ( rfg ) fermi gas.,title="fig : " ] hence in our approach we try to retain as much as possible the relativistic aspects of the problems . in spite of its simplicity , the rfg has the merit of incorporating an exact relativistic treatment , fulfilling the fundamental properties of lorentz covariance and gauge invariance . however , it badly fails to reproduce the electron scattering data , in particular when it is compared with the rosenbluth - separated longitudinal and transverse responses . comparison with electron scattering data must be a guiding principle in selecting reliable models for neutrino reactions . a strong constraint in this connection is represented by the `` superscaling '' analysis of the world inclusive @xmath7 data : in refs . @xcite it has been proved that , for sufficiently large momentum transfers , the reduced cross section ( namely the double differential cross section divided by the appropriate single nucleon factors ) , when represented versus the scaling variable @xmath8 @xcite , is largely independent of the momentum transfer ( first - kind scaling ) and of the nuclear target ( second - kind scaling ) . the simultaneous occurrence of the two kinds of scaling is called susperscaling . moreover , from the experimental longitudinal response a phenomenological quasielastic scaling function has been extracted that shows a clear asymmetry with respect to the quasielastic peak ( qep ) with a long tail extended to positive values of the scaling variable , i.e. , larger energy transfers . on the contrary the rfg model , as well as most models based on impulse approximation , give a symmetric superscaling function with a maximum value 20 - 30% higher than the data @xcite . in this contribution , after recalling the basic formalism for ccqe reactions and their connection with electron scattering , we shall illustrate two models which provide good agreement with the above properties of electron scattering data : one of them , the relativistic mean field ( rmf ) model , comes from microscopic many - body theory , the other , the superscaling approximation ( susa ) model , is extracted from @xmath7 phenomenology . we shall then include the contribution of 2p-2h excitations in the susa model and finally compare our results with the miniboone double differential , single differential and total cross sections . most of the results which will be presented are contained in refs . @xcite and @xcite . charged current quasielastic muonic neutrino scattering @xmath9 off nuclei is very closely related to quasielastic inclusive electron scattering @xmath7 . however two major differences occur between the two processes : 1 . in the former case the probe interacts with the nucleus via the weak force , in the latter the interaction is ( dominantly ) electromagnetic . while the weak vector current is related to the electromagnetic one via the cvc theorem , the axial current gives rise to a more complex structure of the cross sections , with new response functions which can not be related to the electromagnetic ones . as a consequence , while in electron scattering the double - differential cross section can be expressed in terms of two response functions , longitudinal and transverse with respect to three - momentum carried by the virtual photon , for the ccqe process it can be written according to a rosenbluth - like decomposition as @xcite @xmath10_{e_\nu } = \sigma_0 \left [ { \hat v}_{l } r_l + { \hat v}_t r_t + { \hat v}_{t^\prime } r_{t^\prime } \right ] , \label{eq : d2s}\ ] ] where @xmath11 and @xmath12 are the muon kinetic energy and scattering angle , @xmath13 is the incident neutrino energy , @xmath14 is the elementary cross section , @xmath15 are kinematic factors and @xmath16 are the nuclear response functions , the indices @xmath17 referring to longitudinal , transverse , transverse - axial , components of the nuclear current , respectively . the expression ( [ eq : d2s ] ) is formally analogous to the inclusive electron scattering case , but : _ i ) _ the `` longitudinal '' response @xmath18 takes contributions from the charge ( 0 ) and longitudinal ( 3 ) components of the nuclear weak current , which , at variance with the electromagnetic case , are not related to each other by current conservation , _ ii ) _ @xmath18 and @xmath19 have both `` vv '' and `` aa '' components ( stemming from the product of two vector or axial currents , respectively ) , _ iii ) _ a new response , @xmath20 , arises from the interference between the axial and vector parts of the weak nuclear current . + in fig . 2 we show the separate contributions of the three responses in ( [ eq : d2s ] ) , evaluated in the rfg model for the @xmath5 target nucleus , for two different values of the scattering angle . it can be seen that in the forward direction the transverse response dominates over the longitudinal and transverse - axial ones , whereas at higher angles the @xmath21-component becomes negligible and the @xmath22 and @xmath23 responses are almost equal . this cancellation has important consequences on antineutrino - nucleus scattering , where the response @xmath20 has opposite sign . + separate contributions of the rfg longitudinal @xmath24 , transverse @xmath25 and axial - vector interference @xmath26 responses to the double differential @xmath6ccqe cross sections displayed versus the muon kinetic energy at two different angles . the neutrino energy is averaged over the miniboone flux and the axial mass parameter has the standard value . , title="fig : " ] separate contributions of the rfg longitudinal @xmath24 , transverse @xmath25 and axial - vector interference @xmath26 responses to the double differential @xmath6ccqe cross sections displayed versus the muon kinetic energy at two different angles . the neutrino energy is averaged over the miniboone flux and the axial mass parameter has the standard value . , title="fig : " ] 2 . in @xmath7 experiments the energy of the electron is well - known , and therefore the detection of the outgoing electron univoquely determines the energy and momentum transferred to the nucleus . in neutrino experiments the neutrino energy is not known , but distributed over a range of values ( for miniboone from 0 to 3 gev with an average value of about 0.8 gev ) . the cross section must then be evaluated as an average over the experimental flux @xmath27 @xmath28_{e_\nu } \phi(e_\nu ) de_\nu \ , \label{eq : fluxint}\ ] ] which may require to account for effects not included in models devised for quasi - free scattering . this is , for instance , the situation at the most forward scattering angles , where a significant contribution in the cross section comes from very low - lying excitations in nuclei @xcite , as illustrated in fig . 3 : here the double differential cross section is evaluated in the susa model ( see later ) at the miniboone kinematics and the lowest angular bin and compared with the result obtained by excluding the energy transfers lower than 50 mev from the flux - integral ( [ eq : fluxint ] ) . + ( color online ) solid lines ( red online ) : flux - integrated @xmath6ccqe cross sections on @xmath5 calculated in the susa model for a specific bin of scattering angle . dashed lines ( green online ) : a lower cut @xmath29 mev is set in the integral over the neutrino flux . ] + it appears that at these angles 30 - 40% of the cross section corresponds to very low energy transfers , where collective effects dominate . moreover , processes involving meson exchange currents ( mec ) , which can excite both one - particle - one - hole ( 1p1h ) and two - particle - two - hole ( 2p-2h ) states via the exchange of a virtual meson , should also be taken into account , since they lead to final states where no pions are present , classified as `` quasielastic '' in the miniboone experiment . in this section we briefly outline the main ingredients of the rmf and susa model and we illustrate our calculation of the contribution of 2p-2h meson exchange currents . in the rmf model a fully relativistic description of both the kinematics and the dynamics of the process is incorporated . details on the rmf model applied to inclusive qe electron and ccqe neutrino reactions can be found in refs . @xcite . here we simply recall that the weak response functions are given by taking the appropriate components of the weak hadronic tensor , constructed from the single - nucleon current matrix elements @xmath30 where @xmath31 and @xmath32 are relativistic bound - state and scattering wave functions , respectively , and @xmath33 is the relativistic one - body current operator modeling the coupling between the virtual @xmath34-boson and a nucleon . the bound nucleon states are described as self - consistent dirac - hartree solutions , derived by using a lagrangian containing @xmath35 , @xmath36 and @xmath37 mesons @xcite . the outgoing nucleon wave function is computed by using the same relativistic mean field ( scalar and vector energy - independent potentials ) employed in the initial state and incorporates the final state interactions ( fsi ) between the ejected proton and the residual nucleus . the rmf model successfully reproduces the scaling behaviour of inclusive qe @xmath7 processes and , more importantly , it gives rise to a superscaling function with a significant asymmetry , namely , in complete accord with data @xcite . this is a peculiar property associated to the consistent treatment of initial and final state interactions . it has been shown in refs . @xcite that other versions of the rmf model , which deal with the fsi through a real relativistic optical potential , are not capable of reproducing the asymmetry of the scaling function . moreover , contrary to most other models based on impulse approximation , where scaling of the zeroth kind - namely the equality of the longitudinal and transverse scaling functions - occurs , the rmf model provides @xmath21 and @xmath23 scaling functions which differ by typically @xmath38 , the t one being larger . this agrees with the analysis @xcite of the existing @xmath39 separated data , which has shown that , after removing inelastic contributions and two - particle - emission effects , the purely nucleonic transverse scaling function is significantly larger than the longitudinal one . the susa model is based on the phenomenological superscaling function extracted from the world data on quasielastic electron scattering @xcite . the model has been extended to the @xmath40-resonance region in ref . @xcite and to neutral current scattering in ref . @xcite , but here we restrict our attention to the quasielastic charged current case . assuming the scaling function @xmath41 extracted from @xmath7 data to be universal , i.e. , valid for electromagnetic and weak interactions , in @xcite ccqe neutrino - nucleus cross sections have been evaluated by multiplying @xmath41 by the corresponding elementary weak cross section . thus in the susa approach all the nuclear responses in ( [ eq : d2s ] ) are expressed as follows @xmath42 where @xmath43 are the elementary lepton - nucleon responses , @xmath44 and @xmath45 are the fermi energy and momentum , @xmath46 is the number of nucleons ( neutrons in the @xmath6ccqe case ) and @xmath47 in the universal superscaling function , depending only on the scaling variable @xmath8 @xcite . the susa approach provides nuclear - model - independent neutrino - nucleus cross sections and reproduces the longitudinal electron data by construction . however , its reliability rests on some basic assumptions . first , it assumes that the scaling function - extracted from _ @xmath7 data - is appropriate for all of the weak responses involved in neutrino scattering ( charge - charge , charge - longitudinal , longitudinal - longitudinal , transverse and axial ) , and is independent of the vector or axial nature of the nuclear current entering the hadronic tensor . in particular it assumes the equality of the longitudinal and transverse scaling functions ( scaling of the zeroth kind ) , which , as mentioned before , has been shown to be violated both by experiment and by some microscopic models , for example relativistic mean field theory . second , the charged - current neutrino responses are purely isovector , whereas the electromagnetic ones contain both isoscalar and isovector components and the former involve axial - vector as well as vector responses . one then has to invoke a further kind of scaling , namely the independence of the scaling function of the choice of isospin channel so - called scaling of the third kind . finally , the susa approach neglects violations of scaling of first and second kinds . these are known to be important at energies above the qe peak and to reside mainly in the transverse channel , being associated to effects which go beyond the impulse approximation : inelastic scattering , meson - exchange currents and the associated correlations needed to conserve the vector current . the inclusion of these contributions in the susa model is discussed in the next paragraph . meson exchange currents are two - body currents carried by a virtual meson which is exchanged between two nucleons in the nucleus . they are represented by the diagrams in fig . [ fig : mec ] , where the external lines correspond to the virtual boson ( @xmath48 or @xmath34 ) and the dashed lines to the exchanged meson : in our approach we only consider the pion , which is believed to give the dominant contribution in the quasielastic regime . the thick lines in diagrams ( d)-(g ) represent the propagation of a @xmath40-resonance . the explicit relativistic expressions for the current matrix elements can be found , e.g. , in ref . @xcite . two - body meson - exchange currents . ( a ) and ( b ) : `` contact '' , or `` seagull '' diagram ; ( c ) : `` pion - in - flight '' diagram ; ( d)-(g ) : `` @xmath40-mec '' diagram . ] being two - body currents , the mec can excite both one - particle one - hole ( 1p-1h ) and two - particle two - hole ( 2p-2h ) states . in the 1p-1h sector , mec studies of electromagnetic @xmath49 process have been performed for low - to - intermediate momentum transfers ( see , _ e.g. _ , @xcite ) , showing a small reduction of the total response at the quasielastic peak , mainly due to diagrams involving the electroexcitation of the @xmath40 resonance . however in a perturbative scheme where all the diagrams containing one and only one pionic line are retained , the mec are not the only diagrams arising , but pionic correlation contributions , where the virtual boson is attached to one of the two interacting nucleons , should also be considered . these are represented by the same diagrams as in fig . [ fig : mec](d)-(g ) , where now the thick lines are nucleon propagators . only when all the diagrams are taken into account gauge invariance is fulfilled and the full two - body current is conserved . correlation diagrams have been shown to roughly compensate the pure mec contribution @xcite , so that in first approximation we can neglect the 1p-1h sector and restrict our attention to 2p-2h final states . the contribution to the inclusive electron scattering cross section arising from two - nucleon emission via meson exchange current interactions was first calculated in the fermi gas model in refs . @xcite , where sizable effects were found at large energy transfers . in these references a non - relativistic reduction of the currents was performed , while fully relativistic calculations have been developed more recently in refs . it has been found that the mec give a significant positive contribution which leads to a partial filling of the `` dip '' between the quasielastic peak and the analogous peak associated with the excitation of the @xmath40 resonance . moreover , the mec have been shown to break scaling of both first and second kinds @xcite . here we use the fully relativistic model of @xcite , where all the mec many - body diagrams containing two pionic lines that contribute to the electromagnetic 2p-2h transverse response were taken into account . similar results for the 2p-2h mec were obtained in ref . @xcite , where the correlation diagrams were also included . in order to apply the model to neutrino scattering , we observe that in lowest order the 2p2h sector is not directly reachable for the axial - vector matrix elements . hence the mec affect only the transverse polar vector response , @xmath50 . note that , at variance with the 1p-1h sector , where the contribution of the mec diagrams originates from the interference between 1-body and 2-body amplitudes and has therefore no definite sign ( in fact it turns out to be negative due to the dominance of the diagrams involving the @xmath40 ) , the 2p-2h contribution of mec to the nuclear responses is the square of an amplitude , hence it is positive by definition . therefore the net effect of 2p-2h mec to neutrino scattering is to increase the transverse vector response function , as will be illustrated in the next section . in this section we present the predictions of the above models and their comparison with the miniboone data . more results can be found in refs . @xcite and @xcite . in figs . [ fig : cos ] and [ fig : tmu ] the flux - integrated double - differential cross section per target nucleon for the @xmath6ccqe process on @xmath51c is evaluated for the three nuclear models above described : the rmf model and the susa approach with and without the contribution of 2p-2h mec . in fig . [ fig : cos ] the cross sections are displayed versus the muon kinetic energy @xmath11 at fixed scattering energy @xmath12 , in fig . [ fig : tmu ] they are displayed versus @xmath52 at fixed @xmath11 . it appears that the susa predictions systematically underestimate the experimental cross section , the discrepancy being larger at high scattering angle and low muon kinetic energy . the inclusion of 2p-2h mec tends to improve the agreement with the data at low angles , but it is not sufficient to account for the discrepancy at higher angles . the rmf calculation , which , as already mentioned , incorporates violations of scaling of the zeroth kind with a substantial enhancement of the vector transverse response , yields cross sections which are in general larger than the susa ones . in particular , in the region close to the peak in the cross section , the rmf result becomes larger than the one obtained with susa+mec . furthermore , the rmf does better than susa in fitting the shape of the experimental curves versus both the scattering angle and the muon energy : this is partly due to the fact that the rmf is better describing the low - energy excitation region whereas the susa model has no predictive power at very low angles , where the cross section is dominated by low excitation energies and the superscaling ideas are not supposed to apply . concerning the susa+mec results , a possible explanation of the theory / data disagreement is the fact that , as already mentioned , a fully consistent treatment of two - body currents should take into account not only the genuine mec contributions , but also the correlation diagrams that are necessary in order to preserve the gauge invariance of the theory this , however , is not an easy task because in an infinite system like the rfg the correlation diagrams give rise to divergences which need to be regularized @xcite . the divergences arise from a double pole in some of the diagrams , associated to the presence of on - shell nucleon propagators . different prescriptions have been used in the literature in order to overcome this problem @xcite , leading to a substantial model - dependence of the results . in particular in ref . @xcite the divergence has been cured by introducing a parameter @xmath53 which accounts for the finite size of the nucleus ( and therefore the finite time of propagation of a nucleon inside the nucleus ) and the @xmath53-dependence of the contribution of correlation diagrams has been explored . the study has shown that for reasonable values of the parameter the correlations add to the pure mec in the high - energy tail and are roughly of the same order of magnitude , but now contributing to both the longitudinal and the transverse channels . the inclusion of these terms in neutrino reactions is in progress @xcite and is expected to give a further enhancement of the cross sections . flux - integrated double - differential cross section per target nucleon for the @xmath6 ccqe process on @xmath51c evaluated in the rmf ( red ) model and in the susa approach with ( blue line ) and without ( green line ) the contribution of mec and displayed versus the muon kinetic energy @xmath11 for three specific bins of the scattering angle . the data are from miniboone @xcite . ] flux - integrated double - differential cross section per target nucleon for the @xmath6 ccqe process on @xmath51c evaluated in the rmf ( red ) model and in the susa approach with ( blue line ) and without ( green line ) the contribution of mec and displayed versus the muon kinetic energy @xmath11 for three specific bins of the scattering angle . the data are from miniboone @xcite . ] flux - integrated double - differential cross section per target nucleon for the @xmath6 ccqe process on @xmath51c evaluated in the rmf ( red ) model and in the susa approach with ( blue line ) and without ( green line ) the contribution of mec and displayed versus the muon kinetic energy @xmath11 for three specific bins of the scattering angle . the data are from miniboone @xcite . ] flux - integrated double - differential cross section per target nucleon for the @xmath6 ccqe process on @xmath51c evaluated in the rmf ( red ) model and in the susa approach with ( blue line ) and without ( green line ) the contribution of mec and displayed versus the muon scattering angle for three bins of the muon kinetic energy @xmath11 . the data are from miniboone @xcite . ] flux - integrated double - differential cross section per target nucleon for the @xmath6 ccqe process on @xmath51c evaluated in the rmf ( red ) model and in the susa approach with ( blue line ) and without ( green line ) the contribution of mec and displayed versus the muon scattering angle for three bins of the muon kinetic energy @xmath11 . the data are from miniboone @xcite . ] flux - integrated double - differential cross section per target nucleon for the @xmath6 ccqe process on @xmath51c evaluated in the rmf ( red ) model and in the susa approach with ( blue line ) and without ( green line ) the contribution of mec and displayed versus the muon scattering angle for three bins of the muon kinetic energy @xmath11 . the data are from miniboone @xcite . ] the single differential cross sections with respect to the muon kinetic energy and scattering angle , respectively , are presented in figs . [ fig : csvst ] and [ fig : csvscos ] , where the relativistic fermi gas result is also shown for comparison : again it appears that the rmf gives slightly higher cross sections than susa , due to the @xmath39 unbalance , but both models still underestimate the data for most kinematics . the inclusion of 2p-2h excitations leads to a good agreement with the data at high @xmath11 , but strength is still missing at the lower muon kinetic energies ( namely higher energy transfers ) and higher angles . ( color online ) flux - averaged @xmath6ccqe cross section on @xmath5 integrated over the muon kinetic energy and displayed versus the scattering angle . the data are from miniboone @xcite . ] ( color online ) flux - averaged @xmath6ccqe cross section on @xmath5 integrated over the muon kinetic energy and displayed versus the scattering angle . the data are from miniboone @xcite . ] finally , in fig . [ fig : csvsenu ] the total ( namely integrated over over all muon scattering angles and energies ) ccqe cross section per neutron is displayed versus the neutrino energy and compared with the experimental flux - unfolded data . besides the models above discussed , we show for comparison also the results of the relativistic mean field model when the final state interactions are ignored ( denoted as rpwia - relativistic plane wave impulse approximation ) or described through a real optical potential ( denoted as rrop ) . note that the discrepancies between the various models , observed in figs . [ fig : cos ] and [ fig : tmu ] , tend to be washed out by the integration , yielding very similar results for the models that include fsi ( susa , rmf and rrop ) , all of them giving a lower total cross section than the models without fsi ( rfg and rpwia ) . on the other hand the susa+mec curve , while being closer to the data at high neutrino energies , has a somewhat different shape with respect to the other models , in qualitative agreement with the relativistic calculation of @xcite . it should be noted , however , that the result is affected by an uncertainty of about 5% associated with the treatment of the 2p-2h contribution at low momentum transfers and that pionic correlations are not included . ( color online ) total ccqe cross section per neutron versus the neutrino energy . the curves corresponding to different nuclear models are compared with the flux unfolded miniboone data @xcite . ] two different relativistic models , one ( susa ) phenomenological and the other ( rmf ) microscopic , have been applied to the study of charged - current quasielastic neutrino scattering and the impact of 2p-2h meson exchange currents on the cross sections has been investigated . the results can be summarized as follows : 1 . both the susa and the rmf models , in contrast with the relativistic fermi gas , are fitting with good accuracy the longitudinal quasielastic electron scattering response at intermediate to high energy and momentum transfer . + the susa and rmf models give very similar results for the integrated neutrino cross section and both substantially under - predict the miniboone experimental data . however the comparison with the double differential experimental cross section reveals some differences between the two models , which are washed out by the integration . indeed the rmf , although being lower than the data , reproduces better the slopes of the cross section versus the muon energy and scattering angle . this is essentially due to the enhancement of the transverse response , which arises from the self - consistent mean field approach of rmf ( in particular from the consistent treatment of initial and final state interactions ) and is absent in the superscaling approach . 2 . in relativistic or semi - relativistic models final state interactions have been shown to play an essential role for reproducing the shape and size of the electromagnetic response @xcite and can not be neglected , in our scheme , in the study of neutrino interactions . the effect of final state interactions in the susa and rmf models is to lower the cross section , giving a discrepancy with the data larger than the rfg . 3 . in the transverse channel , the analysis of @xmath7 data points to the importance of meson - exchange currents which , through the excitation of two - particle - two - hole states , are partially responsible of filling the `` dip '' region between the qe and @xmath40 peaks . the 2p-2h mec can be even more relevant in the ccqe process , where `` quasielastic '' implies simply that no pions are present in the final state but , due to the large energy region spanned by the neutrino flux , processes involving the exchange of virtual pions can give a sizable contribution . in fact the inclusion of 2p-2h mec contributions yields larger cross sections and accordingly better agreement with the data , although the theoretical curves still lie below the data at high angles and low muon energy . it should be stressed , however , that the present calculation , though exact and fully relativistic , is incomplete . in order to preserve gauge invariance the full two - body current , including not only the mec but also the corresponding correlation diagrams , must be included . these have recently been shown to yield a sizable contribution at high energies in @xmath7 scattering @xcite and are likely to improve the agreement of our models with the miniboone data . 4 . in all our calculations the standard value @xmath541.03 gev / c@xmath3 has been used . it has been suggested that a larger value of the axial mass ( 1.35 gev / c@xmath3 ) would eliminate the disagreement with the data . however the fit was done using a rfg analysis , and more sophisticated nuclear models must be explored before drawing conclusions on the actual value of the axial mass . for instance in ref . @xcite it is shown that the miniboone data can as well be fitted by effectively incorporating some nuclear effects in the magnetic form factor of the bound nucleon , without changing the axial mass . + although our scope here is not to extract a value for the axial mass of the nucleon , but rather to understand which nuclear effects are effectively accounted for by a large axial cutoff parameter , let us mention that a best fit of the rmf and susa results to the miniboone experimental cross section gives an effective axial mass @xmath55 1.5 gev / c@xmath3 and values in the range @xmath56 gev / c@xmath3 yield results compatible with the miniboone data within the experimental errors . a similar analysis in the model including the 2p-2h contribution will be possible only when the above mentioned correlation diagrams will be consistently evaluated @xcite . i would like to thank j.e . amaro , j.a . caballero , t.w . donnelly , j.m . udias and c.w . williamson for the fruitful collaboration which lead to the results reported in this contribution . 9 a. a. aguilar - 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after a short review of the recent developments in studies of neutrino - nucleus interactions , the predictions for double - differential and integrated charged current - induced quasielastic cross sections are presented within two different relativistic approaches : one is the so - called susa method , based on the superscaling behavior exhibited by electron scattering data ; the other is a microscopic model based on relativistic mean field theory , and incorporating final - state interactions .
the role played by the meson - exchange currents in the two - particle two - hole sector is explored and the results are compared with the recent miniboone data .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
numerous independent surveys have demonstrated evidence on both cosmological and galactic scales that about 30% of the matter energy density of the universe consists of non - baryonic , non - luminous matter . a leading candidate for this dark matter is a yet - to - be - discovered weakly interactive massive particle ( wimp ) which could directly interact with detectors based on earth leading to kev - scale nuclear recoils . direct dark matter detection experiments are now probing well - motivated models of extensions to the standard model such as supersymmetry which naturally predict dark matter candidates @xcite . as the exposures of direct dark matter detection experiments continue to improve , they will soon have enough sensitivity to detect neutrinos from several astrophysical sources such as the sun , the atmosphere , and diffuse supernovae @xcite . for example , a 1 kev threshold xe based experiment with a 1 ton - year exposure will detect about 100 @xmath1 solar neutrino events via coherent neutrino - nucleus scattering ( cns ) . in fact for some wimp masses , such neutrino backgrounds can almost perfectly mimic a wimp signal . it has been shown in ref . @xcite that the cns background leads to a strong reduction of the discovery potential of upcoming experiments . therefore , though neither coherent neutrino scattering nor the wimp - nucleus interaction have conclusively been observed yet , the search for discrimination methods to disentangle wimps from neutrino events is a necessity . several methods to improve on the discrimination power between a wimp and a neutrino origin of the observed nuclear recoils have been suggested @xcite , including using the annual modulation signal @xcite or directional detection methods @xcite . in this paper , we propose a new method to reduce the effect of this neutrino background by looking for a possible complementarity between different target nuclei . in addition to discussing the target complementarity for different spin - independent ( si ) targets , for the first time we discuss the prospects for complementarity using spin - dependent ( sd ) targets . this paper is organized as follows . in sec . [ sec : neutfloor ] , we briefly review how to compute the wimp and neutrino event rates and we explain the test statistic that has been used to generate discovery limits . we then compute some discovery limits for several upcoming direct dark matter detection experiments in light of the neutrino background . in sec . [ sec : beyond ] , we show how target complementarity can help on improving the wimp discovery potential by considering both si and sd interactions . we then illustrate the improvement on the discovery limits by combining data from different targets . in sec . [ sec : optimization ] , we explore how one can optimize the relative exposures of the different target materials in a given experiment in order to maximize its discovery potential using the effect of target complementarity . [ cols="^,^,^,^,^,^,^,^ " , ] interestingly , from figs . [ fig : evolution_sys ] and [ fig : evolution_mass ] we can clearly see that the neutrino background will impact the low wimp mass ( below 10 gev / c@xmath0 ) and the high wimp mass ( above 10 gev / c@xmath0 ) regions at very different exposures due to the vastly different rates of neutrino backgrounds seen in fig [ fig : eventrate ] . indeed , the discovery limit evolution with exposure transitions from 1/mt to 1/@xmath2 as soon as the expected neutrino background nears 1 event , which corresponds to exposures of a few kg - years for very - low - threshold experiments in the low - mass region and a few tens of ton - years for the high - mass region for idealized perfect - efficiency experiments ( actual exposure values depend on the target , threshold , and efficiency of a given experiment ) . the saturation regime exists for exposures between 100 kg - year and 10 ton - year and between 10@xmath3 ton - year and 10@xmath4 ton - year for the low and high wimp mass regions respectively . therefore , one can conclude that the next generation of low - threshold experiments focusing on the low wimp mass region could reach well into the @xmath5 regime and even begin to see the effects of saturation , while at high wimp masses the next generation of experiments will at most begin to see the 1/@xmath2 effects from the neutrino background . from fig . [ fig : evolution_sys ] and [ fig : evolution_mass ] , we have seen that the neutrino background can lead to a saturation regime of the discovery potential over certain mass ranges that can span over 2 orders of magnitude in exposure . however , with enough exposure or lower systematics , one can always improve the discovery limit . in the following , we will define an arbitrary neutrino induced discovery limit within the saturation regime , derived from a set of two different exposures ( and energy thresholds ) for the low and high wimp mass regimes . the thresholds are defined such that for the low mass region we expect no @xmath6 neutrino events in the data and that for the high mass region we expect no @xmath1 neutrinos events in the data . the exposures are set such that the low wimp mass regime detects 200 events from @xmath1 ( about 1660 neutrino events total including pep , hep and @xmath7 ) , and the high wimp mass regime detects 400 neutrino events . the corresponding exposures and thresholds for each target discussed in the following are summarized in table [ tab : exposures ] . one can then see , using figure [ fig : evolution_sys ] , that the computed limits are always within the saturation regime of the discovery potential for both the low and the high wimp mass regions . as previously discussed , this is where adding a new observable such as annual modulation @xcite and/or directionality @xcite , or combining data from several experiments , as suggested in sec . [ sec : beyond ] , can lead to the most substantial improvement in the discovery potential . figure [ fig : discovery_limit_si ] , [ fig : discovery_limit_sdp ] and [ fig : discovery_limit_sdn ] show the computed limits for single - target and multi - target - based experiments for both si and sd interactions using the target and neutrino properties shown in tables [ tab : nuclei ] and [ tab : neutrino ] , and the energy thresholds and exposures from table [ tab : exposures ] . it should be noted that these are limits calculated for idealized experiments with perfect efficiency , and no background except for neutrinos , and thus represent the best attainable discovery limit for the listed exposures . in figure [ fig : discovery_limit_si ] ( left panel ) we show the discovery limits of elemental targets considering only the si interaction . one can first notice that , using a binned likelihood function , we have been able to reproduce the results from @xcite . in the si case the equivalent wimp models corresponding to a given neutrino type are only weakly dependent on the considered target ( see sec . [ sec : beyond ] for more details ) , thus all single - nucleus based targets share a very similar discovery limit . on figure [ fig : discovery_limit_si ] ( right panel ) we show the discovery limits for the si interaction considering compound targets such as @xmath8 ( used in cresst ) , @xmath9 ( used in coupp ) , and @xmath10 ( used in pico and picasso ) . since experiments that use @xmath10 and @xmath9 do nt currently measure the recoil energy of the event , we computed limits with and without energy sensitivity , shown as dashed and solid lines respectively . in the case of compound targets , we chose the thresholds by considering the lightest nucleus of the mixture as it has its neutrino spectra the most shifted to the high recoil energies . for example , considering @xmath8 , in order to compute the low threshold part of the discovery limit , we chose a threshold such that there is no pp neutrino events in the data due to the o target , _ i.e _ 25 ev . by choosing this threshold we do not consider the same event rate proportions for the three different targets . this explains the three successive exponential - like falls from a wimp mass of @xmath11 to about @xmath12 for this target . we can apply the same reasoning for the @xmath9 target where we can see two different exponential - like falls due to i and f targets . for @xmath10 this effect is not visible as f and c have very similar atomic masses . note that the discovery limit for cf@xmath13 ( used in experiments such as mimac @xcite and dmtpc @xcite ) is almost identical to the one from a c@xmath14f@xmath15 based experiment with energy sensitivity . therefore , all the following results using @xmath10 are also relevant to cf@xmath13 . figures [ fig : discovery_limit_sdp ] and [ fig : discovery_limit_sdn ] show the discovery limits considering a sd interaction on the proton and on the neutron , respectively . these discovery limits have similar shapes to those derived for the si interaction but their relative amplitudes can be very different . for example , considering a sd interaction on the proton , a si - based experiment and an experiment using @xmath10 have their discovery limits around @xmath16 and @xmath17 in the low mass region , respectively . this difference is due to the nature of the sd interaction . indeed , the neutrino - nucleus cross section is still described by a coherent effect ( evolution in @xmath18 ) while the sd wimp - nucleus cross section depends on the total angular momentum and mean spin contents of the considered target ( see eq . [ eq : sigsd ] and table [ tab : nuclei ] ) . another interesting feature that one can notice from figures [ fig : discovery_limit_sdp ] and [ fig : discovery_limit_sdn ] concerns the multi - target based experiments . indeed , for such experiments , some nuclei from the target material can not lead to a sd interaction as they have @xmath19 . this effect leads to different shapes in the discovery limits between the si and sd cases , as one can see for @xmath10 where @xmath20 and @xmath21 . one may also notice that @xmath8 does not appear on figures [ fig : discovery_limit_sdp ] and [ fig : discovery_limit_sdn ] as it has no sd sensitivity . for these six figures , the discovery limit indicates the region of parameter space where the neutrino background has its highest impact on the reach of upcoming direct detection experiments . indeed , with the considered energy thresholds and exposures , the discovery potential is in the saturation regime over most of the mass range , where a slight increase in sensitivity would be at the cost of a large increase in exposure . this clearly highlights the need for developing alternative strategies to be able to probe dark matter models lying underneath this neutrino - induced bound on the wimp discovery reach . as shown in the previous section , it can be extremely difficult to claim a discovery of dark matter if the true wimp model lies below the neutrino background . however , there are several possible ways to improve the discovery potential , even in the neutrino - induced saturation regime . the first one is to improve the theoretical estimates and experimental measurements of neutrino fluxes as it has been shown in figure [ fig : evolution_sys ] . a second possibility is to add some new observables that could help at disentangling between the wimp and neutrino origin of the observed nuclear recoils . this could be done by searching for annual modulation @xcite and/or by measuring the nuclear recoil direction @xcite , as suggested by upcoming directional detection experiments @xcite . indeed , since the main neutrino background has a solar origin , the directional signature of such events is expected to be drastically different from the wimp induced one @xcite . we are now in position to study the gains in discovery potential that can be achieved by combining data from different target materials in light of the neutrino background . as discussed in the previous section , the effect of neutrino background is notably important at particular wimp masses and cross sections where the wimp and neutrino spectra are similar , both in shape and magnitude . the wimp mass and cross section where this occurs is target - material dependent . one may then wonder whether combining different experiments could alleviate such degeneracies between the wimp and the neutrino hypotheses . in order to investigate this possibility , we determine what wimp models can be mimicked by @xmath22b neutrinos by computing the maximum likelihood distributions under the wimp only hypothesis in the ( @xmath23 , @xmath24 ) plane from fake data containing only nuclear recoils from @xmath22b neutrinos . because of the small differences between the wimp and neutrino energy distributions , the reconstructed wimp masses for each target are slightly different . this is result shown in figure [ fig : complementarity ] ( left panel ) . for a si interaction , the reconstructed cross sections for different targets are all roughly the same as is shown on figure [ fig : complementarity ] ( right panel ) . this is because both neutrino- and wimp - nucleus cross sections evolve as @xmath25 ( eq . [ eq : sigsi ] ) . therefore , we expect the target complementarity in the si case to be fairly weak . however , for a sd interaction the situation is different . as figure [ fig : complementarity ] ( right panel ) clearly indicates , the reconstructed cross sections for the different targets can be different by many orders of magnitude . this is due to the nature of the wimp sd cross section which depends on the angular momentum ( @xmath26 ) , spin content ( @xmath27 ) and isotopic fractions ( @xmath28 ) of the considered targets ( eq . [ eq : sigsd ] ) . thus , as the reconstructed wimp models of a given neutrino for different target nuclei are strongly distinguishable , one could expect to gain in discrimination power between the wimp and the neutrino hypothesis by combining data from several experiments . as a matter of fact , figure [ fig : complementarity ] ( right panel ) can be seen as a way to quantify the target complementarity . indeed , the larger the difference in reconstructed wimp cross sections between two targets is , the greater the discrimination power and improvement in a combined discovery potential becomes . as mentioned above , target complementarity is not very efficient for the si interaction as one can see on figure [ fig : complementarity_si ] ( left ) where we have computed the discovery limits ( solid curves ) for progressive combinations of xe , ge , si , and @xmath10 targets using thresholds taken from table [ tab : exposures ] for the low wimp mass region . the exposures for each individual target have been set such that we expect a total of 1,000 @xmath1 neutrino events equally distributed amongst each experiment . for example , the xe+ge curve has exposures set such that 500 events are detected in xe and 500 in ge . the slight shift to higher wimp masses when adding si to the xe+ge mixture comes from the differences in the reconstructed wimp masses from a @xmath22b signal shown in fig . [ fig : complementarity ] ( left panel ) . @xmath10 is currently used in bubble chamber experiments such as coupp which do not have sensitivity to the recoil energy since they detect events which deposit an energy density above the threshold required to generate bubble nucleation @xcite . we have thus included curves with and without energy sensitivity for this target . for reference , the background - free sensitivity limits for the chosen exposures are also shown as dashed lines . the ratio between the sensitivity and the discovery limits , shown in fig . [ fig : complementarity_si ] ( right panel ) , allows us to quantify the impact of the neutrino background on the wimp discovery potential of these idealized experiments . as expected , for si wimp interactions target complementarity has a modest effect and can at most reduce by a factor of 2 the impact of the neutrino background on the discovery potential . [ fig : complementarity_sdp ] ( left panel ) is similar to fig . [ fig : complementarity_si ] ( left panel ) but for the sd interaction on the proton . from fig . [ fig : complementarity ] ( right ) we see that fluorine targets have a large advantage in sensitivity for sd proton interactions . one can see this in the large improvement in discovery potential when adding @xmath10 to the compliment of targets . one might wonder if it is worth combining @xmath10 with other experiments due to its advantage for this interaction . to consider this , we have added a @xmath10-only discovery and sensitivity limit . one can see that even though one obtains the best background - free sensitivity with @xmath10 , the best discovery limit is obtained by adding other targets which helps at reducing the effect of the neutrino background by about an order of magnitude ( see fig . [ fig : complementarity_sdp ] right panel ) even while the background - free sensitivity is reduced . it is also worth recalling that the curves that consider an experiment using @xmath10 with recoil energy sensitivity have very similar neutrino background implications than cf@xmath13-based experiments such as mimac and dmtpc . finally fig . [ fig : complementarity_sdn ] shows the results of complimentarity for the sd neutron case . for this interaction , the improvement in discovery potential is intermediate between the si and sd proton case . while we have considered only models in which a wimp has one of these three types of nuclear interactions , a true wimp may have non - zero cross sections in all three of these channels . if a potential signal is discovered near the neutrino - induced discovery limit , a more detailed scan of the parameter space would be required to disentangle the possible wimp and neutrino signals . in the previous section , we focused on the effect of target complementarity on the discovery potential of upcoming experiments for a fixed exposure . in the following , we will describe the effect of target complementarity on the dynamics of the discovery limit when increasing the exposure around the saturation regime . + figure [ fig : evolution ] shows the evolution of the discovery limit for xe ( red @xmath29 ) , xe+ge ( blue @xmath30 ) and xe+ge+si ( green @xmath31 ) based experiments considering both si ( left panel ) and sd ( right panel ) interactions for a @xmath11 wimp mass . we can see that in the case of a si interaction , combining xe , ge and si does not allow one to greatly improve on its discovery potential . the green points ( xe+ge+si ) still clearly show a saturation regime in the discovery potential . however , considering a sd interaction on the proton , adding successively ge and si allows one to consequently improve the evolution of the discovery potential with exposure and keep it close to the @xmath5 best - case scenario . by comparing the xe - only and xe+ge+si points , we can see that target complementarity allows one to reach a discovery limit of about @xmath32 with an exposure which is 50 times lower . this clearly highlights the interest of combining data from different experiments to bypass the saturation regime induced by the neutrino background . we considered both the si and the sd interaction on the proton . for the si interaction there is no optimal si fraction because there is no visible effect from target complementarity . however , for the sd interaction on the proton , the discovery limit at @xmath11 has a minimum value for a certain si fraction which depends on the considered exposure . ] the previous results have shown combinations of several single - nucleus based experiments with a fixed relative exposure . in this section , we will address the optimization of several target materials when designing a future experiment that would maximize its discovery potential . we will consider the relative exposures of si and ge crystals in this example since at least one experiment ( supercdms ) can use both of these targets in their future payloads . as suggested by our previous study of the target complementarity , in the context of a si interaction the discovery limit is only slightly affected by changing the si fraction of the total exposure . we thus concentrate on the sd proton interaction . figure [ fig : optimization ] shows the discovery limit for a combination of ge and si - based experiments as a function of the considered si fraction in the total exposure at a fixed wimp mass of 6 gev / c@xmath0 . the considered thresholds for the two experiments target nuclei are taken from table [ tab : exposures ] . we can see that for exposures which are low enough to be in the systematics - dominated regime ( below 0.2 ton - year ) , adding si in the experiment does not improve the discovery potential because the reconstructed cross section for si is about two orders of magnitude higher than the one for ge ( see fig.[fig : complementarity ] ) . however , when the exposure is high enough to be in the discovery potential saturation regime , the complementarity of the two targets leads to an improvement of the combined discovery potential . the optimum si fraction that maximizes the effect of the complementarity of ge and si corresponds to the minimum of these curves which has been marked by a `` @xmath31 '' . we can therefore see that as the exposure increases , the si fraction that maximizes the discovery potential has to be higher . for a 0.4 ton - year experiment it is around 30% but for a 10 ton - year experiment it is around 60% . [ fig : optimization ] suggests that for very high exposures , it is worth tuning the relative exposures of ge and si detectors in order to maximize the discovery potential in light of the neutrino background . since dark matter experiments will soon begin be sensitive to coherent neutrino scattering from solar neutrinos , the search for discrimination methods to disentangle wimp from neutrino events is a necessity . for both spin - independent and spin - dependent targets , we have examined the effect of target complementarity on the discovery potential of single nucleus and multi - target based experiments . in general , we find that combining different targets such as xe and ge will allow for an improvement in the discovery potential . for spin - independent interactions , we show that the similarity between the scaling of the response of wimps and neutrinos with different targets ultimately limits the gain in discovery potential achievable . however , this situation is different for spin - dependent interactions . in this case , the differences in the cross sections of wimps and neutrinos with different targets allows one to bypass the saturation of the discovery potential and allow the search for dark matter to be pursued to lower cross sections . the improvement is most dramatic for spin - dependent wimp - proton interactions . in particular combining c@xmath14f@xmath15 with other targets results in a significant decrease of the impact of the neutrino background on the discovery potential . we have also shown that the effect of target complementarity can be used to optimize the different element fractions for experiments which have more than one target material . in particular for the sd interaction on the proton , there is an optimum si fraction for a ge@xmath29si experiment which maximizes the discovery potential . we have discussed how this fraction depends on the exposure and increases with the exposure . in summary , the results that we have presented in this paper provide valuable input for future direct dark matter searches . we have provided the first suggestion for expanding the reach of wimp dark matter searches in the regime where the neutrino backgrounds are substantial . the methods we have discussed rely only on measuring energy depositions , and can ultimately be complementary to methods that reduce the neutrino background that rely on different experimental signatures such as annual modulation or the direction of the recoiling nucleus . + * acknowledgments* les was supported by nsf grant phy-1417457 . fr , jb and eff were supported by nsf grant no . the authors would like to thank laurent derome and frederic mayet for useful discussions on an earlier version of this work . g. bertone , d. hooper and j. silk , phys . rept . * 405 * , 279 ( 2005 ) [ hep - ph/0404175 ] . l. e. strigari , phys . rept . * 531 * , 1 ( 2013 ) [ arxiv:1211.7090 [ astro-ph.co ] ] . b. cabrera , l. m. krauss , and f. wilczek , phys . lett . * 55 * , 25 ( 1985 ) . a. gtlein , g. angloher , a. bento , c. bucci , l. canonica , a. erb , f. v. feilitzsch and n. f. iachellini _ et al . _ , arxiv:1408.2357 [ hep - ph ] . r. harnik , j . kopp and p. a. n. machado , j. cosmol . . phys . * 1207 * , 026 ( 2012 ) [ arxiv : 0706.3019 [ astro - ph ] ] .
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direct detection dark matter experiments looking for wimp - nucleus elastic scattering will soon be sensitive to an irreducible background from neutrinos which will drastically affect their discovery potential . here
we explore how the neutrino background will affect future ton - scale experiments considering both spin - dependent and spin - independent interactions .
we show that combining data from experiments using different targets can improve the dark matter discovery potential due to target complementarity .
we find that in the context of spin - dependent interactions , combining results from several targets can greatly enhance the subtraction of the neutrino background for wimp masses below 10 gev / c@xmath0 and therefore probe dark matter models to lower cross - sections . in the context of target complementarity , we also explore how one can tune the relative exposures of different target materials to optimize the wimp discovery potential .
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the study of non - zero degree maps between closed , oriented manifolds has become very active over the last few decades @xcite . the existence of a non - zero degree map , @xmath1 , defines a transitive relation on the set of homotopy types of closed , oriented manifolds . whenever such a map exists we say that @xmath2 dominates @xmath3 and write @xmath4 . in this case @xmath2 is at least as complicated as @xmath3 . for example , the induced maps in rational homology are surjective , thus , in particular , the betti numbers of @xmath3 are bounded from above by those of @xmath2 . also , @xmath4 implies that the fundamental group of @xmath2 surjects onto a finite index subgroup of the fundamental group of @xmath3 . in dimension two , the domination relation coincides with the ordering given by the genus , but in higher dimensions it fails to be an ordering . we illustrate this by the following two examples in dimension three . the examples have obvious generalizations to higher dimensions . the two three - manifolds @xmath5 and @xmath6 satisfy @xmath7 and @xmath8 , but fail to be homotopy equivalent . let @xmath2 be a hyperbolic homology three - sphere , and @xmath9 . then @xmath3 has larger first betti number than @xmath2 , and so @xmath10 . we also have @xmath11 since the fundamental group of @xmath3 is infinite cyclic , and so can not surject onto the fundamental group of a closed negatively curved manifold , for example by preissmann s theorem . thus @xmath2 and @xmath3 are not comparable under the domination relation . in this paper , we study domination by products for three - manifolds . this is motivated by the work of lh and the first author in @xcite , where strong restrictions were found for certain manifolds with large universal coverings to be dominated by products . in fact , the results of those papers show that three - manifolds dominated by products can not have hyperbolic or @xmath12-geometry , and must often be prime . however , in this paper we will not use those earlier results , but follow a more direct approach . this is possible since in dimension three the only product manifolds are those with a circle factor , and this gives much stronger constraints than the consideration of arbitrary products . the main result we prove here is the following : [ t : topargument ] a closed , oriented , connected three - manifold @xmath3 is dominated by a product @xmath13 if and only if 1 . either @xmath3 is finitely covered by a product @xmath14 , for some aspherical surface @xmath15 , or 2 . @xmath3 is finitely covered by a connected sum @xmath16 . as usual , the empty connected sum corresponding to @xmath17 is @xmath18 . the proof of theorem [ t : topargument ] falls naturally into two parts . on the one hand , we have to prove that all three - manifolds not listed in the statement of the theorem can not be dominated by products . on the other hand , we have to prove that the manifolds listed in the theorem are indeed dominated by products . this is obvious for manifolds finitely covered by products , but it is not obvious for the connected sums occurring in the second statement . here the proof proceeds by constructing certain maps of non - zero degrees as branched coverings . this construction , which also has a high - dimensional generalization , is of independent interest . previously , many non - trivial results have been proved about the domination relation in dimension three using a variety of tools different from the ones we use here , such as thurston s geometries , gromov s simplicial volume , and the seifert volume . a survey of the state of the art at the beginning of the last decade is given in @xcite . for more recent results , especially related to the issue of finiteness of sets of mapping degrees between three - manifolds , see for example @xcite and the papers quoted there . our proofs here are independent of this earlier work , and in fact clarify certain claims made there , cf . subsection [ ss : w ] below . it is not immediately obvious to what extent theorem [ t : topargument ] really depends on the assumption that the domains of our dominant maps are products , and one could try to replace these products by fibered three - manifolds . for this purpose surface bundles over the circle are not interesting , since every three - manifold is dominated by such a bundle by a result of sakuma @xcite . however , considering non - trivial circle bundles over surfaces we obtain a result parallel to theorem [ t : topargument ] : [ t : topbundle ] a closed , oriented , connected three - manifold @xmath3 is dominated by a non - trivial circle bundle over a surface if and only if 1 . either @xmath3 is finitely covered by a non - trivial circle bundle over some aspherical surface , or 2 . @xmath3 is finitely covered by a connected sum @xmath16 . in section [ s : ess ] we discuss the notion of rational essentialness in the case of three - manifolds . while this is not logically necessary for the proofs of our main results , we find it convenient , following @xcite , to use this concept as an organizing principle . in section [ s : proof1 ] , respectively section [ s : proof2 ] , we then prove theorems [ t : topargument ] and [ t : topbundle ] for rationally essential , respectively inessential , three - manifolds . in section [ s : algargum ] we reformulate these theorems in terms of thurston geometries and in purely algebraic terms . finally , in section [ s : groups ] we determine the three - manifold groups presentable by products , and in section [ s : dis ] we make some further remarks . these last two sections contain two new characterizations of ( aspherical ) seifert manifolds . the obstructions for domination by products found in @xcite are applicable to rationally essential manifolds in the sense of the following definition going back to gromov @xcite : a closed , oriented , connected @xmath19-manifold @xmath3 is called rationally essential if @xmath20)\neq 0\in h_n(b\pi_1(n);{\mathbb{q } } ) \ , \ ] ] where @xmath21 classifies the universal covering of @xmath3 . for three - manifolds , this definition can be interpreted in terms of the kneser - milnor prime decomposition @xcite . recall that this says that a closed oriented connected three - manifold @xmath3 has an essentially unique prime decomposition @xmath22 under the connected sum operation . each prime summand @xmath23 is either aspherical , is @xmath24 , or has finite fundamental group . we now have the following : [ t : ess ] for a closed oriented connected three - manifold @xmath3 the following conditions are equivalent : 1 . @xmath3 is rationally essential , 2 . @xmath3 has an aspherical summand @xmath23 in its prime decomposition , 3 . @xmath3 is not finitely covered by a connected sum @xmath16 , 4 . @xmath3 is compactly enlargeable , 5 . @xmath3 does not admit a metric of positive scalar curvature . the last two items are not relevant to the main results of this paper , so we will only discuss them briefly . a connected sum is rationally essential if and only if at least one of the summands is . since @xmath24 and manifolds with finite fundamental group are not rationally essential , this proves the equivalence of ( @xmath25ess ) and ( asph ) . it is obvious that ( asph ) implies ( nfree ) . for the converse assume that @xmath3 contains no aspherical summands in its prime decomposition , i.e. that @xmath3 has the form @xmath26 where the empty connected sum ( @xmath27 ) denotes the 3-sphere @xmath18 . the summands @xmath28 have infinite cyclic fundamental groups and the summands @xmath29 have finite fundamental groups @xmath30 , @xmath31 . thus , the fundamental group of @xmath3 is the free product @xmath32 where @xmath33 is a free group on @xmath34 generators . we project this free product to the direct product of the @xmath35 to obtain the following exact sequence : @xmath36 by the kurosh subgroup theorem , @xmath37 is a free group @xmath38 . since it has finite index in @xmath39 , we see that @xmath3 has a finite covering whose fundamental group is free . by kneser s prime decomposition theorem and grushko s theorem , we deduce that this covering is a connected sum of @xmath19 copies of @xmath28 . this means that @xmath3 is finitely covered by a connected sum @xmath16 , where @xmath19 is the number of generators of the free group @xmath40 in the exact sequence ( [ eq.1 ] ) . to see that ( asph ) implies ( enl ) it is enough to show that any aspherical three - manifold @xmath3 is compactly enlargeable this was proved by gromov and lawson ( * ? ? ? * theorem 6.1 ) under the assumptions that @xmath39 is residually finite and contains an infinite surface group . it is now known that all three - manifold groups are residually finite @xcite . ( this reference treats only manifolds satisfying thurston s geometrisation conjecture , which has now been verified by perelman @xcite . ) furthermore , if @xmath39 contains no infinite surface group , then @xmath3 is atoroidal , and so is hyperbolic by perelman s work @xcite . since hyperbolic manifolds are compactly enlargeable by ( * ? ? ? 3.3 ) , we conclude that ( asph ) implies ( enl ) . gromov and lawson ( * ? ? ? * theorem 3.7 ) proved that ( enl ) implies ( npsc ) . ( recall that all oriented three - manifolds are spin . ) finally , ( npsc ) implies ( asph ) because @xmath24 has positive scalar curvature , and so do all three - manifolds with finite fundamental group by perelman s proof of the poincar conjecture @xcite . a connected sum of manifolds with positive scalar curvature also has positive scalar curvature by the construction of gromov and lawson ; cf . * theorem 5.4 ) . it was proved by hanke and schick @xcite that ( enl ) implies ( @xmath25ess ) in all dimensions . the converse is not true in dimensions @xmath41 by a recent result of brunnbauer and hanke @xcite . in view of theorem [ t : ess ] , the proofs of theorems [ t : topargument ] and [ t : topbundle ] split into two cases , depending on whether @xmath3 contains an aspherical summand @xmath23 in its prime decomposition , or not . in this section we deal with the case where an aspherical summand does appear . the first part of theorem [ t : topargument ] corresponds to the following statement : [ p : ess ] a rationally essential closed oriented three - manifold @xmath3 is dominated by a product if and only if it is finitely covered by a product @xmath14 , with @xmath15 an aspherical surface . a manifold finitely covered by a product is of course dominated by that product . for the converse assume that a product @xmath13 of a closed , oriented , connected surface @xmath42 with the circle dominates a closed , oriented , connected rationally essential three - manifold @xmath3 and let @xmath43 be a map of non - zero degree . then @xmath13 must be rationally essential , and so @xmath42 is of positive genus . by replacing @xmath3 by a finite covering if necessary , we may assume that @xmath44 is @xmath45-surjective . let @xmath22 be the kneser - milnor prime decomposition of @xmath3 . each prime summand @xmath23 is either aspherical , is @xmath24 , or has finite fundamental group ; cf . @xcite . by theorem [ t : ess ] the rational essentialness of @xmath3 is equivalent to the existence of an aspherical summand @xmath23 . composing @xmath44 with the degree one map @xmath46 collapsing the connected summands other than @xmath23 , we obtain a dominant map @xmath47 between aspherical three - manifolds . this can not factor through @xmath42 , implying that @xmath48 must be non - trivial on the central @xmath49-subgroup generated by the @xmath50 factor . but then @xmath51 is a non - trivial central subgroup in @xmath39 , and so this group is freely indecomposable . thus we may assume that @xmath3 itself is prime and aspherical , for we can either appeal to perelman s proof of the poincar conjecture @xcite to conclude @xmath52 , or we can argue that the assumption @xmath53 depends only on the homotopy type of @xmath3 , which does not change if we replace a manifold by its connected sum with a homotopy sphere . we have shown that @xmath3 is aspherical , and that its fundamental group has infinite center . if @xmath3 is haken , then it follows from a result of waldhausen @xcite that @xmath3 is seifert fibered . in fact , even without the haken condition , @xmath3 must be seifert fibered , by the proof of the seifert fiber space conjecture ( stated in @xcite and proved by casson jungreis @xcite and gabai @xcite ) . therefore , after lifting @xmath44 to a suitable covering space , we may assume that @xmath3 is a circle bundle over an aspherical surface . it remains to show that the euler number of this circle bundle is zero . we will prove this in the following lemma , thereby completing the proof of proposition [ p : ess ] . [ lem ] let @xmath54 be an oriented circle bundle with non - zero euler number over a closed aspherical surface . then every continuous map @xmath55 has degree zero . since @xmath3 is aspherical , we may assume that @xmath42 has positive genus . from the discussion above we may assume that @xmath56 is an element of infinite order in the center of @xmath39 . since on a seifert manifold elements of the center of the fundamental group are , up to taking multiples , fibers of seifert fibrations , cf . @xcite and @xcite , we may assume that @xmath56 is a multiple of the homotopy class of the fiber in @xmath3 . ( the fibration of @xmath3 is unique , cf . thus the composition @xmath57 kills the homotopy class of the @xmath50-factor in @xmath13 . since @xmath15 is aspherical , this implies that @xmath57 is homotopic to a map that factors through the projection @xmath58 . by the homotopy lifting property of @xmath54 , the homotopy of @xmath57 can be lifted to a homotopy of @xmath44 , so we may assume that @xmath59 for some continuous map @xmath60 . since @xmath54 has non - zero euler number , @xmath61 induces the zero map on @xmath62 , and the fundamental class of @xmath15 is not in the image . as @xmath45 is surjective on @xmath63 , the equation @xmath59 shows that @xmath64 . now consider the pullback of @xmath3 under @xmath65 : @xmath66 the map @xmath55 factors through @xmath67 as follows : @xmath68 for any pullback of an oriented bundle , the degree of the map between total spaces is the same as the degree of the map of base spaces under which the bundle is pulled back . in our situation this says that the degree of @xmath69 equals the degree of @xmath65 , which vanishes . thus @xmath44 factors through a degree zero map , and we finally have @xmath70 . the next proposition covers the first part of theorem [ t : topbundle ] . [ p : essbd ] a rationally essential closed oriented three - manifold @xmath3 is dominated by a non - trivial circle bundle over a surface if and only if it is finitely covered by a non - trivial circle bundle over some aspherical surface . let @xmath71 be a map of non - zero degree , with @xmath2 a non - trivial circle bundle over a surface @xmath72 of genus @xmath73 . after replacing @xmath3 by a suitable covering , we may assume that @xmath44 is @xmath45-surjective . since @xmath3 is assumed to be rationally essential , @xmath74 must be infinite , and so @xmath75 . this means that @xmath2 is aspherical and we have a non - trivial central extension @xmath76 the prime decomposition of @xmath3 contains an aspherical summand @xmath23 by theorem [ t : ess ] . composing @xmath44 with the degree one map @xmath46 collapsing the connected summands other than @xmath23 , we obtain a dominant map @xmath77 between aspherical three - manifolds . this can not factor through @xmath72 , implying that @xmath48 must be non - trivial on the central @xmath49-subgroup generated by the circle fibers in @xmath2 . but then @xmath51 is a non - trivial central subgroup in @xmath39 , and so @xmath3 is prime and therefore irreducible and aspherical itself . as in the proof of proposition [ p : ess ] we conclude that @xmath3 is seifert fibered . after replacing @xmath2 and @xmath3 by suitable coverings , we may assume that @xmath3 is also a circle bundle . it remains to prove that it has non - trivial euler class . now @xmath48 sends the element of @xmath74 represented by the circle fibers in @xmath2 to a non - trivial element of the center of @xmath39 . this group is torsion - free , so this non - trivial element has infinite order . some multiple of it is the fiber of a seifert fibration of @xmath3 , cf . @xcite . as mentioned before , we may assume that this seifert fibration is a circle bundle . since the fiber in @xmath2 has finite order in homology because the euler class of @xmath2 was non - zero , it follows that the circle fiber in @xmath3 , being , up to taking multiples , the image under @xmath78 of the circle fiber in @xmath2 , also has finite order in homology , and so the euler class of @xmath3 must be non - zero . in this section we prove theorems [ t : topargument ] and [ t : topbundle ] in the case of rationally inessential manifolds , i. e. those with no aspherical summand in their prime decomposition . the proof is constructive , exhibiting certain dominant maps as branched coverings . the second part of theorem [ t : topargument ] corresponds to the following statement : [ p : iness ] every rationally inessential three - manifold is dominated by a product . since we have shown in the proof of theorem [ t : ess ] that rationally inessential three - manifolds are finitely covered by connected sums of copies of @xmath24 , it suffices to prove the following : [ p : branchcov ] let @xmath79 be a closed , oriented surface of genus @xmath19 . for every @xmath19 the manifold @xmath80 is a @xmath45-surjective branched double covering of @xmath16 . @xmath81 at 300 87 @xmath82 at 590 87 .,title="fig:",width=453 ] the @xmath83-torus @xmath84 is a branched double covering of @xmath85 with four branch points . we denote this branched covering , which is the quotient map for an involution on @xmath84 , by @xmath86 ; see figure [ f : branchcov1 ] . ( the letter @xmath87 stands either for `` pillowcase '' , or for the weierstrass @xmath88-function . ) we multiply @xmath87 by the identity map on @xmath50 to obtain a branched double covering @xmath89 this is the case @xmath90 in the claim . at 251 143 , with two branched points , is an annulus in @xmath84.,title="fig:",width=302 ] now let @xmath91 be a @xmath83-ball in @xmath85 that contains exactly two branch points of @xmath87 in its interior , as shown in figure [ f : branchcov2 ] , and let @xmath92 be an interval in @xmath50 . the product @xmath93 is a @xmath94-ball @xmath95 in @xmath28 . the preimage of this ball under @xmath96 is @xmath97 where @xmath98 is an annulus in @xmath84 ; see figure [ f : branchcov2 ] . we remove this @xmath95 from @xmath28 and its preimage from @xmath99 to obtain a branched double covering @xmath100 where @xmath101 . taking the double of we obtain a branched double covering @xmath102 which is @xmath45-surjective by construction . this gives the case @xmath103 in the claim . finally note that , for arbitrary @xmath19 , the connected sum @xmath16 is an @xmath104-sheeted unramified covering of @xmath105 . taking the fiber product with , we obtain the desired @xmath45-surjective branched double covering of @xmath16 by @xmath80 . this completes the proof . proposition [ p : branchcov ] together with proposition [ p : ess ] completes the proof of theorem [ t : topargument ] . for theorem [ t : topbundle ] we need the following : [ p : inessbundles ] every rationally inessential three - manifold is dominated by a non - trivial circle bundle over a surface . this , together with proposition [ p : essbd ] , completes the proof of theorem [ t : topbundle ] . since every rationally inessential three - manifold is finitely covered by some @xmath16 by the proof of theorem [ t : ess ] , proposition [ p : inessbundles ] is a consequence of the following statement . [ p : branchcovbundles ] for every @xmath19 the connected sum @xmath16 admits a @xmath45-surjective branched double covering by a non - trivial circle bundle over a surface . for @xmath17 , the empty connected sum is , by convention , the three - sphere @xmath18 , which , via the hopf fibration , is a non - trivial circle bundle over @xmath85 . pulling back the hopf fibration under a branched double cover @xmath106 , we obtain the desired double branched cover of @xmath18 . for @xmath90 we prove that the total space @xmath2 of the circle bundle with euler number @xmath107 over @xmath84 is a @xmath45-surjective branched double covering of @xmath28 . start by considering @xmath2 as the mapping torus of the linear torus diffeomorphism given by the matrix @xmath108 and recall that the double branched cover @xmath109 in figure [ f : branchcov1 ] is the quotient map for the involution @xmath110 since @xmath111 commutes with @xmath112 , it induces a fiber - preserving involution , also denoted @xmath111 , of the mapping torus @xmath113 . the quotient @xmath114 is the mapping torus of the diffeomorphism of @xmath115 induced by @xmath112 . this diffeomorphism is orientation - preserving , and so @xmath116 . the projection @xmath117 given by the quotient map for @xmath111 is the desired @xmath45-surjective double branched cover . on every fiber it coincides with @xmath87 . to deal with the case @xmath118 , we revert to thinking of @xmath2 as a circle bundle over @xmath84 , and we fiber sum @xmath19 copies of this circle bundle to obtain a circle bundle with euler number @xmath119 over @xmath79 . we can perform this fiber sum in such a way that the branched double covering maps @xmath120 on the different summands fit together to give the desired @xmath45-surjective branched double covering of @xmath16 . recall that the circles of the circle fibration of @xmath2 over @xmath84 are contained in the fibers of the mapping torus projection @xmath121 . pick one such fiber , and thicken it to an annulus @xmath98 contained in a fiber of @xmath61 whose image under @xmath87 is a disk in @xmath85 containing precisely two branch points of @xmath87 , as shown in figure [ f : branchcov2 ] . a fibered neighbourhood of our circle fiber in @xmath2 is the product of @xmath98 with an interval in @xmath50 , and the image under @xmath120 of this fibered neighbourhood in @xmath28 is a three - ball @xmath95 . now we can perform the connected sum of two copies of @xmath28 along this @xmath95 , and simultaneously fiber sum two copies of @xmath2 by removing the fibered neighbourhood and gluing the boundary tori in a fiber - preserving way that matches up the branch loci . this completes the proof for @xmath103 , and the general case follows by iterating the construction . we now reformulate theorems [ t : topargument ] and [ t : topbundle ] and their proofs to obtain equivalent formulations in terms of thurston geometries and in terms of purely algebraic properties of fundamental groups . the following is the geometric reformulation of theorem [ t : topargument ] . [ t : geoargument ] a closed , oriented , connected three - manifold @xmath3 is dominated by a product @xmath13 if and only if 1 . either @xmath3 possesses one of the geometries @xmath122 or @xmath123 , or 2 . @xmath3 is a connected sum of manifolds possessing the geometries @xmath124 or @xmath18 . let @xmath3 be a closed oriented three - manifold dominated by a product @xmath13 . if the prime decomposition of @xmath3 contains an aspherical summand , then we have seen in the proof of theorem [ t : topargument ] that @xmath3 itself is aspherical , and is finitely covered by a product @xmath14 , with @xmath15 of positive genus . in addition , @xmath3 is seifert fibered since its finite covering @xmath14 is , cf . moreover , @xmath3 carries the same thurston geometry as this covering , namely the @xmath122 geometry if @xmath15 has genus one , or the @xmath123 geometry if the genus of @xmath15 is at least @xmath83 . conversely , every manifold with one of these geometries is indeed finitely covered , and , therefore , dominated by a product @xmath14 , cf . @xcite . if the prime decomposition of @xmath3 does not contain an aspherical summand , then each prime summand is either @xmath24 , with geometry @xmath125 , or has finite fundamental group , and thus carries the @xmath18 geometry by the work of perelman @xcite . for all connected sums with only these summands we have proved in the proof of theorem [ t : topargument ] that they are dominated by products . finally , we give an algebraic formulation , in terms of properties of the fundamental group of the target . [ t : algargument ] a closed , oriented , connected three - manifold @xmath3 is dominated by a product @xmath13 if and only if 1 . either @xmath39 is virtually @xmath126 , for some aspherical surface @xmath15 , or 2 . @xmath39 is virtually free . if @xmath3 is a closed oriented three - manifold dominated by a product @xmath13 , and the prime decomposition of @xmath3 contains an aspherical summand , then we have seen in the proof of theorem [ t : topargument ] that @xmath39 is virtually @xmath126 . conversely , if @xmath3 has a finite covering @xmath127 with fundamental group @xmath126 , then this covering is prime as its fundamental group is freely indecomposable . since @xmath15 is not @xmath85 , it follows that @xmath127 is irreducible and aspherical @xcite . thus @xmath127 is homotopy equivalent to @xmath14 , proving that @xmath3 is dominated by a product . if the prime decomposition of @xmath3 does not contain an aspherical summand , then we have seen that @xmath39 is virtually free . conversely , if @xmath3 has virtually free fundamental group , then it is finitely covered by a three - manifold with free fundamental group . kneser s prime decomposition theorem and grushko s theorem imply that this covering must be a connected sum of copies of @xmath28 , where the number of summands is the number of generators of its fundamental group . the analogous reformulations can also be carried out for theorem [ t : topbundle ] . the geometric formulation is : [ t : geobundle ] a closed , oriented , connected three - manifold @xmath3 is dominated by a non - trivial circle bundle over a surface if and only if 1 . either @xmath3 possesses one of the geometries @xmath128 or @xmath129 , or 2 . @xmath3 is a connected sum of manifolds possessing the geometries @xmath124 or @xmath18 . we only have to prove the equivalence between the first cases of this theorem and of theorem [ t : topbundle ] . in one direction , if @xmath3 has one of the geometries @xmath128 or @xmath129 , then it is finitely covered by a non - trivial circle bundle over an aspherical surface @xcite . conversely , if @xmath3 is finitely covered by a non - trivial circle bundle over an aspherical surface , then it is a seifert manifold carrying the same thurston geometry as this finite covering @xcite . the algebraic version of theorem [ t : topbundle ] reads as follows . [ t : algbundle ] a closed , oriented , connected three - manifold @xmath3 is dominated by a non - trivial circle bundle over a surface if and only if 1 . either @xmath39 has a finite index subgroup @xmath130 which fits into a central extension @xmath131 with non - zero euler class for some aspherical surface @xmath15 , or 2 . @xmath39 is virtually free . again we only have to prove the equivalence between the first cases of this theorem and of theorem [ t : topbundle ] . in one direction , if @xmath3 is finitely covered by a non - trivial circle bundle over an aspherical surface , then its fundamental group has a finite index subgroup admitting the required central extension . conversely , if @xmath39 has a finite index subgroup @xmath130 fitting into such a central extension , then the corresponding finite covering has to be prime , irreducible and aspherical , and is therefore homotopy equivalent to the total space of the corresponding circle bundle over @xmath15 . as an algebraic counterpart of our topological results about domination by products for three - manifolds we now want to determine which fundamental groups of three - manifolds are presentable by products . first we recall the definition : [ repgroupsdef](@xcite ) an infinite group @xmath130 is _ presentable by a product _ if there is a homomorphism @xmath132 onto a subgroup of finite index , such that both factors @xmath133 have infinite image @xmath134 . without loss of generality one can replace each @xmath133 by its image in @xmath130 under the restriction of @xmath112 , so that one can assume the factors @xmath133 to be subgroups of @xmath130 and @xmath112 to be multiplication in @xmath130 . it is obvious that a group with infinite center @xmath135 is presentable by a product just take @xmath136 and @xmath137 . the property of ( not ) being presentable by a product is preserved under passage to finite index subgroups . this property was introduced in @xcite and further studied in @xcite because , according to @xcite , it is a property that the fundamental groups of rationally essential manifolds dominated by products must have . for a closed three - manifold @xmath2 with infinite fundamental group the following three properties are equivalent : 1 . @xmath74 is presentable by a product , 2 . @xmath74 has a finite index subgroup with infinite center , 3 . @xmath2 is a seifert manifold . it is clear that ( 3 ) implies ( 2 ) . the converse is the celebrated seifert fiber space conjecture , the final cases of which were resolved by casson - jungreis @xcite and by gabai @xcite . as noted above , it is also clear that ( 2 ) implies ( 1 ) for any group . we now prove the converse for three - manifold groups . by ( 9.2 ) the only non - trivial free product that is presentable by a product is @xmath138 , which is virtually @xmath49 and so satisfies ( 2 ) . thus , we may assume that @xmath74 is freely indecomposable , and @xmath2 is prime . if @xmath74 is not virtually @xmath49 , then @xmath2 is irreducible and aspherical by the sphere theorem , cf . @xcite . in particular , @xmath74 is torsion - free . by ( 3.2 ) , a torsion - free group @xmath130 which is presentable by a product has one of the following properties : * either @xmath130 has a finite index subgroup with infinite center , or * some finite index subgroup splits as a direct product of infinite groups . applying this to our @xmath74 , we have to see that the second alternative in fact implies the first . it is a theorem of epstein @xcite that if the fundamental group of a closed three - manifold splits as a direct product of infinite groups , then one of the factors has to be infinite cyclic . but then this factor is central in the whole group . [ [ section ] ] the main result of @xcite was that for rationally essential manifolds , in any dimension , domination by a product implies that the fundamental group is presentable by a product . theorem [ t : topargument ] shows that the converse is not true already in dimension three : seifert manifolds carrying one of the geometries @xmath128 or @xmath129 are aspherical and have fundamental groups presentable by products , but are not dominated by products . ( these are the only counterexamples to the converse in dimension three . ) the geometry @xmath129 has another interesting feature relevant to our discussion : @xmath129 is quasi - isometric to @xmath123 , cf . * iv.48 ) . compact manifolds with the latter geometry are finitely covered by products , whereas those with the former geometry are not even dominated by products , although the fundamental groups are presentable by products in both cases . it was noted in ( * ? ? ? 10.2 ) that presentability by products is not a quasi - isometry invariant property of finitely generated groups . this , together with the contrast between manifolds with the geometries @xmath129 and @xmath123 , shows that domination by products can not be detected by coarse methods , neither at the level of groups nor at the level of universal coverings of aspherical manifolds . one of the standard characterizations of closed seifert manifolds is through the property of being finitely covered by circle bundles . in the rationally essential case this can be weakened by replacing finite coverings by arbitrary dominant maps : our discussion in section [ s : proof1 ] shows that there are no maps of non - zero degree between trivial and nontrivial circle bundles over aspherical surfaces . this statement already appeared in the work of wang twenty years ago , see ( * ? ? ? * theorem 2 ) . however , the proof given there is hard to follow . in particular , there is no argument there for the case covered by our lemma [ lem ] . at the corresponding place in the proof , compare @xcite , in particular equation ( iii ) , wang seems to argue that a group that is presentable by a product must itself be a product , which is of course false . the fundamental groups of seifert manifolds with non - zero euler number are presentable by products , but are not virtually products . j. hempel , _ residual finiteness for @xmath94-manifolds _ , in _ combinatorial groups theory and topology _ , ed . s. m. gersten and j. r. stallings , annals of math . studies vol . 111 , princeton univ . press 1987 .
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we determine which three - manifolds are dominated by products .
the result is that a closed , oriented , connected three - manifold is dominated by a product if and only if it is finitely covered either by a product or by a connected sum of copies of @xmath0 .
this characterization can also be formulated in terms of thurston geometries , or in terms of purely algebraic properties of the fundamental group .
we also determine which three - manifolds are dominated by non - trivial circle bundles , and which three - manifold groups are presentable by products .
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we address one of the fundamental problems in uncertainty quantification ( uq ) : the mapping of the probability distribution of a random variable through a nonlinear function . let us assume that we are concerned with a specific physical or engineering model which is computationally expensive . the model is defined by the map @xmath0 . it takes a parameter @xmath1 as input , and produces an output @xmath2 , @xmath3 . in this paper we restrict ourselves to a proof - of - principle one - dimensional case . let us assume that @xmath1 is a random variable distributed with probability density function ( pdf ) @xmath4 . the uncertainty quantification problem is the estimation of the pdf @xmath5 of the output variable @xmath6 , given @xmath4 . formally , the problem can be simply cast as a coordinate transformation and one easily obtains @xmath7 where @xmath8 is the jacobian of @xmath9 . the sum over all @xmath10 such that @xmath11 takes in account the possibility that @xmath12 may not be injective . if the function @xmath12 is known exactly and invertible , eq.([py ] ) can be used straightforwardly to construct the pdf @xmath13 , but this is of course not the case when the mapping @xmath12 is computed via numerical simulations . several techniques have been studied in the last couple of decades to tackle this problem . generally , the techniques can be divided in two categories : intrusive and non - intrusive @xcite . intrusive methods modify the original , _ deterministic _ , set of equations to account for the stochastic nature of the input ( random ) variables , hence eventually dealing with stochastic differential equations , and employing specific numerical techniques to solve them . classical examples of intrusive methods are represented by polynomial chaos expansion @xcite , and stochastic galerkin methods @xcite . on the other hand , the philosophy behind non - intrusive methods is to make use of the deterministic version of the model ( and the computer code that solves it ) as a black - box , which returns one deterministic output for any given input . an arbitrary large number of solutions , obtained by sampling the input parameter space , can then be collected and analyzed in order to reconstruct the pdf @xmath13 . the paradigm of non - intrusive methods is perhaps best represented by monte carlo ( mc ) methods @xcite : one can construct an ensemble of input parameters @xmath14 ( @xmath15 typically large ) distributed according to the pdf @xmath16 , run the corresponding ensemble of simulations @xmath17 , and process the outputs @xmath18 . mc methods are probably the most robust of all the non - intrusive methods . their main shortcoming is the slow convergence of the method , with a typical convergence rate proportional to @xmath19 . for many applications quasi - monte carlo ( qmc ) methods @xcite are now preferred to mc methods , for their faster convergence rate . in qmc the pseudo - random generator of samples is replaced by more uniform distributions , obtained through so - called quasi - random generators @xcite . it is often said that mc and qmc do not suffer the ` curse of dimensionality'@xcite , in the sense that the convergence rate ( but not the actual error ! ) is not affected by the dimension @xmath20 of the input parameter space . therefore , they represent the standard choice for large dimensional problems . on the other hand , when the dimension @xmath20 is not very large , collocation methods @xcite are usually more efficient . collocation methods recast an uq problem as an interpolation problem . in collocation methods , the function @xmath21 is sampled in a small ( compared to the mc approach ) number of points ( ` collocation points ' ) , and an interpolant is constructed to obtain an approximation of @xmath12 over the whole input parameter space , from which the pdf @xmath13 can be estimated . the question then arises on how to effectively choose the collocation points . recalling that every evaluation of the function @xmath12 amounts to performing an expensive simulation , the challenge resides in obtaining an accurate approximation of @xmath22 with the least number of collocation points . + as the name suggests , collocation methods are usually derived from classical quadrature rules @xcite . the type of pdf @xmath23 can guide the choice of the optimal quadrature rule to be used ( i.e. , gauss - hermite for a gaussian probability , gauss - legendre for a uniform probability , etc . @xcite ) . furthermore , because quadratures are associated with polynomial interpolation , it becomes natural to define a global interpolant in terms of a lagrange polynomial @xcite . also , choosing the collocation points as the abscissas of a given quadrature rule makes sense particularly if one is only interested in the evaluation of the statistical moments of the pdf ( i.e. , mean , variance , etc . ) @xcite . on the other hand , there are several applications where one is interested in the approximation of the full pdf @xmath22 . for instance , when @xmath12 is narrowly peaked around two or more distinct values , its mean does not have any statistical meaning . in such cases one can wonder whether a standard collocation method based on quadrature rules still represents the optimal choice , in the sense of the computational cost to obtain a given accuracy . from this perspective , a downside of collocation methods is that the collocation points are chosen a priori , without making use of the knowledge of @xmath21 acquired at previous interpolation levels . for instance , the clenshaw - curtis ( cc ) method uses a set of points that contains nested subset , in order to re - use all the previous computations , when the number of collocation points is increased . however , since the abscissas are unevenly spaced and concentrated towards the edge of the domain ( this is typical of all quadrature rules , in order to overcome the runge phenomenon @xcite ) , it is likely that the majority of the performed simulations will not contribute significantly in achieving a better approximation of @xmath22 . stated differently , one would like to employ a method where each new sampling point is chosen in such a way to result in the fastest convergence rate for the approximated @xmath22 , in contrast to a set of points defined a priori . as a matter of fact , because the function @xmath12 is unknown , a certain number of simulations will always be redundant , in the sense that they will contribute very little to the convergence of @xmath5 . the rationale for this work is to devise a method to minimize such a redundancy in the choice of sampling points while achieving fastest possible convergence of @xmath5 . clearly , this suggests to devise a strategy that chooses collocation points _ adaptively _ , making use of the knowledge of the interpolant of @xmath21 , which becomes more and more accurate as more points are added . a well known adaptive sampling algorithm is based on the calculation of the so - called hierarchical surplus ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * see e.g ) . this is defined as the difference , between two levels of refinement , in the solution obtained by the interpolant . although this algorithm is quite robust , and it is especially efficient in detecting discontinuities , it has the obvious drawback that it can be prematurely terminated , whenever the interpolant happens to exactly pass through the true solution on a point where the hierarchical surplus is calculated , no matter how inaccurate the interpolant is in close - by regions ( see figure [ fig : hierarchical_example ] for an example ) . the goal of this paper is to describe an alternative strategy for the adaptive selection of sampling points . the objective in devising such strategy is to have a simple and robust set of rules for choosing the next sampling point . the paper is concerned with a proof - of - principle demonstration of our new strategy , and we will focus here on one dimensional cases and on the case of uniform @xmath4 only , postponing the generalization to multiple dimensions to future work . it is important to appreciate that the stated goal of this work is different from the traditional approach followed in the overwhelming majority of works that have presented sampling methods for uq in the literature . indeed , it is standard to focus on the convergence of the nonlinear unknown function @xmath21 , trying to minimize the interpolation error on @xmath21 , for a given number of sampling points . on the other hand , we will show that the convergence rates of @xmath21 and of its cumulative distribution function can be quite different . our new strategy is designed to achieve the fastest convergence on the latter quantity , which is ultimately the observable quantity of an experiment . the paper is organized as follows . in section 2 we define the mathematical methods used for the construction of the interpolant and show our adaptive strategy to choose a new collocation points . in section 3 we present some numerical examples and comparisons with the clenshaw - curtis collocation method , and the adaptive method based on hierarchical surplus . finally , we draw our conclusions in section 4 . in section 3 , we compare our method with the cc method , which is the standard appropriate collocation method for a uniform @xmath23 . here , we recall the basic properties of cc , for completeness . the clenshaw - curtis ( cc ) quadrature rule uses the extrema of a chebyshev polynomial ( the so - called ` extrema plus end - points ' collocation points in @xcite ) as abscissas . they are particularly appealing to be used as collocation points in uq , because a certain subset of them are nested . specifically , they are defined , in the interval @xmath24 $ ] as : @xmath25 one can notice that the the set of @xmath26 points is fully contained in the set of @xmath27 points ( with @xmath28 an arbitrary integer , referred to as the level of the set ) . in practice this means that one can construct a nested sequence of collocation points with @xmath29 , re - using all the previous evaluations of @xmath12 . collocation points based on quadratures are optimal to calculate moments : on the left - hand side is a label , such that @xmath30 is the mean , @xmath31 is the variance , and so on . on the right - hand side it is an exponent . ] @xmath32 where we used the identity relation , @xmath33 it is known that integration by quadrature is very accurate ( for smooth enough integrand ) , and the moments can be readily evaluated , without the need to construct an interpolant : @xmath34 where the weights @xmath35 can be computed with standard techniques ( see , e.g. @xcite ) . the interpolant for the cc method is the lagrange polynomial . the hierarchical surplus algorithm is widely used for interpolation on sparse grids . it is generally defined as the difference between the value of an interpolant at the current and previous interpolation levels @xcite . the simplest algorithm prescribes a certain tolerance and looks for all the point at the new level where the hierarchical surplus is larger than the tolerance . the new sampling points ( at the next level ) will be the neighbors of the points where this condition is met . in one - dimension , the algorithm is extremely simple because the neighbors are only two points , that one can define in such a way that cells are always halved . in this work , we compare our new method with a slightly improved version of the hierarchical surplus algorithm . the reason is because we do not want our comparisons to be dependent on the choice of an arbitrary tolerance level , and we want to be able to add new points two at the time . hence , we define a new interpolation level by adding only the two neighbors of the point with the largest hierarchical surplus . all the previous hierarchical surpluses that have been calculated , but for which new points have not been added yet are kept . the pseudo - code of the algorithm follows . the interpolant is understood to be piece - wise linear interpolation , and the grid is @xmath36 $ ] . calculate the interpolant on the grid @xmath37 . define @xmath38 and add them on the grid calculate the interpolant on the new grid calculate the hierarchical surplus on the last two entries of @xmath39 and store them in the vector @xmath40 find the largest hierarchical surplus in @xmath40 , remove it from @xmath40 and remove the corresponding @xmath10 from @xmath41 append the two neighbors to @xmath41 and add them to the grid we use a multiquadric biharmonic radial basis function ( rbf ) with respect to a set of points @xmath42 , with @xmath43 , defined as : @xmath44 where @xmath45 are free parameters ( referred to as shape parameters ) . the function @xmath21 is approximated by the interpolant @xmath46 defined as @xmath47 the weights @xmath48 are obtained by imposing that @xmath49 for each sampling point in the set , namely the interpolation error is null at the sampling points . this results in solving a linear system for @xmath50 of the form @xmath51 , with @xmath52 a real symmetric @xmath53 matrix . we note that , by construction , the linear system will become more and more ill - conditioned with increasing @xmath15 , for fixed values of @xmath54 . this can be easily understood because when two points become closer and closer the corresponding two rows in the matrix @xmath52 become less and less linearly independent . to overcome this problem one needs to decrease the corresponding values of @xmath54 . in turns , this means that the interpolant @xmath46 will tend to a piece - wise linear interpolant for increasingly large @xmath15 . we focus , as the main diagnostic of our method , on the cumulative distribution function ( cdf ) @xmath55 , which is defined as @xmath56 where @xmath57 . as it is well known , the interpretation of the cumulative distribution function is that , for a given value @xmath58 , @xmath59 is the probability that @xmath60 . of course , the cdf @xmath55 contains all the statistical information needed to calculate any moment of the distribution , and can return the probability density function @xmath13 , upon differentiation . moreover , the cdf is always well defined between 0 and 1 . the following two straightforward considerations will guide the design of our _ adaptive selection strategy_. a first crucial point , already evident from eq . ( [ py ] ) , is whether or not @xmath21 is bijective . when @xmath21 is bijective this translates to the cdf @xmath55 being continuous , while a non - bijective function @xmath21 produces a cdf @xmath55 which is discontinuous . it follows that intervals in @xmath10 where @xmath21 is constant ( or nearly constant ) will map into a single value @xmath61 ( or a very small interval in @xmath62 ) where the cdf will be discontinuous ( or ` nearly ' discontinuous ) . secondly , an interval in @xmath10 with a large first derivative of @xmath21 will produce a nearly flat cdf @xmath55 . this is again clear by noticing that the jacobian @xmath63 in eq . ( [ py ] ) ( @xmath64 in one dimension ) is in the denominator , and therefore the corresponding @xmath13 will be very small , resulting in a flat cdf @xmath55 . + loosely speaking one can then state that regions where @xmath21 is flat will produce large jumps in the cdf @xmath55 and , conversely , regions where the @xmath21 has large jumps will map in to a nearly flat cdf @xmath55 . from this simple considerations one can appreciate how important it is to have an interpolant that accurately capture both regions with _ very large _ and _ very small _ first derivative of @xmath21 . moreover , since the cdf @xmath55 is an integrated quantity , interpolation errors committed around a given @xmath62 will propagate in the cdf for all larger @xmath62 values . for this reason , it is important to achieve a global convergence with interpolation errors that are of the same order of magnitude along the whole domain . + the adaptive section algorithm works as follows . we work in the interval @xmath36 $ ] ( every other interval where the support of @xmath21 is defined can be rescaled to this interval ) . we denote with @xmath42 the sampling set which we assume is always sorted , such that @xmath65 . we start with 3 points : @xmath66 , @xmath67 , @xmath68 . for the robustness and the simplicity of the implementation we choose to select a new sampling point always at equal distance between two existing points . one can decide to limit the ratio between the largest and smallest distance between adjacent points : @xmath69 ( with @xmath70 ) , where @xmath71 is the distance between the points @xmath72 and @xmath73 . this avoids to keep refining small intervals when large intervals might still be under - resolved , thus aiming for the above mentioned global convergence over the whole support . at each iteration we create a list of possible new points , by halving every interval , excluding the points that would increase the value of @xmath74 above the maximum desired ( note that @xmath74 will always be a power of 2 ) . we calculate the first derivative of @xmath46 at these points , and alternatively choose the point with largest / smallest derivative as the next sampling point . notice that , by the definition of the interpolant , eq . ( [ interpolant ] ) , its first derivative can be calculated exactly as : @xmath75 without having to recompute the weights @xmath48 . at each iteration the shape parameters @xmath45 are defined at each points , as @xmath76 , i.e. they are linearly rescaled with the smallest distance between the point @xmath73 and its neighbors . the pseudo - code of the algorithm follows . @xmath77 exclude points in @xmath78 such that @xmath79 calculate @xmath80 through ( [ first_derivative ] ) at @xmath81 alternatively choose @xmath78 with largest / smallest values of @xmath82 as new collocation point calculate new weights @xmath48 in this section we present and discuss four numerical examples where we apply our adaptive selection strategy . in this work we focus on a single input parameter and the case of constant probability @xmath83 in the interval @xmath84 $ ] , and we compare our results against the clenshaw - curtis , and the hierarchical surplus methods . we denote with @xmath85 the interpolant obtained with a set of @xmath86 points ( hence the iterative procedure starts with @xmath87 ) . a possible way to construct the cdf @xmath55 from a given interpolant @xmath85 would be to generate a sample of points in the domain @xmath24 $ ] , randomly distributed according to the pdf @xmath16 , collecting the corresponding values calculated through eq . ( [ interpolant ] ) , and constructing their cdf . because here we work with a constant @xmath16 , it is more efficient to simply define a uniform grid in the domain @xmath88 $ ] where to compute @xmath85 . in the following we will use , in the evaluation of the cdf @xmath55 , a grid in @xmath89 with @xmath90 points equally spaced in the interval @xmath91 $ ] , and a grid in @xmath92 with @xmath93 points equally spaced in the interval @xmath24 $ ] . we define the following errors : @xmath94 where @xmath95 denotes the l@xmath96 norm . it is important to realize that the accuracy of the numerically evaluated cdf @xmath55 will always depend on the binning of @xmath97 , i.e. the points at which the cdf is evaluated . as we will see in the following examples , the error @xmath98 saturates for large @xmath99 , which thus is an artifact of the finite bin size . we emphasize that , differently from most of the previous literature , our strategy focuses on converging rapidly in @xmath100 , rather than in @xmath101 . of course , a more accurate interpolant will always result in a more accurate cdf , however the relationship between a reduction in @xmath102 and a corresponding reduction in @xmath98 is not at all trivial . this is because the relation between @xmath16 and @xmath13 is mediated by the jacobian of @xmath21 , and it also involves the bijectivity of @xmath12 . + finally , we study the convergence of the mean @xmath103 , see equation [ eq : mean ] , and the variance @xmath104 , which is defined as @xmath105 these will be calculated by quadrature for the cc methods , and with an integration via trapezoidal method for the adaptive methods . + we study two analytical test cases : * case 1 : @xmath106 ; * case 2 : @xmath107 ; and two test cases where an analytical solution is not available , and the reference @xmath21 will be calculated as an accurate numerical solution of a set of ordinary differential equations : * case 3 : lotka - volterra model ( predator - prey ) ; * case 4 : van der pol oscillator . while case 1 and 2 are more favorable to the cc method , because the functions are smooth and analytical , hence a polynomial interpolation is expected to produce accurate results , the latter two cases mimic applications of real interest , where the model does not produce analytical results , although @xmath21 might still be smooth ( at least piece - wise , in case 4 ) . in this case @xmath21 is a bijective function , with one point ( @xmath108 ) where the first derivative vanishes . figure [ fig : case_1_f ] shows the function @xmath21 ( top panel ) and the corresponding cdf @xmath55 ( bottom panel ) , which in this case can be derived analytically . hence , we use the analytical expression of cdf @xmath55 to evaluate the error @xmath98 . the convergence of @xmath98 and @xmath102 is shown in figure [ fig : case_1_err ] ( top and bottom panels , respectively ) . here and in all the following figures blue squares denote the new adaptive selection method , red dots are for the cc methods , and black line is for the hierarchical surplus method . we have run the cc method only for @xmath109 ( i.e. the points at which the collocation points are nested ) , but for a better graphical visualization the red dots are connected with straight lines . one can notice that the error for the new adaptive method is consistently smaller than for the cc method . from the top panel , one can appreciate the saving in computer power that can be achieved with our new method . although the difference with cc is not very large until @xmath110 , at @xmath111 there is an order of magnitude difference between the two . it effectively means that in order to achieve the same error @xmath112 , the cc method would run at least twice the number of simulations . the importance of focusing on the convergence of the cdf , rather than on the interpolant , is clear in comparing our method with the hierarchical surplus method . for instance , for @xmath113 , the two methods have a comparable error @xmath101 , but our method has achieved almost an order of magnitude more accurate solution in @xmath55 . effectively , this means that our method has sampled the new points less redundantly . in this case @xmath21 is an anti - symmetric function with zero mean . hence , any method that chooses sampling points symmetrically distributed around zero would produce the correct first moment @xmath103 . we show in figure [ fig : case_1_sigma ] the convergence of @xmath114 , as the absolute value of the different with the exact value @xmath115 , in logarithmic scale . blue , red , and black lines represent the new adaptive method , the cc , and the hierarchical surplus methods , respectively ( where again for the cc , simulations are only performed where the red dots are shown ) . the exact value is @xmath116 . as we mentioned , the cc method is optimal to calculate moments , since it uses quadrature . although in our method the error does not decrease monotonically , it is comparable with the result for cc . in this case the function @xmath21 is periodic , and it presents , in the domain @xmath36 $ ] three local minima ( @xmath117 ) and three local maxima ( @xmath118 ) . the function and the cdf @xmath55 are shown in figure [ fig : case_2_f ] ( top and bottom panel , respectively ) . figure [ fig : case_2_err ] shows the error for this case ( from now on the same format of figure [ fig : case_1_err ] will be used ) . the first consideration is that the hierarchical surplus method is the less accurate of the three . second , @xmath102 is essentially the same for the cc and the new method , up to @xmath119 . for @xmath120 the cc methods achieve a much accurate solution as compared to the new adaptive method , whose error has a much slower convergence . however , looking at the error in the cdf in top panel of figure [ fig : case_2_err ] , the two methods are essentially equivalent . this example demonstrates that , in an uq framework , the primary goal in constructing a good interpolant should not be to minimize the error of the interpolant with respect to the true @xmath21 , but rather to achieve the fastest possible convergence on the cdf @xmath121 . although , the two effects are intuitively correlated , they are not into a linear relationship . in other words , not all sample points in @xmath10 count equally in minimizing @xmath98 . the convergence of @xmath122 ( exact value @xmath123 ) and @xmath114 ( exact value @xmath124 ) is shown in figures [ fig : case_2_mu ] and [ fig : case_2_sigma ] , respectively . it is interesting to notice that our method presents errors that are always smaller than the cc method , although the errors degrade considerably in the regions between two cc points , where the two adaptive methods yield comparable results . the lotka - volterra model @xcite is a well - studied model that exemplifies the interaction between two populations ( predators and preys ) . this case is more realistic than cases 1 and 2 , as the solution of the model can not be written in analytical form . as such , both the @xmath21 and the cdf @xmath55 used to compute the errors are calculated numerically . we use the following simple model : @xmath125 where @xmath126 and @xmath127 denote the population size for each species ( say , horses and lions ) as function of time . the ode is easily solved in matlab , with the ` ode45 ` routine , with an absolute tolerance set equal to @xmath128 . we use , as initial conditions , @xmath129 , and we solve the equations for @xmath130 $ ] . clearly , the solution of the model depends on the input parameter @xmath10 . we define our test function @xmath21 to be the result of the model for the @xmath131 population at time @xmath132 : @xmath133 the resulting function @xmath21 , and the computed cdf @xmath55 are shown in figure [ fig : case_3_f ] ( top and bottom panel , respectively ) . we note that , although @xmath21 can not be expressed as an analytical function , it is still smooth , and hence it does not present particular difficulties in being approximated through a polynomial interpolant . indeed the error @xmath102 undergoes a fast convergence both for the adaptive methods and for the cc method ( figure [ fig : case_3_err ] ) . once again , the new adaptive method is much more powerful than the cc method in achieving a better convergence rate , and thus saving computational power , while the hierarchical surplus method is the worst of the three . convergence of @xmath122 and @xmath114 are shown in figures [ fig : case_3_mu ] and [ fig : case_3_sigma ] , respectively . similar to previous cases , the cc presents a monotonic convergence , while this is not the case for the adaptive methods . only for @xmath120 , the cc method yields much better results than the new method . our last example is the celebrated van der pol oscillator@xcite , which has been extensively studied as a textbook case of a nonlinear dynamical system . in this respect this test case is very relevant to uncertainty quantification , since real systems often exhibit a high degree of nonlinearity . similar to case 3 , we define our test function @xmath21 as the output of a set of two odes , which we solve numerically with matlab . the model for the van der pol oscillator is : @xmath134 the initial conditions are @xmath135 , @xmath136 . the model is solved for time @xmath137 $ ] , and the function @xmath21 is defined as @xmath138 the so - called nonlinear damping parameter is rescaled such that for @xmath36 $ ] , it ranges between 50 and 250 . the function @xmath21 and the corresponding cdf @xmath55 are shown in figure [ fig : case_4_f ] . this function is clearly much more challenging than the previous ones . it is divided in two branches , where it takes values @xmath139 and @xmath140 , and it presents discontinuities where it jumps from one branch to the other . correspondingly , cdf @xmath55 presents a flat plateau for @xmath141 , which is the major challenge for both methods . in figure [ fig : case_4_err ] we show the errors @xmath102 and @xmath98 . the overall convergence rate of the cc and the new method is similar . for this case , the hierarchical surplus method yields a better convergence , but only for @xmath142 . as we commented before , the mean @xmath122 has no statistical meaning in this case , because the output is divided into two separate regions . the convergence for @xmath114 is presented in figure [ fig : case_4_sigma ] . we have presented a new adaptive algorithm for the selection of sampling points for non - intrusive stochastic collocation in uncertainty quantification ( uq ) . the main idea is to use a radial basis function as interpolant , and to refine the grid on points where the interpolant presents large and small first derivative . + in this work we have focused on 1d and uniform probability @xmath16 , and we have shown four test cases , encompassing analytical and non - analytical smooth functions , which are prototype of a very wide class of functions . in all cases the new adaptive method improved the efficiency of both the ( non - adaptive ) clenshaw - curtis collocation method , and of the adaptive algorithm based on the calculation of the hierarchical surplus ( note that the method used in this paper is a slight improvement of the classical algorithm ) . the strength of our method is the ability to select a new sampling point making full use of the interpolant resulting from all the previous evaluation of the function @xmath21 , thus seeking the most optimal convergence rate for the cdf @xmath55 . we have shown that there is no one - to - one correspondence between a reduction in the interpolation error @xmath102 and a reduction in the cdf error @xmath98 . for this reason , collocation methods that choose the distribution of sampling points a priori can perform poorly in attaining a fast convergence rate in @xmath98 , which is the main goal of uq . moreover , in order to maintain the nestedness of the collocation points the cc method requires larger and larger number of simulations ( @xmath143 moving from level @xmath28 to level @xmath144 ) , which is in contrast with our new method where one can add one point at the time . + we envision many possible research directions to further investigate our method . the most obvious is to study multi - dimensional problems . we emphasize that the radial basis function is a mesh - free method and as such we anticipate that this will largely alleviate the curse of dimensionality that afflicts other collocation methods based on quadrature points ( however , see @xcite for methods related to the construction of sparse grids , which have the same aim ) . moreover , it will be interesting to explore the versatility of rbf in what concerns the possibility of choosing an optimal shape parameter @xmath54 @xcite . recent work @xcite investigated the role of the shape parameter @xmath54 in interpolating discontinuous functions , which might be very relevant in the context of uq , when the continuity of @xmath21 can not be assumed a priori . finally , a very appealing research direction , would be to simultaneously exploit quasi - monte carlo and adaptive selection methods for extremely large dimension problems . a. a. and c. r. are supported by fom project no . 12cser058 and 12pr304 , respectively . we would like to remember dr.ir . j.a.s . witteveen ( @xmath145 2015 ) for the useful discussions we had about uncertainty quantification . 43 natexlab#1#1[1]`#1 ` [ 2]#2 [ 1]#1 [ 1]http://dx.doi.org/#1 [ ] [ 1]pmid:#1 [ ] [ 2]#2 , , ( ) . , , , ( ) . , , , , , , in : , , pp . . , , , ( ) . , , , , ( ) . , , , , in : , gcms 11 , , , , pp . . , , , ( ) . , , , . , , , , . , , ( ) . , , , , . , , , , . , , , ( ) . , , ( ) . , , , in : , , , pp . . , , , ( ) . , , , ( ) . , , , ( ) . , , , , ( ) . , , , ( ) . , , ( ) . , , ( ) . , , , ( ) . , , , ( ) . , , , in : , , , pp . . , , , ( ) . , , ( ) . , , , ( ) . , , in : , , , pp . . , , ( ) . , , , ( ) . , , , , , , ( ) . , , , , ( ) , , , ( ) . , , , ( ) . , , , ( ) . , , , ( ) . ( in black ) goes exactly through the red straight line at the points @xmath146 . calculating the piece - wise linear interpolant between two ( @xmath147 ) , three ( @xmath148 ) , and five ( @xmath149 ) points would result in a null hierarchical surplus on these points.,width=10 ] ( top ) @xmath102 ( bottom ) as function of number of sampling points @xmath15 . blue squares : new adaptive selection method . red dots : clenshaw - curtis . black curve : adaptive method based on hierarchical surplus.,width=10 ] ( top ) @xmath102 ( bottom ) as function of number of sampling points @xmath15 . blue squares : new adaptive selection method . red dots : clenshaw - curtis . black curve : adaptive method based on hierarchical surplus.,width=10 ] ( top ) @xmath102 ( bottom ) as function of number of sampling points @xmath15 . blue squares : new adaptive selection method . red dots : clenshaw - curtis . black curve : adaptive method based on hierarchical surplus.,width=10 ] ( top ) @xmath102 ( bottom ) as function of number of sampling points @xmath15 . blue squares : new adaptive selection method . red dots : clenshaw - curtis . black curve : adaptive method based on hierarchical surplus.,width=10 ]
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we present a simple and robust strategy for the selection of sampling points in uncertainty quantification .
the goal is to achieve the fastest possible convergence in the cumulative distribution function of a stochastic output of interest .
we assume that the output of interest is the outcome of a computationally expensive nonlinear mapping of an input random variable , whose probability density function is known .
we use a radial function basis to construct an accurate interpolant of the mapping .
this strategy enables adding new sampling points one at a time , _
adaptively_. this takes into full account the previous evaluations of the target nonlinear function .
we present comparisons with a stochastic collocation method based on the clenshaw - curtis quadrature rule , and with an adaptive method based on hierarchical surplus , showing that the new method often results in a large computational saving .
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dielectric spectroscopy has provided many informations to the investigation of complex materials , and especially to the field of glass transition phenomenon ( glass - forming systems ) during the last decades . this technique gives direct access to the polarization of molecular dipoles allowing the study of relaxation processes . it can probe the molecular structure and dynamics of several materials like liquids , polymer composites , colloids , porous materials , or ferroelectric crystals @xcite . it covers a wide range of frequency , typically from @xmath0 to @xmath2 hz , going up to @xmath3 hz with coaxial techniques @xcite . dielectric liquids of small molecules like the glass - forming glycerol @xcite or glass - forming polymers @xcite have been extensively studied . dielectric spectroscopy gives direct access to relaxation times@xcite like rheology , but it is a rotational relaxation one , not a translational . nevertheless , it has been showed that on molecular liquids at equilibrium , these two relaxation times have the same temperature dependence@xcite , at least above the glass transition . recent progress in the glass transition were provided by direct measurement of a dynamical growing length scale@xcite . this technique has always been improved , e.g. , the important work done regarding the electrodes in order to measure charged liquids even at low frequency @xcite . dielectric spectroscopy has also been coupled to scattering techniques such as x - rays @xcite or neutrons @xcite . furthermore , recent theories on aging and out of equilibrium physics predict interesting low frequency properties of the response and fluctuations , either during the relaxation of the system towards equilibrium after a quench of the control parameter @xcite , or when the system is driven by external fields @xcite . in order to test these models it is important to measure the response of the system as a function of time after the beginning of the time dependent phenomenon@xcite . single frequency measurements will force to repeat the experiment many times for each selected frequency making the experiment extremely long and difficult . in those cases multi - frequency measurements are absolutely necessary . the standard technique to perform multi - frequency response functions is to drive the system with a white noise signal . the main disadvantage of this approach is to require long averages before reaching reliable response measurements ( see section [ procedure ] ) . indeed , the lower frequencies need long time sampling . to overcome this problem we present an innovative setup that allows simultaneous impedance measurements at several frequencies over a typical range of @xmath4 hz with low noise and high sensibility . this has been obtained with a careful choice of the input signal , of the current amplifier , and an appropriate correction , which allow us to measure high capacitances with very small losses ( > @xmath5 is the typical order of magnitude ) . this means that our system is particularly useful for the study of relaxation processes in polymer films which have very high resistivity . the length of the measurement time window is imposed only by the smallest frequency that one needs to resolve because all other frequencies are measured simultaneously . the paper is organized as follows . in section [ bkg ] we give some general , basic background on the dielectric spectroscopy . section [ principle ] describes the experimental principle , [ setup ] the experimental setup , and [ procedure ] the measurement procedure . application of the technique to the study of some polymer films is presented in section [ samples ] . we conclude in section [ sumcon ] . dielectric spectroscopy investigates the frequency dependent dielectric properties of a material @xcite . it is based on the interaction of an external field with the electric dipole moment of the sample , often expressed by the complex dielectric permittivity or dielectric constant : @xmath6 where @xmath7 is the dielectric displacement and @xmath8 the electric field , both dependent on the angular frequency @xmath9 . @xmath10 can be expressed as @xmath11 being @xmath12 the stored permittivity , which is related to the stored energy by the sample , and @xmath13 the loss permittivity , which indicates the loss energy . their loss tangent is @xmath14 where @xmath15 is the phase shift between @xmath7 and @xmath8 . the dielectric constant @xmath16 depends on the dielectric material , the frequency , the temperature , age . the orientation polarization of dipoles due to applied disturbances allows the investigation of the dielectric relaxation of the material in the frequency range below @xmath17hz . moreover , if those dipoles are of molecular origin , the dielectric spectroscopy allows to investigate molecular motions . the experimental techniques and procedures for dielectric spectroscopy depend on the frequency range of interest . at low and intermediate frequencies ( @xmath18 hz ) the sample can be modeled as a parallel plate capacitor where the sample is the dielectric insulator in between the two electrodes @xcite . assuming no edge capacitances , the capacitance of an ideal parallel plate capacitor reads @xmath19 where @xmath20 is the distance between electrodes , @xmath21 the surface occupied by the sample and @xmath22 the vacuum permittivity . using eq . [ eq - permittivity2 ] in eq . [ eq - permittivity1 ] , we see that the impedance @xmath23 of the sample is modeled by a resistor @xmath24 in parallel with a capacitor @xmath25 and it can be expressed as : @xmath26 with @xmath27 and @xmath28 . the complex impedance @xmath23 can be measured by imposing the voltage @xmath29 and measuring the current @xmath30 flowing through : @xmath31 as @xmath32 is low on pure dielectric systems this means that the resistive part of the impedance is quite high and the in - phase current becomes hardly detectable . so far we described the ideal capacitor , but in the real experiments there are other parameters to take into account . on the one hand , corrections due to the measurement system like the influence of the extra capacitance in the cables . on the other hand , sample preparation must be very careful to avoid difficulties , such as sample geometry , bad electric contacts between electrodes and sample , edge capacitances , or electrode polarization . sample geometry has to be well known . we do not discuss these standard experimental details but we focus on the description of the features of the electronics , the driving signal , and the data analysis that we use in our experimental setup . our goal is to measure @xmath10 simultaneously in a frequency range covering 4 orders of magnitude in @xmath9 . as dielectric losses are in general small in pure dielectric ( such as polymers ) , any spurious effect must be taken into account in order to have a good accuracy on the wide frequency range . to define notation and to clarify the principle of our technique let us recall basic electronic concepts . in order to measure @xmath33 we use a current amplifier whose entrance impedance is just the sample impedance @xmath23 as sketched in fig . [ fig - circuit ] . in order to analyze the drawbacks of this scheme let scheme of the working principle of the main amplifier of our device . ] us assume first that the amplifier u in fig . [ fig - circuit ] is an ideal one . thus , we suppose that the bias currents at both entrances are zero ( @xmath34 ) , and that the open - loop amplification of u is infinite for all frequencies . this implies that @xmath35 because the positive entrance is at the ground potential . therefore , the impedance @xmath36 ( given by the parallel of the capacitances of the cables and of the amplifier entrance ) does not play any role in this case . from the standard circuit analysis , the current balance between the current flowing in the feedback impedance @xmath37 ( @xmath38 ) and the entrance current we obtain @xmath39 where @xmath40 is the reference and @xmath41 the output potentials ( see fig . [ fig - circuit ] ) . we have omitted the obvious dependence on @xmath9 . thus , the impedance @xmath42 can be calculated as @xmath43 nevertheless , considering a real amplifier , ( i.e. with a finite open loop gain ) one has to take into account in the current balance ( fig . [ fig - circuit ] ) that @xmath44 . thus , @xmath45 can not be neglected and the current balance is : @xmath46 where we have considered that the random output current @xmath47 takes generically into account all the sources of noise produced by the entrance bias currents , the voltage noise , and the component thermal noise . in order to get @xmath42 we first multiply eq . [ eq - real ] by the conjugate complex @xmath48 of @xmath49 and we take an average over several realizations : @xmath50 where we took into account that @xmath47 is uncorrelated with @xmath40 thus @xmath51 . we consider the transfer functions of both the signal and the correction : @xmath52 @xmath53 using eq . [ eq - real_2 ] , and the definitions of @xmath54 and @xmath55 , the impedance of the sample can be calculated as @xmath56 the knowledge of @xmath57 is very important in order to have reliable estimations of the dielectric constant at relatively high frequencies when the amplifier open loop gain begin to decrease and the phase shifts are not negligible with respect to the one that we want to measure . in our experimental setup we actually measure @xmath57 as we describe in the next section . the real scheme of our device is presented in fig . [ circuitall ] . the characteristics of each component are presented in table [ table1 ] . the source signal @xmath58 is sent to a first operational amplifier @xmath59 . this amplifier is a voltage follower or a non - inverting buffer . this leads to decouple the impedance of the source and the sample with a negligible additional noise . for that , we choose the lme 49990 amplifier , which has very low voltage noise . the output of @xmath59 is sent to the sample via @xmath60 which is the equivalent to @xmath61 in fig . [ fig - circuit ] . note that @xmath62 is very useful in order to avoid self oscillations of the operational amplifier @xmath59 when large @xmath63 are used ( see appendix[app ] ) . the signal sent to the sample is @xmath40 as defined in the previous section . .[table1 ] technical details of the components of the circuit of fig . [ circuitall ] . u are the different amplifiers of types lme 49990 ( low voltage noise , 1.4 nv/@xmath64 at 10 hz ) or ad549 ( low current noise , 0.5@xmath65a@xmath66 , @xmath67 to 10 hz , bias current 30 fa ) ; r are resistances . [ cols="^,^,^,^ " , ] scheme of our real circuit . @xmath59 , @xmath68 , and @xmath69 are low noise voltage amplifiers . @xmath70 and @xmath71 are low noise current amplifiers . details of the components are on table [ table1 ] . ] @xmath40 is also sent to a secondary buffer amplifier @xmath68 ( lme 49990 ) , and the output of this amplifier is used as the reference signal @xmath72 ( assuming @xmath73 because the phase - shift of @xmath68 is negligible in our frequency range ) . the current flowing through the sample can be very small if the sample impedance is very high so this part of the circuit uses amplifiers with a low current noise and bias current ( ad549 ) . there are two acquisition paths : in the main one , the voltage from the sample goes through a first amplifier @xmath70 with @xmath74 . this amplifier is the main one of the circuit and is equivalent to the one modeled in fig . [ fig - circuit ] and by eq . [ eq - real ] . it is followed by @xmath69 ( lme 49990 ) which amplifies the voltage by a factor 4 taken into account in data treatment . the second path is the measurement of the negative entrance potential of amplifier @xmath75 , that is to say @xmath57 in eq . [ eq - real ] . this signal also goes through a buffer amplifier @xmath71 and is amplified 4 times by @xmath76 , getting @xmath77 ( i.e. @xmath78 ) . in order to accurately measure impedances , we have to take into account all the contributions from the different components in the electronic device . the impedance of the cables @xmath45 between the sample and the amplifiers has to be considered @xmath79 where @xmath80 f is the capacitance of 0.5 m long cables we used . the entrance capacitance @xmath81 f of @xmath75 can be neglected . we made careful choice of good quality cables between the sample and the amplifier other cables have less sensitive contribution to the noise in the circuit in order to reduce noise in the signal . we also have to take into account the feedback impedance of the amplifiers @xmath82 where @xmath83 is the feedback resistance and @xmath84 f its parallel capacitance ( fig . [ fig - circuit ] ) . this capacitance is negligible compared to the impedance of @xmath85 in our frequency range . the signals @xmath72 , @xmath77 , and @xmath86 are sent to a national instrument pxi-4472 acquisition card ( 24 bits resolution ) . data acquisition is performed via a labview home - made program and data treatment using matlab to compute the transfer functions @xmath87 ( eq . [ eq - transferinput ] ) and @xmath88 ( eq . [ eq - transfercorr ] ) using standard fft algorithm . the transfer functions of the three amplifiers composed by @xmath68 , @xmath69 , and ( @xmath68+@xmath76 ) are measured in order to correct the phase of the three measured signals . in order to perform a simultaneous and precise measurement of @xmath42 in a wide frequency range using the device described in the previous section we have optimized the frequency dependence of the signal @xmath40 . indeed the standard technique employed to perform a response measurement on a wide band is to excite the system with an almost white noise in the band of interest and to measure simultaneously the input and output signals . then , by taking fft of these two signals input and output , one can compute the transfer function @xmath54 by a very long average of the response function on many configurations of the noise . this system although very practical and adaptable on almost all cases presents several drawbacks in the case of a circuit like the one presented in the previous sections . the first is that in the circuit of fig . [ circuitall ] the high frequencies are amplified much more than the low ones . therefore , one has to compensate for this effect . the second drawback is using a random noise at the input , its phase is rapidly changing and many averages are required in order to smooth this phase noise and to have a reliable measurement . in order to avoid these difficulties and reduce the number of averages in our system we apply the following method . we first fix a measurement interval @xmath89 and a sampling frequency @xmath90 , where @xmath91 is an integer . then , we generate a signal composed by a sum of sinusoids whose pulsations @xmath92 are harmonics of the slowest frequency @xmath93 , specifically @xmath94 where @xmath95 are the amplitudes in volts . the amplitudes are chosen to be @xmath96 because favoring low frequencies allows us to compensate the increasing gain of the amplifier at high frequencies . these kinds of signal are numerically generated and recorded in the computer memory which is then sampled by the same clock at the sampling frequency @xmath97 , which is also used for sampling the acquired signals @xmath40 , @xmath41 , and @xmath57 . in such a way all phase noises are suppressed as each frequency contains a fix integer number of acquired samples . we use two different devices to generate the signal either a national instrument pxi 5411-card generator or an agilent 33500b series waveform generator . as an example we plot in fig . [ fig - psdfreq ] the spectra of @xmath72 , @xmath86 , and @xmath77 signals obtained using @xmath98 f. in this specific example we select twenty - five frequencies @xmath99 in a logarithmic scale between a minimum value of @xmath100 hz and @xmath101 hz . the sampling rate is @xmath102 sample / s . the peak of the noise at about 3 khz is explained in the appendix[app ] . notice that the amplitude of the correction signal is not negligible above @xmath103 hz ( fig . [ fig - psdfreq]c ) and it is important to keep it into account in the calculation of @xmath42 . to summarize , this electronics combined with the described data analysis can measure with a good accuracy @xmath104 and @xmath105 f. , ( b ) @xmath86 , and ( c ) @xmath77 signals . circles point to the chosen frequencies . ] pf(@xmath106 ) , @xmath107 nf(@xmath108 ) and @xmath109 nf ( @xmath110 ) are plotted . the losses of the @xmath111 pf(@xmath106 with error bars ) and @xmath107nf@xmath112 capacitances are plotted in ( b ) , whereas those of the @xmath109 nf in fig.[fig - capatest]b . the capacitance of @xmath111 pf has very small losses and this allows us to fix the limits and the losses of our apparatus ( blue line with error bars in b ) . in ( b ) we also plot the value of two pure resistances of @xmath113 m@xmath114 ( @xmath115)and @xmath116 g@xmath117 . these two resistances have of course a small parallel capacitance of about @xmath118 pf shown in ( a ) @xmath119 and @xmath120 , which also fixes the accuracy of our system . in ( b ) we also plot the measured resistance ( @xmath121 ) when @xmath42 is the parallel of the @xmath107nf capacitance with the @xmath113 m@xmath114 resistance . notice that for each @xmath42 the whole frequency interval is covered by only two simultaneous measurements in two overlapping four order of magnitude intervals : [ @xmath122 hz ] and [ @xmath123 hz ] ] nf capacitance of fig.[fig_all_z]a . its losses are plotted in ( b@xmath110 ) together with the pure resistance of @xmath116 g@xmath117 . we also plot the measured total resistance @xmath106 of @xmath42 composed by the parallel of the @xmath109 nf with the @xmath116 g@xmath114 resistance . the dashed line represent the results of the measurement without using the correction signal @xmath57 in the calculation . the whole measurement in the range 0.1 hz-1000 hz takes less than 60 s ( i.e. 6 periods of the smallest frequency ) . ] from the measured impedance ( eq . [ eq - impedance ] ) we can calculate the resistance and the capacitance of the dielectric as @xmath124^{-1 } \label{eq - resistance}\ ] ] @xmath125 in order to check the efficiency of our device in the chosen frequency window , we use several impedances @xmath42 whose values have been checked by standard apparatuses @xcite . in fig.[fig_all_z]a we plot the measured values of the capacitances and in fig.[fig_all_z ] b ) those of the measured resistances . this plot illustrates the whole dynamics of our system ranging from @xmath107pf ( at @xmath118pf the error is large but the measurement can be still used ) to @xmath109 nf and larger if a smaller @xmath126 is used @xcite . the @xmath111 pf capacitance is an air capacitance with extremely small losses and this allows us to determine the losses of our circuit as a function of frequency which are plotted in fig.[fig_all_z]b . in order to appreciate more precisely the accuracy we plot in fig.[fig - capatest]a ) the measured values of the @xmath109 nf capacitance and fig.[fig - capatest]b its losses in the frequency range @xmath127 . we see that the fluctuations of the measurement are less than @xmath128 in all the frequency range . in fig.[fig - capatest]b we compare the losses of the @xmath109 nf capacitance with a measured resistance @xmath129 g@xmath114 . the capacitance @xmath63 has an almost constant value , whereas its losses increase , which correspond to a decrease of @xmath24 as a function of @xmath9 ( see eq.[eq - losses ] and fig.[fig - capatest]b ) . when @xmath63 is mounted in parallel to @xmath130 then the capacitance @xmath63 is unchanged whereas the measured resistance is now the parallel of @xmath131 which is constant at low frequencies up to around 10 hz , and then , decreases as a function of the increasing frequency due to the capacitive losses @xcite . it is important to stress that , using our method , we need to average less than @xmath132 s , in order to measure simultaneously the values of @xmath133 and @xmath63 on 4 orders of magnitude in frequency , plotted in fig . [ fig - capatest ] . as already mentioned , standard techniques use a white noise in the bandwidth of interest . by applying such a signal to our circuit , the result is much more noisy ( even impossible to find correct values ) than that obtained using our specifically chosen input signal described in the previous section ( see eq.[eq - sin ] ) . in fig . [ fig - capatest ] we can also see the importance of the correction signal @xmath57 ( see fig . [ fig - psdfreq]c ) in order to properly calculate the values of @xmath133 and @xmath63 in the whole chosen bandwidth . finally we stress that because of the limited memory of our voltage generator we always measure simultaneously at most 4 order of magnitude in frequency in the range @xmath134 . furthermore the accuracy is not constant as a function of frequency . it is @xmath135 below @xmath118 hz and above @xmath107 khz about @xmath136 in the range @xmath127 in the case of a pure capacitance , from the values of @xmath133 and @xmath63 we can calculate both @xmath137 and @xmath32 : @xmath138 @xmath139 with @xmath140 . therefore , @xmath141 ( eq . [ eq - tandelta ] ) can be calculated as @xmath142 which is independent of the geometry of the sample . we check our experimental device with well - known polymer films . as already mentioned , our experimental device allows to perform simultaneous measurement of impedance at several frequencies in a range from @xmath0 to @xmath101 hz with low noise and high sensitivity . first of all , we investigate an extrusion film makrofol^^ de 1 - 1 000000 ( from bayer ) based on makrolon^^ polycarbonate ( pc ) of 125 @xmath143 m of thickness and with a glass transition temperature ( @xmath144 ) of about 150 @xmath145c . the sample consist of a pc film of 18 cm of diameter . we measure the impedance and compute @xmath63 and @xmath133 from 22 to 167 @xmath145c . fig . [ fig - pc ] shows @xmath133 and @xmath63 at 22 @xmath145c . @xmath133 presents huge values at low frequencies decreasing with frequency . @xmath63 presents a small dependence in frequency . from @xmath63 we calculate the dielectric constant @xmath16 ( eq . [ eq - permittivity1 ] ) to be similar to the value found in the literature@xcite ( @xmath146 ) . @xmath147 ( eq . [ eq - tandelta2 ] ) as a function of temperature is presented in fig . [ fig - pctandelta ] . a peak corresponding to the segmental relaxation of pc ( glass transition ) appears from low to high frequencies when increasing temperature ( well - known in polymers , see e.g. ref . ) . the advantage here is that one can do easily multi - frequency dynamical measurements changing the rate of the temperature ramp . m of thickness and @xmath148 cm of diameter as a function of frequency at room temperature . ] of a pc film of 125 @xmath143 m of thickness and 18 cm of diameter as a function of frequency at different temperatures . ] c at 1 , 11 , and 91 hz . both magnitudes are normalized to their values just after the quench ( time @xmath149 ) : @xmath150 ( 1 hz ) , @xmath151 ( 11 hz ) , and @xmath152 ( 91 hz ) for the resistance ; @xmath153 f ( 1 hz ) , @xmath154 f ( 11 hz ) , and @xmath155 f ( 91 hz ) for the capacitance . ] then we investigate a film of poly(vinyl - acetate ) ( pvac ) in order to study the aging phenomenon . the pvac ( @xmath156c ) is cycled between 19 and 53@xmath145c , with a very fast thermal quench ( around 30@xmath145c / min ) . the electrodes are made out of thin films of aluminum ( 150 nm ) deposited by evaporation over a sapphire disc . such crystal plate is chosen for its high thermal conductivity because heating and cooling are done through the substrate . the polymer is dissolved in chloroform , then poured over the lower electrode . after careful drying of the solvent , the second aluminum thin film is deposited over the dielectric . connexions are fixed on the electrodes thanks to conductive glue . the thickness of the dielectric is around @xmath157 , such that the overall capacitance is a few nf in order to match the amplifier s impedance . [ fig - pvacr ] shows resistance and capacitance of pvac above and below @xmath144 as a function of frequency . the @xmath158-relaxation appears in the measurement range around 53@xmath145c but not at lower temperature ( 19@xmath145c ) . [ fig - pvacr2 ] shows the time evolution of @xmath133 and @xmath63 after a fast thermal quench from 53@xmath145c to 19@xmath145c at 1 , 11 , and 91 hz . the relaxation of @xmath133 and @xmath63 are due to the aging of the polymer below the glass transition going toward equilibrium . this technique allows to work on a time range limited at small times by the speed of the quench ( typically one minute ) . our device is absolutely necessary in this kind of dynamical studies . we present here a technique for impedance measurements , specially designed for highly resistive samples and time dependent phenomena . indeed it allows us to measure the complex dielectric constant simultaneously in a wide frequency range . the electronics combined with the described data analysis allow to measure with a good accuracy @xmath104 and @xmath105 f. this technique is based on two main elements . first of all , a home - made amplifier specially built to measure small phase shifts in the whole frequency range of interest . the proposed method based on a correction signal is very efficient for obtaining accurate values of both on the real and the imaginary part of the impedance , even in the cases of very small phase shift . secondly , the reference signal is carefully chosen to give optimal accuracy over the whole frequency range , which is typically @xmath4 hz . we use a sum of sinus with decreasing intensity at high frequency in order to keep a constant signal - to - noise ratio with frequency . the chosen frequencies of the input signal are exact subharmonic of the sampling frequency , which is used to generate the input signal and to sample the outputs . compared to the use of a white noise , this signal has well - defined phases , ensuring the good phase accuracy we need . furthermore , measuring known frequencies allows shorter acquisition times than using white noise in order to get high precision on the impedances . then , this device is particularly useful for time evolving impedances studies , especially in the field of aging of glass - forming polymers . four orders of magnitude in frequency , chosen in the range of @xmath4 hz , can be simultaneously investigated over typical time range from one minute to arbitrary long time . we also present here few results on two polymer samples : a pc film evolving to the glass transition with temperature , and a pvac sample evolving with time after a fast quench . this last result demonstrates the rich field of non - equilibrium physics that can be probed using this device . r. prez - aparicio acknowledges the funding by solvay . caroline crauste - thibierge and sergio ciliberto thank the founding by erc grant 267687 outeflucop . we also thank debjani bagchi who made use of the setup presented here . in this appendix we give an explanation of the noise peak in the output spectrum of fig . [ fig - psdfreq ] . the amplification of the ad549 is very well approximated at low frequencies by @xmath159 with @xmath160 and @xmath161 . the analysis of the circuit of fig . [ circuitall ] shows that the spectrum of the output signal @xmath162 is related to that at the input @xmath163 : @xmath164 where @xmath165 ^ 2 \omega^2 \label{eq_denominator } \end{split}\ ] ] here we have used the approximation @xmath166 and @xmath167 ( see fig . [ fig - circuit ] ) . these are very good for the purpose of understanding the maximum of the noise . this spectrum has a resonance at @xmath168}$ ] which is rather broad because of the strong damping term @xmath169 . inserting the numerical value we see that the resonance is at about @xmath170 and the broadening a few hz . these values are in agreement with the enhancement of the noise in the power spectrum of fig . [ fig - psdfreq]b , if we consider that the noise of the input signal , fig . [ fig - psdfreq]a , has a wide bandwidth with a slow decrease as function of frequency . in fig . [ fig - simul ] we plot an output spectrum computed from eq . [ eq_power_spectrum ] using a realistic input spectrum , which is plotted in the same figure . we see that the output spectrum is very close to the one measured with @xmath171f . 22ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop _ ( , , ) link:\doibase 10.1080/001075100181259 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.77.318 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.1470235 [ * * , ( ) ] link:\doibase 10.1021/ma992039y [ * * , ( ) ] \doibase doi : 10.1016/0032 - 3861(67)90021 - 3 [ * * , ( ) ] link:\doibase 10.1021/jp9640989 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.104.165703 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.3678324 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.4746992 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.1150528 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.1876992 [ * * , ( ) ] \doibase http://dx.doi.org/10.1103/physreve.55.3898 [ * * , ( ) ] http://iopscience.iop.org/0295-5075/46/5/637 [ * * , ( ) ] \doibase http://dx.doi.org/10.1103/physreve.63.012503 [ * * , ( ) ] \doibase http://dx.doi.org/10.1103/physrevlett.103.040601 [ * * , ( ) ] http://iopscience.iop.org/0295-5075/53/4/511 [ * * , ( ) ] \doibase http://dx.doi.org/10.1103/physrevlett.88.257202 [ * * , ( ) ] http://iopscience.iop.org/0295-5075/63/4/603 [ * * , ( ) ] @noop _ _ ( , , )
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we present an innovative technique which allows the simultaneous measurement of the dielectric constant of a material at many frequencies , spanning a four orders of magnitude range chosen between @xmath0 hz and @xmath1 hz .
the sensitivity and accuracy are comparable to those obtained using standard single frequency techniques .
the technique is based on three new and simple features : a ) the precise real time correction of the amplification of a current amplifier ; b ) the specific shape of the excitation signal and its frequency spectrum ; and c ) the precise synchronization between the generation of the excitation signal and the acquisition of the dielectric response signal .
this technique is useful in the case of relatively fast dynamical measurements when the knowledge of the time evolution of the dielectric constant is needed .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
one of the most remarking properties of quantum mechanics is that the state of a quantum system changes not only via the deterministic evolution given by the schrdinger equation but also when it is measured . although the system can be measured directly , in many situations it is probed by an auxiliary system ( ancilla ) that is then detected , providing information about the system of interest . this is a typical situation in cavity qed where atoms and light interact and the detection of either of them alters the state of the other . in the microwave regime , for example , it is usually the atoms that behave as a probe for the field . this configuration has led to a number of experiments on fundamental aspects of quantum mechanics including the measurement of the decoherence of a cat state @xcite , qnd measurement of single photons @xcite , and the observation of quantum jumps of light @xcite . this property becomes particularly interesting for quantum open systems where the environment that the system is coupled to plays the role of a natural bona fide ancilla . the time evolution of a single quantum system can therefore be probed by directly monitoring this reservoir . this dynamics is going to be stochastic , as the evolution is conditioned on the measurements results , and can be mathematically described in terms of quantum trajectories , which are related to different physical ways to monitor the environment and to extract information about the system . the monitoring of the field leaving a damped cavity mode provides a good example of this stochastic dynamics . for instance , if a photodetector is used to collect the output of the cavity , the dynamics is better described in terms of quantum jumps where each click in the detector corresponds to lowering the number of photons inside the cavity by one . however , a completely different dynamics is found if the same propagating field is combined with a local oscillator in a beam splitter and a homodyne measurement is performed , in which case the time evolution of the damped cavity field is appropriately described by a continuous stochastic trajectory ( quantum state diffusion ) . all these trajectories present very interesting scenarios that allow for the production of non - classical states of the cavity field as well as the protection of the purity of the cavity mode or of the entanglement shared by two or more modes . note that if the lost photons are ignored , or the measurements averaged out , one obtains the usual master equation dynamics for the decoherence process that is responsible , for example , for rapidly turning superpositions of coherent states into mixtures @xcite . for this reason , each monitoring scheme described above is said to represent an unravelling of the master equation in terms of stochastic trajectories . note that the unravellings above represent only a limited set of the possibilities to measure the environment . in fact the master equation can be mathematically unravelled in infinite ways in terms of stochastic trajectories @xcite and therefore one could envisage more general ways of monitoring the environment rather than only simple photodetection and homodyning . this freedom in defining unravellings has been recently explored in the context of entanglement decay and protection @xcite , where monitoring schemes that combine different decay channels play a crucial role on the recovery of the mixed state entanglement dynamics in terms of trajectories @xcite and on the protection of entanglement conditioned on measurement outcomes @xcite . it would be interesting then to propose realistic experimental scenarios where this variety of unravellings could be explored . while the monitoring of the field emitted by a leaky cavity mode or by a decaying atom is within current experimental feasibility , this is not necessarily the most efficient way and certainly not the most complete one to generate different unravellings . this scheme proves to be very limited in two very interesting situations : non - zero reservoir temperatures and unravellings that combine detections and no - detections ( jumps and no - jumps ) . the first case presents two problems : first how to determine when a very large reservoir loses a single photon to the system and second how to distinguish a photon that comes from the system from one that already exists in the reservoir . in the second case , one faces the problem of physically superposing a click with a no - click in the detector . these limitations severely hinder the exploration of quantum trajectories in these systems . even though the direct monitoring of a natural thermal environment remains a challenge , in cavity qed one can artificially engineer this and other reservoirs that could produce different kinds of quantum trajectories when measured . in the microwave regime , for example , beams of atoms crossing the cavity can be used to mimic a thermal dissipative reservoir for the cavity field @xcite . the posterior detection of these atoms produces quantum trajectories for the cavity field , as analyzed in @xcite both in terms of jumps and continuous diffusion processes . however , these schemes also present limitations to the production of different unravellings . for example , while the combination of different channels can be easily accomplished within the optical detection scenario by having the photons for each channel arriving at different ports of a beam splitter @xcite , the situation for atomic detection seems far more complex . in all previous proposals to engineer thermal reservoirs for cavities using atomic beams , decay and excitation channels correspond to two - level atoms entering the cavity either in the ground or in the excited state . a combined detection , which is utterly important for the applications proposed in @xcite , would then require some kind of interaction between the atoms after they cross the cavity , which albeit not impossible , seems rather challenging . in this paper we show that simple modifications to these previous proposals lead to an alternative scheme to engineer a thermal reservoir in the context of microwave cavity qed that accommodates more general detections , hence enabling the simulation of a wide set of different classes of trajectories for a monitored atomic reservoir . in particular , we address the limitations raised in the two previous paragraphs . we then propose a complementary experimental setup where we invert the roles , letting the cavity field play the reservoir for the atoms . we rely on the purcell effect to channel the atomic decays into the modes of a lossy cavity and then use the optical detection proposed in @xcite to monitor the reservoir in different ways . in this last case , we introduce an additional measurement possibility to implement the proposal in @xcite that would circumvent the challenging task of collecting broadly emitted photons from atomic decay . we will start by briefly recalling the usual way to simulate a thermal reservoir for a cavity field using a sequence of two - level atoms as discussed in @xcite . the atoms , prepared in either the ground or excited states , cross a cavity where they interact resonantly via a jaynes - cummings hamiltonian for a short enough time . by short time , we mean a time interval @xmath0 much smaller than the inverse of the vacuum rabi frequency of the system @xmath1 such that the probability of an atom making a transition is small . under this model , one can then show that if the atoms are not detected after interacting with the cavity , the cavity field ( of mode annihilation operator @xmath2 ) will evolve according to a master equation of the form @xmath3 \rho + \gamma_+ \mathcal{d}[a^\dagger ] \rho,\ ] ] where @xmath4 , and the relationship between the rates @xmath5 and @xmath6 is given by the ratio between the flux of atoms initially prepared in the ground ( @xmath7 ) and excited state ( @xmath8 ) . a thermal bath is obtained by setting this ratio to be @xmath9 , where @xmath10 is the average number of photons in the modes with energy @xmath11 in the reservoir . if one now considers that the atoms are detected after the interaction , then the evolution of the cavity field is conditioned on the result of this measurement and better described by quantum trajectories . in this case , if an atom enters in the ground ( excited ) state and is detected in the excited ( ground ) state then the cavity field will be modified by a jump operator @xmath12 ( @xmath13 ) corresponding to the annihilation ( creation ) of a photon in the cavity . note that these jumps will be rare since , under our assumption of small interaction time , the atoms will most probably remain in their original state , in which case the cavity field evolution will correspond to the application of a no - jump operator @xmath14 . if the state of the atom is known before and after the cavity then a quantum jump unravelling corresponding to the operators @xmath15 will be produced for the evolution of the cavity field @xcite . note , as mentioned before , that in order to produce any other jump - like unravelling which necessarily combines the operators @xmath16 and @xmath17 , the atoms would need to interact after going through the cavity which is a rather difficult experimental task . we now present an alternative scheme to engineer a thermal reservoir that naturally allows more general detections . the central idea is to encapsulate in a single atom the interactions with the cavity that correspond to the evolution with @xmath16 and @xmath17 . for that we will use three - level atoms with levels organised in a cascade , as shown in fig . [ fig1 ] , and selected in such a way that the energy difference between the intermediate level ( @xmath18 ) and the lower and upper levels ( respectively @xmath19 and @xmath20 ) are close enough to the energy of the chosen cavity mode so that these transitions can be tuned into resonance through the application of external fields , but only one at a time . the atoms are prepared in the @xmath21 state before entering the cavity and in the first stage , the external field shifts the @xmath18 level such that the transition @xmath22 becomes resonant with the cavity frequency @xmath23 . the system then evolves under the jaynes - cummings hamiltonian @xmath24 for a short time @xmath25 . up to second order on @xmath26 the initial atom - field state @xmath27 will be transformed to @xmath28 where @xmath29 is the coupling constant , @xmath30 the initial field state and the tilde indicates that the state is not normalised . if , now , the other transition @xmath31 is tuned to resonance by changing the external field , the hamiltonian will be @xmath32 and , again , making the short time expansion will lead to @xmath33 considering now a time interval @xmath34 that is large enough so that @xmath35 atoms ( @xmath36 ) cross the cavity and yet the probability of occurrence of a quantum jump remains very small , one can recover the master equation result by tracing out the atomic system . this becomes more evident if one identifies the rates as @xmath37 and @xmath38 , where @xmath39 is the number of atoms entering the cavity per unit of time . therefore , the same atom can emulate at the same time both the dissipative and the excitation reservoirs . transition , then with @xmath40 transition . rotations between levels @xmath41 and @xmath42 after the interaction with the cavity and before atomic detection produces different unravellings of the dynamics in terms of quantum jumps.,width=321 ] if instead of tracing out the atoms one measures their levels , then the evolution of the state of the cavity field obeys a quantum trajectory given by the no - jump operator @xmath43 if the atom is detected in level @xmath18 or the jump @xmath44 ( @xmath45 ) if the atom is detected in level @xmath20 ( @xmath19 ) , that corresponds to the usual monitoring discussed previously . the advantage of encoding both jumps in the same atom is that it allows us to explore different unravellings by introducing simple modifications in the detection scheme . if one makes a unitary transformation between levels @xmath46 and @xmath47 after the atoms cross the cavity , for example , then a detection of each state will correspond to a different unravelling in which the jump operators are combinations of @xmath2 and @xmath48 . for instance , in the particular case that @xmath49 , a @xmath50 pulse between levels @xmath19 and @xmath20 produces jump operators proportional to the field quadratures @xmath51 and @xmath52 where the phase @xmath53 is given by the chosen rotation . in fact , since one can apply unitary operations also involving level @xmath18 then it is clear that any unravelling of the type @xmath54 , where @xmath55 is a unitary matrix acting on the vectorial space of the jumps and @xmath56 is the vector @xmath57 , can be produced . note that the unitary @xmath55 is obtained by individually rotating the atoms after they have already interacted with the cavity mode therefore preserving the master equation structure when the atoms are ignored . in the scheme described above , the detection of the atom modifies the quadratures of the field affecting , in principle , all fock states . for example , if the field is initially constrained to fock states @xmath58 , the detection of the atom in its middle ( @xmath18 ) or highest ( @xmath20 ) levels will preserve the initial subspace , but the detection of the atom in its ground state will expand the effective subspace of the cavity mode to include fock state @xmath59 . this can be avoided by choosing a selective interaction @xcite for the @xmath60 transition in which case the creation of an extra photon in the cavity can be set to occur only if the cavity is empty , hence preserving the initial effective subspace and thus fully mimicking a spin - half system for the cavity field @xcite . inverting the roles of the previous session , now , the three level atom is stationary and interacts with two orthogonal cavity modes such that the @xmath61 ( @xmath60 ) transition generates photons circularly polarized to the right ( left ) associated to the annihilation operator @xmath62 ( @xmath63 ) of the cavity mode . we assume a very lossy cavity ( of decay rate @xmath64 ) made of a nearly perfect mirror and a semi - transparent one ( see fig . [ fig2 ] ) . if the rabi frequencies of each transition are large enough then the purcell effect will channel the atomic decay in the cavity modes , and the cavity asymmetry will guarantee that the photons will always leak in a well defined direction . each atomic transition has to be coupled to its respective cavity mode with different coupling constants @xmath65 and @xmath66 , and a @xmath67-polarized classical field of strength @xmath68 has to drive the @xmath69 transition as shown in fig . [ fig2 ] . the relation between the involved rates should be @xmath70 , where @xmath71 and @xmath72 are the natural atomic linewidths . for the scheme to work , much greater typically reads @xmath73 where this particular choice has already been made to fit the analysis that follows . and @xmath65 ) and the cavity decay ( @xmath64 ) will induce an effective atomic decay via the purcell effect . photons from the right and left polarized modes leaving the cavity will then correspond to @xmath74 and @xmath75 transitions , respectively . the corresponding effective decay rates @xmath76 and @xmath77 are assumed to be much larger than the natural spontaneous decay rates @xmath71 and @xmath72.,width=321 ] the overall master equation that describes the dynamics of the whole system is given by : @xmath78 + \gamma \mathcal{d}[\sigma_-^{(eg)}]\rho + \gamma \mathcal{d}[\sigma_-^{(ie)}]\rho + \kappa \mathcal{d}[a],\end{aligned}\ ] ] where @xmath79 with @xmath80 and @xmath81 being the lowering and raising operators for the @xmath82 transition . being @xmath83 there is no rabi oscillation between the corresponding atomic transition and the cavity mode which acts just as a channel for the atomic decay . in other words , as soon as a photon is transferred from the atom to the cavity mode it leaks through the semi - transparent mirror . the effective atomic decay rate is then given by @xmath84 as long as @xmath85 . much in the same way , the effective decay rate for the @xmath61 transition is given by @xmath86 . now , if the external driving field @xmath68 is still much smaller than this effective decay rate , then level @xmath20 can be adiabatically eliminated and the combined effect of the classical field and the strong decay rate @xmath77 will be to generate an effective incoherent pump between levels @xmath19 and @xmath18 of rate @xmath87 . in this way , the scheme produces a decay and an excitation reservoir for the qubit composed of levels @xmath88 and @xmath42 with rates @xmath89 and @xmath90 , respectively . in particular , when the effective rates @xmath91 are the same , _ i.e. _ @xmath92 , one obtains an infinite temperature environment . note that there is still another degree of freedom available if the conditions are not perfectly matched that is the eventual detuning between cavity modes and atomic transitions . for example , let us say that @xmath93 . if @xmath94 , then the effective decay rate is corrected by @xmath95 @xcite . under the above approximations and assuming that the effective rates are much larger than the natural atomic linewidths ( @xmath96 ) , the unconditioned dynamics of the ( two - level ) atom can be finally written as : @xmath97\rho + \gamma_+ \mathcal{d}[\sigma_+]\rho,\end{aligned}\ ] ] where the superscript @xmath98 has been dropped for simplicity . if the photons leaving the cavity are detected , then the atomic dynamics will follow a stochastic dynamics conditioned on the measurement outcomes as described by the quantum trajectory approach . equivalently to the previous session , different detection techniques will lead to different unravellings . for example , if a @xmath99 plate is placed right after the leaking mirror , then each circular polarization is converted into orthogonal linear polarizations , @xmath100 ( @xmath101 ) . these photons are then sent into a polarized beam splitter that separates the linear polarizations into two different propagation modes where detectors are placed to collect them . in this way , the click in the @xmath102 ( @xmath103 ) path identifies a transition @xmath22 ( @xmath60 ) which corresponds to the `` usual '' photodetection unravelling with the jumps @xmath104 and @xmath105 applied to the atomic state . if no photon is detected in the @xmath106 interval , then the no - jump @xmath107 operator is applied to the system . a more interesting situation occurs if the @xmath99 plate is taken out of the setup . in this case , the photons propagating after the pbs will correspond to linear combinations of the photons leaking from the atom , implementing an unravelling with the jumps @xmath108 and @xmath109 ( similar to the quadrature unravelling described in the previous section ) . we have shown in @xcite that if two qubits initially share an entangled state , then this kind of monitoring performed on both subsystems can preserve the entanglement in the system . the above setup is therefore a way to implement the entanglement protection ideas proposed in @xcite . before concluding , let us just remark that the scheme here described for cavity qed could also be applied to other experiments involving three level systems and harmonic oscillators . natural candidates where quantum jumps have been recently observed or proposed are superconducting qubits @xcite and nanoresonators @xcite . finally , in conclusion , we have proposed two experimental ways to produce a wide range of unravellings of a master equation evolution in cavity qed systems . in the first case , three - level atoms are used to simulate a thermal dissipative reservoir and the preparation and posterior detection of these atoms in different basis produce unravellings that correspond to different combinations of the natural jump operators and even between these and the no - jump one . this is a particular case in which `` no - click '' and `` click '' can be physically combined to generate a new class of unravellings . later , we also show how to invert the roles and use the purcell effect to channel the detection of the spontaneous decay and of the incoherent pump ( or even their combination ) for driven three - level atoms . this work adds to the existing class of proposed cavity qed experiments , expanding the range of quantum effects that can be explored in this experimentally successful system . we would like to acknowledge the support from center for quantum technologies at the national university of singapore . arrc acknowledges financial support by the australian research council centre of excellence program and mfs acknowledges the support of the national research foundation and the ministry of education of singapore . englert b .- g . and morigi g. , _ five lectures on dissipative master equations _ in _ coherent evolution in noisy environments _ , edited by buchleitner a. and hornberger k. , vol . 611 of _ lecture notes in physics _ ( springer berlin / heidelberg ) 2002 pp .
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the experimental observation of quantum jumps is an example of single open quantum systems that , when monitored , evolve in terms of stochastic trajectories conditioned on measurements results . here
we present a proposal that allows the experimental observation of a much larger class of quantum trajectories in cavity qed systems .
in particular , our scheme allows for the monitoring of engineered thermal baths that are crucial for recent proposals for probing entanglement decay and also for entanglement protection .
the scheme relies on the interaction of a three - level atom and a cavity mode that interchangeably play the roles of system and probe .
if the atom is detected the evolution of the cavity fields follows quantum trajectories and vice - versa .
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the first step in describing the interaction between many particles is to determine their pair potential or the forces among a single pair . if the governing physical equations are linear ( like for gravity or electrostatics ) , this approach yields a quantitatively reliable description of the physical system considered , based on the linear superposition principle . however , if nonlinearities are present , linear superposition of pair potentials is no longer accurate and nonadditivity gives rise to many - body effects . these latter effects can lead , e.g. , to a strengthening or weakening of the total force acting on a particle surrounded by more than a single other one , a change of sign of that force , or the appearance of stable or unstable configurations . many - body effects appear in rather diverse systems such as nuclear matter , superconductivity @xcite , colloidal suspensions @xcite , quantum - electrodynamic casimir forces @xcite , polymers @xcite , nematic colloids @xcite , and noble gases with van der waals forces acting among them @xcite . each of these systems is characterized by a wide range of time and length scales . integrating out the degrees of freedom associated with small scales ( such as the solvent of colloidal solutes or polymers ) for fixed configurations of the large particles , generates effective interactions among the latter , which are inherently not pairwise additive . this is the price to be paid for achieving a reduced description of a multicomponent system . driven by these effective interactions the large particles of the system may exhibit collective behavior of their own ( like aggregation or phase separation , see refs . @xcite and @xcite and references therein ) , which can be described much easier if it is governed by pair potentials . in order to be able to judge whether this ansatz is adequate one has to check the relative magnitude of genuine many - body forces . in this paper we assess the quantitative influence of such many - body effects on critical casimir forces ( ccfs ) @xcite . these long - ranged forces arise as a consequence of the confinement of the order parameter fluctuations in a critical fluid @xcite . they have been analyzed paradigmatically by studying the effective interaction between a _ single _ colloidal particle and a homogeneous @xcite or inhomogeneous @xcite container wall as well as between _ two isolated _ colloidal particles @xcite upon approaching the critical point of the solvent . here we add one sphere to the sphere - wall configuration , which is the simplest possibility to study many - body forces . ( the wall mimics a third , very large sphere . ) in order to be able to identify the latter ones one has to resort to a theoretical scheme which allows one to compute the forces between individual pairs and the three - body forces on the same footing . since these forces are characterized by universal scaling functions , which depend on the various geometrical features of the configuration and on the thermodynamic state , we tackle this task by resorting to field - theoretic mean field theory ( mft ) , which captures the universal scaling functions as the leading contribution to their systematic expansion in terms of @xmath1 spatial dimensions . experience with corresponding previous studies for simple geometries tells that this approach does yield the relevant qualitative features of the actual universal scaling functions in @xmath2 ; if suitably enhanced by renormalization group arguments these results reach a semi - quantitative status . we point out that even within this approximation the numerical implementation of this corresponding scheme poses a severe technical challenge . thus at present this approach appears to be the only feasible one to explore the role of many - body critical casimir forces within the full range of their scaling variables . accordingly we consider the standard landau - ginzburg - wilson hamiltonian for critical phenomena of the ising bulk universality class , which is given by @xmath3 = \int_v{\rm d}^d\mathbf{r } \left\ { \frac{1}{2}\left ( \nabla\phi \right)^2 + \frac{\tau}{2}\phi^2 + \frac{u}{4!}\phi^4 \right\ } ~,\ ] ] with suitable boundary conditions ( bcs ) . in the case of a binary liquid mixture near its consolute ( demixing ) point , the order parameter @xmath4 is proportional to the deviation of the local concentration of one of the two species from the critical concentration . @xmath5 is the volume accessible to the fluid , @xmath6 is proportional to the reduced temperature @xmath7 , and the coupling constant @xmath8 stabilizes the statistical weight @xmath9 in the two - phase region , i.e. , for @xmath10 . close to the bulk critical point @xmath11 the bulk correlation length @xmath12 diverges as @xmath13 , where @xmath14 in @xmath2 and @xmath15 in @xmath16 , i.e. , within mft @xcite . the two non - universal amplitudes @xmath17 are of molecular size ; they form the universal ratio @xmath18 for @xmath2 and @xmath19 for @xmath16 @xcite . the bcs reflect the generic adsorption preference of the confining surfaces for one of the two components of the mixture . for the critical adsorption fixed point @xcite , the bc at each of the confining surfaces is either @xmath20 or @xmath21 , to which we refer as @xmath22 or @xmath23 , respectively . within mft the equilibrium order parameter distribution minimizes the hamiltonian in eq . for the aforementioned bcs , i.e. , @xmath24/\delta\phi = 0 $ ] . far from any boundary the order parameter approaches its constant bulk value @xmath25 for @xmath10 or @xmath26 for @xmath27 . @xmath28 is a non - universal bulk amplitude and @xmath29 ( for @xmath16 ) is a standard critical exponent . in the following we consider the reduced order parameter @xmath30 . the remainder of this paper is organized as follows . in sec . [ section_the_system ] we define the system under consideration and the scaling functions for the ccfs as well as the normalization scheme . in sec . [ section_results ] we present the numerical results obtained for the universal scaling functions of the ccfs , from which we extract and analyze the many - body effects . in sec . [ section_conclusion ] we summarize our results and draw some conclusions . we study the normal and the lateral ccfs acting on two colloidal particles immersed in a near - critical binary liquid mixture and close to a homogeneous , planar substrate . we focus on the critical concentration which implies the absence of a bulk field conjugate to the order parameter [ see eq . ] . the surfaces of the colloids and of the substrate are considered to exhibit a strong adsorption preference for one of the two components of the confined liquid leading to @xmath31 bcs . the forces are calculated using the full three - dimensional numerical analysis of the appropriate mft as given by eq . . specifically , we consider two three - dimensional spheres of radii @xmath32 and @xmath33 with bcs @xmath34 and ( @xmath35 ) , respectively , facing a homogeneous substrate with bc @xmath36 at sphere - surface - to - substrate distances @xmath37 and @xmath38 , respectively ( see fig . [ system_sketch ] ) . the coordinate system @xmath39 is chosen such that the centers of the spheres are located at @xmath40 and @xmath41 so that the distance between the centers , projected onto the @xmath42-axis , is given by @xmath43 . the bcs of the whole system are represented by the set @xmath44 , where @xmath45 , @xmath35 , and @xmath46 can be either @xmath47 or @xmath48 . it is important to point out that we discuss colloidal particles with the shape of a hypercylinder @xmath49 where @xmath50 are the semiaxes ( or radii ) of the hypercylinder and @xmath51 , @xmath52 , @xmath53 . if @xmath54 and @xmath55 , the hypercylinder reduces to a hypersphere . the generalization of @xmath56 to values larger than 3 is introduced for technical reasons because @xmath57 is the upper critical dimension for the relevance of the fluctuations of the order parameter . these fluctuations lead to a behavior different from that obtained from the present mft which ( apart from logarithmic corrections @xcite ) is valid in @xmath16 . we consider two hypercylinders in @xmath16 with @xmath58 and @xmath59 . the two colloids are taken to be parallel along the fourth dimension with infinitely long hyperaxes in this direction . considering hypercylinders , which are translationally invariant along the @xmath60axis , allows us to minimize @xmath61 $ ] numerically using a three dimensional finite element method in order to obtain the spatially inhomogeneous order parameter profile @xmath62 for the geometries under consideration ( see fig . [ system_sketch ] ) . and @xmath33 immersed in a near - critical binary liquid mixture ( not shown ) and close to a homogeneous , planar substrate at @xmath63 . the two colloidal particles with bcs @xmath34 and @xmath64 are located at sphere - surface - to - substrate distances @xmath37 and @xmath38 , respectively . the substrate exhibits bc @xmath36 . the lateral distance between the centers of the spheres along the @xmath42-direction is given by @xmath43 , while the centers of both spheres lie in the plane @xmath65 . in the case of four spatial dimensions the figure shows a cut of the system , which is invariant along the fourth direction , i.e. , the spheres correspond to parallel hypercylinders with one translationally invariant direction , which is @xmath66 . ] in the case of an upper critical demixing point of the binary liquid mixture at the critical concentration , @xmath67 corresponds to the disordered ( i.e. , mixed ) phase of the fluid , whereas @xmath68 corresponds to the ordered ( i.e. , phase separated ) phase . the meaning of the sign is reversed for a lower critical point . in the following we assume an upper critical point . the normal ccf @xmath69 acting on sphere @xmath70 in the presence of sphere @xmath71 ( @xmath72 and @xmath73 ) along the @xmath74-direction takes the scaling form @xmath75 where @xmath76 , @xmath77 , @xmath78 , @xmath79 , and @xmath80 ( i.e. , @xmath81 for @xmath27 and @xmath82 for @xmath10 ) . equation describes the singular contribution to the normal force emerging upon approaching @xmath11 . @xmath83 is the force per length of the hypercylinder due to its extension in the translationally invariant direction . in the spirit of a systematic expansion in terms of @xmath84 around the upper critical dimension we study the scaling functions @xmath85 within mft as given by eq . for hypercylinders in @xmath16 , which captures the correct scaling functions in @xmath16 up to logarithmic corrections occurring in @xmath86 @xcite , which we do not take into account here . since mft renders the leading contribution to an expansion around @xmath16 , geometrical configurations with small @xmath87 , @xmath88 , or @xmath89 are not expected to be described reliably by the present approach due to the dimensional crossover in narrow slit - like regions , which is not captured by the @xmath84 expansion . the colloidal particles will also experience a lateral ccf @xmath90 , for which it is convenient to use the scaling form @xmath91 where @xmath92 ( i.e. , @xmath93 for @xmath27 and @xmath94 for @xmath10 ) . note that the choice of @xmath95 as the scaling variable does not depend on the type of particle the force acts on . equation also describes the singular contribution to the lateral force near @xmath11 . the total ccf acting on particle @xmath70 is @xmath96 where @xmath97 and @xmath98 are the unit vectors pointing in @xmath42- and @xmath74-direction , respectively . due to symmetry all other components of the ccf are zero . as a reference configuration we consider a single spherical colloid of radius @xmath99 with bc @xmath100 at a surface - to - surface distance @xmath101 from a planar substrate with bc @xmath36 . this colloid experiences ( only ) a normal ccf @xmath102 in the following we normalize the scaling functions @xmath103 and @xmath104 by the amplitude @xmath105 of the ccf acting at @xmath11 on a single colloid for @xmath106 bcs at a surface - to - surface distance @xmath107 . accordingly , in the following we consider the normalized scaling functions @xmath108 with @xmath109 and @xmath110 . experimentally it can be rather difficult to obtain @xmath105 . a standard alternative way to normalize is to take the more easily accessible amplitude @xmath111 for the ccf at @xmath11 between two parallel plates with @xmath106 bcs , which is given within mft by ( see ref . @xcite and references therein ) @xmath112 ^ 4}{u } \simeq -283.61 u^{-1 } ~,\ ] ] where @xmath113 is the elliptic integral of the first kind . within mft the amplitude @xmath105 can be expressed in terms of @xmath111 : @xmath114 equation allows for a practical implementation of the aforementioned normalization , which eliminates the coupling constant @xmath115 , which is unknown within mft . we calculate the normal and lateral forces directly from the numerically determined order parameter profiles @xmath62 by using the stress tensor which , within the ginzburg - landau approach , is given by @xcite @xmath116 ~,\ ] ] with @xmath117 . the first index of the stress tensor denotes the direction of a force , the second index denotes the direction of the normal vector of the surface upon which the force acts . therefore one has @xmath118 where @xmath119 is a hypersurface enclosing particle @xmath70 , @xmath120 is the @xmath121-th component ( to be summed over ) of its unit outward normal , and @xmath122 is the length of the @xmath123-dimensional hyperaxis of @xmath124 . in particular we focus on the normal and lateral ccfs acting on colloid ( 2 ) for the configuration shown in fig . [ system_sketch ] , for @xmath125 , and with the binary liquid mixture at its critical concentration . in the following analysis we consider colloid ( 1 ) to be fixed in space at a sphere - surface - to - substrate distance @xmath126 , equally sized colloids ( i.e. , @xmath127 ) , and fixed @xmath128 bc for the substrate . we proceed by varying either the vertical ( @xmath74-direction ) or the horizontal ( @xmath42-direction ) position of colloid ( 2 ) by varying either @xmath38 or @xmath129 , respectively . we also consider different sets of bcs for the colloids . in the following results the numerical error is typically less than @xmath130 , unless explicitly stated otherwise . in fig . [ d2_z ] we show the behavior of the normalized [ eq . ] scaling function @xmath131 of the normal ccf acting on colloid ( 2 ) with @xmath132 bcs close to a homogeneous substrate with @xmath128 bc and in the presence of colloid ( 1 ) with @xmath133 bc . the scaling functions are shown as functions of the scaling variable ratio @xmath134 , i.e. , for @xmath27 . the various lines correspond to distinct fixed values of @xmath135 as the sphere - surface - to - substrate distance in units of the sphere radius . thus fig . [ d2_z ] shows the temperature dependence of the normal ccf on colloid 2 for three different values of @xmath38 and for colloid ( 1 ) fixed in space . from fig . [ d2_z ] ( a ) one can see that , for colloid ( 1 ) with @xmath136 bc and for any given value of @xmath137 , the scaling function of the normal ccf acting on colloid ( 2 ) with @xmath138 bc changes sign upon varying @xmath135 . due to the change of sign of @xmath139 , for any value of @xmath137 there is a certain value @xmath140 at which the normal ccf acting on colloid ( 2 ) vanishes . for sphere - surface - to - substrate distances sufficiently large such that @xmath141 , colloid ( 2 ) is pushed away from the substrate due to the dominating repulsion between the two colloids in spite of the attraction by the substrate , whereas for @xmath142 it is pulled to the substrate due to the dominating attraction between it and the substrate this implies that , in the absence of additional forces , levitation of colloid ( 2 ) ( i.e. , zero total normal force ) at height @xmath143 is not stable against perturbations of the sphere - surface - to - substrate distance . on the other hand , upon varying temperature , any distance @xmath38 can become a stable levitation position for colloid ( 2 ) with @xmath144 bc in the presence of colloid ( 1 ) with @xmath133 bc [ see fig . [ d2_z ] ( b ) , according to which each scaling function corresponding to a certain value of @xmath38 exhibits a zero so that , at this zero , increasing ( decreasing ) @xmath38 at fixed temperature leads to an attraction ( repulsion ) to ( from ) the substrate ] . in this case the attraction between the two colloids is dominating for large sphere - surface - to - substrate distances @xmath135 , while the repulsion between colloid ( 2 ) and the substrate dominates for small values of @xmath135 . of the normal ccf acting on colloid ( 2 ) with bcs @xmath145 in ( a ) and @xmath144 in ( b ) . the scaling functions are shown for @xmath27 as functions of the scaling variable ratio @xmath146 for three fixed values of the scaling variable @xmath77 : @xmath147 ( black lines ) , 1.5 ( red lines ) , and 2 ( green lines ) , while @xmath148 for all curves in ( a ) and ( b ) so that @xmath149 . for @xmath33 fixed the three curves correspond to three different vertical positions of colloid ( 2 ) with colloid ( 1 ) fixed in space . as expected , the forces become overall weaker upon increasing @xmath135 . @xmath150 @xmath151 implies that the colloid is attracted to ( repelled from ) the substrate along the @xmath74-direction . ] figure [ d2_x ] shows the behavior of the normalized scaling function @xmath152 of the lateral ccf acting on colloid ( 2 ) in the presence of colloid ( 1 ) having the same bc , i.e. , @xmath153 . the scaling functions are shown as functions of the scaling variable ratio @xmath154 . from figs . [ d2_x ] ( a ) and ( b ) one can infer that @xmath155 . therefore colloid ( 2 ) is always attracted towards colloid ( 1 ) which has the same bc . hence the substrate does not change the sign of the lateral ccf as compared with the attractive lateral ccf in the absence of the confining substrate . however , the shapes of the scaling functions for @xmath156 bcs [ fig . [ d2_x ] ( a ) ] differ from the ones for @xmath157 bcs [ fig . [ d2_x ] ( b ) ] ; without the substrate , they are identical . in the former case and in contrast to the latter one , the scaling functions exhibit minima above @xmath11 , which is reminiscent of the shape of the corresponding scaling functions in the absence of the substrate . of the lateral ccf acting on colloid ( 2 ) facing a homogeneous substrate with @xmath128 bc and in the presence of colloid ( 1 ) with @xmath158 bcs . the scaling functions are shown for @xmath27 as functions of the scaling variable ratio @xmath154 for three fixed values of the scaling variable @xmath77 : @xmath147 ( black curves ) , 1.5 ( red curves ) , and 2 ( green curves ) , while @xmath148 for all curves in ( a ) and ( b ) so that @xmath149 . for @xmath33 fixed the three curves correspond to three different vertical positions of colloid ( 2 ) with colloid ( 1 ) fixed in space . as expected , the forces become overall weaker upon increasing @xmath135 . @xmath159 implies that colloid ( 2 ) is attracted towards colloid ( 1 ) . two different sets of @xmath160 bcs are considered : @xmath156 in ( a ) and @xmath157 in ( b ) . ] in fig . [ l_x ] we show the results obtained for the normalized scaling functions @xmath161 of the lateral ccf acting on colloid ( 2 ) . in fig . [ l_x](a ) the scaling function is shown as function of the scaling variable ratio @xmath154 ; the black , red , and green curves correspond to @xmath162 , @xmath163 , and @xmath164 , respectively . in the absence of the substrate , the ccf between two colloids with opposite bcs is repulsive . however , as shown in fig . [ l_x](a ) , in the presence of the substrate with @xmath128 bc , there is a change of sign in the scaling function of the lateral ccf . this implies that the lateral ccf acting between the two colloids changes from being attractive to being repulsive ( or reverse ) upon decreasing ( increasing ) the reduced temperature . thus temperature allows one to control both the strength and the sign of the lateral ccf in the case of two colloids with opposite bcs being near a wall . the at first sight unexpected lateral attraction between two colloids with opposite bcs in the presence of the substrate ( i.e. , @xmath165 ) can be understood as follows . in the absence of the two colloids , the order parameter profile @xmath166 is constant along any path within a plane @xmath167 because in this case @xmath168 . since the substrate area is much larger than the surface areas of the colloids , one can regard the immersion of these colloidal spheres as a perturbation of this profile . in figs . [ l_x](a ) and ( b ) , the region within which the scaling function is negative ( corresponding to an attractive force ) indicates that under these circumstances [ i.e. , when the colloids are sufficiently away from each other ; see fig . [ l_x](b ) ] the perturbation generated by the presence of the spheres decreases upon decreasing the lateral distance between them . this causes the colloids to move towards each other in order to weaken the perturbation by reducing its spatial extension ; this amounts to an attraction , i.e. , @xmath165 . on the other hand , when they are sufficiently close to each other the pairwise interaction between the two colloids dominates and the total lateral ccf is positive ( i.e. , repulsive ) . in fig . [ l_x](c ) we show how the equilibrium lateral distance @xmath169 measured in units of @xmath170 varies as function of temperature , i.e. , @xmath170 for fixed @xmath99 . of the lateral ccf acting on colloid ( 2 ) . both colloids are taken to have the same size ( @xmath171 ) and the same sphere - surface - to - substrate distance ( @xmath172 ) . the scaling function is shown for @xmath27 as function of the scaling variable ratio @xmath154 . @xmath165 @xmath151 implies that colloid ( 2 ) is attracted ( repelled ) by colloid ( 1 ) . black , red , and green curves correspond to lateral distances @xmath173 , 1.5 , and 2 , respectively , between the surfaces of the colloids [ see fig . [ system_sketch ] ] . the zero @xmath174 of @xmath175 implies that for any lateral distance @xmath129 there is a reduced temperature such that for @xmath176 this distance represents an equilibrium lateral distance between the two colloids , provided they are located at equal sphere - surface - to - substrate distances . ( b ) @xmath175 as function of the reduced surface - to - surface distance @xmath79 between the two colloids for @xmath177 ( blue line ) , 3.873 ( yellow line ) , 4.472 ( magenta line ) , and 5 ( purple line ) . each curve in ( b ) corresponds to the vertical dashed line with same color in panel ( a ) . the change of sign of the scaling function ( from positive to negative ) as the distance between the colloids is increased indicates that the lateral ccf changes from repulsive to attractive , which means that there is a lateral position @xmath169 corresponding to a stable equilibrium point . in ( c ) we show how this equilibrium position , measured in units of @xmath178 , varies as function of temperature ( i.e. , @xmath178 ) for fixed @xmath33 . note that @xmath169 is not proportional to @xmath178 , since @xmath179 is not a constant . as a guide to the eye the four data points are connected by straight lines.,title="fig : " ] of the lateral ccf acting on colloid ( 2 ) . both colloids are taken to have the same size ( @xmath171 ) and the same sphere - surface - to - substrate distance ( @xmath172 ) . the scaling function is shown for @xmath27 as function of the scaling variable ratio @xmath154 . @xmath165 @xmath151 implies that colloid ( 2 ) is attracted ( repelled ) by colloid ( 1 ) . black , red , and green curves correspond to lateral distances @xmath173 , 1.5 , and 2 , respectively , between the surfaces of the colloids [ see fig . [ system_sketch ] ] . the zero @xmath174 of @xmath175 implies that for any lateral distance @xmath129 there is a reduced temperature such that for @xmath176 this distance represents an equilibrium lateral distance between the two colloids , provided they are located at equal sphere - surface - to - substrate distances . ( b ) @xmath175 as function of the reduced surface - to - surface distance @xmath79 between the two colloids for @xmath177 ( blue line ) , 3.873 ( yellow line ) , 4.472 ( magenta line ) , and 5 ( purple line ) . each curve in ( b ) corresponds to the vertical dashed line with same color in panel ( a ) . the change of sign of the scaling function ( from positive to negative ) as the distance between the colloids is increased indicates that the lateral ccf changes from repulsive to attractive , which means that there is a lateral position @xmath169 corresponding to a stable equilibrium point . in ( c ) we show how this equilibrium position , measured in units of @xmath178 , varies as function of temperature ( i.e. , @xmath178 ) for fixed @xmath33 . note that @xmath169 is not proportional to @xmath178 , since @xmath179 is not a constant . as a guide to the eye the four data points are connected by straight lines.,title="fig : " ] of the lateral ccf acting on colloid ( 2 ) . both colloids are taken to have the same size ( @xmath171 ) and the same sphere - surface - to - substrate distance ( @xmath172 ) . the scaling function is shown for @xmath27 as function of the scaling variable ratio @xmath154 . @xmath165 @xmath151 implies that colloid ( 2 ) is attracted ( repelled ) by colloid ( 1 ) . black , red , and green curves correspond to lateral distances @xmath173 , 1.5 , and 2 , respectively , between the surfaces of the colloids [ see fig . [ system_sketch ] ] . the zero @xmath174 of @xmath175 implies that for any lateral distance @xmath129 there is a reduced temperature such that for @xmath176 this distance represents an equilibrium lateral distance between the two colloids , provided they are located at equal sphere - surface - to - substrate distances . ( b ) @xmath175 as function of the reduced surface - to - surface distance @xmath79 between the two colloids for @xmath177 ( blue line ) , 3.873 ( yellow line ) , 4.472 ( magenta line ) , and 5 ( purple line ) . each curve in ( b ) corresponds to the vertical dashed line with same color in panel ( a ) . the change of sign of the scaling function ( from positive to negative ) as the distance between the colloids is increased indicates that the lateral ccf changes from repulsive to attractive , which means that there is a lateral position @xmath169 corresponding to a stable equilibrium point . in ( c ) we show how this equilibrium position , measured in units of @xmath178 , varies as function of temperature ( i.e. , @xmath178 ) for fixed @xmath33 . note that @xmath169 is not proportional to @xmath178 , since @xmath179 is not a constant . as a guide to the eye the four data points are connected by straight lines.,title="fig : " ] in order to determine the preferred arrangement of the colloids , we have also analyzed the direction of the total ccf @xmath180 acting on colloid ( 2 ) [ see eq . ] for several spatial configurations and bcs . for @xmath156 bcs we have found that the colloids tend to aggregate laterally in such a way that several particles with @xmath22 bc , facing a substrate with the same bc , can be expected to form a monolayer on the substrate . on the other hand , for the case of @xmath157 bcs , we have found that the colloids can be expected to aggregate on top of each other so that a collection of colloids with such bcs is expected to form three - dimensional sessile clusters . these tendencies become more pronounced upon approaching the critical point ( see fig . [ direction_resulting_force_d2 ] ) . similar results have been found by soyka et al . @xcite in experiments using chemically patterned substrates . for them the authors have found indeed that colloids with @xmath23 bc distributed over those parts of the substrate with the same bc [ which is equivalent to @xmath156 bcs ] aggregate and form a single layer . moreover , they have found that colloids distributed over parts of the substrate with opposite bc [ corresponding to @xmath157 bcs ] form three - dimensional clusters . and @xmath157 bcs , with @xmath181 . the black rectangles represent the substrate and blue circles represent colloid ( 1 ) while black , red , and green circles represent colloid ( 2 ) with @xmath182 , and 2 , respectively . the centers of all colloids lie in the plane @xmath65 . ] we have determined the many - body force acting on particle @xmath183 by subtracting from the total force @xmath184 [ see eq . ] the sum of the pairwise forces acting on it , i.e. , the colloid - colloid ( cc ) and the colloid - substrate ( cs ) forces . accordingly the many - body ccf @xmath185 acting on colloid ( @xmath70 ) is given by ( see fig . [ system_sketch ] ) @xmath186 where @xmath187 with @xmath188 is the pairwise colloid - colloid force ( acting on colloid 2 @xmath23 or 1 @xmath22 with 2 having the larger @xmath42-coordinate ) expressed in terms of the absolute value @xmath189 of the force between two colloids at surface - to - surface distance @xmath190 in free space . @xmath191 is the ccf between the substrate and a single colloid @xmath183 . we have studied both the normal @xmath192 and the lateral @xmath193 many - body ccfs which are characterized by corresponding scaling functions [ compare eqs . and ] : @xmath194 and @xmath195 in fig . [ mb_l_z ] we show the normalized [ see eqs . and ] scaling functions @xmath196 of the many - body normal ccf acting on colloid ( 2 ) . this figure reveals similar results for @xmath156 [ fig . [ mb_l_z ] ( a ) ] and @xmath197 [ fig . [ mb_l_z ] ( b ) ] bcs . in these cases , each mb scaling function exhibits both a maximum and at least one minimum , the former one appearing for smaller values of the scaling variable @xmath198 ( i.e. , at temperatures closer to @xmath11 ) . for a certain range of temperatures close to @xmath11 , as the distance @xmath129 between the colloids increases , the many - body normal ccf changes from attractive to repulsive . this shows that for each temperature within this range there is a lateral distance @xmath199 for which the many - body contribution to the normal force acting on colloid ( 2 ) is zero . this means that under such conditions the sum of pairwise forces provides a quantitatively reliable description of the total force acting on colloid ( 2 ) . for temperatures sufficiently far from @xmath11 , the many - body normal ccf is always attractive with a monotonic dependence on @xmath198 . here , as in figs . [ d2_z ] - [ l_x ] , the ccfs decay exponentially for @xmath200 . as expected , the many - body effects are more pronounced if the colloids are closer to each other and/or closer to the substrate . indeed for situations in which the colloids are close to each other [ see , e.g. , the black curves in figs . [ mb_l_z ] ( a ) and ( b ) ] we have found that when the normal many - body ccf reaches its maximal strength , corresponding to the minimum of the scaling function @xmath201 at @xmath202 , the relative contribution of the many - body ccf reaches @xmath0 of the strength of the total normal ccf . for larger distances between the colloids [ see , e.g. , the green curves in figs . [ mb_l_z ] ( a ) and ( b ) ] this relative contribution is smaller ( around @xmath203 for @xmath204 and @xmath205 for @xmath206 ) . of the many - body normal ccf acting on colloid ( 2 ) for @xmath207 and @xmath208 . the scaling functions are shown as functions of the scaling variable ratio @xmath209 for the sets of bcs @xmath156 in ( a ) and @xmath197 in ( b ) . the black , red , and green lines correspond to @xmath210 , and @xmath211 , respectively . figures [ d2_z](a ) and [ mb_l_z](b ) allow a direct comparison between the full ccf and the corresponding many - body contribution ( note the different scales of the ordinates . ) ] we are not aware of results for the quantum - electrodynamic casimir interactions which are obtained along the same lines as our ccf analysis above . nonetheless , in order to assess the significance of our results we compare them with the results in ref . @xcite , which is the closest comparable study which we have found in the literature . therein the authors study theoretically two dielectric spheres immersed in ethanol while facing a plate . depending on the kind of fluid and on the materials of the spheres and of the plate as well as on the distances involved , also the quantum - electrodynamic casimir force can be either attractive or repulsive . it is well known @xcite that for two parallel plates with permittivities @xmath212 and @xmath213 separated by a fluid with permittivity @xmath214 and without further boundaries , the quantum - electrodynamic casimir force is repulsive if @xmath215 within a suitable frequency range . in ref . @xcite it is stated that this also holds for two spheres immersed in a fluid . the authors of ref . @xcite analyze the effect of nonadditivity for the above system by studying the influence of an additional , adjacent substrate on the equilibrium separation @xmath56 between two nanometer size dielectric spheres . to this end , they consider two spheres of different materials with the same radii @xmath207 and the same surface - to - plate distances @xmath216 and analyze how the lateral equilibrium distance @xmath217 between the spheres depends on @xmath101 . by comparing the equilibrium distance @xmath217 with that in the absence of the substrate , @xmath218 , they find that @xmath217 increases or decreases ( depending on the kind of materials of the spheres ) by as much as @xmath219 as the distance from the plate varies between @xmath220 and @xmath221 . they also find that `` the sphere - plate interaction changes the sphere - sphere interaction with the same sign as @xmath101 becomes smaller '' , which means that if the sphere - plate force is repulsive ( attractive ) , the sphere - sphere one will become more repulsive ( attractive ) upon decreasing the distance from the plate @xmath101 . by construction these changes are genuine many - body contributions . in the case of two chemically different spheres , the sign of the many - body force contribution ( i.e. , whether it is attractive or repulsive ) agrees with the sign of the stronger one of the two individual sphere - plate interactions . in fig . [ comparing ] we show schematically the system considered in ref . @xcite [ ( a ) and ( b ) ] and the system considered here [ ( c ) , ( d ) , and ( e ) ] . for the quantum - electrodynamic casimir effect , the dielectric spheres are represented by circles of equal radii , with the green one corresponding to a polystyrene sphere and the red one to a silicon sphere . the semi - infinite plates are represented by gray and yellow rectangles for teflon and gold , respectively . the whole configuration is immersed in ethanol which , for simplicity , is not shown in the figure . the dashed arrows indicate the direction of the strongest of the two pairwise sphere - substrate forces , while the solid arrows indicate the direction of the lateral many - body force . the directions of the arrows in figs . [ comparing ] ( a ) and ( b ) are chosen as to illustrate the findings in ref . @xcite , according to which the sign of the _ lateral _ many - body force is the same as the one of the strongest _ normal _ pairwise sphere - plate force : attractive in ( a ) and repulsive in ( b ) . also in the case of the critical casimir forces , depicted in figs . [ comparing ] ( c ) , ( d ) , and ( e ) , we represent the colloids by circles and the laterally homogeneous semi - infinite substrate by rectangles . the orange filling represents the @xmath22 bc while the blue filling represents the @xmath23 bc . again , the dashed arrows indicate the direction of the stronger one of the two pairwise ( normal ) colloid - substrate forces , while the solid arrows indicate the direction of the lateral many - body contribution to the ccf . as one can infer from fig . [ mb_x ] , the _ lateral _ many - body ccf acting on colloid ( 2 ) is always attractive for the given geometrical configuration , regardless of the bcs . and alluding to the system studied in ref . @xcite , the circles represent the projections of dielectric spheres with equal radii , the green one corresponding to polystyrene and the red one to silicon ; the rectangles represent semi - infinite plates with their surfaces perpendicular to the @xmath222 plane , the gray and the yellow one being teflon and gold , respectively . the system is immersed in ethanol , which is not indicated in the figure . in ( a ) and ( b ) each dashed arrow indicates the direction of the stronger one of the two corresponding pairwise forces between the dielectric spheres and the plate , which turns out to determine the direction of the many - body lateral force acting on the spheres : if the stronger one of the two pairwise forces is attractive [ repulsive ] , the lateral many - body force will also be attractive [ repulsive ] ( see ref . also in the case of the critical casimir interaction [ ( c ) , ( d ) , and ( e ) ] , the circles and rectangles represent projections of spherical colloids and of homogeneous substrates , respectively : orange and blue indicate @xmath22 and @xmath23 bcs , respectively . in ( c ) and ( d ) the pairwise normal forces between each of the two spheres and the substrate are equal : attractive in ( c ) and repulsive in ( d ) . in ( e ) the two pairwise normal forces have opposite directions with the repulsive one being the stronger one @xcite . the corresponding dashed arrows have the same meaning as in ( a ) and ( b ) . according to fig . [ mb_x ] the many - body lateral ccfs are attractive for all three cases ( c ) , ( d ) , and ( e ) . the comparison shows that the systems in ( a ) and ( c ) behave similarly . however , the behavior of system ( b ) has no counterpart for ccfs [ see ( d ) and ( e ) ] . in this figure all surface - to - surface distances equal the sphere radius , which in our notation corresponds to @xmath223 . ] we can also compare our results with those for two atoms close to the surface of a planar solid body . mclachlan @xcite has tackled this problem by treating the solid as a uniform dielectric . by using the image method he derived an expression for the many - body corrections to the pairwise interaction energies , i.e. , the atom - atom ( london ) and the atom - surface energies , in order to obtain the total interaction energy between the two atoms close to the surface . qualitatively , he found that the leading contribution of the many - body correction leads to a repulsion if the atoms are side by side , i.e. , at equal surface - to - substrate distances . rauber et al . @xcite used mclachlan s approach to study the electrodynamic screening of the van der waals interaction between adsorbed atoms and molecules and a substrate . the latter plays a role which is `` analogous to that of the third body in the three - body interaction between two particles embedded in a three - dimensional medium '' . the van der waals interaction between the two atoms at equal distances from the substrate is altered by the presence of the solid substrate and this perturbation is given by @xcite @xmath224 where @xmath225 is the distance between the atoms , @xmath121 is the height above the image plane , which is the same for both atoms , and @xmath226 . the coefficients are given by @xmath227 and @xmath228 with @xmath229 / \left [ \epsilon(i\zeta ) + 1 \right ] ~,\ ] ] where @xmath230 is an imaginary frequency , @xmath231 is the polarizability of the atoms ( with the dimension of a volume ) , and @xmath232 is the dielectric function of the solid ( i.e. , the substrate ) . the lateral force due to the perturbation potential given by eq . , which is the analogue of the many - body contribution to the lateral ccf , follows from differentiating @xmath233 with respect to @xmath225 : @xmath234 ~.\end{gathered}\ ] ] in figure [ mclachan_fig ] we plot the lateral force given by eq . as function of the distance @xmath225 between the two atoms for several ( equal ) distances @xmath121 above the substrate . we use the values provided in ref . @xcite for the coefficients @xmath235 and @xmath236 : @xmath237 ev@xmath238()@xmath239 and @xmath240 ev@xmath238()@xmath239 , which correspond to ne , and @xmath241 ev@xmath238()@xmath239 and @xmath242 ev@xmath238()@xmath239 , which correspond to ar . as one can infer from fig . [ mclachan_fig ] , the many - body contribution to the lateral van der waals force is always repulsive and , as the two atoms approach the substrate , its strength increases . on the other hand , in the case of the many - body contribution to the lateral ccf , we have found that it is attractive for all bcs considered , if the surface - to - surface distances between the spheres and the sphere - surface - to - substrate distances are equal to each other and to the radius of the spheres ( see fig . [ mb_x ] ) . further , we can _ quantitatively _ compare our results with those from refs . @xcite and @xcite . to this end , we assign values to the geometrical parameters characterizing the configuration of the two atoms close to the substrate and compare the results of refs . @xcite and @xcite with those for similar configurations in our model . for example , estimating the many - body contribution to the lateral van der waals force for a configuration of two atoms close to a substrate corresponding to the configuration associated with the black curve the black curve corresponds to a function of temperature but there is no temperature dependence of the van der waals force between the atoms . therefore the comparison has to be carried out by choosing a certain value of @xmath243 . in fig . [ mb_x ] ( i.e. , @xmath244 in the case of the ccf and @xmath245 in the case of the two atoms ) , one obtains from eq . a value for the relative contribution of the many - body force to the lateral force which corresponds to ca . this is comparable with the relative contribution of the many - body force to the lateral ccf sufficiently close to @xmath11 , although in the case of the van der waals force it is repulsive ( fig . [ mclachan_fig ] ) while in the case of the ccf ( fig . [ mb_x ] ) it is attractive . considering the decay of the many - body contribution to the normal ccf as function of the surface - to - surface distance @xmath129 between the spheres ( for @xmath216 ) , we can compare the corresponding decay of the normal many - body force @xmath246 given by the potential @xmath233 in eq . . for small separations @xmath225 between the atoms the many - body contribution to the normal van der waals force increases as @xmath247 , while for large separations it decays as @xmath248 . by analyzing the data shown in fig . [ mb_l_z ] one finds that for @xmath249 the scaling function of the many - body contribution to the normal ccf decays slower than @xmath250 . this means that in this temperature regime the many - body contribution to the normal ccf is much more long ranged than the corresponding contribution to the normal van der waals force in the case of two atoms close to a surface . for fixed @xmath225 the many - body contribution to the lateral van der waals force decays as @xmath251 upon increasing the distance of both atoms from the substrate whereas the normal force on a single atom decays as @xmath252 . as a final remark we point out that we have not found a completely stable configuration for the two colloids ( fig . [ system_sketch ] ) : whenever there is a stable position in the horizontal ( vertical ) direction , the force is nonzero in the vertical ( horizontal ) direction . for example , consider a vertical path with @xmath253 in figs . [ d2_z](b ) and [ d2_x](b ) . from the first one can see that along this path the normal ccf changes from being repulsive to being attractive as the sphere - surface - to - substrate distance for colloid ( 2 ) is increased , implying that there is a vertical position of colloid ( 2 ) in which the normal ccf is zero . however , according to fig . [ d2_x](b ) the lateral ccf is always attractive regardless of the vertical position of colloid ( 2 ) . this means that there is a configuration which is stable only in the normal direction . accordingly , a dumbbell configuration with a rigid thin fiber between the two colloids can levitate over the substrate . whether this configuration is stable with respect to a vertical tilt remains as an open question . of the lateral many - body ccf acting on colloid ( 2 ) for @xmath207 and @xmath254 . the scaling function is shown as function of the scaling variable ratio @xmath255 . the black , red , and green lines correspond to the bcs @xmath157 , @xmath197 , and @xmath156 , respectively . for all three bcs the many - body contribution is not monotonic as function of temperature . quantitatively the green , red , and black curves here should be compared with the black curves in figs . [ d2_x](a ) , [ d2_x](b ) , and [ l_x](a ) , respectively . however , for the data shown in _ this _ figure the error bars ( not shown ) due to limits of the numerical accuracy are between 10@xmath256 and 15@xmath256 . this is the main reason why we refrain from showing what would be an instructive plot such as @xmath257 as a function of @xmath258 for various values of @xmath259 and @xmath170 ( as we did in fig . [ l_x ] ) , which would allow for a direct comparison with the case of atoms . ] , derived from the expression for the excess potential given in refs . @xcite and @xcite [ eq . ] . the forces are plotted as functions of the lateral separation @xmath225 between the two atoms for several equal vertical distances @xmath121 of the atoms from the substrate . the curves correspond to two sets of values for the coefficients @xmath235 and @xmath236 in eq . @xcite , corresponding to ne ( solid lines ) and ar ( dashed lines ) . ] we now turn our attention to the case in which the colloids are vertically aligned with respect to a planar , homogeneous substrate , i.e. , when their centers have the same coordinates in both the @xmath42 and the @xmath260 direction ( see fig . [ vertical_sketch ] ) . we focus on the normal ccf acting on colloid ( 1 ) when the system is immersed in a near - critical binary liquid mixture at its critical concentration . as before we consider @xmath31 bcs corresponding to a strong adsorption preference for one of the two components of the confined liquid . in particular , we consider two three - dimensional spheres of radii @xmath32 and @xmath33 with bcs @xmath34 and ( @xmath35 ) , respectively , facing a laterally homogeneous substrate with bc @xmath36 . colloid ( 1 ) is positioned at a sphere - surface - to - substrate distance @xmath37 and colloid ( 2 ) is at a surface - to - surface distance @xmath261 from colloid ( 1 ) ( see fig . [ vertical_sketch ] ) . the coordinate system @xmath39 is chosen such that the centers of the spheres are located at @xmath262 and @xmath263 so that the distance between the centers , along the @xmath74-axis , is given by @xmath264 . as before , the bcs of the system as a whole are represented by the set @xmath44 , where @xmath45 , @xmath35 , and @xmath46 can be either @xmath47 or @xmath48 . and @xmath33 immersed in a near - critical binary liquid mixture ( not shown ) and close to a homogeneous , planar substrate at @xmath63 . the colloidal particle ( 1 ) with bc @xmath34 is located vertically at the sphere - surface - to - substrate distance @xmath37 , whereas the colloidal particle ( 2 ) with bc @xmath64 is located vertically at the surface - to - surface distance @xmath261 between the spheres . the substrate exhibits bc @xmath36 . the vertical distance between the centers of the spheres along the @xmath74-direction is given by @xmath264 , while the centers of both spheres lie on the vertical axis @xmath265 . in the case of four spatial dimensions the figure shows a three - dimensional cut of the system , which is invariant along the fourth direction , i.e. , the spheres correspond to parallel hypercylinders with one translationally invariant direction , which is @xmath66 [ see eq . ] . ] the normal ccf @xmath266 acting on colloid ( 1 ) along the @xmath74-direction takes the scaling form @xmath267 where @xmath268 ( i.e. , @xmath269 for @xmath27 and @xmath270 for @xmath10 ) , @xmath76 , @xmath271 , and @xmath272 ; @xmath273 . equation describes the singular contribution to the normal force emerging upon approaching @xmath11 . @xmath274 is the force on a hypercylinder divided by its extension in the translationally invariant direction [ see eq . ] . we use the same reference system as the one described by eq . in order to normalize the scaling function defined in eq . according to @xmath275 we calculate the many - body contribution to the normal ccf acting on particle ( 1 ) @xmath276 by subtracting from the total force the sum of the pairwise forces acting on it , i.e. , the colloid - colloid and the colloid - substrate forces [ see eq . ] . this many - body force takes the scaling form @xmath277 in fig . [ mb_vertical_z ] we show the normalized [ see eqs . , , and ] scaling functions @xmath278 of the many - body contribution to the normal ccf acting on colloid ( 1 ) as functions of the scaling variable ratio @xmath279 for two spherical colloids of the same size ( @xmath207 ) . keeping the surface - to - surface distance between the spheres fixed at @xmath280 , we vary the sphere - surface - to - substrate distance @xmath37 for several bcs : @xmath156 in ( a ) , @xmath281 in ( b ) , @xmath197 in ( c ) , and @xmath157 in ( d ) . from figs . [ mb_vertical_z](a ) and ( d ) one can infer that if the colloids have symmetric bcs , the scaling function of the many - body normal ccf acting on colloid ( 1 ) is negative ( i.e. , it is directed towards the substrate ) for any value of @xmath198 . on the other hand , for non - symmetric bcs between the colloids [ figs . [ mb_vertical_z](b ) and ( c ) ] , the many - body contribution to the normal ccf acting on colloid ( 1 ) is positive for any value of @xmath198 . the apparent change of sign in figs . [ mb_vertical_z](b ) and ( c ) is likely to be an artifact occurring within the error bars due to numerical imprecision . the relative contribution of the many - body ccf to the total force is between 10@xmath256 and 15@xmath256 . we point out that this configuration with the the two colloids vertically aligned with respect to the substrate allows for a wide range of interesting aspects which will be further explored in future works . we have investigated critical casimir forces ( ccfs ) for a system composed of two equally sized spherical colloids ( @xmath207 ) immersed in a near - critical binary liquid mixture and close to a laterally homogeneous substrate with @xmath128 boundary condition ( bc ) ( see fig . [ system_sketch ] ) . by denoting the set of bcs of the system as @xmath44 , where @xmath282 corresponds to the bc at colloid @xmath283 and @xmath46 to the bc at the substrate , we have first focused on the total normal and lateral forces acting on one of the colloids [ labeled as `` colloid ( 2 ) '' ] for several geometrical configurations of the system and various combinations of bcs at the colloids . both the normal and the lateral forces are characterized by universal scaling functions [ eqs . and , respectively ] , which have been studied in the one - phase region of the solvent as functions of @xmath284 and @xmath285 . @xmath129 is the surface - to - surface distance between the two colloids , and @xmath178 is the bulk correlation length of the binary mixture in the mixed phase . we have used mean - field theory together with a finite element method in order to calculate the order parameter profiles , from which the stress tensor renders the normalized scaling functions associated with the ccfs . for the scaling function of the total normal ccf acting on colloid ( 2 ) with @xmath145 bc , in the presence of colloid ( 1 ) with @xmath286 , we have found ( fig . [ d2_z](a ) ) that the scaling function changes sign for a fixed value of @xmath284 as the distance @xmath38 between colloid ( 2 ) and the substrate increases , signaling the occurrence of an unstable mechanical equilibrium configuration of a vanishing normal force . for the total normal ccf acting on colloid ( 2 ) with @xmath144 bc and in the presence of colloid ( 1 ) with @xmath133 , we have found ( fig . [ d2_z](b ) ) that the force changes sign upon changing the temperature . for this combination of bcs , the equilibrium configuration of colloid ( 2 ) is stable in the normal direction . without a substrate , at the critical composition of the solvent the ccf between two @xmath22 spheres is identical to the one between two @xmath23 ones . this degeneracy is lifted by the presence of the substrate as one can infer from the comparison of the scaling functions for the lateral ccfs for @xmath156 and @xmath157 bcs [ see figs . [ d2_x](a ) and ( b ) , respectively ] . in the first case , the shape of the scaling function resembles that of the two colloids far away from the substrate , with a minimum at @xmath287 . in the second case this minimum at @xmath288 disappears . these substrate - induced changes are more pronounced if the two spheres are close to the substrate ( fig . [ d2_x ] ) . without a substrate the ccf between spheres of opposite bcs is purely repulsive . in the presence of a substrate the corresponding lateral ccf for @xmath197 bcs turns attractive for large lateral distances @xmath129 ( fig . [ l_x ] ) , which is a pure many - body effect . we have also studied the direction of the total ccf acting on colloid ( 2 ) for various spatial configurations and bcs in order to assess the preferred arrangement of the colloids . for @xmath156 bcs we have found that they tend to aggregate laterally . in this case a collection of colloids with @xmath22 bcs , facing a substrate with the same bc , are expected to form a monolayer on the substrate . for the situation of @xmath157 bcs , we have found that the colloids are expected to aggregate on top of each other . this indicates that a set of several colloids with such bcs are expected to form three - dimensional clusters . these tendencies are enhanced upon approaching the critical point ( fig . [ direction_resulting_force_d2 ] ) . by calculating the pairwise colloid - colloid and colloid - substrate forces and subtracting them from the total force , we have determined the pure many - body contribution to the force acting on colloid ( 2 ) . for the scaling functions associated with the normal many - body ccfs we have found the interesting feature of a change of sign at fixed temperature upon varying the lateral position of colloid ( 2 ) ( fig . [ mb_l_z ] ) . this implies that , for a given temperature , there is a lateral position where the normal many - body ccf is zero , in which case the sum of pairwise forces provides a quantitatively reliable description of the interactions of the system . as expected we have found that the contribution of the many - body ccfs to the total force is large if the colloid - colloid and colloid - substrate distances are small , as well as if the binary liquid mixture is close to its critical point . we have compared our results with corresponding ones for quantum - electrodynamic casimir interactions . to this end we have referred to the results in ref . @xcite for two dielectric spheres immersed in ethanol and facing a plate . these authors analyze the influence of the distance @xmath289 from the plate on the equilibrium separation @xmath217 between the spheres , which are subject to quantum - electrodynamic casimir forces . they find that the lateral many - body force is attractive ( repulsive ) if the stronger one of the two normal sphere - plate forces is attractive ( repulsive ) [ figs . [ comparing](a ) and ( b ) ] . on the other hand , in the case of ccfs we have found that for a configuration in which the surface - to - surface distance between the colloids is equal to the sphere - surface - to - substrate ones and equal to the radius of the spheres , the many - body contribution to the lateral ccf is always attractive , regardless of the bcs ( fig . [ mb_x ] ) . we have also compared our results with the corresponding ones for the case of two atoms close to the planar surface of a solid body . in this respect we have referred to the mclachlan model @xcite for the many - body contribution to the van der waals potential [ see eq . and fig . [ mclachan_fig ] ] and the results from ref . @xcite . from this comparison we have found that if the two atoms are fixed at the same distance from the surface of the solid body , the normal many - body contribution to the total van der waals force decays with the atom - atom distance @xmath225 as @xmath248 for large atom - atom distances . this decay is much faster than the decay we estimate for the many - body contribution to the normal ccf , which within a suitable range appears to be slower than @xmath250 . furthermore , we have found that the many - body contribution to the lateral van der waals force is repulsive while the corresponding many - body ccf is attractive regardless of the set of bcs . finally we have considered the configuration in which the two colloids are vertically aligned with respect to the substrate ( fig . [ vertical_sketch ] ) . we have calculated the many - body contribution to the normal ccf acting on colloid ( 1 ) for two spherical colloids of the same size ( @xmath207 ) keeping the sphere - surface - to - surface distance @xmath280 fixed ( fig . [ mb_vertical_z ] ) . we have varied the sphere - surface - to - substrate distance @xmath37 for several bcs and have found that if the colloids have the same bcs , the many - body contribution to the normal ccf is directed towards the substrate [ figs . [ mb_vertical_z](a ) and ( d ) ] , whereas for colloids with opposite bcs , the many - body contribution to the normal ccf is directed away from the substrate [ figs . [ mb_vertical_z](b ) and ( c ) ] . we have found that the contribution of the many - body ccf to the total force is between 10@xmath256 and 15@xmath256 . t.g.m . would like to thank s. kondrat for valuable support with the computational tools used to perform the numerical calculations . s.d . thanks m. cole for providing ref . @xcite .
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within mean - field theory we calculate the scaling functions associated with critical casimir forces for a system consisting of two spherical colloids immersed in a binary liquid mixture near its consolute point and facing a planar , homogeneous substrate . for several geometrical arrangements and boundary conditions we analyze the normal and the lateral critical casimir forces acting on one of the two colloids .
we find interesting features such as a change of sign of these forces upon varying either the position of one of the colloids or the temperature . by subtracting the pairwise forces from the total force we are able to determine the many - body forces acting on one of the colloids .
we have found that the many - body contribution to the total critical casimir force is more pronounced for small colloid - colloid and colloid - substrate distances , as well as for temperatures close to criticality , where the many - body contribution to the total force can reach up to @xmath0 .
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within the growing field of sociophysics ( see @xcite for the defining paper , and @xcite-@xcite for an impression of the recent state of the art ) , two- and three states galam opinion models and their modifications ( @xcite , @xcite ) play a guiding role in analyzing the process of opinion spreading in communities . these models are centered around the _ local majority rule _ ( l.m.r . ) , which is applied either in a deterministic or a probabilistic way . in the basic _ deterministic _ case , supporters of the ( two or three ) opinions present in a community are randomly distributed over all possible groups of a fixed size . within each group the members adopt the opinion that has the majority in that group , after which all group members are recollected again . in case there is no majority in a group , its members stick to their own opinion ( i.e. , _ neutral _ treatment ; the _ probabilistic _ treatment in case of a tie assigns opinions to the group members according to a certain probability distribution ) . repeated application of this principle generates what is called _ randomly localized dynamics with a local majority rule _ ( @xcite ) . in the basic _ probabilistic _ case , the community members are divided among groups of various sizes according to some probability distribution , and within each group all members adopt one of the possible opinions with a certain probability . in the basic deterministic two states opinion model , fast dynamics occurs in which the opinion that originally has the majority eventually will obtain complete presence at the cost of the alternative opinion . in the probabilistic two states opinion model , the final outcome depends on the probability distributions for group sizes and local adaptation . eventually the state of the community can be either a polarization on the initial majority or minority , or a perfect consensus on both opinions ( see @xcite , which unifies basic probabilistic two states opinion models ) . in @xcite a three states opinion model is introduced in which the community members are randomly distributed over groups of size 3 . within each group the l.m.r . is applied , with the additional rule that in case of a tie all members of the group adopt one of the three opinions according to some probability distribution . it is shown that the dynamics leads to fast polarization on one of the three opinions , which may be an opinion that initially has a minor presence in the community . to add more realism to opinion models , in @xcite the basic deterministic two states galam opinion model is extended by the introduction of so - called contrarians . contrarian _ is a community member who , instead of the opinion it should adopt under the l.m.r . , switches to the alternative opinion . depending on the density of contrarians as well as on group size , their presence either leads to a stabilization of the opinion dynamics in which one opinion ( the one with the lower density of contrarians ) dominates the other , to an equilibrium in which neither opinion dominates ( in case both opinions have equal densities of contrarians ) , or ( in case of relatively large densities of contrarians for both opinions ) to a dynamics in which the dominating opinion constantly alternates between the two opinions . the incorporation of contrarians in opinion dynamics models is a step towards an explanation of the `` hung elections '' outcome in several recent voting events ( the u.s.a . presidential elections in 2000 , the german elections in 2002 and 2005 , as well as the 2006 italian elections , see @xcite , @xcite ) , in which one strategy achieved only a very narrow gain at the cost of another strategy . the origin of contrarian behaviour as well as its implications have been the focus of various other recent studies , see @xcite , @xcite , @xcite , @xcite . in a recent paper ( @xcite ) the basic deterministic two states galam model has been modified by introducing opinion supporters that express what in politics ( and other games ) is called _ inflexible behavior_. an inflexible community member is a supporter that under all conditions sticks to its opinion . under this terminology supporters that switch opinion when in the minority then classify as _ floaters _ , and we shall use this distinction in what follows . in @xcite the effect of inflexible behavior on opinion dynamics is studied for the case that opinion supporters repeatedly meet in groups of fixed size 3 . it is shown that a small density of inflexibles for only one of the two opinions allows for the existence of two local attractors . one of these local attractors is a mixed one , on which both opinions are present and on which the opinion that is supported by inflexibles is a minority . the other attractor is a single state attractor , on which the opinion that is supported by inflexibles has complete majority , i.e. , its density equals 1 , the other opinion being absent . due to the presence of these two attractors , the outcome of the opinion dynamics thus depends on the initial condition , the basin of attraction for the mixed local attractor being relatively small compared to that for the single state attractor . if the density of inflexibles is sufficiently large ( approximately 17@xmath0 ) , the mixed attractor disappears and the single state attractor becomes global . in case both opinions have small and equal densities of inflexibles there are two mixed local attractors . these two attractors are symmetrically situated with regard to a separator on which both opinions are present with density @xmath1 . a change in the density of inflexibles for one of the opinions breaks this symmetry , and a sufficiently large increase may lead to a global attractor on which the opinion with the larger density of inflexibles has the majority , see @xcite . recently , the effect of inflexibles and floaters on opinion dynamics has also been studied in @xcite and @xcite . in this paper we combine the approaches presented in @xcite and @xcite , by allowing for groups composed of inflexibles as well as contrarian and non - contrarian opinion supporters . for clarity we restrict ourselves to groups of fixed size at most 3 . for both opinions we assume fixed densities for the inflexibles . also , we consider the contrarians to be part of the floaters , i.e. , in a given group the contrarians first determine their opinion according to the l.m.r . , and subsequently change to become a floater ( not necessarily a contrarian ) for the alternative opinion ( which thus may be the opinion that the contrarian initially was supporting ) . the presence of contrarians for each opinion is quantitatively expressed as a fixed fraction of the density of floaters of the respective opinion . in case of a tie in groups of size 2 we apply the neutral treatment . after an opinion update , all supporters for both opinions are recollected and then are redistributed again , either as an inflexible or as a non - contrarian or contrarian floater , according to the fixed densities for inflexibles and the fixed fractions of contrarians for the two opinions . we study qualitative characteristics of the opinion dynamics generated by repeated updates . in particular we study changes in the number of equilibria , and changes from monotone to alternating dynamics , due to changes in parameter combinations . a detailed mathematical treatment of the generated opinion dynamics , for groups of arbitrary size , shall be given in a forthcoming paper ( @xcite ) . + + _ notation _ we denote the two opinions by @xmath2 and @xmath3 . the densities of inflexibles for the @xmath2 and @xmath3 opinion are denoted by @xmath4 and @xmath5 respectively , with @xmath6 as well as @xmath7 , and in addition @xmath8 . since the roles of the @xmath2 and @xmath3 opinion are interchangeable in deriving the opinion dynamics , we may without loss of generality assume that @xmath9 , and we shall do so in what follows . the fraction of contrarians among the @xmath2 floaters is denoted by @xmath10 , and @xmath11 denotes the fraction of contrarians among the @xmath3 floaters , with both @xmath12 and @xmath13 . the size of the groups in which opinion supporters meet is denoted by @xmath14 . the density of the @xmath2 opinion at time @xmath15 ( or after @xmath16 updates ) shall be denoted as @xmath17 . note that for given @xmath4 and @xmath5 the density @xmath17 necessarily lies in the interval @xmath18 $ ] ( independent of @xmath14 , @xmath10 or @xmath11 ) . with @xmath19 we denote the function that determines the density of the @xmath2 opinion after application of the l.m.r . followed by the switch of the contrarians . thus , @xmath20 . setting @xmath21 , @xmath22 then gives the density obtained from @xmath17 when the l.m.r . is applied without being followed by the switch of the contrarians . in the appendix tables are given , presenting all possible group compositions in terms of inflexibles and non - contrarian and contrarian floaters for group sizes @xmath23 to 3 , together with the effects of the l.m.r . and the opinion changes of contrarians . it is assumed that the community is sufficiently large to allow for the derivation of the density of each possible group composition in the ensemble of all groups of a fixed size from the densities in the community of the constituents of a group . from these tables the expressions for @xmath24 are obtained . + with @xmath25 we denote the dynamics generated by repeated application of @xmath26 in subsequent timesteps . furthermore , @xmath27 denotes an asymptotically stable equilibrium for @xmath28 , and @xmath29 refers to an asymptotically stable periodic point . + we now turn to the treatment of the opinion dynamics for group sizes @xmath30 and @xmath31 . the case @xmath23 resembles a community in which each member is unaffected by other community members in determining its opinion , and the only changes in opinion come from the contrarians . the contributions to the @xmath2 density after application of the local majority rule is obtained from the second column in table 1 in appendix [ subsection : t1 ] . this column obviously is equal to the first one , since in groups of size @xmath32 local majority is automatically obtained , but is without effect on the opinion densities . these contributions are : @xmath4 for the @xmath2 inflexibles , and @xmath33 for the ( non - contrarian and contrarian ) @xmath2 floaters . their sum is @xmath34 , and we obtain that @xmath35 consequently , each @xmath36 $ ] is a neutrally stable equilibrium for the opinion dynamics . in case only ( non - contrarian and contrarian ) floaters are involved both @xmath4 and @xmath5 are equal to 0 , and we restrict ourselves to the contributions from the second , third , fifth and sixth line in the table . since the l.m.r . leaves each group of size 1 unaffected , a switch by a contrarian in this case necessarily implies a change to the opinion it initially does not support . thus , here also a contribution to the @xmath2 density comes from the group that initially consists of only @xmath3 contrarians , as these will turn into @xmath2 floaters . in this case we obtain for the contribution to the @xmath2 density : @xmath37 the effect of both inflexibles and non - contrarian as well as contrarian floaters is obtained by adding all the expressions in the last column : the contributions @xmath4 due to the invariant density of @xmath2 inflexibles , @xmath38 from the non - contrarian @xmath2 floaters , and @xmath39 from the contrarian @xmath3 floaters . this yields : @xmath40 it follows that if @xmath41 , then @xmath42 is the unique equilibrium for the opinion dynamics @xmath43 . due to its linearity as a function of @xmath44 , expression ( [ eq : l=1general ] ) implies that the dynamical characteristics of this equilibrium are governed solely by the frequencies of the contrarians . the equilibrium is asymptotically stable if and only if @xmath45 . for @xmath46 the equilibrium is approached monotonically , with an increase in the @xmath2 density if and only if its initial value is less then the equilibrium value . for @xmath47 , the function @xmath48 is constant and equals @xmath49 ; the opinion dynamics then reaches its equilibrium in one iteration . for @xmath50 , the equilibrium is approached alternately . for @xmath51 , i.e. , both @xmath52 and @xmath53 , the equilibrium equals @xmath54 and is neutrally stable ; each @xmath36 $ ] different from @xmath54 generates a neutrally stable cycle of length 2 . + on the equilibrium , the @xmath2 opinion has the majority if and only if the inequality @xmath55 holds . thus , for an opinion to achieve the majority it is required that it is being supported by a sufficiently large density of inflexibles , and/or a sufficiently small frequency of contrarians among the floaters . + given densities @xmath4 and @xmath5 of inflexibles for the two opinions , a change in the frequencies of contrarians from 0 into small values @xmath10 and @xmath11 causes the bifurcation from a collection of neutrally stable equilibria for @xmath56 into a unique stable equilibrium for @xmath57 . the opinion which has the majority on this equilibrium is determined by inequality ( [ eq : l=1majority ] ) . in case @xmath58 , the opinion with the smaller frequency of contrarians obtains the majority . conversely , given different frequencies @xmath10 and @xmath11 of contrarian floaters for the two opinions , in the absence of inflexibles the dynamics @xmath59 has @xmath60 as its unique stable equilibrium , on which the opinion with the smaller frequency of contrarians has the majority . fixing sufficiently small densities @xmath4 and @xmath5 of both opinions as inflexibles , this equilibrium slightly shifts but leaves the majority unaltered . in case @xmath61 , in the absence of inflexibles the equilibrium @xmath62 equals @xmath1 , and the introduction of small densities of inflexibles for both opinions changes this equilibrium into one on which the opinion with the larger density of inflexibles takes the majority . figure [ cap : cap1alt ] illustrates these conclusions . figure [ cap : l=1overview ] gives a qualitative overview of the outcomes of the possible opinion dynamics @xmath63 . + figure [ cap : l=1overview ] : an overview of the opinion dynamics of @xmath64 , for values @xmath4 and @xmath5 as indicated , and with @xmath10 and @xmath11 in each pane on the horizontal and vertical axis , respectively , both ranging between 0 and 1 . in each pane the line with negative slope @xmath47 is drawn , and possibly an additional dashed line of positive or zero slope . on the line @xmath47 the function @xmath65 is constant , and the corresponding values of @xmath10 and @xmath11 separate between monotone and alternating dynamics , with the monotone dynamics occurring if @xmath46 , i.e. , below the line . the dashed line , if present , gives the values @xmath66 for which the equilibrium of the opinion dynamics equals @xmath1 , and is determined by the expression @xmath67 . opinion @xmath2 obtains the majority if ( and only if ) @xmath68 holds , i.e. , if @xmath69 and @xmath70 lies above the dashed line . the panes for values @xmath71 for which @xmath72 represent degenerate cases , in the sense that these values allow for only one density @xmath73 for the @xmath2 opinion to occur . in groups of size 2 the number of members that support the @xmath2 or @xmath3 opinion may be equal , in which case a tie occurs . we shall deal with the neural treatment in case of a tie , in which each supporter keeps its own opinion . + appendix [ subsection : t2 ] shows the table for this case . we obtain @xmath75 which is obvious , since in groups of size 2 no majorities can occur , and , in case of a tie , the neutral application of the local majority rule does not have any effect . incorporating the effect of non - contrarian as well as contrarina floaters , the table yields that @xmath76 thus , for groups of size 2 the effect of the neutral application of the local majority rule and the contrarians is the same as for groups of size 1 . group size 3 is the smallest value of @xmath14 for which the local majority rule becomes effective due to possible group compositions in which a majority of one of the two opinions occurs . as a consequence , the generated dynamics allows for features different from those for group sizes 1 and 2 . careful bookkeeping based on the table in appendix [ subsection : t3 ] yields that @xmath78 + for clarity we start the analysis of the generated opinion dynamics with the symmetric case of equal densities of inflexibles and equal fractions of contrarians for both opinions . taking @xmath58 and @xmath80 , we obtain that @xmath81 as an illustration to expression [ eq : l=3symm ] , figure [ cap : cap3alt2 ] shows a collection of graphs of @xmath82 as function of @xmath34 , for @xmath83 and several values of @xmath84 . from expression ( [ eq : l=3symm ] ) the analysis of the generated opinion dynamics is straightforward . we give an overview . + symmetry considerations imply that the dynamics @xmath88 has @xmath89 as an equilibrium , for any choice of @xmath90 $ ] and @xmath91 $ ] . in addition to parameter combinations @xmath92 and @xmath84 for which this equilibrium is unique and stable , there are combinations which allow for an unstable repelling equilibrium @xmath93 in combination with two other , asymptotically stable , equilibria , or with two asymptotically stable periodic points of minimal period 2 . details for these possibilities to appear are derived in appendix [ subsection : l=3symm ] , here we confine ourselves to the outcome . + let the _ critical curves _ @xmath94 and @xmath95 be defined as follows : @xmath96\times[0,1]:(3 - 4\alpha)(1 - 2\gamma)=2\},\ ] ] and @xmath97\times[0,1]:(3 - 4\alpha)(1 - 2\gamma)=-2\}.\ ] ] figure [ cap : cap4alt11 ] shows the curves @xmath94 and @xmath95 in the @xmath98-parameter space . on @xmath94 the derivative @xmath99 equals 1 , whereas on @xmath95 this derivative equals -1 . the two corner areas in figure [ cap : cap4alt11 ] enclosed by either @xmath94 or @xmath95 are the regions of parameter combinations for which @xmath1 is unstable ; outside these regions ( including the curves ) @xmath1 is the unique asymptotically stable equilibrium for @xmath88 , independent of the initial condition . the lower left corner region is the area for which the dynamics @xmath88 has two asymptotically stable equilibria @xmath100 . given parameter combinations @xmath98 in this region , the opinion dynamics eventually will stabilize on an equilibrium on which the opinion with the initial majority will have maintained its majority . in case @xmath101 , this equilibrium is mixed ; if neither inflexibles nor contrarians are present for both opinions , i.e. @xmath102 , the equilibrium is a single state attractor with only one opinion present . these results generalize those obtained in @xcite for the case of equal densities of inflexibles and no contrarians for both opinions . for parameter combinations in the upper left corner region in figure [ cap : cap4alt11 ] , the dynamics has two attracting periodic points of period 2 . here an initial majority does not guarantee the eventual majority , since the dynamics is such that both opinions alternately switch between minority and majority . thus , if both opinions are being supported by equal densities @xmath92 of inflexibles and equal fractions @xmath84 of contrarians among the floaters , for an opinion to obtain the majority it is necessary that @xmath92 as well as @xmath84 are sufficiently small , and that it has the initial majority . also , with increasing @xmath92 ( @xmath84 ) , the maximum value of @xmath84 ( @xmath92 ) for which a majority is attainable decreases . if no inflexibles are present , the fraction of contrarians among the floaters must be less than approximately @xmath103 ( @xmath104 ) for a majority to be realizable , and if the fraction of contrarians among the floaters equals 0 , the density of inflexibles must be less than @xmath105 . + if in the parameter space a combination @xmath98 approaches from within a corner towards one of the two critical curves , then the two additional equilibria or periodic points approach towards @xmath89 ; a withdrawal in the parameter space results in the opposite movement of the additional equilibria or periodic points . it follows that when passing through @xmath94 , the dynamics @xmath88 undergoes a supercritical pitchfork bifurcation , and when passing through @xmath95 the dynamics undergoes a period doubling bifurcation ( flip bifurcation ) . we now return to the general expression ( [ eq : l=3 ] ) and give an overview of the possible outcomes of the dynamics @xmath106 . the analytical background is given in appendix [ subsection : l=3general ] . we distinguish several cases . 1 . @xmath47 . + for @xmath10 and @xmath11 such that @xmath47 , the function @xmath107 is quadratic in @xmath34 . the corresponding opinion dynamics @xmath108 has a unique stable equilibrium in the interval @xmath109 $ ] . for @xmath110 , the function @xmath111 becomes constant and equals @xmath112 ; it allows for a unique stable equilibrium @xmath113 , on which opinion @xmath2 has the majority if and only if @xmath114 . the following two figures distinguish between parameter combinations @xmath4 , @xmath5 and @xmath10 for which the @xmath2 opinion obtains either the majority or minority in equilibrium ( figure [ cap : equimajor ] ) , and for which the equilibrium is approached monotonically or alternately ( figure [ cap : switch ] ) . it follows that with increasing value of @xmath10 , the region of parameter combinations @xmath115 for which opinion @xmath2 obtains the majority , decreases . in addition , if @xmath116 , the @xmath2 opinion can obtain the majority for any value of @xmath4 , provided that @xmath5 is sufficiently small ; if @xmath117 , @xmath4 must be sufficiently large and @xmath5 sufficiently small for an @xmath2 majority to occur . + + the different panes , distinguished by different values of @xmath10 , have @xmath4 on the horizontal axis and @xmath5 on the vertical one . each pane shows the line @xmath72 and in addition the dashed line @xmath118 of parameter values for which the equilibrium value @xmath119 equals @xmath1 . below the dashed lines the equilibrium value lies above @xmath1 , i.e. , opinion @xmath2 then obtains the majority the region @xmath120 is not involved in the analysis.,width=755,height=604 ] + the different panes show in white the region of parameters @xmath71 for which the equilibrium @xmath119 is approached monotonically ; the black regions indicate parameter combinations for which the equilibrium is approached alternately . for @xmath121 , the derivative of @xmath122 in the equilibrium equals 0 for all parameter values @xmath71 , and the equilibrium is reached in one iteration . the line @xmath72 refers again to the degenerate cases for which the opinion dynamics of density @xmath34 is restricted to a single density @xmath123 . parameter combinations @xmath71 for which @xmath120 are not involved in the analysis.,width=755,height=604 ] 2 . + the expression @xmath125 for determining the equilibria is @xmath126 the number of solutions is determined by its discriminant , which is denoted by + @xmath127 . the expression for the discriminant is derived in appendix [ subsection : l=3general ] ; here we discuss its implications . + for parameter combinations @xmath128 such that @xmath129 , the equation @xmath130 has a unique real solution . if @xmath131 , there are three real solutions . however , these solutions do not necessarily have to belong to the interval @xmath109 $ ] ( but if a solution lies in this interval , it clearly is an equilibrium for the dynamics @xmath106 ) . if @xmath132 there are three real solutions , of which at least two coincide ; if this happens in the interval @xmath133 $ ] , the parameter combination is at a bifurcation point , discriminating between dynamics with either a unique equilibrium or three equilibria . if at the bifurcation point exactly two of the three solutions coincide , the coinciding solutions form a semistable equilibrium . + figure [ cap : p3selectionalt ] shows a collection of signplots for the discriminant , for values @xmath4 and @xmath5 as indicated , and with @xmath10 and @xmath11 for each signplot between 0 and 1 . in addition the outcome of the analysis for parameter combinations @xmath134 is included , as well as the results of the analysis for combinations @xmath135 . figure [ cap : p3part3 ] in appendix [ subsection : l=3general ] shows an extended version of figure [ cap : p3selectionalt ] that includes more values of @xmath4 and @xmath5 . it shows that for both @xmath4 and @xmath5 larger than 0.3 , there is not much qualitative change in the signplots of the determinant @xmath136 . + , width=755,height=680 ] + figure [ cap : p3selectionalt ] : a collection of panes , for values @xmath4 and @xmath5 as indicated , and @xmath10 and @xmath11 for each pane ranging between 0 and 1 , with @xmath10 on the horizontal axis and @xmath11 on the vertical axis . in each pane the signplot of the discriminant @xmath137 is shown for points @xmath138 for which @xmath139 . white areas represent the parameter combinations with a positive discriminant ( i.e. , combinations for which the corresponding opinion dynamics has a unique equilibrium ) , and in dark regions the discriminant is negative ( the corresponding opinion dynamics then has 3 different equilibria , but not necessarily in the interval @xmath109 $ ] ) . in each pane , on the line @xmath47 a density plot of the equilibrium density @xmath140 is given . an increasing gray level indicates an increasing equilibrium value , with graylevel 0 ( black ) representing a density 0 , and graylevel 1 ( white ) representing density 1 . furthermore , in panes for which @xmath79 holds , on the line @xmath61 in the black regions ( i.e. , a negative discriminant ) in white the points are indicated for which the equilibrium @xmath93 for @xmath106 is unstable ; other points on the lines @xmath61 ( for @xmath79 ) indicate parameter combinations for which @xmath93 is stable ( as follows from figure [ cap : cap4alt11 ] ) . + + the discriminant becomes singular for parameter combinations @xmath141 with @xmath47 . in approaching such parameter combinations for which @xmath142 , the value of @xmath137 goes to @xmath143 . for @xmath144 , the limit generically equals @xmath145 when this point is approached from the region @xmath146 ; the limit equals @xmath143 in case it is approached from the other side , i.e. , from the region @xmath147 . ( in case @xmath148 is approached along the zero set of @xmath136 , i.e. , in each pane in figures [ cap : p3selectionalt ] and [ cap : p3part3 ] along the boundary that distinguishes between the white and black regions and touches with the line @xmath47 , the limit clearly equals 0 . ) + our further discussion of the opinion dynamics @xmath106 is based on figures [ cap : p3selectionalt ] and [ cap : p3part3 ] . instead of a detailed analytical treatment , we continue with a number of characteristic outcomes of the opinion dynamics . a first characteristic that draws attention in figure [ cap : p3selectionalt ] is the existence of a wedge - shaped region of parameter combinations @xmath149 with negative discriminant for sufficiently small values of all four parameters . for the cases with both @xmath79 and @xmath61 within this region , we already found the existence of two attracting equilibria , symmetrically positioned with respect to a third , unstable equilibrium @xmath1 . we therefore expect also to find a similar pattern of three equilibria in @xmath109 $ ] for deviations from such symmetric cases within the wedge - shaped region . in @xcite it has been derived that this is indeed the case in the absence of contrarians , i.e. , for parameter combinations for which @xmath21 , and for @xmath4 and @xmath5 sufficiently small . figure [ cap : qallsmall ] , which shows a number of graphs of functions @xmath150 for relatively small values @xmath4 , @xmath5 , @xmath10 and @xmath11 , implies the same pattern : in case the determinant @xmath136 is negative , the opinion dynamics has two attracting equilibria that are separated by an unstable one . the two attracting equilibria differ with respect to the opinion by which they are dominated . by leaving the wedge - shaped area , a bifurcation in the opinion dynamics occurs on its boundary @xmath151 . generically , when moving from inside the wedge - shaped area towards this boundary , the unstable equilibrium and one of the two stable equilibria move towards each other , and at the bifurcation point merge ( thus causing a supercritical saddle - node bifurcation ) . once the boundary has been crossed , the region of parameters with a positive discriminant is entered , and the dynamics is left with one attracting equilibrium . on this equilibrium opinion @xmath2 dominates if the upper part of the boundary is crossed , i.e. , when @xmath152 ; opinion @xmath3 has the majority when the right - hand side of the boundary is passed , on which @xmath153 holds . this is also illustrated in figure [ cap : qallsmall ] . the occurrence of such a bifurcation may lead to a drastic change in the outcome of the opinion dynamics : whereas inside the wedge - shaped region the outcome of the opinion dynamics depends on the initial condition , outside the wedge - shaped area the opinion dynamics will end on the unique equilibrium , independent of the initial condition . at the bifurcation point at the endpoint of the sharp region of the wedge - shaped area a supercritical pitchfork bifurcation occurs , in which the three equilibria merge together into one attracting equilibrium . figure [ cap : qallsmall ] : four panes of graphs of functions @xmath154 for relatively small values @xmath4 , @xmath5 , @xmath10 and @xmath11 , with @xmath71 as indicated below each pane . in each of the four panes , @xmath10 and @xmath11 satisfy @xmath155 . the color coding of the graphs is as follows : red corresponds to @xmath156 , for green @xmath157 , for blue @xmath158 , for purple @xmath159 , and for magenta @xmath160 . i.e. , in the corresponding panes in figure [ cap : p3selectionalt ] we traverse the line @xmath155 from its upper left point on the @xmath156 axis to its lower right point on the @xmath161 axis , thus passing through regions with positive , zero as well as negative discriminant . above each pane the densities for opinion @xmath2 are presented , as obtained by the corresponding opinion dynamics @xmath162 , with initial density @xmath89 . + + the white regions in figure [ cap : p3selectionalt ] ( and also in figure [ cap : p3part3 ] ) are formed by the parameter combinations for which the discriminant @xmath137 is positive . the corresponding opinion dynamics then have a unique equilibrium , which ( for the parameter combinations in figure [ cap : p3selectionalt ] ) is approached monotonically . figure [ cap : psmallposdis ] shows a number of graphs @xmath163 for parameter combinations with a positive discriminant . the figure indicates that for small values of @xmath10 and large values of @xmath11 opinion @xmath2 dominates in equilibrium , and that the dominion shift towards the alternative opinion if the fraction of contrarians among the @xmath2 floaters increases and that among the @xmath3 floaters decreases . four panes of graphs of functions @xmath154 for relatively small values @xmath4 and @xmath5 , and with @xmath10 and @xmath11 satisfying @xmath164 . the values of @xmath10 and @xmath11 are indicated below each of the four panes . the color coding of the graphs is as follows : red corresponds to @xmath156 , for green @xmath160 , for blue @xmath165 , for purple @xmath121 , and for magenta @xmath166 . i.e. , for given @xmath71 , we traverse the line @xmath164 from its upper left point on the @xmath156 line to its lower right point on the @xmath161 line . the discriminant @xmath136 for the exposed parameter values is positive , indicating a unique equilibrium for the corresponding opinion dynamics.,title="fig:",width=642,height=642 ] + for given parameters @xmath4 and @xmath5 , crossing the boundary of the white area in any direction away from the lower left corner leads to the occurrence of a saddle - node bifurcation , now however outside the domain @xmath18 $ ] of the functions @xmath167 ( maintaining an attracting equilibrium in the domain ) . therefore in the black region thus entered , the opinion dynamics also is characterized by a unique attracting equilibrium . proceeding towards the upper right corner , the line of parameter combinations @xmath138 satisfying @xmath47 is crossed . on this line the discriminant @xmath137 is singular , and the corresponding opinion dynamics have been analyzed in [ sect : general].1 . + in the black area in the upper right corner , for equal and sufficiently small values @xmath58 and sufficiently large and equal values @xmath80 it has been derived earlier that @xmath168 has a unique unstable equilibrium @xmath93 , which causes the convergence of the dynamics towards an attracting periodic orbit of period 2 . neither of the two opinions then achieves the definite majority . the values @xmath92 and @xmath84 for which this occurs have been derived in [ subsection:4.1 ] , and are represented in figures [ cap : p3selectionalt ] and [ cap : p3part3 ] by white line segments in the upper right corners . for these parameter combinations the discriminant of the equation @xmath169 is negative and thus has three different solutions , of which two are situated outside the domain @xmath109 $ ] . continuity arguments imply that this behavior will be maintained for parameter combinations sufficiently close to these line segments . figure [ cap : psmallupperright ] illustrates this . if the parameter combinations are sufficiently far removed from these line segments but @xmath10 and @xmath11 are still relatively large ( i.e. , for given @xmath4 and @xmath5 , in the upper right corner ) , the dynamics will converge alternately to a unique equilibrium . i.e. , by moving away from the manifold determined by the constraints @xmath79 and @xmath61 with large values @xmath10 and @xmath11 , a flip bifurcation occurs in which the attracting periodic 2 orbit collapses to an attracting equilibrium point . this is illustrated by figure [ cap : psmallupperrightconv ] . on the attractor the dominion shifts towards opinion @xmath3 with increasing @xmath10 and decreasing @xmath11 . figure [ cap : psmallupperright ] : four panes of graphs of functions @xmath154 for relatively small values @xmath4 and @xmath5 , and with @xmath10 and @xmath11 satisfying @xmath170 . the values of @xmath10 and @xmath11 are indicated below each of the four panes . the color coding of the graphs is as follows : red corresponds to @xmath171 , for green @xmath172 , for blue @xmath173 , for purple @xmath174 , and for magenta @xmath52 . i.e. , for given @xmath71 , we traverse the line @xmath170 from its upper left point on the @xmath53 line to its lower right point on the @xmath52 line . above each pane the densities for opinion @xmath2 are presented , as obtained by the corresponding opinion dynamics @xmath162 , with initial density @xmath89 . figure [ cap : psmallupperrightconv ] : four panes of graphs of functions @xmath154 for relatively small values @xmath4 and @xmath5 , and with @xmath10 and @xmath11 satisfying @xmath175 . the values of @xmath10 and @xmath11 are indicated below each of the four panes . the color coding of the graphs is as follows : red corresponds to @xmath166 , for green @xmath171 , for blue @xmath172 , for purple @xmath173 , and for magenta @xmath52 . i.e. , for given @xmath71 , we traverse the line @xmath175 from its upper left point on the @xmath53 line to its lower right point on the @xmath52 line . above each pane the densities for opinion @xmath2 are presented , as obtained by the corresponding opinion dynamics @xmath162 , with initial density @xmath89 . + + we end our discussion by presenting some additional opinion dynamics @xmath162 for parameter combinations from both the regions with positive and negative discriminant . we consider cases with @xmath176 and with @xmath177 , i.e. , a case with small @xmath4 and intermediate @xmath5 , and one with both @xmath4 and @xmath5 intermediate . the corresponding graphs of @xmath150 are represented in figures [ cap : pasmallbbig ] and [ cap : pabintmed ] , for several values of @xmath10 and @xmath11 . all cases allow for a unique attracting equilibrium . high values of both @xmath10 and @xmath11 lead to alternating convergence . furthermore , a decrease in the fraction of contrarians among the floaters of an opinion increases the density of this opinion in equilibrium . figure [ cap : pasmallbbig ] : the left column shows four panes of graphs of functions @xmath178 , for different combinations of @xmath10 and @xmath11 . the color coding for the different rows is given in the following table , with in the second to fifth column the values of @xmath10 : + + [ [ subsection : l=3symm ] ] @xmath77 : analysis for the fully symmetric case @xmath79 and @xmath61 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ the derivative ( with respect to @xmath34 ) @xmath179 of the function @xmath180 as given by expression ( [ eq : l=3symm ] ) in the equilibrium @xmath93 equals @xmath181 . there are two additional equilibria @xmath182\ ] ] if and only if @xmath183 and @xmath184 . if the two additional equilibria exist they are symmetrically positioned on opposite sides of @xmath1 , and asymptotically stable ; the equilibrium @xmath1 then is unstable , with @xmath185 . + for @xmath92 and @xmath84 such that @xmath186 and @xmath187 , the equilibrium @xmath1 also is unstable , with @xmath188 . in this case the dynamics @xmath189 has two asymptotically stable periodic points @xmath190 of minimal period 2 , symmetrically positioned with respect to @xmath1 : @xmath191.\ ] ] 1 . @xmath110 : expression ( [ eq : l=3 ] ) equals @xmath112 , and allows for a unique stable equilibrium @xmath113 , on which opinion @xmath2 has the majority if and only if @xmath114 . @xmath193 , @xmath47 : the function @xmath194 is quadratic in @xmath34 . the discriminant + @xmath127 for the equation @xmath125 equals @xmath195 . the opinion dynamics @xmath108 has a unique equilibrium + @xmath196 + + @xmath197 + + in the interval @xmath109 $ ] . the derivative in the equilibrium equals + + @xmath198 . 3 . @xmath124 . the expression @xmath125 for determining the equilibria now is @xmath199 its discriminant is @xmath200 with + @xmath201 , + @xmath202 , + @xmath203 , + @xmath204 , + and + @xmath205 , + + @xmath206 . + + figure [ cap : p3part3 ] shows , in three parts , a signplot of the discriminant for this case . in addition the results of the analysis for combinations @xmath135 is included . figures _ a , b , and c _ shows a collection of panes with signplots of the discriminant @xmath136 , for values @xmath4 and @xmath5 as indicated , and @xmath10 and @xmath11 for each pane ranging between 0 and 1 , with @xmath124 ; @xmath10 is on the horizontal axis and @xmath11 on the vertical axis . the white areas represent the parameter combinations with a positive discriminant , and in dark regions the discriminant is negative . on the line @xmath47 in each pane the discriminant becomes singular . in panes for which @xmath79 holds , on the line @xmath61 in the black regions ( i.e. , a negative discriminant ) in white the points are indicated for which the equilibrium @xmath93 for @xmath106 is unstable ; other points on the lines @xmath61 ( for @xmath79 ) indicate parameter combinations for which @xmath93 is stable ( as follows from figure [ cap : cap4alt11]).,title="fig:",width=529,height=604 ] + 999 s. galam , y. geffen , y. shapiro , math . j. sociology 9 , 1 , ( 1982 ) s. galam , physica a , 333 , 453 - 460 ( 2004 ) s. galam , europhys . , 70 , 6 , 705 - 711 ( 2005 ) s. galam , f. jacobs , physica a , 381 , 366 - 376 ( 2007 ) s. galam , qual . quant , 41 , 579 - 589 ( 2007 ) s. gekle , l. peliti , and s. galam , eur . j. b 45 , 569 - 575 ( 2005 ) f. jacobs , s. galam , _ in preparation _ de la lama , j.m . lpez , and h. s. wio , europhys . 72 , 5 , 851 - 857 ( 2005 ) m. mobilia , a. petersen , and s. redner , j. stat . p08029 ( 2007 ) j.j . schneider , international jpurnal of modern physics c , vol . 15 , 5 , 659 - 674 ( 2004 ) d. stauffer , j.s . s martins , physica a , 334 , 558 - 565 ( 2004 ) h.s . wio , m.s . de la lama , and j. m. lpez , physica a , 371 , 108 - 111 ( 2006 ) l. zhong , d .- f . zheng , b. zheng , and p.m. hui , physical review , e 72 , 026134 ( 2005 )
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we assume a community whose members adopt one of two opinions .
each member appears as an inflexible , or as a non - contrarian or contrarian floater .
an inflexible adheres to its opinion , whereas a floater may change into a floater of the alternative opinion .
the occurrence of this change is governed by the local majority rule : members meet in groups of a fixed size , and a floater changes provided its opinion has the minority in its group .
a non - contrarian floater keeps the opinion it adopts under the local majority rule , whereas a contrarian floater adopts the alternative opinion .
we determine the dynamics generated by repeated opinion changes , and study its dependence on the densities of inflexibles and the frequencies of contrarians among the floaters .
we restrict ourselves to groups of size at most 3 .
20 pt * two opinions dynamics generated by inflexibles and non - contrarian and contrarian floaters * + + f. jacobs + + _ section theoretical biology + institute of biology + leiden university + kaiserstraat 63 + nl-2311 gp leiden + the netherlands + + e - mail address : [email protected]_ + + s. galam
+ + _ centre de recherche en epistmologie applique ( crea ) + ecole polytechnique + 1 rue descartes + 75005 paris + france + + e - mail address : [email protected]_ + + author for correspondence : f. jacobs + + * running title : * two opinions dynamics + + * acknowledgement * f. jacobs appreciates the support of the research underlying this paper by cost grant cost - stsm - p10 - 01215 .
* keywords : * sociomathematics , sociophysics , opinions dynamics , local majority rule , contrarian behaviour , floating behaviour + + * note concerning the current version of the text : due to the large sizes of some figures , the accompanying captions to these figures do not fit together with the figures on the same page . therefore , these captions are placed in the main text , instead of in the sequence of figures at the end of the text . *
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You are an expert at summarizing long articles. Proceed to summarize the following text:
currently a number of intense mid - infrared light sources are being developed @xcite , spurred on by their uses in sub - attosecond pulse generation @xcite , strong - field holography @xcite and laser - induced electron diffraction @xcite . the low frequency and high intensity of these new sources mean that the tunneling picture is an appropriate framework for describing how these light sources interact with atoms and molecules . in this work we deal with the process of dissociative tunneling ionization in molecules , where a static electric field tunnel ionizes an electron , after which the nuclei dissociate . to our knowledge this is the first work on the theory of dissociative tunneling ionization . in the theory we treat the nuclear and electronic degrees freedom on an equal footing and fully quantum mechanically . the reflection principle @xcite is often used to describe the process of dissociative ionization . this principle can be applied within the framework of the born - oppenheimer ( bo ) approximation to relate the nuclear kinetic energy release ( ker ) spectrum to the nuclear wave function . it was formulated as early as 1928 @xcite , and later put on a more rigorous foundation @xcite ( see also references therein for a list of early uses ) . in time - dependent cases where the time - scale of the electric field is shorter than that of nuclear motion , one assumes that the electrons make an instantaneous frank - condon transition to a dissociative electronic state . the probability distribution in the new electronic state is then the absolute value squared of the initial nuclear wave function times some dipole coupling factor . in case this dipole coupling factor is almost constant , the wave packet that enters the dissociative state is practically identical to the initial nuclear wave function . classical energy conservation then dictates that the nuclear ker spectrum can be obtained by reflecting the nuclear wave function in the dissociative potential curve . this is the regime considered mostly in the literature . in the time - independent tunneling case , the electronic ionization rate takes the same role as the dipole coupling factor in the time - dependent case , that is , it multiplies the nuclear wave function before it enters the continuum . however , this electronic rate has an exponential dependence on the internuclear coordinate and can by no means be considered constant ( as was also pointed out in ref . it is therefore essential to consider the effect of this additional factor on the ker spectrum ; the spectrum can not be found simply by reflection of the nuclear wave function . imaging of the nuclear wave function is made possible through the reflection principle , by applying it in reverse on a measured ker spectrum @xcite . this is often referred to as coulomb explosion imaging . in the tunneling case the exponential dependence of the electronic rate on the internuclear coordinate means that the product of the electronic rate and the nuclear wave function is essentially different from the bare nuclear wave function , and the electronic rate therefore needs to be included to image the nuclear wave function based on the ker spectrum . in ref . @xcite it was demonstrated that the bo approximation breaks down for weak fields . in this case the weak - field asymptotic theory ( wfat ) @xcite provides us with accurate results for the ker spectrum . the paper is organized as follows . in sec . [ sec : theory ] the theory for dissociative tunneling ionization of homonuclear molecules is outlined . we derive an exact expression for the ker spectrum and a corresponding expression in the bo approximation . section [ sec:1d - calculation ] exemplifies the theory with numerical reduced dimensionality calculations . numerically exact ker spectra are compared to ker spectra obtained in the bo approximation using the reflection principle . imaging of the nuclear part of the wave function from the ker spectrum is demonstrated . section [ sec : conclusion - outlook ] concludes the paper . atomic units @xmath0 are used throughout . we consider a three - body system consisting of two heavy nuclei with masses @xmath1 and charges @xmath2 , and one electron with mass @xmath3 and charge @xmath4 . in the center - of - mass frame these have coordinates @xmath5 and @xmath6 fulfilling @xmath7 . let us introduce the reduced masses @xmath8 effective charge @xmath9 and jacobi coordinates @xmath10 we assume that the orientation of the internuclear axis @xmath11 is fixed in space . we also assume that the field is directed along the @xmath12-direction , @xmath13 and choose to consider @xmath14 for definiteness . due to the azimuthal symmetry of the molecule only the polar angle @xmath15 between @xmath11 and @xmath16 matters . this @xmath15 angle takes the role as an external parameter , and we omit explicit reference to it in the following . with these assumptions we can write the time - independent schrdinger equation ( se ) within the single - active - electron approximation as @xmath17 \psi({\mathbf{r}},r ) & = 0,\label{eq : schrodinger}\end{aligned}\ ] ] where the effective @xmath18 potential describes how the nuclei interact with each other and the effective @xmath19 potential describes how the electron interacts with the nuclei . for a system with several electrons the @xmath18 potential represents the bo potential of the system without the active electron . in this work , we assume that @xmath18 is monotonically decreasing , i.e. , it corresponds to a purely dissociative bo curve . we assume that the nuclei can not pass through each other . this gives the boundary condition @xmath20 and we consider eq . ( [ eq : schrodinger ] ) in the interval @xmath21 . we also impose outgoing - wave boundary conditions in the electronic coordinate @xmath22 , the exact form of these will be specified below . with these boundary conditions the wave function we seek as a solution of eq . ( [ eq : schrodinger ] ) is a siegert state @xcite , with a complex energy @xmath23 , where @xmath24 is the ionization rate , and it is normalized by @xmath25 the outgoing - wave boundary condition in the electronic coordinate means that the solution we seek to eq . ( [ eq : schrodinger ] ) describes tunneling of the electron . this tunneling is followed by dissociation of the nuclei for the considered class of strictly dissociative potentials @xmath18 . in the following we will describe the energy distribution of the dissociated nuclei . our aim is to describe the energy distribution of the nuclei , i.e. , the ker spectrum , after the molecule is ionized by tunneling of the electron . to this end we need to consider the problem in the @xmath26 limit , where the electron is far away from the nuclei . in this limit we assume that the electron - nuclear interaction potential takes the form @xmath27 where @xmath28 is the total charge of the remaining core system . this assumption makes our problem separable in electron and nuclear coordinates in this asymptotic region . by seeking the partial solutions in the form @xmath29 , eq . ( [ eq : schrodinger ] ) can be written as the separated equations @xmath30 f({\mathbf{r}},k ) & = 0,\label{eq : as_x_eq}\\ \left[- \frac{1}{2 m } { \frac { d^2 } { d { r } ^2 } } + { u}(r ) - e_{r}\right ] g(r , k ) & = 0,\label{eq : as_r_eq } \end{aligned}\ ] ] with separation constants given by @xmath31 where we assume @xmath32 and @xmath33 is the wave number for the state @xmath34 . equation ( [ eq : zero_bc ] ) amounts to @xmath35 we choose the continuum solutions of eq . ( [ eq : as_r_eq ] ) to be real and normalized by @xmath36 the conditions eqs . ( [ eq : zero_bc_g])-([eq : g_norm ] ) completely specify the nuclear problem eq . ( [ eq : as_r_eq ] ) . the electronic problem eq . ( [ eq : as_x_eq ] ) has a potential consisting of a coulomb term and a linear field term . this problem is separable in parabolic coordinates @xcite , which we will therefore use . first we introduce mass - scaled quantities @xmath37 then the following form of the parabolic coordinates is introduced ( as in ref . @xcite ) [ eq : parab_coord ] @xmath38 with this choice of coordinates a potential barrier forms in the @xmath39 coordinates and therefore @xmath39 takes the role as the tunneling coordinate. in the asymptotic region @xmath40 eq . ( [ eq : as_x_eq ] ) has a solution that is a linear combination of partial solutions of the form @xcite @xmath41 where the outgoing - wave @xmath42 is given by @xmath43\right),\label{eq : f_eta}\end{aligned}\ ] ] @xmath44 is the ionization channel function defined by @xmath45 \phi_\nu(\xi,\varphi ) & = \beta_\nu \phi_\nu(\xi,\varphi),\label{eq : xi_phi_ad_eq}\end{aligned}\ ] ] and @xmath46 is a set of parabolic quantum numbers labeling the different ionization channels , see fig . [ fig : parab_coord ] . with our choice of @xmath14 the potential in eq . ( [ eq : xi_phi_ad_eq ] ) goes to infinity as @xmath47 goes to infinity , so the parabolic channels @xmath48 are purely discrete . . the gray paraboloid is the same for a smaller value of @xmath39 . the electron is ionized in the negative @xmath12 direction due to its negative charge , given that the electric field points in the positive @xmath12-direction . the @xmath44 states [ eq . ( [ eq : xi_phi_ad_eq ] ) ] live in the constant @xmath39 paraboloids . the colors in the blue / red surface illustrates an example of the nodal structure of such a @xmath44 state . the curvature of the paraboloids means that the @xmath44 states are bound . this means that @xmath39 is the only coordinate where we have to consider the wave function at infinity , i.e. , @xmath39 is the tunneling coordinate . for large @xmath39 the polar angle @xmath15 , which specifies the orientation of the molecule , does not matter for the asymptotic form of the wave function in parabolic coordinates , though it matters for the size of the coefficients [ eq . ( [ eq : spectrum_ampl_def ] ) ] . ] the full wave function can be expressed as a linear combination , discrete in @xmath48 , continuous in @xmath49 , of the @xmath50 products , @xmath51 where the asymptotic expansion coefficient @xmath52 can be calculated by @xmath53 @xmath54 indicates integration w.r.t . the coordinates @xmath47 and @xmath55 over their full range . note that the polar angle @xmath15 , which we suppressed in the notation , only enters eq . ( [ eq : spectrum_ampl_def ] ) through the wave function @xmath56 . the ker dissociation spectrum into the channel @xmath48 is defined in terms of these expansion coefficients by @xmath57 this is the main observable of interest . by inserting eq . ( [ eq : f_eta ] ) and eq . ( [ eq : spectrum_ampl_def ] ) and assuming @xmath58 to be real , which is approximately the case for small @xmath59 , we obtain @xmath60 the exact ker spectrum in the channel @xmath48 can thus be obtained by projecting the wave function on the channel state @xmath44 , and further projecting this on the continuum states @xmath34 of the @xmath18 potential . the total ker spectrum can then be obtained by summing over all the channels @xmath61 in the @xmath62 limit the total rate can be obtained by @xmath63 now that we have a recipe for finding the exact ker spectrum , we consider some approximations for ease of predictions and gain in physical insight . we first consider the bo approximation , which appears in the limit @xmath64 . in this limit @xmath65 , and the wave function takes the form @xmath66 . the electronic and nuclear part of bo wave function fulfills the bo equations @xmath67 \psi_e({\mathbf{r}};r ) & = 0,\label{eq : bo_elec_eq}\\ \left[- \frac{1}{2 m } { \frac { d^2 } { d { r } ^2 } } + u(r)+e_e(r;f ) - e_{\text{bo}}(f)\right ] \chi(r ) & = 0.\label{eq : bo_nuc_eq } \end{aligned}\ ] ] we impose zero boundary condition for the nuclear wave function @xmath68 and the following normalizations @xmath69 in the asymptotic limit @xmath40 the electronic eq . ( [ eq : bo_elec_eq ] ) takes the same form as eq . ( [ eq : as_x_eq ] ) , and it can be written in parabolic coordinates in the same manner . the electronic wave function then takes the outgoing - wave form @xmath70 where @xmath42 is from eq . ( [ eq : f_eta ] ) and @xmath44 are solutions of eq . ( [ eq : xi_phi_ad_eq ] ) , with @xmath58 replaced by @xmath71 in both . the asymptotic coefficient @xmath72 defines the ionization amplitude in channel @xmath48 @xcite . the partial electronic ionization rate is given by @xmath73 by considering the flux of the electron probability through a surface at large negative @xmath12 , one can show that in the weak field limit the total electronic rate @xmath74 is given as a sum over @xmath48 of all the partial electronic rates . by inserting the bo wave function into eq . ( [ eq : spec_expr ] ) we obtain @xmath75 the separation of electronic and nuclear coordinates in the bo approximation means that this expression for the ker spectrum does not contain any explicit reference to electronic coordinates , as opposed to the expression for the exact spectrum eq . ( [ eq : spec_expr ] ) . equation ( [ eq : spec_bo ] ) is similar to a result previously put forward in the literature [ eq . ( 1 ) of ref . @xcite ] , except that the correct complex ionization amplitude @xmath72 was taken as @xmath76 . in the cases we have considered , the phase variations of @xmath72 are sufficiently small that they can be safely neglected , explaining the successful use of the aforementioned replacement in ref . @xcite , but this is not generally true . to evaluate the integral in eq . ( [ eq : spec_bo ] ) we will use the reflection principle @xcite . at the heart of the reflection principle lies an important mathematical component which we denote the reflection approximation @xcite . this approximation amounts to setting @xmath77 which is exact in the @xmath78 limit . in eq . ( [ eq : g_as_delt ] ) @xmath79 is the classical turning point potentials , so there is only one classical turning point . ] for the @xmath34 function defined by @xmath80 in order to determine the derivative @xmath81 the form of the dissociative @xmath18 potential must be known . inserting eq . ( [ eq : g_as_delt ] ) in eq . ( [ eq : spec_bo ] ) yields @xmath82 this result shows that using the reflection approximation in conjunction with the bo approximation we obtain a ker spectrum that is expressed as a product of a jacobian factor , the electronic rate and the field - dressed nuclear wave function [ eq . ( [ eq : bo_nuc_eq ] ) ] . this is a lot simpler to calculate than evaluating either integrals in eqs . ( [ eq : spec_expr ] ) or ( [ eq : spec_bo ] ) , and is easily reversed to give a way to image the field - dressed nuclear wave function , and it is applicable to any molecule with a dissociative bo curve . the exact electronic rate @xmath83 is often not available , since finding it requires solving the electronic problem eq . ( [ eq : bo_elec_eq ] ) , which is a highly non - trivial task for many systems . in such cases the weak - field asymptotic theory ( wfat ) @xcite can be employed to obtain the rate . wfat is an analytic theory which expresses the ionization rate in terms of properties of the field - free state . it is applicable in the weak - field limit . let @xmath84 and @xmath85 denote the adiabatic eigenvalues and eigenfunctions solving eq . ( [ eq : xi_phi_ad_eq ] ) for @xmath86 with the field - free electronic energy @xmath87 replacing @xmath58 . ref . @xcite provides analytic expressions for these quantities . in terms of these the asymptotic field - free electronic wave function can be written @xmath88 where @xmath89 the electronic wfat rate is then given by @xcite @xmath90 where the field factor @xmath91 is defined by @xmath92 and the asymptotic coefficients @xmath93 can be found from the electronic wave function by inversion of eq . ( [ eq : field_free_elec_wf ] ) @xmath94 wfat can also be applied for the exact state , and not just in the bo approximation as above . in this section we will give the pertaining formulas . let , as before , @xmath95 and @xmath96 denote the adiabatic eigenvalues and eigenfunctions solving eq . ( [ eq : xi_phi_ad_eq ] ) for @xmath86 now with the field - free energy @xmath97 . in terms of these the asymptotic field - free wave function can be written @xcite @xmath98 where @xmath99 the wfat @xcite yields the following expression for the ker spectrum @xmath100 where the field factor @xmath101 is given by @xmath102 and the field - free asymptotic coefficients @xmath103 can be found by inversion of eq . ( [ eq : field_free_wf ] ) @xmath104 solving eq . ( [ eq : schrodinger ] ) in 3d is a computationally heavy task , so we have used a 1d model to illustrate our central points . in this section we compare exactly calculated ker spectra with those obtained through the bo approximation , eq . ( [ eq : spec_result_bo ] ) , and the wfat , eq . ( [ eq : spec_wfat ] ) , within this 1d model . in the following we will consider a model of h@xmath105 as an example . the potentials we consider are thus @xmath106 with @xmath107 and @xmath108 . the interaction between the nuclei and the electrons @xmath109 is described by a soft - core coulomb potential . the function @xmath110 is chosen in such a way that the bo potential of this potential reproduces the bo potential energy curve of 3d h@xmath105 @xcite . we use the method described in ref . @xcite to solve the 1d equivalent of eq . ( [ eq : schrodinger ] ) given by @xmath111 \psi(z , r ) & = 0.\label{eq : schrodinger_1d}\end{aligned}\ ] ] in the 1d model the index @xmath48 , which describes what happens in the paraboloids of constant @xmath39 transversal to @xmath12 , is of no meaning , and it hence does not appear in any of the 1d equivalents of the 3d equations . the 1d equivalent of the exact ker spectrum eq . ( [ eq : spec_expr ] ) is @xmath112 equations ( [ eq : spec_bo ] ) and ( [ eq : spec_result_bo ] ) apply to the 1d case with appropriately redefined quantities . the wfat expressions eqs . ( [ eq : rate_wfat_elec ] ) , ( [ eq : field_factor_elec ] ) and ( [ eq : spec_wfat ] ) , ( [ eq : full_field_factor ] ) are the same as in the 1d case , but the asymptotic coefficients are now found from @xmath113 and @xmath114 in eq . ( [ eq:1d_asymp_coeff ] ) , @xmath115 denotes the field - free 1d electronic bo wave function . ( light blue shaded area in the lower @xmath116 bo curve ) is multiplied by the electronic rate @xmath117 ( dashed purple line ) and reflected in the dissociative @xmath18 bo curve to give a ker spectrum ( solid blue line in upper right corner , [ eq . ( [ eq : spec_result_bo ] ) ] ) , using the relation @xmath118 to translate @xmath49 into @xmath119 . this is compared to the exact ker spectrum @xmath120 ( red dashed line , [ eq . ( [ eq : spec_expr_1d ] ) ] ) . a field strength of @xmath121 was used for this calculation . the solid gray line in the lower part of the figure shows the field - free nuclear wave function @xmath122 . the surface plot in the upper part of the figure shows the continuum states @xmath34 of the @xmath18 potential , these are solutions of eq . ( [ eq : as_r_eq ] ) . ] figure [ fig : spec_1d_ground ] illustrates how the bo approximation can be used in conjunction with the reflection principle to determine the ker spectrum . the figure shows a calculation for the ground state of the h@xmath105 model at @xmath121 . the field dressed nuclear wave function @xmath123 is multiplied by the electronic rate @xmath117 . the exponential dependence of the electronic rate @xmath117 on the internuclear coordinate means that the product @xmath124 ( see [ eq . ( [ eq : spec_result_bo ] ) ] ) has its maximum at a value of @xmath125 , which is significantly different from the maximum of the bare nuclear wave function at @xmath126 . this in turn means that the transition to the continuum which is determined by the product @xmath124 and not the bare nuclear wave function is far from vertical in @xmath119 with respect to the initial nuclear wave function , and the spectrum peaks at a lower energy around @xmath127 and not at @xmath128 . using wfat within the bo approximation we can make a statement about in which direction the maximum of the spectrum shifts when the field is varied . in these approximations the main dependence of the electronic rate on the field is contained in the exponent @xmath129 , see eq . ( [ eq : field_factor_elec ] ) . the electronic energy @xmath87 , in terms of which @xmath130 is defined , generally depends very much on the system considered . in the case of h@xmath105 it is a monotonically increasing function of @xmath119 , since when the two potential wells around each of the nuclei start to overlap the electron is more tightly bound . this in turn means that the electronic rate is an increasing function of @xmath119 , as can also be seen in fig . [ fig : spec_1d_ground ] . when the strength of the field increases the exponent @xmath129 grows , but at the same time the slope of this exponent with respect to @xmath119 decreases , since @xmath131 is multiplied by a smaller number . the smaller slope means that the location of the maximum of the product @xmath124 is shifted less from the maximum of @xmath123 as the field strength increases , and conversely , as the field strength is decreased the maximum of the product @xmath124 is shifted more towards larger @xmath119 . these shifts are directly reflected in the spectrum , which is given as the reflection of the @xmath124 product in the bo and reflection approximations . figure [ fig : spec_1d ] shows ker spectra obtained using as initial state the first vibrationally exited state of h@xmath105 . we have chosen to show these results as they are for the lowest state with a non - trivial nodal structure in @xmath119 . in the figure two different field strengths are considered . in the top panel we see that the nodal structure of the nuclear wave function is reflected in the ker spectrum , although one peak is a lot larger than the other . this asymmetry can be understood in the bo approximation , see eq . ( [ eq : spec_result_bo ] ) , as due to the fact that the electronic rate @xmath117 has an exponential dependence on @xmath119 . in the wfat it can be understood as resulting from the exponential dependence of the field factor [ eq . ( [ eq : full_field_factor ] ) ] on @xmath49 . for the lower field strength the structures at @xmath132 are not visible as the ker spectrum falls below the numerical precision limit of our calculation . ( [ eq : spec_expr_1d ] ) ] . dashed dotted ( blue ) line : bo combined with reflection principle [ eq . ( [ eq : spec_result_bo ] ) ] . short dashed ( green ) line : wfat [ 1d equivalent of eq . ( [ eq : spec_wfat ] ) ] . the insets show the normalized ker spectra on a linear scale . the critical field for use of bo [ eq . ( [ eq : f_bo ] ) ] is for h@xmath105 : @xmath133 . ( a ) @xmath134 and @xmath135 . ( b ) @xmath136 and @xmath137 . ] for the large field strength [ fig . [ fig : spec_1d](a ) ] we see that the bo ker spectrum has a shape much closer to the exact ker spectrum than for the lower field strength . also the maximum value of the bo ker spectrum is more than an order of magnitude closer to the maximum value of the exact ker spectrum for the larger field strength . this can be understood on the basis of the retardation argument provided in ref . @xcite : the bo approximation is expected to hold as long as the electron is close enough to the nuclei that the time it takes for the electron to go to its present location from the nuclei is shorter than the time it takes for the nuclei to move . a typical electron velocity can be estimated as @xmath138 , where @xmath139 is the equilibrium internuclear distance , which for h@xmath105 is @xmath140 . a typical time scale for the nuclear motion can be estimated as @xmath141 , where @xmath142 is obtained by expanding the bo potential around @xmath139 to second order @xmath143 . using these estimates ref . @xcite defines a critical distance @xmath144 such that for @xmath145 we expect bo to work well , while for @xmath146 we expect it to break down . since the magnitude of the wave function is essentially unchanged after the tunneling , the bo approximation is expected to work well when the outer turning point is within this @xmath147 distance , so a critical field @xmath148 can be estimated , such that the bo approximation is expected to give good results for larger fields , but fail for smaller fields . the two field strengths of fig . [ fig : spec_1d ] lies on either side of this critical field , which for the system under consideration is @xmath133 . as we increase the field strength further the bo gives even better results . for the lower field strength where bo fails we can apply the wfat , see sec . [ sec : full_wfat ] . in fig . [ fig : spec_1d ] we see that the shape of the wfat ker spectrum indeed is closer to the exact ker spectrum than the bo ker spectrum for the weaker field strength , and it is also closer in magnitude to the maximum value . for the larger field strength the wfat ker spectrum is further from the exact ker spectrum in both shape and magnitude . ( upper right corner , [ eq . ( [ eq : spec_expr_1d ] ) ] ) at @xmath121 the magnitude of the asymptotic wave function has been found by reversing the reflection principle , giving @xmath149 , using the relation @xmath118 to translate @xmath49 into @xmath119 . from this , the field - dressed nuclear wave function has been imaged by dividing with the electronic rate @xmath117 and normalizing . in the lowest part of the plot , the short dashed ( purple ) line shows this imaging using the exact electronic rate @xmath74 , the long dashed ( red ) line shows it using the bo wfat approximation @xmath150 [ eq . ( [ eq : rate_wfat_elec ] ) ] . the solid gray line shows the field - free nuclear wave function @xmath122 . the shaded ( light blue ) area shows the field - dressed nuclear wave function @xmath123 . the surface plot in the upper part of the figure shows the continuum states @xmath34 . ] the field dressed nuclear wave function can be imaged from a measurement of the ker spectrum by inverting eq . ( [ eq : spec_result_bo ] ) for fields sufficiently large that the bo approximation applies . to demonstrate this we have taken the exact ker spectrum from our calculation at @xmath121 for the first vibrationally exited state and divided it by the jacobian factor and the electronic rate to obtain an image of the nuclear density . since an experimental ker spectrum is typically not known on an absolute scale , we have then normalized this quantity . in a calculation on a more complicated system than the one considered here the exact electronic rate is often not available , so we also show the result using the wfat approximation for the electronic rate [ eq . ( [ eq : rate_wfat_elec ] ) ] . the results are compared to the nuclear wave function known from the calculation in our model in fig . [ fig : reconstructed_chi ] . they do not agree perfectly , but the nodal structure is correctly reproduced . for smaller field strengths where the bo is not applicable this type of imaging is not possible . the ker spectrum , however , does give us access to the asymptotic wave function , as it is the norm square of the expansion coefficients of this , see eq . ( [ eq : wf_expansion ] ) . for the cases we have looked at , the phase of the asymptotic coefficient @xmath151 varies very little over the range where it has support . in our model we have access to the full wave function , and this we show in fig . [ fig : wf ] . the imaging through the 1d equivalent of eq . ( [ eq : wf_expansion ] ) would only give access to the part at large negative @xmath152 . in the classically allowed region at large negative @xmath152 the maximum of the wave function follows a classical trajectory . this is a prediction of the wkb theory , which applies as long we are not too close to the turning line . the classical trajectories can be found using newton s second law @xmath153 [ eq : clas_traj_newton ] a tempting choice of initial condition for the differential eqs . ( [ eq : clas_traj_newton ] ) would be to choose the @xmath154 values at the intersection of the outer turning line and the maximum ridge of the wave function , with zero velocity in both @xmath152 and @xmath119 direction . however , the wkb fails near the turning line , and therefore we can not expect the wave function to follow a classical trajectory here . instead we have chosen as initial condition some point at the maximum of the wave function at a large negative @xmath152 value away from the turning line . the influence of the @xmath155 potential can be neglected for sufficiently large negative @xmath152 , in this region we can write the separated energy conservation equations @xmath156 [ eq : clas_traj_energy ] the initial velocities have then been determined from eqs . ( [ eq : clas_traj_energy ] ) , using the real part of the total ( quantum ) energy for @xmath157 and the @xmath49 at which the ker spectrum @xmath120 [ eq . ( [ eq : spec_expr_1d ] ) ] peaks . the classical trajectories shown in fig . [ fig : wf ] were found using such initial conditions , and then propagated inwards . from fig . [ fig : wf ] it can be seen that contrary to the exact wave function , the position of the ridge of the bo wave function in @xmath119 does not change with @xmath152 . this is expected as the bo approximation appears in the limit of infinite nuclear mass , so classical motion in the nuclear coordinate is not possible . the asymptotic wave function that we can image using eq . ( [ eq : wf_expansion ] ) is therefore a non - bo wave function . it might seem strange that the bo is able to give the correct ker spectrum when the spectrum is the norm square of the expansion coefficients of the asymptotic wave function , and the bo gives a wrong description of this asymptotic wave function . however , the fact that the bo wave function does not obtain a probability current ( or velocity in the classical picture ) in the @xmath119-direction does not alter its projection on the continuum states . the important point is whether the bo wave function is similar to the exact wave function as it emerges at the outer turning line after tunneling , and this is the case if the turning line is within the critical bo distance @xmath147 [ eq . ( [ eq : z_bo ] ) ] . in fig . [ fig : wf ] we also see that for the larger field strength the tunneling is completed before the critical bo distance is reached , contrary to at the smaller field strength . we see that for the large field strength the electronic and full turning lines agree quite well in the region where most of the wave function is localized , but for the smaller field strength they do not . for @xmath158 . solid purple lines : full turning lines @xmath159 . the long dashed red line shows for each @xmath12 the @xmath119 at which the wave function @xmath160 has its maximum . the solid pink line shows a classical trajectory [ eq . ( [ eq : clas_traj_newton ] ) ] . the black dot at the end of the classical trajectory is the exit point @xmath161 determined from the maximum of the spectrum @xmath162 ( see main text ) . the short dashed lines are the simple straight line estimates for the tunneling and initial classical motion described around eq . ( [ eq : directions ] ) . ] one can notice that a phenomenon reminiscent of light refraction occurs for the wave function around the turning line in fig . [ fig : wf ] . it is evident , that the direction in which the maximum of the wave function moves changes noticeably at the turning line , when the wave function escapes from the classically forbidden tunneling region into the classically allowed region . the change of direction is due to the two different types of motion involved . when the wave function emerges from the tunneling region it has essentially zero average velocity in the @xmath119 direction . this means that we can apply the reflection principle in reverse on the spectrum to find the @xmath163 coordinate at which the maximum of the wave function emerges from the tunneling region by the relation @xmath164 , where @xmath162 is the value of @xmath49 for which the spectrum @xmath120 has its maximum . the @xmath12 value corresponding to this @xmath163 can then by found by considering the turning line @xmath165 . in fig . [ fig : wf_max ] we see that near the turning line the location of the wave function ridge differs from the classical trajectory . this is expected , since the prediction that the wave function ridge should follow a classical trajectory comes from wkb theory , which fails near the turning line . nevertheless , we can roughly describe the dissociative tunneling ionization process in two steps . first the system tunnels from the central region around @xmath166 to the exit point @xmath161 . this motion can roughly be described by a straight line from the maximum of the nuclear wave function @xmath123 that has the largest @xmath119 value , since this is the maximum that will dominate the tunneling , to the exit point . notice that this tunneling is not simply the electron tunneling out , but a correlated process involving both the electronic and nuclear degrees of freedom . in the classically allowed region the initial direction of the wave function from the exit point can be found from the classical trajectory : the initial slope of the classical trajectory that starts at the exit point @xmath161 with zero velocity in both @xmath12 and @xmath119 directions can be found to be @xmath167 this is not exactly the trajectory that describes the motion of the wave function ridge , but it is quite close . these two directions are different as they come from different types of motion , and hence we see the refraction - like phenomenon at the turning line . we have formulated theory for the dissociative tunneling ionization process , and derived exact formulas for the ker spectrum , as well as approximations in the framework of the bo and reflection approximations . we have demonstrated that the reflection principle can be used in conjunction with the bo approximation to image the field - dressed nuclear wave function from the ker spectrum . for weaker fields , where the bo approximation fails , the wfat can be used to find the ker spectrum . we have also demonstrated a qualitative difference between asymptotic bo and exact wave functions , as the latter shows classical motion in the nuclear coordinate , whereas the former does not move at all due to the infinite nuclear mass of the bo approximation . around the turning line the wave function exhibits a behavior similar to refraction of light . this work was supported by the erc - stg ( project no . 277767-tdmet ) , and the vkr center of excellence , quscope . the numerical results presented in this work were performed at the centre for scientific computing , aarhus http://phys.au.dk / forskning / cscaa/. o. i. t. acknowledges support from the ministry of education and science of russia ( state assignment no . 3.679.2014/k ) . 29ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1126/science.1218497 [ * * , ( ) ] \doibase http://dx.doi.org/10.1038/ncomms7611 [ * * , ( ) ] link:\doibase 10.1103/physrevx.5.021034 [ * * , ( ) ] link:\doibase 10.1364/optica.2.000623 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.98.013901 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.111.033002 [ * * , ( ) ] link:\doibase 10.1126/science.1198450 [ * * , ( ) ] link:\doibase 10.1103/physreva.84.043420 [ * * , ( ) ] link:\doibase 10.1038/nature10820 [ * * , ( ) ] link:\doibase 10.1103/physrev.32.858 [ * * , ( ) ] \doibase http://dx.doi.org/10.1063/1.1679721 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.108.073202 [ * * , ( ) ] link:\doibase 10.1103/physreva.90.063408 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.92.163004 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.82.3416 [ * * , ( ) ] link:\doibase 10.1103/physreva.58.426 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.111.153003 [ * * , ( ) ] link:\doibase 10.1103/physreva.84.053423 [ * * , ( ) ] link:\doibase 10.1103/physreva.89.013421 [ * * , ( ) ] link:\doibase 10.1103/physrev.56.750 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.79.2026 [ * * , ( ) ] link:\doibase 10.1103/physreva.82.023416 [ * * , ( ) ] @noop _ _ ( , ) link:\doibase 10.1103/physreva.87.043426 [ * * , ( ) ] link:\doibase 10.1103/physreva.53.2562 [ * * , ( ) ] link:\doibase 10.1103/physreva.67.043405 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.98.253003 [ * * , ( ) ] link:\doibase 10.1103/physreva.91.013408 [ * * , ( ) ]
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we present a theoretical study of the dissociative tunneling ionization process .
analytic expressions for the nuclear kinetic energy distribution of the ionization rates are derived .
a particularly simple expression for the spectrum is found by using the born - oppenheimer ( bo ) approximation in conjunction with the reflection principle .
these spectra are compared to exact non - bo _ ab initio _ spectra obtained through model calculations with a quantum mechanical treatment of both the electronic and nuclear degrees freedom . in the regime where the bo approximation is applicable imaging of the bo nuclear wave function
is demonstrated to be possible through reverse use of the reflection principle , when accounting appropriately for the electronic ionization rate . a qualitative difference between the exact and bo wave functions in the asymptotic region of large electronic distances
is shown .
additionally the behavior of the wave function across the turning line is seen to be reminiscent of light refraction . for weak fields , where the bo approximation does not apply ,
the weak - field asymptotic theory describes the spectrum accurately .
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the ubiquitous camassa holm ( ch ) equation @xcite @xmath8 where @xmath9 is a constant , has been extensively studied due to its many intriguing properties . the aim of this paper is to construct a metric that renders the flow generated by the camassa holm equation lipschitz continuous on a function space in the conservative case . to keep the presentation reasonably short , we restrict the discussion to properties relevant for the current study . more precisely , we consider the initial value problem for with periodic initial data @xmath10 . since the function @xmath11 satisfies equation with @xmath12 , we can without loss of generality assume that @xmath13 vanishes . for convenience we assume that the period is @xmath14 , that is , @xmath15 for @xmath16 . the natural norm for this problem is the usual norm in the sobolev space @xmath6 as we have that @xmath17 ( by using the equation and several integration by parts as well as periodicity ) for smooth solutions @xmath3 . even for smooth initial data , the solutions may develop singularities in finite time and this breakdown of solutions is referred to as wave breaking . at wave breaking the @xmath18 and @xmath19 norms of the solution remain finite while the spatial derivative @xmath20 becomes unbounded pointwise . this phenomenon can best be described for a particular class of solutions , namely the multipeakons . for simplicity we describe them on the full line , but similar results can be described in the periodic case . multipeakons are solutions of the form ( see also @xcite ) @xmath21 let us consider the case with @xmath22 and one peakon @xmath23 ( moving to the right ) and one antipeakon @xmath24 ( moving to the left ) . in the symmetric case ( @xmath25 and @xmath26 ) the solution @xmath3 will vanish pointwise at the collision time @xmath27 when @xmath28 , that is , @xmath29 for all @xmath30 . clearly the well - posedness , in particular , lipschitz continuity , of the solution is a delicate matter . consider , e.g. , the multipeakon @xmath31 defined as @xmath32 , see figure [ fig : peakcol ] . for simplicity , we assume that @xmath33 . then , we have @xmath34 and the flow is clearly not lipschitz continuous with respect to the @xmath18 norm . , which vanishes at @xmath27 , for @xmath35 ( on the left ) and @xmath36 ( on the right ) . the solid curve depicts the multipeakon solution given by @xmath32.,title="fig:",width=257 ] , which vanishes at @xmath27 , for @xmath35 ( on the left ) and @xmath36 ( on the right ) . the solid curve depicts the multipeakon solution given by @xmath32.,title="fig:",width=257 ] [ fig : peakcol ] our task is here to identify a metric , which we will denote by @xmath2 for which conservative solutions satisfy a lipschitz property , that is , if @xmath3 and @xmath4 are two solutions of the camassa holm equation , then @xmath37\ ] ] for any given , positive @xmath38 . for nonlinear partial differential equations this is in general a quite nontrivial issue . let us illustrate it in the case of hyperbolic conservation laws @xmath39 in the scalar case with @xmath40 , @xmath16 , it is well - known @xcite that the solution is @xmath41-contractive in the sense that @xmath42 in the case of systems , i.e. , for @xmath43 with @xmath44 it is known @xcite that @xmath45 for some constant @xmath46 . more relevant for the current study , but less well - known , is the recent analysis @xcite of the hunter saxton ( hs ) equation @xmath47 or alternatively @xmath48 which was first introduced in @xcite as a model for liquid crystals . again the equation enjoys wave breaking in finite time and the solutions are not lipschitz in term of convex norms . the hunter saxton equation can in some sense be considered as a simplified version of the camassa holm equation , and the construction of the semigroup of solutions via a change of coordinates given in @xcite is very similar to the one used here and in @xcite for the camassa holm equation . in @xcite the authors constructed a riemannian metric which renders the conservative flow generated by the hunter saxton equation lipschitz continuous on an appropriate function space . for the camassa holm equation , the problem of continuation beyond wave breaking has been considered by bressan and constantin @xcite and holden and raynaud @xcite ( see also xin and zhang @xcite and coclite , karlsen , and holden @xcite ) . both approaches are based on a reformulation ( distinct in the two approaches ) of the camassa holm equation as a semilinear system of ordinary differential equations taking values in a banach space . this formulation allows one to continue the solution beyond collision time , giving either a global conservative solution where the energy is conserved for almost all times or a dissipative solution where energy may vanish from the system . local existence of the semilinear system is obtained by a contraction argument . going back to the original function @xmath3 , one obtains a global solution of the camassa holm equation . in @xcite , bressan and fonte introduce a new distance function @xmath49 which is defined as a solution of an optimal transport problem . they consider two multipeakon solutions @xmath50 and @xmath51 of the camassa holm equation and prove , on the intervals of times where no collisions occur , that the growth of @xmath52 is linear ( that is , @xmath53 for some fixed constant @xmath46 ) and that @xmath52 is continuous across collisions . it follows that @xmath54 for all times @xmath55 that are not collision times and , in particular , for almost all times . by density , they construct solutions for any initial data ( not just the multipeakons ) and the lipschitz continuity follows from . as in @xcite , the goal of this article is to construct a metric which makes the flow lipschitz continuous . however , we base the construction of the metric directly on the reformulation of the equation which is used to construct the solutions themselves , and we use some fundamental geometrical properties of this reformulation ( relabeling invariance , see below ) . the metric is defined on the set @xmath56 which includes configurations where part of the energy is concentrated on sets of measure zero ; a natural choice for conservative solutions . in particular , we obtain that the lipschitz continuity holds for all times and not just for almost all times as in @xcite . let us describe in some detail the approach in this paper , which follows @xcite quite closely in setting up the reformulated equation . let @xmath57 denote the solution , and @xmath58 the corresponding characteristics , thus @xmath59 . our new variables are @xmath58 , @xmath60 where @xmath61 corresponds to the lagrangian velocity while @xmath62 could be interpreted as the lagrangian cumulative energy distribution . in the periodic case one defines @xmath63 then one can show that @xmath64_0^\xi , \end{aligned } \right.\ ] ] is equivalent to the camassa holm equation . global existence of solutions of is obtained starting from a contraction argument , see theorem [ th : global ] . the issue of continuation of the solution past wave breaking is resolved by considering the set @xmath56 ( see definition [ def : d ] ) which consists of pairs @xmath65 such that @xmath66 if @xmath67 and @xmath68 is a positive radon measure with period one , and whose absolutely continuous part satisfies @xmath69 . with three lagrangian variables @xmath70 versus two eulerian variables @xmath65 , it is clear that there can be no bijection between the two coordinate systems . if two lagrangian variables correspond to one and the same solution in eulerian variables , we say that the lagrangian variables are relabelings of each other . to resolve the relabeling issue we define a group of transformations which acts on the lagrangian variables and lets the system of equations invariant . we are able to establish a bijection between the space of eulerian variables and the space of lagrangian variables when we identify variables that are invariant under the action of the group . this bijection allows us to transform the results obtained in the lagrangian framework ( in which the equation is well - posed ) into the eulerian framework ( in which the situation is much more subtle ) . to obtain a lipschitz metric in eulerian coordinates we start by constructing one in the lagrangian setting . to this end we start by identifying a set @xmath71 ( see definition [ def : f ] ) that leaves the flow invariant , that is , if @xmath72 then the solution @xmath73 of with @xmath74 will remain in @xmath71 , i.e. , @xmath75 . next , we identify a subgroup @xmath76 , see definition [ def : g ] , of the group of homeomorphisms on the unit interval , and we interpret @xmath76 as the set of relabeling functions . from this we define a natural group action of @xmath76 on @xmath71 , that is , @xmath77 for @xmath78 and @xmath79 , see definition [ def : group ] and proposition [ prop : action ] . we can then consider the quotient space @xmath80 . however , we still have to identify a unique element in @xmath71 for each equivalence class in @xmath80 . to this end we introduce the set @xmath81 , see , of elements in @xmath71 for which @xmath82 and @xmath83 . this establishes a bijection between @xmath80 and @xmath81 , see lemma [ lemma : bijection ] , and therefore between @xmath81 and @xmath56 . finally , we define a semigroup @xmath84 on @xmath81 ( definition [ def : bars ] ) , and the next task is to identify a metric that makes the flow @xmath85 lipschitz continuous on @xmath81 . we use the bijection between @xmath81 and @xmath56 to transport the metric from @xmath81 to @xmath56 and get a lipschitz continuous flow on @xmath56 . in @xcite , the authors define the metric on @xmath81 by simply taking the norm of the underlying banach space ( the set @xmath81 is a nonlinear subset of a banach space ) . they obtain in this way a metric which makes the flow continuous but not lipschitz continuous . as we will see ( see remark [ rem : compmetric ] ) , this metric is stronger than the one we construct here and for which the flow is lipschitz continuous . in @xcite , for the hunter saxton equation , the authors use ideas from riemannian geometry and construct a semimetric which identifies points that belong to the same equivalence class . the riemannian framework seems however too rigid for the camassa holm equation , and we have not been able to carry out this approach . however , we retain the essential idea which consists of finding a semimetric which identifies equivalence classes . instead of a riemannian metric , we use a discrete counterpart . note that this technique will also work for the hunter saxton and will give the same metric as in @xcite . a natural candidate for a semimetric which identifies equivalence classes is ( cf . ) @xmath86 which is invariant with respect to relabeling . however , it does not satisfy the triangle inequality . nevertheless it can be modified to satisfy all the requirements for a metric if we instead define , see definition [ def : alberto ] , the following quantity @xmath87 where the infimum is taken over all finite sequences @xmath88 which satisfy @xmath89 and @xmath90 . one can then prove that @xmath91 is a metric on @xmath81 , see lemma [ lemma : distance ] . finally , we prove that the flow is lipschitz continuous in this metric , see theorem [ th : stab ] . to transfer this result to the eulerian variables we reconstruct these variables from the lagrangian coordinates as in @xcite : given @xmath92 , we define @xmath66 by ( see definition [ def : ftoe ] ) @xmath93 for any @xmath94 such that @xmath95 , and @xmath96 . we denote the mapping from @xmath71 to @xmath56 by @xmath97 , and the inverse restricted to @xmath81 by @xmath98 . the natural metric on @xmath56 , denoted @xmath2 , is then defined by @xmath99 for two elements @xmath100 in @xmath56 , see definition [ def : dd ] . the main theorem , theorem [ th : main ] , then states that the metric @xmath2 is lipschitz continuous on all states with finite energy . in the last section , section [ sec : topology ] , the metric is compared with the standard norms . two results are proved : the mapping @xmath101 is continuous from @xmath6 into @xmath56 ( proposition [ prop : cont1 ] ) . furthermore , if @xmath102 is a sequence in @xmath56 that converges to @xmath65 in @xmath56 . then @xmath103 in @xmath7 and @xmath104 ( proposition [ prop : cont2 ] ) . the problem of lipschitz continuity can nicely be illustrated in the simpler context of ordinary differential equations . consider three differential equations : @xmath105 straightforward computations give as solutions @xmath106 we find that @xmath107 thus we see that in the regular case we get a lipschitz estimate with constant @xmath108 uniformly bounded as @xmath55 ranges on a bounded interval . in the second case we get a lipschitz estimate uniformly valid for all @xmath109 . in the final example , by restricting attention to strictly increasing solutions of the ordinary differential equations , we achieve uniqueness and continuous dependence on the initial data , but without any lipschitz estimate at all near the point @xmath110 . we observe that , by introducing the riemannian metric @xmath111 an easy computation reveals that @xmath112 let us explain why this metric can be considered as a riemannian metric . the euclidean metric between the two points is then given @xmath113 where the infimum is taken over all paths @xmath114\to{\mathbb{r}}$ ] that join the two points @xmath115 and @xmath116 , that is , @xmath117 and @xmath118 . however , as we have seen , the solutions are not lipschitz for the euclidean metric . thus we want to measure the infinitesimal variation @xmath119 in an alternative way , which makes solutions of equation lipschitz continuous . we look at the evolution equation that governs @xmath119 and , by differentiating with respect to @xmath120 , we get @xmath121 and we can check that @xmath122 let us consider the real line as a riemannian manifold where , at any point @xmath16 , the riemannian norm is given by @xmath123 for any tangent vector @xmath124 in the tangent space of @xmath125 . from , one can see that at the infinitesimal level , this riemannian norm is exactly preserved by the evolution equation . the distance on the real line which is naturally inherited by this riemannian is given by @xmath126 where the infimum is taken over all paths @xmath114\to{\mathbb{r}}$ ] joining @xmath115 and @xmath127 . it is quite reasonable to restrict ourselves to paths that satisfy @xmath128 and then , by a change of variables , we recover the definition . the riemannian approach to measure a distance between any two distinct points in a given set ( as defined in ) requires the existence of a smooth path between points in the set . in the case of the hunter saxton ( see @xcite ) , we could embed the set we were primarily interested in into a convex set ( which is therefore connected ) and which also could be regularized ( so that the riemannian metric we wanted to use in that case could be defined ) . in the case of the camassa holm equation , we have been unable to construct such a set . however , there exists the alternative approach which , instead of using a smooth path to join points , uses finite sequences of points , see . we illustrate this approach with equation . we want to define a metric in @xmath129 which makes the semigroup of solutions lipschitz stable . given two points @xmath130 , we define the function @xmath131 as @xmath132 \frac{\bar{x}-x}{x^{1/2}}&\text { if $ x<\bar{x}$}. \end{cases}\ ] ] the function @xmath133 is symmetric and @xmath134 if and only if @xmath135 , but @xmath133 does not satisfy the triangle inequality . therefore we define ( cf . ) @xmath136 where the infimum is taken over all finite sequences @xmath137 such that @xmath138 and @xmath139 . then , @xmath140 satisfies the triangle inequality and one can prove that it is also a metric . given @xmath141 such that @xmath142 , we denote @xmath143 and @xmath144 the solution of with initial data @xmath145 and @xmath146 , respectively . after a short computation , we get @xmath147 hence , @xmath148 so that @xmath149 and the semigroup of solutions to is a contraction for the metric @xmath140 . it follows from the definition of @xmath133 that , for @xmath150 with @xmath151 , we have @xmath152 it implies that @xmath153 satisfies @xmath154 where @xmath155 , which is also the definition of the riemann integral , so that @xmath156 and the metric we have just defined coincides with the riemannian metric we have introduced . note that if we choose @xmath157 then does not hold ; we have instead @xmath158 , which is the triangle inequality . thus , for @xmath159 as defined by with @xmath133 replaced by @xmath160 , we get @xmath161 it is also possible to check that , for @xmath160 , we can not get that @xmath162 for any constant @xmath46 for any @xmath145 and @xmath146 and @xmath163 $ ] ( for a given @xmath38 ) , so that the definition of @xmath160 is inappropriate to obtain results of stability for . the camassa holm equation for @xmath12 reads @xmath164 and can be rewritten as the following system nonzero , equation is simply replaced by @xmath165 . ] @xmath166 we consider periodic solutions of period one . next , we rewrite the equation in lagrangian coordinates . therefore we introduce the characteristics @xmath167 we introduce the space @xmath168 defined as @xmath169 functions in @xmath168 map the unit interval into itself in the sense that if @xmath3 is periodic with period 1 , then @xmath170 is also periodic with period 1 . the lagrangian velocity @xmath61 reads @xmath171 we will consider @xmath172 and @xmath61 periodic . we define the lagrangian energy cumulative distribution as @xmath173 for all @xmath55 , the function @xmath62 belongs to the vector space @xmath174 defined as follows : @xmath175 equip @xmath174 with the norm @xmath176)}+{\left\vertf_\xi\right\vert}_{l^1([0,1])}.\ ] ] as an immediate consequence of the definition of the characteristics we obtain @xmath177 this last term can be expressed uniquely in term of @xmath61 , @xmath178 , and @xmath62 . we have the following explicit expression for @xmath179 , @xmath180 thus , @xmath181 and , after the change of variables @xmath182 , @xmath183\,d\eta.\end{gathered}\ ] ] we have @xmath184 note that @xmath185 is periodic with period one . then , can be rewritten as @xmath186 where the @xmath55 variable has been dropped to simplify the notation . later we will prove that @xmath178 is an increasing function for any fixed time @xmath55 . if , for the moment , we take this for granted , then @xmath187 is equivalent to @xmath188 where @xmath189 and , slightly abusing the notation , we write @xmath190 the derivatives of @xmath188 and @xmath179 are given by @xmath191 respectively . for @xmath192 $ ] , using the fact that @xmath193 and the periodicity of @xmath185 and @xmath61 , the expressions for @xmath188 and @xmath179 can be rewritten as @xmath194 and @xmath195 thus @xmath196 and @xmath197 can be replaced by equivalent expressions given by and which only depend on our new variables @xmath61 , @xmath62 , and @xmath178 . we obtain a new system of equations , which is at least formally equivalent to the camassa holm equation : @xmath198_0^\xi . \end{aligned } \right.\ ] ] after differentiating we find @xmath199 from and , we obtain the system @xmath200 we can write more compactly as @xmath201 let @xmath202 we equip @xmath203 with the norm of @xmath174 , that is , @xmath204)}+{\left\vertf_\xi\right\vert}_{l^1([0,1])},\ ] ] which is equivalent to the standard norm of @xmath203 because @xmath205)}\leq{\left\vertf\right\vert}_{l^\infty([0,1])}\leq{\left\vertf\right\vert}_{l^1([0,1])}+{\left\vertf_\xi\right\vert}_{l^1([0,1])}$ ] . let @xmath206 be the banach space defined as @xmath207 we derive the following lipschitz estimates for @xmath179 and @xmath188 . [ lem : pq ] for any @xmath208 in @xmath206 , we define the maps @xmath209 and @xmath210 as @xmath211 and @xmath212 where @xmath188 and @xmath179 are given by and , respectively . then , @xmath210 and @xmath209 are lipschitz maps on bounded sets from @xmath206 to @xmath203 . more precisely , we have the following bounds . let @xmath213 then for any @xmath214 , we have @xmath215 and @xmath216 where the constant @xmath217 only depends on the value of @xmath97 . let us first prove that @xmath210 and @xmath209 are lipschitz maps from @xmath218 to @xmath219 . note that by using a change of variables in and , we obtain that @xmath210 and @xmath209 are periodic with period @xmath14 . let now @xmath208 and @xmath220 be two elements of @xmath218 . since the map @xmath221 is locally lipschitz , it is lipschitz on @xmath222 $ ] . we denote by @xmath217 a generic constant that only depends on @xmath97 . since , for all @xmath223 in @xmath224 $ ] we have @xmath225 , we also have @xmath226 it follows that , for all @xmath192 $ ] , @xmath227 and the map @xmath228 which corresponds to the first term in is lipschitz from @xmath218 to @xmath219 and the lipschitz constant only depends on @xmath97 . we handle the other terms in in the same way and we prove that @xmath210 is lipschitz from @xmath218 to @xmath219 . similarly , one proves that @xmath229 is lipschitz for a lipschitz constant which only depends on @xmath97 . direct differentiation gives the expressions for the derivatives @xmath230 and @xmath231 of @xmath179 and @xmath188 . then , as @xmath210 and @xmath209 are lipschitz from @xmath218 to @xmath219 , we have @xmath232 hence , we have proved that @xmath233 is lipschitz for a lipschitz constant that only depends on @xmath97 . we prove the corresponding result for @xmath210 in the same way . the short - time existence follows from lemma [ lem : pq ] and a contraction argument . global existence is obtained only for initial data which belong to the set @xmath71 as defined below . [ def : f ] the set @xmath71 is composed of all @xmath234 such that [ eq : lagcoord ] @xmath235 the set @xmath71 is preserved by the equation , that is , if @xmath73 solves for @xmath163 $ ] with initial data @xmath236 , then @xmath237 for all @xmath163 $ ] . the proof is basically the same as in @xcite . [ th : global ] for any @xmath238 , the system admits a unique global solution @xmath239 in @xmath240 with initial data @xmath241 . we have @xmath237 for all times . let the mapping @xmath242 be defined as @xmath243 given @xmath244 and @xmath245 , we define @xmath218 as before , that is , @xmath246 then there exists a constant @xmath217 which depends only on @xmath97 and @xmath38 such that , for any two elements @xmath247 and @xmath248 in @xmath218 , we have @xmath249 for any @xmath163 $ ] . by using lemma [ lem : pq ] , we proceed using a contraction argument and obtain the existence of short time solutions to . let @xmath38 by the maximal time of existence and assume @xmath250 . let @xmath251 be a solution of in @xmath252 with initial data @xmath253 . we want to prove that @xmath254 from , we get @xmath255 hence , @xmath256 . this identity corresponds to the conservation of the total energy . we now consider a fixed time @xmath257 which we omit in the notation when there is no ambiguity . for @xmath94 and @xmath258 in @xmath224 $ ] , we have @xmath259 because @xmath178 is increasing and @xmath260 . from , we infer @xmath261 and , from , we obtain @xmath262 hence , @xmath263 for some constant @xmath46 . similarly , one prove that @xmath264 and therefore @xmath265 and @xmath266 are finite . since @xmath267 , it follows that @xmath268 and @xmath269 . since @xmath270 , we have that @xmath271 is also finite . thus , we have proved that @xmath272 is finite and depends only on @xmath38 and @xmath273 . let @xmath274 . using the semi - linearity of with respect to @xmath275 , we obtain @xmath276 where @xmath46 is a constant depending only on @xmath277 . it follows from gronwall s lemma that @xmath278 is finite , and this concludes the proof of the global existence . moreover we have proved that @xmath279 for a constant @xmath217 which depends only on @xmath38 and @xmath280 . let us prove . given @xmath38 and @xmath281 , from lemma [ lem : pq ] and , we get that @xmath282 where @xmath217 is a generic constant which depends only on @xmath97 and @xmath38 . using again and lemma [ lem : pq ] , we get that for a given time @xmath163 $ ] , @xmath283 hence , @xmath284 where @xmath285 is defined as in . then , follows from gronwall s lemma applied to . we denote by @xmath76 the subgroup of the group of homeomorphisms on the unit interval defined as follows : [ def : group ] [ def : g ] let @xmath76 be the set of all functions @xmath286 such that @xmath286 is invertible , @xmath287 the set @xmath76 can be interpreted as the set of relabeling functions . note that @xmath288 implies that @xmath289 for some constant @xmath290 . this condition is also almost sufficient as lemma 3.2 in @xcite shows . given a triplet @xmath291 , we denote by @xmath292 the total energy @xmath293 . we define the subsets @xmath294 of @xmath71 as follows @xmath295 the set @xmath296 is then given by @xmath297 we have @xmath298 . we define the action of the group @xmath76 on @xmath71 . we define the map @xmath299 as follows @xmath300 where @xmath301 . we denote @xmath302 . [ prop : action ] the map @xmath303 defines a group action of @xmath76 on @xmath71 . by the definition it is clear that @xmath303 satisfies the fundamental property of a group action , that is @xmath304 for all @xmath79 and @xmath305 , @xmath306 . it remains to prove that @xmath307 indeed belongs to @xmath71 . we denote @xmath308 , then it is not hard to check that @xmath309 , @xmath310 , and @xmath311 for all @xmath312 . by definition we have @xmath313 , and we will now prove that @xmath314 almost everywhere . let @xmath315 be the set where @xmath178 is differentiable and @xmath316 the set where @xmath317 and @xmath286 are differentiable . using rademacher s theorem , we get that @xmath318 . for @xmath319 , we consider a sequence @xmath320 converging to @xmath94 with @xmath321 for all @xmath322 . we have @xmath323 since @xmath286 is continuous , @xmath324 converges to @xmath325 and , as @xmath178 is differentiable at @xmath325 , the left - hand side of tends to @xmath326 , the right - hand side of tends to @xmath327 , and we get @xmath328 for all @xmath329 . since @xmath330 is lipschitz continuous , one - to - one , and @xmath331 , we have @xmath332 and therefore holds almost everywhere . one proves the other identity similarly . as @xmath333 almost everywhere , we obtain immediately that and are fulfilled . that is also satisfied follows from the following considerations : @xmath334 , as @xmath335 is periodic with period @xmath14 . the same argument applies when considering @xmath336 and @xmath337 . as @xmath61 is periodic with period @xmath14 , we can also conclude that @xmath338 . as @xmath288 , one obtains that @xmath339 is bounded , but not equal to @xmath340 . note that the set @xmath218 is invariant with respect to relabeling while the @xmath206-norm is not , as the following example shows : consider the function @xmath341 , and @xmath288 , then this yields @xmath342)}={\left\vertf(\xi)\right\vert}_{l^\infty([0,1])}.\ ] ] hence , the @xmath19-norm of @xmath343 will always depend on @xmath286 . since @xmath76 is acting on @xmath71 , we can consider the quotient space @xmath80 of @xmath71 with respect to the group action . let us introduce the subset @xmath81 of @xmath296 defined as follows @xmath344 it turns out that @xmath81 contains a unique representative in @xmath71 for each element of @xmath80 , that is , there exists a bijection between @xmath81 and @xmath80 . in order to prove this we introduce two maps @xmath345 and @xmath346 defined as follows @xmath347 with @xmath348 and @xmath208 , and @xmath349 with @xmath350 . first , we have to prove that @xmath286 indeed belongs to @xmath76 . we have @xmath351 and this proves . since @xmath291 , there exists a constant @xmath352 such that @xmath353 for almost every @xmath94 and therefore follows from an application of lemma 3.2 in @xcite . after noting that the group action lets the quantity @xmath354 invariant , it is not hard to check that @xmath355 indeed belongs to @xmath296 , that is , @xmath356 where we denote @xmath357 . let us prove that @xmath358 belongs to @xmath81 for any @xmath359 . on the one hand , we have @xmath356 because @xmath360 and @xmath361 as @xmath359 . on the other hand , @xmath362 and , since @xmath193 , we obtain @xmath363 thus @xmath364 . note that the definition of @xmath365 can be rewritten as @xmath366 where @xmath367 denotes the translation of length @xmath368 so that @xmath369 is a relabeling of @xmath370 . we denote by @xmath371 the projection of @xmath71 into @xmath81 given by @xmath372 . one checks directly that @xmath373 . the element @xmath374 is the unique relabeled version of @xmath370 which belongs to @xmath81 and therefore we have the following result . [ lemma : bijection ] the sets @xmath80 and @xmath81 are in bijection . given any element @xmath375\in{{{\ensuremath{\mathcal{f}}}/{g}}}$ ] , we associate @xmath376 . this mapping is well - defined and is a bijection . [ lem : equivpi ] the mapping @xmath377 is equivariant , that is , @xmath378 for any @xmath379 and @xmath380 , we denote @xmath381 , @xmath382 , and @xmath383 . we claim that @xmath384 satisfies and therefore , since @xmath384 and @xmath385 satisfy the same system of differential equations with the same initial data , they are equal . we denote @xmath386 . then we obtain @xmath387(\eta)d\eta.\ ] ] as @xmath388 and @xmath389 for almost every @xmath312 , we obtain after the change of variables @xmath390 , @xmath391(\eta)d\eta.\ ] ] treating similarly the other terms in , it follows that @xmath392 is a solution of . thus , since @xmath392 and @xmath393 satisfy the same system of ordinary differential equations with the same initial conditions , they are equal and is proved . from this lemma we get that @xmath394 [ def : bars ] we define the semigroup @xmath85 on @xmath81 as @xmath395 the semigroup property of @xmath85 follows from . using the same approach as in @xcite , we can prove that @xmath85 is continuous with respect to the norm of @xmath206 . it follows basically of the continuity of the mapping @xmath371 but @xmath371 is not lipschitz continuous and the goal of the next section is to improve this result and find a metric that makes @xmath85 lipschitz continuous . let @xmath396 , we define @xmath397 as @xmath398 note that , for any @xmath396 and @xmath399 , we have @xmath400 it means that @xmath133 is invariant with respect to relabeling . the mapping @xmath133 does not satisfy the triangle inequality , which is the reason why we introduce the mapping @xmath140 . [ def : alberto ] let @xmath396 , we define @xmath401 as @xmath402 where the infimum is taken over all finite sequences @xmath88 which satisfy @xmath403 and @xmath404 . for any @xmath396 and @xmath399 , we have @xmath405 and @xmath140 is also invariant with respect to relabeling . the definition of the metric @xmath401 is the discrete version of the one introduced in @xcite . in @xcite , the authors introduce the metric that we denote here as @xmath406 where @xmath407 where the infimum is taken over all smooth path @xmath408 such that @xmath409 and @xmath410 and the triple norm of an element @xmath174 is defined at a point @xmath370 as @xmath411 where @xmath412 is a scalar function , see @xcite for more details . the metric @xmath406 also enjoys the invariance relabeling property . the idea behind the construction of @xmath140 and @xmath406 is the same : we measure the distance between two points in a way where two relabeled versions of the same point are identified . the difference is that in the case of @xmath140 we use a set of points whereas in the case of @xmath406 we use a curve to join two elements @xmath247 and @xmath248 . formally , we have @xmath413 we need to introduce the subsets of bounded energy in @xmath296 . we denote by @xmath414 the set @xmath415 and let @xmath416 . the important propery of the set @xmath414 is that it is preserved both by the flow , see , and relabeling . let us prove that @xmath417 for @xmath418 so that the sets @xmath419 and @xmath420 are in this sense equivalent . from , we get @xmath421 which implies @xmath422 . by , we get that @xmath423 and therefore @xmath424 . since @xmath425 and @xmath426 , the set @xmath427\mid y_\xi(\xi)\geq\frac12\}$ ] has strictly positive measure . for a point @xmath428 in this set , we get , by , that @xmath429 . hence , @xmath430 and , finally , @xmath431 which concludes the proof of . let @xmath432 be the metric on @xmath420 which is defined , for any @xmath433 , as @xmath434 where the infimum is taken over all finite sequences @xmath435 which satisfy @xmath403 and @xmath404 . [ lem : linfbdj ] for any @xmath433 , we have @xmath436 for some fixed constant @xmath217 which depends only on @xmath97 . first , we prove that for any @xmath433 , we have @xmath437 for some constant @xmath217 which depends only on @xmath97 . by a change of variables in the integrals , we obtain @xmath438 we have @xmath439 from the definition of @xmath420 we get that , for any element @xmath440 , we have @xmath441 . since @xmath442 , from , it follows that @xmath443 . thus , yields @xmath444 we denote by @xmath217 a generic constant which depends only on @xmath97 . the identity will be proved when we prove @xmath445 by using the definition of @xmath81 , we get that @xmath446 let @xmath447 . similar to and , we can conclude that @xmath448 thus we have @xmath449 and analogously @xmath450 . hence , @xmath451 by , we get that @xmath452 then , since @xmath453 we obtain that @xmath454 then , yields @xmath455 from and , and therefore follows . for any @xmath456 , we consider a sequence @xmath457 in @xmath420 such that @xmath403 and @xmath404 and @xmath458 . we have @xmath459 since @xmath460 is arbitrary , we get . from the definition of @xmath140 , we obtain that @xmath461 so that the metric @xmath140 is weaker than the @xmath206-norm . [ lemma : distance ] the mapping @xmath462 is a metric on @xmath420 . the symmetry is embedded in the definition of @xmath133 while the construction of @xmath432 from @xmath133 takes care of the triangle inequality . from lemma [ lem : linfbdj ] , we get that @xmath463 implies that @xmath464 , @xmath465 and @xmath466 . then , the definition of @xmath296 implies that @xmath467 . [ rem : compmetric ] in @xcite , a metric on @xmath81 is obtained simply by taking the norm of @xmath206 . the authors prove that the semigroup is continuous with respect to this norm , that is , given a sequence @xmath468 and @xmath370 in @xmath81 such that @xmath469 , we have @xmath470 . however , @xmath85 is not lipschitz in this norm . from , we see that the distance introduced in @xcite is stronger than the one introduced here . ( the definition of @xmath206 in @xcite differs slightly from the one employed here , but the statements in this remark remain valid ) . we can now prove the lipschitz stability theorem for @xmath85 . [ th : stab ] given @xmath245 and @xmath244 , there exists a constant @xmath217 which depends only on @xmath97 and @xmath38 such that , for any @xmath433 and @xmath163 $ ] , we have @xmath471 by the definition of @xmath432 , for any @xmath456 , there exists a sequences @xmath457 in @xmath420 and functions @xmath472 , @xmath473 in @xmath76 such that @xmath403 , @xmath404 and @xmath474 since @xmath475 for @xmath476 , see , and @xmath477 is preserved by relabeling , we have that @xmath478 and @xmath479 belong to @xmath480 . from the lipschitz stability result given in , we obtain that @xmath481 where the constant @xmath217 depends only on @xmath97 and @xmath38 . introduce @xmath482 and @xmath483 then rewrites as @xmath484 while rewrites as @xmath485 we have @xmath486 and similarly @xmath487 . we consider the sequence in @xmath420 which consists of @xmath488 and @xmath489 . the set @xmath414 is preserved by the flow and by relabeling . therefore , @xmath490 and @xmath489 belong to @xmath420 . the endpoints are @xmath491 and @xmath489 . from the definition of the metric @xmath432 , we get @xmath492 by using the equivariance of @xmath377 , we obtain that @xmath493 hence , by using , that is , the invariance of @xmath133 with respect to relabeling , we get from that @xmath494 after letting @xmath460 tend to zero , we obtain . we now introduce a second set of coordinates , the so called eulerian coordinates . therefore let us first consider @xmath495 . we can define the eulerian coordinates as in @xcite and also obtain the same mappings between eulerian and lagrangian coordinates . for completeness we will state the results here . [ def : d ] the set @xmath56 consists of all pairs @xmath65 such that 1 . @xmath496 , and 2 . @xmath68 is a positive radon measure whose absolute continuous part , @xmath497 , satisfies @xmath498 we can define a mapping , denoted by @xmath98 , from @xmath56 to @xmath499 : [ defl ] for any @xmath65 in @xmath56 , let @xmath500 where @xmath501 then @xmath502 . we define @xmath503 . thus from any initial data @xmath504 , we can construct a solution of in @xmath71 with initial data @xmath505 . it remains to go back to the original variables , which is the purpose of the mapping @xmath97 , defined as follows . [ def : ftoe ] for any @xmath79 , then @xmath65 given by @xmath506 belongs to @xmath56 . we denote by @xmath97 the mapping from @xmath71 to @xmath56 which for any @xmath79 associates the element @xmath66 given by . the mapping @xmath97 satisfies @xmath507 the inverse of @xmath98 is the restriction of @xmath97 to @xmath81 , that is , @xmath508 where the lagrangian variables are defined is represented by the interior of the closed domain on the left . the equivalence classes @xmath375 $ ] and @xmath509 $ ] ( with respect to the action of the relabeling group @xmath510 ) of @xmath370 and @xmath511 , respectively , are represented by horizontal curves . to each equivalence class there corresponds a unique element in @xmath81 and @xmath56 ( the set of eulerian variables ) . the sets @xmath81 and @xmath56 are represented by the vertical curves.,width=377 ] next we show that we indeed have obtained a solution of the ch equation . by a weak solution of the camassa holm equation we mean the following . let @xmath512 . assume that @xmath3 satisfies + ( i ) @xmath513 , + ( ii ) the equations @xmath514 and @xmath515 hold for all @xmath516 . then we say that @xmath3 is a weak global solution of the camassa holm equation . given any initial condition @xmath517 , we denote @xmath518 . then @xmath519 is a weak global solution of the camassa holm equation . after making the change of variables @xmath520 we get on the one hand @xmath521y_\xi(t,\xi)d\xi dt\\{\nonumber } & = -\iint_{{\mathbb{r}}_+\times { \mathbb{r}}}[u(t,\xi)y_\xi(t,\xi)(\phi(t , y(t,\xi)))_t-\phi_\xi(t , y(t,\xi))u(t,\xi)^2]d\xi dt\\ & = \int_{\mathbb{r}}u(0,\xi)\phi(0,y(0,\xi))y_\xi(0,\xi)d\xi\\{\nonumber } & \quad + \iint_{{\mathbb{r}}_+\times { \mathbb{r } } } [ u_t(t,\xi)y_\xi(t,\xi)+u(t,\xi ) y_{\xi t}(t,\xi)]\phi(t , y(t,\xi))d\xi dt \\{\nonumber } & \quad + \iint_{{\mathbb{r}}_+\times { \mathbb{r } } } u^2(t,\xi)\phi_\xi(t , y(t,\xi))d\xi dt\\{\nonumber } & = \int_{\mathbb{r}}u(0,x)\phi(0,x)dx\\{\nonumber } & \quad -\iint_{{\mathbb{r}}_+\times { \mathbb{r } } } ( q(t,\xi)y_\xi(t,\xi)+u_\xi(t,\xi)u(t,\xi))\phi(t , y(t,\xi))d\xi dt , \end{aligned}\ ] ] while on the other hand @xmath522 which shows that is fulfilled . equation can be shown analogously @xmath523\phi(t , y(t,\xi))d\xi dt\\{\nonumber}&= \iint_{{\mathbb{r}}_+\times { \mathbb{r}}}[\frac{1}{2 } u_x^2(t , x)+u^2(t , x)-p(t , x)]\phi(t , x ) dx dt . \end{aligned}\ ] ] in the last step we used the following @xmath524 \mid y_\xi ( t,\xi)>0\ } } u^2y_\xi + \frac{u_\xi^2}{y_\xi}d\xi=\int_0 ^ 1 \nu dx , \end{aligned}\ ] ] the last equality holds only for almost all @xmath55 because for almost every @xmath525 the set @xmath427 \mid y_\xi ( t,\xi)>0\}$ ] is of full measure and therefore @xmath526 which is bounded by a constant for all times . thus we proved that @xmath3 is a weak solution of the camassa holm equation . next we return to the construction of the lipschitz metric on @xmath56 . let @xmath527 note that , by the definition of @xmath85 and , we also have that @xmath528 next we show that @xmath529 is a lipschitz continuous semigroup by introducing a metric on @xmath56 . using the bijection @xmath98 transport the topology from @xmath81 to @xmath56 . [ def : dd ] we define the metric @xmath530 by @xmath531 the lipschitz stability of the semigroup @xmath529 follows then naturally from theorem [ th : stab ] . the stability holds on sets of bounded energy that we now introduce in the following definition . given @xmath244 , we define the subsets @xmath532 of @xmath56 , which corresponds to sets of bounded energy , as @xmath533 on the set @xmath532 , we define the metric @xmath534 as @xmath535 where the metric @xmath432 is defined in . the definition is well - posed as we can check from the definition of @xmath98 that if @xmath536 then @xmath537 . we can now state our main theorem . [ th : main ] the semigroup @xmath538 is a continuous semigroup on @xmath56 with respect to the metric @xmath2 . the semigroup is lipschitz continuous on sets of bounded energy , that is : given @xmath244 and a time interval @xmath539 $ ] , there exists a constant @xmath46 which only depends on @xmath97 and @xmath38 such that , for any @xmath65 and @xmath540 in @xmath532 , we have @xmath541 for all @xmath163 $ ] . first , we prove that @xmath529 is a semigroup . since @xmath85 is a mapping from @xmath81 to @xmath81 , we have @xmath542 where we also use and the semigroup property of @xmath85 . we now prove the lipschitz continuity of @xmath529 . by using theorem [ th : stab ] , we obtain that @xmath543 [ prop : cont1 ] the mapping @xmath544 is continuous from @xmath6 into @xmath56 . in other words , given a sequence @xmath545 converging to @xmath496 , then @xmath546 converges to @xmath547 in @xmath56 . let @xmath548 be the image of @xmath549 given as in and @xmath208 the image of @xmath550 given as in . we will at first prove that @xmath551 converges to @xmath3 in @xmath6 implies that @xmath468 converges against @xmath370 in @xmath206 . denote @xmath552 and @xmath553 , then @xmath554 and @xmath412 are periodic functions . moreover , as @xmath468 , @xmath555 , we have @xmath556 and @xmath557 , where @xmath558 and @xmath354 . by definition [ defl ] , we have that @xmath559 and @xmath560 , and hence @xmath561 by assumption @xmath562 in @xmath563 , which implies that @xmath564 in @xmath19 , @xmath565 in @xmath41 , and @xmath566 . therefore we also obtain that @xmath567 in @xmath19 . we have @xmath568 then , since @xmath564 in @xmath19 , also @xmath569 in @xmath19 and as @xmath3 is in @xmath563 , we also obtain that @xmath570 in @xmath19 . hence , it follows that @xmath571 in @xmath19 . by definition , the measures @xmath572 and @xmath573 have no singular part , and we therefore have almost everywhere @xmath574 hence @xmath575 in order to show that @xmath576 in @xmath577 , it suffices to investigate @xmath578 and @xmath579 as we already know that @xmath566 and therefore @xmath580 and @xmath335 are bounded . since @xmath581 , we have @xmath582 for the second term , let @xmath583 . then for any @xmath584 there exists a continuous function @xmath4 with compact support such that @xmath585 and we can make the following decomposition @xmath586 this implies @xmath587 and analogously we obtain @xmath588 . as @xmath567 in @xmath19 and @xmath4 is continuous , we obtain , by applying the lebesgue dominated convergence theorem , that @xmath589 in @xmath41 , and we can choose @xmath590 so big that @xmath591 hence , we showed , that @xmath592 and therefore , using , @xmath593 combing now , , and , yields @xmath594 in @xmath41 , and therefore also @xmath595 in @xmath41 . because @xmath596 and @xmath597 are bounded in @xmath19 , we also have that @xmath576 in @xmath598 and @xmath595 in @xmath598 . since @xmath580 , @xmath597 and @xmath599 tend to @xmath335 , @xmath185 and @xmath61 in @xmath598 and @xmath600 and @xmath601 , are uniformly bounded , it follows from that @xmath602 once we have proved that @xmath603 converges weakly to @xmath604 , this will imply that @xmath605 in @xmath598 . for any smooth function @xmath606 with compact support in @xmath224 $ ] we have @xmath607 by assumption we have @xmath608 in @xmath598 . moreover , since @xmath567 in @xmath19 , the support of @xmath609 is contained in some compact set , which can be chosen independently of @xmath590 . thus , using lebesgue s dominated convergence theorem , we obtain that @xmath610 in @xmath598 and therefore @xmath611 form we know that @xmath603 is bounded and therefore by a density argument holds for any function @xmath606 in @xmath598 and therefore @xmath605 weakly and hence also in @xmath598 . using now that @xmath612 shows that we also have convergence in @xmath41 . thus we obtained that @xmath613 in @xmath206 . as a second and last step , we will show that @xmath365 is continuous , which then finishes the proof . we already know that @xmath567 in @xmath19 and therefore @xmath614 converges to @xmath615 . thus we obtain as an immediate consequence @xmath616 and hence the same argumentation as before shows that @xmath617 in @xmath19 . moreover , @xmath618 and again using the same ideas as in the first part of the proof , we have that @xmath619 in @xmath41 , which finally proves the claim , because of let @xmath621 and @xmath622 . by the definition of the metric @xmath2 , we have @xmath623 . we immediately obtain that @xmath624 by lemma [ lem : linfbdj ] . the rest can be proved as in ( * ? ? ? * proposition 5.2 ) . * acknowledgments . * k. g. gratefully acknowledges the hospitality of the department of mathematical sciences at the ntnu , norway , creating a great working environment for research during the fall of 2009 .
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we study stability of conservative solutions of the cauchy problem for the periodic camassa holm equation @xmath0 with initial data @xmath1 .
in particular , we derive a new lipschitz metric @xmath2 with the property that for two solutions @xmath3 and @xmath4 of the equation we have @xmath5 . the relationship between this metric and usual norms in @xmath6 and @xmath7
is clarified .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
open quantum systems are at the heart of many physical phenomena from nuclear physics to quantum information theory @xcite . in fact , all `` real '' quantum systems are , to some extent , open systems . interactions with the environment cause decoherence , resulting in non - equilibrium dynamics . it is often simpler to design experiments that probe non - equilibrium physics than it is to design experiments that probe equilibrium physics . conversely , the theoretical toolkit for describing systems in equilibrium is generally much farther developed than that for describing systems in non - equilibrium . ultracold atom systems provide a platform for realizing clean and tunable quantum systems @xcite . over the past few years , much effort has gone into describing non - equilibrium experiments that are accessible , within approximate or exact frameworks , to theory . notable experiments are the equilibration dynamics of one - dimensional bose gases @xcite , the spin dynamics of dipolar molecules in optical lattices with low filling factor @xcite , and the tunneling dynamics of effectively one - dimensional few - fermion systems @xcite . this paper focuses on the latter set of experiments . specifically , the goal of the present work is to describe the tunneling dynamics of few - fermion systems , which are prepared in a well defined quasi - eigenstate ( metastable state ) , into free space . we consider small systems and directly solve the time - dependent schrdinger equation in coordinate space . as we will show , this approach provides a means to quantify the importance of the particle - particle interaction , covering time scales from a fraction of the trap scale to thousands times the trap scale . alternatively , one could adopt a quantum optics perspective and pursue a master equation approach . tunneling is arguably the most quantum phenomenon there is : if the system was behaving classically , tunneling would be absent @xcite . tunneling plays an important role across physics , chemistry and technology . the scanning tunneling microscope @xcite , for example , nicely illustrates how a physics phenomenon , the tunneling of electrons , has been turned into a powerful practical tool ( the imaging of materials ) . the @xmath1-decay , i.e. , the decay of a @xmath2he nucleus from a heavy nucleus , is an example discussed in most undergraduate physics texts ( see , for example , ref . the typical picture is to identify an effective reaction coordinate and to obtain the tunneling rate from a wkb analysis . while powerful , such treatments completely neglect the effect of interactions . interactions also play a crucial role in sorting out under which conditions electrons in light atoms tunnel sequentially or simultaneously @xcite . the two - particle system considered in this work has been realized experimentally and is the possibly simplest scenario that deals with a truly open quantum system ( the atoms can escape to infinity ) in which interactions ( short - range atom - atom interactions ) play a crucial role . as we will show , even for this relatively simple set - up , matching theory and experiment is a non - trivial task . of course , two - particle tunneling has been investigated previously in this and related contexts @xcite . the remainder of this paper is organized as follows . section [ sec_system ] introduces the hamiltonian , the heidelberg experiment and selected simulation details . sections [ sec_molecular ] and [ sec_upper ] discuss the molecular and upper branch tunneling dynamics . for both cases , it is argued that the trapping potential needs to be reparameterized . using the reparameterized trapping potential , numerical simulations for the tunneling dynamics of two distinguishable @xmath0li atoms on the molecular branch and the upper branch are discussed . comparisons with the experimentally measured tunneling rates are presented . finally , sec . [ sec_summary ] summarizes and provides an outlook . simulation details and some technical aspects are relegated to appendices [ app_br][app_flux - anal ] . this section considers a single @xmath0li atom with mass @xmath3 . the atom is assumed to be in the hyperfine state @xmath4 . we consider the three lowest hyperfine states of the @xmath0li atom , referred to as @xmath5 , @xmath6 , and @xmath7 . figure [ fig_br - energy ] li as a function of the magnetic field strength @xmath8 . solid , dashed and dotted lines correspond to states @xmath9 , @xmath10 , and @xmath11 , respectively ( see text for details ) . states @xmath9 and @xmath10 are used in the upper branch experiments @xcite , while states @xmath9 and @xmath11 are used in the molecular branch experiments @xcite . the higher - lying energy states shown by dash - dotted lines are not relevant for the present paper . , scaledwidth=40.0% ] shows the dependence of the hyperfine energy levels on the magnetic field strength @xmath8 . the atom with coordinates @xmath12 is trapped optically in a non - separable potential that is much tighter in the @xmath13-direction ( @xmath14 ) than in the @xmath15-direction @xcite . throughout this work , we do not simulate the motion in the tight transverse confining direction . the transverse trapping frequency does , however , enter into the calculation of the renormalized one - dimensional coupling constant ( see sec . [ subsec_two - body_ham ] ) . evaluating the confinement created by the gaussian laser beam at @xmath16 , the effective one - dimensional single - particle hamiltonian @xmath17 reads @xcite @xmath18 where the trapping potential @xmath19 along the @xmath15-direction depends implicitly on the internal or hyperfine state @xmath20 of the atom through the coefficient @xmath21 , @xmath22 the first term on the right hand side of eq . ( [ vtrap ] ) accounts for the optical confinement . @xmath23 denotes the maximum depth of the trap , @xmath24 a time - dependent parameter [ @xmath25 , and @xmath26 the rayleigh range of the laser beam that produces the confinement . the second term on the right hand side of eq . ( [ vtrap ] ) is linear in @xmath15 and makes the tunneling possible . @xmath27 is the bohr magneton and @xmath28 depends on the hyperfine state , magnetic field strength and magnetic field gradient @xmath29 , @xmath30 here , @xmath31 is a dimensionless parameter close to @xmath32 ( see below for details ) . table [ table - exp ] .parameters from refs . @xcite that define the trapping potential . since the energy of the two - particle system on the molecular branch is smaller than the energy of the two - particle system on the upper branch , the @xmath33 value for the molecular branch is chosen to be smaller than that for the upper branch ; this guarantees that the tunneling rates for the two experiments have roughly comparable orders of magnitude . the harmonic oscillator units are defined in terms of @xmath34hz , corresponding to @xmath35j , @xmath36 m , and @xmath37s , or @xmath38 , @xmath39 , and @xmath40 . in an alternative levitation measurement , the magnetic field gradient was found to be @xmath41g / m @xcite . [ cols="^,^",options="header " , ] due to the dependence of @xmath42 on the state ( through the wkb energy ) , the wkb rates @xmath43 for the three states vary by about a factor of 6 for cases ( a ) and ( b ) . for the parameters considered in table [ table_app_wkb ] and in the main text , the wkb rate for the ground state is smaller than that obtained through the full time propagation , with the ratio @xmath44 depending on the exact shape of the trap . for the excited states , in contrast , the wkb rates are larger than those obtained through the full time propagation . 49ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty https://books.google.com/books?id=dkcjpwaacaaj[__ ] ( , ) https://books.google.com/books?id=znjvhah8qa4c[__ ] ( , ) link:\doibase 10.1103/revmodphys.80.885 [ * * , ( ) ] http://stacks.iop.org/0034-4885/75/i=4/a=046401 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.80.1215 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.82.1225 [ * * , ( ) ] link:\doibase 10.1126/science.1257026 [ * * , ( ) ] link:\doibase 10.1038/nature12483 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.108.075303 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.111.175302 [ * * , ( ) ] https://books.google.com/books?id=acj0nqeacaaj[__ ] ( , ) link:\doibase 10.1103/revmodphys.86.1127 [ * * , ( ) ] https://books.google.com/books?id=yfo3rnt3bkec[__ ] ( , ) link:\doibase 10.1126/science.1163439 [ * * , ( ) ] link:\doibase 10.1073/pnas.1201345109 [ * * , ( ) ] link:\doibase 10.1103/physreva.87.043626 [ * * , ( ) ] http://stacks.iop.org/0953-4075/44/i=19/a=195301 [ * * , ( ) ] http://stacks.iop.org/0953-4075/42/i=4/a=044018 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.108.115302 [ * * , ( ) ] link:\doibase 10.1103/physreva.88.043633 [ * * , ( ) ] link:\doibase 10.1103/physreva.91.041601 [ * * , ( ) ] https://books [ _ _ ] , ( , ) link:\doibase 10.1103/physrev.38.2082.2 [ * * , ( ) ] @noop ( ) link:\doibase 10.1103/physrevlett.110.135301 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.81.938 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.110.203202 [ * * , ( ) ] link:\doibase 10.1103/physreva.84.043619 [ * * , ( ) ] http://stacks.iop.org/1367-2630/7/i=1/a=192 [ * * , ( ) ] http://stacks.iop.org/1367-2630/11/i=7/a=073031 [ * * , ( ) ] link:\doibase 10.1103/physreva.89.023603 [ * * , ( ) ] @noop ( ) . link:\doibase 10.1103/physreva.70.042709 [ * * , ( ) ] m. l. wall , k. r. a. hazzard , and a. m. rey , " effective many - body parameters for atoms in nonseparable gaussian optical potentials , phys . a * 92 * , 013610 ( 2015 ) . https://books.google.com/books?id=i-40vaxqrj0c[__ ] , graduate texts in contemporary physics ( , ) https://books.google.com/books?id=t_7hg08x7cmc[__ ] , international series of monographs on physics ( , ) @noop * * , ( ) \doibase http://dx.doi.org/10.1063/1.448136 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/0021-9991(91)90137-a [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/0375-9474(80)90509-6 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/0009-2614(86)80262-7 [ * * , ( ) ] link:\doibase 10.1103/physreva.37.973 [ * * , ( ) ] link:\doibase 10.1103/physreva.43.68 [ * * , ( ) ] link:\doibase 10.1103/physreva.91.043607 [ * * , ( ) ] @noop * * , ( ) \doibase http://dx.doi.org/10.1063/1.3126363 [ * * , ( ) ]
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we present one - dimensional simulation results for the cold atom tunneling experiments by the heidelberg group [ g. zrn _ et al .
_ , phys . rev
. lett . * 108 * , 075303 ( 2012 ) and g. zrn _
et al . _ ,
phys .
rev . lett . * 111 * , 175302 ( 2013 ) ] on one or two @xmath0li atoms confined by a potential that consists of an approximately harmonic optical trap plus a linear magnetic field gradient . at the non - interacting particle level , we find that the wkb ( wentzel - kramers - brillouin ) approximation may not be used as a reliable tool to extract the trapping potential parameters from the experimentally measured tunneling data .
we use our numerical calculations along with the experimental tunneling rates for the non - interacting system to reparameterize the trapping potential . the reparameterized trapping potentials serve as input for our simulations of two interacting particles . for two interacting ( distinguishable ) atoms on the upper branch ,
we reproduce the experimentally measured tunneling rates , which vary over several orders of magnitude , fairly well . for infinitely strong interaction strength
, we compare the time dynamics with that of two identical fermions and discuss the implications of fermionization on the dynamics . for two attractively - interacting atoms on the molecular branch
, we find that single - particle tunneling dominates for weakly - attractive interactions while pair tunneling dominates for strongly - attractive interactions .
our first set of calculations yields qualitative but not quantitative agreement with the experimentally measured tunneling rates .
we obtain quantitative agreement with the experimentally measured tunneling rates if we allow for a weakened radial confinement .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
one of the most exciting yet observationally challenging scientific objectives of the large area telescope ( lat ) on board the _ fermi gamma - ray space telescope _ @xcite , is the indirect detection of particle dark matter @xcite . however , limited gamma - ray statistics make diffuse signals arising from the pair - annihilation of dark matter difficult to differentiate from astrophysical processes . the limitation of using a diffuse signal to search for non - standard emission stems from difficulties in controlling the instrumental background and formulating a rigorous model for the astrophysical diffuse foregrounds . an intriguing excess of microwave radiation in the wmap data has been uncovered by @xcite and @xcite . the morphology and spectrum of the wmap haze indicates a hard electron - positron injection spectrum spherically distributed around the galactic center . while the origin of this haze need not be related to _ new _ particle physics , the possibility that the wmap haze corresponds to synchrotron radiation of stable leptons produced by dark matter has been explored in several studies ( see e.g. * ? ? ? a potentially conclusive way to determine whether the wmap haze originates from a population of energetic leptons is to observe gamma - rays produced by inverse compton up - scattering ( ic ) of photons in the interstellar galactic radiation field ( isrf ) . recently , @xcite ( hereafter d09 ) examined the lat gamma - ray sky and reported an excess emission morphologically similar to the wmap haze . d09 s observations suggest a confirmation of the _ haze hypothesis _ : that excess microwave emission stems from relativistic electron synchrotron with a spherical source distribution and a hard injection spectrum . in the type 2 " and type 3 " fits of d09 , the excess was claimed over a best - fit background using spatial templates which employed the gas map of @xcite ( sfd ) to trace gamma - ray emission from @xmath0 decay , and the 408 mhz haslam synchrotron map to trace ic emission from galactic cosmic ray electrons . the spatial templates ( plus an isotropic component obtained by mean - subtracting the residual skymap ) were used to fit the observed gamma - ray sky in energy bins spanning 2 - 100 gev . this analysis uncovered a residual gamma - ray emission above and below the galactic center with a morphology and spectrum similar to that found in the wmap dataset @xcite . in this @xmath1 , we test the following assumptions used in d09 for the removal of astrophysical foregrounds at gamma - ray energies : * that line of sight ism maps are adequate tracers for the morphology of @xmath0 emission , and * that the 408 mhz synchrotron map is an adequate tracer for the morphology of the galactic ic emission . assumption ( 1 ) entails neglecting the morphology of galactic cosmic - ray sources , since the observed @xmath0 emission results from the line - of - sight integral of the gas density ( `` target '' ) times the cosmic - ray density ( `` beam '' ) . assumption ( 2 ) neglects the difference between the morphology of the isrf and the galactic magnetic fields . on theoretical grounds , we expect that any detailed galactic cosmic - ray model would predict _ systematic deviations _ from the templates used in d09 . utilizing the galactic cosmic - ray propagation code galprop , we find that the procedure based on spatial templates creates deviations comparable to the amplitude of the d09 residual . furthermore , we find that these deviations are morphologically similar to the fermi haze . we thus conclude that the determination of an excess gamma - ray diffuse emission can not reliably be assessed from the spatial template proxies used in the type 2 " and type 3 " fits of d09 . we stress that our results do not claim that there is no `` haze '' in the fermi data . in particular , the systematic effects we study here are not relavent to explain the puzzling excess emission in the `` type 1 '' fit of d09 , which employes fermi - lat data in the 1 - 2 gev range as a proxy for the morphology of the @xmath0 component . we comment on this `` type 1 '' approach in section [ sec : discussion ] . employing the cosmic ray propagation code galprop ( v 50.1p ) @xcite , we compute the line - of - sight emission for galactic synchrotron , ic and @xmath0 decay predicted by a galprop model that is consistent with all cosmic ray and photon observations ( see * ? ? ? * for further detail ) . except where noted , we employ standard parameters given by the galdef file 599278 throughout this work . a large uncertainty in the propagation of cosmic rays relates to the intensity and orientation of galactic magnetic fields as the intensity of synchrotron radiation varies with the square of the local magnetic field intensity . in our default simulation we assume a magnetic field of random orientation and an intensity that exponentially decays in both @xmath2 and @xmath3 with scale radii of 10 kpc and 2 kpc respectively , normalized to 5 @xmath4 g at the solar position @xcite . to determine the accuracy of the d09 spatial templates for astrophysical foreground emission , we generate line - of - sight skymaps for the input gas density , as well as the outputs of emission due to @xmath0 decay , synchrotron and ic . note that the gas density maps we employ here differ from the sfd map used in d09 . most notably , the sfd map traces dust , while our map traces galactic gas . the difference between these approaches is expected to be small , but might introduce additional systematic deviations . by dividing , pixel by pixel , the line - of - sight map for @xmath0 decay by the input gas map , and the map of ic emission by the synchrotron map , we can assess the size of any systematic effects produced by assumptions ( 1 ) and ( 2 ) of section [ sec : introduction ] . we normalize each map over pixels of @xmath5b@xmath5 @xmath6 5@xmath7 , using equal area weighting to determine the normalization constant . this is equivalent to the masking procedure of d09 - though we do not mask out the galactic plane in our plots . we select several regions of the sky for which we provide numerical analyses of each map we present , with a background normalized as in d09 . we first evaluate d09 s claim of an excess emission in the southern galactic plane between -30@xmath7 @xmath8 b @xmath8 -10@xmath7 and @xmath5l@xmath5 @xmath8 15@xmath7 , indicated as the _ d09 haze_. we add the symmetric _ d09 northern haze _ region ( 10@xmath7 @xmath8 b @xmath8 30@xmath7 and @xmath5l@xmath915 ) . d09 defines their haze to begin at 10@xmath7 , but only masked the region @xmath5b@xmath5 @xmath8 5@xmath7 . in order to determine the importance of this choice , we include both northern and southern regions following 5@xmath7 @xmath8 @xmath5b@xmath5 @xmath8 25@xmath7 and @xmath5l@xmath5 @xmath8 15@xmath7 , which we denote as the _ inner haze _ region . since d09 models the morphology of the haze with a bivariate gaussian that decays exponentially in both latitude and longitude , we further consider a map weighting the value at each unmasked pixel using a bivariate gaussian of 25@xmath7 in latitude and 15@xmath7 in longitude , and dubbed the _ gaussian haze_. finally , in order to ascertain the variation of each map , we consider the galactic anticenter region 5@xmath10b@xmath9 25@xmath7 and @xmath5l@xmath5 @xmath6 170 . in figure [ pi0divgas ] we show the normalized line - of - sight skymap for @xmath0 decay divided by the normalized line - of - sight gas density input into our galprop simulations . the results are shown on a linear scale ( left ) and a logarithmic scale ( right ) , and both are smoothed using a gaussian of 2@xmath7 width . we find that the resultant skymap displays significant deviations from unity , with factors of approximately two above and below the galactic center , to values of about 0.3 near the galactic anti - center . in table [ tab : regions ] ( top row ) , we provide both the average value in the d09 background region , as well as the numeric ratios in each of our defined regions . while this map for @xmath0 decay is taken at a test energy of 1 gev , the variation of the ratio across four decades in energy ( 0.1 gev - 1 tev ) is less than 2% . while we find a deviation between the d09 haze region and the d09 background of only 4% , we find a difference of 15% between the southern inner haze region and the d09 background . these changes stem from the removal of a slight deficit in @xmath0 decay at approximately 30@xmath7 , and more importantly , the addition of increased emission between 5 - 10@xmath7 . because the galactic gas distribution is not symmetric above and below the galactic plane , we expect some north - south variation , and we indeed find substantially higher values in the northern hemisphere . finally , we calculate the ratio of the entire skymap weighted by the bivariate gaussian employed in d09 . we find a 17% excess in this measurement , which implies that using the gas map to account for the emission from @xmath0 decay in the fermi - lat signal would create a residual structure that could be fit by a bivariate gaussian with a mean intensity 17% as large as the overall neutral pion gamma - ray emission . since the latter vastly dominates the gamma - ray sky at low energies , this is a very significant effect between 0.1 and a few gev . as remarked in @xcite , the comparison of preliminary galactic gamma - ray diffuse models with the lat source - subtracted sky data implies residuals which are likely much smaller than the systematic effect we point out here . in figure [ icsdivsync ] we show the normalized line - of - sight ratio of ic emission at four energies ( 3.55 gev , 7.08 gev , 14.1 gev and 35.5 gev ) divided by synchrotron emission at 408 mhz . the maps are again smoothed using a gaussian of 2@xmath7 , although in this case the intrinsic discreteness of galprop simulation grids makes the effect of this smoothing minimal . we note a pronounced structure above and below the galactic center which corresponds to an excess emission at low energies , but a deficit of emission at high energies . returning to table [ tab : regions ] ( rows 2 - 5 ) , we note that the ratios of ic and synchrotron emission remain relatively constant in all of the sky regions we consider . this occurs for two reasons : ( 1 ) the galprop model is nearly symmetric around the galactic plane , and ( 2 ) synchrotron emission dies off quickly at high latitudes , making the deviation just as prominant in the d09 regions as the inner regions . the morphology of these skymaps , however , has a pronounced energy dependence . at 3.55 gev , there is a 16% overabundance in ic emission above and below the galactic center , while at 35.5 gev , this changes to an approximately 5% deficit in ic emission . at all energies we see a deficit in ic near the galactic anti - center of between 5 - 17% . the systematic effect stemming from assumption ( 2 ) would thus play a role in the determination of a gamma - ray haze spectrum , since the spatial template would change significantly with energy . the morphology of these systematic effects is intuitively reasonable . both ic and synchrotron emission depend on the same input population of high energy electrons ( here modeled according to the results of @xcite ) . thus , the morphology of the ic to synchrotron ratio depends primarily on the morphology of the energy density in the interstellar - radiation field ( isrf ) compared to the galactic magnetic field ( though we note that electrons at a given energy do not upscatter photons to a single energy , as the isrf is composed of photons across a large range of frequencies ) . the magnetic field model we employ falls off sharply at high latitudes , but only weakly with changing radii , explaining its relative brightness near the galactic anti - center , and dimness at high galactic latitude . we note that the ratio of @xmath0 emission to the input gas map is fairly independent of the galprop propagation setup , as it depends primarily on the cosmic - ray density throughout space , which is constrained by local observations of cosmic ray fluxes and primary - to - secondary ratios . while the ratio of ic to synchrotron emission depends on the same source of high energy leptons , the morphology depends on the morphology of the isrf compared to the magnetic fields . thus , one important feature which greatly affects the ratio of ic to synchrotron emission is the assumed morphology of the galactic magnetic fields , which are highly uncertain away from the sun s local neighborhood . in figure [ icsdivsync.blargeradii ] , we show the resulting ratio of ic to synchrotron emission for a magnetic field which decays exponentially with scale radii of 20.0 kpc in r and 2.0 kpc in z , tuned to a magnitude of 5 @xmath4 g at the solar position . we note that this magnetic field setup increases the haze structure above and below the galactic pole by comparatively decreasing synchrotron radiation in that region , creating ratios greater than two in the galactic center and a gaussian deviation ranging from 1.54 at 3.55 gev to 1.40 at 35.5 gev . furthermore , this setup decreases the height of the haze structure to approximately 45@xmath7 , which is in good agreement with the observed haze in d09 . in table [ tab : regions ] ( rows 6 - 9 ) we show the resulting haze ratios at each energy cutoff , finding ratios between 1.4 - 1.6 depending on the given energy level and region . another feature which may artificially yield greater uniformity in the ratio of the ic to synchrotron emission is the height of the diffusion zone . for small diffusion boxes , our high latitude gamma ray emission would be dominated by local cosmic rays , decreasing the variability of our results . however , this modeling trick can not be used to support the haze hypothesis , since the conjecture states that a flux of high energy leptons exists between 10@xmath7 to 30@xmath7 above and below the galactic center . a smaller diffusion region such as that at 2 kpc , will shut off emission at more than 14@xmath7 latitude above the galactic plane , surpressing almost all emission from the haze region . while a complete study of the diffuse emission detected by fermi - lat in terms of foreground templates is outside the scope of this @xmath1 , work is currently ongoing that will also include the possible contribution to gamma radiation of nearby structures such as loop i @xcite which are not modeled within galprop . whether improved spatial templates could conclusively pinpoint an excess diffuse emission in the fermi - lat data is hard to assess , as changes in the spatial morphology of each input will influence the best fit intensities of each component . however , we note that if the gas and synchrotron templates used in d09 were appropriate matches to @xmath0 and ic emission , we would expect ratios very close to unity across the entire skymap . we instead find significant deviations with an intensity comparable to the fermi haze at low energies . moreover , these deviations are not randomly distributed across the sky , but instead contain a structure resembling the bivariate gaussian reported by d09 . thus , our results point to the possibility that the fermi haze determined by the `` type 2 '' and `` type 3 '' templates of d09 is the result of systematic effects in the spatial template fitting procedure as opposed to the existence of a new source class . we have shown that the use of the gas map would create a best match bivariate gaussian intensity approximately 17% as strong as the @xmath0 decay amplitude . since we expect @xmath0 decay to dominate the diffuse gamma - ray sky below 10 gev , this error can explain a large fraction of the d09 haze . similarly , ic due to ordinary galactic cosmic rays is expected to show an excess of approximately 20% in the haze region , with large variances depending on the assumed magnetic field model . at high energies ( @xmath6 10 gev ) , @xmath0 decay is weaker , and thus a 15% error is not expected to dominate astrophysical ic or haze signals . furthermore , some magnetic field models create synchrotron templates which would overestimate ic emission in this region . thus , the remaining amplitude of a d09 haze may in fact be larger or smaller than reported in this region . we note additional errors may be present at high energies both due to low photon counts as well as instrumental effects such as cosmic ray contamination . in addition , nearby cosmic rays such as those in the giant radio feature loop i can fake a diffuse gamma - ray emission in the direction of the galactic center unrelated to a wmap haze counterpart . d09 additionally adopts another template , dubbed `` type i '' , where they use the fermi - lat data in the 1 - 2 gev range as a proxy for the morphology of @xmath0 emission . using this template , d09 finds an excess emission that becomes more prominent at higher energies , indicating a residual with a harder spectrum than @xmath0-decay emission . they note that this excess has a morphology comparable with those found by their astrophysical template fits , except with a more peaked structure that may be due to the subtraction of the 1 - 2 gev components of the ics and bremsstrahlung maps . this emission template does not claim to subtract any @xmath11-ray foregrounds except for those due to @xmath0-decay , but notes that the residual has a morphology which is peculiar for models such as ics of high energy astrophysical leptons . as this template does not make use of either the synchrotron or sfd maps , it falls outside the scope of our analysis . we agree that the emission is likely not due to @xmath0 , and it is difficult to construct an astrophysical source distribution which is more pronounced towards high latitudes than near the galactic plane . any galactic electron sources should lie close to the galactic plane , and would primarily upscatter starlight to gev energies . in figure [ fig : isrf ] , we show the isrf used in our galprop models integrated over the line - of - sight . we see that the isrf is also strongest along the galactic plane , extended in longitude rather than latitude by a ratio from 5 - 4 at 12.4ev to 9 - 1 at 0.0124ev . we further note that the isrf dims by between 63 - 72% between 10@xmath7 - 30@xmath7 latitude . this decay is itself slightly stronger than the decay in diffuse emission used in the d09 gaussian template . this isrf is morphologically identical to the ic morphologies obtained by comvolving the input isrf with a isotropic and monochromatic input electron spectrum . since this spectrum is much broader than the haze determined by d09 , and falls slightly more quickly at high galactic latitudes , the input source class would have to be significantly peaked above and below the galactic plane , with very little extent along the plane . furthermore , the source class must have a flux at 4.25 kpc above the galactic center , which is almost equvalent to the flux at 1.5 kpc above the galactic center , in order that the product of the lepton flux and isrf fall off by a ratio comparable to the gaussian haze . the characteristics of this input source spectrum is not similar to those of expected galactic sources . this problem night be overcome by several ad hoc changes in the isrf around the galactic center region , or by changes in the convection currents and diffusion constants away from the galactic plane @xcite . however , these same changes will greatly alter the assumed morphology of astrophysical ics as well , possibly eliminating the need for an extra diffuse component . in summary , we showed that significant systematic effects make it difficult to reliably assess a diffuse gamma - ray emission in the region associated to the wmap haze . a fully self - consistent galactic cosmic - ray model is necessary to model the astrophysical diffuse emission from the galaxy and to compare it to the fermi - lat data . we thank troy porter and andy strong for feedback regarding galprop models , as well as gregory dobler and douglas finkbeiner for useful discussions regarding the template construction in d09 . tl is supported by a gaann fellowship by the department of education . sp is supported by an oji award from the us department of energy ( doe contract defg02 - 04er41268 ) , and by nsf grant phy-0757911 . ccccccccc model i d & background & & d09 haze & d09 n. haze & inner haze & n. inner haze & gaussian haze & anticenter + + + * @xmath0 decay * & 0.97 & & 1.04 & 1.15 & 1.15 & 1.21 & 1.17 & 0.79 + + * base ic - 3.55 gev * & 0.95 & & 1.16 & 1.16 & 1.16 & 1.16 & 1.16 & 0.83 + * base ic - 7.08 gev * & 0.96 & & 1.09 & 1.09 & 1.10 & 1.09 & 1.10 & 0.86 + * base ic - 14.1 gev * & 0.98 & & 1.03 & 1.03 & 1.04 & 1.04 & 1.05 & 0.89 + * base ic - 35.5 gev * & 1.00 & & 0.94 & 0.94 & 0.95 & 0.95 & 0.96 & 0.95 + + * alt . ic - 3.55 gev * & 0.88 & & 1.47 & 1.46 & 1.57 & 1.56 & 1.54 & 0.62 + * alt . ic - 7.08 gev * & 0.89 & & 1.42 & 1.41 & 1.53 & 1.51 & 1.49 & 0.65 + * alt . ic - 14.1 gev * & 0.90 & & 1.38 & 1.37 & 1.49 & 1.48 & 1.46 & 0.68 + * alt . ic - 35.5 gev * & 0.92 & & 1.31 & 1.30 & 1.43 & 1.41 & 1.40 & 0.74 + + + + , e. a. , berenji , b. , bertone , g. , bergstrm , l. , bloom , e. , bringmann , t. , chiang , j. , cohen - tanugi , j. , conrad , j. , edmonds , y. , edsj , j. , godfrey , g. , hughes , r. e. , johnson , r. p. , lionetto , a. , moiseev , a. a. , morselli , a. , moskalenko , i. v. , nuss , e. , ormes , j. f. , rando , r. , sander , a. j. , sellerholm , a. , smith , p. d. , strong , a. w. , wai , l. , wang , p. , & winer , b. l. 2008 , journal of cosmology and astro - particle physics , 7 , 13
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recent claims of a gamma - ray excess in the diffuse galactic emission detected by the fermi large area telescope made use of spatial templates from the interstellar medium ( ism ) column density and the 408 mhz sky as proxies for neutral pion and inverse compton ( ic ) gamma - ray emission , respectively .
we identify significant systematic effects in this procedure that can artificially induce an additional diffuse component with a morphology strikingly similar to the claimed gamma - ray haze .
to quantitatively illustrate this point we calculate sky - maps of the ratio of the gamma - ray emission from neutral pions to the ism column density , and of ic to synchrotron emission , using detailed galactic cosmic - ray models and simulations . in the region above and below the galactic center ,
the ism template underestimates the gamma - ray emission due to neutral pion decay by approximately 20% .
additionally , the synchrotron template tends to under - estimate the ic emission at low energies ( few gev ) and to over - estimate it at higher energies ( tens of gev ) by potentially large factors that depend crucially on the assumed magnetic field structure of the galaxy .
the size of the systematic effects we find are comparable to the size of the claimed `` fermi haze '' signal . we thus conclude that a detailed model for the galactic diffuse emission is necessary in order to conclusively assess the presence of a gamma - ray excess possibly associated to the wmap haze morphology .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
a charge placed in a polarizable medium is screened . dielectric theory describes the phenomenon by the induction of a polarization around the charge carrier . the induced polarization will follow the charge carrier when it is moving through the medium . the carrier together with the induced polarization is considered as one entity ( see fig.[fig_scheme1 ] ) . it was called a _ polaron _ by l. d. landau @xcite . the physical properties of a polaron differ from those of a band - carrier . a polaron is characterized by its _ binding ( or self- ) energy _ @xmath0 , an _ effective mass _ @xmath1 and by its characteristic _ response _ to external electric and magnetic fields ( e. g. dc mobility and optical absorption coefficient ) . if the spatial extension of a polaron is large compared to the lattice parameter of the solid , the latter can be treated as a polarizable continuum . this is the case of a _ large ( frhlich ) _ polaron . when the self - induced polarization caused by an electron or hole becomes of the order of the lattice parameter , a _ smll ( holstein ) _ polaron can arise . as distinct from large polarons , small polarons are governed by short - range interactions . [ [ the - polaron - radius .- large - polarons - vs - small - polarons ] ] the polaron radius . large polarons vs small polarons + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + consider the lo phonon field with frequency @xmath2 interacting with an electron . denote by @xmath3 the quadratic mean square deviation of the electron velocity . in the electron - phonon interaction is weak , the electron can travel a distance @xmath4 during a time @xmath5characteristic for the lattice period , because it is the distance within which the electron can be localized using the phonon field as measuring device . from the uncertainty relations it follows @xmath6 at weak coupling @xmath7 is a measure of the polaron radius @xmath8 . to be consistent , the polaron radius @xmath8 must be considerably larger than the lattice parameter @xmath9(this is a criterion of a large polaron ) . experimental evaluation of the polaron radius leads to the follwing typical values : @xmath10 for alkali halides , @xmath11 for silver halides , @xmath12 for ii - vi , ii - v semiconductors . the continuum approximation is not satisfied for transition metal oxides ( nio , cao , mno ) , in other oxides ( uo@xmath13nbo@xmath14 ) . for those solids the small polaron concept is used . in some substances ( e.g. perovskites ) some intermediate region between large and small polarons is realized . [ [ the - coupling - constant ] ] the coupling constant @xcite + + + + + + + + + + + + + + + + + + + + + + + + + + + + + consider the case of _ strong electron - phonon interaction _ in a polar crystal . the electron of mass @xmath15 is then localized and can - to a first approximation - be considered as a static charge distribution within a sphere with radius @xmath16 the medium is characterized by an average dielectric constant @xmath17which will be defined below . the potential energy of a sphere of radius @xmath18uniformly charged with the charge @xmath19 in a vacuum is ( see eq . ( 8.6 ) of ref . @xcite)@xmath20 the potential energy of a uniformly charged sphere in a medium with the high - frequency dielectric constant @xmath21 is@xmath22 this is the potential energy of the _ self - interaction _ of the charge @xmath19 uniformly spread over the sphere of radius @xmath18 in a medium with the dielectric constant @xmath21 . in a medium with an inertial polarization field ( due to lo phonons ) , the potential energy of the uniformly charged sphere is@xmath23 where @xmath24 is the static dielectric constant . the polaron effect is then related to the change of the potential energy of the interaction of the charged sphere due to the inertial polarization field . this change is the potential energy @xmath25 of the uniformly charged sphere in the presence of the inertial polarization field minus the potential energy of the _ self - interaction _ @xmath26of the charge @xmath19 uniformly spread over the sphere in a medium without the inertial polarization:@xmath27 with@xmath28 the electron distribution in a sphere may be non - uniform , what may influence the numerical coefficient in eqs . ( [ u1 ] ) to ( [ upol ] ) . in this connection one can use the estimate @xcite@xmath29 the restriction of the electron in space requires its de broglie wave length to be of the order @xmath30so that its kinetic energy is of the order @xmath31minimizing the total energy with respect to @xmath18leads to @xmath32 wherefrom the binding energy is @xmath33 for _ weak coupling _ , one can neglect the kinetic energy of the electron . taking the polaron radius according to ( [ i2 ] ) , @xmath34 the binding energy is@xmath35 we note that @xmath36 following the conventions of the field theory , the self energy at weak coupling is written as @xmath37 therefore the so - called frhlich polaron coupling constant is @xmath38 for the average dielectric constant one shows that @xmath39 where @xmath21 and @xmath24 are , respectively , the electronic and the static dielectric constant of the polar crystal . the difference @xmath40 arises because the ionic vibrations occur in the infrared spectrum and the electrons in the shells can follow the conduction electron adiabatically . [ [ polaron - mobility ] ] polaron mobility + + + + + + + + + + + + + + + + here we give a simple derivation leading to the gross features of the mobility behaviuor , especially its temperature dependence . the key idea is that the mobility will change because the number of phonons in the lattice , with which the polaron interacts , is changing with temperature . the phonon density is given by @xmath41 the mobility for large polaron is proportional to the inverse of the number of phonons : @xmath42 and for low temperatures @xmath43 @xmath44 the mobility of continuum poarons decreases with increasing temperature following an exponential law . the slope of the straight line in @xmath45 vs @xmath46 is characterized by the lo phonon frequesncy . systematic study performed , in particular , by frhlich and kadanoff , gives @xmath47 the small polaron will jump from ion to ion under the influence of optical phonons . the lerger the numver of phonons , the lerger the mobility . the behaviuor of the small polaron is the opposite of that of the large polaron . one expects : @xmath48 for low temperatures @xmath43 one has : @xmath49 the mobility of small polaron is thermally activated . systematic analysis within the small - polaron theory shows that @xmath50 with @xmath51 frhlich proposed a model hamiltonian for the large polaron through which its dynamics is treated quantum mechanically ( frhlich hamiltonian ) . the polarization , carried by the longitudinal optical ( lo ) phonons , is represented by a set of quantum oscillators with frequency @xmath52 , the long - wavelength lo - phonon frequency , and the interaction between the charge and the polarization field is linear in the field @xcite:@xmath53 where @xmath54 is the position coordinate operator of the electron with band mass @xmath55 , @xmath56 is its canonically conjugate momentum operator ; @xmath57 and @xmath58 are the creation ( and annihilation ) operators for longitudinal optical phonons of wave vector @xmath59 and energy @xmath60 . the @xmath61 are fourier components of the electron - phonon interaction @xmath62 the strength of the electron phonon interaction is expressed by a dimensionless coupling constant @xmath63 , which is defined as : @xmath64 in this definition , @xmath21 and @xmath24 are , respectively , the electronic and the static dielectric constant of the polar crystal . in table [ table1 ] the frhlich coupling constant is given for a few solids are not well established . ] . [ c]@llllllmaterial & @xmath63 & ref . & material & @xmath63 & ref . + insb & 0.023 & @xcite & agcl & 1.84 & @xcite + inas & 0.052 & @xcite & ki & 2.5 & @xcite + gaas & 0.068 & @xcite & tlbr & 2.55 & @xcite + gap & 0.20 & @xcite & kbr & 3.05 & @xcite + cdte & 0.29 & @xcite & bi@xmath65sio@xmath66 & 3.18 & @xcite + znse & 0.43 & @xcite & cdf@xmath67 & 3.2 & @xcite + cds & 0.53 & @xcite & kcl & 3.44 & @xcite + @xmath63-al@xmath67o@xmath68 & 1.25 & @xcite & csi & 3.67 & @xcite + agbr & 1.53 & @xcite & srtio@xmath68 & 3.77 & @xcite + @xmath63-sio@xmath67 & 1.59 & @xcite & rbcl & 3.81 & @xcite + in deriving the form of @xmath61 , expressions ( [ eq_1b ] ) and ( [ eq_1c ] ) , it was assumed that ( i ) the spatial extension of the polaron is large compared to the lattice parameter of the solid ( continuum approximation ) , ( ii ) spin and relativistic effects can be neglected , ( iii ) the band - electron has parabolic dispersion , ( iv ) in line with the first approximation it is also assumed that the lo - phonons of interest for the interaction , are the long - wavelength phonons with constant frequency @xmath52 . the model , represented by the hamiltonian ( [ eq_1a ] ) ( which up to now could not been solved exactly ) has been the subject of extensive investigations , see , e. g. , refs . @xcite . in what follows the key approaches of the frhlich - polaron theory are briefly reviewed with indication of their relevance for the polaron problems in nanostructures . here some insight will be given in the type of transformation that might be useful to study the frhlich hamiltonian ( [ eq_1a ] ) . for this purpose the hamiltonian will be treated for a particle with infinite mass @xmath69 ( which is at @xmath70@xmath71 which can be transformed into the following expression with shifted phonon operators:@xmath72 to determine the eigenstates of this hamiltonian , one can perform a unitary transformation which produces the following shift of the phonon operators:@xmath73 the transformation@xmath74 \label{s}\ ] ] is canonical : @xmath75 = s^{-1}\ ] ] and has the desired property:@xmath76 the transformed hamiltonian is now:@xmath77 the eigenstates of the hamiltonian contain an integer number of phonons @xmath78the eigenenergies are evidently:@xmath79 this expression is divergent at it is often the case in field theory of point charges are considered . a transformation of the type @xmath80 has been of great interest in developing weak coupling theory as shown below . for actual crystals @xmath63-values typically range from @xmath81 ( insb ) up to @xmath82 to @xmath83 ( alkali halides , some oxides ) , see table 1 . a weak - coupling theory of the polaron was developed originally by frhlich @xcite . he derived the first weak - coupling perturbation - theory results : @xmath84 and @xmath85 expressions ( [ eq_5a ] ) and ( [ eq_5b ] ) are rigorous to order @xmath63 . inspired by the work of tomonaga on quantum electrodynamics ( q. e. d. ) , lee , low and pines ( llp ) @xcite analyzed the properties of a weak - coupling polaron starting from a formulation based on canonical transformations ( cp . the results of the subsection [ sec - shift]).as hown by them , the unitary transformation@xmath86 where @xmath87 is a `` c''-number representing the _ total system momentum _ allows to eliminate the electron co - ordinates from the system . intuitively one might guess this transformation by writing the exact wave function in the form@xmath88 it is plausible that the bloch factor @xmath89 attaches the system to the electron as origin of the co - cordinates . after this transformation the hamiltonian ( [ eq_1a ] ) becomes:@xmath90 if , for the sake of simplicity , the case of total momentum equal to zero is considered , this expression becomes:@xmath91 the first term of this hamiltonian is the correlation energy term involving different values for @xmath59 and @xmath92if one diagonalizes the second and the trird term of the hamiltonian ( [ hcorr ] ) ( this can be done exactly by means of the `` shifted - oscillator canonical transformation '' @xmath80 ( [ s ] ) ) , the result of llp is found . the expectation value of the first term is zero for the wave function @xmath93 therefore one is sure to obtain a variational result . it is remarkable that merely extracting the @xmath94 term from the expression@xmath95 eliminates the divergency from the problem ( cp . with the case @xmath96 ) and is equivalent to the sophisticated theory by lee , low and pines ( llp ) , which corresponds thus to neglect of the term ( [ corr ] ) . the details of the llp theory are given in appendix 1 . the explicit form for the energy is now@xmath97 this self energy is no longer divergent . the divergence is elmininated by the quantum cut - off occurring at @xmath98 for the self energy the llp result is equivalent to the perturbation result . the effective mass however is now given by@xmath99 a result , which follows if one considers the case @xmath100 and which is also exact for @xmath101 . however , the llp effective mass is different from the perturbation result if @xmath63 insreases . the llp approximation has often been called intermediate - coupling approximation . however its range of validity is the same as that of perturbation theory to order @xmath63 . the significance of the llp approximation consists of the flexibility of the canonical transformations together with the fact that it puts the frhlich result on a variational basis . to order @xmath102 , the analytical expressions for the coefficients are @xmath102 : @xmath103 for the energy and @xmath104 for the polaron mass @xcite . at present the following weak - coupling expansions are known : for the energy @xcite @xmath105 and for the polaron mass @xcite @xmath106 historically , the strong coupling limit was studied before all other treatments ( landau , pekar @xcite ) . although it is only a formal case because the actual crystals seems to have @xmath63 values smaller than 5 , it is very interesting because it contains some indication of the intermediate coupling too : approach the excitations from the strong coupling limit and extrapolate to intermediate coupling is interesting because it is expected that some specific strong coupling properties survive at intermediate coupling . in what follows , a treatment , equivalent to that of pekar , but in second quantization and written with as much analogy to the llp treatment as possible is given . we start from the frhlich hamiltonian ( [ eq_1a ] ) . at strong coupling one makes the assumption ( a produkt - ansatz ) for the polaron wave - function@xmath107 where @xmath108 is the electron - component of the wave function ( @xmath109the field - component of the wave function @xmath110 ( @xmath111 parametrically depends on @xmath112 . the produkt - ansatz ( [ eq_2a ] ) or born - oppenheimer approximation implies that the electron adiabatically follows the motion of the atoms , while the field can not follow the instantaneous motion of the electron . frhlich showed that the approximation ( [ eq_2a ] ) leads to results , which are only valid for sufficiently large @xmath113 , i. e. in the strong - coupling regime . a more systematic analysis of strong - coupling polarons based on canonical transformations applied to the hamiltonian ( [ eq_1a ] ) was performed in refs . @xcite . the expectation value for the energy is now:@xmath114 \left\vert f\right\rangle\ ] ] with @xmath115 we wish to minimize @xmath116 but also @xmath117 \left\vert f\right\rangle\ ] ] has to be minimized . this expression will be minimized if @xmath118 is the ground state wave function of the shifted oscullator - type hamiltonian . as we can diagonalize this hamiltonian exactly:@xmath119 we can apply a canonical transformation similar to ( [ s]):@xmath120 , \ ] ] which has the property : @xmath121 the transformed hamiltonian is now : @xmath122 s\\ & = \sum_{\mathbf{k}}\hbar\omega_{\mathrm{lo}}a_{\mathbf{k}}^{\dag } a_{\mathbf{k}}-\sum_{\mathbf{k}}\frac{\left\vert v_{k}\right\vert ^{2}\left\vert \rho_{\mathbf{k}}\right\vert ^{2}}{\hbar\omega_{\mathrm{lo}}}.\end{aligned}\ ] ] the phonon vacuum @xmath123 provides a minimum:@xmath124 s\left\vert 0\right\rangle = -\sum_{\mathbf{k}}\frac{\left\vert v_{k}\right\vert ^{2}\left\vert \rho_{\mathbf{k}}\right\vert ^{2}}{\hbar\omega_{\mathrm{lo}}}.\ ] ] hence , the hamiltonian ( [ sc1 ] ) is minimized by the ground state wave function@xmath125 \left\vert 0\right\rangle . \label{scwf}\ ] ] it gives the ground state energy@xmath126 which is still a functional of @xmath127 . the functionals @xmath128 are different for differerent excitations . [ [ ground - state - of - strong - coupling - polarons ] ] ground state of strong - coupling polarons + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + for the ground state one considers a gaussian wave function : @xmath129 with a variational parameter @xmath130.@xmath131 ^{3}\\ & = c^{2}\left ( \frac{\sqrt{\pi}}{\frac{m_{b}\omega_{0}}{\hbar}}\right ) ^{3}=c^{2}\left ( \frac{\pi\hbar}{m_{b}\omega_{0}}\right ) ^{3/2}=1\rightarrow c^{2}=\left ( \frac{m_{b}\omega_{0}}{\pi\hbar}\right ) ^{3/2}\]]@xmath132 for the further use , we introduce a notation @xmath133 such a wave function is consistent with the localization of the electron , which we expect for large @xmath63 . the kinetic energy in ( [ sc2 ] ) for this function is calculated using the representation of the operator @xmath134 @xmath135:@xmath136@xmath137 \\ & = 3\frac{\hbar^{2}}{2m_{b}}c_{1}^{2}\frac{m_{b}\omega_{0}}{\hbar}\int_{-\infty}^{\infty}dx\left ( 1-\frac{m_{b}\omega_{0}}{\hbar}x^{2}\right ) \exp\left ( -\frac{m_{b}\omega_{0}}{\hbar}x^{2}\right ) \\ & = = 3\frac{\hbar^{2}}{m_{b}}c_{1}^{2}\frac{m_{b}\omega_{0}}{\hbar}\int _ { 0}^{\infty}dx\left ( 1-\frac{m_{b}\omega_{0}}{\hbar}x^{2}\right ) \exp\left ( -\frac{m_{b}\omega_{0}}{\hbar}x^{2}\right ) \\ & = 3\frac{\hbar^{2}}{m_{b}}c_{1}^{2}\frac{m_{b}\omega_{0}}{\hbar}\left [ \frac{\sqrt{\pi}}{2\sqrt{\frac{m_{b}\omega_{0}}{\hbar}}}-\frac{m_{b}\omega _ { 0}}{\hbar}\frac{\sqrt{\pi}}{4\sqrt{\left ( \frac{m_{b}\omega_{0}}{\hbar } \right ) ^{3}}}\right ] \\ & = 3\frac{\hbar\omega_{0}}{4}c_{1}^{2}\sqrt{\frac{\pi\hbar}{m_{b}\omega_{0}}}=3\frac{\hbar\omega_{0}}{4}\sqrt{\frac{m_{b}\omega_{0}}{\pi\hbar}}\sqrt { \frac{\pi\hbar}{m_{b}\omega_{0}}}=\frac{3}{4}\hbar\omega_{0}.\end{aligned}\ ] ] the functional@xmath138 \right ) \nonumber\\ & = c^{2}\exp\left ( -\frac{m_{b}\omega_{0}}{\hbar}\frac{\hbar^{2}k^{2}}{4m_{b}^{2}\omega_{0}^{2}}\right ) \int d^{3}r\exp\left ( -\frac{m_{b}\omega_{0}}{\hbar}\left [ \mathbf{r}+i\frac{\hbar}{2m_{b}\omega_{0}}\mathbf{k}\right ] ^{2}\right ) \rightarrow\nonumber\\ \rho_{\mathbf{k}"1s " } & = \exp\left ( -\frac{\hbar k^{2}}{4m_{b}\omega_{0}}\right ) . \label{scrho}\ ] ] the second term in ( [ sc2 ] ) is then @xmath139 the variational energy ( [ sc2 ] ) thus becomes@xmath140 putting @xmath141 one obtains@xmath142@xmath143 substituting ( [ sc4 ] ) in ( [ sc3 ] ) , we find the ground state energy of the polaron @xmath0 ( calculated with the energy of the uncoupled electron - phonon system as zero energy ) : @xmath144@xmath145 the strong - coupling mass of the polaron , resulting again from the approximation ( [ eq_2a ] ) , is given @xcite as : @xmath146 more rigorous strong - coupling expansions for @xmath0 and @xmath1 have been presented in the literature @xcite : @xmath147@xmath148 the strong - coupling ground state energy ( [ sc5 ] ) is lower than the llp ground state energy for @xmath149 [ [ the - excited - states - of - the - polaron - ss - fc - res ] ] the excited states of the polaron : ss , fc , res + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + in principle , excited states of the polaron exist at all coupling . in the general case , and for simplicity for * * @xmath150 , a continuum of states starts at @xmath60 above the ground state of the polaron . this continuum physically corresponds to the scattering of free phonons on the polaron . those scattering.states ( ss ) were studied in @xcite anf for the first time more generally in @xcite are not the only excitations of the polaron . there are also _ internal _ excitation states corresponding to the excitations of the electron in the potential it created itself . by analogy with the excited states of colour centers , the following terminology is used . \(i ) the states where the electron is excited in the potential belonging to the ground state configuration of the lattice are called _ franck - condon _ ( fc ) states \(ii ) excitations of the electron in which the lattice polarization is adapted to the electronic configuration of the excited electron ( which itself then adapts its wave function to the new potential , etc . leading to a self - consistent final state ) , are called _ relaxed excited state _ ( res ) @xcite . [ [ calculation - of - the - lowest - fc - state ] ] calculation of the lowest fc state + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the formalism used until now is well adapted to treat the polaron excitations at strong coupling . the field dependence of the wave function is ( [ scwf ] ) . for the fc state the @xmath128 are the same as for the ground state ( [ scrho ] ) . physically @xmath128 tells us , to what electronic distribution the field is adapted . the electronic part of the excited wave function is @xmath151-like:@xmath152 with a parameter @xmath153 , which is equal to @xmath154@xmath155 ^{2}\\ & = c_{"2p"}^{2}\left ( \frac{\sqrt{\pi}}{\frac{m_{b}\omega_{p}}{\hbar}}\right ) ^{2}\frac{\sqrt{\pi}}{2\left ( \frac{m_{b}\omega_{p}}{\hbar}\right ) ^{3/2}}=c_{"2p"}^{2}\left ( \frac{\pi\hbar}{m_{b}\omega_{p}}\right ) ^{3/2}\frac{\hbar}{2m_{b}\omega_{p}}=1\rightarrow\\ c_{"2p"}^{2 } & = \left ( \frac{m_{b}\omega_{p}}{\pi\hbar}\right ) ^{3/2}\frac{2m_{b}\omega_{p}}{\hbar}\]]@xmath156 we introduce still a notation @xmath157 the fc state energy is , similarly to ( [ sc2]),@xmath158 the kinetic energy term is @xmath159 \nonumber\\ & = -\frac{\hbar^{2}}{2m_{b}}c_{_{"2p"}}^{2}\int_{-\infty}^{\infty}dzz\exp\left ( -\frac{m_{b}\omega_{p}}{2\hbar}z^{2}\right ) \nabla_{z}^{2}\left [ z\exp\left ( -\frac{m_{b}\omega_{p}}{2\hbar}z^{2}\right ) \right ] \nonumber\\ & \times\int_{-\infty}^{\infty}dy\exp\left ( -\frac{m_{b}\omega_{p}}{\hbar } y^{2}\right ) \int_{-\infty}^{\infty}dz\exp\left ( -\frac{m_{b}\omega_{p}}{\hbar}z^{2}\right ) + \frac{2}{4}\hbar\omega_{p}\nonumber\\ & = -\frac{\hbar^{2}}{2m_{b}}c_{2}^{2}\int_{-\infty}^{\infty}dzz\exp\left ( -\frac{m_{b}\omega_{p}}{2\hbar}z^{2}\right ) \nonumber\\ & \times\nabla_{z}\left [ \left ( 1-\frac{m_{b}\omega_{p}}{\hbar}z^{2}\right ) \exp\left ( -\frac{m_{b}\omega_{p}}{2\hbar}z^{2}\right ) \right ] + \frac{1}{2}\hbar\omega_{p}\nonumber\\ & = \frac{\hbar^{2}}{2m_{b}}c_{2}^{2}\int_{-\infty}^{\infty}dzz\exp\left ( -\frac{m_{b}\omega_{p}}{2\hbar}z^{2}\right ) \nonumber\\ & \times\left [ \frac{2m_{b}\omega_{p}}{\hbar}z+\left ( 1-\frac{m_{b}\omega_{p}}{\hbar}z^{2}\right ) \frac{m_{b}\omega_{p}}{\hbar}z\right ] \exp\left ( -\frac{m_{b}\omega_{p}}{2\hbar}z^{2}\right ) \nonumber\\ + \frac{1}{2}\hbar\omega_{p } & = \frac{\hbar^{2}}{2m_{b}}c_{2}^{2}\int_{-\infty}^{\infty}dz\left ( 3-\frac{m_{b}\omega_{p}}{\hbar}z^{2}\right ) \frac{m_{b}\omega_{p}}{\hbar}z^{2}\exp\left ( -\frac{m_{b}\omega_{p}}{\hbar } z^{2}\right ) \nonumber\\ + \frac{1}{2}\hbar\omega_{p } & = \frac{\hbar\omega_{p}}{2}c_{2}^{2}\left [ \frac{3\sqrt{\pi}}{2\left ( \frac{m_{b}\omega_{p}}{\hbar}\right ) ^{3/2}}-\frac{m_{b}\omega_{p}}{\hbar}\frac{3\sqrt{\pi}}{4\left ( \frac{m_{b}\omega_{p}}{\hbar}\right ) ^{5/2}}\right ] + \frac{1}{2}\hbar\omega _ { p}\nonumber\\ & = \frac{\hbar\omega_{p}}{2}c_{2}^{2}\frac{3\sqrt{\pi}}{4\left ( \frac { m_{b}\omega_{p}}{\hbar}\right ) ^{3/2}}+\frac{1}{2}\hbar\omega_{p}\nonumber\\ & = \frac{\hbar\omega_{p}}{2}\frac{2}{\sqrt{\pi}}\left ( \frac{m_{b}\omega _ { p}}{\hbar}\right ) ^{3/2}\frac{3\sqrt{\pi}}{4\left ( \frac{m_{b}\omega_{p}}{\hbar}\right ) ^{3/2}}+\frac{1}{2}\hbar\omega_{p}\nonumber\\ & = \frac{3}{4}\hbar\omega_{p}+\frac{1}{2}\hbar\omega_{p}=\frac{5}{4}\hbar\omega_{p}. \label{sckin}\ ] ] for the fc state , @xmath160 the second term in ( [ sc6 ] ) is precisely ( [ scfield ] ) , @xmath161 and the fc energy ( [ sc6 ] ) becomes@xmath162 the energy of the lowest fc state is , within the produkt - ansatz @xcite : @xmath163 the fact that this energy is positive , is presumably due to the choice of a harmonic potential . the real potential the electron sees is anharmonic , and a bound state may be expected . [ [ calculation - of - res ] ] calculation of res + + + + + + + + + + + + + + + + + + the electronic part of the excited wave function is ( [ scwf2p ] ) with a variational parameter @xmath164which is determined below . the variational res energy is , similarly to ( [ sc2 ] ) , @xmath165 here the kinetic energy term is given by eq . ( [ sckin ] ) . the functional , which is now needed , is @xmath166 \right ) \\ & = c_{"2p"}^{2}\ exp\left ( -\frac{m_{b}\omega_{p}}{\hbar}\frac{\hbar ^{2}k^{2}}{4m_{b}^{2}\omega_{p}^{2}}\right ) \int d^{3}rz^{2}\exp\left ( -\frac{m_{b}\omega_{p}}{\hbar}\left [ \mathbf{r}+i\frac{\hbar}{2m_{b}\omega_{p}}\mathbf{k}\right ] ^{2}\right ) \rightarrow\\end{aligned}\]]@xmath167 ^{2}\right ) \nonumber\\ & = .\exp\left ( -\frac{\hbar k^{2}}{4m_{b}\omega_{p}}\right ) c_{2}^{2}\int_{-\infty}^{\infty}dz\left ( z - i\frac{\hbar}{2m_{b}\omega_{p}}k_{z}\right ) ^{2}\exp\left ( -\frac{m_{b}\omega_{p}}{\hbar}z^{2}\right ) \nonumber\\ & = \exp\left ( -\frac{\hbar k^{2}}{4m_{b}\omega_{p}}\right ) c_{2}^{2}\int_{-\infty}^{\infty}dz\left [ z^{2}-\left ( \frac{\hbar}{2m_{b}\omega_{p}}k_{z}\right ) ^{2}\right ] ^{2}\exp\left ( -\frac{m_{b}\omega_{p}}{\hbar } z^{2}\right ) \nonumber\\ & = \exp\left ( -\frac{\hbar k^{2}}{4m_{b}\omega_{p}}\right ) c_{2}^{2}\left\ { \frac{\sqrt{\pi}}{2\left ( \frac{m_{b}\omega_{p}}{\hbar}\right ) ^{3/2}}-\left ( \frac{\hbar}{2m_{b}\omega_{p}}k_{z}\right ) ^{2}\frac { \sqrt{\pi}}{\left ( \frac{m_{b}\omega_{p}}{\hbar}\right ) ^{1/2}}\right\ } \nonumber\\ & = \left ( 1-\frac{\hbar k_{z}^{2}}{2m_{b}\omega_{p}}\right ) \exp\left ( -\frac{\hbar k^{2}}{4m_{b}\omega_{p}}\right ) . \label{sc8}\ ] ] further , we substitute ( [ sc8 ] ) in the second term in the r.h.s . of eq . ( [ sc7a]):@xmath168\end{aligned}\ ] ] @xmath169 \\ & = -\frac{\alpha\hbar}{\sqrt{\pi}}\sqrt{\omega_{\mathrm{lo}}\omega_{p}}\frac{60 - 20 + 9}{60}=-\frac{49}{60}\frac{\alpha\hbar}{\sqrt{\pi}}\sqrt { \omega_{\mathrm{lo}}\omega_{p}}\ ] ] the variational energy ( [ sc7a ] ) becomes@xmath170 putting @xmath171 one obtains@xmath172 @xmath173 the energy of the res is ( see refs . @xcite ) : @xmath174 the effective mass of the polaron in the res is given @xcite as : @xmath175 the structure of the energy spectrum of the strong - coupling polaron is shown in fig.[fig_statesa ] . the ground state , @xmath176 the ( first ) relaxed excited state ; the franck - condon states ( @xmath177 ) . in fact , both the franck - condon states and the relaxed excited states lie in the continuum and , strictly speaking , are resonances . ] the significance of the strong - coupling large polaron theory is formal only : it allows to test all - coupling theories in the limit @xmath178 . remarkably , the effective electron - phonon coupling strength significantly increases in systems of low dimension and low dimensionality . feynman developed a superior all - coupling polaron theory using his path - integral formalism@xcite . he studied first the self - energy @xmath0 and the effective mass @xmath1 of polarons@xcite . feynman got the idea to formulate the polaron problem into the lagrangian form of quantum mechanics and then eliminate the field oscillators , in exact analogy to q. e. d. ( resulting in ) a sum over all trajectories . the resulting path integral ( here limited to the ground - state properties ) is of the form ( ref . @xcite ) : @xmath179 , \label{eq_8a}\ ] ] where @xmath180 . ( [ eq_8a ] ) gives the amplitude that an electron found at a point in space at time zero will appear at the same point at the ( imaginary ) time @xmath181 . this path integral ( [ eq_8a ] ) has a great intuitive appeal : it shows the polaron problem as an equivalent one - particle problem in which the interaction , non - local in time or retarded , occurs between the electron and itself . subsequently feynman showed how the variational principle of quantum mechanics could be adapted to the path - integral formalism and he introduced a quadratic trial action ( non - local in time ) to simulate ( [ eq_8a ] ) . applying the variational principle for path integrals then results in an upper bound for the polaron self - energy at all @xmath63 , which at weak and strong coupling gives accurate expressions . feynman obtained smooth interpolation between a weak and strong coupling ( for the ground state energy ) . the weak - coupling expansions of feynman for the ground - state energy and the effective mass of the polaron are : @xmath182@xmath183 in the strong - coupling limit feynman found for the ground - state energy energy : @xmath184 and for the polaron mass : @xmath185 over the years the feynman model for the polaron has remained the most successful approach to this problem . the analysis of an exactly solvable ( symmetrical ) 1d - polaron model @xcite , monte carlo schemes @xcite and other numerical schemes @xcite demonstrate the remarkable accuracy of feynman s path - integral approach to the polaron ground - state energy . experimentally more directly accessible properties of the polaron , such as its mobility and optical absorption , have been investigated subsequently . within the path - integral approach , feynman et al . studied later the mobility of polarons@xcite . subsequently the path - integral approach to the polaron problem was generalized and developed to become a powerful tool to study optical absorption , magnetophonon resonance and cyclotron resonance @xcite . in ref . @xcite , a self - consistent treatment for the polaron problem at all @xmath63 was presented , which is based on the heisenberg equations of motion starting from a trial expression for the electron position . it was used to derive the effective mass and the optical properties of the polaron at arbitrary coupling . a variational justification of the approximation used in ref . @xcite ( through a stiltjes continuous fraction ) is reproduced in appendix 2 . in ref . @xcite , using a monte carlo calculation , the ground - state energy of a polaron was derived as @xmath186 where @xmath187 with @xmath188 the free energy per polaron and @xmath189 \ln\left ( 2\pi\beta\right ) $ ] the free energy per electron . the value @xmath190 , used for the actual computation in ref . @xcite , corresponds to @xmath191 ( @xmath192 @xmath193 is the lo phonon energy ) . so , as pointed out in ref . @xcite , the authors of ref . @xcite actually calculated the _ free energy _ @xmath194 , rather than the polaron _ ground - state energy_. to investigate the importance of temperature effects on @xmath195 the authors of ref . @xcite considered the polaron energy as obtained by osaka @xcite , who generalized the feynman @xcite polaron theory to nonzero temperatures:@xmath196 \int_{0}^{\beta_{0}}du\frac{e^{-u}}{\sqrt{d\left ( u\right ) } } , \label{e1}\ ] ] where @xmath197 @xmath198 and@xmath199 this result is variational , with variational parameters @xmath200 and @xmath201 and gives an upper bound to the exact polaron free energy . the results of a numerical - variational calculation of eq . ( [ e1 ] ) are shown in fig . [ pd316826 ] , where the free energy @xmath202 is plotted ( in units of @xmath193 ) as a function of @xmath63 for different values of the lattice temperature . as seen from fig . [ pd316826 ] , ( i ) @xmath202 increases with increasing temperature and ( ii ) the effect of temperature on @xmath194 increases with increasing @xmath63 . [ h ] part1fig3.eps in table [ tablepd316826 ] , the monte carlo results @xcite , @xmath203 , are compared with the free energy of the feynman polaron , @xmath204 calculated in @xcite . the values for the free energy obtained from the feynman polaron model are _ _ lower _ _ than the mc results for @xmath205 and @xmath206 ( but lie within the 1% error of the monte carlo results ) . since the feynman result for the polaron free energy is an upper bound to the exact result , we conclude that for @xmath205 and @xmath206 the results of the feynman model are closer to the exact result than the mc results of @xcite . [ t]|c|c|c|c|@xmath63 & @xmath207 & @xmath208 & @xmath209 ( % ) + @xmath210 & @xmath211 & @xmath212 & @xmath213 + @xmath214 & @xmath215 & @xmath216 & @xmath217 + @xmath218 & @xmath219 & @xmath220 & @xmath221 + @xmath222 & @xmath223 & @xmath224 & @xmath225 + @xmath226 & @xmath227 & @xmath228 & @xmath229 + @xmath230 & @xmath231 & @xmath232 & @xmath233 + @xmath234 & @xmath235 & @xmath236 & @xmath237 + @xmath238 & @xmath239 & @xmath240 & @xmath241 + the analysis of an exactly solvable ( symmetric ) 1d - polaron model was performed in refs . the model consists of an electron interacting with two oscillators possessing the opposite wave vectors : * * * * @xmath59 and -@xmath243the parity operator , which changes @xmath58 and @xmath244 ( and also @xmath245 and @xmath246 ) , commutes with the hamiltonian of the system . hence , the polaron states are classified into the even and odd ones with the eigenvalues of the parity operator @xmath2471 and @xmath2481 , respectively . for the lowest even and odd states , the phonon distribution functions @xmath249 are plotted in fig . 1 , upper panel , at some values of the effective coupling constant @xmath250 of the symmetric model . the value of the parameter@xmath251 for these graphs was taken 1 , while the total polaron momentum @xmath150 . in the weak - coupling case ( @xmath252 ) @xmath249 is a decaying function of @xmath242 . when increasing @xmath250 , @xmath249 acquires a maximum , e.g. at @xmath253 for the lowest even state at @xmath254 . the phonon distribution function @xmath255has the same character for the lowest even and the lowest odd states at all values of the number of the virtual phonons in the ground state . ( as distinct from the higher states ) . this led to the conclusion that the lowest odd state is an internal excited state of the polaron . in ref @xcite , the structure of the polaron cloud was investigated using the diagrammatic quantum monte carlo ( dqmc ) method . in particular , partial contributions of @xmath242-phonon states to the polaron ground state were found as a function of @xmath242 for a few values of the coupling constant @xmath256 see fig . 1 , lower panel . it was shown to gradually evolve from the weak - coupling case ( @xmath257 ) into the strong - coupling regime ( @xmath258 ) . comparion of the lower panel to the upper panel in fig . [ npolarons ] clearly shows that the evolution of the shape and the scale of the distribution of the @xmath242-phonon states with increasing @xmath63 as derived for a large polaron within dqmc method @xcite is _ in remarkable agreement _ with the results obtained within the `` symmetric '' 1d polaron model @xcite . [ h ] part1fig4.eps the mobility of large polarons was studied within various theoretical approaches(see ref . @xcite for the detailed references ) . frhlich @xcite pointed out the typical behavior of the large - polaron mobility @xmath259 which is characteristic for weak coupling . here , @xmath260 , @xmath261 is the temperature . within the weak - coupling regime , the mobility of the polaron was then derived , e. g. , using the boltzmann equation in refs . @xcite and starting from the llp - transformation in ref . @xcite . a nonperturbative analysis was embodied in the feynman polaron theory , where the mobility @xmath262 of the polaron using the path - integral formalism was derived by feynman et al . ( usually referred to as fhip ) as a static limit starting from a frequency - dependent impedance function . for sufficiently low temperature @xmath261 the mobility then takes the form @xcite @xmath263 where @xmath200 and @xmath264 are ( variational ) functions of @xmath63 obtained from the feynman polaron model . using the boltzmann equation for the feynman polaron model , kadanoff @xcite found the mobility , which for low temperatures can be represented as follows : @xmath265 the weak - coupling perturbation expansion of the low - temperature polaron mobility as found using the green s function technique @xcite has confirmed that the mobility derived from the boltzmann equation is exceedingly exact for weak coupling ( @xmath266 ) and at low temperatures ( @xmath267 ) . as shown in ref . @xcite , the mobility of eq . ( [ eq : p24 - 1 ] ) differs by the factor of @xmath268 from that derived using the polaron boltzmann equation as given by eq . ( [ eq : p24 - 1-kadanoff ] ) . in the limit of weak electron - phonon coupling and low temperature , the fhip polaron mobility of eq . ( [ eq : p24 - 1 ] ) differs by the factor of @xmath269 from the previous result @xcite , which , as pointed out in ref . @xcite and in later publications ( see , e.g. , refs.@xcite ) , is correct for @xmath270 . as follows from this comparison , the result of ref . @xcite is not valid when @xmath271 . as argued in ref . @xcite and later confirmed , in particular , in ref . @xcite the above discrepancy can be attributed to an interchange of two limits in calculating the impedance . in fhip , for weak electron - phonon coupling , one takes @xmath272 , whereas the correct order is @xmath273 ( @xmath274 is the frequency of the applied electric field ) . it turns out that for the correct result the mobility at low temperatures is predominantly limited by the absorption of phonons , while in the theory of fhip it is the emission of phonons which gives the dominant contribution as @xmath261 goes to zero @xcite . the analysis based on the boltzmann equation takes into account the phonon emission processes whenever the energy of the polaron is above the emission threshold . the independent - collision model , which underlies the boltzmann - equation approach , however , fails in the strong coupling regime of the large polaron , when the thermal mean free path becomes less than the de broglie wavelength ; in this case , the boltzmann equation can not be expected to be adequate @xcite . in fact , the expression ( [ eq : p24 - 1 ] ) for the polaron mobility was reported to adequately describe the experimental data in several polar materials ( see , e.g. , refs . @xcite ) . experimental work on alkali halides and silver halides indicates that the mobility obtained from eq . ( [ eq : p24 - 1 ] ) describes the experimental results quite accurately @xcite . measurements of mobility as a function of temperature for photoexcited electrons in cubic @xmath275-type bi@xmath65sio@xmath66 are explained well in terms of large polarons within the feynman approach @xcite . the experimental findings on electron transport in crystalline tio@xmath67 ( rutile phase ) probed by thz time - domain spectroscopy are quantitatively interpreted within the feynman model @xcite . one of the reasons for the agreement between theory based on eq . ( [ eq : p24 - 1 ] ) and experiment is that in the path - integral approximation to the polaron mobility , a maxwellian distribution for the electron velocities is assumed , when applying the adiabatic switching on of the frhlich interaction . although such a distribution is not inherent in the frhlich interaction , its incorporation tends to favor agreement with experiment because other mechanisms ( interaction with acoustic phonons etc . ) cause a gaussian distribution . at zero temperature and in the weak - coupling limit , the optical absorption is due to the elementary polaron scattering process , schematically shown in fig.[fig_scheme ] . in the weak - coupling limit ( @xmath276 ) the polaron absorption coefficient was first obtained by v. gurevich , i. lang and yu . firsov @xcite , who started from the kubo formula . their optical - absorption coefficient is equivalent to a particular case of the result of j. tempere and j. t. devreese ( ref . @xcite ) , with the dynamic srtucture factor @xmath277 corresponding to the hartree - fock approximation ( see also @xcite , p. 585 ) . at zero temperature , the absorption coefficient for absorption of light with frequency @xmath274 can be expressed in terms of elementary functions in two limiting cases : in the region of comparatively high polaron densities ( @xmath278 ) @xmath279 and in the low - concentration region ( @xmath280 ) @xmath281 where @xmath282 , @xmath283 is the dielectric permittivity of the vacuum , @xmath275 is the refractive index of the medium , @xmath242 is the concentration of polarons and @xmath284 is the fermi level for the electrons . a step function @xmath285{lll}1 & \mbox{if } & \omega>1,\\ 0 & \mbox{if } & \omega<1 \end{array } \right.\ ] ] reflects the fact that at zero temperature the absorption of light accompanied by the emission of a phonon can occur only if the energy of the incident photon is larger than that of a phonon ( @xmath286 ) . in the weak - coupling limit , according to eqs.([weak ] ) , ( [ weak1 ] ) , the absorption spectrum consists of a one - phonon line . at nonzero temperature , the absorption of a photon can be accompanied not only by emission , but also by absorption of one or more phonons . a simple derivation in ref.@xcite using a canonical transformation method gives the absorption coefficient of free polarons , which coincides with the result ( [ weak1 ] ) of ref . @xcite . the optical absorption of large polarons as a function of the frequency of the incident light is calculated using the canonical - transformation formalism by devreese , huybrechts and lemmens ( dhl ) ref . @xcite . a simple calculation , which is developed below in full detail , gives a result for the absorption coefficient , which is exact to order @xmath287 we start from the hamiltonian of the electron - phonon system interacting with light is written down using the vector potential of an electromagnetic field @xmath288:@xmath289 the electric field is related to the vector potential as@xmath290 within the electric dipole interaction the electric field with frequency @xmath274 is @xmath291@xmath292 when expanding @xmath293 in the hamiltonian , we find@xmath294 where the first term is the kinetic energy of the electron , and the second term describes the interaction of the electron - phonon system with light@xmath295@xmath296 since @xmath288 does not depend on the electron coordinates , the term @xmath297 in ( [ dhl-4a ] ) does not play a role in our description of the optical absorption . the total hamiltonian for the system of a continuum polaron interacting with light is thus@xmath298 where @xmath299 is frhlich s hamiltonian ( [ eq_1a ] ) . the absorption coefficient for absorption of light with frequency @xmath274 by free polarons is proportional to the probability @xmath300 that a photon is absorbed by these polarons in their ground state,@xmath301 here @xmath242 is number of polarons , which are considered as independent from each other , @xmath24 is the permittivity of vacuum , @xmath302 is the velocity of light , @xmath275 is the refractive index of the medium in which the polarons move , @xmath303 is the modulus of the electric field vector of the incident photon . if the incident light can be treated as a perturbation , the transition probability @xmath300 is given by the golden rule of fermi:@xmath304 @xmath305 is the amplitude of the time - dependent perturbation given by ( [ dhl-7 ] ) . * * the ground state wave function of a free polaron is @xmath306 and its energy is @xmath307 the wave functions of all possible final states are @xmath308 and the corresponding energies are @xmath309 . the possible final states are all the excited states of the polaron . the main idea of the present calculation is to avoid the explicit summation over the final polaron states , which are poorly known , by eliminating all the excited state wave functions @xmath118 from the expression ( [ dhl2a ] ) . with this aim , the representation of the @xmath310-function is used:@xmath311 .\ ] ] this leads to:@xmath312 \\ & = 2\operatorname{re}{\displaystyle\sum\nolimits_{f } } { \displaystyle\int\nolimits_{-\infty}^{0 } } dt\exp\left [ -i(\omega+i\varepsilon)t\right ] \left\langle \phi_{0}\left\vert v\right\vert f\right\rangle \left\langle f\left\vert e^{iht}ve^{-iht}\right\vert \phi_{0}\right\rangle .\end{aligned}\ ] ] using the fact that@xmath313 and the notation@xmath314\end{aligned}\ ] ] we find@xmath315 \left\langle \phi_{0}\left\vert v(0)v(t)\right\vert \phi_{0}\right\rangle . \label{dhl8}\ ] ] defining@xmath316 \left\langle \phi_{0}\left\vert v(0)v(t)\right\vert \phi_{0}\right\rangle , \label{dhl-10}\ ] ] one has@xmath317 substituting ( [ dhl-7 ] ) to ( [ dhl-10 ] ) , we find that@xmath318 and hence@xmath319 it is convenient to apply the first llp transformation @xmath320([eq_6a ] ) , which eliminates the electron operators from the polaron hamiltonian:@xmath321 where @xmath322 can be diagonalized exactly and gives rise to the self - energy @xmath323and @xmath324 contains the correlation effects between the phonons . the optical absorption will be calculated here for the total momentum of the system @xmath325 in the llp approximation the explicit form of the matrix element in ( [ dhl-11 ] ) is@xmath326 where @xmath320and @xmath327 are the first ( [ eq_6a ] ) and the second ( [ eq_6b ] ) llp transformations . the application of @xmath320 gives:@xmath328 using @xmath329 we arrive at @xmath330further we recall @xmath331 where @xmath332 and @xmath56 is set 0 ( see appendix 1 ) . this results in @xmath333 then ( [ dhl15 ] ) takes the form@xmath334 here the second llp transformation is given by ( [ eq_6b ] ) with@xmath335 and the vacuum is defined by @xmath336 the calculation of the matrix element ( [ dhl17a ] ) proceeds as follows:@xmath337 further on , we calculate@xmath338 where@xmath339 s_{2}\ ] ] further we use @xmath340:@xmath341 \\ & = \sum_{\mathbf{k}}\left ( \omega_{\mathrm{lo}}+\frac{k^{2}}{2m_{b}}\right ) a_{\mathbf{k}}^{\dag}a_{\mathbf{k}}+\sum_{\mathbf{k}}\frac{\left\vert v_{k}\right\vert ^{2}}{\left ( \omega_{\mathrm{lo}}+\frac{k^{2}}{2m_{b}}\right ) } -2\sum_{\mathbf{k}}\frac{\left\vert v_{k}\right\vert ^{2}}{\left ( \omega_{\mathrm{lo}}+\frac{k^{2}}{2m_{b}}\right ) } \\ & = \sum_{\mathbf{k}}\left ( \omega_{\mathrm{lo}}+\frac{k^{2}}{2m_{b}}\right ) a_{\mathbf{k}}^{\dag}a_{\mathbf{k}}-\sum_{\mathbf{k}}\frac{\left\vert v_{k}\right\vert ^{2}}{\left ( \omega_{\mathrm{lo}}+\frac{k^{2}}{2m_{b}}\right ) } .\end{aligned}\ ] ] the last term can be calculated analytically:@xmath342 @xmath343 the term@xmath344 will be neglected:@xmath345 neglecting @xmath346 , consistent with the llp description , introduces no error in order @xmath63 . therefore ( [ dhl18a ] ) becomes@xmath347{c}\sum_{\mathbf{k}}\mathbf{e}\cdot\mathbf{k}\left ( a_{\mathbf{k}}^{\dagger } a_{\mathbf{k}}+f_{\mathbf{k}}a_{\mathbf{k}}^{\dagger}+f_{\mathbf{k}}^{\ast } a_{\mathbf{k}}+f_{\mathbf{k}}f_{\mathbf{k}}^{\ast}\right ) e^{ih_{0}t}\\ \times\sum_{\mathbf{k}}\mathbf{e}\cdot\mathbf{k}\left ( a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}}+f_{\mathbf{k}}a_{\mathbf{k}}^{\dagger}+f_{\mathbf{k}}^{\ast}a_{\mathbf{k}}+f_{\mathbf{k}}f_{\mathbf{k}}^{\ast } \right ) e^{-ih_{0}t}\end{array } \right\vert 0\right\rangle . \label{dhl19}\ ] ] for * * @xmath150 there is no privileged direction and @xmath348 ( [ dhl19 ] ) reduces to:@xmath349 from the equation of motion for @xmath350@xmath351 = i\left ( \omega_{\mathrm{lo}}+\frac{k^{2}}{2m_{b}}\right ) a_{\mathbf{k}}^{\dag},\ ] ] it is easy now to calculate@xmath352 .\ ] ] the matrix element ( [ dhl17a ] ) now becomes@xmath353 + o(\alpha^{2}).\ ] ] the transition probability ( [ dhl8 ] ) is then given by the expression@xmath354 \\ & = 2\pi\frac{e^{2}}{m_{b}^{2}\omega^{2}}\sum_{\mathbf{k}}\left ( \mathbf{e}\cdot\mathbf{k}\right ) ^{2}\left\vert f_{\mathbf{k}}\right\vert ^{2}\delta\left ( \omega-\omega_{\mathrm{lo}}-\frac{k^{2}}{2m_{b}}\right ) .\end{aligned}\ ] ] using ( [ dhl17c ] ) , we obtain@xmath355 where@xmath356{cc}1 & \text{if\qquad}\frac{\omega}{\omega_{\mathrm{lo}}}>1\\ 0 & \text{if\qquad}\frac{\omega}{\omega_{\mathrm{lo}}}<1 \end{array } \right . .\ ] ] the absorption coefficient ( [ dhl1 ] ) for absorption by free polarons for @xmath357 finally takes the form@xmath358 the behaviuor of @xmath359 ( [ dhl24 ] ) as a function of @xmath274 is as follows . for @xmath360 there is no absorption . the threshold for absorption is at @xmath361from @xmath362 up to @xmath363 increases to a maximum and for @xmath364 the absorption coefficient decreases slowly with increasing @xmath365 experimentally , this one - phonon line has been observed for free polarons in the infrared absorption spectra of cdo - films , see fig.[fig_abscdo ] . in cdo , which is a weakly polar material with @xmath366 , the polaron absorption band is observed in the spectral region between 6 and 20 @xmath262 m ( above the lo phonon frequency ) . the difference between theory and experiment in the wavelength region where polaron absorption dominates the spectrum is due to many - polaron effects . @xmath367 at @xmath368 k. the experimental data ( solid dots ) of ref . @xcite are compared to different theoretical results : with ( solid curve ) and without ( dashed line ) the one - polaron contribution of ref . @xcite and for many polarons ( dash - dotted curve ) of ref . @xcite . the absorption of light by free large polarons was treated in ref . @xcite using the polaron states obtained wihtin the adiabatic strong - coupling approximation , which was considered above in subsection [ strong ] . it was argued in ref . @xcite , that for sufficiently large @xmath63 ( @xmath369 ) , the ( first ) res of a polaron is a relatively stable state , which can participate in optical absorption transitions . this idea was necessary to understand the polaron optical absorption spectrum in the strong - coupling regime . the following scenario of a transition , which leads to a _ zero - phonon peak _ in the absorption by a strong - coupling polaron , can then be suggested . if the frequency of the incoming photon is equal to @xmath370 then the electron jumps from the ground state ( which , at large coupling , is well - characterized by `` @xmath371''-symmetry for the electron ) to an excited state ( `` @xmath151 '' ) , while the lattice polarization in the final state is adapted to the `` @xmath151 '' electronic state of the polaron . in ref . @xcite considering the decay of the res with emission of one real phonon it is demonstrated , that the zero - phonon peak can be described using the wigner - weisskopf formula valid when the linewidth of that peak is much smaller than @xmath372 for photon energies larger than @xmath373 a transition of the polaron towards the first scattering state , belonging to the res , becomes possible . the final state of the optical absorption process then consists of a polaron in its lowest res plus a free phonon . a one - phonon sideband then appears in the polaron absorption spectrum . this process is called _ one - phonon sideband absorption_. the one- , two- , ... @xmath374- , ... phonon sidebands of the zero - phonon peak give rise to a broad structure in the absorption spectrum . it turns that the _ first moment _ of the phonon sidebands corresponds to the fc frequency @xmath375 : @xmath376 to summarize , the polaron optical absorption spectrum at strong coupling is characterized by the following features ( @xmath377 ) : 1 . an intense absorption peak ( `` zero - phonon line '' ) appears , which corresponds to a transition from the ground state to the first res at @xmath378 . 2 . for @xmath379 , a phonon sideband structure arises . this sideband structure peaks around @xmath380 . the qualitative behaviour predicted in ref.@xcite , namely , an intense zero - phonon ( res ) line with a broader sideband at the high - frequency side , was confirmed after an all - coupling expression for the polaron optical absorption coefficient at @xmath381 had been studied @xcite . in what precedes , the low - frequency end of the polaron absorption spectrum was discussed ; at higher frequencies , transitions to higher res and their scattering states can appear . the two - phonon sidebands in the optical absorption of free polarons in the strong - coupling limit were numerically studied in ref . @xcite . the study of the optical absorption of polarons at large coupling is mainly of formal interest because all reported coupling constants of polar semiconductors and ionic crystals are smaller than 5 ( see table 1 ) . [ [ definitions ] ] definitions + + + + + + + + + + + we derive here the linear response of the frhlich polaron , described by the hamiltonian@xmath382 to a spatially uniform , time - varying electric field@xmath383 this field induces a current in the @xmath384-direction @xmath385 the complex function @xmath386 is called the impedance function . the frequency - dependent mobility is defined by @xmath387 for nonzero frequencies ( in the case of polarons the frequencies of interest are in the infrared ) one defines the absorption coefficient @xcite@xmath388 where @xmath283 is the dielectric constant of the vacuum , @xmath275 the refractive index of the crystal , and @xmath302 the velocity of light . in the following the amplitude of the electric field @xmath0 is taken sufficiently small so that linear - response theory can be applied . the impedance function can be expressed via a frequency - dependent conductivity of a single polaron in a unit volume @xmath389 using _ the standard kubo formula _ ( cf . ( 3.8.8 ) from ref . \right\rangle dt . \label{ifkubo}\ ] ] in order to introduce a convenient representation of the impedance function , we give in the next subsection a definition and discuss properties of a scalar product of two operators [ cf . @xcite , chapter 5 ] . [ [ definition - and - properties - of - the - scalar - product ] ] definition and properties of the scalar product + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + for two operators @xmath391 and @xmath392 ( i.e. , elements of the hilbert space of operators ) the scalar product is defined as @xmath393 the notation @xmath394 is used in order to indicate that the operator @xmath395 acts on the operator @xmath396 . the time evolution of the operator @xmath391 is determined by the liouville operator @xmath397:@xmath398 \label{if9}\ ] ] with a commutator @xmath399 $ ] , wherefrom@xmath400 the expectation value in ( [ if8 ] ) is taken over the gibbs ensemble:@xmath401 with the equilibtium density matrix when the electric field is absent @xmath402 one can show that ( [ if8 ] ) defines a positive definite scalar product with the following properties @xmath403 \right\rangle , \label{if11c}\ ] ] and [ cf . ( 5.11 ) of @xcite ] @xmath404 [ [ demonstration - of - the - property - if11a . ] ] demonstration of the property ( [ if11a ] ) . starting from the definition ( [ if8 ] ) and using ( [ if10 ] ) , we obtain@xmath405 substituting here ( [ if10a ] ) with ( [ if10b ] ) , one finds@xmath406 /\mathrm{tr}(e^{-\beta h}).\ ] ] change of the variable @xmath407 allows us to represnt this integral as@xmath408 /\mathrm{tr}(e^{-\beta h}).\ ] ] further , a cyclic permutation of the operators under the trace @xmath409 sign gives @xmath410 /\mathrm{tr}(e^{-\beta h})\\ & = { \displaystyle\int\nolimits_{0}^{\beta } } d\lambda\left\langle e^{\lambda h}be^{-\lambda h}a^{\dag}\right\rangle = { \displaystyle\int\nolimits_{0}^{\beta } } d\lambda\left\langle \left ( e^{\lambda\hbar l}b\right ) a^{\dag}\right\rangle \\ & = { \displaystyle\int\nolimits_{0}^{\beta } } d\lambda\left\langle \left [ e^{\lambda\hbar l}\left ( b^{\dag}\right ) ^{\dag}\right ] a^{\dag}\right\rangle .\end{aligned}\ ] ] according to the definition ( [ if8 ] ) , this finalizes the demonstration of ( [ if11a ] ) . [ [ demonstration - of - the - property - if11b . ] ] demonstration of the property ( [ if11b ] ) . starting from ( [ if11e ] ) and using ( [ if9 ] ) , we obtain@xmath411 a cyclic permutation of the operators under the average @xmath412 sign gives@xmath413 using the commutation of @xmath414 and @xmath415 , one finds@xmath416\right ) ^{\dag } e^{-\lambda h}b\right\rangle .\end{aligned}\ ] ] with the definition ( [ if9 ] ) , this gives @xmath417 b\right\rangle .\ ] ] according to the definition ( [ if8 ] ) , this finalizes the demonstration of ( [ if11b ] ) . [ [ demonstration - of - the - property - if11c . ] ] demonstration of the property ( [ if11c ] ) . starting from ( [ if11 g ] ) and performing a cyclic permutation of the operators under the average @xmath418 we find @xmath419 further we notice that @xmath420 consequently , @xmath421 /\mathrm{tr}(e^{-\beta h})\nonumber\\ & = \frac{1}{\hbar}\mathrm{tr}\left [ e^{-\beta h}a^{\dag}b - a^{\dag}e^{-\beta h}b\right ] /\mathrm{tr}(e^{-\beta h}).\end{aligned}\ ] ] further , a cyclic permutation of the operators in the second term under the trace @xmath409 sign gives@xmath422 /\mathrm{tr}(e^{-\beta h})\\ & = \frac{1}{\hbar}\mathrm{tr}\left [ e^{-\beta h}\left ( a^{\dag}b - ba^{\dag } \right ) \right ] /\mathrm{tr}(e^{-\beta h})\\ & = \frac{1}{\hbar}\left\langle a^{\dag}b - ba^{\dag}\right\rangle = \frac { 1}{\hbar}\left\langle \left [ a^{\dag},b\right ] \right\rangle .\end{aligned}\ ] ] thus , the property ( [ if11c ] ) has been demonstrated . [ [ demonstration - of - the - property - if11d . ] ] demonstration of the property ( [ if11d ] ) . we start from the represntation of the scalar product ( [ if11e ] ) and take a complex conjugate:@xmath423 a cyclic permutation of the operators under the average @xmath412 sign gives then@xmath424 according to the definition ( [ if8 ] ) , this finalizes the demonstration of ( [ if11d ] ) . the above scalar product allows one to represent different dynamical quantities in a rather simple way . for example , let us consider a scalar product @xmath425 b\right\rangle \label{if12a}\\ & = -i{\displaystyle\int\nolimits_{0}^{\infty } } dte^{izt}{\displaystyle\int\nolimits_{0}^{\beta } } d\lambda\left\langle e^{\lambda\hbar l}a^{\dag}e^{-ilt}b\right\rangle \nonumber\\ & = -i{\displaystyle\int\nolimits_{0}^{\infty } } dte^{izt}{\displaystyle\int\nolimits_{0}^{\beta } } d\lambda\left\langle e^{\lambda h}a^{\dag}e^{-\lambda h}e^{-iht/\hbar } be^{iht/\hbar}\right\rangle \nonumber\\ & = -i{\displaystyle\int\nolimits_{0}^{\infty } } dte^{izt}{\displaystyle\int\nolimits_{0}^{\beta } } d\lambda\left\langle e^{iht/\hbar+\lambda h}a^{\dag}e^{-iht/\hbar-\lambda h}b\right\rangle \nonumber\\ & = -i{\displaystyle\int\nolimits_{0}^{\infty } } dte^{izt}{\displaystyle\int\nolimits_{0}^{\beta } } d\lambda\left\langle e^{ih\left ( t - i\lambda\hbar\right ) /\hbar}a^{\dag } e^{-ih\left ( t - i\lambda\hbar\right ) /\hbar}b\right\rangle \nonumber\\ & = -i{\displaystyle\int\nolimits_{0}^{\infty } } dte^{izt}{\displaystyle\int\nolimits_{0}^{\beta } } d\lambda\left\langle e^{il\left ( t - i\lambda\hbar\right ) } a^{\dag } b\right\rangle \label{if12b}\\ & = -i{\displaystyle\int\nolimits_{0}^{\infty } } dte^{izt}{\displaystyle\int\nolimits_{0}^{\beta } } d\lambda\left\langle a^{\dag}(t - i\hbar\lambda)b(0)\right\rangle . \label{if13}\ ] ] [ [ representation - of - the - impedance - function - in - terms - of - the - relaxation - function ] ] representation of the impedance function in terms of the relaxation function + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the impedance function is related to the relaxation function@xmath426 where @xmath427 is the velocity operator , by the following expression : @xmath428 ( @xmath429 [ [ demonstration - of - the - representation - if7 . ] ] demonstration of the representation ( [ if7 ] ) . _ _ apply ( [ if12b ] ) to the relaxation function entering ( [ if6]):@xmath430 and perform the integration by parts using the formula@xmath431@xmath432 this allows us to represent the relaxation function in the form@xmath433 where the expression ( [ if15a ] ) is inserted in the first term . further on , the integral over @xmath250 is taken as follows:@xmath434 \\ & = \frac{1}{\hbar\mathrm{tr}e^{-\beta h}}\mathrm{tr}\left [ \left ( \dot { x}\left ( t\right ) e^{-\beta h}-e^{-\beta h}\dot{x}\left ( t\right ) \right ) \dot{x}\right ] \\ & = \frac{1}{\hbar\mathrm{tr}e^{-\beta h}}\mathrm{tr}\left [ e^{-\beta h}\left ( \dot{x}\dot{x}\left ( t\right ) -\dot{x}\left ( t\right ) \right ) \dot{x}\right ] \\ & = \frac{1}{\hbar}\left\langle \dot{x}\dot{x}\left ( t\right ) -\dot { x}\left ( t\right ) \dot{x}\right\rangle = -\frac{1}{\hbar}\left\langle \dot{x}\left ( t\right ) , \dot{x}\right\rangle .\end{aligned}\ ] ] hence , @xmath435 \right\rangle . \label{if13a}\ ] ] when setting @xmath436 with @xmath437 , we have@xmath438 \right\rangle dt . \label{k}\ ] ] for @xmath439 , we can set@xmath440 \right\rangle dt.\ ] ] multiplying @xmath441 by @xmath442 , we find that@xmath443 \right\rangle dt\right ) \nonumber\\ & = i\frac{e^{2}}{m_{b}\omega}+\frac{e^{2}}{\hbar\omega}\int_{0}^{\infty } e^{i\left ( \omega+i\varepsilon\right ) t}\left\langle \left [ \dot{x}\left ( t\right ) , \dot{x}\right ] \right\rangle dt\nonumber\\ & = i\frac{e^{2}}{m_{b}\omega}+\frac{1}{\hbar\omega}\int_{0}^{\infty } e^{i\left ( \omega+i\varepsilon\right ) t}\left\langle \left [ j_{x}\left ( t\right ) , j_{x}\right ] \right\rangle dt , \label{if13c}\ ] ] where the electric current density is @xmath444 substituting further ( [ if13c ] ) in ( [ if6 ] ) , we arrive at@xmath445 \right\rangle dt,\ ] ] what coincides with the expression of the impedance function ( [ ifcond ] ) through a frequency - dependent conductivity given by the kubo formula ( [ ifkubo ] ) , q.e.d . [ [ application - of - the - projection - operator - technique ] ] application of the projection operator technique + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + using the mori - zwanzig projection operator technique ( cf . @xcite , chapter 5 ) , the relaxation function ( [ if7 ] ) @xmath446 can be represented in a form , which is especially convenient for the application in the theory of the optical absorption of polarons . the projection operator @xmath447 ( @xmath448 ) is defined as @xmath449 with @xmath391 an operator and@xmath450 the projection operator @xmath448 projects an operator onto the space orthogonal to the space containing @xmath451 here we give some examples of the action of the projection operators:@xmath452 \right\rangle = 0,\label{if14f}\\ qx & = ( 1-p)x = x.\label{if14ff}\\ pa_{\mathbf{k } } & = \frac{\dot{x}\left ( \dot{x},a_{\mathbf{k}}\right ) } { \chi}=\frac{\dot{x}\left ( ilx , a_{\mathbf{k}}\right ) } { \chi}=-\frac { i\dot{x}\left ( a_{\mathbf{k}},lx\right ) } { \chi}=-\frac{i\dot{x}}{\chi } \left\langle \left [ a_{\mathbf{k}}^{\dag},x\right ] \right\rangle = 0,\label{if14g}\\ qa_{\mathbf{k } } & = ( 1-p)a_{\mathbf{k}}=a_{\mathbf{k } } \label{if14gg}\ ] ] the projection operators @xmath447 and @xmath453 are idempotent : @xmath454 the liouville operator can be identiaclly represented as @xmath455 . then the operator @xmath456 in the relaxation function ( [ if7 ] ) can be represented as follows:@xmath457 we use the algebraic operator identity:@xmath458 with @xmath459 and @xmath460@xmath461 consequently , the relaxation function ( [ if7 ] ) takes the form@xmath462 the first term in the r.h.s . of ( [ ifmori1 ] ) simplifies as follows:@xmath463 \dot{x}\right ) = \left ( \dot{x},\frac{1}{z}\dot{x}\right)\ ] ] because @xmath464 using the quantity ( [ if14a ] ) we obtain : @xmath465 the second term in the r.h.s . of ( [ ifmori1 ] ) contains the operator@xmath466 which according to the definition of the projection operator @xmath447 ( [ if14d ] ) can be transformed as@xmath467 it is remarkable that this term is exactly expressed in terms of the sought relaxation function ( [ if7 ] ) . substituting ( [ ifmori2 ] ) and ( [ ifmori3 ] ) in ( [ ifmori1 ] ) , we find @xmath468 l\frac{\dot{x}}{\chi}\right ) \phi\left ( z\right ) \rightarrow\nonumber\\ z\phi\left ( z\right ) & = \chi+\left [ \frac{\left ( \dot{x},l\dot { x}\right ) } { \chi}+\frac{1}{\chi}\left ( \dot{x},\frac{lq}{z - lq}l\dot { x}\right ) \right ] \phi\left ( z\right ) .\nonumber\end{aligned}\ ] ] introducring the quantity@xmath469 and the function called the _ memory function _ @xmath470 we represent ( [ ifmori3a ] ) in the form of the equation@xmath471 \phi\left ( z\right ) = \chi.\ ] ] a solition of this equation gives the relaxation function @xmath472 represented within the mori - zwanzig projection operator technique : @xmath473 the memory function ( [ ifmori4 ] ) can be still transformed to another useful form.first of all , we apply the property of a scalar product ( [ if11b ] ) : @xmath474 for any two operators @xmath391 and @xmath392 @xmath475 \right ) \\ & = \frac{\left ( \dot{x},a\right ) } { \chi}\left [ \left ( \dot{x},b\right ) -\frac{\left ( \dot{x},\dot{x}\right ) \left ( \dot{x},b\right ) } { \chi } \right ] \\ & = \frac{\left ( \dot{x},a\right ) } { \chi}\left [ \left ( \dot{x},b\right ) -\left ( \dot{x},b\right ) \right ] = 0,\end{aligned}\ ] ] therefore the first term on the r.h.s . in ( [ ifmori5 ] ) vanishes , and we obtain@xmath476 in this expression , the operator @xmath477 can be represented in the following form , using the fact that @xmath453 is the idempotent operator:@xmath478 \nonumber\\ & = \frac{1}{z}q+\frac{1}{z^{2}}qlq+\frac{1}{z^{3}}qlqlq+ ... \nonumber\\ & = \frac{1}{z}q+\frac{1}{z^{2}}qlq^{2}+\frac{1}{z^{3}}qlq^{2}lq^{2}+ ... \nonumber\\ & = \left [ \frac{1}{z}+\frac{1}{z^{2}}qlq+\frac{1}{z^{3}}qlqqlq+ ... \right ] q\rightarrow\nonumber\\ q\frac{1}{z - lq } & = \frac{1}{z - qlq}q . \label{ifmori7}\ ] ] a new liouville operator can be defined , @xmath479 , which describes the time evolution in the hilbert space of operators , which is orthogonal complement of @xmath451 substituting then ( [ ifmori7 ] ) with the operator @xmath480 into ( [ ifmori6 ] ) , we bring it to the form , which will be used in what follows.@xmath481 for the hamiltonian ( [ eq_1a ] ) we obtain the following quantities:@xmath482 using ( [ if11c ] ) , we find@xmath483 \right\rangle = \frac{i}{m_{b}\hbar}\left\langle \left ( -i\hbar\right ) \right\rangle = \frac{1}{m_{b } } \label{if15a}\ ] ] and@xmath484 \right\rangle = 0 . \label{if15b}\ ] ] substituting ( [ if15a ] ) and ( [ if15b ] ) in ( [ if14 ] ) , one obtains@xmath485 the operator @xmath486 = \nonumber\\ & = -\frac{1}{m_{b}\hbar}\left [ p_{x},\sum_{\mathbf{k}}(v_{k}a_{\mathbf{k}}e^{i\mathbf{k\cdot r}}+v_{k}^{\ast}a_{\mathbf{k}}^{\dag}e^{-i\mathbf{k\cdot r}})\right ] \nonumber\\ & = \frac{i}{m_{b}}\sum_{\mathbf{k}}ik_{x}(v_{k}a_{\mathbf{k}}e^{i\mathbf{k\cdot r}}-v_{k}^{\ast}a_{\mathbf{k}}^{\dag}e^{-i\mathbf{k\cdot r}})\rightarrow\nonumber\\ l\dot{x } & = -\frac{1}{m_{b}}\sum_{\mathbf{k}}k_{x}(v_{k}a_{\mathbf{k}}e^{i\mathbf{k\cdot r}}-v_{k}^{\ast}a_{\mathbf{k}}^{\dag}e^{-i\mathbf{k\cdot r } } ) \label{if15d}\ ] ] does not depend on the velocities . therefore , multiplying both parts of ( [ if15d ] ) with @xmath453 and taking into account ( [ if14ff ] ) and ( [ if14gg ] ) , we obtain@xmath487 what allows us to represent the memory function in the form @xmath488{c}k_{x}(v_{k}a_{\mathbf{k}}e^{i\mathbf{k\cdot r}}-v_{k}^{\ast}a_{\mathbf{k}}^{\dag}e^{-i\mathbf{k\cdot r}}),\\ \frac{1}{z-\mathcal{l}}k_{x}^{\prime}(v_{k^{\prime}}a_{\mathbf{k}^{\prime}}e^{i\mathbf{k}^{\prime}\mathbf{\cdot r}}-v_{k^{\prime}}^{\ast}a_{\mathbf{k}^{\prime}}^{\dag}e^{-i\mathbf{k}^{\prime}\mathbf{\cdot r } } ) \end{array } \right ) \nonumber\\ & = \frac{1}{m_{b}}\sum_{\mathbf{k}}\sum_{\mathbf{k}^{\prime}}k_{x}k_{x}^{\prime}v_{k}v_{k^{\prime}}^{\ast}\left ( \begin{array } [ c]{c}(a_{\mathbf{k}}e^{i\mathbf{k\cdot r}}+a_{\mathbf{k}}^{\dag}e^{-i\mathbf{k\cdot r}}),\\ \frac{1}{z-\mathcal{l}}(a_{\mathbf{k}^{\prime}}e^{i\mathbf{k}^{\prime } \mathbf{\cdot r}}+a_{\mathbf{k}^{\prime}}^{\dag}e^{-i\mathbf{k}^{\prime } \mathbf{\cdot r } } ) \end{array } \right ) . \label{if16}\ ] ] in transition to ( [ if16 ] ) we have used the property of the amplitude ( [ eq_1b ] ) : @xmath489 and taken into account that according to the definition ( [ if8 ] ) , the first operator enters a scalar product in the hermitian conjugate form . introducing the operators @xmath490 we represent the memory function as@xmath491 we notice that @xmath492 . it will be represented through the four relaxation functions:@xmath493 , \label{if18a}\\ \phi_{\mathbf{kk}^{\prime}}^{+\,+}(z ) & = \left ( b_{\mathbf{k}}^{\dag},\frac{1}{z-\mathcal{l}}b_{\mathbf{k}^{\prime}}^{\dag}\right ) , \label{if18b}\\ \phi_{\mathbf{kk}^{\prime}}^{-\,-}(z ) & = \left ( b_{\mathbf{k}},\frac { 1}{z-\mathcal{l}}b_{\mathbf{k}^{\prime}}\right ) , \label{if18c}\\ \phi_{\mathbf{kk}^{\prime}}^{+\,-}(z ) & = \left ( b_{\mathbf{k}}^{\dag},\frac{1}{z-\mathcal{l}}b_{\mathbf{k}^{\prime}}\right ) , \label{if18d}\\ \phi_{\mathbf{kk}^{\prime}}^{-\,+}(z ) & = \left ( b_{\mathbf{k}},\frac { 1}{z-\mathcal{l}}b_{\mathbf{k}^{\prime}}^{\dag}\right ) . \label{if18e}\ ] ] there exist relations between the above relaxation functions . for example , the relaxation function ( [ if18c ] ) , takes the form@xmath494 then the complex conjugate of this relaxation function : @xmath495 ^{\ast } & = i{\displaystyle\int\nolimits_{0}^{\infty } } dte^{-iz^{\ast}t}\left ( e^{i\mathcal{l}t}b_{\mathbf{k}}(0),b_{\mathbf{k}^{\prime}}(0)\right ) ^{\ast}\\ & = i{\displaystyle\int\nolimits_{0}^{\infty } } dte^{-iz^{\ast}t}\left ( b_{\mathbf{k}^{\prime}}(0),e^{i\mathcal{l}t}b_{\mathbf{k}}(0)\right ) , \end{aligned}\ ] ] where the property ( [ if11d ] ) has been used . the property ( [ if11a ] ) gives @xmath495 ^{\ast } & = i{\displaystyle\int\nolimits_{0}^{\infty } } dte^{-iz^{\ast}t}\left ( b_{\mathbf{k}^{\prime}}(0),e^{i\mathcal{h}t/\hbar } b_{\mathbf{k}}(0)e^{-i\mathcal{h}t/\hbar}\right ) \\ & = i{\displaystyle\int\nolimits_{0}^{\infty } } dte^{-iz^{\ast}t}\left ( e^{i\mathcal{h}t/\hbar}b_{\mathbf{k}}^{\dag } ( 0)e^{i\mathcal{h}t/\hbar},b_{\mathbf{k}^{\prime}}^{\dag}(0)\right ) \\ & = i{\displaystyle\int\nolimits_{0}^{\infty } } dte^{-iz^{\ast}t}\left ( e^{i\mathcal{l}t}b_{\mathbf{k}}^{\dag}(0),b_{\mathbf{k}^{\prime}}^{\dag}(0)\right ) \\ & = i{\displaystyle\int\nolimits_{0}^{\infty } } dte^{-iz^{\ast}t}\left ( b_{\mathbf{k}}^{\dag}(t),b_{\mathbf{k}^{\prime}}^{\dag}(0)\right ) = -\phi_{\mathbf{kk}^{\prime}}^{+\,+}(-z^{\ast}),\end{aligned}\ ] ] wherefrom it follows that @xmath496 ^{\ast}. \label{if19c}\ ] ] similarly , the relation @xmath497 ^{\ast } \label{if19d}\ ] ] is proven . [ [ memory - function ] ] memory function + + + + + + + + + + + + + + + in this subsection we indicate which approximations must be made in the calculation of the relaxation functions in order to obtain the fhip results for the impedance function . consider the relaxation function ( [ if18b]):@xmath498 where @xmath499 and perform a partial integration:@xmath500 here we supposed that @xmath501 in the second term in ( [ if19]),@xmath502 because @xmath503 and @xmath504 further on , we have@xmath505 so , we find from ( [ if19])@xmath506 \right\rangle . \label{if19a}\ ] ] the first term in the r.h.s . of this expression can be represented as follows : @xmath507 \right\ } \\ & = \frac{1}{z\hbar}\frac{1}{\mathrm{tr}e^{-\beta h}}\mathrm{tr}\left\ { b_{\mathbf{k}^{\prime}}^{\dag}\frac{1}{l}b_{\mathbf{k}}e^{-\beta h}-e^{-\beta h}\frac{1}{l}b_{\mathbf{k}}b_{\mathbf{k}^{\prime}}^{\dag}\right\ } \\ & = \frac{1}{z\hbar}\left\langle b_{\mathbf{k}^{\prime}}^{\dag}\frac{1}{l}b_{\mathbf{k}}-\frac{1}{l}b_{\mathbf{k}}b_{\mathbf{k}^{\prime}}^{\dag } \right\rangle \\ & = \frac{i}{z\hbar}\left\langle b_{\mathbf{k}^{\prime}}^{\dag}\frac{1}{il}b_{\mathbf{k}}-\frac{1}{il}b_{\mathbf{k}}b_{\mathbf{k}^{\prime}}^{\dag } \right\rangle \\ & = \frac{i}{z\hbar}\left\langle { \displaystyle\int\nolimits_{0}^{\infty } } dte^{ilt}b_{\mathbf{k}}b_{\mathbf{k}^{\prime}}^{\dag}-{\displaystyle\int\nolimits_{0}^{\infty } } dtb_{\mathbf{k}^{\prime}}^{\dag}e^{ilt}b_{\mathbf{k}}\right\rangle \\ & = \frac{i}{z\hbar}\left\langle { \displaystyle\int\nolimits_{0}^{\infty } } dtb_{\mathbf{k}}(t)b_{\mathbf{k}^{\prime}}^{\dag}-{\displaystyle\int\nolimits_{0}^{\infty } } dtb_{\mathbf{k}^{\prime}}^{\dag}b_{\mathbf{k}}(t)\right\rangle \\ & = \frac{i}{z\hbar}{\displaystyle\int\nolimits_{0}^{\infty } } dt\left\langle \left [ b_{\mathbf{k}}(t),b_{\mathbf{k}^{\prime}}^{\dag } ( 0)\right ] \right\rangle .\end{aligned}\ ] ] substituting it in the r.h.s . of ( [ if19a ] ) , we find@xmath508 \right\rangle -\frac{i}{z\hbar}{\displaystyle\int\nolimits_{0}^{\infty } } dte^{izt}\left\langle \left [ b_{\mathbf{k}}(t),b_{\mathbf{k}^{\prime}}^{\dag } ( 0)\right ] \right\rangle \nonumber\\ & = \frac{i}{z\hbar}{\displaystyle\int\nolimits_{0}^{\infty } } dt\left ( 1-e^{izt}\right ) \left\langle \left [ b_{\mathbf{k}}(t),b_{\mathbf{k}^{\prime}}^{\dag}(0)\right ] \right\rangle . \label{if19b}\ ] ] in a similar way one obtains @xmath509 \right\rangle . \label{if19e}\ ] ] inserting the relaxation functions ( [ if19b ] ) , ( [ if19e ] ) , ( [ if19c ] ) and ( [ if19d ] ) , we find the memory function ( [ if18a])@xmath510{c}\left\langle \left [ b_{\mathbf{k}}(t),b_{\mathbf{k}^{\prime}}^{\dag } ( 0)\right ] \right\rangle + \left\langle \left [ b_{\mathbf{k}}(t),b_{\mathbf{k}^{\prime}}(0)\right ] \right\rangle \\ \left [ b_{\mathbf{k}}(t),b_{\mathbf{k}^{\prime}}^{\dag } ( 0)\right ] \right\rangle ^{\ast}-\left\langle \left [ b_{\mathbf{k}}(t),b_{\mathbf{k}^{\prime}}(0)\right ] \right\rangle ^{\ast}\end{array } \right ] \\ & = -\frac{1}{m_{b}}\sum_{\mathbf{k}}\sum_{\mathbf{k}^{\prime}}k_{x}k_{x}^{\prime}v_{k}v_{k^{\prime}}^{\ast}\frac{2}{z\hbar}{\displaystyle\int\nolimits_{0}^{\infty } } dt\left ( 1-e^{izt}\right ) \\ & \times\operatorname{im}\left [ \left\langle \left [ b_{\mathbf{k}}(t),b_{\mathbf{k}^{\prime}}^{\dag}(0)\right ] \right\rangle + \left\langle \left [ b_{\mathbf{k}}(t),b_{\mathbf{k}^{\prime}}(0)\right ] \right\rangle \right ] , \end{aligned}\ ] ] wherefrom @xmath511 with @xmath512 \right\rangle + \left\langle \left [ b_{\mathbf{k}}(t),b_{\mathbf{k}^{\prime}}(0)\right ] \right\rangle \right\ } . \label{if20b}\ ] ] [ [ derivation - of - the - memory - function ] ] derivation of the memory function + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + to calculate the expectation values in eq . ( [ if20b ] ) , we shall make the following approximations ( cf . the liouville operator @xmath480 , which determines the time evolution of the operator @xmath513 , is replaced by @xmath514 where @xmath515 is the liouville operator for free phonons and @xmath516 is the liouville operator for the feynman polaron model @xcite . the frhlich hamiltonian appearing in the statistical average @xmath517 is imilarly replaced by @xmath518with @xmath519 the hamiltonian of free phonons and @xmath520 the hamiltonian of the feynman polaron model . with this approximation , e.g. , the average @xmath521 the time evolution of the free - phonon annihilation operator ( [ if10]),@xmath522 accorting to ( [ if9 ] ) is @xmath523 \exp\left ( -i\omega_{\mathbf{k}}a_{\mathbf{k}}^{+}a_{\mathbf{k}}t\right ) \\ & = \omega_{\mathbf{k}}\exp\left ( i\omega_{\mathbf{k}}a_{\mathbf{k}}^{+}a_{\mathbf{k}}t\right ) \left [ a_{\mathbf{k}}^{+},a_{\mathbf{k}}\right ] a_{\mathbf{k}}\exp\left ( -i\omega_{\mathbf{k}}a_{\mathbf{k}}^{+}a_{\mathbf{k}}t\right ) \\ & = -\omega_{\mathbf{k}}\exp\left ( i\omega_{\mathbf{k}}a_{\mathbf{k}}^{+}a_{\mathbf{k}}t\right ) a_{\mathbf{k}}\exp\left ( -i\omega_{\mathbf{k}}a_{\mathbf{k}}^{+}a_{\mathbf{k}}t\right ) \rightarrow\\ a_{\mathbf{k}}(t ) & = \exp\left ( -i\omega_{\mathbf{k}}t\right ) a_{\mathbf{k}}.\end{aligned}\ ] ] similarly , @xmath524 hence , we have @xmath525 , \ ] ] where @xmath526 ^{-1}$ ] is the average number of phonons with energy @xmath527 . the calculation of the fourier component of the electron density - density correlation function @xmath528 in eq . ( [ if21 ] ) for an electron described by the feynman polaron model is given below following the approach of ref . @xcite . we calculate the correlation function @xmath529 with the feynman trial hamiltonian@xmath530 here , @xmath531 denotes the operator in the heisenberg representation@xmath532 we show that the correlation function @xmath533 depends on @xmath534 rather than on @xmath535 and @xmath536 independently:@xmath537 where @xmath538 . the normal coordinates are the center - of - mass vector @xmath539 and the vector of the relative motion @xmath540:@xmath541{c}\mathbf{r}=\frac{m\mathbf{r}+m_{f}\mathbf{r}_{f}}{m+m_{f}}\\ \mathbf{\rho}=\mathbf{r}-\mathbf{r}_{f}\end{array } \right . \label{tf - eq2}\ ] ] the inverse transformation is:@xmath541{c}\mathbf{r}=\mathbf{r}+\frac{m_{f}}{m+m_{f}}\mathbf{\rho}\\ \mathbf{r}_{f}=\mathbf{r}-\frac{m}{m+m_{f}}\mathbf{\rho}\end{array } \right . \label{tf - eq3}\ ] ] the same transformation as ( [ tf - eq2 ] ) takes place for velocities:@xmath541{c}\mathbf{\dot{r}}=\mathbf{\dot{r}}+\frac{m_{f}}{m+m_{f}}\mathbf{\dot{\rho}}\\ \mathbf{\dot{r}}_{f}=\mathbf{\dot{r}}-\frac{m}{m+m_{f}}\mathbf{\dot{\rho}}\end{array } \right.\ ] ] from ( [ tf - eq2 ] ) we derive the transformation for moments@xmath541{c}\frac{\mathbf{p}}{m}=\frac{\mathbf{p}}{m+m_{f}}+\frac{m_{f}}{m+m_{f}}\frac{\mathbf{p}_{\rho}}{\frac{mm_{f}}{m+m_{f}}}\\ \frac{\mathbf{p}_{f}}{m_{f}}=\frac{\mathbf{p}}{m+m_{f}}-\frac{m}{m+m_{f}}\frac{\mathbf{p}_{\rho}}{\frac{mm_{f}}{m+m_{f}}}\end{array } \right.\]]@xmath542@xmath541{c}\mathbf{p}=\frac{m}{m+m_{f}}\mathbf{p}+\mathbf{p}_{\rho}\\ \mathbf{p}_{f}=\frac{m_{f}}{m+m_{f}}\mathbf{p}-\mathbf{p}_{\rho}\end{array } \right . \label{tf - eq4}\ ] ] the hamiltonian ( [ tf - eq1 ] ) then takes the form@xmath543 with the masses@xmath544 and with the frequency@xmath545 the cartesian coordinates and moments corresponding to the relative motion can be in the standard way expressed in terms of the second quantization operators:@xmath546 in these notations , the hamiltonian ( [ tf - eq5 ] ) takes the form@xmath547 using ( [ tf - eq9 ] ) , we find the operators in the heisenberg representation , ( i ) for the center - of mass coordinates@xmath548 \\ & = x_{j}+i\frac{\sigma}{2m\hbar}\left ( p_{j}^{2}x_{j}-p_{j}x_{j}p_{j}+p_{j}x_{j}p_{j}-x_{j}p_{j}^{2}\right ) \\ & = x_{j}+i\frac{\sigma}{2m\hbar}\left ( p_{j}\left [ p_{j},x_{j}\right ] + \left [ p_{j},x_{j}\right ] p_{j}\right ) \\ & = x_{j}+i\frac{\sigma}{2m\hbar}p_{j}\left ( -2i\hbar\right ) = x_{j}+\frac{\sigma}{m}p_{j},\end{aligned}\]]@xmath549 ( ii ) for the operators @xmath550 and @xmath551@xmath552 using the first formula of ( [ tf - eq3 ] ) we find@xmath553@xmath542@xmath554 we denote@xmath555@xmath556 therefore , we obtain@xmath557 @xmath558 \nonumber\\ & = \exp\left ( -i\mathbf{k\cdot r}-i\frac{\sigma}{m}\mathbf{k\cdot p}\right ) \exp\left ( -ia\mathbf{k\cdot c}e^{-i\bar{\omega}\sigma}-ia\mathbf{k\cdot c}^{\dag}e^{i\bar{\omega}\sigma}\right ) \nonumber\\ & = \prod_{j=1}^{3}\exp\left ( -ik_{j}x_{j}-i\frac{\sigma}{m}k_{j}p_{j}\right ) \exp\left ( -iak_{j}c_{j}e^{-i\bar{\omega}\sigma}-iak_{j}c_{j}^{\dag}e^{i\bar{\omega}\sigma}\right ) . \label{tf - eq14}\ ] ] the disentangling of the exponents is performed using the formula@xmath559 in the case when @xmath560 $ ] commutes with both @xmath391 and @xmath392 , this formula is reduced to@xmath561 } . \label{tf - eq16}\ ] ] we perform the necessary commutations:@xmath562 = \frac{1}{2}k_{j}^{2}\frac{\sigma}{m}\left [ x_{j},p_{j}\right ] = i\frac{\hbar k_{j}^{2}}{2m}\sigma,\]]@xmath563 = \frac{1}{2}a^{2}k_{j}^{2}\left [ c_{j}^{\dag},c_{j}\right ] = -\frac{1}{2}a^{2}k_{j}^{2}\]]@xmath564 = \frac{1}{2}a^{2}k_{j}^{2}\left [ c_{j}^{\dag } , c_{j}\right ] = -\frac{1}{2}a^{2}k_{j}^{2}\]]@xmath542@xmath565@xmath542@xmath566 it follows from eq . ( [ tf - eq16 ] ) that when @xmath560 $ ] commutes with both @xmath391 and @xmath392,@xmath567 } . \label{tf - eq17}\ ] ] using ( [ tf - eq17 ] ) , we find@xmath568 } \\ & = e^{-iak_{j}\cdot c_{j}^{\dag}e^{i\bar{\omega}\sigma}}e^{iak_{j}\cdot c_{j}}e^{a^{2}k_{j}^{2}e^{i\bar{\omega}\sigma}}.\end{aligned}\ ] ] herefrom , we find @xmath569 the correlation function then is@xmath570 since the variables of the center - of mass motion and of the relative motion are averaged independently.@xmath571 with@xmath572 let us consider the auxiliary expectation value@xmath573@xmath574 where @xmath575 are the eigenstates of @xmath576 . the operators @xmath577 act on these states as follows:@xmath578 therefore , we find@xmath579@xmath542@xmath580 and@xmath581 in particular , for @xmath582 we have@xmath583 as a result , the expectation value @xmath584 is@xmath585 , \label{tf - eq19}\ ] ] with@xmath586 using this result , we obtain the expression@xmath587 \\ & = \exp\left [ -n\left ( \omega\right ) a^{2}k^{2}\left ( 1-e^{i\bar{\omega } \sigma}\right ) \left ( 1-e^{-i\bar{\omega}\sigma}\right ) \right ] \\ & = \exp\left [ -4n\left ( \omega\right ) a^{2}k^{2}\sin^{2}\left ( \frac { 1}{2}\omega\sigma\right ) \right ] .\end{aligned}\ ] ] the expectation value @xmath588 is@xmath589 collecting all factors in eq . ( [ tf - eq20 ] ) together , we find@xmath590 \rightarrow.\]]@xmath591 with the function@xmath592 , \label{if22a}\ ] ] where @xmath593 according to ( [ tf - time ] ) , @xmath594 taking @xmath595 in ( [ tf - eq21 ] ) we finally find the fourier component of the electron density - density correlation function @xmath596 which enters eq . ( [ if21 ] ) : @xmath597 \label{if22}\ ] ] finally , the correlation functions in ( [ if20b ] ) reduce to@xmath598 \exp\left ( -i\omega_{\mathbf{k}}t\right ) \exp\left [ -k^{2}d(-t)\right ] , \\ \left\langle b_{\mathbf{k}}^{\dag}(0)b_{\mathbf{k}}(t)\right\rangle & = n(\omega_{\mathbf{k}})\exp\left ( -i\omega_{\mathbf{k}}t\right ) \exp\left [ -k^{2}d(t)\right ] , \\ \left\langle b_{\mathbf{k}}(t)b_{\mathbf{k}}(0)\right\rangle & = 0,\\ \left\langle b_{\mathbf{k}}(0)b_{\mathbf{k}}(t)\right\rangle & = 0.\end{aligned}\ ] ] inserting these equations into eqs . ( [ if20a ] ) and ( [ if20b ] ) , one obtains@xmath599 with@xmath600{c}-\left [ 1+n(\omega_{\mathbf{k}})\right ] \exp\left ( -i\omega_{\mathbf{k}}t\right ) \exp\left [ -k^{2}d(-t)\right ] \\ + n(\omega_{\mathbf{k}})\exp\left ( -i\omega_{\mathbf{k}}t\right ) \exp\left [ -k^{2}d(t)\right ] \end{array } \right\ } .\ ] ] using the property @xmath601 for real vaues of @xmath602one obtains@xmath603 \ } & = \operatorname{im}\exp\left ( i\omega_{\mathbf{k}}t\right ) \exp\left [ -k^{2}d(-t)^{\ast}\right ] \}\\ & = \operatorname{im}\exp\left ( i\omega_{\mathbf{k}}t\right ) \exp\left [ -k^{2}d(t)\right ] \}\end{aligned}\ ] ] and consequently@xmath604 \left\ { \left [ 1+n(\omega_{\mathbf{k}})\right ] \exp\left ( i\omega_{\mathbf{k}}t\right ) + n(\omega_{\mathbf{k}})\exp\left ( -i\omega_{\mathbf{k}}t\right ) \right\ } . \label{if23b}\ ] ] owing to the rotational invariance of @xmath605 and @xmath606we can substitute in ( [ if23b])@xmath607 the resulting expression for @xmath608 \left\ { \left [ 1+n(\omega_{\mathbf{k}})\right ] \exp\left ( i\omega_{\mathbf{k}}t\right ) + n(\omega_{\mathbf{k}})\exp\left ( -i\omega_{\mathbf{k}}t\right ) \right\ } \label{if23c}\ ] ] is identical with eq . ( 35 ) of fhip @xcite . in the case of frhlich polarons , taking into account ( [ eq_1b ] ) , eq . ( [ if23c ] ) simplifies to @xmath609{c}\left [ 1+n(\omega_{\mathrm{lo}})\right ] \exp\left ( i\omega_{\mathrm{lo}}t\right ) \\ + n(\omega_{\mathrm{lo}})\exp\left ( -i\omega_{\mathrm{lo}}t\right ) \end{array } \right\ } \left [ d(t)\right ] ^{-3/2}. \label{if23d}\ ] ] upon substituion of ( [ if6 ] ) and ( [ if15c ] ) into eq . ( [ if5 ] ) , we find the absorption coefficient@xmath610 = -\frac { 1}{n\epsilon_{0}c}\frac{e^{2}}{m_{b}}\operatorname{im}\left [ \frac{1}{\omega-\sigma(\omega)}\right ] \\ & = -\frac{1}{n\epsilon_{0}c}\frac{e^{2}}{m_{b}}\operatorname{im}\left\ { \frac{\omega-\sigma^{\ast}(\omega)}{\left [ \omega-\operatorname{re}\sigma(\omega)\right ] ^{2}+\left [ \operatorname{im}\sigma(\omega)\right ] ^{2}}\right\ } \\ & = \frac{1}{n\epsilon_{0}c}\frac{e^{2}}{m_{b}}\frac{\operatorname{im}\sigma^{\ast}(\omega)}{\left [ \omega-\operatorname{re}\sigma(\omega)\right ] ^{2}+\left [ \operatorname{im}\sigma(\omega)\right ] ^{2}}\rightarrow\end{aligned}\ ] ] @xmath611 ^{2}+\left [ \operatorname{im}\sigma(\omega)\right ] ^{2}}\ . \label{eq : p24 - 2}\ ] ] this general expression was the strating point for a derivation of the theoretical optical absorption spectrum of a single large polaron , at _ all electron - phonon coupling strengths _ by devreese et al . in ref.@xcite . the memory function @xmath612 as given by eq . ( [ if23a ] ) with ( [ if23d ] ) contains the dynamics of the polaron and depends on @xmath63 , temperature and @xmath365following the notation , introduced in ref . @xcite , @xmath613 we reresent eq . ( [ eq : p24 - 2 ] ) in the form used in ref.@xcite:@xmath614 ^{2}+\left [ \operatorname{im}\chi(\omega)\right ] ^{2}}. \label{eq : p24 - 2a}\ ] ] according to ( [ if23a ] ) and ( [ chi0]),@xmath615 s(t ) . \label{if24}\ ] ] in the present notes we limit our attention to the case @xmath377 @xmath616 it was demonstrated in ref.@xcite that the exact zero - temperature limit arises if the limit @xmath617 is taken directly in the expressions ( [ if24 ] ) ( see appendices a and b of ref.@xcite ) . as follows from ( [ if23d ] ) , @xmath618 ^{-3/2}\qquad(\beta\rightarrow\infty ) . \label{if25}\ ] ] accorting to ( [ if22a])@xmath619 \qquad(\beta\rightarrow\infty).\ ] ] using the feynman units ( where @xmath620 and @xmath621 ) , we obtain from ( [ if22b]):@xmath622 and consequently@xmath623 with @xmath624 and according to ( [ if25 ] ) @xmath625 ^{-3/2}\qquad(\beta\rightarrow\infty).\ ] ] from ( [ if24 ] ) one obtains immediately@xmath626 ^{3/2}},\label{if26a}\\ \operatorname{re}\chi(\omega ) & = \frac{2\alpha}{3\sqrt{\pi}}\left ( \frac { v}{w}\right ) ^{3}\operatorname{im}{\displaystyle\int\nolimits_{0}^{\infty } } dt\frac{\left [ 1-\cos(\omega t)\right ] e^{it}}{\left [ r(1-e^{ivt})-it\right ] ^{3/2}}. \label{if26b}\ ] ] in the limit @xmath617 the function @xmath627 was calculated by fhip @xcite . however , to study the optical absorption to the same approximation as fhip s treatment of the impedance , we have also to calculate @xmath628 and use this result in ( [ eq : p24 - 2a ] ) . the calculation of @xmath629 which is a kramers - kronig - type transform of @xmath627 , is a key ingredient in ref.@xcite . the details of those calculations are presented in the appendices a , b and c to ref.@xcite . developing the denominator of both integrals on the right - hand side of ( [ if26a ] ) and ( [ if26b ] ) , the calculations are reduced to the evaluation of a sum of integrals of the type@xmath630 in appendix b to ref.@xcite it is shown how such integrals are evaluated using a recurrence formula . for @xmath627 a very convenient result was found in @xcite:@xmath631 this expression is a finite sum and not and infinite series . fhip gave the first two terms of ( [ if27 ] ) explicitly . using the same recurrence relation it is seen the analytical expression ( see appendix b to ref.@xcite ) , which was found for @xmath628 is far more complicated . to circumwent the difficulty with the numerical treatment of @xmath628 , the corresponding integrals in ( [ if26c ] ) have been transformed in @xcite to integrals with rapildy convergent integrands:@xmath632 e^{i(1+nv)t}}{\left ( r - it\right ) ^{3/2+n}}\nonumber\\ & = -\frac{1}{\gamma(n+\frac{3}{2})}{\displaystyle\int\nolimits_{0}^{\infty } } dx\left [ ( n+\frac{1}{2})x^{n-1/2}e^{-rx}-rx^{n+1/2}e^{-rx}\right ] \nonumber\\ & \times\ln\left\vert \left ( \frac{\left ( 1+nv+x\right ) ^{2}}{\omega ^{2}-\left ( 1+nv+x\right ) ^{2}}\right ) \right\vert ^{1/2}. \label{if28}\ ] ] the integral on the right - hand side of ( [ if28 ] ) is adequate for computer calculations . in appendix c to ref.@xcite some supplementary details of the computation of ( [ if28 ] ) are given . another analytical representation for the memory function ( [ if23a ] ) was derived in ref . @xcite . at weak coupling , the optical absorption spectrum ( [ eq : p24 - 2 ] ) of the polaron is determined by the absorption of radiation energy , which is reemitted in the form of lo phonons . for @xmath633 , the polaron can undergo transitions toward a relatively stable res ( see fig . [ fig_3 ] ) . the res peak in the optical absorption spectrum also has a phonon sideband - structure , whose average transition frequency can be related to a fc - type transition . furthermore , at zero temperature , the optical absorption spectrum of one polaron exhibits also a zero - frequency central peak [ @xmath634 . for non - zero temperature , this central peak smears out and gives rise to an anomalous drude - type low - frequency component of the optical absorption spectrum . for example , in fig . [ fig_3 ] from ref . @xcite , the main peak of the polaron optical absorption for @xmath635 5 at @xmath636 is interpreted as due to transitions to a res . a shoulder at the low - frequency side of the main peak is attributed to one - phonon transitions to polaron - scattering states . the broad structure centered at about @xmath637 is interpreted as a fc band . as seen from fig . [ fig_3 ] , when increasing the electron - phonon coupling constant to @xmath63=6 , the res peak at @xmath638 stabilizes . it is in ref . @xcite that the all - coupling optical absorption spectrum of a frhlich polaron , together with the role of res - states , fc - states and scattering states , was first presented . and @xmath639 . the res peak is very intense compared with the fc peak . the frequency @xmath640 is indicated by the dashed lines.),scaledwidth=70.0% ] recent interesting numerical calculations of the optical conductivity for the frhlich polaron performed within the diagrammatic quantum monte carlo method @xcite , see fig . [ fig_4 ] , fully confirm the essential analytical results derived by devreese et al . in ref . @xcite for @xmath641 in the intermediate coupling regime @xmath642 the low - energy behavior and the position of the res - peak in the optical conductivity spectrum of ref . @xcite follow closely the prediction of ref . there are some minor qualitative differences between the two approaches in the intermediate coupling regime : in ref . @xcite , the dominant ( res ) peak is less intense in the monte - carlo numerical simulations and the second ( fc ) peak develops less prominently . there are the following qualitative differences between the two approaches in the strong coupling regime : in ref.@xcite , the dominant peak broadens and the second peak does not develop , giving instead rise to a flat shoulder in the optical conductivity spectrum at @xmath643 this behavior has been tentatively attributed to the optical processes with participation of two @xcite or more phonons . the above differences can arise also due to the fact that , within the feynman polaron model , one - phonon processes are assigned more oscillator strength and the res tends to be more stable as compared to the monte - carlo result . the nature of the excited states of a polaron needs further study . an independent numerical simulation might be called for . and @xmath257 and to the analytical dsg calculations @xcite ( solid lines ) . _ right - hand panel _ : monte carlo optical conductivity spectra for the intermediate coupling regime ( open circles ) compared to the analytical dsg approach @xcite ( solid lines ) . arrows point to the two- and three - phonon thresholds . ( from ref.@xcite.),scaledwidth=80.0% ] in fig . [ fig_4a ] , monte - carlo optical conductivity spectrum of one polaron for @xmath257 compares well with that obtained in ref . @xcite within the canonical - transformation formalism taking into account correlation in processes involving two lo phonons . the difference between the results of these two approaches becomes less pronounced when decreasing the value of @xmath257 and might be indicative of a possible precision loss , which requires an independent check . for @xmath257 calculated within the monte carlo approach @xcite ( open circles ) and derived using the expansion in powers of @xmath63 up to @xmath102 @xcite ( solid curve).,scaledwidth=70.0% ] the coupling constant @xmath63 of the known ionic crystals is too small ( @xmath644 ) to allow for the experimental detection of sharp res peaks , and the resonance condition @xmath645 can not be satisfied for @xmath646 as shown in ref , @xcite . nevertheless , for @xmath647 the development of res is already reflected in a broad optical absorption peak . such a peak , predicted in ref . @xcite , was identified , e. g. , in the optical absorption of pr@xmath67nio@xmath648 in ref . @xcite . also , the resonance condition can be fulfilled if an external magnetic field is applied ; the magnetic field stabilizes the res , which then can be detected in a cyclotron resonance peak . in this section , we analyze the sum rules for the optical conductivity spectra obtained within the dsg approach @xcite with those obtained within the diagrammatic monte carlo calculation @xcite . the values of the polaron effective mass for the monte carlo approach are taken from ref . @xcite . in tables 3 and 4 , we represent the polaron ground - state @xmath0 and the following parameters calculated using the optical conductivity spectra:@xmath649 where @xmath650 is the upper value of the frequency available from ref . @xcite,@xmath651 where @xmath1 is the polaron mass , the optical conductivity is calculated in units @xmath652 @xmath1 is measured in units of the band mass @xmath55 , and the frequency is measured in units of @xmath52 . the values of @xmath650 are : @xmath653 for @xmath654 1 and 3 , @xmath655 for @xmath656 5.25 and 6 , @xmath657 for @xmath658 7 and 8 . * table 3 . polaron parameters obtained within the diagrammatic monte carlo approach * [ c]|c|c|c|c|@xmath63 & @xmath659 & @xmath660 & @xmath661 + @xmath662 & @xmath663 & @xmath664 & @xmath665 + @xmath666 & @xmath667 & @xmath668 & @xmath669 + @xmath670 & @xmath671 & @xmath672 & @xmath673 + @xmath674 & @xmath675 & @xmath676 & @xmath677 + @xmath678 & @xmath679 & @xmath680 & @xmath681 + @xmath639 & @xmath682 & @xmath683 & @xmath684 + @xmath685 & @xmath686 & @xmath687 & @xmath688 + @xmath689 & @xmath690 & @xmath691 & @xmath692 + @xmath693 & @xmath694 & @xmath695 & @xmath696 + [ c]|c|c|@xmath697 & @xmath698 + @xmath699 & @xmath225 + @xmath700 & @xmath701 + @xmath702 & @xmath703 + @xmath704 & @xmath705 + @xmath706 & @xmath707 + @xmath708 & @xmath709 + @xmath710 & @xmath711 + @xmath712 & @xmath713 + @xmath714 & @xmath715 + * table 4 . polaron parameters obtained within the path - integral approach * [ c]|c|c|c|c|@xmath63 & @xmath716 & @xmath717 & @xmath718 + @xmath662 & @xmath719 & @xmath664 & @xmath665 + @xmath666 & @xmath720 & @xmath721 & @xmath722 + @xmath670 & @xmath723 & @xmath724 & @xmath725 + @xmath674 & @xmath726 & @xmath727 & @xmath728 + @xmath678 & @xmath729 & @xmath730 & @xmath731 + @xmath639 & @xmath732 & @xmath733 & @xmath734 + @xmath685 & @xmath735 & @xmath736 & @xmath737 + @xmath689 & @xmath738 & @xmath739 & @xmath740 + @xmath693 & @xmath741 & @xmath742 & @xmath743 + [ c]|c|c|@xmath744 & @xmath745 + @xmath746 & @xmath225 + @xmath747 & @xmath748 + @xmath749 & @xmath750 + @xmath751 & @xmath752 + @xmath753 & @xmath754 + @xmath755 & @xmath756 + @xmath757 & @xmath758 + @xmath759 & @xmath760 + @xmath761 & @xmath762 + the parameters corresponding to the monte carlo calculation are obtained using the numerical data kindly provided by a. mishchenko . the comparison of the zero frequency moments @xmath661 and @xmath718 with each other and with the value @xmath763 corresponding to the sum rule @xcite@xmath764 shows that @xmath765 is smaller than each of the differences @xmath766 , @xmath767 which appear due to a finite interval of the integration in ( [ 1 ] ) , ( [ 2 ] ) . we analyze also the fulfilment of the ground - state theorem @xcite@xmath768 using the first - frequency moments @xmath769 and @xmath770 . the results of this comparison are presented in fig . [ moments2-f1 ] . the dots indicate the polaron ground - state energy calculated using the feynman variational principle . the solid curve is the value of @xmath771 calculated numerically using the optical conductivity spectra and the ground - state theorem with the dsg optical conductivity @xcite for a polaron,@xmath772 the dashed and the dot - dashed curves are the values obtained using @xmath773 and @xmath774 , respectively:@xmath775 as seen from the figure , @xmath776 to a high degree of accuracy coincides with the variational polaron ground - state energy . both @xmath777 and @xmath778 differ from @xmath779 due to the integration over a finite interval of frequencies . however , @xmath780 and @xmath781 are very close to each other . herefrom , a conclusion follows that for integrals over the finite frequency region characteristic for the polaron optical absorption ( i. e. , except the tails ) , the function @xmath782 ( [ e3 ] ) reproduces very well the function @xmath783 . the form of the frhlich hamiltonian in @xmath275 dimensions is the same as in 3d,@xmath784 except that now all vectors are @xmath275-dimensional . in this subsection , dispersionless longitudinal phonons are considered , i.e. , @xmath785 , and units are chosen such that @xmath786 . the electron - phonon interaction is a representation in second quantization of the electron interaction with the lattice polarization , which in 3d is essentially the coulomb potential @xmath787 @xmath788 is proportional to the fourier transform of this potential , and as a consequence we have in @xmath275 dimensions@xmath789 where @xmath790 is the volume of the @xmath275-dimensional crystal . note that @xmath791 , where @xmath792 is an ( @xmath793)-dimensional vector , can be obtained from @xmath794 , where @xmath795 @xmath796 is an @xmath275-dimensional vector , by summing out one of the dimensions explicitly:@xmath797 inserting eq . ( [ sr2 ] ) into eq . ( [ sr3 ] ) , we have@xmath798 replacing the sum in eq . ( [ sr4 ] ) by an integral , i.e.,@xmath799 we obtain @xmath800 since @xmath801 we have@xmath802 where @xmath803 is eh @xmath804 function . inserting eq . ( [ sr6 ] ) into eq . ( [ sr5 ] ) , we obtain@xmath805 in 3d the interaction coefficient is well known , @xmath806 so that @xmath807 inserting eq . ( [ sr8 ] ) into eq . ( [ sr7 ] ) , we immediately obtain@xmath808 applying eq . ( [ sr7 ] ) @xmath809 times , we further obtain for @xmath810@xmath811 so , the interaction coefficient in @xmath275 dimensions becomes @xcite @xmath812 following the feynman approach @xcite , the upper bound for the polaron ground - state energy can be written down as@xmath813 where @xmath80 is the exact action functional of the polaron problem , while @xmath814 is the trial action functional , which corresponds to a model system where an electron is coupled by an elastic force to a fictitious particle ( i.e. , the model system describes a harmonic oscillator ) . @xmath0 is the ground - state energy of the above model system , and@xmath815 as indicated above , the frhlich hamiltonian in @xmath275 dimensions is the same as in 3d , except that now all vectors are @xmath275-dimensional [ and the coupling coefficient @xmath816 is modified in accordance with eq . ( [ srv ] ) ] . similarly , the only difference of the model system in @xmath275 dimensions from the model system in 3d is that now one deals with an @xmath275-dimensional harmonic oscillator . so , directly following @xcite , one can represent @xmath817 as @xmath818 where@xmath819 } \right\rangle _ { 0}e^{-t}dt , \label{sr14}\]]@xmath820 ^{2}\right\rangle _ { 0}e^{-wt}dt , \label{sr15}\ ] ] @xmath264 and @xmath200 are variational parameters , which should be determined by minimizing @xmath303 of eq . ( [ sr11 ] ) . since the averaging @xmath821 in eq . ( [ sr14 ] ) is performed with the trial action , which corresponds to a harmonic oscillator , components of the electron coordinates , @xmath822 ( @xmath823 ) , in @xmath824 } \right\rangle _ { 0}$ ] separate @xcite:@xmath825 } \right\rangle _ { 0}=\prod\limits_{j=1}^{n}\left\langle e^{ik_{j}\left [ r_{j}\left ( t\right ) -r_{j}\left ( 0\right ) \right ] } \right\rangle _ { 0}. \label{sr16}\ ] ] for the average @xmath826 } \right\rangle _ { 0}$ ] , feynman obtained @xcite@xmath827 } \right\rangle _ { 0}=e^{-k_{j}^{2}d_{0}\left ( t\right ) } , \label{sr17}\ ] ] where@xmath828 inserting eq . ( [ sr16 ] ) with eq . ( [ sr17 ] ) into eq . ( [ sr14 ] ) , we obtain@xmath829 inserting expression ( [ srv ] ) for @xmath788 into eq . ( [ sr19 ] ) and replacing the sum over @xmath59 by an integral [ see ( [ sr4a ] ) ] , we have @xmath830 where @xmath831 is the elemental solid angle in @xmath275 dimensions . since the integrand in eq . ( [ sr20 ] ) depends only on the modulus @xmath832 of @xmath59 , one have simply @xmath833 with@xmath834 so , we obtain for @xmath391 the result@xmath835 like in ref . @xcite , @xmath392 can be easily calculated by noticing that @xmath836 ^{2}\right\rangle _ { 0 } & = \sum_{j=1}^{n}\left\langle \left [ r_{j}\left ( t\right ) -r_{j}\left ( 0\right ) \right ] ^{2}\right\rangle _ { 0}=\sum_{j=1}^{n}\left . \left [ -\frac{\partial^{2}}{\partial k_{j}^{2}}\left\langle e^{i\mathbf{k}\cdot\left [ \mathbf{r}\left ( t\right ) -\mathbf{r}\left ( 0\right ) \right ] } \right\rangle _ { 0}\right ] \right\vert _ { \mathbf{k}=0}\nonumber\\ & = \sum_{j=1}^{n}2d_{0}\left ( t\right ) = 2nd_{0}\left ( t\right ) , \label{ssr1}\ ] ] so that@xmath837 dt\nonumber\\ & = \frac{nw\left ( v^{2}-w^{2}\right ) } { 2}\left [ \frac{w^{2}}{2v^{2}}\frac{1}{v^{2}}+\frac{v^{2}-w^{2}}{2v^{3}}\left ( \frac{1}{w}-\frac{1}{v+w}\right ) \right ] \nonumber\\ & = \frac{n\left ( v^{2}-w^{2}\right ) } { 4v}. \label{ssr3}\ ] ] inserting eq . ( [ srv ] ) with @xmath391 and @xmath392 , given by eqs . ( [ sr22 ] ) and ( [ ssr3 ] ) , together with the ground - state energy of the model system @xcite ( an isotropic @xmath275-dimensional harmonic oscillator),@xmath838 into eq . ( [ sr11 ] ) , we obtain @xmath839 in order to make easier a comparison of @xmath303 for @xmath275 dimensions with the feynman result @xcite for 3d,@xmath840 it is convenient to rewrite eq . ( [ ssr5 ] ) in the form@xmath841 . \label{ssr7}\ ] ] it is worth recalling that the parameters @xmath264 and @xmath200 must be determined by minimizing @xmath303 . thus , in the case of eq . ( [ ssr7 ] ) one has to minimize the expression in the square brackets . the only difference of this expression from the r.h.s . of eq . ( [ ssr6 ] ) is that @xmath63 is multiplied by the factor @xmath842 this means that the minimizing parameters @xmath264 and @xmath200 in @xmath275d at a given @xmath63 will be exactly the same as those calculated in 3d for the frhlich constant as large as @xmath843 : @xmath844 therefore , comparing eq . ( [ ssr7 ] ) to eq . ( [ ssr6 ] ) , we obtain the scaling relation @xcite@xmath845 where @xmath846 is given by eq . ( [ sran ] ) . as discussed in ref . @xcite , the above scaling relation is not an exact relation . it is valid for the feynman polaron energy and also for the ground - state energy to order @xmath63 . the next - order term ( i.e. , @xmath102 ) no longer satisfies eq . ( [ ssr8 ] ) . the reason is that in the exact calculation ( to order @xmath102 ) the electron motion in the different space directions is coupled by the electron - phonon interaction . no such a coupling appears in the feynman polaron model [ see , e.g. , eq . ( [ sr16 ] ) ] ; and this is the underlying reason for the validity of the scaling relation for the feynman approximation . in refs . @xcite , scaling relations are obtained also for the impedance function , the effective mass and the mobility of a polaron . the inverse of the impedance function @xmath847 is given by @xmath848 where the memory function @xmath849 can be expressed as @xcite@xmath850 with @xmath851 and@xmath852@xmath853 e^{t}+n\left ( 1\right ) e^{-it},\]]@xmath854 .\ ] ] here , @xmath181 is the inverse temperature and @xmath855 is the occupation number of phonons with frequency @xmath856 ( recall that in our units @xmath857 ) . as implied from eqs . ( [ ssr9 ] ) and ( [ ssr10 ] ) , scaling of @xmath858 and @xmath847 is determined by scaling of @xmath859 . for an isotropic crystal , since @xmath794 , @xmath860 and @xmath861 do not depend on the direction of @xmath59 , one can write @xmath862 , so that@xmath863 inserting expression ( [ srv ] ) for @xmath788 and replacing the sum over @xmath59 by an integral , we have@xmath864 in particular , for 3d one has from eq . ( [ ssr13])@xmath865 for @xmath275d , eq . ( [ ssr13 ] ) can be rewritten is the form @xmath866 so , the only difference of the expression for @xmath867 from @xmath868 is that @xmath63 is multiplied by @xmath846 . since for the minimizing parameters @xmath264 and @xmath200 , which enter @xmath860 , scaling is determined by the same product @xmath63 with @xmath846 [ see eq . ( [ srvw ] ) ] , we can write @xmath869 so that @xcite@xmath870 and@xmath871 the polaron mass at zero temperature can be obtained from the impedance function as @xcite@xmath872 so that from the scaling relation ( [ ssr16 ] ) for the memory function we also have a scaling relation for the polaron mass @xcite:@xmath873 since the mobility can be obtained from the memory function as @xcite@xmath874 fulfilment of the scaling relation ( [ ssr16 ] ) implies also a scaling relation for the mobility @xcite:@xmath875 in the important particular case of 2d , the above scaling relations take the form @xcite:@xmath876@xmath877@xmath878@xmath879 the fulfilment of the pd - scaling relation @xcite is checked for the path integral monte carlo results @xcite for the polaron free energy . the path integral monte carlo results of ref.@xcite for the polaron free energy in 3d and in 2d are given for a few values of temperature and for some selected values of @xmath287 for a check of the scaling relation , the values of the polaron free energy at @xmath880 are taken from ref . @xcite in 3d ( table i , for 4 values of @xmath63 ) and in 2d ( table ii , for 2 values of @xmath63 ) and plotted in fig . [ sccomp ] , upper panel , with squares and open circles , correspondingly . the pd - scaling relation for the polaron ground - state energy as derived in ref . @xcite reads:@xmath881 in fig . [ sccomp ] , lower panel , the available data for the free energy from ref @xcite are plotted in the following form _ inspired by the l.h.s . and the r.h.s parts of eq . ( 1 ) _ : @xmath882 ( squares ) and @xmath883(open triangles ) . as follows from the figure , t__he path integral monte carlo results for the polaron free energy in 2d and 3d very closely follow the pd - scaling relation of the form given by eq . ( [ e]):__@xmath884 [ h ] part1fig11.eps inspired by the work of tomonaga on quantum electrodynamics ( q. e. d. ) , lee , low and pines ( l.l.p . ) @xcite derived ( [ eq_5a ] ) and @xmath885 from a canonical transformation formulation , which establishes ( [ eq_5a ] ) as a variational upper bound for the ground - state energy . the wave equation corresponding to the frhlich hamiltonian ( [ eq_1a ] ) is @xmath886 we shall take advantage of the fact that the total momentum of the system @xmath887 ( where @xmath888 * * is the momentum of the electron ) is a constant of motion because it commutes with the hamiltonian ( [ eq_1a ] ) indeed,@xmath889 & = [ \mathbf{p},\sum_{\mathbf{k}}(v_{k}a_{\mathbf{k}}e^{i\mathbf{k\cdot r}}+v_{k}^{\ast}a_{\mathbf{k}}^{\dag } e^{-i\mathbf{k\cdot r}})]=\sum_{\mathbf{k}}(v_{k}a_{\mathbf{k}}\left [ \mathbf{p},e^{i\mathbf{k\cdot r}}\right ] + v_{k}^{\ast}a_{\mathbf{k}}^{\dag } \left [ \mathbf{p},e^{-i\mathbf{k\cdot r}}\right ] ) \\ & = \sum_{\mathbf{k}}\hbar\mathbf{k}(v_{k}a_{\mathbf{k}}e^{i\mathbf{k\cdot r}}-v_{k}^{\ast}a_{\mathbf{k}}^{\dag}e^{-i\mathbf{k\cdot r}});\end{aligned}\ ] ] @xmath890 & = \left [ \sum_{\mathbf{k}}\hbar\mathbf{k}a_{\mathbf{k}}^{\dag } a_{\mathbf{k}},\sum_{\mathbf{k}^{\prime}}(v_{k^{\prime}}a_{\mathbf{k}^{\prime}}e^{i\mathbf{k}^{\prime}\mathbf{\cdot r}}+v_{k^{\prime}}^{\ast } a_{\mathbf{k}^{\prime}}^{\dag}e^{-i\mathbf{k}^{\prime}\mathbf{\cdot r}})\right ] = \\ & = \sum_{\mathbf{k}}\hbar\mathbf{k}\left [ a_{\mathbf{k}}^{\dag}a_{\mathbf{k}},(v_{k}a_{\mathbf{k}}e^{i\mathbf{k\cdot r}}+v_{k}^{\ast } a_{\mathbf{k}}^{\dag}e^{-i\mathbf{k\cdot r}})\right ] = \\ & = \sum_{\mathbf{k}}\hbar\mathbf{k}\left\ { v_{k}\left [ a_{\mathbf{k}}^{\dag}a_{\mathbf{k}},a_{\mathbf{k}}\right ] e^{i\mathbf{k\cdot r}}+v_{k}^{\ast}\left [ a_{\mathbf{k}}^{\dag}a_{\mathbf{k}},a_{\mathbf{k}}^{\dag } \right ] e^{-i\mathbf{k\cdot r}}\right\ } = \\ & = -\sum_{\mathbf{k}}\hbar\mathbf{k}\left ( v_{k}a_{\mathbf{k}}e^{i\mathbf{k\cdot r}}-v_{k}^{\ast}a_{\mathbf{k}}^{\dag}e^{-i\mathbf{k\cdot r}}\right ) ; \end{aligned}\ ] ] @xmath891 = \left [ \sum_{\mathbf{k}}\hbar \mathbf{k}a_{\mathbf{k}}^{\dag}a_{\mathbf{k}}+\mathbf{p},h\right ] = 0 . \label{phcomm}\ ] ] because of the commutation ( [ phcomm ] ) , the operators @xmath414 and @xmath892 have a common set of basis functions : @xmath893 and @xmath894 it is possible to transform to a representation in which @xmath892 becomes a c number @xmath87 , and in which the hamiltonian no longer contains the electron coordinates . the unitary ( canonical ) transformation required is @xmath895 , where @xmath896 . \label{eq_6a}\ ] ] derivation of the transformations of the operators . \mathbf{p}\exp\left [ \frac{i}{\hbar}(\mathbf{p}-\sum_{\mathbf{k}}\hbar\mathbf{k}a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}})\mathbf{\cdot r}\right ] \nonumber\\ & = \exp\left [ -\frac{i}{\hbar}(\mathbf{p}-\sum_{\mathbf{k}}\hbar \mathbf{k}a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}})\mathbf{\cdot r}\right ] \left ( -i\hbar\nabla\right ) \exp\left [ \frac{i}{\hbar}(\mathbf{p}-\sum_{\mathbf{k}}\hbar\mathbf{k}a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}})\mathbf{\cdot r}\right ] \nonumber\\ & = \exp\left [ -\frac{i}{\hbar}(\mathbf{p}-\sum_{\mathbf{k}}\hbar \mathbf{k}a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}})\mathbf{\cdot r}\right ] \left\ { \begin{array } [ c]{c}(\mathbf{p}-\sum_{\mathbf{k}}\hbar\mathbf{k}a_{\mathbf{k}}^{\dagger } a_{\mathbf{k}})\exp\left [ \frac{i}{\hbar}(\mathbf{p}-\sum_{\mathbf{k}}\hbar\mathbf{k}a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}})\mathbf{\cdot r}\right ] \\ + \exp\left [ \frac{i}{\hbar}(\mathbf{p}-\sum_{\mathbf{k}}\hbar\mathbf{k}a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}})\mathbf{\cdot r}\right ] \left ( -i\hbar\nabla\right ) \end{array } \right\ } \nonumber\\ & = \mathbf{p}-\sum_{\mathbf{k}}\hbar\mathbf{k}a_{\mathbf{k}}^{\dagger } a_{\mathbf{k}}+\mathbf{p , } \label{llp3a}\ ] ] @xmath898 \left ( \sum_{\mathbf{k}}\hbar\mathbf{k}a_{\mathbf{k}}^{\dag}a_{\mathbf{k}}+\mathbf{p}\right ) \exp\left [ \frac{i}{\hbar}(\mathbf{p}-\sum_{\mathbf{k}}\hbar\mathbf{k}a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}})\mathbf{\cdot r}\right ] \nonumber\\ & = \exp\left [ \frac{i}{\hbar}\sum_{\mathbf{k}}\hbar\mathbf{k}a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}}\mathbf{\cdot r}\right ] \sum_{\mathbf{k}}\hbar\mathbf{k}a_{\mathbf{k}}^{\dag}a_{\mathbf{k}}\exp\left [ -\frac{i}{\hbar } \sum_{\mathbf{k}}\hbar\mathbf{k}a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}}\mathbf{\cdot r}\right ] + s_{1}^{-1}\mathbf{p}s_{1}\nonumber\\ & = \sum_{\mathbf{k}}\hbar\mathbf{k}a_{\mathbf{k}}^{\dag}a_{\mathbf{k}}+\mathbf{p}-\sum_{\mathbf{k}}\hbar\mathbf{k}a_{\mathbf{k}}^{\dagger } a_{\mathbf{k}}+\mathbf{p = p}+\mathbf{p , } \label{llp3b}\ ] ] @xmath899 a_{\mathbf{k}}\exp\left [ \frac{i}{\hbar}(\mathbf{p}-\sum_{\mathbf{k}}\hbar\mathbf{k}a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}})\mathbf{\cdot r}\right ] \nonumber\\ & = \exp\left [ \frac{i}{\hbar}\sum_{\mathbf{k}}\hbar\mathbf{k}a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}}\mathbf{\cdot r}\right ] a_{\mathbf{k}}\exp\left [ -\frac{i}{\hbar}\sum_{\mathbf{k}}\hbar\mathbf{k}a_{\mathbf{k}}^{\dagger } a_{\mathbf{k}}\mathbf{\cdot r}\right ] \nonumber\\ & = \exp\left [ i\mathbf{k}a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}}\mathbf{\cdot r}\right ] a_{\mathbf{k}}\exp\left [ -i\mathbf{k}a_{\mathbf{k}}^{\dagger } a_{\mathbf{k}}\mathbf{\cdot r}\right ] \nonumber\\ & = \exp\left [ i\mathbf{k}a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}}\mathbf{\cdot r}\right ] a_{\mathbf{k}}{\displaystyle\sum\nolimits_{n=0}^{\infty } } \frac{1}{n!}\left ( -i\mathbf{k}a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}}\mathbf{\cdot r}\right ) ^{n}\nonumber\\ & = \exp\left [ i\mathbf{k}a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}}\mathbf{\cdot r}\right ] { \displaystyle\sum\nolimits_{n=0}^{\infty } } \frac{1}{n!}a_{\mathbf{k}}\left ( -i\mathbf{k}a_{\mathbf{k}}^{\dagger } a_{\mathbf{k}}\mathbf{\cdot r}\right ) ^{n}\nonumber\\ & = \exp\left [ i\mathbf{k\cdot r}a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}}\right ] { \displaystyle\sum\nolimits_{n=0}^{\infty } } \frac{1}{n!}\left ( -i\mathbf{k\cdot r}\right ) ^{n}a_{\mathbf{k}}\left ( a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}}\right ) ^{n}\overset{\text{see}(\ast ) } { = } \nonumber\\ & = \exp\left [ i\mathbf{k\cdot r}a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}}\right ] { \displaystyle\sum\nolimits_{n=0}^{\infty } } \frac{1}{n!}[-i\mathbf{k\cdot r}\left ( a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}}+1\right ) ] ^{n}a_{\mathbf{k}}\nonumber\\ & = \exp\left [ i\mathbf{k\cdot r}a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}}\right ] \exp\left [ -i\mathbf{k\cdot r(}a_{\mathbf{k}}^{\dagger } a_{\mathbf{k}}+1)\right ] a_{\mathbf{k}}\nonumber\\ & = a_{\mathbf{k}}\exp\left ( -i\mathbf{k\cdot r}\right ) . \label{llp3c1}\ ] ] here the property was used : @xmath900 it is evident for @xmath901for @xmath902 it is demonstrated as follows : @xmath903 then for @xmath904 the validity of ( [ * ] ) is straightforwardly demonstrated by induction . finally , @xmath905^{\dag}=a_{\mathbf{k}}^{\dag}\exp\left ( i\mathbf{k\cdot r}\right ) . \label{llp3d}\ ] ] using ( [ llp3a ] ) , ( [ llp3b ] ) , ( [ llp3c1 ] ) and ( [ llp3d ] ) , one arrives at @xmath906 where @xmath56 is set @xmath907 . leads to @xmath908at the same time , applying eq . ( [ llp3b ] ) , we obtain @xmath909setting the gauge @xmath910 eliminates the operator @xmath56 from the problem . ] the wave equation ( [ llp1 ] ) takes the form @xmath911 our aim is to calculate for a given momentum @xmath87 the lowest eigenvalue @xmath912 of the hamiltonian ( [ llp3e ] ) . for the low - lying energy levels of the electron @xmath912 may be well represented by the first two terms of a power series expansion in @xmath913where @xmath914 is the effective mass of the polaron . the canonical transformation ( [ eq_6a ] ) formally eliminates the electron operators from the hamiltonian . llp use further a variational method of calculation . the trial wave function is chosen as @xmath915 where @xmath916 is the eigenstate of the unperturbed hamiltonian with no phonons present ( vacuum state ) . specifically , @xmath916 is defined by @xmath917 and the second canonical transformation : @xmath918 , \label{eq_6b}\ ] ] where @xmath919 are treated as variational functions and will be chosen to minimize the energy . the physical significance of eq.([eq_6b ] ) is that it dresses the electron with the virtual phonon field , which describes the polarization . viewed as a unitary transformation , @xmath327 is a _ displacement _ operator on @xmath58 and @xmath350 @xmath920 a_{\mathbf{k}}\exp\left [ \sum_{\mathbf{k}}(a_{\mathbf{k}}^{\dagger}f_{\mathbf{k}}-a_{\mathbf{k}}f_{\mathbf{k}}^{\ast})\right ] \nonumber\\ & = \exp\left [ -(a_{\mathbf{k}}^{\dagger}f_{\mathbf{k}}-a_{\mathbf{k}}f_{\mathbf{k}}^{\ast})\right ] a_{\mathbf{k}}\exp\left [ ( a_{\mathbf{k}}^{\dagger}f_{\mathbf{k}}-a_{\mathbf{k}}f_{\mathbf{k}}^{\ast})\right ] \overset{\text{see}(\ast\ast)}{=}\nonumber\\ & = a_{\mathbf{k}}+\left [ a_{\mathbf{k}},(a_{\mathbf{k}}^{\dagger } f_{\mathbf{k}}-a_{\mathbf{k}}f_{\mathbf{k}}^{\ast})\right ] + \frac{1}{2}\left [ \left [ a_{\mathbf{k}},(a_{\mathbf{k}}^{\dagger}f_{\mathbf{k}}-a_{\mathbf{k}}f_{\mathbf{k}}^{\ast})\right ] , ( a_{\mathbf{k}}^{\dagger } f_{\mathbf{k}}-a_{\mathbf{k}}f_{\mathbf{k}}^{\ast})\right ] + ... \nonumber\\ & = a_{\mathbf{k}}+f_{\mathbf{k } } , \label{llp6a}\ ] ] @xmath921 here the relation was used@xmath922 a\exp\left [ v\right ] = a+\left [ a , v\right ] + \frac { 1}{2}\left [ \left [ a , v\right ] , v\right ] + \frac{1}{3!}\left [ \left [ \left [ a , v\right ] , v\right ] , v\right ] + ... \tag{**}\label{**}\ ] ] further we seek to minimize the expression for the energy , @xmath923 in virtue of ( [ llp6a ] ) and ( [ llp6b ] ) , we obtain : @xmath924 ^{2}}{2m_{b}}\\ & + \sum_{\mathbf{k}}\hbar\omega_{\mathrm{lo}}\left ( a_{\mathbf{k}}^{\dag } + f_{\mathbf{k}}^{\ast}\right ) \left ( a_{\mathbf{k}}+f_{\mathbf{k}}\right ) + \sum_{\mathbf{k}}\left [ v_{k}\left ( a_{\mathbf{k}}+f_{\mathbf{k}}\right ) + v_{k}^{\ast}\left ( a_{\mathbf{k}}^{\dag}+f_{\mathbf{k}}^{\ast}\right ) \right ] \\ & = \frac{{\left [ ( \mathbf{p}-\sum_{\mathbf{k}}\hbar\mathbf{k}a_{\mathbf{k}}^{\dag}a_{\mathbf{k}})-\sum_{\mathbf{k}}\hbar\mathbf{k}\left\vert f_{\mathbf{k}}\right\vert ^{2}-\sum_{\mathbf{k}}\hbar\mathbf{k}\left ( a_{\mathbf{k}}^{\dag}f_{\mathbf{k}}+a_{\mathbf{k}}f_{\mathbf{k}}^{\ast } \right ) \right ] ^{2}}}{2m_{b}}\\ & + \sum_{\mathbf{k}}\hbar\omega_{\mathrm{lo}}\left ( a_{\mathbf{k}}^{\dag } a_{\mathbf{k}}+\left\vert f_{\mathbf{k}}\right\vert ^{2}+a_{\mathbf{k}}^{\dag}f_{\mathbf{k}}+a_{\mathbf{k}}f_{\mathbf{k}}^{\ast}\right ) \\ & + \sum_{\mathbf{k}}\left [ v_{k}\left ( a_{\mathbf{k}}+f_{\mathbf{k}}\right ) + v_{k}^{\ast}\left ( a_{\mathbf{k}}^{\dag}+f_{\mathbf{k}}^{\ast } \right ) \right ] \\ & = h_{0}+h_{1},\end{aligned}\ ] ] where@xmath925 ^{2}+\left [ \sum_{\mathbf{k}}\hbar\mathbf{k}\left\vert f_{\mathbf{k}}\right\vert ^{2}\right ] ^{2}}}{2m_{b}}+\sum_{\mathbf{k}}\left [ v_{k}f_{\mathbf{k}}+v_{k}^{\ast } f_{\mathbf{k}}^{\ast}\right ] \nonumber\\ & + \sum_{\mathbf{k}}\left\vert f_{\mathbf{k}}\right\vert ^{2}\left\ { \hbar\omega_{\mathrm{lo}}-\frac{\hbar\mathbf{k\cdot p}}{m_{b}}+\frac{\hbar ^{2}k^{2}}{2m_{b}}\right\ } + \frac{\hbar^{2}}{m_{b}}\sum_{\mathbf{k}}\mathbf{k}a_{\mathbf{k}}^{\dag}a_{\mathbf{k}}\cdot\sum_{\mathbf{k}^{\prime}}\mathbf{k}^{\prime}\left\vert f_{\mathbf{k}^{\prime}}\right\vert ^{2}\nonumber\\ & + \sum_{\mathbf{k}}a_{\mathbf{k}}\left\ { v_{k}+f_{\mathbf{k}}^{\ast}\left [ \hbar\omega_{\mathrm{lo}}-\frac{\hbar\mathbf{k\cdot p}}{m_{b}}+\frac{\hbar ^{2}k^{2}}{2m_{b}}+\frac{\hbar^{2}\mathbf{k}}{m_{b}}\cdot\sum_{\mathbf{k}^{\prime}}\mathbf{k}^{\prime}\left\vert f_{\mathbf{k}^{\prime}}\right\vert ^{2}\right ] \right\ } \nonumber\\ & + \sum_{\mathbf{k}}a_{\mathbf{k}}^{\dag}\left\ { v_{k}^{\ast}+f_{\mathbf{k}}\left [ \hbar\omega_{\mathrm{lo}}-\frac{\hbar\mathbf{k\cdot p}}{m_{b}}+\frac{\hbar^{2}k^{2}}{2m_{b}}+\frac{\hbar^{2}\mathbf{k}}{m_{b}}\cdot \sum_{\mathbf{k}^{\prime}}\mathbf{k}^{\prime}\left\vert f_{\mathbf{k}^{\prime } } \right\vert ^{2}\right ] \right\ } \nonumber\\ & + \sum_{\mathbf{k}}\hbar\omega_{\mathrm{lo}}a_{\mathbf{k}}^{\dag } a_{\mathbf{k } } ; \label{llp7a}\ ] ] @xmath926 using ( [ llp6 ] ) , we obtain from ( [ llp7 ] ) that@xmath927 ^{2}}}{2m_{b}}+\sum_{\mathbf{k}}\left [ v_{k}f_{\mathbf{k}}+v_{k}^{\ast}f_{\mathbf{k}}^{\ast}\right ] \nonumber\\ & + \sum_{\mathbf{k}}\left\vert f_{\mathbf{k}}\right\vert ^{2}\left\ { \hbar\omega_{\mathrm{lo}}-\frac{\hbar\mathbf{k\cdot p}}{m_{b}}+\frac{\hbar ^{2}k^{2}}{2m_{b}}\right\ } . \label{llp8}\ ] ] we minimize ( [ llp8 ] ) by imposing @xmath928 this results in @xmath929 \cdot}\mathbf{k}\right\ } = 0 \label{llp9}\ ] ] and the appropriate complex conjugate equation for @xmath919 . upon comparing ( [ llp9 ] ) and ( [ llp7a ] ) , we see that the linear terms in @xmath245 and @xmath58 are identically zero if ( [ llp9 ] ) is satisfied , and hence that @xmath930 is diagonal in a representation in which @xmath931 is diagonal . so , the variational calculation by llp is equivalent to the use of ( [ llp7a ] ) as the total hamiltonian provided @xmath932 satisfies ( [ llp9 ] ) . an estimate of the accuracy of the llp variational procedure was obtained by an estimate of the effect of @xmath346using a perturbation theory . now we evaluate the energy of the ground state of the system , which is given by eq . ( [ llp8 ] ) with @xmath932 satisfying eq . ( [ llp9 ] ) . the only preferred direction in this problem is @xmath87 . therefore one may introduce the parameter @xmath933 defined as@xmath934 then eq . ( [ llp9 ] ) leads to @xmath935 \right . , \label{fk}\ ] ] and we obtain the following implicit equation for @xmath933:@xmath936 ^{2}\right . \\ & = \frac{v}{\left ( 2\pi\right ) ^{3}}\int d^{3}k\hbar\mathbf{k}\left ( \frac{\hbar\omega_{\mathrm{lo}}}{k}\right ) ^{2}\frac{4\pi\alpha}{v}\left ( \frac{\hbar}{2m_{b}\omega_{\mathrm{lo}}}\right ) ^{\frac{1}{2}}\left/ \left [ \hbar\omega_{\mathrm{lo}}-\frac{\hbar\mathbf{k\cdot p}}{m_{b}}(1-\eta ) + \frac{\hbar^{2}k^{2}}{2m_{b}}\right ] ^{2}\right . .\end{aligned}\ ] ] let us introduce spherical coordinates with a polar axis along @xmath87 and denote @xmath937:@xmath938 ^{2}\right . \\ & = \frac{\alpha\hbar}{2\pi^{2}}\left ( \frac{\hbar}{2m_{b}\omega _ { \mathrm{lo}}}\right ) ^{\frac{1}{2}}2\pi{\displaystyle\int\nolimits_{-1}^{1 } } dxx{\displaystyle\int\nolimits_{0}^{\infty } } dkk\left/ \left [ 1 - 2\frac{\hbar kpx}{2m_{b}\hbar\omega_{\mathrm{lo}}}(1-\eta)+\frac{\hbar^{2}k^{2}}{2m_{b}\hbar\omega_{\mathrm{lo}}}\right ] ^{2}\right . .\end{aligned}\ ] ] further , we introduce the parameter @xmath939 and a new variable@xmath940 this gives@xmath941 ^{2}\right . \\ & = \frac{\alpha}{\pi}\frac{\left ( 2m_{b}\hbar\omega_{\mathrm{lo}}\right ) ^{1/2}}{p}{\displaystyle\int\nolimits_{-1}^{1 } } dxx{\displaystyle\int\nolimits_{0}^{\infty } } d\kappa\kappa\left/ \left [ ( \kappa - qx)^{2}+(1-q^{2}x^{2})\right ] ^{2}\right . \\ & = \frac{\alpha}{\pi}\frac{\left ( 2m_{b}\hbar\omega_{\mathrm{lo}}\right ) ^{1/2}}{p}{\displaystyle\int\nolimits_{-1}^{1 } } dxx{\displaystyle\int\nolimits_{-qx}^{\infty } } d\kappa\left ( \kappa+qx\right ) \left/ \left [ \kappa^{2}+(1-q^{2}x^{2})\right ] ^{2}\right . \\ & = \frac{\alpha}{\pi}\frac{\left ( 2m_{b}\hbar\omega_{\mathrm{lo}}\right ) ^{1/2}}{p}{\displaystyle\int\nolimits_{-1}^{1 } } dxx\left\ { \begin{array } [ c]{c}-\frac{1}{2\left [ \kappa^{2}+(1-q^{2}x^{2})\right ] } \\ + qx\left [ \frac{\kappa}{2(1-q^{2}x^{2})\left [ \kappa^{2}+(1-q^{2}x^{2})\right ] } + \frac{1}{2(1-q^{2}x^{2})^{3/2}}\arctan\left ( \frac{\kappa } { \left [ 1-q^{2}x^{2}\right ] ^{1/2}}\right ) \right ] \end{array } \right\ } _ { -qx}^{\infty}\\ & = \frac{\alpha}{\pi}\frac{\left ( 2m_{b}\hbar\omega_{\mathrm{lo}}\right ) ^{1/2}}{p}{\displaystyle\int\nolimits_{-1}^{1 } } dxx\left\ { \frac{1}{2}+\frac{qx\pi}{4(1-q^{2}x^{2})^{3/2}}+\frac{q^{2}x^{2}}{2(1-q^{2}x^{2})}+\frac{qx}{2(1-q^{2}x^{2})^{3/2}}\arcsin\left ( qx\right ) \right\ } .\end{aligned}\]]@xmath542@xmath942 so , we have arrived at the equation@xmath943 or equivalently , using the definition ( [ q]),@xmath944 using eqs . ( [ etap ] ) and ( [ fk ] ) , we can simplify the energy ( [ llp8 ] ) as follows:@xmath945@xmath946@xmath947@xmath948@xmath949@xmath950@xmath542@xmath951 the sum over @xmath59 in eq . ( [ en ] ) is calculated as follows:@xmath952 } \\ & = \frac{m_{b}\omega_{\mathrm{lo}}^{2}\alpha}{\pi^{2}}\left ( \frac{\hbar } { 2m_{b}\omega_{\mathrm{lo}}}\right ) ^{\frac{1}{2}}\int d\mathbf{k}\frac { 1}{k^{2}\left ( k^{2}-2\frac{\mathbf{k\cdot p}}{\hbar}(1-\eta)+\frac { 2m_{b}\omega_{\mathrm{lo}}}{\hbar}\right ) } .\end{aligned}\ ] ] for the calculation of this integral , we can use the auxiliary identity@xmath953 ^{2}}. \label{id}\ ] ] setting@xmath954 we find@xmath955 ^{2}}\\ & = \frac{m_{b}\omega_{\mathrm{lo}}^{2}\alpha}{\pi^{2}}\left ( \frac{\hbar } { 2m_{b}\omega_{\mathrm{lo}}}\right ) ^{\frac{1}{2}}\int_{0}^{1}dx\int d\mathbf{k}\frac{1}{\left ( k^{2}-2\frac{\mathbf{k\cdot p}}{\hbar}(1-\eta)x+\frac{2m_{b}\omega_{\mathrm{lo}}}{\hbar}x\right ) ^{2}}\\ & = \frac{m_{b}\omega_{\mathrm{lo}}^{2}\alpha}{\pi^{2}}\left ( \frac{\hbar } { 2m_{b}\omega_{\mathrm{lo}}}\right ) ^{\frac{1}{2}}\int_{0}^{1}dx\int d\mathbf{k}\frac{1}{\left ( \left ( \mathbf{k-}\frac{\mathbf{p}}{\hbar}(1-\eta)x\right ) ^{2}+\frac{2m_{b}\omega_{\mathrm{lo}}}{\hbar}x-\frac{p^{2}}{\hbar^{2}}(1-\eta)^{2}x^{2}\right ) ^{2}}\\ & = \frac{m_{b}\omega_{\mathrm{lo}}^{2}\alpha}{\pi^{2}}\left ( \frac{\hbar } { 2m_{b}\omega_{\mathrm{lo}}}\right ) ^{\frac{1}{2}}\int_{0}^{1}dx\int d\mathbf{k}\frac{1}{\left ( k^{2}+\frac{2m_{b}\omega_{\mathrm{lo}}}{\hbar } x-\frac{p^{2}}{\hbar^{2}}(1-\eta)^{2}x^{2}\right ) ^{2}}\ ] ] as long as @xmath956 is sufficiently small so that no spontaneous emission can occur ( roughly @xmath957 ) , the quantity @xmath958 is supposed to be positive for @xmath959 . therefore , we can use the integral@xmath960 what gives@xmath961 we change the variable @xmath962 what gives@xmath963 and hence@xmath964 as a result , the energy ( [ en ] ) is expressed in a closed form@xmath965 further , we expand the r.h.s . of eq . ( [ llp27-a ] ) to the second order in powers of @xmath447 ( or , what is the same , in powers of @xmath966 ) using the relation@xmath967 what results in@xmath968 = \frac{\alpha}{6}+o\left ( q^{2}\right ) .\ ] ] solving this equation for @xmath933 , we find@xmath969 we also apply the expansion in powers of @xmath966 up to @xmath970 to the energy ( [ llp29]):@xmath971 finally , we have arrived at the llp polaron energy@xmath972 in this derivation it is shown that the approximation used in the evaluation of the function , which determines the polaron mass [ see eqs . ( 40 ) and ( b1 ) from ref . @xcite ] @xmath973 \exp\left [ k^{2}c(\tau)\right ] \label{b1}\ ] ] with @xmath974 is equivalent to an expansion in a continued fraction limited to the first step . moreover , it is proved that the choice of the coefficients of the continued fraction can be justified by a variational principle , at least when @xmath975 is real and positive . expanding the last exponential of eq . ( [ b1 ] ) in a power series leads to @xmath976 \int_{-\infty}^{0}d\tau e^{iz\tau}{\displaystyle\sum\limits_{n=0}^{\infty } } \frac{1}{n!}\left ( \frac{1}{3m^{2}}\right ) ^{n}\nonumber\\ & \times{\displaystyle\sum\limits_{\vec{k}_{1}, ... ,\vec{k}_{n } } } \frac{k_{1}^{2}k_{2}^{2} ... k_{n}^{2}\left\vert f_{k_{1}}\right\vert ^{2}\left\vert f_{k_{2}}\right\vert ^{2} ... \left\vert f_{k_{n}}\right\vert ^{2}}{\gamma_{k_{1}}^{2}\gamma_{k_{2}}^{2} ... \gamma_{k_{n}}^{2}}\nonumber\\ & \times\exp\left [ i(\gamma_{k_{1}}+\gamma_{k_{2}}+ ... \gamma_{k_{n}}^{2})\tau\right ] \nonumber\\ & = -i\exp\left [ -k^{2}c(0)\right ] { \displaystyle\sum\limits_{n=0}^{\infty } } \frac{(3m^{2})^{-n}}{n!}\label{b3}\\ & \times{\displaystyle\sum\limits_{\vec{k}_{1}, ... ,\vec{k}_{n } } } \frac{k_{1}^{2}k_{2}^{2} ... k_{n}^{2}\left\vert f_{k_{1}}\right\vert ^{2}\left\vert f_{k_{2}}\right\vert ^{2} ... \left\vert f_{k_{n}}\right\vert ^{2}}{\gamma_{k_{1}}^{2}\gamma_{k_{2}}^{2} ... \gamma_{k_{n}}^{2}}\frac { 1}{\gamma_{k_{1}}+\gamma_{k_{2}}+ ... +\gamma_{k_{n}}+z}.\nonumber\end{aligned}\ ] ] the multiple sum over the @xmath832 s is in fact an integral with 3n variables . it is possible to change the variables in that one of the new variables is @xmath977 then the multiple sum which appears in the last term of eq . ( [ b3 ] ) is of the following type : @xmath978 where @xmath979 is the result of the integration over the @xmath793 other variables . an expansion of integrals of the type ( [ b5 ] ) into stieltjes continued fractions is known to give good results when @xmath975 is real and not located on the cut of @xmath980 , i.e. , when @xmath981 the first nontrivial step in the continued fraction expansion is @xmath982 with @xmath983 @xmath984 a _ variational principle _ can be established , which gives a rather strong argument in favour of the approximation ( [ b7 ] ) . let us introduce a variational parameter @xmath985 writing @xmath986 performing two steps of the division , this relation becomes @xmath987 with @xmath988 the approximation consists of neglecting the term @xmath989 in eq . ( [ b11 ] ) . as this term is positive [ cf . ( [ b6 ] ) ] , the best approximation is obtained when it is minimum . therefore let us use the freedom in the choice of @xmath985 to minimize the expression ( [ b12 ] ) , @xmath990 which gives @xmath991 or @xmath992 this provides us with the best value of the variational parameter @xmath993 which is @xmath994[cf . ( [ b9 ] ) ] . with this value of @xmath985 and neglecting @xmath989 , the expression ( [ b11 ] ) of the calculated quantity @xmath980 becomes @xmath995 which is the first step ( [ b7 ] ) of a stieltjes continued fraction . to prove that this value of @xmath985 gives a minimum of @xmath989 , let us calculate the second derivative @xmath996 now the parameter @xmath985 is replaced by its expression ( [ b16 ] ) . the relation ( [ b18 ] ) becomes @xmath997 which is positive of @xmath998 , since it follows from relation ( [ b16 ] ) that @xmath999 our approximation is related to that used by feynman which is based on the following inequality : @xmath1000 where the brackets denote the expectation value of the random variable @xmath384 . for instance , @xmath1001 where l(x ) is the non - normalized probability density of @xmath384 . the laplace transform of eq . ( [ b19 ] ) gives @xmath1002 which after integration becomes @xmath1003 the last inequality shows the relation with our procedure . the derivations in the present section are based on ref . the hamiltonian of a system of @xmath242 interacting continuum polarons is given by : @xmath1004 where @xmath1005 represent the position and momentum of the @xmath242 constituent electrons ( or holes ) with band mass @xmath55 ; @xmath1006 denote the creation and annihilation operators for longitudinal optical ( lo ) phonons with wave vector @xmath1007 and frequency @xmath2 ; @xmath1008 describes the amplitude of the interaction between the electrons and the phonons ; and @xmath19 is the elementary electron charge . this hamiltonian can be subdivided into the following parts : @xmath1009 where@xmath1010 is the kinetic energy of electrons,@xmath1011 is the potential energy of the coulomb electron - electron interaction,@xmath1012 is the hamlitonian of phonons , and@xmath1013 is the hamiltonian of the electron - phonon interaction . further on , we use the second quantization formalism for electrons , in which the terms @xmath1014 , @xmath1015 and @xmath1016 are@xmath1017 where @xmath1018 is the symbol of the normal product of operators,@xmath1019 is the fourier component of the coulomb potential , and@xmath1020 is the fourier component of the electron density . the ground state energy of the many - polaron hamiltonian ( [ hmpol ] ) has been studied by l. lemmens , j. t. devreese and f. brosens ( ldb ) @xcite , for weak and intermediate strength of the electron - phonon coupling . they introduce a variational wave function : @xmath1021 where @xmath1022 represents the ground - state many - body wave function for the electron ( or hole ) system and @xmath1023 is the phonon vacuum , @xmath1024 is a many - body unitary operator which determines a canonical transformation for a fermion gas interacting with a boson field : @xmath1025 in ref . @xcite , this canonical transformation was used to establish a many - fermion theory . the @xmath1026 were determined variationally @xcite resulting in @xmath1027 for a system with total momentum @xmath1028 . in this expression , @xmath1029 represents the static structure factor of the constituent interacting many electron or hole system : @xmath1030 the angular brackets @xmath1031 represent the expectation value with respect to the ground state . the many - polaron optical conductivity is the response of the current - density , in the system described by the hamiltonian ( [ hmpol ] ) , to an applied electric field ( along the @xmath384-axis ) with frequency @xmath856 . this applied electric field introduces a perturbation term in the hamiltonian ( [ hmpol ] ) , which couples the vector potential of the incident electromagnetic field to the current - density . within linear response theory , the optical conductivity can be expressed through the kubo formula as a current - current correlation function : @xmath1032 \right\rangle dt.\ ] ] in this expression , @xmath305 is the volume of the system , and @xmath1033 is the @xmath384-component of the current operator @xmath1034 which is related to the momentum operators of the charge carriers : @xmath1035 with @xmath966 the charge of the charge carriers ( @xmath1036 for holes , @xmath1037 for electrons ) and @xmath87 the total momentum operator of the charge carriers . the real part of the optical conductivity at temperature zero , which is proportional to the optical absorption coefficient , can be written as a function of the total momentum operator of the charge carriers as follows : @xmath1038 \right\rangle dt\right\ } . \label{k1}\ ] ] let us integrate over time in ( [ k1 ] ) twice by parts as follows : @xmath1039 \right\rangle e^{i\omega t-\delta t}\\ & = \frac{1}{i\omega-\delta}\left\ { \left . \left\langle \left [ p_{x}\left ( t\right ) , p_{x}\right ] \right\rangle e^{i\omega t-\delta t}\right\vert _ { t=0}^{\infty}-\int_{0}^{\infty}dt\left\langle \left [ \frac{d}{dt}p_{x}\left ( t\right ) , p_{x}\right ] \right\rangle e^{i\omega t-\delta t}\right\ } \\ & = -\frac{1}{i\omega-\delta}\int_{0}^{\infty}dt\left\langle \left [ \frac { d}{dt}\left ( e^{\frac{it}{\hbar}h}p_{x}e^{-\frac{it}{\hbar}h}\right ) , p_{x}\right ] \right\rangle e^{i\omega t-\delta t}\\ & = -\frac{1}{i\omega-\delta}\int_{0}^{\infty}dt\left\langle \left [ \left ( e^{\frac{it}{\hbar}h}\frac{i}{\hbar}\left [ h , p_{x}\right ] e^{-\frac { it}{\hbar}h}\right ) , p_{x}\right ] \right\rangle e^{i\omega t-\delta t}\\ & = -\frac{1}{i\omega-\delta}\int_{0}^{\infty}dt\left\langle \left [ \frac { i}{\hbar}\left [ h , p_{x}\right ] , e^{-\frac{it}{\hbar}h}p_{x}e^{\frac { it}{\hbar}h}\right ] \right\rangle e^{i\omega t-\delta t}\\ & = -\frac{1}{i\omega-\delta}\int_{0}^{\infty}dt\left\langle \left [ f_{x}\left ( 0\right ) , e^{-\frac{it}{\hbar}h}p_{x}e^{\frac{it}{\hbar}h}\right ] \right\rangle e^{i\omega t-\delta t}\\ & = -\left ( \frac{1}{i\omega-\delta}\right ) ^{2}\left\ { \left . \left\langle \left [ f_{x}\left ( 0\right ) , e^{-\frac{it}{\hbar}h}p_{x}e^{\frac{it}{\hbar}h}\right ] \right\rangle e^{i\omega t-\delta t}\right\vert _ { t=0}^{\infty}\right . \\ & \left . -\int_{0}^{\infty}dt\left\langle \left [ f_{x}\left ( 0\right ) , \frac{d}{dt}e^{-\frac{it}{\hbar}h}p_{x}e^{\frac{it}{\hbar}h}\right ] \right\rangle e^{i\omega t-\delta t}\right\}\end{aligned}\ ] ] @xmath1040 \right\rangle + \int_{0}^{\infty}dt\left\langle \left [ f_{x}\left ( 0\right ) , e^{-\frac{it}{\hbar}h}\left ( \frac{i}{\hbar } \left [ h , p_{x}\right ] \right ) e^{\frac{it}{\hbar}h}\right ] \right\rangle e^{i\omega t-\delta t}\right\ } \\ & = -\left ( \frac{1}{i\omega-\delta}\right ) ^{2}\left\ { -\left\langle \left [ f_{x},p_{x}\right ] \right\rangle + \int_{0}^{\infty}dt\left\langle \left [ f_{x}\left ( 0\right ) , e^{-\frac{it}{\hbar}h}f_{x}\left ( 0\right ) e^{\frac{it}{\hbar}h}\right ] \right\rangle e^{i\omega t-\delta t}\right\ } \\ & = -\left ( \frac{1}{i\omega-\delta}\right ) ^{2}\left\ { -\left\langle \left [ f_{x},p_{x}\right ] \right\rangle + \int_{0}^{\infty}dt\left\langle \left [ e^{\frac{it}{\hbar}h}f_{x}\left ( 0\right ) e^{-\frac{it}{\hbar}h},f_{x}\left ( 0\right ) \right ] \right\rangle e^{i\omega t-\delta t}\right\ } \\ & = \frac{1}{\left ( \omega+i\delta\right ) ^{2}}\left\ { -\left\langle \left [ f_{x},p_{x}\right ] \right\rangle + \int_{0}^{\infty}dt\left\langle \left [ f_{x}\left ( t\right ) , f_{x}\left ( 0\right ) \right ] \right\rangle e^{i\omega t-\delta t}\right\ } , \end{aligned}\ ] ] where @xmath1041 $ ] is the operator of the force applied to the center of mass of the electrons . since both @xmath1042 and @xmath1043 are hermitian operators , the average @xmath1044 \right\rangle $ ] is imaginary . hence , for @xmath1045 this term does not give a contribution into @xmath1046 as a result , integrating by parts twice , the real part of the optical conductivity of the many - polaron system is written with a force - force correlation function : @xmath1047 \right\rangle dt\right\ } . \label{ff}\ ] ] the force operator is determined as @xmath1048 = \frac{i}{\hbar}\left [ h_{e}+h_{e - e}+h_{ph}+h_{e - ph},p_{x}\right ] .\ ] ] further , we use the commutators : @xmath1049 = \sum_{\mathbf{k}^{\prime}}\hbar k_{x}^{\prime}\left [ a_{\mathbf{k+q},\sigma } ^{+}a_{\mathbf{k},\sigma},a_{\mathbf{k}^{\prime},\sigma}^{+}a_{\mathbf{k}^{\prime},\sigma}\right ] \\ = \sum_{\mathbf{k}^{\prime}}\hbar k_{x}^{\prime}\left ( \begin{array } [ c]{c}a_{\mathbf{k+q},\sigma}^{+}a_{\mathbf{k},\sigma}a_{\mathbf{k}^{\prime},\sigma } ^{+}a_{\mathbf{k}^{\prime},\sigma}+a_{\mathbf{k+q},\sigma}^{+}a_{\mathbf{k}^{\prime},\sigma}^{+}a_{\mathbf{k},\sigma}a_{\mathbf{k}^{\prime},\sigma}\\ -a_{\mathbf{k}^{\prime},\sigma}^{+}a_{\mathbf{k+q},\sigma}^{+}a_{\mathbf{k}^{\prime},\sigma}a_{\mathbf{k},\sigma}-a_{\mathbf{k}^{\prime},\sigma}^{+}a_{\mathbf{k}^{\prime},\sigma}a_{\mathbf{k+q},\sigma}^{+}a_{\mathbf{k},\sigma}\end{array } \right ) \\ = \sum_{\mathbf{k}^{\prime}}\hbar k_{x}^{\prime}\left ( \delta_{\mathbf{kk}^{\prime}}a_{\mathbf{k+q},\sigma}^{+}a_{\mathbf{k}^{\prime},\sigma}-\delta_{\mathbf{k}^{\prime},\mathbf{k+q}}a_{\mathbf{k}^{\prime},\sigma}^{+}a_{\mathbf{k},\sigma}\right ) \\ = \sum_{\mathbf{k}^{\prime}}\hbar k_{x}^{\prime}\left ( \delta_{\mathbf{kk}^{\prime}}a_{\mathbf{k+q},\sigma}^{+}a_{\mathbf{k},\sigma}-\delta _ { \mathbf{k}^{\prime},\mathbf{k+q}}a_{\mathbf{k+q},\sigma}^{+}a_{\mathbf{k},\sigma}\right ) \\ = a_{\mathbf{k+q},\sigma}^{+}a_{\mathbf{k},\sigma}\sum_{\mathbf{k}^{\prime}}\hbar k_{x}^{\prime}\left ( \delta_{\mathbf{kk}^{\prime}}-\delta _ { \mathbf{k}^{\prime},\mathbf{k+q}}\right ) = -\hbar q_{x}a_{\mathbf{k+q},\sigma}^{+}a_{\mathbf{k},\sigma},\end{gathered}\ ] ] @xmath1050 = -\hbar q_{x}\rho_{\mathbf{q}}.\ ] ] hence , @xmath1051 = 0,$ ] @xmath1052 = 0,$]@xmath1053 & = \sum_{\mathbf{q}}\left ( v_{\mathbf{q}}b_{\mathbf{q}}\left [ \rho_{\mathbf{q}},p_{x}\right ] + v_{\mathbf{q}}^{\ast } b_{\mathbf{q}}^{+}\left [ \rho_{-\mathbf{q}},p_{x}\right ] \right ) \\ & = -\hbar\sum_{\mathbf{q}}q_{x}\left ( v_{\mathbf{q}}b_{\mathbf{q}}\rho_{\mathbf{q}}-v_{\mathbf{q}}^{\ast}b_{\mathbf{q}}^{+}\rho_{-\mathbf{q}}\right ) , \end{aligned}\ ] ] so , the commutator of the hamiltonian ( [ hmpol ] ) with the total momentum operator of the charge carriers leads to the expression for the force@xmath1054 this result for the force operator clarifies the significance of using the force - force correlation function rather than the momentum - momentum correlation function . the operator product @xmath1055 is proportional to @xmath1056 , the charge carrier - phonon interaction strength . this will be a distinct advantage for any expansion of the final result in the charge carrier - phonon interaction strength , since one power of @xmath1056 is factored out beforehand . substituting ( [ force ] ) into ( [ ff ] ) , the real part of the optical conductivity then takes the form : @xmath1057 \rho_{\mathbf{q}}\left ( t\right ) , \left ( v_{-\mathbf{q}^{\prime}}b_{-\mathbf{q}^{\prime}}+v_{\mathbf{q}^{\prime}}^{\ast}b_{\mathbf{q}^{\prime } } ^{+}\right ) \rho_{-\mathbf{q}^{\prime}}\right ] \right\rangle . \label{ff2}\ ] ] up to this point , no approximations other than _ linear response theory _ have been made . the expectation value appearing in the right hand side of expression ( [ ff2 ] ) for the real part of the optical conductivity is calculated now with respect to the ldb many - polaron wave function ( [ psildb ] ) . the unitary operator ( [ u ] ) can be written as @xmath1058 the transformed hamiltonian ( [ h ] ) is denoted as@xmath1059 the momentum operator of an electron @xmath1060 the operator of the total momentum of electrons @xmath87 and the phonon creation and annihilation operators are transformed by the unitary transformation ( [ unit ] ) as follows : @xmath1061 as a result , after the transformation ( [ unit ] ) , the hamiltonian takes the form ( see ref . @xcite ) : @xmath1062 where the terms are@xmath1063@xmath1064 \nonumber\\ & + \frac{\hbar^{2}}{2m_{b}}\sum_{\mathbf{q}}a_{\mathbf{q}}\sum_{\mathbf{k},\sigma}\left ( \mathbf{q}^{2}+2\mathbf{k\cdot q}\right ) a_{\mathbf{k+q},\sigma}^{+}a_{\mathbf{k},\sigma } , \label{h1b}\]]@xmath1065@xmath1066 the exact expression for the real part of the conductivity ( [ ff2 ] ) after the replacement of @xmath1067 by @xmath1068 is transformed to the approximate one @xmath1069{c}e^{\frac{it}{\hbar}h}\left [ v_{\mathbf{q}}b_{\mathbf{q}}+v_{-\mathbf{q}}^{\ast}b_{-\mathbf{q}}^{+}\right ] \rho_{\mathbf{q}}e^{-\frac{it}{\hbar}h},\\ \left ( v_{-\mathbf{q}^{\prime}}b_{-\mathbf{q}^{\prime}}+v_{\mathbf{q}^{\prime}}^{\ast}b_{\mathbf{q}^{\prime}}^{+}\right ) \rho_{-\mathbf{q}^{\prime}}\end{array } \right ] u\right\vert \phi\right\rangle \right\vert \psi_{el}^{\left ( 0\right ) } \right\rangle\end{aligned}\]]@xmath1070{c}e^{\frac{it}{\hbar}\tilde{h}}u^{-1}\left [ v_{\mathbf{q}}b_{\mathbf{q}}+v_{-\mathbf{q}}^{\ast}b_{-\mathbf{q}}^{+}\right ] u\rho_{\mathbf{q}}e^{-\frac{it}{\hbar}\tilde{h}},\\ u^{-1}\left ( v_{-\mathbf{q}^{\prime}}b_{-\mathbf{q}^{\prime}}+v_{\mathbf{q}^{\prime}}^{\ast}b_{\mathbf{q}^{\prime}}^{+}\right ) u\rho_{-\mathbf{q}^{\prime}}\end{array } \right ] \right\vert \phi\right\rangle \right\vert \psi_{el}^{\left ( 0\right ) } \right\rangle\end{aligned}\]]@xmath1071 \right\vert \phi\right\rangle \right\vert \psi_{el}^{\left ( 0\right ) } \right\rangle .\end{aligned}\ ] ] so , we have arrived at the expression@xmath1072 \right\vert \phi\right\rangle \right\vert \psi_{el}^{\left ( 0\right ) } \right\rangle .\end{gathered}\ ] ] since @xmath1073 and @xmath1074 the terms proportional to @xmath1075 vanish after the summation over @xmath1007 : @xmath1076 hence we obtain the real part of the optical conductivity in the form@xmath1077 \right\vert \phi \right\rangle \right\vert \psi_{el}^{\left ( 0\right ) } \right\rangle . \label{res}\ ] ] introducing the factor@xmath1078 \right\vert \phi \right\rangle \right\vert \psi_{el}^{\left ( 0\right ) } \right\rangle , \end{aligned}\ ] ] the optical conductivity can be written as@xmath1079 in the case of a weak electron - phonon coupling , we can neglect in the exponent @xmath1080 of ( [ res ] ) the terms @xmath1081 and @xmath1082 [ i. e. , the renormalized hamiltonian of the electron - phonon interaction ( [ h1a ] ) and ( [ hppe ] ) ] . namely , we replace @xmath1083 in eq . ( [ res ] ) by the hamiltonian @xmath1084 in this case , we find@xmath1085 \right\vert \phi \right\rangle \right\vert \psi_{el}^{\left ( 0\right ) } \right\rangle\end{aligned}\]]@xmath1086 .\end{aligned}\ ] ] the time - dependent phonon operators are@xmath1087 so that we have@xmath1088 \\ & = 2i|v_{\mathbf{q}}|^{2}\delta_{\mathbf{qq}^{\prime}}\operatorname{im}\left [ \left\langle \psi_{el}^{\left ( 0\right ) } \left\vert e^{i\tilde { h}_{e}t/\hbar}\rho_{\mathbf{q}}e^{-i\tilde{h}_{e}t/\hbar}\rho_{-\mathbf{q}}\right\vert \psi_{el}^{\left ( 0\right ) } \right\rangle \left\langle \phi\left\vert b_{\mathbf{q}}b_{\mathbf{q}}^{+}\right\vert \phi\right\rangle \right ] , \end{aligned}\ ] ] where @xmath1089 . taking the expectation value with respect to the phonon vacuum , we find @xmath1090 .\ ] ] the optical conductivity ( [ res ] ) then takes the form:@xmath1091 \label{res2}\ ] ] for an isotropic electron - phonon system , @xmath1092 in 3d can be replaced by @xmath1093 what gives us the result@xmath1094 , \label{resig1}\ ] ] where the two - point correlation function is @xmath1095 the same derivation for the 2d case , provides the expression@xmath1096 , \label{resig1 - 2d}\ ] ] where @xmath391 is the surface of the 2d system . to find the formula for the real part of the optical conductivity in its final form , we introduce the standard expression for the dynamic structure factor of the system of charge carriers interacting through a coulomb potential,@xmath1097 the dynamic structure factor is expressed in terms of the two - point correlation function as follows:@xmath1098@xmath542@xmath1099 where @xmath1100 is the fourier image of @xmath1101:@xmath1102 the function @xmath1103 obeys the following property:@xmath1104 where @xmath1105 is the eigenvalue of the hamiltonian @xmath1106:@xmath1107 herefrom , we find that@xmath1108 from ( [ sp ] ) , for the function@xmath1109 \label{b}\ ] ] the following equality is derived:@xmath1110 \\ & = \operatorname{im}\left [ e^{i\omega_{\mathrm{lo}}t}f^{\ast}\left ( \mathbf{q},t\right ) \right ] \\ & = -\operatorname{im}\left [ e^{-i\omega_{\mathrm{lo}}t}f\left ( \mathbf{q},t\right ) \right ] = -b\left ( \mathbf{q},t\right ) , \end{aligned}\]]@xmath1111 the integral in eq . ( [ resig1 ] ) @xmath1112 = \operatorname{im}\int_{0}^{\infty}dte^{i\omega t-\delta t}b\left ( \mathbf{q},t\right)\ ] ] is then transformed as follows:@xmath1113 \\ & = \frac{1}{2i}\left [ \int_{0}^{\infty}dte^{i\omega t-\delta t}b\left ( \mathbf{q},t\right ) -\int_{-\infty}^{0}dte^{i\omega t+\delta t}b\left ( \mathbf{q},-t\right ) \right ] \\ & = \frac{1}{2i}\left [ \int_{0}^{\infty}dte^{i\omega t-\delta t}b\left ( \mathbf{q},t\right ) + \int_{-\infty}^{0}dte^{i\omega t+\delta t}b\left ( \mathbf{q},t\right ) \right ] \\ & = \frac{1}{2i}\int_{-\infty}^{\infty}dte^{i\omega t-\delta\left\vert t\right\vert } b\left ( \mathbf{q},t\right ) \\ & = \frac{1}{2i}\int_{-\infty}^{\infty}dte^{i\omega t-\delta\left\vert t\right\vert } \frac{1}{2i}\left [ e^{-i\omega_{\mathrm{lo}}t}f\left ( \mathbf{q},t\right ) -e^{i\omega_{\mathrm{lo}}t}f^{\ast}\left ( \mathbf{q},t\right ) \right ] \\ & = -\frac{1}{4}\int_{-\infty}^{\infty}dte^{i\omega t-\delta\left\vert t\right\vert } \left [ e^{-i\omega_{\mathrm{lo}}t}f\left ( \mathbf{q},t\right ) -e^{i\omega_{\mathrm{lo}}t}f\left ( \mathbf{q},-t\right ) \right ] .\end{aligned}\ ] ] we can show that , as far as @xmath1114 is the _ ground _ state , the integral @xmath1115 for positive @xmath856 is equal to zero . let @xmath1116 is the total basis set of the eigenfunctions of the hamiltonian @xmath1117 using these functions we expand @xmath1103:@xmath1118@xmath1119 because for @xmath1120 @xmath1121 is never equal to zero . so , rewriting expression ( [ resig1 ] ) with the dynamic structure factor of the electron ( or hole ) gas results in:@xmath1122 and we obtain @xmath1123 where @xmath1124{c}n / v\;\text{in 3d,}\\ n / a\;\text{in 2d}\end{array } \right.\ ] ] is the density of charge carriers . for an isotropic medium , the dynamic structure factor does not depend on the direction of @xmath1007 , so that @xmath1125 , where @xmath1126 . let us simplify the expression ( [ opticabs ] ) using explicitly the amplitudes of the frhlich electron - phonon interaction . the modulus squared of the frhlich electron - phonon interaction amplitude is given by @xmath1127{l}\dfrac{(\hbar\omega_{\mathrm{lo}})^{2}}{q^{2}}\dfrac{4\pi\alpha}{v}\left ( \dfrac{\hbar}{2m_{b}\omega_{\mathrm{lo}}}\right ) ^{1/2}\text { in 3d}\\ \dfrac{(\hbar\omega_{\mathrm{lo}})^{2}}{q}\dfrac{2\pi\alpha}{a}\left ( \dfrac{\hbar}{2m_{b}\omega_{\mathrm{lo}}}\right ) ^{1/2}\,\text{in 2d,}\end{array } \right.\ ] ] where @xmath63 is the ( dimensionless ) frhlich coupling constant determining the coupling strength between the charge carriers and the longitudinal optical phonons @xcite . in 3d and 2d , respectively , the sums over @xmath1007 is transformed to the integrals as follows:@xmath1128@xmath542 @xmath1129 @xmath542@xmath1130 in the same way , we transform @xmath1131 : @xmath1132 using the feynman units ( @xmath1133 , @xmath621 , @xmath1134 ) , @xmath1135 is@xmath1136 from these expressions , it is clear that the scaling relation @xmath1137 which holds for the one - polaron case introduced in ref . @xcite , is also valid for the many - polaron case if the corresponding 2d or 3d dynamic structure factor is used . the dynamic structure factor @xmath1138 is expressed through the two - point correlation function by eq . ( [ sthroughf ] ) . the correlation function @xmath1100 can be found using the retarded green s function of the density operators @xmath1139 \right\vert \psi_{el}^{\left ( 0\right ) } \right\rangle , \ ] ] where @xmath1140 is the step function . let us consider the more general case of a finite temperature,@xmath1141 \right\rangle , \ ] ] where the average is@xmath1142 the fourier image @xmath1143 of the retarded green s function @xmath1144 is@xmath1145 \right\rangle e^{i\omega t}dt\\ & = -i\int_{0}^{\infty}\left ( \left\langle e^{\frac{it}{\hbar}\tilde{h}_{0}}\rho_{\mathbf{q}}e^{-\frac{it}{\hbar}\tilde{h}_{0}}\rho_{-\mathbf{q}}\right\rangle -\left\langle \rho_{-\mathbf{q}}e^{\frac{it}{\hbar}\tilde { h}_{0}}\rho_{\mathbf{q}}e^{-\frac{it}{\hbar}\tilde{h}_{0}}\right\rangle \right ) e^{i\omega t}dt\end{aligned}\ ] ] the imaginary part of @xmath1143 then is@xmath1146 in the second integral here , we replace @xmath535 by @xmath1147:@xmath1148 as far as the integral over @xmath535 converges ( i. e. , @xmath1149 tends to zero at @xmath1150 ) , we can shift the integration contour to the real axis , what gives us the result@xmath1151 herefrom , we find that@xmath1152@xmath542@xmath1153 so , the equation follows from the analytical properties of the green s functions : @xmath1154 the formula ( [ rel1 ] ) is related to arbitrary temperatures . in the zero - temperature limit ( @xmath617 ) , the factor @xmath1155 ( [ rel1 ] ) turns into the heavicide step function @xmath1156 , what leads to the formula@xmath1157@xmath542@xmath1158 the retarded green function is related to the dielectric function of the electron gas by the following equation : @xmath1159 . \label{diel}\ ] ] within the random phase approximation ( rpa ) , following @xcite , the expression for @xmath1143 is @xmath1160 ^{-1}\hbar p\left ( \mathbf{q},\omega\right ) , \label{dr1}\ ] ] where the polarization function @xmath1161 is ( see , e. g. , p. 434 of @xcite ) @xmath1162 with the fermi distribution function of electrons @xmath1163 . for a finite temperature , the explicit analytic expression for the imaginary part of the structure factor @xmath1164 is obtained ( see @xcite ) , @xmath1165 \right\ } } { 1+\exp\left\ { \beta\left [ \mu - e^{\left ( -\right ) } \left ( q,\omega \right ) \right ] \right\ } } , \nonumber\\ e^{\left ( \pm\right ) } \left ( q,\omega\right ) & \equiv\frac{\left ( \hbar\omega\pm\frac{\hbar^{2}q^{2}}{2m_{b}}\right ) ^{2}}{4\frac{\hbar ^{2}q^{2}}{2m_{b}}},\end{aligned}\ ] ] with the chemical potential @xmath262 . the real part of the structure factor is obtained using the kramers - kronig dispersion relation : @xmath1166 analytical expressions for both real and imaginary parts of @xmath1167 can be written down for the zero temperature ( see @xcite ) , @xmath1168{c}\left [ k_{f}^{2}-\frac{m_{b}^{2}}{\hbar^{2}q^{2}}\left ( \omega-\frac{\hbar q^{2}}{2m_{b}}\right ) ^{2}\right ] \ln\left\vert \frac{\omega-\frac{\hbar q^{2}}{2m_{b}}-\frac{\hbar k_{f}q}{m_{b}}}{\omega-\frac{\hbar q^{2}}{2m_{b}}+\frac{\hbar k_{f}q}{m_{b}}}\right\vert \\ + \left [ k_{f}^{2}-\frac{m_{b}^{2}}{\hbar^{2}q^{2}}\left ( \omega+\frac{\hbar q^{2}}{2m_{b}}\right ) ^{2}\right ] \ln\left\vert \frac{\omega+\frac{\hbar q^{2}}{2m_{b}}+\frac{\hbar k_{f}q}{m_{b}}}{\omega+\frac{\hbar q^{2}}{2m_{b}}-\frac{\hbar k_{f}q}{m_{b}}}\right\vert \\ + 2k_{f}q \end{array } \right\ } , \nonumber\\ \operatorname{im}p_{\mathrm{3d}}\left ( q,\omega\right ) & = -\frac{vm_{b}}{4\pi\hbar^{2}q}\left\ { \begin{array } [ c]{c}\left [ k_{f}^{2}-\frac{m_{b}^{2}}{\hbar^{2}q^{2}}\left ( \omega-\frac{\hbar q^{2}}{2m_{b}}\right ) ^{2}\right ] \theta\left ( k_{f}^{2}-\frac{m_{b}^{2}}{\hbar^{2}q^{2}}\left ( \omega-\frac{\hbar q^{2}}{2m_{b}}\right ) ^{2}\right ) \\ -\left [ k_{f}^{2}-\frac{m_{b}^{2}}{\hbar^{2}q^{2}}\left ( \omega+\frac{\hbar q^{2}}{2m_{b}}\right ) ^{2}\right ] \theta\left ( k_{f}^{2}-\frac{m_{b}^{2}}{\hbar^{2}q^{2}}\left ( \omega+\frac{\hbar q^{2}}{2m_{b}}\right ) ^{2}\right ) \end{array } \right\ } , \label{pri}\ ] ] where @xmath1169 is the fermi wave number . after substituting into eq . ( [ rel1 ] ) the retarded green s function ( [ dr1 ] ) in terms of the polarization function we arrive at the formula@xmath1170@xmath542@xmath1171 ^{2}+\left [ v_{\mathbf{q}}\operatorname{im}p\left ( \mathbf{q},\omega\right ) \right ] ^{2}},\label{sqw}\\ \left . s\left ( \mathbf{q},\omega\right ) \right\vert _ { t=0 } & = -\frac{\hbar}{n}\theta\left ( \omega\right ) \frac{\operatorname{im}p\left ( \mathbf{q,}\omega\right ) } { \left [ 1-v_{\mathbf{q}}\operatorname{re}p\left ( \mathbf{q},\omega\right ) \right ] ^{2}+\left [ v_{\mathbf{q}}\operatorname{im}p\left ( \mathbf{q},\omega\right ) \right ] ^{2}}.\end{aligned}\ ] ] with this dynamic structure factor , the optical conductivity ( [ resigmaff1 ] ) ( in the feynman units ) takes the form@xmath1172 ^{2}+\left [ v_{q}\operatorname{im}p_{\mathrm{3d}}\left ( q,\omega-1\right ) \right ] ^{2}}q^{2}dq . \label{rs3d}\ ] ] correspondingly , in the 2d case we obtain the expression@xmath1173 ^{2}+\left [ v_{q}\operatorname{im}p_{\mathrm{2d}}\left ( q,\omega-1\right ) \right ] ^{2}}q^{2}dq . \label{rs2d}\ ] ] the rpa dynamic structure factor for the electron ( or hole ) system can be separated in two parts , one related to continuum excitations of the electrons ( or holes ) @xmath1174 , and one related to the undamped plasmon branch : @xmath1175 where @xmath1176 is the wave number dependent plasmon frequency and @xmath1177 is the strength of the undamped plasmon branch . in eqs . ( [ rs3d ] ) , ( [ rs2d ] ) , the contribution of the continuum excitations corresponds to the region @xmath1178 where @xmath1179 . the contribution related to the undamped plasmons is provided by a region of @xmath1180 where the equations @xmath541{c}\operatorname{im}p\left ( q\mathbf{,}\omega\right ) = 0\\ 1-v_{q}\operatorname{re}p\left ( q,\omega\right ) = 0 \end{array } \right . . \label{c}\ ] ] are fulfilled simultaneously . using ( [ diel ] ) , we find that eqs . ( [ b ] ) are equivalent to @xmath1181 in the region where @xmath1182 the expression @xmath1183 ^{2}+\left [ v_{q}\operatorname{im}p\left ( q,\omega\right ) \right ] ^{2}}$ ] is proportional to the delta function , which gives a finite contribution to the memory function after the integration over @xmath966 : @xmath1184 ^{2}+\left [ v_{q}\operatorname{im}p\left ( q,\omega\right ) \right ] ^{2}}\right\vert _ { \operatorname{im}p\left ( q\mathbf{,}\omega\right ) = 0}\nonumber\\ & = \frac{1}{\pi v_{q}}\delta\left ( 1-v_{q}\operatorname{re}p\left ( q,\omega\right ) \right ) . \label{vv}\ ] ] using eq . ( [ vv ] ) , the coefficients @xmath1185 in eq . ( [ srpa ] ) can be expressed in terms of the polarization function @xmath1186 as follows:@xmath1184 ^{2}+\left [ v_{q}\operatorname{im}p\left ( q,\omega\right ) \right ] ^{2}}\right\vert _ { \operatorname{im}p\left ( q\mathbf{,}\omega\right ) = 0}\\ & = \left . \frac{1}{\pi v_{q}^{2}\left\vert \frac{\partial}{\partial\omega } \operatorname{re}p\left ( q,\omega\right ) \right\vert } \right\vert _ { \omega=\omega_{\mathrm{pl}}\left ( q\right ) } \delta\left ( \omega -\omega_{\mathrm{pl}}\left ( q\right ) \right)\end{aligned}\]]@xmath542@xmath1187 the contribution derived from the undamped plasmon branch @xmath1188 is denoted in ref . @xcite as the _ ` plasmon - phonon ' contribution_. the physical process related to this contribution is the emission of both a phonon and a plasmon in the scattering process calvani and collaborators have performed doping - dependent measurements of the infrared absorption spectra of the high - t@xmath1192 material nd@xmath1189ce@xmath1190cuo@xmath1191 ( ncco ) . the region of the spectrum examined by these authors ( 50 - 10000 @xmath1193 ) is very rich in absorption features : they observe is a drude - like component at the lowest frequencies , and a set of sharp absorption peaks related to phonons and infrared active modes ( up to about 1000 @xmath1193 ) possibly associated to small ( holstein ) polarons . three distinct absorption bands can be distinguished : the ` _ _ d__-band ' ( around 1000 @xmath1193 ) , the mid - infrared band ( mir , around 5000 @xmath1193 ) and the charge - transfer band ( around 10@xmath1194 @xmath1193 ) . of all these features , the _ d_-band and , at a higher temperatures , the drude - like component have ( hypothetically ) been associated with large polaron optical absorption @xcite . [ h ] part2fig1.eps for the lowest levels of ce doping , the _ d_-band can be most clearly distinguished from the other features . the experimental optical absorption spectrum ( up to 3000 @xmath1193 ) of nd@xmath67cuo@xmath1195 ( @xmath1196 ) , obtained by calvani and co - workers @xcite , is shown in fig . [ p2fig1 ] ( shaded area ) together with the theoretical curve obtained by the present method ( full , bold curve ) and , for reference , the one - polaron optical absorption result ( dotted curve ) . at lower frequencies ( 600 - 1000 @xmath1193 ) a marked difference between the single polaron optical absorption and the many - polaron result is manifest . the experimental _ d_-band can be clearly identified , rising in intensity at about 600 @xmath1193 , peaking around 1000 @xmath1193 , and then decreasing in intensity above that frequency . at a density of @xmath1197 @xmath367 , we found a remarkable agreement between our theoretical predictions and the experimental curve . in refs . @xcite , the experimental results on the optical spectroscopy of la@xmath1198sr@xmath1199mno@xmath68 ( lsmo ) and la@xmath1198ca@xmath1199mno@xmath68 ( lcmo ) thin films in the mid - infrared frequency region are presented . the optical conductivity spectra of lcmo films are interpreted in @xcite in terms of the optical response of small polarons , while the optical conductivity spectra of lsmo films are explained using the large - polaron picture ( see fig . [ p2fig2 ] ) . [ h ] part2fig2.eps the real part of the optical conductivity @xmath1200 is expressed in ref . @xcite by the formula @xmath1201 where @xmath63 is the electron - phonon coupling constant , @xmath1202 is the polaron density , @xmath15 is the electron band mass , @xmath1203 is the lo - phonon frequency , and @xmath1204 is the dynamic structure factor , determined in ref . @xcite through the dielectric function of an electron gas @xmath1205 : @xmath1206 . \label{s1}\ ] ] in ref . @xcite , the other definition for the dynamic structure factor is used , which is equivalent to that given by ( [ s1 ] ) ( see ref . @xcite ) with the factor @xmath242 ( the number of electrons)@xmath1207 here , @xmath1208 denotes the ground state of the electron subsystem ( without the electron - phonon interaction ) , @xmath1209 is the fourier component of the electron density . in ref . @xcite , the dynamic structure factor is calculated within the random - phase approximation ( rpa ) taking into account the coulomb interaction between electrons with the fourier component of the coulomb potential@xmath1210 where @xmath21 is the high - frequency dielectric constant of the crystal . in refs . @xcite , the dynamic structure factor is calculated taking into account the local hubbard electron - electron interaction instead of the coulomb interaction . the local hubbard interaction is used in the small - polaron formalism and describes the potential energy of two electrons on one and the same site ( see , e. g. , refs . @xcite ) . in its simplest form the hubbard interaction is ( see eq . ( 1 ) of ref . @xcite)@xmath1211 where @xmath1024 is the coupling constant of the hubbard interaction , @xmath1212 is the electron occupation number for the @xmath1213-th site . in refs . @xcite , there are no details of the calculation using the interaction term ( [ hub ] ) . the following procedure can be supposed . the transition from the summation over the lattice sites to the integral over the crystal volume @xmath305 is performed taking into account the normalization condition@xmath1214 where the density @xmath1215 is to be determined through @xmath1216 . as far as the lattice cell volume @xmath1217 , the integral @xmath1218 can be written as the sum over the lattice sites:@xmath1219 where @xmath1220 are the vectors of the lattice . therefore , from the equality@xmath1221 we find that@xmath1222 the potential ( [ hub ] ) is then transformed from the sum over sites to the integral:@xmath1223 consequently , in the continuum approach the hubbard model is described by the @xmath310-like interparticle potential @xmath1224 . this development of the approach @xcite performed in refs . @xcite seems to be contradictory by the following reason . for a many - polaron system , both the electron - phonon and electron - electron interactions are provided by the electrostatic potentials . therefore , it would be consistent to consider them both within one and the same approach . namely , the coulomb electron - electron interaction with the potential ( [ c ] ) is relevant for large and small polarons with the frhlich electron - phonon interaction , while the hubbard electron - electron interaction is relevant for small holstein polarons . nevertheless , as recognized in ref . @xcite , this model reproduces the observed shape of the polaron peak quite convincingly and provides a better agreement with the experiment @xcite than the phenomenological approach @xcite and the one - polaron theory @xcite . we consider a system of @xmath242 electrons with mutual coulomb repulsion and interacting with the lattice vibrations following ref . the system is assumed to be confined by a parabolic potential characterized by the frequency parameter @xmath130 . the total number of electrons is represented as @xmath1225 where @xmath1226 is the number of electrons with the spin projection @xmath1227 . the electron 3d ( 2d ) coordinates are denoted by @xmath1228 with @xmath1229 the bulk phonons ( characterized by 3d wave vectors @xmath59 and frequencies @xmath1230 ) are described by the complex coordinates @xmath1231 which possess the property @xcite @xmath1232 the full set of the electron and phonon coordinates are denoted by @xmath1233 and @xmath1234 throughout the present treatment , the euclidean time variable @xmath1235 is used , where @xmath535 is the real time variable . in this representation the lagrangian of the system is @xmath1236 where @xmath1237 is the lagrangian of an electron with band mass @xmath55 in a quantum dot : @xmath1238 @xmath130 is the confinement frequency , @xmath1239 is the potential of a background charge ( supposed to be static and uniformly distributed with the charge density @xmath1240 in a sphere of a radius @xmath1241),@xmath1242 , \label{vb}\ ] ] where @xmath24 is the static dielectric constant of a crystal , @xmath1243 is the potential energy of the electron - electron coulomb repulsion in the medium with the high - frequency dielectric constant @xmath21 : @xmath1244 @xmath1245 is the lagrangian of free phonons : @xmath1246 further , @xmath1247 is the lagrangian of the electron - phonon interaction : @xmath1248 where @xmath128 is the fourier transform of the electron density operator:@xmath1249 @xmath1250 is the amplitude of the electron - phonon interaction . here , we consider electrons interacting with the long - wavelength longitudinal optical ( lo ) phonons with a dispersionless frequency @xmath1251 , for which the amplitude @xmath1250 is @xcite@xmath1252 where @xmath63 is the electron - phonon coupling constant and @xmath305 is the volume of the crystal . we consider a _ canonical _ ensemble , where the numbers @xmath1226 are fixed . the partition function @xmath1253 of the system can be expressed as a path integral over the electron and phonon coordinates : @xmath1254 } , \label{z}\ ] ] where @xmath1255 $ ] is the action functional : @xmath1256 = -\frac{1}{\hbar}\int_{0}^{\hbar\beta}l\left ( \mathbf{\dot{\bar{x}}},\dot{\bar{q}};\mathbf{\bar{x}},\bar{q}\right ) d\tau . \label{sn}\ ] ] the parameter @xmath1257 is inversely proportional to the temperature @xmath261 . in order to take the fermi - dirac statistics into account , the integral over the electron paths @xmath1258 in eq . ( [ z ] ) contains a sum over all permutations @xmath447 of the electrons with the same spin projection , and @xmath1259 denotes the parity of a permutation @xmath447 . the action functional ( [ sn ] ) is quadratic in the phonon coordinates @xmath1260 therefore , _ the path integral over the phonon variables _ in @xmath1261 _ can be calculated analytically _ following ref . let us describe this path integration in detail . first , we introduce the real phonon coordinates through the real and imaginary parts of the complex phonon coordinates @xmath1262 , @xmath1263 . according to the symmetry property ( [ spn ] ) , they obey the equalities@xmath1264@xmath1265{c}\sqrt{2}q_{\mathbf{k}}^{\prime},\;k_{x}\geq0,\\ \sqrt{2}q_{\mathbf{k}}^{\prime\prime},\;k_{x}<0 . \end{array } \right . \label{det1}\ ] ] in this representation , the sum over phonon coordinates @xmath1266 is transformed in the following way using the symmetry property ( [ sp1]):@xmath1267 \\ & = 2\sum_{\substack{\mathbf{k}\\\left ( k_{x}\geq0\right ) } } \left ( q_{\mathbf{k}}^{\prime}\right ) ^{2}+2\sum_{\substack{\mathbf{k}\\\left ( k_{x}<0\right ) } } \left ( q_{\mathbf{k}}^{\prime\prime}\right ) ^{2}\\ & = \sum_{\substack{\mathbf{k}\\\left ( k_{x}\geq0\right ) } } q_{\mathbf{k}}^{2}+\sum_{\substack{\mathbf{k}\\\left ( k_{x}<0\right ) } } q_{\mathbf{k}}^{2}=\sum_{\mathbf{k}}q_{\mathbf{k}}^{2}.\end{aligned}\ ] ] therefore , the phonon lagrangian ( [ lph ] ) with the real phonon coordinates is@xmath1268 the lagrangian of the electron - phonon interaction ( [ leph ] ) with the real phonon coordinates is transformed in the following way using ( [ sp1]):@xmath1269 let us introduce the real forces:@xmath1270{c}\frac{1}{\sqrt{2}}\left ( \frac{2\omega_{\mathbf{k}}}{\hbar}\right ) ^{1/2}\left ( v_{\mathbf{k}}\rho_{\mathbf{k}}+v_{-\mathbf{k}}\rho _ { -\mathbf{k}}\right ) , \;k_{x}\geq0,\\ \frac{1}{i\sqrt{2}}\left ( \frac{2\omega_{\mathbf{k}}}{\hbar}\right ) ^{1/2}\left ( v_{\mathbf{k}}\rho_{\mathbf{k}}-v_{-\mathbf{k}}\rho _ { -\mathbf{k}}\right ) , \;k_{x}>0 . \end{array } \right . \label{realforces}\ ] ] this gives us the lagrangian of the electron - phonon interaction in terms of the real forces and real phonon coordinates:@xmath1271 so , the sum of the lagrangians of phonons and of the electron - phonon interaction is expressed through ordinary real oscillator variables:@xmath1272 the path integration for each phonon mode with the coordinate @xmath1273 is performed independently as described in sec . 2 of ref . @xcite and gives the result@xmath1274 \\ & = \frac{1}{2\sinh\left ( \frac{\beta\hbar\omega_{\mathbf{k}}}{2}\right ) } \\ & \times\exp\left\ { \frac{1}{4\hbar}\int_{0}^{\hbar\beta}d\tau\int _ { 0}^{\hbar\beta}d\tau^{\prime}\frac{\cosh\left [ \omega_{\mathbf{k}}\left ( \left\vert \tau-\tau^{\prime}\right\vert -\hbar\beta/2\right ) \right ] } { \omega_{\mathbf{k}}\sinh\left ( \beta\hbar\omega_{\mathbf{k}}/2\right ) } \gamma_{\mathbf{k}}\left ( \tau\right ) \gamma_{\mathbf{k}}\left ( \tau^{\prime}\right ) , \right\}\end{aligned}\ ] ] where the exponential is the influence functional of a driven oscillator \{@xcite , eq . ( 3.43)}. therefore , the path integral over all phonon modes is@xmath1275 \\ & = \left ( \prod_{\mathbf{k}}\frac{1}{2\sinh\left ( \frac{\beta\hbar \omega_{\mathbf{k}}}{2}\right ) } \right ) \\ & \times\exp\left\ { \frac{1}{4\hbar}\int_{0}^{\hbar\beta}d\tau\int _ { 0}^{\hbar\beta}d\tau^{\prime}\sum_{\mathbf{k}}\frac{\cosh\left [ \omega_{\mathbf{k}}\left ( \left\vert \tau-\tau^{\prime}\right\vert -\hbar\beta/2\right ) \right ] } { \omega_{\mathbf{k}}\sinh\left ( \beta \hbar\omega_{\mathbf{k}}/2\right ) } \gamma_{\mathbf{k}}\left ( \tau\right ) \gamma_{\mathbf{k}}\left ( \tau^{\prime}\right ) .\right\}\end{aligned}\ ] ] here , the product @xmath1276 is the partition function of free phonons , and the exponential is the influence functional of the phonon subsystem on the electron subsystem . this influence functional results from the above described _ elimination of the phonon coordinates _ and is usually written down as @xmath1277 , where @xmath1278 is@xmath1279 } { \omega_{\mathbf{k}}\sinh\left ( \beta\hbar\omega_{\mathbf{k}}/2\right ) } \gamma_{\mathbf{k}}\left ( \tau\right ) \gamma_{\mathbf{k}}\left ( \tau^{\prime}\right ) . \label{phi}\ ] ] the sum over the phonon wave vectors @xmath59 can be simplified as follows:@xmath1280 } { \omega _ { \mathbf{k}}\sinh\left ( \beta\hbar\omega_{\mathbf{k}}/2\right ) } \gamma_{\mathbf{k}}\left ( \tau\right ) \gamma_{\mathbf{k}}\left ( \tau^{\prime}\right ) \\ & = \frac{1}{\hbar}\sum_{\substack{\mathbf{k}\\\left ( k_{x}\geq0\right ) } } \frac{\cosh\left [ \omega_{\mathbf{k}}\left ( \left\vert \tau-\tau^{\prime } \right\vert -\hbar\beta/2\right ) \right ] } { \sinh\left ( \beta\hbar \omega_{\mathbf{k}}/2\right ) } \\ & \times\left [ v_{\mathbf{k}}\rho_{\mathbf{k}}\left ( \tau\right ) + v_{-\mathbf{k}}\rho_{-\mathbf{k}}\left ( \tau\right ) \right ] \left [ v_{\mathbf{k}}\rho_{\mathbf{k}}\left ( \tau^{\prime}\right ) + v_{-\mathbf{k}}\rho_{-\mathbf{k}}\left ( \tau^{\prime}\right ) \right ] \\ & -\frac{1}{\hbar}\sum_{\substack{\mathbf{k}\\\left ( k_{x}<0\right ) } } \frac{\cosh\left [ \omega_{\mathbf{k}}\left ( \left\vert \tau-\tau^{\prime } \right\vert -\hbar\beta/2\right ) \right ] } { \sinh\left ( \beta\hbar \omega_{\mathbf{k}}/2\right ) } \\ & \times\left [ v_{\mathbf{k}}\rho_{\mathbf{k}}\left ( \tau\right ) -v_{-\mathbf{k}}\rho_{-\mathbf{k}}\left ( \tau\right ) \right ] \left [ v_{\mathbf{k}}\rho_{\mathbf{k}}\left ( \tau^{\prime}\right ) -v_{-\mathbf{k}}\rho_{-\mathbf{k}}\left ( \tau^{\prime}\right ) \right ] \\ & = \frac{2}{\hbar}\sum_{\substack{\mathbf{k}\\\left ( k_{x}\geq0\right ) } } \frac{\cosh\left [ \omega_{\mathbf{k}}\left ( \left\vert \tau-\tau^{\prime } \right\vert -\hbar\beta/2\right ) \right ] } { \sinh\left ( \beta\hbar \omega_{\mathbf{k}}/2\right ) } v_{\mathbf{k}}v_{-\mathbf{k}}\\ & \times\left [ \rho_{\mathbf{k}}\left ( \tau\right ) \rho_{-\mathbf{k}}\left ( \tau^{\prime}\right ) + \rho_{-\mathbf{k}}\left ( \tau\right ) \rho_{\mathbf{k}}\left ( \tau^{\prime}\right ) \right ] \\ & = \frac{2}{\hbar}\sum_{\mathbf{k}}\frac{\cosh\left [ \omega_{\mathbf{k}}\left ( \left\vert \tau-\tau^{\prime}\right\vert -\hbar\beta/2\right ) \right ] } { \sinh\left ( \beta\hbar\omega_{\mathbf{k}}/2\right ) } \left\vert v_{\mathbf{k}}\right\vert ^{2}\rho_{\mathbf{k}}\left ( \tau\right ) \rho_{-\mathbf{k}}\left ( \tau^{\prime}\right ) .\end{aligned}\ ] ] herefrom , we find that@xmath1281 } { \sinh\left ( \frac { \beta\hbar\omega_{\mathbf{k}}}{2}\right ) } \rho_{\mathbf{k}}\left ( \tau\right ) \rho_{-\mathbf{k}}\left ( \tau^{\prime}\right ) . \label{phi1}\ ] ] as a result , the partition function of the electron - phonon system ( [ z ] ) factorizes into a product @xmath1282 of the partition function of free phonons with a partition function @xmath1283 of interacting polarons , which is a path integral over the electron coordinates only:@xmath1284 } . \label{zp}\ ] ] the functional @xmath1285 = -\frac{1}{\hbar } \int_{0}^{\hbar\beta}\left [ l_{e}\left ( \mathbf{\dot{\bar{x}}}\left ( \tau\right ) , \mathbf{\bar{x}}\left ( \tau\right ) \right ) -v_{c}\left ( \mathbf{\bar{x}}\left ( \tau\right ) \right ) \right ] d\tau+\phi\left [ \mathbf{\bar{x}}\left ( \tau\right ) \right ] \label{sp}\ ] ] describes the phonon - induced retarded interaction between the electrons , including the retarded self - interaction of each electron . using ( [ le ] ) and ( [ vc ] ) we write down @xmath1286 $ ] explicitly:@xmath1287 & = \frac{1}{\hbar}\int_{0}^{\hbar\beta}\left [ \sum_{\sigma}\sum_{j=1}^{n_{\sigma}}\frac{m_{b}}{2}\left ( \mathbf{\dot{x}}_{j,\sigma}^{2}+\omega_{0}^{2}\mathbf{x}_{j,\sigma}^{2}\right ) \underset{\left ( j,\sigma\right ) \neq\left ( l,\sigma^{\prime}\right ) } { + \sum_{\sigma,\sigma^{\prime}}\sum_{j=1}^{n_{\sigma}}\sum_{l=1}^{n_{\sigma^{\prime}}}}\frac{e^{2}}{2\varepsilon_{\infty}\left\vert \mathbf{x}_{j,\sigma}-\mathbf{x}_{l,\sigma^{\prime}}\right\vert } \right ] d\tau\nonumber\\ & -\sum_{\mathbf{q}}\frac{\left\vert v_{\mathbf{q}}\right\vert ^{2}}{2\hbar^{2}}\int\limits_{0}^{\hbar\beta}d\tau\int\limits_{0}^{\hbar\beta}d\tau^{\prime}\frac{\cosh\left [ \omega_{\mathrm{lo}}\left ( \left\vert \tau-\tau^{\prime}\right\vert -\hbar\beta/2\right ) \right ] } { \sinh\left ( \beta\hbar\omega_{\mathrm{lo}}/2\right ) } \rho_{\mathbf{q}}\left ( \tau\right ) \rho_{-\mathbf{q}}\left ( \tau^{\prime}\right ) . \label{sp1}\ ] ] the free energy of a system of interacting polarons @xmath1288 is related to their partition function ( [ zp ] ) by the equation : @xmath1289 at present no method is known to calculate the non - gaussian path integral ( [ zp ] ) analytically . for _ distinguishable _ particles , the jensen - feynman variational principle @xcite provides a convenient approximation technique . it yields a lower bound to the partition function , and hence an upper bound to the free energy . it can be shown @xcite that the path - integral approach to the many - body problem for a fixed number of identical particles can be formulated as a feynman - kac functional on a state space for @xmath242 indistinguishable particles , by imposing an ordering on the configuration space and by the introduction of a set of boundary conditions at the boundaries of this state space . the resulting variational inequality for identical particles takes the same form as the jensen - feynman variational principle : @xmath1290 where @xmath814 is a model action with the corresponding free energy @xmath1291 . the angular brackets mean a weighted average over the paths@xmath1292 } } { \sum_{p}\frac{\left ( -1\right ) ^{\mathbf{\xi}_{p}}}{n_{1/2}!n_{-1/2}!}\int d\mathbf{\bar{x}}\int _ { \mathbf{\bar{x}}}^{p\mathbf{\bar{x}}}d\mathbf{\bar{x}}\left ( \tau\right ) e^{-s_{0}\left [ \mathbf{\bar{x}}\left ( \tau\right ) \right ] } } . \label{aver}\ ] ] in the zero - temperature limit , the polaron ground - state energy @xmath1293 obeys the inequality following from ( [ jf ] ) with ( [ fvar]):@xmath1294 with@xmath1295 we consider a model system consisting of @xmath242 electrons with coordinates @xmath1233 and @xmath1296 fictitious particles with coordinates @xmath1297 in a harmonic confinement potential with elastic interparticle interactions as studied in refs . the lagrangian of this model system takes the form@xmath1298 the frequencies @xmath1299 @xmath1300 @xmath1301 the mass of a fictitious particle @xmath1302 and the force constant @xmath832 are variational parameters . clearly , this lagrangian is symmetric with respect to electron permutations . performing the path integral over the coordinates of the fictitious particles in the same way as described above for phonons , the partition function @xmath1303 of the model system of interacting polarons becomes a path integral over the electron coordinates:@xmath1304 } , \label{z0}\ ] ] with the action functional @xmath1305 $ ] given by@xmath1306 & = \frac{1}{\hbar}\int_{0}^{\hbar\beta}\sum_{\sigma}\sum_{j=1}^{n_{\sigma}}\frac{m_{b}}{2}\left [ \mathbf{\dot{x}}_{j,\sigma}^{2}\left ( \tau\right ) + \omega ^{2}\mathbf{x}_{j,\sigma}^{2}\left ( \tau\right ) \right ] d\tau\nonumber\\ & -\frac{1}{\hbar}\int_{0}^{\hbar\beta}\sum_{\sigma,\sigma^{\prime}}\sum_{j=1}^{n_{\sigma}}\sum_{l=1}^{n_{\sigma^{\prime}}}\frac{m_{b}\omega^{2}}{4}\left [ \mathbf{x}_{j,\sigma}\left ( \tau\right ) -\mathbf{x}_{l,\sigma^{\prime}}\left ( \tau\right ) \right ] ^{2}d\tau\nonumber\\ & -\frac{k^{2}n^{2}n_{f}}{4m_{f}\hbar\omega_{f}}\int\limits_{0}^{\hbar\beta } d\tau\int\limits_{0}^{\hbar\beta}d\tau^{\prime}\frac{\cosh\left [ \omega _ { f}\left ( \left\vert \tau-\tau^{\prime}\right\vert -\hbar\beta/2\right ) \right ] } { \sinh\left ( \beta\hbar\omega_{f}/2\right ) } \mathbf{x}\left ( \tau\right ) \cdot\mathbf{x}\left ( \tau^{\prime}\right ) , \label{s0}\ ] ] where @xmath1307 is the center - of - mass coordinate of the electrons,@xmath1308 the partition function @xmath1309 [ eq . ( [ z0 ] ) ] for the model system of interacting polarons can be expressed in terms of the partition function @xmath1310 of the model system of interacting electrons and fictitious particles with the lagrangian @xmath1311 [ eq . ( [ lm ] ) ] as follows:@xmath1312 where @xmath1313 is the partition function of fictitious particles,@xmath1314 with the frequency@xmath1315 and d=3(2 ) for 3d(2d ) systems . the partition function @xmath1310 is the path integral for both the electrons and the fictitious particles:@xmath1316 } \label{zm}\ ] ] with the action functional @xmath1317 = -\frac{1}{\hbar}\int_{0}^{\hbar\beta}l_{m}\left ( \mathbf{\dot{\bar{x}}},\mathbf{\dot{\bar{y}}};\mathbf{\bar{x}},\mathbf{\bar { y}}\right ) d\tau , \label{sm}\ ] ] where the lagrangian is given by eq . ( [ lm ] ) . let us consider an auxiliary ghost subsystem with the lagrangian @xmath1318 with two frequencies @xmath264 and @xmath1319 where @xmath264 is given by@xmath1320 the partition function @xmath1321 of this subsystem @xmath1322 \right\ } , \label{zg1}\ ] ] with the action functional@xmath1323 = -\frac{1}{\hbar}\int\limits_{0}^{\hbar\beta}l_{g}\left ( \mathbf{\dot{x}}_{g},\mathbf{x}_{g},\mathbf{\dot{y}}_{g},\mathbf{y}_{g}\right ) \,d\tau\label{sg}\ ] ] is calculated in the standard way , because its lagrangian ( [ ghost ] ) has a simple oscillator form . consequently , the partition function @xmath1321 is@xmath1324 ^{d}}\frac{1}{\left [ 2\sinh\left ( \frac{\beta\hbar w_{f}}{2}\right ) \right ] ^{d}}. \label{zg2}\ ] ] the product @xmath1325 of the two partition functions @xmath1321 and @xmath1326 is a path integral in the state space of @xmath242 electrons , @xmath1296 fictitious particles and two ghost particles with the coordinate vectors @xmath1327 and @xmath1328 the lagrangian @xmath1329 of this system is a sum of @xmath1311 and @xmath1330@xmath1331 the ghost subsystem is introduced because the center - of - mass coordinates in @xmath1329 can be explicitly separated much more transparently than in @xmath1311 . this separation is realized by the linear transformation of coordinates,@xmath541{l}\mathbf{x}_{j,\sigma}=\mathbf{x}_{j,\sigma}^{\prime}+\mathbf{x}-\mathbf{x}_{g},\\ \mathbf{y}_{j\sigma}=\mathbf{y}_{j\sigma}^{\prime}+\mathbf{y}-\mathbf{y}_{g } , \end{array } \right . \label{trans}\ ] ] where @xmath1307 and @xmath1332 are the center - of - mass coordinate vectors of the electrons and of the fictitious particles , correspondingly:@xmath1333 before the transformation ( [ trans ] ) , the independent variables are @xmath1334 with the center - of - mass coordinates @xmath1307 and @xmath1332 determined by eq . ( [ xy ] ) . when applying the transformation ( [ trans ] ) to the centers of mass ( [ xy ] ) , we find that@xmath1335 as seen from eqs . ( [ t1 ] ) , ( [ t2 ] ) , after the transformation ( [ trans ] ) the independent variables are @xmath1336 while the coordinates @xmath1337 obey the equations@xmath1338 in order to find the explicit form of the lagrangian ( [ lmt ] ) after the transformation ( [ trans ] ) , we use the following relations for the quadratic sums of coordinates : @xmath1339 the substitution of eq . ( [ xy ] ) into eq . ( [ lmt ] ) then results in the following 3 terms:@xmath1340 where @xmath1341 and @xmath1342 are lagrangians of non - interacting identical oscillators with the frequencies @xmath264 and @xmath1319 respectively,@xmath1343 , \label{lwa}\\ l_{w_{f}}\left ( \mathbf{\dot{\bar{y}}}^{\prime},\mathbf{\bar{y}}^{\prime } \right ) & = -\frac{m_{f}}{2}\sum_{j=1}^{n_{f}}\left [ \left ( \mathbf{\dot{y}}_{j,\sigma}^{\prime}\right ) ^{2}+w_{f}^{2}\left ( \mathbf{y}_{j,\sigma}^{\prime}\right ) ^{2}\right ] . \label{lwb}\ ] ] the lagrangian @xmath1344 describes the combined motion of the centers - of - mass of the electrons and of the fictitious particles,@xmath1345 with@xmath1346 the lagrangian ( [ lc ] ) is reduced to a diagonal quadratic form in the coordinates and the velocities by a unitary transformation for two interacting oscillators using the following replacement of variables : @xmath1347 with the coefficients @xmath1348 ^{1/2},\quad a_{2}=\left [ \frac{1-\chi}{2}\right ] ^{1/2},\label{a12}\\ \chi & \equiv\frac{\tilde{\omega}^{2}-\tilde{\omega}_{f}^{2}}{\left [ \left ( \tilde{\omega}^{2}-\tilde{\omega}_{f}^{2}\right ) ^{2}+4\gamma^{2}\right ] ^{1/2}},\quad\gamma\equiv k\sqrt{\frac{nn_{f}}{m_{b}m_{f}}}. \label{ch}\ ] ] the eigenfrequencies of the center - of - mass subsystem are then given by the expression @xmath541{c}\omega_{1}=\sqrt{\frac{1}{2}\left [ \tilde{\omega}^{2}+\tilde{\omega}_{f}^{2}+\sqrt{\left ( \tilde{\omega}^{2}-\tilde{\omega}_{f}^{2}\right ) ^{2}+4\gamma^{2}}\right ] } , \\ \omega_{2}=\sqrt{\frac{1}{2}\left [ \tilde{\omega}^{2}+\tilde{\omega}_{f}^{2}-\sqrt{\left ( \tilde{\omega}^{2}-\tilde{\omega}_{f}^{2}\right ) ^{2}+4\gamma^{2}}\right ] } . \end{array } \right . \label{o12}\ ] ] as a result , four independent frequencies @xmath1349 @xmath1350 @xmath264 and @xmath1351 appear in the problem . three of them ( @xmath1349 @xmath1350 @xmath264 ) are the eigenfrequencies of the model system . @xmath1352 is the frequency of the relative motion of the center of mass of the electrons with respect to the center of mass of the fictitious particles ; @xmath1353 is the frequency related to the center of mass of the model system as a whole ; @xmath264 is the frequency of the relative motion of the electrons with respect to their center of mass . the parameter @xmath1351 is an analog of the second variational parameter @xmath264 of the one - polaron feynman model . further , the lagrangian ( [ lc ] ) takes the form @xmath1354 leading to the partition function corresponding to the combined motion of the centers - of - mass of the electrons and of the fictitious particles @xmath1355 ^{d}}\frac{1}{\left [ 2\sinh\left ( \frac{\beta\hbar\omega_{2}}{2}\right ) \right ] ^{d}}. \label{zc}\ ] ] taking into account eqs . ( [ zg2 ] ) and ( [ zc ] ) , we obtain finally the partition function of the model system for interacting polarons @xmath1356 ^{d}\mathbb{\tilde{z}}_{f}\left ( \left\ { n_{\sigma}\right\ } , w,\beta\right ) . \label{z02}\ ] ] here@xmath1357 is the partition function of @xmath1358 non - interacting fermions in a parabolic confinement potential with the frequency @xmath1359 the analytical expressions for the partition function of @xmath1226 spin - polarized fermions @xmath1360 were derived in ref . @xcite . in order to obtain an upper bound to the free energy @xmath1361 , we substitute the model action functional ( [ s0 ] ) into the right - hand side of the variational inequality ( [ jf ] ) and consider the limit @xmath1362:@xmath1363 \nonumber\\ & + \lim_{\beta\rightarrow\infty}\frac{k^{2}n^{2}n_{f}}{4m_{f}\beta\hbar \omega_{f}}\int\limits_{0}^{\hbar\beta}d\tau\int\limits_{0}^{\hbar\beta}d\tau^{\prime}\frac{\cosh\left [ \omega_{f}\left ( \left\vert \tau -\tau^{\prime}\right\vert -\hbar\beta/2\right ) \right ] } { \sinh\left ( \beta\hbar\omega_{f}/2\right ) } \left\langle \mathbf{x}\left ( \tau\right ) \cdot\mathbf{x}\left ( \tau^{\prime}\right ) \right\rangle _ { s_{0}}\nonumber\\ & -\lim_{\beta\rightarrow\infty}\sum_{\mathbf{q}}\frac{\left\vert v_{\mathbf{q}}\right\vert ^{2}}{2\hbar^{2}\beta}\int\limits_{0}^{\hbar\beta } d\tau\int\limits_{0}^{\hbar\beta}d\tau^{\prime}\frac{\cosh\left [ \omega_{\mathrm{lo}}\left ( \left\vert \tau-\tau^{\prime}\right\vert -\hbar\beta/2\right ) \right ] } { \sinh\left ( \beta\hbar\omega_{\mathrm{lo}}/2\right ) } \mathcal{g}\left ( \mathbf{q},\tau-\tau^{\prime}|\left\ { n_{\sigma}\right\ } , \beta\right ) . \label{varfun}\ ] ] here , @xmath1364 is the energy of @xmath242 non - interacting fermions in a parabolic confinement potential with the confinement frequency @xmath264,@xmath1365 where @xmath1366 is the spin of an electron , @xmath1367 is the lower partly filled or empty level for @xmath1226 electrons with the spin projection @xmath1366 . the first term in the curly brackets of eq . ( [ 2n ] ) ( the upper line ) is the number of electrons at fully filled energy levels , while the second term ( square brackets ) is the number of electrons at the next upper level ( which can be empty or filled partially ) . the energy levels of a 3d oscillator are degenerate , so that @xmath1368 is the degeneracy of the @xmath275-th energy level . the parameter @xmath1369 is the number of electrons at all fully filled levels . the summation in eq . ( [ ew ] ) is performed explicitly , what gives us the result@xmath1370 .\ ] ] in eq . ( [ varfun ] ) , @xmath1371 is the two - point correlation function for the electron density operators:@xmath1372 the averages @xmath1373 are calculated using the generating function method : @xmath1374 \right\rangle _ { s_{0}}\right\vert _ { \substack{\mathbf{\xi}=0,\\\mathbf{\eta}=0 } } , \label{b2}\]]@xmath1375 } { \omega_{i}\sinh\left ( \hbar\beta\omega_{i}/2\right ) } . \label{xtxs}\ ] ] substituting this expression into eq . ( [ varfun ] ) and performing integrations over @xmath536 and @xmath1366 analytically , we obtain the result@xmath1376 } { \sinh\left ( \beta\hbar\omega_{f}/2\right ) } \left\langle \mathbf{x}\left ( \tau\right ) \cdot\mathbf{x}\left ( \tau^{\prime}\right ) \right\rangle _ { s_{0}}\nonumber\\ & = \frac{3\hbar\gamma}{4}\sum_{i=1}^{2}\frac{a_{i}^{2}}{\omega_{f}^{2}-\omega_{i}^{2}}\left [ \frac{\coth\left ( \beta\omega_{i}/2\right ) } { \omega_{i}}-\frac{\coth\left ( \beta\omega_{f}/2\right ) } { \omega_{f}}\right ] , \end{aligned}\ ] ] and in the zero - temperature limit we have@xmath1377 } { \sinh\left ( \beta\hbar\omega_{f}/2\right ) } \left\langle \mathbf{x}\left ( \tau\right ) \cdot\mathbf{x}\left ( \tau^{\prime}\right ) \right\rangle _ { s_{0}}\nonumber\\ & = \frac{3\hbar\gamma}{4}\sum_{i=1}^{2}\frac{a_{i}^{2}}{\omega_{f}^{2}-\omega_{i}^{2}}\left ( \frac{1}{\omega_{i}}-\frac{1}{\omega_{f}}\right ) .\end{aligned}\ ] ] the average @xmath1378 is transformed , using the described above operations with the ghost subsystem , @xmath1379 and taking into account the first of equations ( [ tosum ] ) @xmath1380 consequently , averaging the left - hand side of eq . ( [ b5 ] ) on the model action functional @xmath1381 one obtains @xmath1382 the term @xmath1383 is expressed using the virial theorem through the ground - state energy @xmath1364 of @xmath242 independent 3d fermion oscillators with the frequency @xmath264 and with the mass @xmath55 , @xmath1384 two other terms in eq . ( [ trr ] ) are [ cf . ( [ xtxs ] ) ] : @xmath1385 so , we obtain @xmath1386 the averaging of the background - charge potential gives us the result@xmath1387 \right\ } , \\ \eta & \equiv\varepsilon_{\infty}/\varepsilon_{0},\;\;a\equiv\frac{\hbar } { 4m_{b}n}\left ( \sum\limits_{i=1}^{2}\frac{a_{i}^{2}}{\omega_{i}}+\frac { n-1}{w}\right ) , \nonumber\end{aligned}\ ] ] where @xmath1388 is the one - particle distribution function of fermions ( the distribution functions are considered in more details in the next subsection ) . collecting all terms together , we arrive at the variational functional@xmath1389 + \frac{3}{2}\left ( \omega_{1}+\omega_{2}-\omega_{f}\right ) \right . \nonumber\\ + \left . \frac{3}{4}\left ( \omega_{0}^{2}-\omega_{1}^{2}-\omega_{2}^{2}+\omega_{f}^{2}\right ) \sum_{i=1}^{2}\frac{a_{i}^{2}}{\omega_{i}}+\frac{3\gamma^{2}}{4\omega_{f}}\sum_{i=1}^{2}\frac{a_{i}^{2}}{\omega _ { i}\left ( \omega_{i}+\omega_{f}\right ) } \right\ } \nonumber\\ + \left\langle u_{b}\left ( \mathbf{\bar{x}}\right ) \right\rangle _ { s_{0}}+e_{c}+e_{e - ph } , \label{evar}\ ] ] where @xmath1390 and @xmath1391 are the coulomb and polaron contributions , respectively : @xmath1392 , \label{ec}\]]@xmath1393 the correlation function ( [ ciuchif ] ) is calculated analytically in the next subsection . with this correlation function , the variational ground - state energy is calculated and minimized numerically . the two - point correlation function ( [ ciuchif ] ) is represented as the following path integral:@xmath1394 } \rho_{\mathbf{q}}\left ( \tau\right ) \rho_{-\mathbf{q}}\left ( 0\right ) . \label{g1}\ ] ] we observe that @xmath1395 can be rewritten as an average within the model action @xmath1396 $ ] of interacting electrons and fictitious particles:@xmath1397 } \nonumber\\ & \times\rho_{\mathbf{q}}\left ( \tau\right ) \rho_{-\mathbf{q}}\left ( 0\right ) . \label{g2}\ ] ] indeed , one readily derives that the elimination of the fictitious particles in ( [ g2 ] ) leads to ( [ g1 ] ) . the representation ( [ g2 ] ) allows one to calculate the correlation function @xmath1398 in a much simpler way than through eq . ( [ g1 ] ) , using the separation of the coordinates of the centers of mass of the electrons and of the fictitious particles . this separation is performed for the two - point correlation function ( [ g2 ] ) by the same method as it has been done for the partition function ( [ zm ] ) . as a result , one obtains@xmath1399 \right\rangle _ { s_{c}}}{\left\langle \exp\left [ i\mathbf{q\cdot}\left ( \mathbf{x}_{g}\left ( \tau\right ) -\mathbf{x}_{g}\left ( \sigma\right ) \right ) \right ] \right\rangle _ { s_{g } } } , \label{fact2}\ ] ] where @xmath1400 is the time - dependent correlation function of @xmath242 non - interacting electrons in a parabolic confinement potential with the frequency @xmath264 , @xmath1401 the action functional @xmath1402 $ ] is related to the lagrangian @xmath1403 [ eq . ( [ lwa])]@xmath1404 = \frac{1}{\hbar}\int \limits_{0}^{\hbar\beta}l_{w}\left ( \mathbf{\dot{\bar{x}}},\mathbf{\bar{x}}\right ) \,d\tau . \label{sw}\ ] ] the averages in ( [ fact2 ] ) are calculated using feynman s method of generating functions @xcite . namely , according to @xcite , the average @xmath1405 \equiv\left\langle \exp\left\ { \frac{i}{\hbar}\int\limits_{0}^{\beta}f\left ( \tau\right ) x_{\tau}\,d\tau\right\ } \right\rangle _ { s_{\omega } } , \label{gosc}\ ] ] where @xmath1406 is the action functional of a one - dimensional harmonic oscillator with the frequency @xmath856 and with the mass @xmath15 , results in @xmath1405 = \exp\left\ { -\frac{1}{4m\hbar\omega } \int\limits_{0}^{\beta}d\tau\int\limits_{0}^{\beta}d\sigma\frac{\cosh\left [ \omega\left ( \left\vert \tau-\sigma\right\vert -\beta/2\right ) \right ] } { \sinh\left ( \beta\omega/2\right ) } f\left ( \tau\right ) f\left ( \sigma\right ) \right\ } . \label{gosc1}\ ] ] the diagonalization procedure for the lagrangian @xmath1407 ( [ lc ] ) allows us to represent that lagrangian as a sum of lagrangians of independent harmonic oscillators , what gives the following explicit expressions for averages in eq . ( [ fact2]):@xmath1408 \right\rangle _ { s_{c}}\\ & = \exp\left\ { -\frac{\hbar q^{2}}{nm_{b}}\left [ \sum_{i=1}^{2}a_{i}^{2}\frac{\sinh\left ( \frac{\omega_{i}\left\vert \tau-\sigma\right\vert } { 2}\right ) \sinh\left ( \frac{\omega_{i}\left ( \hbar\beta-\left\vert \tau-\sigma\right\vert \right ) } { 2}\right ) } { \omega_{i}\sinh\left ( \frac{\beta\hbar\omega_{i}}{2}\right ) } \right ] \right\ } , \\ & \left\langle \exp\left [ i\mathbf{q\cdot}\left ( \mathbf{x}_{g}\left ( \tau\right ) -\mathbf{x}_{g}\left ( \sigma\right ) \right ) \right ] \right\rangle _ { s_{g}}\\ & = \exp\left [ -\frac{\hbar q^{2}}{nm_{b}}\frac{\sinh\left ( \frac { w\left\vert \tau-\sigma\right\vert } { 2}\right ) \sinh\left ( \frac{w\left ( \hbar\beta-\left\vert \tau-\sigma\right\vert \right ) } { 2}\right ) } { w\sinh\left ( \frac{\beta\hbar w}{2}\right ) } \right ] .\end{aligned}\ ] ] as seen from the formula ( [ gqtild ] ) , @xmath1409 is the time - dependent correlation function of @xmath242 non - interacting fermions in a parabolic confinement potential with the frequency @xmath264 . let us consider first of all a system of @xmath242 identical spin - polarized oscillators with the lagrangian @xmath1410 the corresponding hamiltonian is @xmath1411 @xmath1412 a set of eigenfunctions of the one - particle hamiltonian @xmath1413 is determined as follows : @xmath1414 where @xmath1415 @xmath1416 is the @xmath275-th eigenfunction of a one - dimensional oscillator with the frequency @xmath1417 the hamiltonian ( [ 2n ] ) can be written down in terms of the annihilation @xmath1418 and creation @xmath1419 operators : @xmath1420 the many - particle quantum states in the representation of `` occupation numbers '' are written down as @xmath1421 where @xmath1422 is the number of particles in the @xmath1423-th one - particle quantum state . the states @xmath1424 are defined as the eigenstates of the operator of the number of particles in the @xmath1423-th state @xmath1425 : @xmath1426 let us determine a set of quantum states with a _ finite _ total number of particles @xmath1427 as follows : @xmath1428 further on , we use the basis set of quantum states ( [ 8 ] ) for the derivation of the partition function , of the density function and of the two - point correlation function . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ * partition function * _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the density matrix of the _ canonical _ hibbs ensemble is @xmath1429 the partition function of this ensemble is the trace of the density matrix on the set of quantum states ( [ 8 ] ) : @xmath1430 _ { \sum_{\mathbf{n}}n_{\mathbf{n}}=n}. \label{9}\ ] ] this expression can be written down also in the form @xmath1431 where @xmath1432{c}1,\quad j = k\\ 0,\quad j\neq k \end{array } \right.\ ] ] is the delta symbol . let us introduce the generating function for the partition function in the same way as in ref . @xcite : @xmath1433 @xmath1434 ^{n_{\mathbf{n}}}\right\ } . \label{gf1}\ ] ] _ _ _ _ _ _ _ _ _ _ * fermions * _ _ _ _ _ _ _ _ _ _ for fermions , the number @xmath1422 can take only values @xmath1435 and @xmath1436 hence , for fermions ( denoted by the index @xmath1437 ) , we obtain : @xmath1438 .\ ] ] since the @xmath275-th level of a 3d oscillator is degenerate with the degeneracy @xmath1439 we find that the generating function @xmath1440 is given by@xmath1441 ^{g\left ( n\right ) } . \label{equiv1}\ ] ] _ _ _ _ _ _ _ _ * bosons * _ _ _ _ _ _ _ _ for bosons ( denoted by the index @xmath392 ) , @xmath1442 the summations over @xmath1443 in eq . ( [ gf1 ] ) gives : @xmath1444 ^{g\left ( n\right ) } . \label{gf1a}\ ] ] the results ( [ gf1 ] ) and ( [ gf1a ] ) prove ( for the partition function ) the equivalence of the path - integral approach for identical particles @xcite and of the second - quantization method . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ * integral representation * _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ let us use the fourier representation for the delta symbol : @xmath1445 d\theta , \label{delta}\ ] ] where @xmath284 is an arbitrary constant . substituting eq . ( [ delta ] ) into eq . ( [ 10 ] ) we obtain @xmath1446 d\theta\\ & = \frac{1}{2\pi}\int\limits_{0}^{2\pi}d\theta\exp\left [ -in\left ( \theta - i\zeta\right ) \right ] \sum_{\left\ { n_{\mathbf{n}}\right\ } } \exp\left ( -\beta\sum_{\mathbf{n}}\varepsilon_{\mathbf{n}}n_{\mathbf{n}}+i\sum_{\mathbf{n}}n_{\mathbf{n}}\left ( \theta - i\zeta\right ) \right ) \\ & = \frac{1}{2\pi}\int\limits_{0}^{2\pi}d\theta\exp\left ( -in\theta -n\zeta\right ) \xi\left ( \beta , e^{i\theta+\zeta}\right ) \quad \longrightarrow\end{aligned}\]]@xmath1447 . \label{qq}\ ] ] the partition function for a finite number of particles can be obtained from the generation function also by the inversion formula @xcite @xmath1448 } e^{-in\theta}d\theta . \label{211}\ ] ] let us denote in eq . ( [ qq ] ) : @xmath1449 in these notations , eqs . ( [ qq ] ) and ( [ 211 ] ) are _ identical_. for the numerical calculation , it is more convenient to choose in eq . ( [ delta ] ) the interval of the integration over @xmath1450 as @xmath1451 $ ] instead of @xmath1452 , $ ] what gives : @xmath1453 with the function @xmath1454 . \label{fin}\ ] ] the aforesaid method of derivation of the partition function [ eqs . ( [ delta ] ) to ( [ qq ] ) ] is heuristically useful , because it allows a simple generalization to spin - mixed systems with various polarization distributions . the two - point density - density correlation function in the operator formalism is@xmath1455 where @xmath1456 is the density operator in the heisenberg representation : @xmath1457 in the second - quantization representation , @xmath1456 is @xmath1458 . \label{rhoqt}\ ] ] after substituting eq . ( [ rhoqt ] ) into ( [ g ] ) , we find that @xmath1459 \left\langle \hat{a}_{\mathbf{n}}^{+}\hat { a}_{\mathbf{n}^{\prime}}\hat{a}_{\mathbf{m}}^{+}\hat{a}_{\mathbf{m}^{\prime}}\right\rangle . \label{gt1}\ ] ] the operator @xmath1460 has non - zero diagonal matrix elements in the basis of quantum states @xmath1461 only in the cases @xmath541{c}\mathbf{n}=\mathbf{n}^{\prime}\\ \mathbf{m}=\mathbf{m}^{\prime}\end{array } \right . \quad\mathrm{or}\quad\left\ { \begin{array } [ c]{c}\mathbf{n}=\mathbf{m}^{\prime}\\ \mathbf{m}=\mathbf{n}^{\prime}\end{array } \right . . \label{cond}\ ] ] hence , the average @xmath1462 is not equal to zero only when the condition ( [ cond ] ) is fulfilled . this allows us to write down the average ( [ av2 ] ) as @xmath1463 here , the notation is used for the average occupation number @xmath1464 : @xmath1465@xmath1466 _ { \sum_{\mathbf{n}}n_{\mathbf{n}}=n}.\end{aligned}\ ] ] in the same way as eq . ( [ 9 ] ) , the average ( [ nav ] ) can be written down in the form @xmath1467 _ { \sum _ { \mathbf{n}^{\prime}}n_{\mathbf{n}^{\prime}}=n}.\ ] ] @xmath1468@xmath1469 since @xmath1470 @xmath1471 depends only on @xmath275 . ( [ zu ] ) , we can write @xmath1472 as @xmath1473 } { \frac{1}{u}\exp\left ( \beta\varepsilon_{n}-i\theta\right ) + 1}d\theta\nonumber\\ & = \frac{1}{2\pi\mathbb{z}_{i}\left ( \beta|n\right ) } \int\limits_{-\pi } ^{\pi}\frac{\xi\left ( \beta , ue^{i\theta}\right ) } { u^{n-1}}\frac{\exp\left [ -i\theta\left ( n-1\right ) -\beta\varepsilon_{n}\right ] } { 1+u\exp\left ( i\theta-\beta\varepsilon_{n}\right ) } d\theta . \label{co}\ ] ] the averages @xmath1474 for @xmath1475 can be also expressed in terms of the integral representation : @xmath1476 _ { \sum_{\mathbf{n}^{\prime}}n_{\mathbf{n}^{\prime}}=n,\quad\mathbf{m}\neq\mathbf{n}}\\ & = \frac{1}{\mathbb{z}_{i}\left ( \beta|n\right ) \beta^{2}}\frac{\delta ^{2}\mathbb{z}_{i}\left ( \beta|n\right ) } { \delta\varepsilon_{\mathbf{m}}\delta\varepsilon_{\mathbf{n}}}\\ & = \frac{1}{\mathbb{z}_{i}\left ( \beta|n\right ) \beta^{2}}\frac{\delta^{2}}{\delta\varepsilon_{\mathbf{m}}\delta\varepsilon_{\mathbf{n}}}\left ( \frac{1}{2\pi}\int\limits_{0}^{2\pi}d\theta\exp\left [ \ln\xi\left ( \beta , e^{i\theta+\zeta}\right ) -n\zeta - in\theta\right ] \right ) \quad\rightarrow\end{aligned}\ ] ] we obtain the integral representation for the average of the product of operators @xmath1477 for @xmath1478 : @xmath1479 \left [ \exp\left ( \beta\varepsilon_{\mathbf{m}}-\zeta -i\theta\right ) + 1\right ] } d\theta . \label{num2}\ ] ] let us introduce the notation @xmath1480 which formally coincides with the fermi distribution function of the energy @xmath1481 with the chemical potential @xmath1482 using this notation , the averages ( [ num1 ] ) and ( [ num2 ] ) can be written down in the form @xmath1483 we can develop the aforesaid procedure for the average of a product of any number of operators @xmath1484 where all quantum numbers @xmath1485 are _ different_. the result is : @xmath1486 it should be emphasized , that all expressions above [ including eq . ( [ gen ] ) ] are derived for a _ canonical _ hibbs ensemble ( i. e. , for a fixed number of particles ) and for both closed - shell and open - shell systems . let us substitute the average ( [ av3 ] ) into the correlation function @xmath1487 : @xmath1488 \\ & \times\left ( \delta_{\mathbf{n}^{\prime}\mathbf{n}}\delta_{\mathbf{m}^{\prime}\mathbf{m}}\left\langle \hat{n}_{\mathbf{n}}\hat{n}_{\mathbf{m}}\right\rangle + \delta_{\mathbf{m}^{\prime}\mathbf{n}}\delta_{\mathbf{n}^{\prime}\mathbf{m}}\left\langle \hat{n}_{\mathbf{n}}\left ( 1-\hat { n}_{\mathbf{m}}\right ) \right\rangle \right)\end{aligned}\ ] ] @xmath1489@xmath1490 \left\langle \hat{n}_{\mathbf{n}}\left ( 1-\hat{n}_{\mathbf{m}}\right ) \right\rangle . \label{gq}\ ] ] the matrix elements @xmath1491 has the following form @xmath1492 where @xmath1493 is the matrix element of a one - dimensional oscillator with the frequency @xmath264 : @xmath1494{l}n_{>}\equiv\max\left ( n , m\right ) ; \\ n_{<}\equiv\min\left ( n , m\right ) , \end{array } \right . \label{mel1}\ ] ] @xmath1495 , @xmath1496 are the quantum states of the one - dimensional oscillator with the frequency @xmath264 , @xmath1497 is the generalized laguerre polynomial . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ * system with mixed spins * _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the correlation functions for a system with mixed spins can be explicitly derived by the generalization of eqs . ( [ zi5 ] ) and ( [ gen ] ) to the case of the particles with the non - zero spin . we use the fact , that the derivation of eqs . ( [ zi5 ] ) and ( [ gen ] ) , performed in this section , does not depend on the concrete form of the energy spectrum @xmath1498 hence , in the formulae , derived above , the replacement should be made : @xmath1499 where @xmath1366 is the electron spin projection . consequently , the matrix elements @xmath1500 are replaced by @xmath1501 taking into account eqs . ( [ repl ] ) and ( [ repl1 ] ) , the two - point correlation function ( [ gq ] ) becomes @xmath1502 \left\langle \hat{n}_{\mathbf{n},\sigma}\left ( 1-\hat{n}_{\mathbf{m},\sigma } \right ) \right\rangle . \label{gq1}\ ] ] the averages @xmath1503 and @xmath1504 are , respectively , one - particle and two - particle distribution functions,@xmath1505 the one - electron distribution function @xmath1506 is the average number of electrons with the spin projection @xmath1366 at the @xmath275-th energy level , while the two - electron distribution function @xmath1507 is the average product of the numbers of electrons with the spin projections @xmath1366 and @xmath1508 at the levels @xmath275 and @xmath1509 these functions are expressed through the following integrals [ see ( [ num1a ] ) , ( [ num2a])]:@xmath1510{c}\frac{1}{2\pi\mathbb{z}_{f}\left ( n_{\sigma},w,\beta\right ) } \int \limits_{-\pi}^{\pi}f\left ( \varepsilon_{n},\theta\right ) f\left ( \varepsilon_{n\mathbf{^{\prime}}},\theta\right ) \phi\left ( \theta , \beta , n_{\sigma}\right ) d\theta,\;\text{if } \sigma^{\prime}=\sigma;\\ f_{1}\left ( n,\sigma|n_{\sigma},\beta\right ) f_{1}\left ( n,\sigma^{\prime } |n_{\sigma^{\prime}},\beta\right ) , \;\text{if } \sigma^{\prime}\neq\sigma \end{array } \right . \label{num2a-1}\ ] ] with the notations@xmath1511 , \label{finn}\]]@xmath1512 the function @xmath1513 formally coincides with the fermi - dirac distribution function of the energy @xmath1481 with the chemical potential @xmath1514 here we consider the zero - temperature limit , for which the integrals ( [ num1a-1 ] ) and ( [ num2a-1 ] ) can be calculated analytically . the result for the one - electron distribution function is@xmath1515{cc}1 , & n < l_{\sigma};\\ 0 , & n > l_{\sigma};\\ \frac{n_{\sigma}-n_{l_{\sigma}}}{g_{l_{\sigma } } } , & n = l_{\sigma}. \end{array } \right . \label{p1}\ ] ] according to ( [ p1 ] ) , @xmath1367 is the number of the lowest open shell , and @xmath1516{ccc}\frac{1}{2}\left ( n+1\right ) \left ( n+2\right ) & & \left ( 3d\right ) , \\ n+1 & & \left ( 2d\right ) . \end{array } \right.\ ] ] is the degeneracy of the @xmath275-th shell . @xmath1517 is the number of electrons in all the closed shells with the spin projection @xmath1518@xmath1519{ccc}\frac{1}{6}l_{\sigma}\left ( l_{\sigma}+1\right ) \left ( l_{\sigma}+2\right ) & & \left ( 3d\right ) , \\ \frac{1}{2}l_{\sigma}\left ( l_{\sigma}+1\right ) & & \left ( 2d\right ) . \end{array } \right . \label{nls}\ ] ] the two - electron distribution function @xmath1520 at @xmath377 takes the form@xmath1521{cc}\left . f_{1}\left ( n,\sigma|\beta , n_{\sigma}\right ) \right\vert _ { \beta\rightarrow\infty}\left . f_{1}\left ( n^{\prime},\sigma^{\prime}|\beta , n_{\sigma^{\prime}}\right ) \right\vert _ { \beta\rightarrow\infty } , & n\neq n^{\prime}\;\mathrm{or}\;\sigma\neq\sigma^{\prime}\\ 1 , & \sigma=\sigma^{\prime}\;\mathrm{and}\;n = n^{\prime}<l_{\sigma};\\ 0 , & \sigma=\sigma^{\prime}\;\mathrm{and}\;n = n^{\prime}>l_{\sigma};\\ \frac{n_{\sigma}-n_{l_{\sigma}}}{g_{l_{\sigma}}}\frac{n_{\sigma}-n_{l_{\sigma } } -1}{g_{l_{\sigma}}-1 } , & \sigma=\sigma^{\prime}\;\mathrm{and}\;n = n^{\prime } = l_{\sigma}. \end{array } \right . \label{p2}\ ] ] in summary , we have obtained the following expression for @xmath1522:@xmath1523 \nonumber\\ & \times\left [ f_{1}\left ( n,\sigma|\left\ { n_{\sigma}\right\ } , \beta\right ) -f_{2}\left ( n,\sigma;n^{\prime},\sigma|\left\ { n_{\sigma } \right\ } , \beta\right ) \right ] . \label{g4}\ ] ] this formula is valid for both closed and open shells . the correlation functions derived in this subsection are used both for the calculation of the ground - state energy and , a shown below , for the calculation of the optical conductivity of an @xmath242-polaron system in a quantum dot . the correlation function given by eq . ( [ g4 ] ) can be subdivided as@xmath1524 with@xmath1525 \nonumber\\ & \times\left [ f_{1}\left ( n,\sigma|\left\ { n_{\sigma}\right\ } , \beta\right ) -f_{2}\left ( n,\sigma;n^{\prime},\sigma|\left\ { n_{\sigma } \right\ } , \beta\right ) \right ] , \end{aligned}\]]@xmath1526 in accordance with the subdivision ( [ subd ] ) of the correlation function , we subdivide the coulomb and polaron contributions:@xmath1527 we have numerically checked whether the polaron contribution per particle @xmath1528 tends to a finite value at @xmath1529 . in figs . [ polcon3 ] and [ polcon4 ] , we have plotted the polaron contributions @xmath1528 as a function of @xmath242 for a quantum dot in zno and in a polar medium with @xmath1530 , @xmath1531 , respectively . has a physical meaning , the plots for @xmath1532 in figs . 3 and 4 start from @xmath1533 . the total polaron contribution @xmath1391 for @xmath1534 is plotted below , in fig . 9 . ] as seen from fig . [ polcon3 ] , the polaron contribution @xmath1535 in zno as a function of @xmath242 oscillates taking expressed maxima for @xmath242 corresponding to the closed shells @xmath1536 . there exist kinks of @xmath1528 at @xmath242 corresponding to the half - filled shells , but these kinks are extremely small . in the case of the medium with @xmath1537 @xmath1538 for @xmath1539 ( what corresponds to the density @xmath1540 @xmath367 ) , the polaron contribution @xmath1528 oscillates taking maximal values at the numbers of fermions , which correspond to the closed shells for a spin - polarized system with parallel spins . in figs . [ polcon3 ] and [ polcon4 ] , the dashed curves are the envelopes for local maxima ( closed shells ) and local minima of @xmath1541we see that when these envelopes are _ extrapolated _ to larger number of fermions , the distance between the envelopes decreases . therefore , the magnitude of the variations of @xmath1542 related to the shell filling diminishes with increasing @xmath242 , and it is safe to suppose that in the limit of large @xmath1543 the envelopes tend to one and the same value . that value corresponds to the _ homogeneous _ ( bulk ) limit @xmath1544 . [ h ] part2fig3.eps [ h ] part2fig4.eps in fig . [ polcon ] , we compare the polaron contribution @xmath1535 calculated within our variational path - integral method for different numbers of fermions with the polaron contribution to the ground - state energy per particle for a polaron gas in bulk , calculated ( i ) in ref . @xcite within an intermediate - coupling approach ( the thin solid curve ) , ( ii ) in ref . @xcite , using a variational approach developed first in ref . @xcite . as seen from this figure , our all - coupling variational method provides lower values for the polaron contribution than those obtained in refs . the difference between the polaron contribution calculated within our method and that of ref . @xcite is smaller at low densities and increases in magnitude with increasing density . the difference between the polaron contribution calculated within our method and that of ref . @xcite very slightly depends on the density . the result of ref . @xcite becomes closer to our result only at high densities . [ h ] part2fig5.eps in ref . @xcite we have extended the memory - function approach to a system of arbitrary - coupling interacting polarons confined to a parabolic confinement potential . the optical conductivity relates the current @xmath1545 per electron to a time - dependent uniform electric field @xmath1546 in the framework of linear response theory . further on , the fourier components of the electric field are denoted by @xmath1547 @xmath1548 and the similar denotations are used for other time - dependent quantities . the electric current per electron @xmath1545 is related to the mean electron coordinate response @xmath1549 by @xmath1550 and hence@xmath1551 within the linear - response theory , both the electric current and the coordinate response are proportional to @xmath1552:@xmath1553 where @xmath1554 is the conductivity per electron . because we treat an isotropic electron - phonon system , @xmath1555 is a scalar function . it is determined from the time evolution of the center - of - mass coordinate : @xmath1556 the symbol @xmath1557 denotes an average in the _ real - time _ representation for a system with action functional @xmath80:@xmath1558 } \left ( \bullet\right ) \left . \left\langle \mathbf{\bar{x}}_{0}\left\vert \hat{\rho}\left ( t_{0}\right ) \right\vert \mathbf{\bar{x}}_{0}^{^{\prime}}\right\rangle \right\vert _ { t_{0}\rightarrow-\infty } , \label{average}\ ] ] where @xmath1559 is the density matrix before the onset of the electric field in the infinite past @xmath1560 . the corresponding action functional is@xmath1561 = \int\limits_{-\infty}^{t}\left [ l_{e}\left ( \mathbf{\dot{\bar{x}}}\left ( t\right ) , \mathbf{\bar{x}}\left ( t\right ) , t\right ) -l_{e}\left ( \mathbf{\dot{\bar{x}}}^{\prime}\left ( t\right ) , \mathbf{\bar{x}}^{\prime}\left ( t\right ) , t\right ) \right ] dt^{\prime } -i\hbar\phi\left [ \mathbf{\bar{x}}\left ( t\right ) , \mathbf{\bar{x}}^{\prime}\left ( t\right ) \right ] , \label{s1}\ ] ] where @xmath1562 is the lagrangian of @xmath242 interacting electrons in a time - dependent uniform electric field @xmath1546 @xmath1563 the influence phase of the phonons @xmath1564 & = -\sum_{\mathbf{q}}\frac{\left\vert v_{\mathbf{q}}\right\vert ^{2}}{\hbar^{2}}\int\limits_{-\infty}^{t}ds\int\limits_{-\infty}^{s}ds^{\prime}\left [ \rho_{\mathbf{q}}\left ( s\right ) -\rho_{\mathbf{q}}^{\prime}\left ( s\right ) \right ] \nonumber\\ & \times\left [ t_{\omega_{\mathbf{q}}}^{\ast}\left ( s - s^{\prime}\right ) \rho_{\mathbf{q}}\left ( s^{\prime}\right ) -t_{\omega_{\mathbf{q}}}\left ( s - s^{\prime}\right ) \rho_{\mathbf{q}}^{\prime}\left ( s^{\prime}\right ) \right]\end{aligned}\ ] ] describes both a retarded interaction between different electrons and a retarded self - interaction of each electron due to the elimination of the phonon coordinates . this functional contains the free - phonon green s function : @xmath1565 the equation of motion for @xmath1549 is@xmath1566 where @xmath1567 is the average force due to the electron - phonon interaction , @xmath1568 the two - point correlation function @xmath1569 should be calculated from eq . ( [ average ] ) using the exact action ( [ s1 ] ) , but like for the free energy above , this path integral can not be calculated analytically . instead , we perform an approximate calculation , replacing the two - point correlation function in eq . ( [ fph ] ) by @xmath1570 where @xmath1571 $ ] is the action functional with the optimal values of the variational parameters for the model system considered in the previous section in the presence of the electric field @xmath1546 . the functional @xmath1572 $ ] is quadratic and describes a system of coupled harmonic oscillators in the uniform electric field @xmath1573 . this field enters the term @xmath1574 in the lagrangian , which only affects the center - of - mass coordinate . hence , a shift of variables to the frame of reference with the origin at the center of mass @xmath541{l}\mathbf{x}_{n}\left ( t\right ) = \mathbf{\tilde{x}}_{n}\left ( t\right ) + \mathbf{r}\left ( t\right ) , \\ \mathbf{x}_{n}^{\prime}\left ( t\right ) = \mathbf{\tilde{x}}_{n}^{\prime } \left ( t\right ) + \mathbf{r}\left ( t\right ) , \end{array } \right . \label{tr1}\ ] ] results in@xmath1575 } . \label{tr2}\ ] ] this result ( [ tr2 ] ) is valid for any _ quadratic _ model action @xmath1576 the applicability of the parabolic approximation is confirmed by the fact that a self - induced polaronic potential , created by the polarization cloud around an electron , is rather well described by a parabolic potential whose parameters are determined by a variational method . for weak coupling , our variational method is at least of the same accuracy as the perturbation theory , which results from our approach at a special choice of the variational parameters . for strong coupling , an interplay of the electron - phonon interaction and the coulomb correlations within a confinement potential can lead to the assemblage of polarons in multi - polaron systems . our choice of the model variational system is reasonable because of this trend , apparently occurring in a many - polaron system with arbitrary @xmath242 for a finite confinement strength . the correlation function @xmath1577 corresponds to the model system in the absence of an electric field . for @xmath1578 this function is related to the imaginary - time correlation function @xmath1579 described in the previous section:@xmath1580 using the transformation ( [ tr1 ] ) and the relation ( [ tr3 ] ) , we obtain from eq . ( [ fph])@xmath1581 } \mathcal{g}\left ( \mathbf{q},i\left ( t - s\right ) |\left\ { n_{\sigma}\right\ } , \beta\right ) ds . \label{f1}\ ] ] within the linear - response theory , we expand the function @xmath1582 } $ ] in eq . ( [ f1 ] ) as a taylor series in @xmath1583 $ ] up to the first - order term . the zeroth - order term gives no contribution into @xmath1567 due to the symmetry of @xmath1584 and of @xmath1585 with respect to the inversion @xmath1586 . in this approach , the cartesian coordinates of the force @xmath1587 become @xmath1588 \nonumber\\ & \times\operatorname{im}\left [ t_{\omega_{\mathrm{lo}}}^{\ast}\left ( t - s\right ) \mathcal{g}\left ( \mathbf{q},i\left ( t - s\right ) |\left\ { n_{\sigma}\right\ } , \beta\right ) \right ] ds . \label{aa}\ ] ] further on , we perform the fourier expansion:@xmath1589 in eq . ( [ aa ] ) , we make the replacement@xmath1590 what gives@xmath1591 \operatorname{im}\left [ t_{\omega _ { \mathrm{lo}}}^{\ast}\left ( \tau\right ) \mathcal{g}\left ( \mathbf{q},it|\left\ { n_{\sigma}\right\ } , \beta\right ) \right ] \\ & = \sum_{k=1}^{3}\sum_{\mathbf{q}}\frac{2\left\vert v_{\mathbf{q}}\right\vert ^{2}q_{j}q_{k}}{n\hbar}\frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega r_{k}\left ( \omega\right ) e^{-i\omega t}\int\limits_{0}^{\infty}d\tau\;\left ( 1-e^{i\omega\tau}\right ) \operatorname{im}\left [ t_{\omega_{\mathrm{lo}}}^{\ast}\left ( \tau\right ) f_{\mathbf{q}}\left ( \tau\right ) \right ] \\ & = \frac{1}{2\pi}\int_{-\infty}^{\infty}d\omega f_{j}\left ( \omega\right ) e^{-i\omega t},\end{aligned}\ ] ] where the fourier component of the force is@xmath1592 r_{k}\left ( \omega\right ) . \label{bb}\ ] ] the expression ( [ bb ] ) can be written down as@xmath1593 where @xmath1594 are components of the tensor@xmath1595 . \label{t}\ ] ] in the abstract tensor form , eq . ( [ tt ] ) is@xmath1596 in particular , for the isotropic electron - phonon interaction and in the absence of the magnetic field , the tensor @xmath1597 is proportional to the unity tensor @xmath1598,@xmath1599 where @xmath1600 is the scalar memory function:@xmath1601 .\ ] ] let us perform the fourier transformation of the equation of motion ( [ eqmotion]):@xmath1602 with eq . ( [ tt2 ] ) , this equation takes the form@xmath1603@xmath542@xmath1604 \mathbf{r}\left ( \omega\right ) = e\mathbf{e}\left ( \omega\right ) . \label{eqm3}\ ] ] comparing eqs . ( [ j2 ] ) and ( [ eqm3 ] ) between each other , we find that@xmath1604 \frac{\sigma\left ( \omega\right ) } { ie\omega } \mathbf{e}\left ( \omega\right ) = e\mathbf{e}\left ( \omega\right ) , \ ] ] so that@xmath1605 ^{-1}.\ ] ] in the case when eq . ( [ scalar ] ) is valid , we obtain the conductivity in the scalar form@xmath1606 } .\ ] ] the real part of the conductivity is@xmath1607 } { m_{b}\left\ { \left [ \left ( \omega^{2}-\omega _ { 0}^{2}\right ) -\operatorname{re}\chi\left ( \omega\right ) \right ] ^{2}+\left [ \operatorname{im}\chi\left ( \omega\right ) \right ] ^{2}\right\ } } \\ & = -\frac{e^{2}\omega}{m_{b}}\frac{\operatorname{im}\chi\left ( \omega\right ) } { \left [ \left ( \omega^{2}-\omega_{0}^{2}\right ) -\operatorname{re}\chi\left ( \omega\right ) \right ] ^{2}+\left [ \operatorname{im}\chi\left ( \omega\right ) \right ] ^{2}}.\end{aligned}\ ] ] in summary , the optical conductivity can be expressed in terms of the memory function @xmath1600 ( cf . ^{2}+\left [ \operatorname{im}\chi\left ( \omega\right ) \right ] ^{2 } } , \label{kw}\ ] ] where @xmath1600 is given by @xmath1609 . \label{hi}\ ] ] it is worth noting that the optical conductivity ( [ kw ] ) differs from that for a translationally invariant polaron system both by the explicit form of @xmath1600 and by the presence of the term @xmath1610 in the denominator . for @xmath1611 the optical conductivity tends to a @xmath310-like peak at @xmath1612@xmath1613 for a translationally invariant system @xmath1614 , and this weak - coupling expression ( [ limit ] ) reproduces the central peak of the polaron optical conductivity @xcite . the further simplification of the memory function ( [ hi ] ) is performed in the following way . with the frhlich amplitudes of the electron - phonon interaction , we transform the summation over @xmath1007 to the integral and use the feynman units ( @xmath1133 , @xmath1134 , @xmath621 ) , in which @xmath1615 . we also use the fact that in an isotropic crystal , @xmath1616 . as a result , we find @xmath1617 \\ & = \frac{2\sqrt{2}\alpha}{3\pi n}\int_{0}^{\infty}q^{2}dq\int\limits_{0}^{\infty}dt\left ( e^{i\omega t}-1\right ) \operatorname{im}\left [ t_{\omega_{\mathrm{lo}}}^{\ast}\left ( t\right ) \mathcal{g}\left ( q , it|\left\ { n_{\sigma}\right\ } , \beta\right ) \right ] .\end{aligned}\ ] ] in the zero - temperature case , @xmath1618 and we arrive at the expression@xmath1619 . \label{hi1}\ ] ] substituting the two - point correlation function @xmath1620 with the one - electron ( [ p1 ] ) and the two - electron ( [ p2 ] ) distribution functions into the memory function ( [ hi1 ] ) and expanding @xmath1621 in powers of @xmath1622 , @xmath1623 and @xmath1624 , the integrations over @xmath966 and @xmath535 in eq . ( [ hi1 ] ) are performed analytically . the similar transformations are performed also in the 2d case . as a result , the memory function ( [ hi ] ) is represented in the unified form for 3d and 2d interacting polarons : @xmath1625 \right\vert _ { \beta\rightarrow\infty } \right . \right . \nonumber\end{aligned}\]]@xmath1626{c}\frac{1}{\omega-\omega_{\mathrm{lo}}-\left [ p_{1}\omega_{1}+p_{2}\omega _ { 2}+\left ( p_{3}-m+n\right ) w\right ] + \mathrm{i}\varepsilon}-\frac { 1}{\omega+\omega_{\mathrm{lo}}+p_{1}\omega_{1}+p_{2}\omega_{2}+\left ( p_{3}-m+n\right ) w+\mathrm{i}\varepsilon}\\ + \mathcal{p}\left ( \frac{2}{\omega_{\mathrm{lo}}+p_{1}\omega_{1}+p_{2}\omega_{2}+\left ( p_{3}-m+n\right ) w}\right ) \end{array } \right ) \nonumber\\ & \times\sum\limits_{l=0}^{m}\sum\limits_{k = n - m+l}^{n}\frac{\left ( -1\right ) ^{n - m+l+k}\gamma\left ( p_{1}+p_{2}+p_{3}+k+l+\frac{3}{2}\right ) } { k!l!}\left ( \frac{1}{wa}\right ) ^{l+k}\nonumber\\ & \left . \times\binom{n+d-1}{n - k}\binom{2k}{k - l - n+m}\right ] \nonumber\end{aligned}\]]@xmath1627{c}\frac{1}{\omega-\omega_{\mathrm{lo}}-\left ( p_{1}\omega_{1}+p_{2}\omega _ { 2}+p_{3}w\right ) + \mathrm{i}\varepsilon}-\frac{1}{\omega+\omega _ { \mathrm{lo}}+p_{1}\omega_{1}+p_{2}\omega_{2}+p_{3}w+\mathrm{i}\varepsilon}\\ + \mathcal{p}\left ( \frac{2}{\omega_{\mathrm{lo}}+p_{1}\omega_{1}+p_{2}\omega_{2}+p_{3}w}\right ) \end{array } \right ) \right . \nonumber\\ & \times\sum\limits_{m=0}^{\infty}\sum\limits_{n=0}^{\infty}\sum \limits_{\sigma,\sigma^{\prime}}\left . f_{2}\left ( n,\sigma;m,\sigma ^{\prime}|\left\ { n_{\sigma}\right\ } , \beta\right ) \right\vert _ { \beta\rightarrow\infty}\nonumber\\ & \times\sum\limits_{k=0}^{n}\sum\limits_{l=0}^{m}\frac{\left ( -1\right ) ^{k+l}\gamma\left ( p_{1}+p_{2}+p_{3}+k+l+\frac{3}{2}\right ) } { k!l!}\left ( \frac{1}{wa}\right ) ^{k+l}\nonumber\\ & \left . \times\binom{n+d-1}{n - k}\binom{m+d-1}{m - l}\right ] \right\ } , \label{mf}\ ] ] where @xmath1628 is the dimensionality of the space , @xmath1629 denotes the principal value , @xmath391 is defined as @xmath1630 /n$ ] , @xmath1631 and @xmath264 are the eigenfrequencies of the model system , @xmath1632 and @xmath1633 are the coefficients of the canonical transformation which diagonalizes the model lagrangian ( [ lm ] ) . the shell filling schemes for an @xmath242-polaron system in a quantum dot can manifest themselves in the spectra of the optical conductivity . in fig.[spectra ] , optical conductivity spectra for @xmath1634 polarons are presented for a quantum dot with the parameters of cdse : @xmath1635 @xmath1636 @xcite and with different values of the confinement energy @xmath1637 . the electron band mass @xmath55 and @xmath1638 have the numerical value of 1 . this means that the unit of length is the effective bohr radius @xmath1639 , while the unit of energy is the effective hartree @xmath1640 . ] in this case , the spin - polarized ground state changes to the ground state satisfying hund s rule with increasing @xmath1637 in the interval @xmath1641 . [ h ] part2fig6.eps in the inset to fig.[spectra ] , the first frequency moment of the optical conductivity @xmath1642 as a function of @xmath1637 shows a _ discontinuity _ , at the value of the confinement energy corresponding to the change of the shell filling schemes from the spin - polarized ground state to the ground state obeying hund s rule . this discontinuity might be observable in optical measurements . the shell structure for a system of interacting polarons in a quantum dot is clearly revealed when analyzing the addition energy and the first frequency moment of the optical conductivity in parallel . in fig [ moments ] , we show both the function @xmath1643 and the addition energy@xmath1644 for interacting polarons in a 3d cdse quantum dot . [ h ] part2fig7.eps as seen from fig [ moments ] , distinct peaks appear in @xmath1645 and @xmath1646 at the magic numbers corresponding to closed - shell configurations at @xmath1647 and to half - filled - shell configurations at @xmath1648 . we see that each of the peaks of @xmath1649 corresponds to a peak of the addition energy . the filling patterns for a many - polaron system in a quantum dot can be therefore determined from the analysis of the first moment of the optical absorption for different numbers of polarons . in the present section , the ground - state properties of a translation invariant @xmath242-polaron system are theoretically studied in the framework of the variational path - integral method for identical particles , using a further development @xcite of the model introduced in refs . @xcite . in order to describe a many - polaron system , we start from the translation invariant @xmath242-polaron hamiltonian @xmath1650 where @xmath15 is the band mass , @xmath19 is the electron charge , @xmath52 is the longitudinal optical ( lo ) phonon frequency , and @xmath61 are the amplitudes of the frhlich electron - lo - phonon interaction@xmath1651 with the electron - phonon coupling constant @xmath1652 the high - frequency dielectric constant @xmath1653 and the static dielectric constant @xmath1654 and consequently @xmath1655 in the expression ( [ units ] ) , @xmath1656 is the effective hartree @xmath1657 where @xmath1658 is the effective bohr radius . the partition function of the system can be expressed as a path integral over all electron and phonon coordinates . the path integral over the phonon variables can be calculated analytically @xcite . feynman s phonon elimination technique for this system is well known and leads to the partition function , which is a path integral over the electron coordinates only:@xmath1659 where @xmath1660 denotes the set of electron coordinates , and @xmath1661 denotes the path integral over all the electron coordinates , integrated over equal initial and final points , i.e. @xmath1662 throughout this paper , imaginary time variables are used . the effective action for the @xmath242-polaron system is retarded and given by@xmath1663 note that the electrons are fermions . therefore the path integral for the electrons with parallel spin has to be interpreted as the required antisymmetric projection of the propagators for distinguishable particles . we below use units in which @xmath1133 , @xmath1664 , and @xmath1134 . the units of distance and energy are thus the effective polaron radius @xmath1665 ^{1/2}$ ] and the lo - phonon energy @xmath60 . for distinguishable particles , it is well known that the jensen - feynman inequality @xcite provides a lower bound on the partition function @xmath1666 ( and consequently an upper bound on the free energy @xmath1437)@xmath1667@xmath1668 for a system with real action @xmath80 and a real trial action @xmath1576the many - body extension ( ref.@xcite ) of the jensen - feynman inequality , requires that the potentials are symmetric with respect to all particle permutations , and that the exact propagator as well as the model propagator are defined on the same state space . within this interpretation we consider the following generalization of feynman s trial action@xmath1669 with the variational frequency parameters @xmath1670 . using the explicit forms of the exact ( [ eq : spol ] ) and the trial ( [ eq : s0 ] ) actions , the variational inequality ( [ varineq ] ) takes the form@xmath1671 in the zero - temperature limit ( @xmath617 ) , we arrive at the following upper bound for the ground - state energy @xmath1672 of a translation invariant @xmath242-polaron system@xmath1673 with@xmath1674 where @xmath1675 is the energy of @xmath242 spin - polarized fermions confined to a parabolic potential with the confinement frequency @xmath856 , @xmath1676 is the coulomb energy of the electrons with parallel spins , @xmath1677 is the coulomb energy of the electrons with opposite spins , @xmath1678 is the electron - phonon energy of the electrons with parallel spins , and @xmath1679 is the electron - phonon energy of the electrons with opposite spins . here , we discuss some results of the numerical minimization of @xmath1680 with respect to the three variational parameters @xmath200 , @xmath264 , and @xmath856 . the frhlich constant @xmath63 and the coulomb parameter @xmath1681 characterize the strength of the electron - phonon and of the coulomb interaction , obeying the physical condition @xmath1682 [ see ( [ units ] ) ] . the optimal values of the variational parameters @xmath1683@xmath201and @xmath856 are denoted @xmath1684@xmath1685and @xmath1686 , respectively . the optimal value of the total spin was always determined by choosing the combination @xmath1687 for fixed @xmath1688which corresponds to the lowest value @xmath1689 of the variational functional@xmath1690 [ h ] part2fig8.eps in fig . [ trinv - pd ] , the phase diagrams analogous to the bipolaron phase diagram of ref . @xcite are plotted for an @xmath242-polaron system in bulk with @xmath1691 and @xmath1692 . the area where @xmath1693 is the non - physical region . for @xmath1694 , each sector between a curve corresponding to a well defined @xmath242 and the line indicating @xmath1695 shows the stability region where @xmath1696 , while the white area corresponds to the regime with @xmath1697 . when comparing the stability region for @xmath1533 from fig . [ trinv - pd ] with the bipolaron phase diagram of ref . @xcite , the stability region in the present work starts from the value @xmath1698 ( instead of @xmath1699 in ref . the width of the stability region within the present model is also larger than the width of the stability region within the model of ref . @xcite . also , the absolute values of the ground - state energy of a two - polaron system given by the present model are smaller than those given by the approach of ref . @xcite . the difference between the numerical results of the present work and of ref . @xcite is due to the following distinction between the used model systems . the model system of ref . @xcite consists of two electrons interacting with two fictitious particles and with each other through quadratic interactions . but the trial hamiltonian given by eq . ( 6 ) of ref . @xcite is not symmetric with respect to the permutation of the electrons . it is only symmetric under the permutation of the pairs electron + fictitious particle . as a consequence , this trial system is only applicable if the electrons are distinguishable , i.e. have opposite spin . in contrast to the model of ref . @xcite , the model used in the present paper is described by the trial action ( 9 ) , which is fully symmetric with respect to the permutations of the electrons , as is required to describe identical particles . the phase diagrams for @xmath1700 demonstrate the existence of stable multipolaron states ( see ref . @xcite ) . as distinct from ref . @xcite , here the ground state of an @xmath242-polaron system is investigated supposing that the electrons are fermions . as seen from fig . [ trinv - gse ] , for @xmath1700 , the stability region for a multipolaron state is narrower than the stability region for @xmath1533 , and its width decreases with increasing @xmath242 . [ h ] part2fig9.eps a consequence of the fermi statistics is the dependence of the polaron characteristics and of the total spin of an @xmath242-polaron system on the parameters ( @xmath1701@xmath242 ) . in fig . [ trinv - gse ] , we present the ground - state energy per particle , the confinement frequency @xmath1686 and the total spin @xmath80 as a function of the coupling constant @xmath63 for @xmath1702 and for a different numbers of polarons . the ground - state energy turns out to be a continuous function of @xmath63 , while @xmath1686and @xmath80 reveal jumps . for @xmath1533 ( the case of a bipolaron ) , we see from fig . [ trinv - gse ] that the ground state has a total spin @xmath1703 for all values of @xmath63 , i. e. , the ground state of a bipolaron is a singlet . this result is in agreement with earlier investigations on the large - bipolaron problem ( see , e. g. , @xcite ) . in summary , using the extension of the jensen - feynman variational principle to the systems of identical particles , we have derived a rigorous upper bound for the free energy of a translation invariant system of @xmath242 interacting polarons . the developed approach is valid for an arbitrary coupling strength . the resulting ground - state energy is obtained taking into account the fermi statistics of electrons . spherical shells of charged particles appear in a variety of physical systems , such as fullerenes , metallic nanoshells , charged droplets and neutron stars . a particularly interesting physical realization of the spherical electron gas is found in multielectron bubbles ( mebs ) in liquid helium-4 . these mebs are 0.1 @xmath262 m 100 @xmath262 m sized cavities inside liquid helium , that contain helium vapor at vapor pressure and a nanometer - thick electron layer anchored to the surface of the bubble @xcite . they exist as a result of equilibrium between the surface tension of liquid helium and the coulomb repulsion of the electrons @xcite . recently proposed experimental schemes to stabilize mebs @xcite have stimulated theoretical investigation of their properties . we describe the dynamical modes of an meb by considering the motion of the helium surface ( ripplons ) and the vibrational modes of the electrons together . in particular , we analyze the case when the ripplopolarons form a wigner lattice @xcite . first , we derive the lagrangian of interacting ripplons and phonons within a continuum approach . the shape of the surface of a bubble is described by the function @xmath1704 where @xmath1705 is the deformation of the surface from a sphere with radius @xmath1706 the deformation can be expanded in a series of spherical harmonics @xmath1707 with amplitudes @xmath1708 @xmath1709 we suppose that the amplitudes are small in such a way that @xmath1710 the ripplon contribution ( @xmath1711 ) to the kinetic energy of an meb , and the contributions to the potential energy due to the surface tension ( @xmath1712 ) and due to the pressure ( @xmath1713 ) were described in ref . @xcite : @xmath1714{l}t_{\text{r}}=\dfrac{\rho}{2}r_{\text{b}}^{3}{\displaystyle\sum\limits_{l=1}^{\infty } } { \displaystyle\sum\limits_{m =- l}^{l } } \dfrac{1}{l+1}\left| \dot{q}_{lm}\right| ^{2},\\ u_{\sigma}=4\pi\sigma r_{\text{b}}^{2}+\dfrac{\sigma}{2}{\displaystyle\sum\limits_{l=1}^{\infty } } { \displaystyle\sum\limits_{m =- l}^{l } } \left ( l^{2}+l+2\right ) \left| q_{lm}\right| ^{2},\\ u_{\text{v}}=\dfrac{4\pi}{3}pr_{\text{b}}^{3}+pr_{\text{b}}{\displaystyle\sum\limits_{l=1}^{\infty } } { \displaystyle\sum\limits_{m =- l}^{l } } \left| q_{lm}\right| ^{2}. \end{array } \label{usp}\ ] ] here @xmath1715 kg / m@xmath1716 is the density of liquid helium , @xmath1717 j / m@xmath1718 is its surface tension , and @xmath1719 is the difference of pressures outside and inside the bubble . expanding the surface electron density @xmath1720 in a series of spherical harmonics with amplitudes @xmath1721 @xmath1722 the kinetic energy of the motion of electrons can be written as @xmath1723 where @xmath1724 is the bare electron mass and @xmath242 is the number of electrons . finally , the electrostatic energy ( @xmath1725 ) of the deformed meb with a non - uniform surface electron density ( [ n1 ] ) is calculated using the maxwell equations and the electrostatic boundary conditions at the surface . the result is : @xmath1726 with the dielectric constant of liquid helium @xmath1727 . the last term in eq . ( [ uc ] ) describes the ripplon - phonon mixing . only ripplon and phonon modes which have the same angular momentum couple to each other . after the diagonalization of the lagrangian of this ripplon - phonon system , we arrive at the eigenfrequencies : @xmath1728 ^{2}+4\gamma^{2}\left ( l\right ) } \right ] \right\ } ^{1/2 } , \label{fab}\ ] ] where @xmath1729 is the bare ripplon frequency , @xmath1730 \right\ } ^{1/2 } , \label{wr}\ ] ] while @xmath1731 is the bare phonon frequency , @xmath1732 and @xmath1733 describes the ripplon - phonon coupling : @xmath1734 the interaction energy between the ripplons and the electrons in the multielectron bubble can be derived from the following considerations : ( i ) the distance between the layer electrons and the helium surface is fixed ( the electrons find themselves confined to an effectively 2d surface anchored to the helium surface ) and ( ii ) the electrons are subjected to a force field , arising from the electric field of the other electrons . for a spherical bubble , this electric field lies along the radial direction and equals @xmath1735 a bubble shape oscillation will displace the layer of electrons anchored to the surface . the interaction energy which arises from this , equals the displacement of the electrons times the force @xmath1736 acting on them . thus , we get for the interaction hamiltonian @xmath1737 here @xmath1738 is the radial displacement of the surface in the direction given by the spherical angle @xmath274 ; and @xmath1739 is the ( angular ) position operator for electron @xmath1740 . the displacement can be rewritten using ( [ radius ] ) and we find @xmath1741 using the relation @xmath1742 the interaction hamiltonian can be written in the suggestive form @xmath1743 with the electron - ripplon coupling amplitude for a meb given by @xmath1744 substituting @xmath1745 into ( [ hint ] ) , we get @xmath1746 ( \hat{a}_{\ell , m}+\hat{a}_{\ell ,- m}^{+}).\nonumber\end{aligned}\ ] ] in this expression , we consider the limit of a bubble so large that the surface becomes flat on all length scales of interest . hence we let @xmath1747 but keep @xmath1748 a constant . this means we have to let @xmath1749 as well . in this limit , @xmath1750,\ ] ] and @xmath1751 varies locally as a plane wave with wave vector @xmath1752 . the wave function @xmath1753 is furthermore normalized with respect to integration over the surface ( with total area @xmath1754 ) . thus , we get in the locally flat approximation @xmath1755 or @xmath1756 this corresponds in the limit of large bubbles to the interaction hamiltonian expected for a flat surface . in their treatment of the electron wigner lattice embedded in a polarizable medium such as a semiconductors or an ionic solid , fratini and qumerais @xcite described the effect of the electrons on a particular electron through a mean - field lattice potential . the ( classical ) lattice potential @xmath1757 is obtained by approximating all the electrons acting on one particular electron by a homogenous charge density in which a hole is punched out ; this hole is centered in the lattice point of the particular electron under investigation and has a radius given by the lattice distance @xmath1758 . within this particular mean - field approximation , the lattice potential can be calculated from classical electrostatics and we find that for a 2d electron gas it can be expressed in terms of the elliptic functions of first and second kind , @xmath1759 and @xmath1760 , @xmath1761 \right . \nonumber\\ & \left . + \left ( d+r\right ) \mathop{\rm sgn}\left ( d - r\right ) k\left [ -\frac{4rd}{\left ( d - r\right ) ^{2}}\right ] \right\ } . \label{potential}\ ] ] here , @xmath54 is the position vector measured from the lattice position . we can expand this potential around the origin to find the small - amplitude oscillation frequency of the electron lattice : @xmath1762 with the confinement frequency @xmath1763 in the mean - field approximation , the hamiltonian for a ripplopolaron in a lattice on a _ locally flat _ helium surface is given by @xmath1764 where @xmath1765 is the electron position operator . now that the lattice potential has been introduced , we can move on and include effects of the bubble geometry . if we restrict our treatment to the case of large bubbles ( with @xmath1766 electrons ) , then both the ripplopolaron radius and the inter - electron distance @xmath1758 are much smaller than the radius of the bubble @xmath1767 . this gives us ground to use the locally flat approximation using the auxiliary model of a ripplonic polaron in a planar system described by ( [ h1ri ] ) , but with a modified ripplon dispersion relation and an modified pressing field . we find for the modified ripplon dispersion relation in the meb : @xmath1768 where @xmath1767 is the equilibrium bubble radius which depends on the pressure and the number of electrons . the bubble radius is found by balancing the surface tension and the pressure with the coulomb repulsion . the modified electron - ripplon interaction amplitude in an meb is given by @xmath1769 the effective electric pressing field pushing the electrons against the helium surface and determining the strength of the electron - ripplon interaction is @xmath1770 to study the ripplopolaron wigner lattice at finite temperature and for any value of the electron - ripplon coupling , we use the variational path - integral approach @xcite . this variational principle distinguishes itself from rayleigh - ritz variation in that it uses a trial action functional @xmath1771 instead of a trial wave function . the action functional of the system described by hamiltonian ( [ h1ri ] ) , becomes , after elimination of the ripplon degrees of freedom , @xmath1772\right\ } + \sum_{\mathbf{q}}\left\vert m_{q}\right\vert ^{2}\nonumber\\ & \times\displaystyle\int\limits_{0}^{\hbar\beta}d\tau\displaystyle\int \limits_{0}^{\hbar\beta}d\sigma g_{\omega(q)}(\tau-\sigma)e^{i\mathbf{q}\cdot\lbrack\mathbf{r}(\tau)-\mathbf{r}(\sigma ) ] } , \label{sri}\ ] ] with @xmath1773}{\sinh(\beta\hbar\nu/2)}}}.\ ] ] in preparation of its customary use in the jensen - feynman inequality , the action functional ( [ sri ] ) is written in imaginary time @xmath1774 with @xmath180 where @xmath261is the temperature . we introduce a quadratic trial action of the form @xmath1775 \nonumber\\ & -{\displaystyle{\frac{mw^{2}}{4\hbar}}}\displaystyle\int\limits_{0}^{\hbar\beta}d\tau\displaystyle\int\limits_{0}^{\hbar\beta}d\sigma g_{w}(\tau-\sigma)\mathbf{r}(\tau)\cdot\mathbf{r}(\sigma ) . \label{s0ri}\ ] ] where @xmath1776 and @xmath274 are the variationally adjustable parameters . this trial action corresponds to the lagrangian @xmath1777 from which the degrees of freedom associated with @xmath539 have been integrated out . this lagrangian can be interpreted as describing an electron with mass @xmath1778 at position @xmath54 , coupled through a spring with spring constant @xmath1779 to its lattice site , and to which a fictitious mass @xmath1780 at position @xmath539 has been attached with another spring , with spring constant @xmath374 . the relation between the spring constants in ( [ l0 ] ) and the variational parameters @xmath1781 is given by @xmath1782 based on the trial action @xmath1771 , feynman s variational method allows one to obtain an upper bound for the free energy @xmath1437 of the system ( at temperature @xmath261 ) described by the action functional @xmath80 by minimizing the following function : @xmath1783 with respect to the variational parameters of the trial action . in this expression , @xmath1291 is the free energy of the trial system characterized by the lagrangian @xmath1784 , @xmath1785 is the inverse temperature , and the expectation value @xmath1786 is to be taken with respect to the ground state of this trial system . the evaluation of expression ( [ jfri ] ) is straightforward though lengthy . we find @xmath1787 + \displaystyle{2 \over\beta}\ln\left [ 2\sinh\left ( \displaystyle{\beta\hbar\omega_{2 } \over2}\right ) \right ] } \nonumber\\ & -{\displaystyle{\frac{2}{\beta}}}\ln\left [ 2\sinh\left ( { \displaystyle{\frac{\beta\hbar w}{2}}}\right ) \right ] -{\displaystyle{\frac { \hbar}{2}}}\sum_{i=1}^{2}a_{i}^{2}\omega_{i}\coth\left ( { \displaystyle{\frac { \beta\hbar\omega_{i}}{2}}}\right ) \nonumber\\ & -{\displaystyle{\frac{\sqrt{\pi}e^{2}}{d}}}e^{-d^{2}/(2d)}\left [ i_{0}\left ( { \displaystyle{\frac{d^{2}}{2d}}}\right ) + i_{1}\left ( { \displaystyle{\frac{d^{2}}{2d}}}\right ) \right ] \label{fri}\\ & -{\displaystyle{\frac{1}{2\pi\hbar\beta}}}\int_{1/r_{b}}^{\infty}dqq|m_{q}|^{2}\int_{0}^{\hbar\beta/2}d\tau{\displaystyle{\frac{\cosh [ \omega(q)(\tau-\hbar\beta/2)]}{\sinh[\beta\hbar\omega(q)/2]}}}\nonumber\\ & \times\exp\left [ -{\textstyle{\frac{\hbar q^{2}}{2m_{e}}}}\sum_{j=1}^{2}a_{j}^{2}{\textstyle{\frac{\cosh(\hbar\omega_{j}\beta/2)-\cosh[\hbar \omega_{j}(\tau-\beta/2)]}{\omega_{j}\sinh(\hbar\omega_{j}\beta/2)}}}\right ] .\nonumber\end{aligned}\ ] ] in this expression , @xmath1788 and @xmath1789 are bessel functions of imaginary argument , and @xmath1790@xmath1791 finally , @xmath1352 and @xmath1353 are the eigenfrequencies of the trial system , given by @xmath1792 .\ ] ] optimal values of the variational parameters are determined by the numerical minimization of the variational functional @xmath1437 as given by expression ( [ fri ] ) . the lindemann melting criterion @xcite states in general that a crystal lattice of objects ( be it atoms , molecules , electrons , or ripplopolarons ) will melt when the average motion of the objects around their lattice site is larger than a critical fraction @xmath1793 of the lattice parameter @xmath1758 . it would be a strenuous task to calculate from first principles the exact value of the critical fraction @xmath1793 , but for the particular case of electrons on a helium surface , we can make use of an experimental determination . grimes and adams @xcite found that the wigner lattice melts when @xmath1794 , where @xmath804 is the ratio of potential energy to the kinetic energy per electron . at temperature @xmath261 the average kinetic energy in a lattice potential @xmath1757 is @xmath1795 and the average distance that an electron moves out of the lattice site is determined by @xmath1796 from this we find that for the melting transition in grimes and adams experiment @xcite , the critical fraction equals @xmath1797 . this estimate is in agreement with previous ( empirical ) estimates yielding @xmath1798 @xcite , and we shall use it in the rest of this section . within the approach of fratini and qumerais @xcite , the wigner lattice of ( ripplo)polarons melts when at least one of the two following lindemann criteria are met : @xmath1799@xmath1800 where @xmath540 and @xmath1801 are , respectively , the relative coordinate and the center of mass coordinate of the model system ( [ l0 ] ) : if @xmath54 is the electron coordinate and @xmath539 is the position coordinate of the fictitious ripplon mass @xmath1780 , this is @xmath1802 the appearance of two lindemann criteria takes into account the composite nature of ( ripplo)polarons . as follows from the physical sense of the coordinates @xmath540 and @xmath1801 , the first criterion ( [ lind1 ] ) is related to the melting of the ripplopolaron wigner lattice towards a ripplopolaron liquid , where the ripplopolarons move as a whole , the electron together with its dimple . the second criterion ( [ lind2 ] ) is related to the dissociation of ripplopolarons : the electrons shed their dimple . the path - integral variational formalism allows us to calculate the expectation values @xmath1803 and @xmath1804 with respect to the ground state of the variationally optimal model system . we find @xmath1805 \left ( \omega_{1}^{2}-\omega_{2}^{2}\right ) } } } \nonumber\\ & \times\left [ \omega_{2}^{4}(\omega_{1}^{2}-w^{2})\coth(\hbar\omega _ { 1}\beta/2)/\omega_{1}\right . \nonumber\\ & \left . + \omega_{1}^{4}(w^{2}-\omega_{2}^{2})\coth(\hbar\omega_{2}\beta/2)/\omega_{2}\right ] , \label{rcms}\]]@xmath1806 . \label{rhori}\ ] ] numerical calculation shows that for ripplopolarons in an meb the inequality @xmath1807 is fulfilled ( @xmath1808 to @xmath1809 ) so that the strong - coupling regime is realized . owing to this inequality , we find from eqs . ( [ rcms]),([rhori ] ) that @xmath1810 so , the destruction of the ripplopolaron wigner lattice in an meb occurs through the dissociation of ripplopolarons , since the second criterion ( [ lind2 ] ) will be fulfilled before the first ( [ lind1 ] ) . the results for the melting of the ripplopolaron wigner lattice are summarized in the phase diagram shown in fig . [ phased1 ] . for every value of @xmath242 , pressure @xmath1719 and temperature @xmath261 in an experimentally accessible range , this figure shows whether the ripplopolaron wigner lattice is present ( points above the surface ) or molten ( points below the surface ) . below a critical pressure ( on the order of 10@xmath1194 pa ) the ripplopolaron solid will melt into an electron liquid . this critical pressure is nearly independent of the number of electrons ( except for the smallest bubbles ) and is weakly temperature dependent , up to the helium critical temperature 5.2 k. this can be understood since the typical lattice potential well in which the ripplopolaron resides has frequencies of the order of thz or larger , which correspond to @xmath1811 k. the new phase that we predict , the ripplopolaron wigner lattice , will not be present for electrons on a flat helium surface . at the values of the pressing field necessary to obtain a strong enough electron - ripplon coupling , the flat helium surface is no longer stable against long - wavelength deformations @xcite . multielectron bubbles , with their different ripplon dispersion and the presence of stabilizing factors such as the energy barrier against fissioning @xcite , allow for much larger electric fields pressing the electrons against the helium surface . the regime of @xmath242 , @xmath1719 , @xmath261 parameters suitable for the creation of a ripplopolaron wigner lattice lies within the regime that would be achievable in recently proposed experiments aimed at stabilizing multielectron bubbles @xcite . the ripplopolaron wigner lattice and its melting transition might be detected by spectroscopic techniques @xcite probing for example the transverse phonon modes of the lattice @xcite . j. t. devreese , in _ lectures on the physics of highly correlated electron systems vii , _ edited a. avella and f. mancini proceedings of the 7th training course in the physics of correlated electron systems&high - t@xmath1812 superconductors , vietri sul mare , italy , october 14 - 16 , 2002 , aip , melville ( 2003 ) , pp . 3 - 56 . the spectra of the infrared - active lo ( and to ) phonons in @xmath63-al@xmath67o@xmath68 ( sapphire ) contain six modes . the values of the lo and to phonon frequencies and of the high - frequency dielectric constants @xmath1813 , @xmath1814 are taken from ref . @xcite . using these parameters and the electron band mass @xmath1815 as estimated in ref . @xcite , the effective value of the electron - phonon coupling constant @xmath63 in al@xmath67o@xmath68 has been calculated as @xmath1816 where @xmath1817 is the polarization vector of the @xmath1740-th lo - phonon branch , @xmath59 is the phonon wave vector , @xmath1818 denote the angular averaging , and the coupling constants @xmath1819 for each branch are obtained using the method @xcite . the resulting value of the polaron coupling constant is @xmath1820 the spectra of the lo ( and to ) phonons in @xmath63-sio@xmath67 contain ten modes . the values of the lo and to phonon frequencies are taken from refs . @xcite . using these frequencies and the value @xmath1821 from ref . @xcite for the high - frequency dielectric constant , the effective value of the electron - phonon coupling constant @xmath63 in sio@xmath67 has been calculated using the method of ref . @xcite as indicated in ref . we use the estimated value of the electron band mass @xmath1822 as in refs . @xcite . the resulting value of the polaron coupling constant is @xmath1823 . s. i. pekar , _ issledovanija po ekektronnoj teorii kristallov _ , gostekhizdat , moskva , 1951 ( in russian ) [ german translation : _ untersuchungen ber die elektronentheorie der kristalle _ , akademie verlag , berlin , 1951 ] . j. t. devreese and r. evrard , in _ proceedings of the british ceramic society _ * 10 * , 151 ( 1968 ) . reprinted in : _ path integrals and their applications in quantum , statistical , and solid state physics _ , edited by g. j. papadopoulos and j. t. devreese , nato asi series b , physics , vol . 34 , plenum , new york , 1977 , pp . 344 - 357 .
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based on a course presented at the international school of physics enrico fermi , clxi course,.``polarons in bulk materials and systems with reduced dimensionality '' , varenna , italy , 21.6 .
- 1.7.2005 .
in the present course , an overview is presented of the fundamentals of continuum - polaron physics , which provide the basis of the analysis of polaron effects in ionic crystals and polar semiconductors .
these lecture notes deal with large , or continuum , polarons , as described by the frhlich hamiltonian .
the emphasis is on the polaron optical absorption , with detailed mathematical derivations .
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when electron collides elastically with open - shell atom or molecule , they can exchange their spins . thus , spin polarization of the electron beam is in general reduced after scattering with unpolarized open - shell targets . we can obtain more precise information of the scattering process by studying this depolarization @xcite , which is difficult to observe in usual experiment with spin - averaging procedure . collisions of spin - polarized electrons with atoms have been studied for long years ( see hegemann et al.@xcite and references therein ) . in contrast , number of experiment on electron molecule system is limited . ratliff et al.@xcite measured rate constants for electron exchange in elastic electron collisions with o@xmath0 @xmath3 and no @xmath4 molecules in thermal energies . their spin - exchange rate constants are substantially smaller than those in electron hydrogen - atom or alkali - metal - atom collisions . hegemann et al.@xcite studied exchange process in elastic electron collisions with o@xmath0 @xmath3 and no @xmath4 molecules and na @xmath5 atoms . they measured ratio of spin - polarizations in electron beams before and after collisions , i.e. , polarization fraction , which is directly related to the spin - exchange differential cross sections . as in the work of ratliff et al.@xcite , hegemann et al.@xcite confirmed that the exchange cross sections of electron molecule collisions are much smaller than those of electron collisions with atoms . although absolute value of spin - exchange cross section is small , the degree of spin - exchange in electron o@xmath0 collisions becomes larger at 100 degrees with collision energies between 8 and 15 ev compared to the other angles and energies , which they attributed to the existence of the o@xmath6 @xmath2 resonance . theoretical study of spin - exchange in electron o@xmath0 collisions was performed by da paixao et al.@xcite . they used the schwinger multichannel method with the three lowest electronic states of o@xmath0 in their model , and confirmed that spin - exchange cross section is small in electron o@xmath0 @xmath3 elastic scatterings . although exchange cross sections are small for electron collisions with randomly oriented o@xmath0 , they observed large depolarization at some scattering angles when electrons were scattered from spatially oriented o@xmath0 molecules . the profile of depolazation as a function of scattering angle depends strongly on molecular orientation . based on these resuls , da paixao et al.@xcite explained for the first time that the experimental exchange cross section in electron - molecule collisions is small because averaging over molecular orientation washes out depolarization effects . fullerton et al.@xcite , nordbeck et al.@xcite and wste et al.@xcite used the r - matrix method to calculate polarization fractions in electron o@xmath0 collisions . the calculations of fullerton et al.@xcite and nordbeck et al.@xcite employed the fixed bond approximation with t - matrix elements obtained by the nine - state r - matrix calculation of noble and burke @xcite , whereas wste et al.@xcite used vibrational averaging of t - matrices to include the effect of nuclear motion . the fixed bond r - matrix calculations of fullerton et al.@xcite and nordbeck et al.@xcite confirmed the results of da paixao et al.@xcite . agreement with experimental results at energies from 10 to 15 ev is marginal , as in the calculation of da paixao et al.@xcite . the vibrational averaging procedure of wste et al.@xcite improved agreement with the experimental results in this energy region . machado et al.@xcite applied the schwinger variational iterative method combined with the distorted - wave approximation and obtained similar elastic e - o@xmath0 polarization fractions to those of fullerton et al.@xcite . other than electron o@xmath0 collisions , theoretical work of spin exchange in electron molecule collisions is scarce . da paixao et al.@xcite calculated polarization fractions in electron no @xmath4 elastic collisions as they did in electron o@xmath0 collisions @xcite . calculated exchange effect was small in e - no elastic collisions , in agreement with the experimental results of hegemann et al.@xcite . sartori et al.@xcite studied spin - exchange in the superelastic electron collisions with h@xmath0 c @xmath7 state using the schwinger multichannel method . large depolarization was observed in their results , intermediate between depolarizations in e - na and e - o@xmath0 collisions . recently , fujimoto et al.@xcite performed the iterative schwinger variational calculation of spin - exchange effect in elastic electron c@xmath0o @xmath8 collisions . they found modest depolarization near resonances , however , spin - exchange effect was very small in other energy region . recently , we have studied electron o@xmath0 scatterings by the r - matrix method with improved molecular orbitals and increased number of target states@xcite compared to the previous theoretical studies . our results are in good agreement with the previous experimental results . since the previous theoretical polarization fractions of elastic e - o@xmath0 collisions at energies between 10 and 15ev agree not so well with the experimental results , it would be interesting to examine how spin - exchange cross section will change in this energy region by our improved calculational parameters . at the same time , it is important to understand general behaviour of spin - exchange cross sections , polarization fractions in other words , in elastic electron molecule collisions . until now , spin - exchange effect in low - energy electron molecule elastic scattering has been studied only for no and c@xmath0o molecules other than o@xmath0 . thus it is desirable to study spin - exchange in other electron - molecule scattering systems as well . in this work , we study spin - exchange in electron o@xmath0 collisions with the same calculational parameters as we used in our previous works@xcite . in addition , we calculate spin - exchange cross section in elastic electron collisions with b@xmath0 , s@xmath0 and si@xmath0 molecules . these b@xmath0 , s@xmath0 and si@xmath0 are stable homo - nuclear molecules with @xmath9 symmetry in their ground states , as in o@xmath0 molecule . in this paper , details of the calculations are presented in section 2 , and we discuss the results in section 3 comparing our results with previous theoretical and available experiments . then the summary is given in section 4 . in this work , we consider elastic scattering of spin - polarized electrons from randomly oriented unpolarized molecules . when the incident electrons have spin polarization @xmath10 and the scattered electrons have polarization @xmath11 , with the polarization direction perpendicular to the scattering plane , the polarization fraction , the ratio of @xmath10 and @xmath11 , is a measure of spin - exchange and is related to the spin - flip differential cross section ( dcs ) @xmath12 as @xcite , @xmath13 here @xmath14 is the dcs obtained by unpolarized electrons . the dcss of @xmath14 and @xmath12 are evaluated by the spin - specific scattering amplitude @xcite , @xmath15 where @xmath16 and @xmath17 specify the states of the target molecule as well as the scattering electron in the initial and final channels , respectively . @xmath18 and @xmath19 are the initial and final wavenumber of the electron , @xmath20 is the rotation matrix with the euler angles @xmath21 representing orientation of the target molecule in the laboratory frame . the electron is scattered to the direction @xmath22 in the laboratory frame in this expression . the t - matrix elements @xmath23 are prepared for all possible spin @xmath24 of the electron - molecule system as well as all irreducible representation @xmath25 of the symmetry of the system . we used @xmath26 in the r - matrix calculations . since the target molecules have triplet spin symmetry in this work , we only include @xmath27 and @xmath28 in our calculations . the matrix element @xmath29 relates the spherical harmonics @xmath30 to the real spherical harmonics @xmath31 . the explicit expression of @xmath29 can be found in our previous paper @xcite . in this paper , we consider elastic scattering of electron from molecule with triplet spin symmetry . then , @xmath14 and @xmath12 are expressed by the spin specific amplitude @xmath32 as @xcite , @xmath33 and @xmath34 here summations over channel indices are omitted for notational simplicity . note that the expression of the spin - flip dcs contains interference of amplitudes with different spin multiplicities . since the target molecules are randomly oriented , these dcss are averaged over all possible molecular orientations in space . the t - matrix elements @xmath23 were obtained by a modified version of the polyatomic programs in the uk molecular r - matrix codes @xcite . general procedure of calculation is almost the same as in our previous works@xcite . since the r - matrix method itself has been described extensively in the literature @xcite and references therein , we do not repeat general explanation of the method here . in this work , elastic electron collisions with o@xmath0 , b@xmath0 , s@xmath0 and si@xmath0 molecules were studied . for electron o@xmath0 scattering , we used the same parameter set as we used in the previous works@xcite . specifically , we employed the equilibrium bond length of 2.300 a@xmath35 for o@xmath0 , the r matrix radius of 10 a@xmath35 . the angular quantum number of the scattering electron was included up to @xmath36=5 . the atomic basis set for bound molecular orbitals , number of the target states included in the model as well as choice of the configurations in the inner region calculation were the same . for the electron b@xmath0 , s@xmath0 and si@xmath0 scatterings , we included 14 , 13 and 15 target electronic states in the r - matrix calculation , respectively . symmetries and spin - multiplicities of these states are given in table [ tab1 ] . these target states were represented by valence configuration interaction wave functions constructed by the state averaged complete active space scf ( sa - casscf ) orbitals . fixed - bond approximation was employed with internuclear distances of 3.036 , 3.700 and 4.400 a@xmath35 for b@xmath0 , s@xmath0 and si@xmath0 , respectively . although we study only elastic scattering in this work , we included these excited target states to improve quality of the r - matrix calculations . also , by including these excited states , we can suppress artificial structure coming from pseudo - resonance . in this study , the sa - casscf orbitals were obtained by calculations with molpro suites of programs @xcite . the target orbitals of b@xmath0 , s@xmath0 and si@xmath0 were constructed from the cc - pvtz basis set@xcite . the radius of the r - matrix sphere was chosen to be 13 a@xmath35 , which is larger than the r - matrix sphere used in the electron o@xmath0 calculation . we need this extended r - matrix sphere to avoid overlap of b@xmath0 , s@xmath0 and si@xmath0 molecular orbitals with the r - matrix boundary . in order to represent the scattering electron , we included diffuse gaussian functions up to @xmath36 = 4 , with 13 functions for @xmath36 = 0 , 11 functions for @xmath36 = 1 , 10 functions for @xmath36 = 2 , 8 functions for @xmath36 = 3 , 6 functions for @xmath36 = 4 . exponents of these diffuse gaussians were taken from faure et al . @xcite . the construction of the configuration state functions ( csfs ) for the electron - molecule system is the same as in our previous e - o@xmath0 papers @xcite . two different kind of ( @xmath37)-electron configurations are included , where @xmath38 is a number of electrons in the target molecule . the first type of the ( @xmath37)-electron csfs is constructed from @xmath38 target molecular orbitals ( mos ) plus one continuum orbital . the second type of csfs is constructed from the @xmath37 target mos . these target mos are just the sa - casscf orbitals , whereas the continuum orbitals are obtained by orthogonalization of the diffuse gaussian functions to the target mos @xcite . since only the continuum orbitals have overlap with the r - matrix sphere , the first type of csfs mainly contributes the cross sections . however , the second type of csfs is also important , as it is crucial to describe resonance . for reference , we show orbital set used in the e - b@xmath0 calculation in table [ tab2 ] . the orbital sets for e - s@xmath0 and e - si@xmath0 scatterings are very similar . more detailed explanation can be found in our previous paper@xcite . in this section , we show excited state energies of b@xmath0 , s@xmath0 and si@xmath0 molecules . since o@xmath0 energies have been shown in our previous paper@xcite , we do not discuss them here . in table [ tab3 ] , calculated excitation energies of b@xmath0 molecule are compared with full configuration interaction ( fci ) vertical excitation energies of hald et al.@xcite . although they employed different basis set , aug - cc - pvdz , and shorter internuclear distance of 3.005a@xmath35 , our casscf values agree reasonably well with their results . in table [ tab4 ] , our casscf energies of s@xmath0 molecule are compared with mrd ci vertical excitation energies of hess et al.@xcite and mrci adiabatic excitation energies of kiljunen et al.@xcite . in this case , our results agree well with the previous calculations for the lowest two excitations . for excitation energies to the three higher states , deviations become larger because kiljunen et al.@xcite studied adiabatic excitation energies whereas we calculated vertical excitation energies . in table [ tab5 ] , calculated energies of si@xmath0 molecule are compared with mrd ci vertical excitation energies of peyerimhoff and buenker @xcite . since they employed shorter internuclear distance of 4.3 a@xmath35 compared to 4.4 a@xmath35 of our calculation , precise comparison is difficult . however , our casscf results agree reasonably well with their results . in figure [ fig1 ] ( a ) , integral cross sections ( icss ) for elastic electron collision with o@xmath0 molecules are shown . the sharp peak around 0.2 ev comes from the o@xmath6 @xmath39 resonance . also , @xmath2 resonance causes a small rise of cross section around 13ev . the details of these e - o@xmath0 icss were discussed in the previous paper@xcite , however , they are shown here for comparison with the icss of the electron b@xmath0 , s@xmath0 and si@xmath0 collisions . in figure [ fig1 ] ( b ) , elastic icss for electron b@xmath0 collisions are shown . in this case , very large cross section is observed near zero energy , about @xmath40 , compared to @xmath41 in the e - o@xmath0 collisions . the partial cross sections of @xmath42 and @xmath43 symmetries equally contribute to this enhancement . there is a broad peak around 3 ev , which comes from @xmath44 symmetry partial cross section . an analysis of configuration state functions ( csfs ) suggests that this peak is related to the configuration @xmath45 , which is the ground state b@xmath0 with a scattering electron attached to the @xmath46 orbital . by removing this @xmath47 configuration from the r - matrix calculation , this peak vanishes from the icss . in figure [ fig2 ] ( a ) , icss for elastic electron scattering with s@xmath0 molecules are shown . the magnitude of the ics increases from @xmath48 at zero energy to @xmath49 at 10 ev , then it decreases to @xmath50 at 15 ev . although the magnitudes are different , the profiles of the @xmath42 and @xmath43 symmetry partial cross sections are very similar to those partial cross sections in the e - o@xmath0 elastic collision . a broad peak is observed around 4.5 ev , which comes from the @xmath51 symmetry partial cross section . the csf analysis suggests that this peak is related to the configuration @xmath52 , and likely belongs to the s@xmath6 @xmath51 resonance . the @xmath53 symmetry partial cross section has also a small rise around 6 ev ( not shown in the figure ) , its contribution to the total ics is small . two anomalous structures are observed in the ics , a kink at 2.7 ev and a cusp at 5 ev . the former kink belongs to the @xmath54 partial cross section , whereas the cusp at 5 ev comes from the @xmath51 symmetry . we analyzed the csfs and found that the kink at 2.7 ev is likely related to a resonance with configuration @xmath55 , which is obtained from an attachment of the scattering electron to the excited @xmath56 , @xmath57 and @xmath58 states of s@xmath0 with configuration @xmath59 . the position of the cusp coincides with the s@xmath0 b@xmath60 state , thus it is associated with opening of this excitation channel . in figure [ fig2 ] ( b ) , icss for elastic electron scattering with si@xmath0 molecules are shown . the magnitude of the ics is about @xmath61 between 0 and 15 ev . there are two sharp peaks below 1 ev . the peak at 0.55 ev is from the @xmath39 symmetry partial cross sections and the other peak at 0.12 ev is from the @xmath44 symmetry . we checked the csfs of the @xmath39 and @xmath44 symmetry calculations and found that the configuration @xmath62 has dominant contribution to these resonances . in figure [ fig3 ] , calculated polarization fractions ( pfs ) for elastic electron o@xmath0 collisions are shown for scattering energies of 5 , 10 , 12 and 15 ev with the previous theoretical results of fullerton et al.@xcite , machado et al.@xcite , da paixo et al.@xcite and wste et al.@xcite . these theoretical results are also compared with the experimental values of hegemann et al.@xcite in the figure . as we can see from eq.[eq1 ] , deviation of pf from unity is a measure of spin - exchange . our e - o@xmath0 pfs are close to 1 at all scattering energies , indicating the degree of spin - exchange is relatively small . for 5 ev , our results are very similar to the previous r - matrix results of fullerton et al.@xcite . the results of machado et al.@xcite are also similar , but smaller at low angles below 30 degrees . our pfs at 10ev are slightly smaller than the results of fullerton et al.@xcite and machado et al.@xcite . the pfs of da paixao et al.@xcite at 10 ev are smaller than our results in all angles , especially 120 - 180 degrees . our calculation can not reproduce the drop of experimental pfs at 10 ev at 100 degrees . the pfs of wste et al.@xcite have a minimum at this position and their value at 100 degree is the closest to the experimental result , although there is still some deviation in magnitude . at collision energy of 12 ev , the results of our calculation , wste et al.@xcite and fullerton et al.@xcite have similar angular behaviour , though our results are smaller than the others at all angles . all of these three theoretical results agree reasonably well with the experimental results . for 15ev , our pfs are larger than the results of the other theoretical calculations in all scattering angles and are closer to the experimental results . the pfs of fullerton et al.@xcite and machado et al.@xcite are very similar in shape and magnitude , whereas the results of da paixao et al.@xcite are slightly smaller at higher angles above 110 degrees . the deviation of our pfs from the previous theoretical results is the largest around 90 - 110 degrees , where there is a dip in the profile . although the magnitude of the pfs are different , the shape of the our pf profiles itself is similar to the previous calculations . in figure [ fig4 ] , the pfs for elastic electron o@xmath0 collisions are shown as a function of energy at a scattering angle of 100 degrees . our result has a minimum at 13 ev , however , it is located at 15 ev and 12 ev in the result of fullerton et al.@xcite and wste et al.@xcite , respectively . the magnitude of the pf at the minimum is larger in wste et al.@xcite than in our calculation and fullerton et al.@xcite . for comparison of the e - o@xmath0 pfs with the pfs of electron b@xmath0 , s@xmath0 and si@xmath0 collisions in the following figures , the pfs of elastic electron o@xmath0 collisions are again shown in the figure [ fig5 ] ( a ) for collision energies of 3 , 5 , 7 , 10 and 15 ev . the depolarization , i.e. , deviation of pf from 1 , is only prominent at 10 and 15 ev where the @xmath2 resonance exists as shown in fig.[fig1 ] ( a ) . in order to check the relation of the @xmath2 resonance and the pfs at 15 ev , we have carried out the r - matrix calculation with modified configurations , removing the @xmath63 configuration from the original calculation . by this procedure we can suppress the effect of the resonance . the results in the fig . [ fig5 ] ( a ) indicates that the pfs become very close to 1 by removing the configuration of the @xmath2 resonance . in figure [ fig5 ] ( b ) , calculated pfs for elastic electron b@xmath0 collisions are shown for scattering energies of 3 , 5 , 7 , 10 and 15 ev . for most of the scattering energies and angles , the depolarization in electron b@xmath0 collision is larger than that in electron o@xmath0 collisions . between 80 and 180 degrees , the magnitude of the pfs are about 0.8 - 0.9 at all energies . in contrast , the e - o@xmath0 pfs are larger than 0.9 . the e - b@xmath0 pfs show large depolarization effect at scattering energies of 3 and 5 ev , which are close to the @xmath44 resonance . in order to understand the origin of the large depolarizations at 3 and 5 ev , we excluded the effect of the b@xmath6 @xmath44 resonance around 3.5 ev and re - calculated the e - b@xmath0 pfs . specifically , we removed @xmath45 configuration from the r - matrix calculation and erased the @xmath44 resonance . as shown in the fig . [ fig5 ] ( b ) , the effect of the resonance on the pfs is evident . with the resonance effect , the lowest value of the pf at 3 ev is about 0.7 at 90 degrees , but it becomes about 0.95 without the resonance contribution . also , the depolarizations at 5 and 7 ev become less pronounced when we remove the effect of the resonance . the calculated pfs for elastic electron s@xmath0 collisions are shown in figure [ fig6 ] ( a ) . in general , the degree of depolarization is smaller than the e - b@xmath0 case , but is larger than the e - o@xmath0 case . the profiles of the pfs at 7 , 10 and 15 ev look similar to each other . however , the pfs at 3 and 5 ev behave differently . the degree of depolarization is larger at forward angles for 3 ev case , however , it is larger at backward angles at 5 ev . to understand the effect of resonance on electron s@xmath0 pfs , we removed the @xmath52 configuration and erased the @xmath51 resonance at 4.5ev , then re - calculated the pfs . the results are shown in the same figure . as in the case of electron b@xmath0 pfs , the degree of depolarization becomes smaller when we removed the resonance effect . the calculated pfs for elastic electron si@xmath0 collisions are shown in figure [ fig6 ] ( b ) . in this case , relatively large depolarization is observed for 3 ev at 100 degrees . the depolarization becomes smaller as the collision energy increases , however , some degree of depolarization remains near 80 and 180 degrees . the effect of resonance on the electron si@xmath0 pfs were examined by removing the @xmath62 configuration , which is responsible for the @xmath39 and @xmath44 resonances at 0.12 and 0.55 ev , respectively . by removing this configuration , these two sharp peaks in the icss disappear . the pfs without the resonances are shown in the fig . [ fig6 ] ( b ) . for 3 ev case , the depolarization becomes smaller at all angles . however , the decrease of depolarization is not so large compared to the cases of e - b@xmath0 and e - s@xmath0 collisions . for 5 ev case , the depolarization at 85 degrees becomes larger , though it becomes smaller at backward angles . thus , in this case , the association of the resonances with the depolarization is not so straightforward as in the e - o@xmath0 , b@xmath0 and s@xmath0 cases . as we show in the figures [ fig5 ] and [ fig6 ] , existence of resonance and behaviour of pf is closely related each other . when resonance exists at some energy , relatively large depolarization is observed compared to the other energies . also , when the resonance is artificially removed by deleting specific configuration in the r - matrix calculation , depolarization becomes smaller in general . in case of electron si@xmath0 collisions , this trend is partly broken at 5 ev around 80 degrees , however , depolarization generally becomes smaller at the other region after removing the resonance effect . the association of resonance and pf has been discussed in the previous theoretical and experimental papers @xcite , and we have confirmed this association more clearly by explicitly studying the effect of resonance on the pf of four different electron - molecule systems . even if collision occurs away from the resonance energy , some degree of depolarization is observed in all cases of e - o@xmath0 , b@xmath0 , s@xmath0 and si@xmath0 collisions . in e - b@xmath0 case , depolarization is relatively large even outside of the resonance energy region . in contrast , the pfs in e - o@xmath0 collisions are very close to 1 when collision energy is distant from the resonance energy . the degree of depolarization in e - s@xmath0 and si@xmath0 collisions is intermediate between e - b@xmath0 and e - o@xmath0 depolarizations . it is unclear why different degree of depolarization is seen in these four electron - molecule collisions when collision energy is distant from resonance . the extent of molecular orbitals may be related to this difference , as discussed by sartori et al.@xcite on the electron h@xmath0 superelastic collisions . for the electron o@xmath0 elastic scattering , the previous theoretical and experimental pfs are available for comparison with our results . our low energy pfs at 5 and 10 ev are similar to the other theoretical results . however , the pfs at 12ev are smaller than the previous results at all angles , and our pfs at 15ev are much closer to unity than the other theoretical results as shown in fig.[fig3 ] . the reason of these deviations can be attributed to the shift of o@xmath6 @xmath2 resonance position . in our calculation , the position of the @xmath2 resonance peak is located around 13.0 ev , whereas it is around 14.1 ev in the previous r - matrix calculation . as discussed in wste et al.@xcite , the position of the @xmath2 resonance is sensitive to the internuclear distance . in this work , the internuclear length is fixed to be 2.3 a0 and it is the same as in the calculation of fullerton et al.@xcite . so the choice of basis set , molecular orbitals and number of target states is important to the difference in the position of the resonance . probably the position of resonance is stabilized by inclusion of more target states in the present r - matrix calculations compared to the previous calculations of fullerton et al.@xcite , machado et al.@xcite and da paixo et al.@xcite . in figure [ fig1 ] ( b ) , sharp increase of cross section is observed in electron b@xmath0 elastic scattering near zero energy . this increase of cross section is similar to the case of electron polar - molecule collision , although b@xmath0 molecule has no dipole moment . the cross section of electron co@xmath0 elastic collision also has similar sharp increase near zero energy , and several experimental and theoretical works have been performed to understand this behaviour . morrison analyzed this problem and suggested that this behaviour is related to the existence of a virtual state @xcite . morgan has shown that the correlation and polarization effect is important for this sharp peak @xcite . similar mechanism may exist for the electron b@xmath0 elastic collisions . for the electron o@xmath0 elastic collisions , we put larger number of target electronic states and better quality molecular orbitals in the r - matrix calculations , and obtained improved results around @xmath2 resonance region compared to the previous fixed - bond calculations . however , the results of wste et al.@xcite agree better with the experimental pfs at 10 ev . they achieved this good agreement by vibrational averaging of the t - matrices to include the effect of the nuclear motion . by extending the present r - matrix calculation to include the vibrational effect using vibrational averaging procedure or the non - adiabatic r - matrix method , we may obtain better agreement with the experimental pfs . also , inclusion of nuclear motion effect may improve the quality of the calculations on electron b@xmath0 , s@xmath0 and si@xmath0 elastic collisions . we have calculated the polarization fractions ( pfs ) on low - energy elastic collisions of spin - polarized electrons with open - shell molecules , o@xmath0 , b@xmath0 , s@xmath0 and si@xmath0 , all of them having @xmath9 symmetry in their ground states . as in our previous works , we employed the fixed - bond r - matrix method based on state - averaged complete active space scf orbitals . our pfs for electron o@xmath0 collisions agree better with the previous experimental result , especially around the @xmath2 resonance , compared to the previous theoretical calculations . larger spin - exchange effect is observed in the electron b@xmath0 and si@xmath0 collisions than in the e - o@xmath0 collisions . in e - s@xmath0 collisions , degree of depolarization is similar to the e - o@xmath0 collisions . in all four electron - molecule collisions , the pfs deviate larger from 1 near resonances . this association of resonance and pf was explicitly confirmed by the r - matrix calculations removing configurations responsible for the resonance . panel ( a ) : the elastic integrated cross sections ( icss ) of electron scattering by o@xmath0 molecules . panel ( b ) : the elastic icss of electron scattering by b@xmath0 molecules . thick full lines represent total cross sections . the partial cross sections are represented by thin lines . symmetries with minor contributions are not shown in the figure . ] panel ( a ) : the elastic icss of electron scattering by s@xmath0 molecules . panel ( b ) : the elastic icss of electron scattering by si@xmath0 molecules . other details are the same as in the figure [ fig1 ] . ] polarization fractions of electron o@xmath0 elastic scattering . panel ( a ) : 5ev , ( b ) 10ev , ( c ) 12ev , ( d ) 15ev . our results are shown as thick full lines . experimental results of hegemann et al.@xcite are shown as open circles with error bars and theoretical pfs of fullerton et al.@xcite , machado et al.@xcite , da paixao et al.@xcite , and wste et al.@xcite are shown as thin lines . ] panel ( a ) : polarization fractions ( pfs ) of electron o@xmath0 elastic scattering . panel ( b ) : pfs of electron b@xmath0 elastic scattering . calculated pfs are shown as thick lines . thin lines with symbols represent pfs without the effect of resonances , obtained by the r - matrix calculation with modified configurations ( see text for details ) . ]
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the spin - exchange effect in spin - polarized electron collisions with unpolarized open - shell molecules , o@xmath0 , b@xmath0 , s@xmath0 and si@xmath0 , has been studied by the r - matrix method with the fixed - bond approximation .
all of these molecules have @xmath1 symmetry in their ground states .
usual integrated cross sections with unpolarized electrons has also been studied .
we used the complete active space self consistent field orbitals and put more than 10 target electronic states in the r - matrix models . in electron o@xmath0 elastic collisions ,
calculated polarization fractions agree well with the experimental results , especially around the @xmath2 resonance . in e - b@xmath0 , s@xmath0 and si@xmath0 elastic collisions ,
larger spin - exchange effect is observed compared to the e - o@xmath0 elastic collisions . in all four cases ,
spin - exchange effect becomes prominent near resonances .
this association of resonance and magnitude of the spin - exchange effect was studied by explicitly removing the resonance configurations from the r - matrix calculations . in general , spin - exchange effect is larger in e - b@xmath0 collisions than in e - s@xmath0 and si@xmath0 collisions , and is smallest in e - o@xmath0 collisions .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
a degree @xmath7 polynomial - like mapping is a degree @xmath7 proper holomorphic map @xmath8 , where @xmath9 and @xmath10 are topological disks and @xmath9 is compactly contained in @xmath11 . this definition captures the behaviour of a polynomial in a neighbourhood of its filled julia set . the filled julia set is defined in the polynomial - like case as the set of points which do not escape the domain . the external class of a polynomial - like map is the ( conjugacy classes of ) the map which encodes the dynamics of the polynomial - like map outside the filled julia set . the external class of a degree @xmath7 polynomial - like map is a degree @xmath7 real - analytic orientation preserving and strictly expanding self - covering of the unit circle : the expansivity of such a circle map implies that all the periodic points are repelling , and in particular not parabolic . in order to avoid this restriction , in @xcite we introduce an object , which we call _ parabolic - like mapping _ , to describe the parabolic case . a parabolic - like mapping is thus similar to a polynomial - like mapping , but with a parabolic external class ; that is to say , the external map has a parabolic fixed point . a parabolic - like map can be seen as the union of two different dynamical parts : a polynomial - like part and a parabolic one , which are connected by a dividing arc . [ definitionparlikemap ] * ( parabolic - like maps ) * a parabolic - like map of degree @xmath7 is a 4-tuple ( @xmath12 ) where * @xmath9 and @xmath11 are open subsets of @xmath13 , with @xmath14 and @xmath15 isomorphic to a disc , and @xmath9 not contained in @xmath11 , * @xmath16 is a proper holomorphic map of degree @xmath7 with a parabolic fixed point at @xmath17 of multiplier 1 , * @xmath18 \rightarrow \overline { u}$ ] is an arc with @xmath19 , forward invariant under @xmath20 , @xmath21 on @xmath22 $ ] and on @xmath23 $ ] , and such that @xmath24 @xmath25 ) \subseteq u \setminus u ' , \,\,\,\,\,\,\gamma(\pm 1 ) \in \partial u.\ ] ] it resides in repelling petal(s ) of @xmath26 and it divides @xmath9 and @xmath11 into @xmath27 and @xmath28 respectively , such that @xmath29 ( and @xmath30 ) , @xmath31 is an isomorphism and @xmath32 contains at least one attracting fixed petal of @xmath26 . we call the arc @xmath33 a _ dividing arc_. in @xcite we extend the theory of polynomial - like maps to parabolic - like maps , and we straighten degree @xmath0 parabolic - like maps to members of the family of quadratic rational maps with a parabolic fixed point of multiplier @xmath34 at infinity and critical points at @xmath34 and @xmath35 , which is @xmath36\,|\,p_a(z)=z+\frac{1}{z}+a\}.\ ] ] more precisely , we prove the following : every degree @xmath0 parabolic - like mapping @xmath37 is hybrid equivalent to a member of the family @xmath2 . moreover , if @xmath38 is connected , this member is unique . note that @xmath39= \ { p_a,\,\,p_{-a}\},$ ] since the involution @xmath40 conjugates @xmath41 and @xmath42 , interchanging the roles of the critical points . we refer to a member of the family @xmath2 as one of the representatives of its class . the family @xmath2 is typically parametrized by @xmath43 , which is the multiplier of the free fixed point @xmath44 of @xmath45 . the connectedness locus of @xmath2 is called @xmath6 . if @xmath46 is a family of degree @xmath0 parabolic - like maps with parameter space @xmath47 , calling @xmath48 the connectedness locus of @xmath49 , by the uniqueness of the straightening we can define a map @xmath50 @xmath51 which associates to each @xmath52 the multiplier of the fixed point @xmath44 of the member @xmath39 $ ] hybrid equivalent to @xmath53 . in this paper we will prove that if the family @xmath49 is _ analytic _ and _ nice _ ( see def . [ def ] and [ nf ] ) , _ the map @xmath54 extends to a map defined on the whole of @xmath55 ( see [ external ] ) , whose restriction to @xmath48 , under suitable conditions ( see def . [ properfam ] ) is a ramified covering of @xmath4 ( see thm . [ bigthm])_. the reason why the map @xmath54 covers @xmath4 , instead of the whole of @xmath6 , resides in the definition of analytic family of parabolic - like mappings , and it will be explained in section [ anfam ] . [ m ] [ m1 ] as an application , we will show that the connectedness locus of the family @xmath56 ( see fig.[m ] ) presents @xmath0 baby @xmath6 ( see fig . + the results in this paper were developed during the author s ph.d . so the author would like to thank her former advisor , carsten lunde petersen , for suggesting the idea of parabolic - like mapping and for his help , roskilde university and universit paul sabatier for their hospitality , and roskilde university , the anr-08-jcjc-0002 founded by the agence nationale de la recherche and the marie curie rtn 035651-cody for their financial support during her ph.d . in this section we define an analytic family of parabolic - like maps and its connectedness locus , nice families of parabolic - like maps , and we give an example of nice analytic family of parabolic - like maps . then we give a review of the straightening theorem , an overview of this paper and we state the main result . [ def ] let @xmath47 , @xmath57 and let @xmath58 be a family of degree @xmath7 parabolic - like mappings . set @xmath59 , @xmath60 , @xmath61 , and @xmath62 . then @xmath49 is a degree @xmath7 analytic family of parabolic - like maps if the following conditions are satisfied : 1 . @xmath63 , @xmath64 , @xmath65 and @xmath66 are domains in @xmath67 ; 2 . the map @xmath68 is holomorphic in @xmath69 . all the parabolic - like maps in the family have the same number of attracting petals in the filled julia set . for all @xmath70 let us call @xmath71 the parabolic - fixed point of @xmath72 , and let us set @xmath73 , and @xmath74 . define @xmath75 an analytic family of parabolic - like mappings is _ nice _ if there exists a holomorphic motion of the dividing arcs @xmath76 and there exists a holomorphic motion of the ranges @xmath77 which is a piecewise @xmath21-diffeomorphism with no cusps in @xmath78 ( for every fixed @xmath52 ) , and @xmath79 . a nice family is basically endowed by definition with a holomorphic motion of a fundamental annulus ( see section [ holmot ] ) . we did not require analytic families to have these properties , because the concept of parabolic - like map is local . on the other hand , since all the maps in an analytic family of parabolic - like maps have the same number of attracting petals in its filled julia set , it follows from the holomorphic parameter dependence of fatou coordinates ( see appendix in @xcite ) , that in many cases there is a holomorphic motion of the dividing arcs ( however , in individual cases further detail might be required according to circumstances ) . moreover , since the concept of parabolic - like map is local , in many cases it is not difficult to construct a holomorphic motion of the ranges for an analytic family of parabolic - like mappings . the definition of analytic family of parabolic - like maps is valid for any degree . however , since in this paper we are interested in proving that , under suitable conditions , the map @xmath54 defined in the introduction is a ramified covering between @xmath48 and @xmath80 , in the remainder we will restrict our attention to degree @xmath0 nice analytic families of parabolic - like maps . all the maps of an analytic family of parabolic - like maps have the same number of attracting petals in their filled julia set , and each ( maximal ) attracting petal requires a critical point in its boundary . hence , if @xmath49 is a degree @xmath0 analytic family of parabolic - like maps , either for each @xmath70 the map @xmath72 has no attracting petals in @xmath81 , or for each @xmath70 the map @xmath72 has exactly one attracting petal in @xmath81 . consider now the family @xmath2 . the @xmath82-part of a parabolic - like mapping requires ( at least ) one attracting petal , and for all the members of the family @xmath2 with @xmath83 the parabolic fixed point has parabolic multiplicity @xmath34 . so a parabolic - like restriction of @xmath45 , with @xmath84 has no attracing petals in the filled julia set . on the other hand , @xmath85 has a parabolic fixed point of parabolic multiplicity @xmath0 and the julia set of @xmath86 is the common boundary of the immediate parabolic basins , so a parabolic - like restriction of @xmath86 has exactly one attracting petal in its filled julia set . so , if all the members an analytic family of degree @xmath0 parabolic - like mappings @xmath49 have exactly one attracting petal in the filled julia set , they are all hybrid conjugate to the map @xmath85 , and @xmath87 ( but this case is not really interesting ) . on the other hand , if all the members of @xmath49 have no petals in the filled julia set , there is no @xmath70 such that @xmath72 is hybrid conjugate to the map @xmath85 , and finally the image of @xmath48 under the map @xmath54 is not the whole of @xmath6 , but it belongs to @xmath4 . this is the case we are interested in . consider the family of cubic polynomials @xmath88 the maps belonging to this family have a parabolic fixed point at @xmath89 of multiplier @xmath34 , and critical points at @xmath90 and @xmath91 . call @xmath92 the connectedness locus for this family . let @xmath93 denote the bttcher coordinates for @xmath94 tangent to the identity at infinity , call @xmath95 the co - critical point of @xmath96 and let @xmath97 be the conformal rappresentation of @xmath98 given by @xmath99 define @xmath100 as the open set bounded by the external rays of angle @xmath101 and @xmath102 ( see @xcite ) . in this section we are going to prove that the family @xmath103 yields to a nice family of parabolic - like mappings . let us construct a parabolic - like restriction for every member of the family @xmath105 . call @xmath106 the immediate basin of attraction of the parabolic fixed point @xmath89 . then @xmath107 belongs to @xmath106 , while @xmath96 does not belong to @xmath106 . let @xmath108 be the riemann map normalized by setting @xmath109 and @xmath110 , and let @xmath111 be its inverse . by the carathodory theorem the map @xmath112 extends continuously to @xmath113 . note that @xmath114 . let @xmath115 be a @xmath116 periodic point in the first quadrant , such that the hyperbolic geodesic @xmath117 connecting @xmath115 and @xmath118 separates the critical value @xmath119 from the parabolic fixed point @xmath120 . let @xmath121 be the jordan domain bounded by @xmath122 , union the arcs up to potential level @xmath34 of the external rays landing at @xmath123 and @xmath124 , together with the arc of the level @xmath34 equipotential connecting this two rays around @xmath96 ( see fig . [ cubicbcnnew ] ) . let @xmath125 be the connected component of @xmath126 containing @xmath127 and the dividing arcs @xmath128 be the fixed external rays landing at the parabolic fixed point @xmath127 and parametrized by potential . then ( @xmath129 ) is a parabolic - like map of degree @xmath0 ( see fig . [ cubicbcnnew ] ) . for every @xmath104 the parabolic fixed point @xmath127 of @xmath94 has parabolic multiplicity @xmath34 , and ( @xmath129 ) is a parabolic - like map with no attracing petals in its filled julia set . by the construction we gave , it follows easily that @xmath105 restricts to an analytic family of parabolic - like mappings . since external rays move holomorphically , to prove that this analytic family of parabolic - like maps is nice it suffices to show that the boundaries of @xmath121 move holomorphically with the parameter ( by construction the motion defines a piecewise @xmath21-diffeomorphisms with no cusps in @xmath78 ) . let us start by proving that the basin of attraction @xmath106 of @xmath127 depends holomorphically on the parameter . call @xmath132 the maximal attracting petal in @xmath106 , and let @xmath133 be fatou coordenates for @xmath94 normalized by sending the critical point @xmath107 to @xmath34 . since the family @xmath134 depends holomorphically on @xmath131 , @xmath135 depends holomorphically on @xmath131 and the extended fatou coordenates to the whole parabolic basin @xmath136 depend holomorphically on @xmath131 . on the other hand , let @xmath137 be extended fatou coordinates for the map @xmath116 , normalized by sending the critical point to @xmath34 . since the riemann map @xmath138 is a holomorphic conjugacy between @xmath94 and @xmath116 , @xmath139 are fatou coordinates for @xmath94 . since @xmath140 , we have that @xmath141 . hence the riemann map @xmath138 depends holomorphically on @xmath131 . so ( fixing a base point @xmath142 ) the dynamical holomorphic motion @xmath143 ( holomorphic in @xmath78 ) induces a dynamical holomorphic motion @xmath144 ( holomorphic in @xmath78 ) , which extends by the @xmath52-lemma to a dynamical holomorphic motion of @xmath145 . since @xmath146 moves holomorphically , the points @xmath147 and @xmath148 and the arc @xmath149 defined in [ dynamicalconstruction ] depend holomorphically on @xmath131 . since equipotentials and external rays move holomorphically , for every @xmath104 the set @xmath150 moves holomorphically . hence the family @xmath130 restricts to a degree @xmath0 nice analytic family of parabolic - like maps . in @xcite we proved that a degree @xmath0 parabolic - like map is hybrid conjugate to a member of the family @xmath2 by changing its external class into the class of @xmath151 ( see theorem 6.3 in @xcite ) and showing that a parabolic - like map is holomorphically conjugate to a member of the family @xmath2 if and only if its external class is given by the class of @xmath116 ( see proposition 6.2 in @xcite ) . we defined a ( quasiconformal ) conjugacy between two parabolic - like maps @xmath152 and @xmath153 to be a ( quasiconformal ) homeomorphism between ( appropriate ) restrictions of @xmath154 and @xmath155 which conjugates dynamics on @xmath156 ( see def . @xmath157 in @xcite ) . let us review how we changed the external class of a degree @xmath0 parabolic - like map @xmath20 into the class of @xmath116 . as first step , we constructed a homeomorphism @xmath158 , quasiconformal everywhere but at the parabolic fixed point , between a fundamental annulus @xmath159 of @xmath20 and a fundamental annulus @xmath160 of @xmath116 . then we defined on @xmath161 an almost complex structure @xmath162 by pulling back the standard structure by @xmath158 . in order to obtain on @xmath154 a bounded and invariant ( under a map coinciding with @xmath20 on @xmath163 ) almost complex structure @xmath164 we replaced @xmath20 with @xmath116 on @xmath82 , and spread @xmath162 by the dynamics of this new map @xmath165 ( and kept the standard structure on @xmath38 ) . finally , by integrating @xmath164 we obtained a parabolic - like map hybrid conjugate to @xmath20 and with external map @xmath116 . in this paper we want to perform this surgery for nice analytic families of degree @xmath0 parabolic - like maps , and prove that the map @xmath166 induced by the family of hybrid conjugacies extends to a continuous map @xmath167 which under suitable conditions restricts to a branched covering of @xmath168 . we will start by defining a family of quasiconformal maps , depending holomorphically on the parameter , between a fundametal annulus of @xmath116 and fundamental annuli @xmath169 of @xmath170 . in analogy with the polynomial - like setting we will call this family a _ holomorphic tubing_. in order to construct a holomorphic tubing , fixed a @xmath171 , we will start by constructing a quasiconformal homeomorphism @xmath158 between @xmath172 and @xmath173 ( see section [ psidiff ] ) and a dynamical holomorphic motion @xmath174 ( see section [ holmot ] ) . hence we will obtain a holomorphic tubing by composing the inverse of @xmath158 with the holomorphic motion ( see section [ a ] ) . by tubing , we will extend the map @xmath54 to the whole of @xmath55 ( see section [ external ] ) . we will prove that the map @xmath54 is continuous ( see section [ con ] ) , holomorphic on the interior of @xmath48 ( see section [ anal ] ) and with discrete fibers ( see section [ d ] ) . finally , we will prove that , on compact subsets of @xmath55 , the map @xmath54 is a degree @xmath175 branched covering ( see section [ final ] ) . by defining proper families of parabolic - like maps we wil give the condition under which , for each neighborhood @xmath11 of @xmath34 , @xmath176 is a compact subset of @xmath55 ( see section [ p ] ) . this implies the following result : [ bigthm ] given a proper family of parabolic - like maps @xmath177 , the map @xmath178 is a degree @xmath175 branched covering . more precisely , for every neighborhood @xmath11 of @xmath34 in @xmath13 ( with @xmath179 ) there exists a neighborhood @xmath180 of @xmath181 in @xmath182 such that the map @xmath183 is a degree @xmath175 branched covering . in this section we will first , fixed a @xmath171 , construct a fundamental annulus for @xmath116 and one for @xmath184 , and recall the quasiconformal @xmath21-diffeomorphism @xmath158 between these fundamental annuli . then we will construct a holomorphic motion of the ranges of the nice analytic family of parabolic - like maps , and by this holomorphic motion derive fundamental annuli for @xmath72 from the fundamental annulus of @xmath184 . finally we will obtain a holomorphic tubing by composing the inverse of @xmath158 with the holomorphic motion . the map @xmath185 is an external map of every member of the family @xmath2 ( see prop . 4.2 in @xcite ) . let @xmath186 ( where @xmath187 for an @xmath188 , and @xmath189 ) be a degree @xmath0 covering extension ( this is , an extension such that @xmath186 is a degree @xmath0 covering and there exists a dividing arc which devides @xmath190 and @xmath190 into @xmath191 and @xmath191 respectively , such that @xmath192 is a topological quadrilateral ; see def . 5.2 in @xcite ) . choose @xmath171 . let @xmath193 be an external map of @xmath184 , @xmath26 be its parabolic fixed point and define @xmath194 , @xmath195 ( where @xmath196 is an external equivalence between @xmath184 and @xmath193 ) . let @xmath197 and @xmath198 be repelling petals for the parabolic fixed point @xmath26 which intersect the unit circle and @xmath199 be fatou coordinates for @xmath193 with axis tangent to the unit circle at the parabolic fixed point @xmath26 . let @xmath200 and @xmath201 be repelling petals which intersect the unit circle for the parabolic fixed point @xmath120 of @xmath116 , and let @xmath202 be fatou coordinates for @xmath116 with axis tangent to the unit circle at @xmath34 . define @xmath203 and @xmath204 . define @xmath205 and @xmath206 . we call _ fundamental annulus for @xmath116 _ the topological annulus @xmath207 . let @xmath209 be a homeomorphism coinciding with @xmath210 on @xmath211 , quasiconformal on @xmath212 ( where @xmath213 is the parabolic fixed point of @xmath184 ) and real - analytic diffeomorphism on @xmath214 ( see claim @xmath215 in the proof of thm . @xmath216 in @xcite ) . define @xmath217 , @xmath218 , and @xmath219 . consider @xmath220 define @xmath221 , @xmath222 , and the _ fundamental annulus _ @xmath223 . let @xmath224 be a quasiconformal map which coincides with @xmath210 on @xmath211 ( see the proof claim @xmath225 in thm . @xmath216 in @xcite ) . define a map @xmath226 as follows : @xmath227 the map @xmath158 is a homeomorphism , quasiconformal on @xmath228 , so the map @xmath229 is a homeomorphism , quasiconformal on @xmath230 . define for all @xmath70 the set @xmath231 . then the set @xmath232 is a topological annulus . define the map @xmath233 as follows : @xmath234 since @xmath235 and @xmath236 are holomorphic motions with disjoint images on @xmath237 , and @xmath238 is a degree @xmath7 covering , @xmath239 is a holomorphic motion with basepoint @xmath240 . since @xmath57 , by the slodkowski s theorem we can extend @xmath239 to a holomorphic motion @xmath241 . in particular we obtain a holomorphic motion of @xmath242 . for every @xmath70 , define @xmath243 , and @xmath244 . define for every @xmath70 the map @xmath245 as follows : @xmath246 and the set @xmath247 . finally , define for all @xmath70 the set @xmath248 . then the set @xmath249 is a topological annulus , and we call it _ fundamental annulus of @xmath72_. the holomorphic motion @xmath241 restricts to a holomorphic motion @xmath250 which respects the dynamics . define @xmath251 . we call the map @xmath252 a * holomorphic tubing*. a holomorphic tubing is not a holomorphic motion , since @xmath253 , but nevertheless it is quasiconformal in @xmath78 for every fixed @xmath70 and holomorphic in @xmath52 for every fixed @xmath254 . let us now straighten the members of the family @xmath255 to members of the family @xmath2 . for every @xmath70 define on @xmath256 the beltrami form @xmath257 as follows : @xmath258 for every @xmath52 the map @xmath259 is quasiconformal , hence @xmath260 on every compact subset of @xmath55 . on @xmath261 the beltrami form @xmath262 is obtained by spreading @xmath263 by the dynamics of @xmath72 , which is holomorphic , while on @xmath264 the beltrami form @xmath262 is constant for all @xmath265 ( by construction of the map @xmath266 ) . thus @xmath267 , which is bounded . by the measurable riemann mapping theorem ( see @xcite ) for every @xmath70 there exists a quasiconformal map @xmath268 such that @xmath269 . finally , for every @xmath70 the map @xmath270 is a parabolic - like map hybrid conjugate to @xmath72 and holomorphically conjugate to a member of the family @xmath2 ( see prop . 6.2 in @xcite ) . [ locboundil ] note that for every @xmath70 , the dilatation of the integrating map @xmath271 is equal to the dilatation of the holomorphic tubing @xmath259 . so the family of integrating maps @xmath272 has locally bounded dilatation . let us lift the tubing @xmath259 . define @xmath273 , @xmath274 , @xmath275 and @xmath276 . hence @xmath277 and @xmath278 are degree @xmath0 covering maps , and we can lift the tubing @xmath259 to @xmath279 ( such that @xmath280 on @xmath281 ) . define recursively @xmath282 , @xmath283 , @xmath284 and @xmath285 . hence @xmath286 and @xmath287 are degree @xmath0 covering maps , and we can lift the tubing to @xmath288 ( such that @xmath289 on @xmath290 ) . in the case @xmath81 is connected , we can lift the tubing @xmath259 to the whole of @xmath291 . if @xmath81 is not connected , the maximum domain we can lift the tubing @xmath259 to is @xmath292 , such that @xmath293 contains the critical value of @xmath72 . note that the extension is still quasiconformal in @xmath78 . consider the map @xmath294 ( defined in section [ intro ] ) which associates to each @xmath295 the multiplier of the fixed point of the map @xmath45 hybrid equivalent to @xmath53 . in this section , we will first extend the map @xmath54 to the whole of @xmath55 ( see section [ external ] ) , then prove that the map @xmath296 is continuous at the boundary of @xmath48 ( see prop . [ con ] ) and that it depends analytically on @xmath52 for @xmath297 ( see prop . finally , we will prove that the map @xmath54 has discrete fibers ( see prop . [ discretness ] ) let @xmath252 be a holomorphic tubing for the nice analytic family of parabolic - like maps @xmath49 . call @xmath298 the critical point of @xmath72 and let @xmath265 be such that @xmath299 , @xmath300 . lift the holomorphic tubing @xmath259 to @xmath301 ( see section [ li ] ) . we can therefore extend the map @xmath54 by setting : @xmath302 @xmath303 where @xmath304 is the canonical isomorphism between the complement of @xmath6 and the complement of the unit disk ( see @xcite ) . since the tubing @xmath259 has locally bounded dilatation , the map @xmath305 is quasiregular on @xmath306 for any open @xmath307 . an indifferent periodic point @xmath308 for @xmath309 , is called _ persistent _ if for each neighborhood @xmath310 of @xmath308 there exists a neighborhood @xmath311 of @xmath240 such that , for every @xmath312 the map @xmath53 has in @xmath310 an indifferent periodic point @xmath313 of the same period and multiplier ( see @xcite ) . let @xmath1 be an family of parabolic - like mappings . for all @xmath70 , the parabolic fixed point is persistent . since all the other indifferent periodic points are non persistent , in the remainder we will call them _ indifferent periodic points _ without further notation . [ neutparametervalues ] the indifferent parameter values for a family of parabolic - like mappings belong to @xmath314 . the proof follows the proof of prop . 11 in @xcite . since for all @xmath315 the critical point @xmath298 of @xmath72 belongs to @xmath316 , the map @xmath72 is hyperbolic . assume that for @xmath317 the map @xmath318 has an indifferent periodic point @xmath319 of period @xmath320 , and assume first @xmath321 . by the implicit function theorem there exist @xmath322 neighborhoods of @xmath240 and @xmath319 respectively , with @xmath323 , where the indifferent cycle and the critical point move holomorphically with the parameter , and where the multiplier map @xmath324 is a holomorphic non constant map . set @xmath325 . by taking a restrictions if necessary , we can assume @xmath240 is the only parameter in @xmath326 for which @xmath72 has in @xmath327 an indifferent periodic point . let @xmath328 be a sequence in @xmath326 converging to @xmath240 , such that for all @xmath265 , @xmath329 . hence for all @xmath265 , there exists a @xmath330 such that @xmath331 we can assume @xmath332 independent of @xmath52 by choosing a subsequence . let us define for all @xmath333 the sequence @xmath334 since @xmath335 is a family of analytic maps bounded on any compact subset of @xmath326 , it is a normal family . let @xmath336 be a subsequence converging to some function @xmath337 . then @xmath338 for all @xmath265 , hence @xmath339 and for all @xmath333 , @xmath340 . but in @xmath326 there are parameters @xmath341 for which @xmath342 is a repelling periodic point , and thus it can not attract the sequence @xmath343 . in the case @xmath344 , let @xmath345 be a neighborhood of @xmath346 , let @xmath347 , @xmath348 be a branched covering of @xmath345 branched at @xmath127 for some neighborhood @xmath349 of @xmath127 , and repeat the previous argument . in this section we prove that the map @xmath296 is continuous on the boundary of @xmath48 . [ 7 ] suppose @xmath350 , with @xmath351 . if the maps @xmath352 and @xmath353 are quasiconformally conjugate , then @xmath354 . let @xmath355 and @xmath356 be parabolic - like restrictions of @xmath352 and @xmath353 respectively ( for the construction of a parabolic - like restriction of members of the family @xmath2 see the proof of prop . 4.2 in @xcite ) , and let @xmath357 be a quasiconformal conjugacy between them . if @xmath358 is of measure zero ( where @xmath359 , and @xmath360 is the parabolic basin of attraction of infinity , see section 1 in @xcite ) , then @xmath361 is a hybrid conjugacy and the result follows from prop . 6.5 in @xcite let @xmath358 be not of measure zero . define on @xmath362 the following beltrami form : @xmath363 since @xmath361 is quasiconformal , @xmath364 . therefore for @xmath365 we can define on @xmath362 the family of beltrami form @xmath366 , and @xmath367 . the family @xmath368 depends holomorphically on @xmath369 . let @xmath370 be the family of integrating maps fixing @xmath371 , @xmath34 and @xmath127 . hence the family @xmath372 depends holomorphically on @xmath369 , @xmath373 and @xmath374 . the family of holomorphic maps @xmath375 has the form @xmath376 ( since it is a family of quadratic rational maps with a parabolic fixed point at @xmath377 with preimage at @xmath89 and a critical point at @xmath378 ) and it depends holomorphically on @xmath369 . therefore the map @xmath379 is holomorphic , hence it is either an open or constant , and @xmath380 . if @xmath381 is open , there exists a neighborhood @xmath326 of @xmath127 such that @xmath382 . hence the map @xmath381 is constant , so for all @xmath369 , @xmath383 . in particular @xmath384 , and @xmath385 . finally the map @xmath386 is a quasiconformal conjugacy between @xmath352 and @xmath353 with @xmath387 on @xmath358 . so @xmath352 and @xmath353 are hybrid equivalent , and the result follows by prop . 6.5 in @xcite . [ conti ] the map @xmath296 is continuous at any point @xmath388 , and moreover @xmath389 . in order to prove continuity at any point @xmath388 , we have to show that for every sequence @xmath390 converging to @xmath391 there exists a subsequence @xmath392 such that @xmath393 converges to @xmath394 . let us start by proving that @xmath395 . let @xmath396 be a sequence of indifferent parameters converging to @xmath240 . hence there exists a sequence @xmath397 such that , for each @xmath398 , @xmath399 is hybrid conjugate to @xmath400 by some quasiconformal map @xmath401 . the sequence @xmath401 is a sequence of quasiconformal maps with locally bounded dilatation ( see remark [ locboundil ] ) , hence it is precompact in the topology of uniform convergence on compact subsets of @xmath402 ( see @xcite ) . therefore there exists a subsequence @xmath403 which converges to some quasiconformal limit map @xmath404 , which conjugates @xmath184 to some @xmath405 , so @xmath406 . for all @xmath398 the map @xmath399 has an indifferent periodic point , hence @xmath400 has an indifferent periodic point , thus @xmath407 and finally @xmath408 . since the map @xmath184 is hybrid conjugate to @xmath409 and quasiconformally conjugate to @xmath405 , and @xmath410 , by prop.[7 ] @xmath411 . let now @xmath390 be a sequence converging to @xmath391 . since the sequence @xmath412 is precompact , there exists a subsequence @xmath413 such that @xmath414 converges to some limit map @xmath415 which conjugates @xmath184 to @xmath416 so @xmath417 . finally , since @xmath395 and @xmath184 is quasiconformally conjugate to both @xmath418 and @xmath409 , by prop.[7 ] @xmath419 . the proof of the analycity of the map @xmath54 on the interior of @xmath420 ( see [ an ] ) follows the proof lyubich gave in the setting of polynomial - like mappings ( see @xcite ) . we will prove that the map @xmath54 is holomorphic on hyperbolic components first , and then on queer components . to prove that @xmath54 is holomorphic on queer components , we first need the following proposition . [ invariantline ] let @xmath421 be a queer component of @xmath422 . then for every @xmath423 , @xmath424 admits an invariant beltrami form with positive support . in particular , @xmath425 . choose @xmath426 and set @xmath427 . let us start by proving that there exists a dynamical holomorphic motion @xmath428 with base point @xmath429 , holomorphic in @xmath78 . let @xmath430 be an attracting petal of @xmath86 containing the critical value @xmath431 , and let @xmath432 be the incoming fatou coordinates of @xmath86 normalized by @xmath433 . for @xmath423 , let @xmath434 be an attracting petal of @xmath45 and let @xmath435 be the incoming fatou coordinates of @xmath45 with @xmath436 . the map @xmath437 is a conformal conjugacy between @xmath86 and @xmath45 . defining @xmath438 as the connected component of @xmath439 containing @xmath430 , and @xmath440 as the connected component of @xmath441 containing @xmath434 , we can lift the map @xmath442 to @xmath443 . since @xmath444 and @xmath445 are connected ( where @xmath444 and @xmath445 are the complements of the basin of attraction of the parabolic fixed point for @xmath86 and @xmath45 respectively ) , by interated lifting we can extend @xmath446 to @xmath447 . the map @xmath448 is a holomorphic conjugacy between @xmath86 and @xmath45 , and since the family @xmath2 is a family of holomorphic maps depending holomorphic on the parameter and so fatou coordenates depend holomorphically on the parameter , the family @xmath449 depends holomorphically on the parameter . hence @xmath450 is a dynamical holomorphic motion with base point @xmath429 , holomorphic in @xmath78 . since for every @xmath423 all the periodic points of @xmath45 ( but the parabolic fixed point ) are repelling , @xmath451 is nowhere dense . hence @xmath452 and thus by the @xmath52-lemma we can extend @xmath448 to @xmath453 . note that @xmath454 still conjugates dynamics , and it is conformal on @xmath455 . define @xmath456 . by construction , @xmath457 on @xmath455 . on the other hand , if @xmath458 on @xmath362 , by the weyl s lemma for every @xmath423 the map @xmath454 is holomorphic , hence for all @xmath423 the maps @xmath45 are conformally equivalent . therefore @xmath459 on @xmath444 , and thus @xmath460 on @xmath444 . in particular , this implies @xmath461 . [ an ] the map @xmath3 depends analytically on @xmath52 for @xmath297 . let us start by proving that , for every hyperbolic component @xmath462 , there exists a hyperbolic component @xmath463 such that @xmath464 is a holomorphic map . by the implicit function theorem and prop . [ neutparametervalues ] , all the parameter values in @xmath465 are hyperbolic . hence for @xmath466 , @xmath72 has an attracting cycle , thus @xmath467 has an attracting cycle and @xmath463 . since @xmath271 is conformal on @xmath468 , calling @xmath469 the multiplier maps on @xmath470 respectively , @xmath471 . hence on @xmath465 we can write the map @xmath54 as @xmath472 . since @xmath72 is holomorphic in both @xmath52 and @xmath78 , and by the implicit function theorem the attracting cycle moves holomorphically on @xmath465 , the map @xmath473 is holomorphic . for the same reason , @xmath474 is holomorphic as well . since @xmath474 has degree @xmath34 ( see @xcite ) , it is conformal and then @xmath54 is holomorphic . let now @xmath475 be a queer component in @xmath420 , and let @xmath476 . since @xmath477 we can lift the holomorphic motion @xmath478 constructed in [ holmot ] to @xmath479 ( as we did for the holomorphic tubing , see [ li ] ) . since for all @xmath480 , @xmath81 is nowhere dense , by the @xmath52-lemma we can extend @xmath481 to a dynamical holomorphic motion @xmath482 . let @xmath483 be the member of the family @xmath2 hybrid conjugate to @xmath184 , let @xmath484 be a hybrid conjugacy between them and set @xmath485 . note that , for all @xmath480 , the map @xmath486 is a quasiconformal conjugacy between @xmath483 and @xmath72 . define on @xmath487 the following family of beltrami forms : @xmath488 the family @xmath489 is a family of @xmath483-invariant beltrami forms depending holomorphically on @xmath52 . by prop . [ invariantline ] , for every @xmath480 , on @xmath490 @xmath491 . let @xmath492 be the family of integrating maps fixing @xmath493 and @xmath371 , then the family @xmath494has the form @xmath495 , where @xmath496 depends holomorphically on the parameter . finally , for every @xmath480 , the map @xmath497 is a hybrid conjugacy between @xmath72 and @xmath498 , hence @xmath498 is the straightening of @xmath72 and the map @xmath499 is holomorphic . set @xmath500 . [ discretness ] for every @xmath501 , @xmath502 is discrete . this follows the proof of lemma @xmath503 in @xcite . let us assume there exists a @xmath504 such that there exists a sequence @xmath505 and @xmath506 . the map @xmath54 is quasiregular on @xmath507 and holomorphic on @xmath420 , hence @xmath508 ( or @xmath509 in a queer component of @xmath510 for which @xmath54 is constant , and then we can replace @xmath509 with a boundary point ) . note that , for every @xmath265 , the maps @xmath511 are hybrid equivalent to @xmath512 by some hybrid equivalence @xmath513 . let us assume that for all @xmath52 in a neighborhood of @xmath514 , @xmath515 ( in other case , take a nice analytic family of parabolic - like restrictions for which the assumption holds ) . for every @xmath52 , call @xmath71 the parabolic fixed point of @xmath53 , @xmath298 its critical point and @xmath516 its critical value . consider @xmath514 as the base point of a holomorphic motion @xmath517 , extend it by the slodkowski s theorem and then restrict it to a holomorphic motion @xmath518 . define for every @xmath265 the map @xmath519 as follows : @xmath520 where the maps @xmath521 are as in [ holmot ] and the sets @xmath522 are constructed in [ li ] . the proof of prop . 6.4 in @xcite shows that , for every @xmath265 , the map @xmath523 is continuous and hence quasiconformal . therefore , for every @xmath265 , @xmath523 restricts to a hybrid equivalence between @xmath512 and @xmath511 . consider on @xmath487 the family of beltrami forms @xmath524 . note that trivially @xmath523 integrates @xmath525 , and for some subsets @xmath526 of @xmath527 and @xmath528 of @xmath529 , @xmath530 . on the other hand , define on @xmath487 the family of beltrami forms @xmath257 as follows : @xmath531 where for every @xmath52 the map @xmath532 which defines the sets @xmath533 and spreads @xmath257 is defined as follows : @xmath534 note that @xmath535 and @xmath532 coincide on @xmath536 , hence for every @xmath537 . therefore , for all @xmath265 , @xmath538 . the family @xmath539 depends holomorphically on @xmath52 , because @xmath481 is a holomorphic motion , on @xmath540 it is constant and on @xmath541 it is spread by the dynamics of @xmath512 ( which does not depends on @xmath52 ) . let @xmath542 be the holomorphic family of integrating maps mapping @xmath543 to @xmath71 , @xmath544 to @xmath298 and @xmath545 to @xmath516 . the family @xmath546 is a holomorphic family of parabolic - like mappings hybrid equivalent to @xmath512 . for all @xmath265 , @xmath547 ( since they are solutions of the same family of beltrami form and they coincide on @xmath548 points ) , and therefore on the @xmath526 , @xmath549 . choose a neighborhood @xmath550 of @xmath514 in @xmath55 and an open set @xmath327 in @xmath551 such that the maps @xmath552 and @xmath72 are well - defined in @xmath553 . since @xmath554 on @xmath555 , and @xmath556 converges to @xmath514 , @xmath557 on @xmath558 . this is impossible , because in @xmath550 there are @xmath52 for which @xmath72 has disconnected julia set ( since @xmath559 ) , while for all @xmath52 , @xmath552 has connected julia set ( since it is equivalent to @xmath512 ) . in this chapter , we will first prove that for any closed and connected subset @xmath561 of @xmath500 , if @xmath562 is compact , then @xmath499 is a proper map of degree @xmath7 ( prop . [ th ] ) , and then that @xmath563 is a @xmath7-fold branched covering ( prop . [ cov ] ) . finally , we will prove that , under certain conditions ( def . [ properfam ] ) , for every neighborhood @xmath11 of the root of @xmath6 ( without specifications , we consider a neighborhood open ) , the set @xmath176 is compact in @xmath55 . this implies theorem [ bigthm ] . [ th ] let @xmath561 be a closed and connected subset of @xmath567 , and @xmath562 . if @xmath475 is compact , then there exist neighborhoods @xmath180 of @xmath561 in @xmath567 and @xmath568 of @xmath475 in @xmath55 such that @xmath569 is a proper map of degree @xmath7 , where , for every @xmath570 , @xmath571 . the proof follows the analogous one in @xcite . let @xmath572 be a closed neighborhood of @xmath475 in @xmath55 with @xmath573 ( it exists since @xmath475 compact ) . hence @xmath574 , @xmath575 is proper and @xmath576 . call @xmath180 the connected component of @xmath577 which contains @xmath561 , and set @xmath578 . then @xmath579 , hence the map @xmath580 is proper . since @xmath54 is continuous and @xmath180 is connected , @xmath568 is the union of connected components . set @xmath581 . the restriction @xmath582 is then a proper map between connected sets , thus it has a degree @xmath583 , and since @xmath54 has discrete fibers , for every @xmath584 ( see @xcite pg . 136 ) : @xmath585 ( note that @xmath586 ) . therefore @xmath569 has a degree @xmath587 and in particular for all @xmath588 the map @xmath580 is continuous , and by the previous proposition it is a proper surjective map of degree @xmath7 . let @xmath589 , and let @xmath590 be a neighborhood of @xmath591 in @xmath180 such that for all @xmath592 and @xmath593 , @xmath594 is a regular point ( such a @xmath590 exists since the fiber of @xmath54 are finite ) . hence @xmath595 , with the @xmath596 disjoint and @xmath597 . if @xmath598 , @xmath591 is a regular point , and for all @xmath599 , @xmath600 is a homeomorphism . if @xmath601 , we want to show that for every @xmath602 there exists a neighborhood @xmath603 of @xmath604 such that @xmath605 , with the @xmath606 disjoint and @xmath607 homeomorphism this is clear because all the points in @xmath608 different from the preimage of @xmath591 are regular points . as we saw in [ anfam ] , the range @xmath567 of the map @xmath54 is not the whole of @xmath13 but a proper subset , because there is no @xmath70 for which @xmath72 is hybrid equivalent to @xmath609 . hence @xmath610 . however , we could hope that @xmath611 is the only point of @xmath6 missing from @xmath567 , or in other words , that as @xmath612 @xmath613 or @xmath614 . indeed this is the case under appropriate conditions ( e.g. the following one ) . assume @xmath475 is not compact in @xmath55 . then there exists a sequence @xmath622 such that @xmath623 as @xmath624 . on the other hand , for all @xmath265 , @xmath625 . let @xmath626 be a subsequence converging to some parameter @xmath627 . since @xmath561 is compact , the limit point @xmath627 belongs to @xmath628 . this is a contradiction , because @xmath49 is a proper family of parabolic - like mappings . therefore @xmath475 is compact in @xmath55 . hence if @xmath49 is a proper family of parabolic - like mappings , @xmath619 a neighborhood of @xmath611 , @xmath620 , and @xmath621 , by prop . [ cov ] there exist neighborhoods @xmath568 of @xmath475 in @xmath55 and @xmath180 of @xmath561 in @xmath567 such that the restriction @xmath569 is a @xmath629-fold branched covering . next proposition tells us how to compute the degree @xmath629 of the covering . let @xmath49 be a proper family of parabolic - like mappings , @xmath619 a neighborhood of @xmath611 ( with @xmath630 ) , @xmath620 , and @xmath180 a neighborhood of @xmath561 given by prop . [ th ] . then calling @xmath298 the critical point of @xmath72 and @xmath631 , the degree @xmath629 of the branched covering @xmath632 is equal to the number of times @xmath633 turns around @xmath127 as @xmath52 describes @xmath634 . let us remember that for every @xmath172 the map @xmath635 has two critical points : @xmath120 and @xmath378 . after a change of coordenates we can assume @xmath120 is the first critical point attracted by @xmath371 . hence for all @xmath636 , @xmath378 is the critical point of the parabolic - like restriction of @xmath45 ( see the proof of prop . 4.2 in @xcite ) . the proof follows the analogous one in @xcite . let @xmath298 be the critical point of @xmath72 . choose @xmath240 such that @xmath637 . let @xmath409 be the member of the family @xmath2 hybrid equivalent to @xmath184 . therefore @xmath638 , hence @xmath639 . this means that the multiplicity of @xmath240 as zero of the map @xmath640 is the multiplicity of @xmath240 as zero of the map @xmath641 . this last one is @xmath642 . let @xmath1 be a proper family of parabolic - like mappings . for every @xmath70 , @xmath72 is the restriction of some map @xmath643 . consequently , @xmath55 is the restriction of the parameter plane of the maps @xmath643 , call it @xmath644 . call @xmath645 the connectedness locus of @xmath643 , hence @xmath646 let @xmath49 be a proper family of parabolic - like mappings . if the map @xmath647 is a homeomorphism , and @xmath648 , then @xmath54 extends to a homeomorphism @xmath649 for a unique @xmath650 . more generally , if the the map @xmath647 is a degree @xmath629 branched covering , and @xmath648 , then map @xmath54 extends continuously to @xmath651 for exactly @xmath652 points in @xmath653 . let @xmath49 be a proper family of parabolic - like mappings for which the map @xmath647 is a degree @xmath629 covering and @xmath648 . since @xmath49 is a proper family , as @xmath618 , @xmath654 . we will prove that for every @xmath655 , @xmath656 , and that @xmath657 is a discrete set . then by continuity , @xmath658 . the original family @xmath643 has a persistent parabolic fixed point of multiplier @xmath34 and it depends holomorphically on @xmath52 . take a succession @xmath659 such that @xmath660 , and call @xmath661 the limit of the @xmath662 in @xmath644 . since for every @xmath332 , @xmath662 is a hyperbolic parameter , the limit @xmath661 is a hyperbolic or indifferent parameter . so , if @xmath663 , @xmath664 presents a degree @xmath0 parabolic - like restriction and @xmath665 . since @xmath666 , @xmath667 . let us prove now that the set @xmath657 is discrete . note that this is the set of parameters for which the parabolic fixed point @xmath71 of @xmath643 has parabolic multiplicity @xmath668 , where @xmath265 is the multiplicity of @xmath71 for @xmath669 . then , in a neighborhood of @xmath71 we can consider @xmath643 as @xmath670 with @xmath671 and @xmath672 holomorphic in @xmath52 . hence the set @xmath673 for which @xmath674 is a discrete set . [ [ the - parameter - plane - of - the - family - c_azzaz2z3-presents - baby - m_1 ] ] the parameter plane of the family @xmath675 presents baby-@xmath6 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ let us show that the family of parabolic - like mappings @xmath130 is proper . call @xmath676 the connectedness locus of @xmath105 . the finite boundaries of @xmath55 are the external rays of angle @xmath101 and @xmath102 , which can not intersect the connectedness locus @xmath676 in other point than the landing point , if they land . since these rays land at @xmath677 ( see @xcite ) , for @xmath678 either @xmath679 , hence @xmath613 , or @xmath680 , hence @xmath614 . finally , by the relation @xmath681 between external rays in dynamical and parameter plane , the degree of @xmath682 is @xmath34 . therefore @xmath92 presents a baby @xmath6 . by symmetry , we can repeat the construction for the family @xmath683 , where @xmath684 is the open set bounded by the external rays of angle @xmath685 and @xmath686 . hence the connectedness locus of the family @xmath687 presents two baby @xmath6 , namely in the connected component bounded by the external rays of angle @xmath101 and @xmath102 , and in the connected component bounded by the external rays of angle @xmath685 and @xmath686 ( see fig . [ m ] and [ m1 ] in the introduction ) . 13 k. astala , _ elliptic partial differential equations and quasiconformal mappings in the plane _ , princeton univ . press , ( 2008 ) . l. ahlfors , lectures on quasiconformal mappings , second edition . ams university lecture series , vol . b. branner & j. h. hubbard , the iteration of cubic polynomials ii : patterns and parapatterns , _ acta math._,157 ( @xmath688 ) , no . 1 - 2 , 2348 . a. douady & c. j. earle , conformally natural extension of homeomorphisms of the circle , _ acta math._,169 ( @xmath689 ) , no . 3 - 4 , 229 - 325 . a. douady & j. h. hubbard , on the dynamics of polynomial - like mappings , _ ann . sci . cole norm . sup._,(4 ) , vol.@xmath690 ( @xmath691 ) , 287 - 343 . o. forster , _ lectures on riemann surfaces _ , springer , ( 1981 ) . a. hatcher , _ algebraic topology _ , cambridge univ . press , ( 2002 ) . l. hrmander , _ the analysis of linear partial differential operators i _ , springer - verlag , ( 1983 ) . j. hubbard , _ teichmller theory , volume 1 : teichmller theory _ , matrix editions , ( 2006 ) . l. lomonaco , parabolic - like maps , arxiv:1111.7150 . l. lomonaco , _ parabolic - like maps _ , imfufa tekst , ( 2013 ) . m. lyubich , _ conformal geometry and dynamics of quadratic polynomials _ , www.math.sunysb.edu/ mlyubich / book.pdf . j. milnor , _ dynamics in one complex variable _ , annals of mathematics studies , ( 2006 ) . j. milnor , on rational maps with two critical points , _ experimental mathematics _ , ( 4 ) , vol . 9 ( 2000 ) , 481 - 522 . r. ma , p. sad & d. sullivan , on the dynamics of rational maps , _ ann . cole norm . sup._,(4 ) , vol.@xmath692 ( @xmath693 ) , 193 - 217 . d. sullivan , quasiconformal homeomorphisms and dynamics iii , _ ann . cole norm . sup._,(4 ) , vol.@xmath692 ( @xmath693 ) , 193 - 217 . s. nakane , capture components for cubic polynomials with parabolic fixed points , _ academic reports fac . eng . tokio polytech . univ._,(1 ) , vol.@xmath694 ( @xmath695 ) , 33 - 41 . c. petersen & l. tan , branner - hubbard motions and attracting dynamics , _ dynamics on the riemann sphere _ , ( 45 - 70 ) , eur . math . ( @xmath696 ) . m. shishikura , bifurcation of parabolic fixed points , _ the mandelbrot set , theme and variations , _ ( 325 - 363 ) , _ london math . lecture note ser . , 274 _ cambridge univ . press , ( 2000 ) .
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in this paper we study families of degree @xmath0 parabolic - like mappings @xmath1 ( as we defined in @xcite ) .
we prove that the hybrid conjugacies between a nice analytic family of degree @xmath0 parabolic - like mappings and members of the family @xmath2 induce a continuous map @xmath3 , which under suitable conditions restricts to a ramified covering from the connectedness locus of @xmath1 to the connectedness locus @xmath4 of @xmath2 . as an application
, we prove that the connectedness locus of the family @xmath5 presents baby @xmath6 .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the determination of a quantum state is a highly nontrivial problem . a wave function itself , or more generally a density operator , is not an observable and can not be measured directly . one can see the state only through measurable quantities , which are related to the matrix elements of the density operator . it is therefore an interesting and important issue to discuss how to infer a given quantum state from a list of observed quantities . such a problem is called state reconstruction or state tomography @xcite . recent advances in technology have been enabling us to engineer variety of peculiar quantum states to explore fundamental aspects of quantum mechanics . generation of highly quantum states is one of the key elements for the realization of the ideas of quantum information @xcite . it is clear that such issues can not go without supports by the state reconstruction technology . probing nontrivial correlations in a quantum system also helps us to clarify intrinsic characters of various many - body systems . from a practical point of view , direct accesses to target systems are often limited . in such a case , we have to explore a way to probe the target in an indirect way . scattering has always been considered a very powerful way to investigate many physical systems in a wide range of fields of physics , from elementary - particle physics to condensed - matter physics . loosely speaking , all the physical processes to access targets can be regarded as scattering processes . in this paper , we focus on the state tomography of a qubit ( spin-1/2 particle ) via scattering , in a simple 1d setup . we send a probe qubit to the fixed target qubit and see the state of the probe after its scattering off the target . its scattering data contain the information on the target state before the scattering , from which we reconstruct the state . in order to reconstruct the qubit state , three independent scattering data are required . one possibility is to make use of the spin degree of freedom of the probe . a collection of three transmission / reflection probabilities with three different sets of the initial and final spin states of the probe provides sufficient information for the reconstruction of the target qubit state . notice here that in our setup , there is an additional ( spatial ) degree of freedom available in the scheme for the tomography of the spin , i.e. , the momentum of the probe . it will be shown that this degree of freedom can be utilized to optimize " ( in the sense prescribed below ) the scheme . it is interesting to see that the _ spatial _ degree of freedom can play a central role for the tomography of the _ spin_. it will be demonstrated that three scattering data required for the qubit tomography are available by arranging different incident momenta and scattering directions , with the initial and final spin states of the probe being fixed . suppose that a qubit @xmath0 is fixed at @xmath1 on a 1d line . we are going to discuss the reconstruction of its state , which is in general a mixed state @xmath2 , through the scattering data of a probe qubit @xmath3 off the target @xmath0 . we assume that the following hamiltonian describes the scattering process : @xmath4 here @xmath5 and @xmath6 are respectively the position and the momentum of the probe qubit @xmath3 of mass @xmath7 , and the potential produced by the fixed qubit @xmath0 is assumed to be represented by the delta function . when @xmath3 is scattered by @xmath0 , their spins interact with each other through the heisenberg - type interaction with a positive coupling constant @xmath8 , with @xmath9 being the pauli operators acting on qubit @xmath10 . see fig . [ fig : scattering picture ] . is sent with a fixed wave number @xmath11 from the left to a target qubit @xmath0 fixed at @xmath1 , scattered by the delta - shaped potential produced by the target qubit @xmath0 with a spin - spin interaction of the heisenberg type , and detected on the right or on the left with spin - sensitive detectors.,scaledwidth=48.0% ] let us start by solving the scattering problem of the hamiltonian ( [ eqn : hamiltonian ] ) . the probe @xmath3 is sent from the left ( @xmath12 ) to the target @xmath0 with a fixed incident wave number @xmath13 , and scattered . the scattering ( s ) matrix element is given by @xcite @xmath14 where @xmath15 is the eigenstate of the free hamiltonian @xmath16 and @xmath17 that of the total hamiltonian @xmath18 , both belonging to the same eigenvalue @xmath19 , with @xmath20 denoting the spin degrees of freedom of two qubits . the coordinate representation of the latter reads as @xcite @xmath21 where @xmath22 is the retarded green function in 1d . it is easy to calculate the source term at @xmath23 , @xmath24 with a dimensionless parameter @xmath25 recall that the heisenberg - type coupling @xmath26 is rewritten as @xmath27 in terms of the projection operators @xmath28 onto the spin - singlet and -triplet eigenspaces , respectively . this allows us to evaluate the inverse operator as @xmath29 we therefore obtain @xmath30 where the scattering matrices responsible for transmission and reflection , @xmath31 and @xmath32 , read @xmath33 here the coefficients @xmath34 and @xmath35 coincide with the reflection and transmission amplitudes calculated separately for the spin - singlet(triplet ) eigenspace , in which the interaction hamiltonian is given by a scalar ( i.e. , not spin - dependent ) potential @xmath36 for the spin - singlet case or by @xmath37 for the spin - triplet case . this implies that the problem can be reduced to an ordinary scattering problem of a spinless particle . in the tomographic schemes we are going to discuss in the following , the probe @xmath3 is sent with its spin polarized to the direction specified by a unit vector @xmath38 , and we see the probability of @xmath3 being transmitted or reflected with its spin rotated to @xmath39 . such probabilities are given by @xmath40 where @xmath41 and @xmath42 are the incident and final spin states of @xmath3 , which are expressed in the bloch - sphere formalism as @xmath43 and @xmath44 is to be substituted by @xmath31 or @xmath32 depending on whether @xmath3 is transmitted or reflected . the target qubit @xmath0 is in general in a mixed state @xmath2 , which is characterized by a bloch vector @xmath45 such that @xmath46 the transmission probability for a given set @xmath47 reads as @xmath48 + c_t(\omega)\bm{v}\cdot({\bm n}_f\times{\bm n}_i)\end{aligned}\ ] ] with @xmath49 these coefficients are plotted in fig . [ fig_transmission_coeff ] as functions of the incident wave number @xmath11 of the probe qubit @xmath3 . similarly , the probability for the reflected case reads @xmath50+c_r(\omega ) \bm{v}\cdot({\bm n}_f\times{\bm n}_i)\end{aligned}\ ] ] with @xmath51 which are plotted in fig . [ fig_reflection_coeff ] . [ cols= " > " , ] observe that the transmission and reflection probabilities @xmath52 and @xmath53 are both spherically symmetric in spin space . this reflects the symmetry of the heisenberg coupling @xmath54 : no preferred direction is present in the system . let us now discuss the tomography of the state @xmath2 , through the scattering data @xmath55 . our objective is to determine the three independent components of vector @xmath45 , which exactly corresponds to the complete specification of the state @xmath2 , see ( [ target initial state ] ) . from an experimental point of view , this means that we need to arrange three independent experimental setups . as is clear from ( [ transmission_probability ] ) and ( [ reflection_probability ] ) , the scattering probabilities @xmath55 explicitly depend on the initial and detection orientations @xmath38 and @xmath39 of the spin of the probe qubit @xmath3 as well as on the parameter @xmath56 containing the incident wave number @xmath11 and the coupling constant @xmath8 ; we can consider different strategies for the tomography , by properly choosing these parameters . in the next subsections , we discuss two of such tomographic strategies , in which the spatial and spin degrees of freedom of @xmath3 play different roles . in the first approach presented in sec . [ first strategy ] , a central role is played by the spin degree of freedom of @xmath3 : we tune the incident and detection orientations @xmath38 and @xmath39 of its spin , while the incident wave number @xmath11 is fixed . in order to completely reconstruct @xmath2 , we arrange the orientations @xmath38 and @xmath39 in three independent ways . for instance , we can choose three different detection orientations @xmath39 with the orientation of the incident spin @xmath38 being fixed , or we can tune both of them at the same time . the remaining degree of freedom , the wave vector @xmath11 , also plays an active role : we can optimize " in the sense described below the tomographic scheme by tuning it appropriately . in sec . [ par : second strategy ] , we discuss another approach , in which we fully make use of the spatial degree of freedom @xmath11 . the tomography of the _ spin _ state @xmath2 is possible through the _ spatial _ degree of freedom . we collect three independent scattering data for different incident wave numbers @xmath11 and scattering directions ( i.e. , transmission or reflection ) when the incident and detection orientations of the spin @xmath38 and @xmath39 are fixed , from which we reconstruct the spin state @xmath2 . furthermore , one can follow the same criterion introduced for the first strategy in order to optimize " the tomographic scheme . in this section , we reconstruct the initial state of the target qubit @xmath0 by tuning the incident and detection orientations @xmath38 and @xmath39 of the spin of the probe qubit @xmath3 with a fixed incident wave number @xmath11 . in particular , we discuss two possible examples of this first tomographic strategy . let us first fix the orientation of the spin of the incident qubit @xmath3 , @xmath38 , and choose three different detection orientations @xmath57 , which can be chosen to be orthogonal to each other . for instance , choosing @xmath58 perpendicular to @xmath38 , @xmath59 would be a natural choice as a reference frame . the transmission probabilities associated to these detection orientations with a fixed wave number @xmath11 read @xmath60 where @xmath61 are the three independent components of the target vector @xmath45 along the axes of the reference system @xmath62 introduced above and @xmath63 the transmission probability when the spin of @xmath3 is detected along direction @xmath64 . the three components of the vector @xmath45 are readily obtained by inverting the relation ( [ trasmission_first_strategy ] ) as @xmath65\ ] ] with @xmath66 which completes the reconstruction of the state @xmath2 . observe that the only condition needed to be satisfied in order to invert the matrix in ( [ trasmission_first_strategy ] ) to obtain @xmath67 in ( [ m_t ] ) is @xmath68 , which can be considered as a self - consistency condition for the present tomographic scheme . another example of this first strategy , which is simpler from a computational point of view , is detecting the spin @xmath3 oriented in the same direction as the incident spin , @xmath69 . in this case , the transmission probability reads @xmath70 from which we immediately obtain @xmath71 by choosing three different orientations for @xmath72 , we gain the complete information on the vector @xmath45 , that is , on the state @xmath2 . observe that also in this case the self - consistency condition reads @xmath68 . similar schemes are available with reflection probabilities @xmath53 . notice however that , if we choose @xmath69 as we did in the above two schemes , no information on @xmath45 is attainable from the reflection probability @xmath53 , see ( [ reflection_probability ] ) , where @xmath45 disappears from @xmath53 for @xmath69 . a possible solution to this problem is to flip the orientations @xmath73 in the choice of the reference system @xmath62 in ( [ eq : refsystemf ] ) for the former scheme , while measuring the reflected @xmath3 in @xmath74 direction instead of @xmath38 for the latter . in this way , the schemes with the reflection probabilities work similarly to the ones with the transmission probabilities , with the same self - consistency condition needed for the tomographic reconstruction of the initial state of the target qubit , @xmath2 . the reason why these tomographic schemes that make use of the spin degree of freedom of @xmath3 work is the following . let @xmath75 denote the state with spin parallel ( anti - parallel ) to the orientation @xmath38 of the incident probe spin @xmath3 . then , the state of @xmath0 is in general expressed as @xmath76 and the spin state of the probe @xmath3 after the transmission by @xmath0 becomes ( apart from the normalization ) @xmath77 a similar expression is available for the reflection case . observe that the component @xmath78 of @xmath2 is associated to @xmath79 in @xmath80 , @xmath81 to @xmath82 , and so on : the spin state @xmath2 is more or less `` transferred '' to @xmath3 after the scattering . this is due to the heisenberg coupling @xmath54 , which `` swaps '' the states between @xmath0 and @xmath3 . this is why we can see the spin state of @xmath0 by looking at the spin state of @xmath3 . until now , we have shown how the initial state of the target qubit @xmath0 can be reconstructed by sending a probe qubit @xmath3 with a fixed wave numebr @xmath11 . a natural question would then arise : can we use the spatial degree of freedom of @xmath3 to optimize the above tomographic schemes in the sense that possible errors in the scattering data can least affect the determination of the state ? it is actually possible . for instance , one would be able to reduce the effects of possible errors in the observations of the probabilities @xmath55 on the reconstructed vector @xmath45 , by properly tuning the incident wave number @xmath11 . observe that ( [ v components ] ) and ( [ component of v along n ] ) are linear mappings between the bloch sphere of radius @xmath83 and the probability space , associated to the scattering data @xmath55 . the reconstructed vector @xmath45 is least sensitive to the errors in the observed probabilities @xmath55 , when the volume of this probability space is maximum . stated differently , under such a condition , the probabilities @xmath55 are the most sensitive to the bloch vector @xmath45 to be reconstructed , and one can do a better tomography . for the first scheme , in which we tune the detection orientation @xmath39 of the spin of the probe @xmath3 , the volume associated to the probabilities is maximum when the jacobian of the map ( [ v components ] ) , which is given by the determinant @xmath84 is minimum . for the second scheme , in which the incident and detection orientations of the probe spin @xmath3 are the same @xmath69 , the coefficient multiplying the scattering probability @xmath52 in ( [ component of v along n ] ) , @xmath85 is to be minimized . in fig . [ fig : first strategy optimization ] , @xmath86^{1/3}$ ] and @xmath87 are plotted as functions of the incident wave number @xmath11 of the probe @xmath3 . they become minimum at @xmath88 and @xmath89 , respectively . in particular , observe that , at @xmath90 , the formula for the tomography ( [ component of v along n ] ) is reduced to @xmath91 from which it immediately follows that the probability @xmath52 ranges @xmath92 which is the maximum in this scheme . ) between the probability space and the bloch sphere , @xmath93 given in ( [ eqn : detm ] ) ( solid ) , and the coefficient @xmath87 ( dashed ) , as functions of the incident wave number @xmath11 of the probe @xmath3 , which are both to be minimized by tuning @xmath11 for the optimizations of the two approaches of strategy i , presented in the text . @xmath86^{1/3}$ ] is actually plotted instead of @xmath93 , to better compare it with @xmath87 . @xmath86^{1/3}$ ] is minimum at @xmath94 , while @xmath87 at @xmath89 ( indicated by dots ) . ] the same analysis can be applied to the schemes with reflection probabilities , for which the optimal momenta are @xmath95 and @xmath89 for the two schemes , respectively . the _ spatial _ degree of freedom of @xmath3 can itself play a fundamental role for the tomographic reconstruction of the target _ spin _ state @xmath2 . we fix the incident and detection orientations @xmath96 and @xmath97 of the qubit @xmath3 and tune the wave number @xmath11 to collect sufficient number of scattering data required for the tomography of @xmath2 . given in ( [ determinant__second strategy ] ) , as a function of @xmath98 and @xmath99 . only the contours in the range @xmath100 are shown . @xmath101 takes its minimum value @xmath102 at @xmath103 in the unit @xmath104 ( indicated by a dot).,scaledwidth=40.0% ] let us fix @xmath39 perpendicular to @xmath38 and select three different incident wave numbers @xmath105 . a collection of the three transmission probabilities with these wave numbers yields @xmath106 where @xmath107 , and @xmath108 are the three components of the vector @xmath45 with respect to the reference frame fixed by @xmath109 . this relation can not be inverted , since two columns of the matrix are proportional to each other , irrespectively of the choice of @xmath110 . this is due to the fact that , once the orientations @xmath38 and @xmath39 are fixed , only two components of the vector @xmath45 along the directions specified by @xmath111 and @xmath112 are involved in the transmission probability ( [ transmission_probability ] ) , while it is insensitive to the other component of @xmath45 perpendicular to the plane spanned by these directions . the same happens if we collect the reflection probabilities ( [ reflection_probability ] ) in a similar way . this problem however can be overcome by combing the transmission and reflection probabilities . for instance , @xmath113 which is inverted as @xmath114\end{aligned}\ ] ] with an inverse matrix @xmath115 , when @xmath116 is nonvanishing and finite . this condition is fulfilled by @xmath98 and @xmath99 that are both finite and different from zero as well as different from each other . in this way , we can reconstruct the _ spin _ state @xmath2 by making use of the _ spatial _ degree of freedom of @xmath3 . note that only two experimental setups are actually needed to collect the three scattering data : the two probabilities @xmath117 and @xmath118 are obtained at the same time in a single setup , by sending the probe @xmath3 with @xmath98 and seeing whether it is transmitted or reflected with its spin orientated to @xmath39 . similarly to the first strategy discussed in the previous subsection , the present scheme is optimized by appropriately choosing the two incident wave numbers @xmath98 and @xmath99 of the probe @xmath3 . under the same criterion as the one for the first strategy [ maximizing the volume of the probability space associated by the linear mapping ( [ good_secondstrategy1 ] ) to the bloch sphere of radius @xmath83 , to which the vector @xmath45 belongs ] , the optimal choice of @xmath119 is found by minimizing the quantity @xmath101 from ( [ determinant__second strategy ] ) . see fig . [ fig : optimal momenta second strategy ] , where the optimal set of wave numbers for the present strategy is found at @xmath103 in the unit @xmath104 . in this paper , we have discussed the state reconstruction / tomography of a fixed qubit through the scattering data of a probe qubit off the target . we have presented two different strategies for the tomography , in which the spin and spatial degrees of freedom of the probe qubit play different roles . the first strategy makes use of the spin degree of freedom of the probe . the spin state of the target is more or less transferred to the probe spin during the scattering , and therefore , we can infer the spin state of the target through the state tomography of the probe spin . the other degree of freedom , the momentum of the probe , can be utilized to optimize the tomographic scheme . the spatial degree of freedom can also play a central role for the state tomography of the target spin . in the second strategy , three scattering data required for the state tomography of a target spin are collected by choosing different incident wave numbers and scattering directions ( transmitted or reflected ) , with the incident and detection orientations of the probe spin being fixed . this tomographic scheme can be optimized also by appropriately tuning the set of the incident wave numbers . the strategies introduced in this paper for a single fixed qubit can be generalized to the tomography of multiple qubits . in particular , the detection of entanglement would be an important task in the light of quantum information . in order to reconstruct the state of @xmath120 spins , we need @xmath121 different experimental setups . imagine , for instance , how to choose 15 different sets of orientations of the incident and detection spin states of the probe qubit for the two - qubit tomography . if the number of the target qubits grows , tuning the probe spin to the different orientations required for the first strategy would become more and more difficult . in such a case , the second strategy may provide a way out of this problem . it would be worth exploring such a potential of the scheme , which would be an interesting future subject . this work was done during a.dp.s stay at waseda university under the support by a special coordination fund for promoting science and technology from the ministry of education , culture , sports , science and technology , japan . it is also supported by the bilateral italian - japanese projects ii04c1af4e on `` quantum information , computation and communication '' of the italian ministry of education , university and research , by the joint italian - japanese laboratory on `` quantum information and computation '' of the italian ministry for foreign affairs , by the grant - in - aid for young scientists ( b ) ( no . 21740294 ) from the ministry of education , culture , sports , science and technology , japan , and by the grant - in - aid for scientific research ( c ) from the japan society for the promotion of science . m. a. nielsen and i. l. chuang , _ quantum computation and quantum information _ ( cambridge university press , cambridge , 2000 ) ; _ the physics of quantum information : quantum cryptography , quantum teleportation , quantum computation _ , edited by d. bouwmeester , a. zeilinger , and a. ekert ( springer , berlin , 2000 ) .
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we discuss the state tomography of a fixed qubit ( a spin-1/2 target particle ) , which is in general in a mixed state , through 1d scattering of a probe qubit off the target .
two strategies are presented , by making use of different degrees of freedom of the probe , spin and momentum .
remarkably , the spatial degree of freedom of the probe can be useful for the tomography of the qubit .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
fluctuations of physical quantities near zero frequency have been investigated by several authors since the papers of @xcite and @xcite . a general theory on the fluctuation - dissipation theorem , which will be the starting point of this paper , was developed by @xcite . to the best of our knowledge , a concrete expression for the low - frequency spectrum of fluctuations of magnetic fields in a thermal plasma was obtained for the first time by @xcite . they found a peak around @xmath1 magnetic fluctuation which was interpreted as the evanescent energy component of electromagnetic fluctuations `` screened '' in a plasma below the plasma frequency . the impact of such result into the cosmic microwave background was then investigated by @xcite . although in these two references the authors claim that the fluctuations were rigorously computed , several approximations were indeed made and they were not able to get an unique formula covering the low- and high - frequency spectrum . the aim of this paper is to reevaluate the derivation of the spectrum of magnetic fluctuations avoiding approximations in the low - frequency region and also in the transition between the low- and the high - frequency spectrum . several different behaviors between our results and the previous one , mainly in the low - frequency part of the spectrum , are found and discussed . the fluctuation - dissipation theorem developed by @xcite is able to deal with the thermal fluctuations inside a plasma in or near thermal equilibrium . the expression for the magnetic field fluctuation in an homogeneous isotropic non - magnetized equilibrium plasma was obtained by @xcite looking at waves in such a plasma . in an electron - positron plasma , for example , the magnetic fluctuations in wave number and frequency space is given by @xmath2 \omega^2}\end{aligned}\ ] ] where @xmath3 is the boltzmann constant and @xmath4 and @xmath5 are , respectively , the plasma and the collisional frequencies . here @xmath6 , and @xmath7 being @xmath8 the particle density inside the plasma , @xmath9 is the atomic number of these constituents , @xmath10 and @xmath11 are respectively the charge and the mass of the electron , and @xmath12 is the lorentz factor given by @xmath13 . this result can be integrated in @xmath14 to get ( the fourier transform ) @xmath15 and the magnetic energy density is @xmath16 . the integral of eq . ( [ original - formula ] ) over wave numbers shows a high wave number divergence . according to @xcite , this is expected since the derivation is based on classical fluid equations of motion and the constant collision frequency @xmath5 is independent of @xmath17 . however , they prefer to carry on their analyzes in the simpler phenomenological approach . in order to overcome the large @xmath17 dependence , they first take the limit @xmath18 and then they integrate over @xmath17 to infinity , which corresponds to the vanishing cross section of collisions as @xmath19 . this is a very delicate point and we will turn back to this point in section [ general ] . for both the high frequency and high wave number limits the authors emphasized that the expression of eq . ( [ original - formula ] ) has a substantial value only where @xmath20 . the combined high - frequency and high wave - number limits were get by letting @xmath18 . the expression for the low - frequency spectrum was obtained by breaking up the @xmath17 integral into two intervals , by introducing a cutoff value @xmath21 , with @xmath22 . in the integration from 0 to @xmath21 , @xmath5 was kept finite while in the integral from @xmath21 it was used the approximation @xmath18 . the expressions obtained for the high and low parts of the spectrum was , respectively : @xmath23 and @xmath24 where @xmath25 is the heaviside step function , @xmath26 , @xmath27 , and @xmath28 . finally the zero frequency limit of the magnetic fluctuations is give by @xmath29 at this point the frequency spectral intensity was plotted for a temperature @xmath30 k by requiring that the value of @xmath21 ( or @xmath31 ) provide a smooth behavior at the joint between the low - frequency spectrum and the black - body spectrum . the choice was @xmath32 or ( @xmath33 . the result for other values of the temperature were presented in another paper by @xcite . the main claims by these authors was that the intensity of the spectrum does not vary sensitively with @xmath21 and that near @xmath34 the spectrum goes like @xmath35 . let us now show our results . we have integrated eq . ( [ original - formula ] ) over @xmath17 analytically , without any approximation , by the partial fractions technique . the exact result for the indefinite integral over the wave number is : @xmath36\right\}\end{aligned}\ ] ] with the following definitions : @xmath37 is the debye length , @xmath38 , @xmath39 , @xmath40 , @xmath41 , @xmath42 , @xmath43 , @xmath44 , @xmath45 , @xmath46 , @xmath47 , @xmath48 , and @xmath49 where @xmath50 . note that the general result shows only a linear divergence in @xmath17 restricted to the first term of eq . ( [ indef_integ ] ) . this term , however , can not be simply discarded by doing the limit @xmath51 before we integrate over @xmath17 , as did by @xcite , since it plays a very important role when the strict limit @xmath52 ( which we prefer to indicate from now on by @xmath53 ) is to be considered , even when @xmath17 is large . indeed , if we discard it for all large values of @xmath17 it can be shown that the limit @xmath54 of @xmath55 will be negative . therefore , our result for the definite integral can be put in the form @xmath56 where @xmath57 the term @xmath58 will be taken as @xmath59 with @xmath60 as large as we want . this will render the confrontation with the previous result easier . we note that for @xmath61 @xmath62 , @xmath63 . these three terms can be written as functions of the original parameters , considering that @xmath64 } + \omega(\omega^2 + \eta^2 -\omega_p^2)\right]^{1/2}\ ] ] and @xmath65 } - \omega ( \omega^2 + \eta^2 -\omega_p^2)\right]^{1/2}\ ] ] compared to eqs . ( [ tajima_high])-([zero_lim_tajima ] ) it is immediately evident how our result is different from those of equations , showing a much more complicated dependence of the frequency spectrum on the variable @xmath66 , and on the parameters @xmath4 and @xmath5 , which dependencies on plasma temperature are shown , respectively , in figures [ fig - omega ] . our full result @xmath55 is shown in fig . [ fig - full ] , where both the normalized ( gray curve ) and non normalized ( green curve ) spectral intensities are given , for @xmath67 @xmath62 e @xmath68 k. and normalized @xmath69 spectral intensities showing the different components of the spectrum , @xmath58 , @xmath70 and @xmath71 , as given by eq . ( [ our - limit ] ) . ] in this figure we have also plotted each one of the terms that contribute to @xmath55 are shown in different colors . the deep we see in this figure near @xmath72 tends to disappear as @xmath0 goes to high values ; as @xmath0 decreases up to @xmath73 @xmath62 , the ordinate of the deep tends to zero . the detail of the non normalized spectral intensity near @xmath74 is shown in fig . [ fig - detail ] . it shows naturally a smooth behavior between the low - frequency spectrum and the blackbody spectrum , which is constructed by hand in @xcite . .,height=302 ] if we compare this graph to the correspondent one shown in fig . 1(a ) of the paper of @xcite , we see that our result , for @xmath75 , is a classical blackbody radiation spectrum which goes to zero at @xmath76 while their blackbody spectrum has a much greater width ( more than two orders of magnitude ) . another important qualitative difference between the non approximate and the approximate results is that we found a very peculiar oscillations in @xmath55 for the @xmath77 region of the spectrum , as can be seen from figure [ fig - detail - k10 ] . these oscillations occur in an @xmath78 region where the classical blackbody spectrum still have a significant value ; there is however , in this case , a strong interference in the total spectrum @xmath55 , eq . ( [ our - result ] ) , due to a change of sign of the function @xmath79 . such a kind of behavior was found just for cut - off values of the order of @xmath80 @xmath62 . for values of @xmath0 greater than this one such fluctuations disappear . in any case , this feature confirm our statement that the result can vary sensitively with @xmath0 , contrary to what was sustain by @xcite . @xmath62 . ] finally , we have studied the behavior of @xmath81 by varying @xmath66 e @xmath5 . the result is shown in figure [ fig - surf6 ] . notice that just the peak of the zero - frequency plasma spectrum depends on @xmath5 ( the blackbody part remains unchanged ) . indeed , the spectral intensity varies two orders of magnitude by varying @xmath5 by two orders too , namely , it goes from @xmath82 , for @xmath83 , to @xmath84 , for @xmath85 . thus , our result indicates that , when one goes backwards in time , temperature grows , dynamo action is enhanced ( since @xmath5 goes down ) , and the resonance peak of the zero - frequency plasma peak goes down . and in this paper we have computed the spectrum of magnetic fluctuations of an homogeneous cosmic plasma avoiding any approximations . several different behaviors between our results and the previous one obtained by @xcite , mainly in the low - frequency part of the spectrum , are found and discussed . it is important to stress that the exact result indicates that the peak of the zero - frequency spectrum can indeed vary sensitively with the cut - off value @xmath86 . in the light of this new result , and following the papers of @xcite and @xcite , the problem of establishing an upper limit for fractal space dimensionality from cobe data can be revisited . our results can still be improved towards cosmological applications by computing the fourier transformed volume element @xmath87 in terms of curved riemannian space embedded into general relativistic spacetime . this will allow us to address the problem of dynamo action in einstein cosmology . the dynamo effect in plasmas is a competitive effect between convection of the cosmological fluid and the plasma resistivity . this is the reason why is so interesting to consider the relation between the dynamo action @xmath12 and the plasma resistivity and its frequency . recently , some new investigations on this directions were addressed by @xcite , by using the magnetic field correlation tensor in space of negative curvature , and by @xcite in the case of positive curvature . also recently , @xcite has obtained a constraint on dynamo action from cobe data , using two - dimensional spatial sections of negative curvature of friedmann universe , based on the general relativistic magnetohydrodynamic equation derived by @xcite . in near future , by using the formula of @xcite , @xmath88 where @xmath12 is the magnetic field growth rate ( dynamo action ) , we expect to compute @xmath12 in terms of the magnetic plasma resistivity ( @xmath5 ) and plasma frequency ( @xmath4 ) . in this way we shall estimate the variation of dynamo action in terms of @xmath5 and @xmath4 . the authors would like to thank garcia de andrade for useful comments .
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magnetic fluctuations in a non - magnetized gaseous plasma is revisited and calculated without approximations , based on the fluctuation - dissipation theorem . it is argued that the present results are qualitative and quantitative different form previous one based on the same theorem . in particular , it is shown that it is not correct that the spectral intensity does not vary sensitively with @xmath0 .
also the simultaneous dependence of this intensity on the plasma and on the collisional frequencies are discussed .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
quantum phase transitions between a fermi liquid and magnetic phases have been a subject of experimental and theoretical investigations for several decades @xcite . the transition to a spin - density - wave ( sdw ) in particular is relevant to problems of current interest . in the cuprates , daou _ et al . _ @xcite argued that the fermi surface change associated with this transition is a key to understanding anomalous normal state properties . recent studies in the pnictides @xcite , heavy fermion materials @xcite and organic superconductors @xcite focus on the relation between the sdw and superconductivity . in fact , strong experimental similarities between quantum critical behavior in the organics , pnictides and cuprates have been pointed out recently @xcite . the electron - doped cuprates @xcite have provided an early example where quantum critical behavior has been inferred from the temperature dependence of resistivity at low temperature . it was measured to be linear @xcite from 35mk to 10k in pr@xmath11ce@xmath12cuo@xmath13 ( pcco ) at doping @xmath14 . more recent transport @xcite and thermopower measurements @xcite also suggest the presence of a quantum critical point at a similar doping . the precise nature of the quantum critical point remains however unclear , as thoroughly discussed in ref . . for example , it has also been suggested that the quantum critical point coincides with the onset of superconductivity in the overdoped regime @xcite . . the arrows gives examples of pseudo - nesting conditions , namely of points such that @xmath15 is equal to the antiferromagnetic wave vector . the @xmath0 ordering wave - vector is defined with respect to the @xmath16 coordinate system . in the field theory approach introduced later , we work in the rotated ( @xmath17 ) coordinate system . ] here we study quantum critical behavior associated with the transition between a sdw phase and a fermi liquid when the wave vector @xmath0 of the sdw connects hot spots with parallel fermi velocities . the two fermi surfaces connected by @xmath0 in this case are tangent to each other , as shown in fig . [ bzf ] . on a spherical fermi surface , the fermi wave vector would satisfy the condition @xmath18 . we call this pseudo - nesting @xcite . we will explain why the qcp can be located at , or close to , the pseudo - nesting filling @xmath19 . this occurs naturally in the one - band hubbard model for the electron - doped cuprates and we will perform some of our calculations specifically for this case , although the frequency and temperature dependencies that we find are valid more generally . the methods that we describe below can be applied to electron - doped cuprates because these materials are described by a hubbard model in an intermediate coupling regime where one can neglect effects induced by mott physics @xcite . the theory of hertz @xcite and millis@xcite has formed the basis for much of the work on quantum critical phenomena . in this approach , fermions are integrated out and an effective bosonic theory for the collective modes is studied using standard renormalization group methods that can be taken to high order @xcite . it has been pointed out by abanov and chubukov @xcite that for a commensurate sdw at the upper critical dimension , namely @xmath20 for @xmath4 , all the coefficients of the bosonic theory are singular so that one must treat simultaneously the bosonic collective modes and the fermions . metlitski and sachdev @xcite have reexamined this problem and obtained the non - fermi liquid behavior at the hot spots , and shown that the bosonic sdw spectrum does not obey dynamic scaling with @xmath4 but instead that a super - power - law form is obtained . they have also thoroughly discussed the failure of the @xmath21 expansion at higher order , leading to a strong - coupling problem . more recently , it has been argued @xcite that non - fermi liquid corrections are also important away from the hot spots . an alternate approach is the self - consistent - renormalized theory of moriya @xcite . it is in the universality class of the spherical model and as such its critical behavior will not be exactly that expected for the o(3 ) model . nevertheless , it can be accurate away from the critical point and provide leading order estimates for the exponents . the two - particle - self - consistent ( tpsc ) theory @xcite is a related approach that has critical behavior similar to that of moriya , including logarithmic corrections @xcite . it has the advantage that although it is an approximate solution to the hubbard model , it is quantitatively very close to benchmark quantum monte carlo results @xcite . it is non - perturbative , does not include any phenomenological parameters , has internal consistency checks , and satisfies a number of exact results . previous theoretical studies have mostly been done for the case where the quantum critical point occurs when the fermi velocities at the hot spots that are connected by the sdw are not parallel , unlike the case of parallel velocities we consider here ( see fig . [ bzf ] ) . such a case of parallel fermi velocities is generic in one dimension , but at first sight appears as an accident in two dimensions , because upon translation by the @xmath0 sdw wave vector , the fermi surfaces touch at only one point . if the surfaces were flat , we would recover the case of perfect nesting encountered in one dimension . the curvature here provides a cutoff , and so we refer to the situation with parallel fermi velocities as `` pseudo - nesting '' @xcite . such fermi surfaces have also been studied in three dimensions @xcite , but in the presence of this pseudo - nesting the spin susceptibility is singular in two dimensions @xcite and the analysis must be redone . altshuler , ioffe and millis @xcite first looked at the case where the instability is at @xmath22 , hence connects parallel segments of the fermi surface , but the sdw wavelength is not commensurate with the lattice . they found that the transition is weakly first order with an intermediate scaling regime when the sdw wavelength is close to @xmath0 . the scaling regime was obtained in a systematic expansion in a number proportional to the inverse number of fermion flavors . krotkov and chubukov @xcite found different results for the self - energy . here we consider only the commensurate case . some of our results differ from those of previous authors because they overlooked the significance of the umklapp process shown by the top double arrow in fig . [ bzf ] . we use two different approaches . we obtain finite temperature results appropriate for the sdw quantum critical point of electron - doped cuprates using tpsc . then a field - theory for the spin - fermion model allows us to find the finite - frequency zero temperature results and some finite temperature results . the results of both approaches are consistent . we do not , however , consider the possibility of a first order transition @xcite . the rest of this paper is organized as follows . in sec . [ sec : model ] we present the model along with general arguments suggesting why one should expect the antiferromagnetic quantum critical point to be located close to pseudo - nesting , namely at a filling where the @xmath0 wave vector connects parts of the fermi surface that are tangent , or equivalently with parallel fermi velocities . [ sec : tfinite ] contains the finite temperature results . they are obtained with tpsc , which is described in sec . [ sub : tpsc ] . analytical results for the behavior at the qcp are illustrated with numerical examples appropriate for electron - doped cuprates in the subsections of sec . [ sub : analytical ] . the critical behavior is obtained in sec . [ sub : criticaltpsc ] . zero temperature finite - frequency results are treated in sec . [ sec : fieldtheory ] with field theoretical methods . the lagrangian appears in sec . [ sub : lagrangian ] followed by sections on the polarization bubble ( spin susceptibility ) [ sub : polarizationft ] , on the electron self - energy [ sub : selfft ] and on the irrelevance of the quartic term [ sub : scaling ] . an appendix on vertex corrections [ sec : vertex ] appears after the summary in sec.[sec : summary ] . consistency between tpsc and field theory results are pointed out in the field theory sec . [ sec : fieldtheory ] . in this section , we introduce the model and give generic arguments why we expect the quantum critical point to often be located close to the filling where translation by the antiferromagnetic wave vector leads to fermi surfaces that are tangent to each other , as illustrated in fig . [ bzf ] . while the frequency and temperature dependencies that we find do not depend on details of the model , specific numerical examples at finite temperature calculations will be performed on the two - dimensional @xmath23 hubbard model on the square lattice at weak to intermediate coupling . the model is given by @xmath24 where @xmath25 are the hopping integrals , @xmath26 are the site index , @xmath27 is the spin label , @xmath28 and @xmath29 are the particle creation and annihilation operators . doubly occupied sites cost an energy @xmath30 and @xmath31 . units are such that @xmath32 , @xmath33 and lattice spacing is unity . the kinetic energy of a single - particle excitation in momentum space is obtained from@xmath34 with the sum over @xmath35 running over all neighbors of any of the sites @xmath36 . the chemical potential @xmath37 is chosen so that we have the required density . one can explain on general grounds the filling where the qcp is likely to occur . figure [ fig : suscepatqcp ] displays the lindhard function @xmath38 along the @xmath39 direction for different fillings . its maximum is at @xmath0 so it is symmetric in @xmath40 and @xmath39 . there are two remarkable features . first , below a certain doping @xmath19 , the maximum value is almost independent of filling @xcite , and second it falls rapidly as soon as the filling exceeds @xmath19 . the filling @xmath19 corresponds to the point where the fermi surfaces joined by @xmath0 touch instead of intersecting . a @xmath41 change in filling leads to almost @xmath42 drop in value of the susceptibility . if we consider a simple stoner criterion for the transition , we would conclude that if @xmath30 takes the value @xmath43 , then the qcp would be close to this filling @xmath19 . this does not require fine tuning because the value of @xmath30 that should enter the stoner criterion is the value renormalized by kanamori - brckner screening @xcite . this renormalized value becomes essentially @xmath30 independent when @xmath30 becomes of the order of the bandwidth because the two - body wave function creates a cusp to minimize double - occupancy @xcite and the renormalized interaction can not become larger . this maximum renormalized value in tpsc , @xmath44 , takes a value @xcite near @xmath45 . in addition , in tpsc the value of @xmath44 self - consistently adjusts itself to the value necessary to prevent a finite temperature phase transition on the sdw side of the qcp . although , at sufficiently low temperature , details will start to matter and one needs to start to tune the value of @xmath30 to find the qcp precisely at @xmath19 , there is an intermediate temperature scale that can be quite broad where fine tuning is unnecessary . point as a function of doping for @xmath46 , @xmath47 and @xmath48 , values that are appropriate for electron - doped cuprates . the rapid fall with filling larger than @xmath49 , close to the critical filling , is apparent.,title="fig : " ] + in this section , we use the non - perturbative two - particle self - consistent ( tpsc ) approach @xcite . this approach respects the pauli principle , the mermin - wagner theorem and conversation laws . it also contains quantum fluctuations in crossed channels that lead to kanamori - brckner screening . @xcite it is valid in the weak to intermediate coupling regime @xmath50 and not too deep in the renormalized classical regime where a pseudogap is observed . it has been benchmarked on quantum monte carlo calculations on the hubbard model . @xcite . tpsc has been shown to be in the @xmath51 universality class of the @xmath52 model @xcite . it has the same critical behavior as moriya theory and hence has the same logarithmic corrections @xcite . these logarithms have the same functional form as those of the renormalization group asymptotically close to the quantum critical point , but in tpsc and in moriya theory the mode - mode coupling term does not flow , hence the corrections may differ in the details from the renormalization group @xcite . quantum critical behavior of the susceptibility and of the self - energy in the closely related spin - fermion model has been discussed by abanov et al . @xcite . it has been argued from detailed comparisons of numerical calculations with experiment @xcite that strong - coupling physics is not important for electron - doped cuprates , at least not too close to half - filling . hence , tpsc is appropriate to study these compounds . it gives a satisfactory description of arpes data @xcite , and the temperature @xmath53 , where the pseudogap seen in arpes opens up experimentally , corresponds to that where the antiferromagnetic correlation length coincides with the thermal de broglie wavelength @xcite , as predicted for two - dimensional precursors of three - dimensional long - range order @xcite . hence , all the numerical results are presented in units where @xmath54 , @xmath55 , @xmath32 ( with @xmath56 component of spin defined by @xmath57 ) for values of the hubbard model hopping parameters appropriate for electron - doped cuprates , namely second and third nearest - neighbor hopping @xmath47 and @xmath48 @xcite . interaction strengths @xmath46 and @xmath58 , again in the range appropriate for electron - doped cuprates , @xcite will be considered . we first present the formalism and then give analytical and numerical results for the qcp . given the hubbard model parameters , tpsc has no adjustable parameter . irreducible vertices are obtained self - consistently and in such a way that the pauli principle and conservation laws are obeyed . the formal derivation is given in refs . . here , we simply present the equations that are solved . in tpsc , the retarded spin @xmath59 and charge @xmath60 susceptibilities are written as@xmath61 where @xmath62 is the non - interacting retarded lindhard function at wave vector @xmath63 and angular frequency @xmath64@xmath65 here , @xmath66 is the fermi function @xmath67 , @xmath68 is the temperature and @xmath69 is the total number of sites . the effective spin interaction @xmath44 is evaluated without adjustable parameter using the _ _ ansatz _ _ @xcite@xmath70 with the local - moment sum rule that follows from the fluctuation - dissipation theorem@xmath71 where @xmath72 and @xmath73 is the double occupancy . we dropped the site index using translational invariance and we used the pauli principle to write @xmath74 similarly the irreducible vertex @xmath75 entering @xmath76 is found using a sum - rule that is the analog of eqn.([sum_rule ] ) for spin : @xmath77 the crossing symmetric self - energy is obtained from @xmath78 g_{\sigma } ^{\left ( 1\right ) } ( k+q).\label{self}%\ ] ] the superscript @xmath79 reminds us that we are at the second level of approximation . @xmath80 is the same green s function as that used to compute the susceptibilities @xmath81 . charge fluctuations @xmath76 are included in numerical calculations but they are neglected in the analytical results because they are small . since the self - energy is constant at the first level of approximation , this means that @xmath80 is the non - interacting green s function with the chemical potential that gives the correct filling . this chemical potential @xmath37 is slightly different from the one that we must use in @xmath82 to obtain the same density @xcite . unless otherwise specified , all the numerical results below are obtained using the matsubara frequency version of equations ( [ chisp ] ) to ( [ self ] ) without any approximation , hence they are valid at arbitrary distance from the quantum critical point . we begin below with the ornstein - zernicke form of the spin susceptibility that is usually valid when the correlation length is large . the case where there is perfect nesting leads us naturally to the pseudo - nesting condition relevant for this paper . the situation , however , is not as simple as usual since the ornstein - zernicke form for the spin susceptibility is incorrect in our case , as we will explain . the self - energy is treated at the end of this section . [ [ general - case ] ] general case + + + + + + + + + + + + when the correlation length is large , one usually assumes that the denominator of the spin susceptibility can be expanded around the wave vectors @xmath83 , where the maxima in @xmath84 occur in @xmath85dimensions . one then obtains @xmath86 where @xmath63 is measured with respect to the wave vector @xmath87 where the spin suseptibility is maximum ( @xmath0 in our case ) . defining @xmath88 as the value of the interaction at the mean - field sdw transition , the other quantities in the previous expression are @xmath89 in the expression for the spin susceptibility , the denominators are expanded around the @xmath90 wave vector . to obtain analytical results for the imaginary part of the self - energy @xmath91 in eqn.([self ] ) we use the spectral representation for the susceptibilities and for the green s function , perform the sum of the internal matsubara frequency and then the analytical continuation neglecting the charge fluctuations , to obtain @xmath92 \chi_{sp}^{\prime\prime } \left ( \mathbf{q}_{\vert},q_{\perp}\left ( \mathbf{k}_{f}+\mathbf{q}% _ { d},\omega,\omega^{\prime}\right ) ; \omega^{\prime}\right ) \label{imsigmareel}%\]]where @xmath93 , the component of @xmath63 parallel to the fermi velocity @xmath94 , is obtained from the solution of the equation @xmath95 for all fermi wave vectors , where @xmath96 the above equation reduces to @xmath97 where @xmath98 is the fermi velocity in the hot region , i.e. where @xmath99 in the asymptotic form of the spin susceptibility eqn.([chi_as ] ) , the wave vector appears only in the form @xmath100 so that keeping this general form in the equation for the self - energy eqn.([imsigmareel ] ) , we obtain @xmath101 \chi_{sp}^{\prime\prime } \left ( \mathbf{q}_{\vert},(\omega+\omega')/v_f ; \omega^{\prime},t\right ) . \label{selfwithchipp}\ ] ] normally , one expects @xmath102 to be a temperature independent constant of the order of the lattice spacing and @xmath103 to be a constant of the order of the fermi energy . in the case of perfect nesting , or of pseudo - nesting , this is not the case . [ [ perfect - nesting ] ] perfect nesting + + + + + + + + + + + + + + + although the case we are interested in does not correspond to perfect nesting , understanding that case first will facilitate our task later . there is perfect nesting when the equality @xmath104 is satisfied for all wave vectors , with @xmath83 the nesting wave vector . this case was treated by virosztek and ruvalds@xcite . the quantities @xmath105 and @xmath106 that are usually assumed temperature independent , here become temperature dependent . we show this below . for perfect nesting , the lindhard function becomes@xmath107 so that changing to an energy integral we have@xmath108 where @xmath109 is the density of states . the real part at zero frequency on the other hand is given by@xmath110 in two dimensions , there is a well known logarithmic divergence of the density of states @xmath111 at the van hove singularity . neglecting this logarithmic divergence that appears only for a special filling in the hole - doped case , we take @xmath111 as a constant . in that case , integrating by part and replacing the upper bound by infinity in the convergent integral we are left with @xmath112 where @xmath113 is a temperature independent constant . we also have @xmath114 these results suggest that the quantity @xmath106 defined by eqn.([gamma_0 ] ) scales as @xmath115 following ref . @xmath116 we move on to demonstrate analytically that @xmath105 in eqn.([xsi02 ] ) scales as @xmath117 . the @xmath118 dependence , fundamentally comes from the second derivative of @xmath119 in eqn.([xhi1log ] ) . we shall now make this argument more rigorous . keeping for a while a general notation where @xmath36 is some direction in the brillouin zone , and @xmath63 is measured with respect to the center of the zone , one can write @xmath120 where @xmath121 measuring @xmath63 with respect to @xmath83 we evaluate the above second derivative at @xmath122 as before , for perfect nesting we have @xmath123 the last equation shows that @xmath124 scales as @xmath125 times a function of @xmath126 in the integrals , the derivatives of the type @xmath127 will not introduce singular terms in temperature . hence , replacing them by some average value in the brillouin zone , we can change the integration variable to energy and the most singular terms in temperature will come from @xmath128 neglecting the energy dependence of the density of states , we are left with@xmath129 using the definition of @xmath105 , eqn.([xsi02 ] ) , and the result for @xmath106 eqn.([gamma0avecxsi0 ] ) above , we have that @xmath130 [ [ pseudo - nesting ] ] pseudo - nesting + + + + + + + + + + + + + + touches the antiferromagnetic brillouin zone . the important integration region is over a rectangle of thickness @xmath68 and width @xmath131 . the critical chemical potential @xmath132 where the antiferromagnetic zone boundary touches @xmath133 on the fermi surface is the solution of @xmath134 . this corresponds to a filling @xmath135 for @xmath136 and @xmath137 . for @xmath46 , the critical filling that we find , @xmath138 , is slightly larger . , title="fig : " ] + in this subsection we show that , for the pseudo - nesting case , the main contribution to the lindhard function at @xmath139 has the same form as in the perfect nesting case except for a temperature dependant prefactor . the previous calculation illustrates that the main contribution to the quantities of interest , @xmath105 and @xmath140 come from nested regions of the fermi surface . in the pseudo - nesting case @xcite illustrated in fig . [ fig : integrationdomain ] , the fermi surface displaced by @xmath139 just touches the original fermi surface , with the fermi velocities of the two surfaces that are parallel at the touching point . as in the perfect nesting case , the most important contribution to the integral for the lindhard function around @xmath139 comes from the regions in @xmath141-space connected by @xmath139 , with a width around the fermi surface that corresponds to an energy range @xmath142 . now , imagine that we divide the integral over @xmath143 near one of those points of the fermi surface , for example @xmath144 , into two components , @xmath145 parallel to the fermi velocity , i.e. perpendicular to the fermi surface , and @xmath146 parallel to the fermi surface at @xmath147 . for @xmath143 near that region of the fermi surface , we can write in two dimensions @xmath148 where we have measured wave vectors with respect to @xmath147 and used the fact that @xmath149 the quantity @xmath150 measures the curvature of the fermi surface . from that approximation we have @xmath151 and thus @xmath152 since the terms in @xmath153 cancel out in that expression , this approximation is valid if the next term in the series of @xmath154 is negligible compared with the first one , namely if @xmath155 and since @xmath156 , we have the following upper bound for the temperature , @xmath157 now , for the fermi function , over a region around @xmath147 , we have @xmath158 since the derivative of the fermi function is even in @xmath159 , @xmath160 the region where this is valid is given by the condition @xmath161 which is satisfied if @xmath162 . therefore , from and , the lindhard function takes the form @xmath163 where @xmath164 is a domain such that @xmath165 while @xmath166 is the domain such that @xmath167 @xmath168 the integration over @xmath166 thus gives a factor proportional to @xmath169 . the integral over @xmath170 can be transformed into an integral over energy in the same way as the perfect nesting case , with a constant density of states determined by the fermi velocity . the domain delimited by @xmath164 and @xmath166 is depicted in fig . [ fig : integrationdomain ] . overall then , the final result will be that @xmath171 where the @xmath172 prefactor comes from the @xmath173 integration . a similar reasoning leads to@xmath174 which means that the regular temperature independent term represented here by @xmath175 dominates at low temperature . repeating the same analogous arguments for @xmath176 we find that @xmath177 which implies from the definition of @xmath178 eqn.([gamma_0 ] ) and eqs.([chi(1)0]-[chi(1)0 ] ) that @xmath179the same result for @xmath106 , within logarithmic corrections , as if we had perfect nesting . in three dimensions , the correction compared to perfect nesting is determined by the area spanned by @xmath180 , proportional to @xmath181 , hence we would have had @xmath182 and again @xmath183 results of numerical calculations shown in fig . [ fig : xsi0_and_gamma_0 ] confirm the power law temperature dependencies found above . at the actual filling @xmath19 where the fermi surface is tangent to the antiferromagnetic zone boundary , the power laws extend to low temperature . and @xmath178 evaluated at @xmath135 obtained from the condition that the fermi surface is tangent to the antiferromagnetic zone boundary , as explained in the caption of fig . [ fig : integrationdomain ] . ] and @xmath178 evaluated at @xmath135 obtained from the condition that the fermi surface is tangent to the antiferromagnetic zone boundary , as explained in the caption of fig . [ fig : integrationdomain ] . ] given that @xmath102 and @xmath106 are now temperature dependent , we should check whether the small @xmath63 and small @xmath64 expansion of the denominator that lead to eqn.([chi_as ] ) is still valid . normally , the expansion is of the form @xmath184 with @xmath185 a constant . since the function falls on a scale @xmath186 the higher order term @xmath187 can be neglected . however , our case is different . the coefficients of the expansion in powers of @xmath63 are singular at @xmath188 . for example we have @xmath189 where the @xmath190 comes from counting the powers of @xmath68 associated with derivatives in eqs.([d2chidq2initial ] ) to ( [ d2chidq2 ] ) and the @xmath172 from the restriction to the @xmath191 integral as usual . knowing the scaling of @xmath105 , we can rewrite @xmath192 so that we are left with@xmath193 where @xmath194 is a constant . the susceptibility will preserve a scaling form as a function of @xmath195 and @xmath196 if the scaling exponent is @xmath3 . indeed , in that case @xmath197 and since @xmath198 @xmath199 , the susceptibility becomes @xmath200 with @xmath201 constants . each higher power of @xmath202 has an additional power of @xmath203 coming from the additional derivatives of the non - interacting susceptibility and the scaling form is preserved to all orders . for frequency , there are also higher order terms , @xmath204 etc . hence we have the general scaling form @xmath205 we check in the following section that this is consistent with the tpsc self - consistency condition . the quantum critical behavior has been thoroughly studied in ref . for the case where the ornstein - zernicke form is valid . this analysis does not apply here because of the more general form of the spin susceptibility obtained in the previous section . nevertheless , it is interesting to note that with the ornstein - zernicke form , one obtains @xmath206 for both perfect and pseudo nesting . hence , simply taking into account the temperature dependence of @xmath178 , we recover @xmath3 scaling , namely@xmath207 note that with a temperature independent value for @xmath178 we recover the more usual result @xcite @xmath4 . the physics of the result for the correlation length is however quite different from the calculation with the ornstein - zernicke form . indeed , in the latter case , it is the self - consistency relation eqn.([sum_rule ] ) that determines the temperature dependence of the correlation length . in the present case , we found that temperature dependence in the previous section without invoking the self - consistency relation . we will show in sec.([sub : scaling ] ) that indeed in our case , the self - consistency relation leads to irrelevant corrections to the temperature dependence of the correlation length . the scaling of the correlation length can be obtained from numerical calculations by plotting , for example , the inverse of the width of the real part of the spin susceptibility at zero frequency measured at various fractions of the maximum value . for @xmath58 , the critical doping corresponds to @xmath135 where the fermi surface is tangent to the antiferromagnetic zone boundary . for that case , [ fig : chi_sp](a ) shows that whether we measure the width at half - maximum or at some other fraction of the maximum , that width scales essentially as @xmath2 , with small deviations probably coming from the fact that we have not reached the asymptotic regime . we also show on this figure the correlation length @xmath208 obtained from the ornstein - zernicke form eqn.([xsi2 ] ) using @xmath44 from the self - consistency relation eqn.([sum_rule ] ) . deviations from the @xmath2 power law occur if we measure the width of the spin susceptibility too far in the tails , i.e. for a small fraction of the maximum ( not shown ) . as demonstrated in fig . [ fig : chi_sp](b ) , deviations from @xmath2 also occur at low temperature for a value of @xmath30 ( @xmath209 in the present example ) where the critical point does not occur precisely when the fermi surface is tangent to the antiferromagnetic zone boundary . at various fractions of the maximum height . on ( a ) , for @xmath58 , the critical doping corresponds to @xmath135 , where the fermi surface is tangent to the antiferromagnetic zone boundary . the temperature scale is too small to detect possible logarithmic corrections . the results are consistent with @xmath3 . on ( b ) , deviations from @xmath2 occur at low temperature because , for the chosen value @xmath46 , the calculation is at the critical doping @xmath210 , slightly away from the point where the fermi surface is tangent to the antiferromagnetic zone boundary . also shown , @xmath208 obtained from the ornstein - zernicke form eqn.([xsi2 ] ) using @xmath44 from the self - consistency relation eqn.([sum_rule ] ) ] at various fractions of the maximum height . on ( a ) , for @xmath58 , the critical doping corresponds to @xmath135 , where the fermi surface is tangent to the antiferromagnetic zone boundary . the temperature scale is too small to detect possible logarithmic corrections . the results are consistent with @xmath3 . on ( b ) , deviations from @xmath2 occur at low temperature because , for the chosen value @xmath46 , the calculation is at the critical doping @xmath210 , slightly away from the point where the fermi surface is tangent to the antiferromagnetic zone boundary . also shown , @xmath208 obtained from the ornstein - zernicke form eqn.([xsi2 ] ) using @xmath44 from the self - consistency relation eqn.([sum_rule ] ) ] to conclude this section , we show that one can easily obtain the temperature scaling for two more quantities . first , from the general form for the susceptibility eqn.([suscep_simple ] ) used above , the static susceptibility at @xmath0 scales as @xmath211 which can be checked directly numerically , or more simply deduced from the temperature dependent results for @xmath212 , eqn.([xsi_qcp ] ) , and @xmath213 , eqn.([xsi_0 ] ) . behave as @xmath7 over the accessible temperature range . calculations are done with @xmath58 , @xmath47 and @xmath48 at the quantum critical filling @xmath135 . the lower figures are the corresponding results for @xmath46 , @xmath47 and @xmath48 at the quantum critical point @xmath214 . since in that case the fermi surface is not exactly tangent to the antiferromagnetic zone boundary , the @xmath7 behavior near @xmath215 is recovered only if the temperature is high enough that details of the fermi surface can not be resolved . the black lines on ( b ) and ( d ) are the curves defined by @xmath216 where @xmath217 is the component of @xmath218 parallel to @xmath219 at a given angle @xmath220 . ] behave as @xmath7 over the accessible temperature range . calculations are done with @xmath58 , @xmath47 and @xmath48 at the quantum critical filling @xmath135 . the lower figures are the corresponding results for @xmath46 , @xmath47 and @xmath48 at the quantum critical point @xmath214 . since in that case the fermi surface is not exactly tangent to the antiferromagnetic zone boundary , the @xmath7 behavior near @xmath215 is recovered only if the temperature is high enough that details of the fermi surface can not be resolved . the black lines on ( b ) and ( d ) are the curves defined by @xmath216 where @xmath217 is the component of @xmath218 parallel to @xmath219 at a given angle @xmath220 . ] behave as @xmath7 over the accessible temperature range . calculations are done with @xmath58 , @xmath47 and @xmath48 at the quantum critical filling @xmath135 . the lower figures are the corresponding results for @xmath46 , @xmath47 and @xmath48 at the quantum critical point @xmath214 . since in that case the fermi surface is not exactly tangent to the antiferromagnetic zone boundary , the @xmath7 behavior near @xmath215 is recovered only if the temperature is high enough that details of the fermi surface can not be resolved . the black lines on ( b ) and ( d ) are the curves defined by @xmath216 where @xmath217 is the component of @xmath218 parallel to @xmath219 at a given angle @xmath220 . ] behave as @xmath7 over the accessible temperature range . calculations are done with @xmath58 , @xmath47 and @xmath48 at the quantum critical filling @xmath135 . the lower figures are the corresponding results for @xmath46 , @xmath47 and @xmath48 at the quantum critical point @xmath214 . since in that case the fermi surface is not exactly tangent to the antiferromagnetic zone boundary , the @xmath7 behavior near @xmath215 is recovered only if the temperature is high enough that details of the fermi surface can not be resolved . the black lines on ( b ) and ( d ) are the curves defined by @xmath216 where @xmath217 is the component of @xmath218 parallel to @xmath219 at a given angle @xmath220 . ] finally , the nuclear magnetic resonance ( nmr ) relaxation rate @xmath221 can be obtained from the two - dimensional version of the moriya formula @xmath222 where @xmath223 is proportional to the hyperfine matrix element . taking this as a constant and using the general scaling form eqn.([chispscaling ] ) , a simple change of integration variable shows that @xmath224 however , the integral over momenta @xmath225 also contains contributions far from the peak in the susceptibility . there the susceptibility is essentially temperature independent . there is thus a korringa contribution @xmath226 that is dominant at low temperature . to find the scaling of the self - energy , we use the scaling form of the susceptibility eqn.([chispscaling ] ) to rewrite the self - energy eqn.([selfwithchipp ] ) in the form@xmath92 \frac{1}{\sqrt{t}}g\left(\frac{q_{\vert}}{t},\frac{(\omega+\omega')/v_f}{t};\frac{\omega}{t}\right ) % \ ] ] specializing to two dimensions and remembering the scaling of the bose and fermi functions with frequency and temperature , we change integration variables to @xmath227 and @xmath228 and we are left with @xmath229 where @xmath230 is a scaling function . at various angles along the fermi surface . the result of a power law fit is shown on ( b ) and ( d ) . the dashed lines on ( a ) and ( c ) correspond to the fitted power laws . at the hot spot located at @xmath215 , @xmath231 scales as @xmath232 . the frequency range is small because of the low temperature saturation shown on the next figure . we have verified that the crossover from @xmath232 to the fermi liquid regime @xmath233 occurs on a broader angular scale when the temperature is higher , as expected from the results of fig . [ fig : self_t_tpsc ] . calculations are done at @xmath234 , @xmath47 and @xmath48 at @xmath58 and @xmath235 for ( a ) and ( b ) and @xmath46 , @xmath214 for ( c ) and ( d ) . ] at various angles along the fermi surface . the result of a power law fit is shown on ( b ) and ( d ) . the dashed lines on ( a ) and ( c ) correspond to the fitted power laws . at the hot spot located at @xmath215 , @xmath231 scales as @xmath232 . the frequency range is small because of the low temperature saturation shown on the next figure . we have verified that the crossover from @xmath232 to the fermi liquid regime @xmath233 occurs on a broader angular scale when the temperature is higher , as expected from the results of fig . [ fig : self_t_tpsc ] . calculations are done at @xmath234 , @xmath47 and @xmath48 at @xmath58 and @xmath235 for ( a ) and ( b ) and @xmath46 , @xmath214 for ( c ) and ( d ) . ] at various angles along the fermi surface . the result of a power law fit is shown on ( b ) and ( d ) . the dashed lines on ( a ) and ( c ) correspond to the fitted power laws . at the hot spot located at @xmath215 , @xmath231 scales as @xmath232 . the frequency range is small because of the low temperature saturation shown on the next figure . we have verified that the crossover from @xmath232 to the fermi liquid regime @xmath233 occurs on a broader angular scale when the temperature is higher , as expected from the results of fig . [ fig : self_t_tpsc ] . calculations are done at @xmath234 , @xmath47 and @xmath48 at @xmath58 and @xmath235 for ( a ) and ( b ) and @xmath46 , @xmath214 for ( c ) and ( d ) . ] at various angles along the fermi surface . the result of a power law fit is shown on ( b ) and ( d ) . the dashed lines on ( a ) and ( c ) correspond to the fitted power laws . at the hot spot located at @xmath215 , @xmath231 scales as @xmath232 . the frequency range is small because of the low temperature saturation shown on the next figure . we have verified that the crossover from @xmath232 to the fermi liquid regime @xmath233 occurs on a broader angular scale when the temperature is higher , as expected from the results of fig . [ fig : self_t_tpsc ] . calculations are done at @xmath234 , @xmath47 and @xmath48 at @xmath58 and @xmath235 for ( a ) and ( b ) and @xmath46 , @xmath214 for ( c ) and ( d ) . ] the latter result can be checked numerically at @xmath236 where we expect @xmath237 . [ fig : self_t_tpsc](a ) and [ fig : self_t_tpsc](c ) displays the temperature dependent scattering rate for various angles @xmath220 along the fermi surface for @xmath58 and @xmath235 and @xmath46 and @xmath214 , respectively . the line @xmath238 is horizontal in the brillouin zone appearing in the inset . in fig . [ fig : self_t_tpsc](a ) , at the hot spot located at @xmath239 , we recover the predicted result , @xmath7 . this is best illustrated in fig . [ fig : self_t_tpsc](b ) by a plot of the local slope of the preceding log - log plot . as we move away from the hot spot , fermi liquid behavior appears to be recovered . there are well known logarithmic corrections in two dimensions @xcite that may explain why we seem to deviate from exactly @xmath240 . one also notices that the @xmath7 behavior occurs over a wider range of angles when the temperature is high . this is easily understood from figs . [ fig : self_t_tpsc](b ) and [ fig : self_t_tpsc](d ) that illustrates how temperature affects the domain where the pseudo - nesting occurs in the spin susceptibility . the solid black line in figs . [ fig : self_t_tpsc](b ) and [ fig : self_t_tpsc](d ) is defined by @xmath216 where @xmath217 is the component of @xmath218 parallel to @xmath219 at a given angle @xmath220 . [ fig : self_t_tpsc](c ) and figs . [ fig : self_t_tpsc](d ) are for @xmath241 . since at this quantum critical point the fermi surface is not tangent to the antiferromagnetic zone boundary , the @xmath7 behavior occurs near @xmath239 only at high enough temperature . at low temperature , deviations become apparent . when @xmath242 , the scaling form for the self - energy eqn.([self - scaling ] ) predicts @xmath243 . however , at zero temperature , or when @xmath242 , the analytical approach taken above fails because the expansion of the spin susceptibility in @xmath196 and @xmath195 is no longer justified and we can not expect the latter result to be correct . nevertheless , tpsc can be solved numerically . to set the stage for the next section where calculations are performed analytically directly at zero temperature , we show in fig . [ fig : self_w_tpsc ] the result of the numerical calculations for @xmath242 for two values of the interaction strength at a doping near @xmath19 . on figs . [ fig : self_w_tpsc](a ) and [ fig : self_w_tpsc](b ) , @xmath58 while @xmath46 on figs . [ fig : self_w_tpsc](c ) and [ fig : self_w_tpsc](d ) . at the hot spot for @xmath58 , the scaling of the imaginary part of the self energy is very close to the expected result @xmath232 . for @xmath46 , there is a larger discrepancy with the predicted scaling because at @xmath214 the fermi surface does not touch @xmath133 and thus the present theory does not apply anymore at low temperature . away from the hot spot , fermi liquid behavior is recovered . again , logarithmic corrections are inaccessible from the numerical solution of the full tpsc equations because of the limited range of available frequencies : scaling is no - longer valid at frequency of the order of the fermi energy , while at low frequency there is a saturation arising from the finite temperature . this saturation is illustrated in fig . [ fig : self_saturation_tpsc ] . we discuss analytically the @xmath188 regime in the following section where logarithmic corrections are found . the @xmath6 temperature dependence of the static @xmath0 spin susceptibility obtained above will also be recovered . at the hot spot for two different temperatures . the saturation at low frequency occurs at higher frequency when the temperature is higher . calculations are done with @xmath46 , @xmath47 and @xmath48 at the quantum critical filling @xmath244 appropriate for electron - doped cuprates.,title="fig : " ] + in this section we study the properties of fermionic excitations close to the hot spots within the field - theoretic framework of a spin - fermion model . this effective low - energy theory describes fermions with a parabolic dispersion ( represented by fields , @xmath245 ) interacting with sdw fluctuations ( represented by a o(3 ) vector field , @xmath246 ) . as shown in fig . [ bzf ] there are four hot - spots on the fermi surface which are connected by the sdw wave - vector @xmath247 . earlier studies@xcite of the spin - fermion model in the present context did not include the umklapp processes properly and we show in the following that a correct treatment of these terms modifies the results drastically . we start with the two patch ( denoted by @xmath248 ) model @xcite in the rotated @xmath249 coordinate system . the umklapp contributions will be discussed later . in order to simplify the notation , we have rescaled our coordinates to get rid of the fermi velocity and curvature of the fermi - surface . the corresponding lagrangian takes the form @xmath250 here we have promoted each fermion field to have @xmath69 flavors ( the flavor index is suppressed ) . the yukawa - coupling , @xmath251 , is chosen to be @xmath252 . as a result of this , the couplings of all the bosonic terms in the last line above are scaled by a factor of @xmath69 as they will appear naturally upon integrating out the fermion fields . we do nt include the kinetic energy of the boson , @xmath253 , as this is an irrelevant term @xcite . the bare fermion propagator is given by , @xmath254 the fermi surfaces are located at @xmath255 and @xmath256 for patch @xmath257 and @xmath258 respectively . from fig . [ bzf ] we immediately observe that @xmath259 and also @xmath260 connect two more points in the bz . however , it is convenient to fold back the points within the bz , which effectively gives rise to two more patches . these can be described by rotating the original patches by @xmath261 . let us denote @xmath262 by @xmath263 . then , the equations of these two additional fermi surfaces are given by @xmath264 . physically , these two scattering processes are very different since in the former case , the @xmath265fluctuation scatters fermions that disperse strongly in the direction transverse to the fermi surface while in the latter case , they disperse strongly in the tangential direction . this will have interesting consequences in the behavior of the electron self energy as a function of the external frequency . the rest of this section is organized as follows : in section [ rpa ] , we compute the rpa contributions to the sdw propagator including both direct as well as umklapp processes . we then use the dressed bosonic propagator to evaluate the fermion self - energy in section [ fse ] at leading order in @xmath21 . [ rpa ] at @xmath188 , the one loop polarization bubble ( fig . [ pol ] ) for the two - patch theory is given by , @xmath266,\\ % \textnormal{i.e . } \\ \pi(q)&=&n[\pi_0(q_\tau,{\bf q})+\pi_0(q_\tau,-{\bf q})],\end{aligned}\ ] ] where we are working with imaginary frequencies @xmath267 and @xmath268 denote the three sdw - polarizations . after performing the integrals , we obtain @xcite . the internal solid lines in the loop correspond to the free fermion propagators ( different patches denoted by @xmath248 ) and the external wavy lines correspond to the boson @xmath269 . ] @xmath270=\frac{\lambda^2}{2\pi } \sqrt{e_{{\bf q}}+\sqrt{e_{{\bf q}}^2 + |q_\tau|^2}}. \label{bubble}\ ] ] where , @xmath271 in the rpa propagator obtained after bubble summation , the @xmath272 contribution determines the location of the quantum critical point . it is thus convenient to add and subtract this component to make the integrals convergent . from now on , we include @xmath272 in the definition of the bubble . to make connection with results of the previous sections , we also quote the results for the bubble at finite temperature . in this case , @xmath273 where the contour @xmath124 has to be chosen appropriately . therefore , the above integral simplifies to , @xmath274\ ] ] on integrating the above equation by parts , we obtain , @xmath275}\sqrt{\omega+\sqrt{\omega^2 + |q_\tau|^2 } } , \\ \pi_0(q)&=&\sqrt{t } f\bigg(\frac{|q_\tau|}{t},\frac{e_{\bf q}}{t}\bigg ) , \label{scalingpi0t}\end{aligned}\ ] ] where , @xmath276}\sqrt{y+\sqrt{y^2+\frac{|q_\tau|^2}{t^2}}}\ ] ] as found in ref . . the rpa propagator for the sdw fluctuations is then given by , @xmath277 } , % endmodified% \label{renprop}\ ] ] where we have included the rpa contribution arising from all four hot spots on the fermi surface and @xmath278 the terms ( @xmath279 ) are not equal to ( @xmath280 ) as was incorrectly assumed by the authors of ref . . at the quantum critical filling @xmath281 , zero matsubara frequency @xmath282 but finite temperature , the @xmath283 term is negligible compared to the contribution of the bubbles @xmath284 . using the scaling form eqn.([scalingpi0 t ] ) and keeping only terms linear in @xmath225 in @xmath285 ( eqn . [ eofq ] ) , one recovers the zero frequency limit of the previous result eqn.([chispscaling ] ) for the spin susceptibility . naively doing the analytical continuation in frequency , the full scaling form would also follow . for the rest of the computations , we consider @xmath188 and carefully take into account logarithmic corrections that were beyond the reach of the previous calculation . before we proceed to evaluate the fermionic self - energy , let us compute the forms of the real and imaginary parts of the retarded polarization bubble at @xmath188 , @xmath286 , where @xmath287 . for the imaginary part , we obtain from eqn . ( [ bubble ] ) , @xmath288 , \\ \textnormal{im}\pi_r({\bf q},\omega)&=&\frac{-\lambda^2}{2\sqrt{2}\pi } \bigg[\theta ( \omega - e_{{\bf q}})\sqrt { \omega - e_{{\bf q}}}-\theta(- \omega - e_{{\bf q}})\sqrt{- \omega - e_{{\bf q}}}\bigg],\end{aligned}\ ] ] which is chosen in a way such that @xmath289 . the real part can also be obtained from eqn . ( [ bubble ] ) or from the kramers - kronig relation @xmath290.\end{aligned}\ ] ] these results agree with those of refs . . the @xmath291 limit calculated in ref . also agrees with the above . [ fse ] we are interested in evaluating the electron self energy ( fig . [ se ] ) at @xmath188 , @xmath292 , which at leading order in @xmath21 is given by , @xmath293 at leading order in @xmath21 . the @xmath265 propagator includes the one loop bubbles evaluated earlier . ] after analytic continuation @xmath294 we obtain the following expression for the imaginary part of the retarded self - energy @xmath295 \text{im } d_r(\q,\omega-\xi^\mp_{\k-\q}),\ ] ] where @xmath296 and @xmath297 . at @xmath188 and @xmath298 we are left with @xmath299 \text{im } d_r(\q,\omega-\xi^-_{-\q } ) \notag \\ & = & \frac{3\lambda^2}{4\pi^2 } \int dq_y \int_{-q_y^2}^{-q_y^2 + \omega } dq_x \ , \text{im } d_r(\q,\omega-\xi^-_{-\q } ) \notag \\ & \overset{\omega\rightarrow0}{\approx } & \frac{3\lambda^2 \omega}{4\pi^2 } \int dq_y \ , \text{im } d_r(\q,\omega-\xi^-_{-\q } ) \big|_{q_x \to -q_y^2 } \label{eq1}\end{aligned}\ ] ] note that @xmath300 for @xmath301 . in terms of @xmath302 , this can be rewritten as , @xmath303 where @xmath304 is the total retarded rpa bubble including direct and umklapp terms and we have ignored the @xmath305 term in the denominator . for @xmath306 , we get @xmath307 , @xmath308 , @xmath309 and @xmath310 . therefore , re(im)@xmath311 are given by , @xmath312\nonumber\\ \text{im } \pi_r^{tot}(\q,\omega)=-\frac{\lambda^2}{4\pi } \bigg[\sqrt{2 \omega-3q_y^2 } \ , \theta(2 \omega-3q_y^2)&+&\sqrt{2 \omega+q_y^2 } \ , \theta(2 \omega+q_y^2)\nonumber \\ -\sqrt{-3q_y^2 - 2 \omega } \ , \theta(-3q_y^2 - 2 \omega)&-&\sqrt{q_y^2 - 2 \omega } \ , \theta(q_y^2 - 2 \omega ) \nonumber\\ + \sqrt{2 \omega+2q_y - q_y^4 } \ , \theta(2 \omega+2q_y - q_y^4)&+&\sqrt{2 \omega-2q_y - q_y^4 } \ , \theta(2 \omega-2q_y - q_y^4)\nonumber\\ -\sqrt{2q_y - q_y^4 - 2 \omega } \ , \theta(2q_y - q_y^4 - 2 \omega)&-&\sqrt{-2q_y - q_y^4 - 2 \omega } \ , \theta(-2q_y - q_y^4 - 2 \omega ) \bigg]\nonumber\\\end{aligned}\ ] ] as a starting point , we can drop the @xmath313 terms in the limit of small @xmath64 and retain only the @xmath39 terms in the @xmath314 contributions that we take into account . this is a consistent way of handling these terms , since if typical @xmath315 , then @xmath313 is smaller than @xmath39 , so that it is justified to drop these terms . since the integrand is an even function of @xmath39 , we integrate only over @xmath316 . then , the expression for self energy reduces to , @xmath317\nonumber\\ a&=&\frac{\sqrt{2 \omega+2q_y}+\sqrt{2 \omega-2q_y}+\sqrt{2 \omega-3q_y^2}+\sqrt{2 \omega+q_y^2}}{\bigg[\frac{1}{\lambda^2 } q_y^2+\frac{\sqrt{2q_y+2 \omega}+\sqrt{2 \omega-2q_y}+\sqrt{3q_y^2 + 2 \omega}+\sqrt{2 \omega - q_y^2}}{4\pi}\bigg]^2+\frac{[\sqrt{2 \omega+2q_y}+\sqrt{2 \omega-2q_y}+\sqrt{2 \omega-3q_y^2}+\sqrt{2 \omega+q_y^2}]^2}{16\pi^2 } } \nonumber\\ b&=&\frac{\sqrt{2 \omega+2q_y}-\sqrt{2q_y-2 \omega}+\sqrt{2 \omega-3q_y^2}+\sqrt{2 \omega+q_y^2}}{\bigg[\frac{1}{\lambda^2 } q_y^2+\frac{\sqrt{2q_y+2 \omega}+\sqrt{2q_y-2 \omega}+\sqrt{3q_y^2 + 2 \omega}+\sqrt{2 \omega - q_y^2}}{4\pi}\bigg]^2+\frac{[\sqrt{2 \omega+2q_y}-\sqrt{2q_y-2 \omega}+\sqrt{2 \omega-3q_y^2}+\sqrt{2 \omega+q_y^2}]^2}{16\pi^2 } } \nonumber\\ c&=&\frac{\sqrt{2 \omega+2q_y}-\sqrt{2q_y-2 \omega}+\sqrt{2 \omega+q_y^2}}{\bigg[\frac{1}{\lambda^2 } q_y^2+\frac{\sqrt{2q_y+2 \omega}+\sqrt{2q_y-2 \omega}+\sqrt{3q_y^2 + 2 \omega}+\sqrt{3q_y^2 - 2 \omega}+\sqrt{2 \omega - q_y^2}}{4\pi}\bigg]^2+\frac{[\sqrt{2 \omega+2q_y}-\sqrt{2q_y-2 \omega}+\sqrt{2 \omega+q_y^2}]^2}{16\pi^2 } } \nonumber\\ d&=&\frac{\sqrt{2q_y+2 \omega}-\sqrt{2q_y-2 \omega}-\sqrt{q_y^2 - 2 \omega}+\sqrt { q_y^2 + 2 \omega}}{\bigg[\frac{1}{\lambda^2 } q_y^2+\frac{\sqrt{2q_y+2 \omega}+\sqrt{2q_y-2 \omega}+\sqrt{3q_y^2 + 2 \omega}+\sqrt{3q_y^2 - 2 \omega}}{4\pi}\bigg]^2+\frac{[\sqrt{2q_y+2 \omega}-\sqrt{2q_y-2 \omega}-\sqrt{q_y^2 - 2 \omega}+\sqrt{q_y^2 + 2 \omega}]^2}{16\pi^2}}\nonumber\\ \label{reg}\end{aligned}\ ] ] at small frequencies the dominant contribution to the imaginary part of the self - energy comes from the second integral between @xmath318 and @xmath319 , which scales as @xmath320 . the contributions from the other regions scale as @xmath321 and thus are negligible at low frequencies . the correct prefactor can be obtained by expanding the numerator and the denominator of integrand @xmath113 for small frequencies @xmath64 and retaining only the largest terms , which gives @xmath322 one then integrates over @xmath39 to obtain , @xmath323 we note here that the self - energy is less singular compared to earlier works where umklapp scattering was not taken into account , in which case the self - energy scales as @xmath324.@xcite . , calculated numerically from the second line of eqn . , shown as a red solid line . the black dashed line is the asymptotic result eqn . . for comparison , the blue dash - dotted line indicates @xmath232 . right : different contributions to @xmath325 . the red line shows the dominant contribution , denoted by @xmath113 in the main text . black dashed line : asymptotic result eqn . . the other three curves show the sub - leading contributions @xmath326 and @xmath327 , which scale as @xmath321 . ] so far we havent addressed an important issue : what happens if we include the renormalization of the boson - fermion vertex ? in the two - patch theory originally considered by altschuler et al . @xcite , the one - loop correction to this vertex was found to be logarithmically singular . however in appendix [ appver ] , we show that the full four - patch theory does not have this singularity . based on our discussion so far , we see that the additional umklapp terms considered in our work play a very crucial role at the critical point . in the absence of these contributions , the self - energy was more singular than what we have found here . moreover , the vertex correction was also found to be singular . however , here we have shown that the singular behavior is washed out when we include the additional scattering contributions . in the tpsc approach we imposed two - particle self - consistency in the form of a sum - rule that is similar to the spherical model . in the present field - theory approach , this amounts to imposing @xmath328 where the expectation value is taken with respect to the fermions and bosons . we have argued that the @xmath3 scaling does not come from the self - consistency condition . to confirm this result , in this subsection we obtain the scaling of the quartic term in the boson lagrangian . integrating out the fermions , the polarization operator @xmath329 appears in the quadratic term of the boson lagrangian . emphasizing the scaling only , this term is symbolically written as @xmath330 this is the most relevant quadratic term . integrating out the large wave number modes for @xmath331 and rescaling @xmath225 and @xmath64 such that @xmath332 , @xmath333 with @xmath334 , returns the new cutoff @xmath335 to its original value @xmath336 . invariance of the quadratic term written in terms of the prime variables then imposes that @xmath337 . the effect of this tree level scaling on the quartic term is that @xmath338 this in turn means that @xmath339 scales to zero and is thus irrelevant . we have argued that for bare interaction strengths @xmath30 in the intermediate coupling range , commensurate sdw fluctuations at @xmath0 and band parameters similar to those of electron - doped cuprates , the antiferromagnetic quantum critical point naturally occurs close to the filling where the fermi surface points joined by @xmath0 are nearly tangent to each other . as long as the temperature or frequency are not too low , the limiting case of tangent fermi surfaces describes the physics . in this pseudo - nesting situation , the fermi liquid behavior breaks down . quasiparticles still exist but the self - energy and spin susceptibility , for example , are different from those predicted by fermi liquid theory . we considered this problem at zero temperature , or for frequencies larger than temperature , using a field - theoretical model of gapless collective bosonic modes ( sdw fluctuations ) interacting with fermions . the imaginary part of the retarded fermionic self - energy close to the hot spots scales as @xmath9 . this is less singular than earlier predictions of the form @xmath10 . the difference arises from the effects of umklapp terms that were not included in previous studies . at finite temperature , we have used tpsc to study this problem and have obtained numerical results for the one - band hubbard model with band parameters and interaction strength appropriate for electron - doped cuprates . neglecting logarithmic corrections , we found analytically and numerically that the correlation length @xmath1 scales like @xmath2 , namely @xmath3 instead of the naive @xmath4 . the static spin susceptibility @xmath5 scales like @xmath6 , and the correction @xmath8 to the korringa nmr relaxation rate is subdominant . nmr experiments are difficult in electron - doped cuprates . we also found that the imaginary part of the self - energy at the hot spot scales like @xmath7 . the latter result and the @xmath9 frequency dependence of the self - energy should be experimentally verifiable with angle - resolved photoemission spectroscopy ( arpes ) in electron - doped cuprates . recent transport measurements in these compounds @xcite have found a @xmath7 behavior of the resistivity above the quantum critical point at the end of the overdoped side of the superconducting dome . while there may be a relation with the above result if antiferromagnetic fluctuations disappear at the same time , one must also be careful not to equate scattering rate with resistivity because in a simple picture it is the inverse of the scattering rates that are averaged over the fermi surface . this suggests that in the resistivity , fermi liquid behavior of the cold spots should short - circuit the non - fermi liquid behavior of the hot spots . @xcite we thank erez berg , andrey chubukov , lev ioffe and max metlitski for useful discussions . this research was supported by the national science foundation under grant dmr-1103860 ( d.c . , s.s . ) , by the austrian science fund ( fwf)-erwin schrdinger fellowship j 3077-n16 ( m.p . ) , by a muri grant from afosr ( s.s . ) and by nserc , the tier i canada research chair program ( a .- m . m.s.t is grateful to the harvard physics department , cifar , and the center for materials theory at rutgers university for support during the writing of this work . partial support was also provided by the mit - harvard center for ultracold atoms . computer intensive calculations were performed on computers provided by cfi , mels , calcul qubec and compute canada . [ appver ] in this appendix , we compute the 1-loop correction to the boson - fermion vertex , which is defined as , @xmath340 where we are working again with imaginary frequencies . the expression for the diagram in fig . [ bf ] can be written as , @xmath341 we now use the identity @xmath342 twice to simplify the above expression . then on defining @xmath343 , we have , @xmath344\nonumber\\ \bigg[\frac{1}{-i ( l_\tau+p_\tau)-(l_x+p_x)+(l_y+p_y)^2}\bigg]\bigg[\frac{1}{{\bf l}^2+r+[\pi_0({\bf l})+\pi_0(-{\bf l})+\pi_0({\bf \tilde{l}})+\pi_0(-{\bf \tilde{l}})]}\bigg].\end{aligned}\ ] ] let us now evaluate this for zero external momenta @xmath345 at the critical point and check for singularities . the expression reduces to , @xmath346\bigg[\frac{1}{-i l_\tau - l_x+l_y^2}\bigg]\nonumber\\\bigg[\frac{1}{{\bf l}^2+\pi_0({\bf l},l_\tau)+\pi_0(-{\bf l},l_\tau)+\pi_0({\bf \tilde{l}},l_\tau)+\pi_0(-{\bf \tilde{l}},l_\tau)}\bigg ] . \label{renver}\end{aligned}\ ] ] the above integrand has a very complex structure . let us therefore analyze the ( non-)singular nature of this diagram by power counting . one needs to be careful as the @xmath347propagator has many combinations of powers of the internal momenta . we begin by rescaling the variables as , @xmath348 eqn . [ renver ] then takes the form , @xmath349\bigg[\frac{1}{-i l_\tau'-l_x'+1 } \bigg ] \nonumber \\ & & \times \frac{1}{l^2+[lf(l_x',l_\tau')+lf(-l_x',l_\tau')+\sqrt{l}g_+(l , l_x',l_\tau')+\sqrt{l}g_-(l , l_x',l_\tau')]/2\pi } , \\ f(l_x',l_\tau')&=&\sqrt{(1/2-l_x')+\sqrt{(1/2-l_x')^2 + l_\tau'^2 } } , \\ g_\pm((l , l_x',l_\tau')&=&\sqrt{(l^3l_x'^2/2\pm1)+\sqrt{(l^3l_x'^2/2\pm1)^2 + l^2l_\tau'^2}}\end{aligned}\ ] ] all we need to do now is to check whether this expression ( which is so far exact ) is singular in the ir and uv . in the ir , we can ignore the momentum dependence of the fermionic green s functions compared to @xmath350 in the denominator . moreover , @xmath351 and @xmath352 in this small momentum limit . therefore , the above expression reduces to , @xmath353 where @xmath354 is a small cutoff . but the above expression is convergent , so that there are no ir singularities . let us now check for uv singularities . we proceed by introducing a characteristic lower cutoff @xmath336 which is large , but finite , such that the integration runs from @xmath336 to @xmath355 . then , in the limit of these large momenta , we have , @xmath356\bigg[\frac{1}{-i l_\tau'-l_x ' } \bigg]\frac{1}{l^2+[l\sqrt{\sqrt{l_x'^2 + l_\tau'^2}+l_x'}+2l^2l_x']/2\pi } , \nonumber \\\end{aligned}\ ] ] where we have ignored the contribution from @xmath357 compared to @xmath358 and @xmath359 compared to @xmath360 . we first want to do the integral over @xmath361 and @xmath362 . it is more convenient to change to polar coordinates , @xmath363 . however , we estimate @xmath364 and eliminate the @xmath220 dependence , which does not give rise to any singularities . then , @xmath365/2\pi } .\ ] ] we also notice that @xmath366 in the uv . therefore , ignoring the @xmath367 term , we obtain , @xmath368 both the @xmath369 and @xmath370 integrals can be performed easily to verify that @xmath371 is convergent in the uv . therefore , the 1-loop vertex correction is non - singular both in the uv and in the ir .
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many correlated materials display a quantum critical point between a paramagnetic and a spin - density wave ( sdw ) state .
the sdw wave vector connects points , so - called hot spots , on opposite sides of the fermi surface .
the fermi velocities at these pairs of points are in general not parallel . here
we consider the case where pairs of hot spots coalesce , and the wave vector @xmath0 of the sdw connects hot spots with parallel fermi velocities . using the specific example of electron - doped cuprates , we first show that kanamori screening and generic features of the lindhard function make this case experimentally relevant .
the temperature dependence of the correlation length , the spin susceptibility and the self - energy at the hot spots are found using the two - particle - self - consistent theory and specific numerical examples worked out for band and interaction parameters characteristic of the electron - doped cuprates .
while the curvature of the fermi surface at the hot spots leads to deviations from perfect nesting , the pseudo - nesting conditions lead to drastic modifications of the temperature dependence of these physical observables : neglecting logarithmic corrections , the correlation length @xmath1 scales like @xmath2 , namely @xmath3 instead of the naive @xmath4 , the @xmath0 static spin susceptibility @xmath5 like @xmath6 , and the imaginary part of the self - energy at the hot spots like @xmath7 . the correction @xmath8 to the korringa nmr relaxation rate is subdominant .
we also consider this problem at zero temperature , or for frequencies larger than temperature , using a field - theoretical model of gapless collective bosonic modes ( sdw fluctuations ) interacting with fermions .
the imaginary part of the retarded fermionic self - energy close to the hot spots scales as @xmath9 .
this is less singular than earlier predictions of the form @xmath10 .
the difference arises from the effects of umklapp terms that were not included in previous studies .
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following the seminal work of markowitz @xcite , the problem of optimization of an investor s portfolio based on different criteria and market assumptions are being studied by several authors . in the mean - variance optimization approach , as done by markowitz , either the expected value of the portfolio wealth is optimized by keeping the variance fixed or the variance is minimized by keeping the expectation fixed . though markowitz s mean - variance approach to portfolio is immensely useful in practice , its scope is limited by the fact that only gaussian distributions are completely determined by their first two moments . in a pioneering work merton @xcite , @xcite has introduced the utility maximization to the optimal portfolio selection . merton s approach is based on applying the method of stochastic optimal control via an appropriate hamilton - jacobi- bellman ( hjb ) equation . the corresponding optimal dynamic portfolio allocation can also be obtained from the same equation . although this approach has greater mathematical tractability but does not capture the tradeoff between maximizing expectation and minimizing variance of portfolio value . there is another approach , namely risk sensitive optimization , where a tradeoff between the long run expected growth rate and the asymptotic variance is captured in an implicit way . the aforesaid utility maximization method can be employed to study the risk - sensitive optimization by choosing a parametric family of exponential utility functions . in such optimization , an appropriate value of the parameter is to be chosen by the investor depending on the investors degree of risk tolerance . we refer @xcite , @xcite , @xcite , @xcite for this criterion under the geometric brownian motion ( gbm ) market model . risk sensitive optimization of portfolio value in a more general type of market is also studied by various authors . jump diffusion model is one such generalization , which captures the discontinuity of asset dynamics . empirical results support such models @xcite . terminal utility optimization problem under such a model assumption is studied in @xcite . in all these references , it is assumed that the market parameters , i.e. , the coefficients in the asset price dynamics , are either constant or deterministic functions of time . we study a class of models where these parameters are allowed to be finite state pure jump processes . we call each state of the coefficients as a regime and the dynamics as a regime switching model . the regime switching can be of various types . it is known that for a markov switching model , the sojourn or holding times in each state are distributed as exponential random variables , whereas the holding time can be any positive random variable for the semi - markov case . thus the class of semi - markov processes subsumes the class of markov chains . there are some statistical results in the literature ( see @xcite and the references therein ) , which emphasize the advantage of the applicability of semi - markov switching models over simple homogeneous markov switching models . it is mainly useful to deal with the impact of a changing environment , which exhibits duration dependence . to understand this , consider a market situation where , if the volatility of certain stock price remains low for longer than certain duration , then that observation discourages increasingly more traders to trade on that depending on the length of the duration . in that case , this type of duration dependence mass - trading behaviour might cause further low volume trading resulting in lack of volatility boost . in this type of market behaviour , the density function of holding time of low volatility regime should exhibit heavier tail than exponential . it is important to note that , a markov chain either time homogeneous or in - homogeneous does not exhibit such age dependent transition , whereas a generic semi - markov process may exhibit this phenomenon . this motivate us to consider age - dependent transition of the regimes . risk sensitive portfolio optimization in a gbm model with markov regimes is studied in @xcite whereas @xcite studies the same problem in a semi - markov modulated gbm model . in @xcite the market parameters , @xmath0 , @xmath1 and @xmath2 are driven by a finite - state semi - markov process @xmath3 , where @xmath1 and @xmath2 denote the drift and volatility parameters of @xmath4-th asset in the portfolio . strictly speaking , the assumption that all the parameters from different assets are governed by a single semi - markov process is rather restrictive . ideally those could be driven by independent or correlated processes in practice . although two independent markov processes jointly becomes a markov process , the same phenomena is not true for semi - markov processes . for this reason the case of independent regimes are important where regimes are not markov . in general a pure jump process need not be a semi - markov process . in particular the class of age - dependent processes ( as in @xcite ) is much wider than the type of age independent semi - markov processes studied in @xcite . in a recent paper @xcite , option pricing is studied in a switching market where the regimes are assumed to be an age - dependent process . an age - dependent process @xmath5 on a finite state space @xmath6 is specified by its instantaneous transition rate @xmath7 , which is a collection of measurable functions @xmath8 where @xmath9 and @xmath10 . indeed , embedding @xmath11 in @xmath12 , an age dependent process , @xmath13 on @xmath11 is defined as the strong solution to the following system of stochastic integral equations @xmath14}{\displaystyle\int\limits}_{\mathbb{r } } h_{\lambda}(x_{u- } , y_{u-},z)\wp(du , dz),\\ y_t & = y_0 + t- { \displaystyle\int\limits}_{(0,t ] } { \displaystyle\int\limits}_{\mathbb{r } } g_{\lambda}(x_{u- } , y_{u-},z ) \wp(du , dz ) , \end{array}\right\}\ ] ] where @xmath15 is the poisson random measure with intensity @xmath16 , independent of @xmath17 , and @xmath18 and for every @xmath19 , @xmath20 , using a strict total order @xmath21 on @xmath22 . we refer to @xcite for a proof that @xmath7 indeed represents instantaneous transition rate of @xmath13 . in this paper we consider a regime switching jump diffusion model of a financial market , where an observed euclidean space valued pure jump process drives the regimes of every asset . further , we assume that every component of that pure jump process is an age dependent semi - markov process and the components are independent . we study the finite horizon portfolio optimization via the risk sensitive criterion under the above market assumption . the optimization problem is solved by studying the corresponding hjb equation where we employ the technique of separation of variables to reduce the hjb equation to a system of linear first order pdes containing some non - local terms . in the reduced equation , the nature of non - locality is such that the standard theory of integro - pde is not applicable to establish the existence and uniqueness of the solution . in this paper , to show well - posedness of this pde , a volterra integral equation(ie ) of the second kind is obtained and then the existence of a unique @xmath23 solution is shown . then it is proved that the solution to the ie is a classical solution to the pde under study . the uniqueness of the pde is proved by showing that any classical solution also solves the ie . in the uniqueness part , we use conditioning with respect to the transition times of the underlying process . besides , we also obtain the optimal portfolio selection as a continuous function of time and underlying switching process . the expression of this function does not involve the functional parameter @xmath7 . thus the optimal selection is robust . our approach of solving the pde also enables us to develop a robust numerical procedure to compute the optimal portfolio wealth using a quadrature method . the rest of the paper is organized as follows . in the next section we give a rigorous description of the model of a financial market dynamics and then derive the wealth process of an investor s portfolio . the problem of optimizing the portfolio wealth under the risk sensitive criterion on the finite time horizon is also stated in section 2 . in section 3 we have established a characterization of the optimal wealth using the corresponding hamilton - jacobi - bellman equation . an optimal portfolio strategy is also shown to exist in the class of markov feedback control . furthermore , an optimal feedback control is produced as a minimizer of a certain functional associated with the hjb equation . we illustrate the theoretical results by performing numerical experiments with an example and obtain some relevant results in section 4 . section 5 contains some concluding remarks . the proofs of certain important lemmata are given in an appendix . let @xmath11 denote a finite subset @xmath24 . without loss of generality , we choose @xmath25 . consider for each @xmath26 , @xmath27 a continuously differentiable function with @xmath28 and @xmath29 assume that for each @xmath30 , @xmath31 denotes a finite borel measure on @xmath24 . let @xmath32\times \mathcal{x}^{n+1 } \rightarrow [ 0,\infty)$ ] , @xmath33\times \mathcal{x}^{n+1 } \rightarrow \mathbb{r}$ ] , and @xmath34\times \mathcal{x}^{n+1 } \rightarrow ( 0,\infty)^{1\times m_1}$ ] be continuous functions of the time variable for each @xmath35 , where @xmath36 and @xmath37 are positive integers . we also consider a collection of measurable functions @xmath38 for each @xmath39 , @xmath40 . we further introduce some more notations . fix @xmath41 and @xmath42 $ ] and we denote @xmath43_{1 \times n}$ ] , and @xmath44_{n \times m_1}$ ] , where @xmath45 is the @xmath46-th component of @xmath2 function . for each @xmath47 , we denote @xmath48_{n \times m_2}$ ] . we use @xmath49^*$ ] to denote transpose of a vector . let @xmath50 be a complete probability space on which @xmath51 @xmath11 valued random variables @xmath52 , @xmath51 non negative random variables @xmath53 , a standard @xmath36-dimensional brownian motion @xmath54^{*}\}_{t \ge 0}$ ] , @xmath37 poisson random measures @xmath55 on @xmath56 with intensity @xmath57 , for @xmath58 , and another set of poisson random measures @xmath59 on @xmath56 with intensities @xmath60 defined such that all the random variables , processes and measures are independent . we denote the compensated measures by @xmath61 for @xmath58 and @xmath62 for @xmath63 . for each @xmath64 , let @xmath65 be the solution to with @xmath66 replaced by @xmath67 , @xmath68 by @xmath69 , @xmath70 by @xmath71 and @xmath72 by @xmath73 . in other words @xmath74}{\displaystyle\int\limits}_{\mathbb{r } } h^l(x^l_{u- } , y^l_{u-},z_0)\wp^l(du , dz_0)\text{\label{1}}\\ y_t^l & = & y_0^l+t- { \displaystyle\int\limits}_{(0,t ] } { \displaystyle\int\limits}_{\mathbb{r } } g^l(x^l_{u- } , y^l_{u-},z_0 ) \wp^l(du , dz_0)\text{\label{2}},\end{aligned}\ ] ] where @xmath75 and @xmath76 . we denote the tuple @xmath77 by @xmath78 and @xmath79 by @xmath80 . hence , @xmath81 , @xmath82 and @xmath13 are independent . the process @xmath83 is a time homogeneous strong markov process . let the filtration @xmath84 be the right continuous augmentation of the filtration generated by @xmath85 such that @xmath86 contains all the @xmath87-null sets . we consider a frictionless market consisting of @xmath51 assets whose prices are denoted by @xmath88 and @xmath89 and are traded continuously . we model the hypothetical state of the assets at time @xmath90 by the pure jump process @xmath5 . the state of the asset indicates its mean growth rate and volatility . we assume @xmath91 thus the corresponding asset is ( locally ) risk free , which refers to the money market account with the floating interest rate @xmath92 at time @xmath90 corresponding to regime @xmath93 . the other @xmath94 asset prices are assumed to be given by the following stochastic differential equation @xmath95 , \\ s_0^l & = s_l , \quad s_l \geq 0 , ~ l=1,2,\ldots , n. \nonumber\end{aligned}\ ] ] these prices correspond to @xmath94 different risky assets . therefore , @xmath1 represents the growth rate of the @xmath4-th asset and @xmath96 the volatility matrix of the market . here we further assume the following . * assumptions * : * for each @xmath39 and @xmath40 , we assume @xmath97 . * for each @xmath39 and @xmath40 , we further assume @xmath98 . * let @xmath99 denote the diffusion matrix . assume that there exist a @xmath100 such that for each @xmath90 and @xmath93 , @xmath101 , where @xmath102 denotes the euclidean norm . the next lemma asserts the existence and uniqueness of the solution to the sde ( [ eq : sde ] ) . the proof is deferred to the appendix . [ sdesol ] under the assumption _ * ( a2 ) * _ the equation ( [ eq : sde ] ) has a strong solution , which is adapted , a.s . unique and an rcll process . we note that _ * ( a1 ) * _ and _ * ( a2 ) * _ follow for the special case where @xmath103 by _ * ( a3 ) * _ the diffusion matrix @xmath104 is uniformly positive definite , which ensures that @xmath104 is invertible . we will use this condition in section 3 . this condition also implies that @xmath105 . consider an investor who is employing a self - financing portfolio of the above @xmath51 assets starting with a positive wealth . if the portfolio at time @xmath90 comprises of @xmath106 number of units of @xmath107 asset for every @xmath108 , then for each @xmath109 the value of the portfolio at time @xmath90 is given by @xmath110 we allow @xmath106 be real valued , i.e. , borrowing from the money market and short selling of assets are allowed . we further assume that @xmath111 is a @xmath112 adapted , rcll process for each @xmath4 . then the self - financing condition implies that @xmath113 if @xmath106 are such that @xmath114 remains positive , we can set @xmath115 , the fraction of investment in the @xmath4-th asset . then we have @xmath116 and hence @xmath117 . we call @xmath118^{*}$ ] as the portfolio strategy of risky assets at time @xmath90 . then the wealth process , @xmath119 , now onward denoted by @xmath120 , takes the form @xmath121 thus we would consider the following sde for the value process , @xmath122 u_t^l\right ) dt \\ { \nonumber } & \quad + v_t^{u } \sum\nolimits_{l=1}^{n } \sum\nolimits_{j=1}^{m_1 } \sigma_{lj}(t , x_t ) ~u_t^l dw_t^j \\ { \nonumber } & \quad + v_{t-}^{u } \sum\nolimits_{l=1}^{n } \sum\nolimits_{j=1}^{m_2}u_{t-}^l { \displaystyle\int\limits}_{\mathbb{r } } \eta_{lj}(z_j ) n_j(dt , dz_j ) \\ { \nonumber}&= v_t^{u } ( ~r(t , x_t ) + b(t , x_t)u_t ) dt + v_t^{u } u_t^{*}\sigma(t , x_t)dw_t \nonumber \\ & \quad + v_{t-}^{u } \sum\nolimits_{j=1}^{m_2 } \int_{\mathbb{r } } \left[u_{t-}^ { * } \eta(z)\right]_j n_j(dt , dz_j),\end{aligned}\ ] ] where @xmath123_{1 \times m_2}$ ] . note that , some additional assumptions on @xmath124 are needed for ensuring a strong solution of . [ rem1 ] as before , we need to assume that @xmath125 is such that for each @xmath58 , and @xmath126 , @xmath127_j > -1 $ ] to ensure a positive solution to . for some technical reasons we require a stronger condition on @xmath125 . we would require that the the portfolio should be chosen from @xmath128_j \geq -1+\delta , \forall j , z\ } ~\quad \text{for some}~ 0<\delta\leq 1.\end{aligned}\ ] ] it is clear from the definition and above derivation that @xmath129 , the portfolio wealth process , is a controlled process . let @xmath130 be a non empty convex set , denoting the range of portfolio . the range is determined based on investment restrictions . for example , @xmath131 in the case of unrestricted short selling . the restrictions on short selling makes @xmath132 , where @xmath133 for @xmath134 . clearly , @xmath135 for @xmath134 , correspond to no short selling . [ defi1 ] an rcll and adapted process @xmath136}$ ] is said to be _ admissible _ portfolio strategy if : * the process @xmath124 takes values from the convex set @xmath137 , where @xmath138 is as in , * has an almost sure unique strong solution , * @xmath139}\|u_t(\omega)\|<\infty$ ] . [ solstate ] under * ( a1 ) * and with admissible control @xmath124 , ( i ) the sde has an almost sure unique positive strong solution , ( ii ) the solution has finite moments of all positive and negative orders , which are also bounded on @xmath140 $ ] uniformly in @xmath124 . \(i ) we first note that , since @xmath141 and satisfies definition [ defi1](iii ) , @xmath142_j)|<\max\left(|\ln \delta| , c\|\eta_{\cdot j}(z_j)\|\right),\ ] ] where @xmath143}\|u_t(\omega)\|$ ] and @xmath144 is the @xmath46-th column of the matrix @xmath145 . again using * ( a1 ) * and the finiteness of measure @xmath31 , the integration of rhs with respect to @xmath146 has finite expectation . this implies that @xmath147_j)\,n_j(ds , dz_j)<\infty$ ] . therefore in the similar line of proof of lemma [ sdesol ] , we can show under the assumption * ( a1 ) * and admissibility of @xmath124 , ( [ eq : pov ] ) has an a.s . unique positive rcll solution , which is an adapted process , and the solution is given by @xmath148_j)\,n_j(ds , dz_j ) \bigg ] . \end{aligned}\ ] ] ( ii ) we first consider the first order moment . to prove for each @xmath90 , @xmath149 has a bounded expectation , we first note that the rhs can be written as a product of a log - normal random variable and @xmath150_j)\,n_j(ds , dz_j)\right)$ ] , where both are conditionally independent , given the process @xmath124 . we further note that the log - normal random variable has bounded parameters on @xmath140 $ ] uniformly in @xmath124 . therefore it is sufficient to check if @xmath151 , \end{aligned}\ ] ] is bounded on @xmath140 $ ] , for all @xmath152 . using finiteness of @xmath31 and * ( a1 ) * , one can show that the above expectation is bounded . thus @xmath153 has bounded expectation on @xmath140 $ ] , uniformly in @xmath124 . now for moments of general order , we note that for any @xmath154 can also be written in a similar form of where each of the integrals inside the exponential would be multiplied by the constant @xmath155 . thus the rest of the proof follows in a similar line of that of first order case , given above . our goal is to study risk sensitive optimal control problem on the above wealth process . we would see in the next section that , in order to obtain a classical solution to the corresponding hjb equation to be defined shortly , certain regularity of the conditional c.d.f of holding time of @xmath78 is needed . we devote the next subsection to establishing some smoothness of relevant density functions . we define the function @xmath156 $ ] as @xmath157 and let @xmath158 and for each @xmath159 with @xmath160 for all @xmath161 and @xmath162 . set @xmath163 we assume further conditions on the transition rate so that the unconditional transition probability matrix is irreducible . + * assumption : * * ( a4 ) * the matrix @xmath164 is irreducible . from the definition of @xmath165 and the assumptions on @xmath7 , we observe @xmath166 , for all @xmath167 . we also note that @xmath168 hold for all @xmath169 . now assume that @xmath170 denotes the time of @xmath94-th transition of the @xmath4-th component of @xmath78 , whereas @xmath171 and @xmath172 . for a fixed @xmath90 , let @xmath173 . hence @xmath174 and @xmath175 . it is shown in @xcite that @xmath176 is the conditional c.d.f of the holding time of @xmath177 and @xmath178 is the conditional probability that @xmath177 transits to @xmath46 given the fact that it is at @xmath161 for a duration of @xmath162 . let @xmath179 the remaining life of @xmath4-th component i.e. , the time period from time @xmath90 after which the @xmath4-th component of @xmath78 would have the first transition . note that @xmath180 is independent of every component of @xmath13 other than @xmath4-th one . we denote the conditional c.d.f of @xmath180 given @xmath181 and @xmath182 as @xmath183 . it is important to note that this c.d.f does not depend on @xmath90 mainly because @xmath184 is time - homogeneous . we also notice that @xmath185 is the duration of stagnancy of @xmath186 at present state before it moves to another . from now we denote @xmath187 by @xmath188 and the corresponding conditional expectation as @xmath189 . let @xmath190 be the component of @xmath78 , where the subsequent transition happens . therefore , @xmath191 represents the conditional probability of observing next transition to occur at the @xmath4-th component given that @xmath192 and @xmath193 . we find the expressions of the c.d.f and the probability defined above and obtain some properties in the following lemma . the proof is deferred to the appendix . in order to state the lemma , we introduce some notations . we define an open set @xmath194 and a linear operator @xmath195 where dom(@xmath196 ) , the domain of @xmath196 is the subspace of @xmath197 such that for each @xmath198 dom(@xmath196 ) above limit exists for every @xmath199 and @xmath200 . [ theo1 ] consider @xmath201 as given above . + ( i ) for each @xmath4 , @xmath202 ( ii ) let @xmath203 be the conditional c.d.f of @xmath180 given @xmath204 and @xmath205 . then @xmath206 and is @xmath207 in @xmath0 variable . \(iii ) @xmath208 is differentiable with respect to @xmath0 . \(iv ) @xmath209 and @xmath191 are in _ dom_(@xmath196 ) . @xmath210 \(v ) @xmath211 . in this paper we consider a risk sensitive optimization criterion of terminal portfolio wealth corresponding to a portfolio @xmath124 , that is given by @xmath212 \nonumber \\ \nonumber & = -\left(\frac{2}{\theta}\right ) ~\ln\mathbb{e}\left [ ( v_t^u)^{-\frac{\theta}{2 } } ~\biggr|~ x_0=x , y_0=y , v^u_0=v\right],\end{aligned}\ ] ] which is to be maximized over all admissible portfolio strategies with constant risk aversion parameter @xmath213 . since logarithm is increasing , it suffices to consider the following cost function @xmath214,\ ] ] which is to be minimized . for all @xmath215 , let @xmath216,\\ { \varphi}_\theta(t , x , y , v ) : = \inf_{u}\tilde{j}_\theta^{u , t}(t , x , y , v ) , \end{array}\right\}\ ] ] where the infimum is taken over all admissible strategies as in definition [ defi1 ] . hence , @xmath217 represents the optimal cost . we also define the following class of functions @xmath218 [ defig ] let @xmath219 be such that for every @xmath220 the following hold : * @xmath221 is twice continuously differentiable with respect to @xmath222 for all @xmath223 and @xmath224 is in dom(@xmath196 ) for each @xmath225,@xmath93 , * for fixed @xmath226 , @xmath227 , * for each @xmath228 , @xmath229 is in @xmath230 . let @xmath136}$ ] be an admissible strategy such that it has the following form @xmath231 for some measurable @xmath232 . we call such controls as markov feedback control . then the augmented process @xmath233}$ ] is markov where , @xmath234 are as in , , ( [ eq : pov ] ) . we note that for any measurable @xmath232 , the equation may not have a strong solution . however , we will show the existence of a markov feedback control which is optimal and under which has an a.s . unique strong solution . let @xmath235 be the infinitesimal generator of @xmath236}$ ] , and @xmath224 be a @xmath237 function with compact support , then we have @xmath238 \frac{\partial } { \partial v } { \varphi}(t , x , y , v ) \nonumber\\ & \quad + \frac{1}{2 } v^2 \left[\tilde u^{*}(t , x , y , v)a(t , x)\tilde u(t , x , y , v ) \right]\frac{\partial^2 } { \partial v^2}{\varphi}(t , x , y , v)\nonumber \\ & \quad + \sum\nolimits_{j=1}^{m_2 } { \displaystyle\int\limits}_{\mathbb{r } } \!\left[{\varphi}\left(t , x , y , v\left(1 + [ \tilde u^{*}(t , x , y , v ) \eta(z)]_j\right)\right ) - { \varphi}(t , x , y , v ) \right]\ , \nu_j(dz_j ) \nonumber \\ & \quad + \sum_{l=0}^{n}\sum_{j\neq x^l}\lambda^l_{x^lj}(y^l)\left[{\varphi}(t , r^l_jx , r^l_0y , v ) - { \varphi}(t , x , y , v)\right],\end{aligned}\ ] ] where the linear operator @xmath239 is given by @xmath240 , @xmath63 , @xmath241 and @xmath242 is the standard basis of @xmath243 . for a given @xmath244 , by abuse of notation , we write @xmath245 , when @xmath246 for all @xmath247 . we consider the following hjb equation @xmath248 with the terminal condition @xmath249^{n+1 } , \quad v>0.\end{aligned}\ ] ] we now define a classical solution to the problem - . [ defcl ] we say @xmath250 is a classical solution to - if @xmath220 and for all @xmath251 , @xmath252 satisfies - . we look for a solution to - of the form @xmath253 where @xmath254 . clearly the left hand side of is in class @xmath255 . we will establish the following result in first two subsections . [ theo2 ] the cauchy problem - has a unique classical solution , @xmath256 , of the form . substitution of ( [ eq : rstransf ] ) into , yields @xmath257 + h_\theta(t , x)\psi(t , x , y)=0,\end{aligned}\ ] ] for each @xmath199 with the condition @xmath258 where the map @xmath259\times \mathcal{x}^{n+1}\rightarrow \mathbb{r}$ ] is given by @xmath260,\ ] ] the infimum of a family of continuous functions @xmath261 + \frac{1}{2 } \left(-\frac{\theta}{2}\right)\left(-\frac{\theta}{2}-1\right)\left[u^{*}~a(t , x)~u \right ] \nonumber \\ & \quad + \sum\nolimits_{j=1}^{m_2 } { \displaystyle\int\limits}_{\mathbb{r } } \!\left(\left(1 + [ u^ { * } \eta(z)]_j\right)^{\left(-\frac{\theta}{2}\right)}- 1 \right)\ , \nu_j(dz_j).\nonumber\end{aligned}\ ] ] it is important to note that the linear first order equation is nonlocal due to the presence of the term @xmath262 in the equation . it implies that @xmath263 depends on the value of @xmath264 at the point @xmath265 , which does not lie in the neighbourhood of @xmath266 . we now define a classical solution to - below . [ deffcl ] we say @xmath267 is a classical solution to - if @xmath268 dom@xmath269 and for all @xmath199 , @xmath252 satisfies - . it is interesting to note that other than the terminal condition , no additional boundary conditions are imposed . the remaining parts of the boundary is @xmath270\}$ ] . we note from that , @xmath271 , for all @xmath42 $ ] . hence @xmath272 does not cross the boundary . thus the pde would have no solution for any boundary condition which is not obtained from the terminal condition . [ theo3 ] the cauchy problem - has a unique classical solution . note that theorem [ theo2 ] may be treated as a corollary of theorem [ theo3 ] in view of the substitution ( [ eq : rstransf ] ) and subsequent analysis . thus it suffices to establish theorem [ theo3 ] . we establish theorem [ theo3 ] in the subsection [ lfoe ] via a study of an integral equation which is presented in subsection [ tvie ] . the following result would be useful to establish well - posedness of - . [ hcontinuityrs ] consider the map @xmath259\times \mathcal{x}^{n+1}\rightarrow \mathbb{r}$ ] , given by , . then under _ * ( a3 ) * _ , we have * @xmath273 is continuous , negative valued and bounded below ; * @xmath274 is @xmath23 in both @xmath275 and @xmath276 for each @xmath93 ; * for every @xmath277 , there exists a unique @xmath278 such that @xmath279 . and @xmath280\times \mathcal{x}^{n+1}\to \mathbb{a}_1 $ ] is continuous in @xmath90 ; * @xmath281 is admissible . \(i ) we recall that , @xmath282 , the range of portfolio includes the origin . therefore @xmath283 thus @xmath273 is negative valued . by the continuity assumptions on @xmath284 and @xmath285 , for fixed @xmath124 and each @xmath286 , @xmath92 , @xmath287 and @xmath104 are bounded on @xmath140 $ ] . let @xmath288 be such that @xmath289}\{|r(t , x)| , ||b(t , x)|| , ||a(t , x)||\ } \leq m.\ ] ] we also observe that for each @xmath290 , @xmath291_j\right)^{-\frac{\theta}{2}}-1)\ , \nu_j(dz_j ) & \geq & -\sum_j { \displaystyle\int\limits}_{\mathbb{r } } \nu_j(dz_j ) \\ & = & -\sum_j\nu_j(\mathbb{r})>-\infty , \end{aligned}\ ] ] using the finiteness of the measure @xmath31 . also , * ( a3 ) * gives @xmath292 hence by using the above mentioned bounds , we can write , @xmath293 , where @xmath294 since @xmath295 is independent of @xmath90 and @xmath296 as @xmath297 , @xmath298 is bounded below . now we will show that for fixed @xmath90 and @xmath93 , @xmath299 is a strictly convex function of variable @xmath244 . for fixed @xmath90 and @xmath93 , let @xmath300 denote the hessian matrix for @xmath301 . then @xmath302-th element of @xmath300 , @xmath303_j\right)^{-\frac{\theta}{2}-2}\ , \nu_j(dz_j).\end{aligned}\ ] ] since @xmath124 is in @xmath304 , @xmath305_j$ ] is bounded below by a positive @xmath306 . hence , in addition to that using * ( a3 ) * , there exists @xmath307 such that @xmath308 is a positive definite matrix and this proves the strict convexity of @xmath299 on variable @xmath124 . therefore @xmath309\right)$ ] is a non - empty convex compact set . hence , @xmath310 is a compact - valued correspondence . since @xmath273 is negative , from , we can write @xmath311 we also note that @xmath312 is jointly continuous . since @xmath310 is continuous , then it follows from the maximum theorem ( @xcite , th . @xmath313 ) that @xmath314 is continuous with respect to @xmath277 . hence ( i ) is proved . \(ii ) follows from the continuity of @xmath314 . \(iii ) the set of minimizers is defined by @xmath315 again by using ( @xcite , th . @xmath313 ) , @xmath316 is upper semi - continuous . since @xmath299 is strictly convex in @xmath124 , for each @xmath317 $ ] and @xmath286 there exist only one element in @xmath318 . by abuse of notation , we denote that element by @xmath318 itself . since a single - valued upper semi - continuous correspondence is continuous , @xmath318 is a continuous function . \(iv ) since @xmath319 is continuous in @xmath90 , there exists a positive constant @xmath320 such that @xmath321 for all @xmath42 $ ] , @xmath322 . thus @xmath323 is bounded . since @xmath323 does not depend on @xmath225 , the lipschitz conditions of theorem 1.19 of @xcite are satisfied . again since @xmath323 is bounded , all growth conditions are also satisfied . therefore definition [ defi1](ii ) is satisfied and this completes the proof . in order to study - we consider the following integral equation with the previous notations @xmath324dr , \end{aligned}\ ] ] for all @xmath325 , where @xmath326 with each component being @xmath327 . .2 in equation is a volterra integral equation of second kind . we note that the boundary of @xmath328 has many facets . for @xmath329 , we directly obtain from , @xmath330 . hence no additional terminal conditions are required . although the values of @xmath264 in facets @xmath270\}$ ] are not directly followed but can be obtained by solving the integral equation on the facets . [ iesolnrs ] ( i ) the integral equation has a unique solution in @xmath331 , and ( ii ) the solution is in the _ dom_(@xmath196 ) . \(i ) we first observe that the solution to the integral equation is a fixed point of the operator @xmath332 , where @xmath333dr.\end{aligned}\ ] ] it is easy to check that for each @xmath334 is continuous . now since @xmath335 by proposition [ hcontinuityrs](i ) , @xmath336f_{\tau^l|l}(r|x , y)dr]|\\ & \leq & \sum_{i=1}^{n}p_{t , x , y}(\ell(t)=l)\int_{0}^{t - t}e^{h_\theta(t , t+r , x)}\sum_{j\neq x^l}p^l_{x^lj}(y^l+r)f_{\tau^l|l}(r|x , y)dr\|\psi-\tilde{\psi}\|\\ & < & k_1\|\psi-\tilde{\psi}\|,\end{aligned}\ ] ] where @xmath337 . since @xmath338 is strictly less than @xmath327 , implies that @xmath339,for all @xmath340 . hence @xmath341 . therefore , @xmath332 is a contraction . thus a direct application of banach fixed point theorem ensures the existence and uniqueness of the solution to . \(ii ) we denote the unique solution by @xmath264 . next we show that @xmath342 dom@xmath269 . to this end , it is sufficient to show that @xmath343 . the first term of @xmath344 is in dom@xmath269 , which follows from lemma [ theo1 ] ( iv ) and proposition [ hcontinuityrs ] ( ii ) . now to show that the remaining term @xmath345dr,\end{aligned}\ ] ] is also in the dom(@xmath196 ) for any @xmath346 , we need to check if the following limit @xmath347,\end{aligned}\ ] ] exists and , the limit is continuous in @xmath348 . if the limit exists , the limiting value is clearly @xmath349 . by a suitable substitution of variables in the integral , the expression in the above limit can be rewritten , using , as @xmath350 by lemma [ theo1 ] ( iv ) , @xmath351 is in dom@xmath269 . thus @xmath352 is bounded on @xmath353 $ ] by a positive constant @xmath354 . hence by the mean value theorem on @xmath351 , the integrand of the first integral of is uniformly bounded . therefore , using the bounded convergence theorem , the integral converges as @xmath355 . the second integral of converges as the integrand is continuous at @xmath356 . now we compute @xmath357\big)dr\\ & & -\sum_{j\neq x^l}p^l_{x^lj}(y^l)\psi(t , r^l_jx , r^l_0y)f_{\tau^l|l}(0|x , y),\end{aligned}\ ] ] using lemma [ theo1 ] ( iii ) . from we know @xmath358 , therefore @xmath349 can be rewritten using as @xmath359\beta_l(t , x , y)-\sum_{j\neq x^l}p^l_{x^lj}(y^l)\psi(t , r^l_jx , r^l_0y)f_{\tau^l|l}(0|x , y).\end{aligned}\ ] ] clearly is in @xmath197 . hence @xmath360 is in the dom(@xmath196 ) . hence the right hand side of is in the dom(@xmath196 ) for any @xmath361 . thus ( ii ) holds . [ ivpuniqrs ] the unique solution to also solves the initial value problem - . let @xmath264 be the solutions of the integral equation . then by substituting @xmath329 in , follows . using the results from the proof of lemma [ theo1 ] , proposition [ iesolnrs ] , lemma [ theo1](iv ) and , we have @xmath362[1-f_{\tau^l|l}(t - t|x , y)]e^{h_\theta(t , t , x)}\\ & -\sum_{l=0}^{n}p_{t , x , y}(\ell(t)=l)\big[f_{\tau^l|l}(0|x , y)(f_{\tau^l|l}(v|x , y)-1)\big]\\ & \times e^{h_\theta(t , t , x)}-h_\theta(t , x)\sum_{l=0}^{n}p_{t , x , y}(\ell(t)=l)[1-f_{\tau^l|l}(t - t|x , y)\\ & \times e^{h_\theta(t , t , x)}+\sum_{l=0}^{n}\big[\sum_r f_{\tau^r}(0|x^r , y^r)p_{t , x , y}(\ell(t)=l)-f_{\tau^l}(0|x^l , y^l)\big]\beta_l(t , x , y)\\ & + \sum_{l=0}^{n}p_{t , x , y}(\ell(t)=l)\big(-h_\theta(t , x)+f_{\tau^l|l}(0|x , y)\big)\beta_l(t , x , y)\\ & -\sum_{j\neq x^l}p^l_{x^lj}(y^l)\psi(t , r^l_jx , r^l_0y)f_{\tau^l|l}(0|x , y)\big).\end{aligned}\ ] ] using the equality in lemma [ theo1 ] ( v ) , the right hand side of above equation can be rewritten as @xmath363-h_\theta(t , x)\psi(t , x , y)\\ & = -\sum_l\sum_{j\neq x^l}{\lambda}^l_{x^lj}(y^l)\big[\psi(t , r^l_jx , r^l_0y)-\psi(t , x , y)\big]-h_\theta(t , x)\psi(t , x , y).\end{aligned}\ ] ] hence @xmath264 satisfies . [ classicalrs ] let @xmath264 be a classical solution to - . then @xmath264 solves the integral equation . if the pde has a classical solution @xmath264 , then @xmath264 is also in the domain of @xmath364 , where @xmath364 is the infinitesimal generator of @xmath365 . then we have from @xmath366 consider @xmath367 then by it s formula , @xmath368 where @xmath369 is a local martingale with respect to @xmath370 , the usual filtration generated by @xmath371 . thus from @xmath372 is a local martingale . from definition of @xmath373 , @xmath374}n_t<\|\psi\|e^{\|h_\theta\|t}$ ] a.s . thus @xmath372 is a martingale . therefore by using , we obtain @xmath375.\end{aligned}\ ] ] hence using the markov property of @xmath184 and * ( a4 ) * , @xmath376,~\forall ( t , x , y)\in\bar{\mathscr{d}}.\ ] ] by conditioning on the component of @xmath78 where the transition happens , @xmath377}|\ell(t)]]\\ & = & \sum_{l=0}^{n}p_{t , x , y}(\ell(t)=l)\mathbb{e}_{t , x , y}[e^{[\int_{t}^{t}h_\theta(s , x_s)ds]}|\ell(t)=l ] \end{aligned}\ ] ] where @xmath190 is described in subsection 2.4 after * ( a4)*. next by conditioning on @xmath180 we rewrite @xmath378\\ { \nonumber}&=&\mathbb{e}_{t , x , y}[\mathbb{e}_{t , x , y}[e^{\int_{t}^{t}h_\theta(s , x_s)ds}|\ell(t)=l,\tau^l(t)]|\ell(t)=l]\\ { \nonumber}&=&p_{t , x , y}(\tau^l(t)>t - t|\ell(t)=l)e^{\int_{t}^{t } h_\theta(s , x)ds}\\ & & + \int_{0}^{t - t}\mathbb{e}_{t , x , y}[e^{\int_{t}^{t}h_\theta(s , x_s)ds}|\ell(t)=l,\tau^l(t)=r ] f_{\tau^l|l}(r|x , y ) dr . \end{aligned}\ ] ] since @xmath379 is constant on @xmath380 provided @xmath381 , the above expression is equal to @xmath382e^{h_\theta(t , t , x)}+\int_{0}^{t - t}e^{h_\theta(t , t+r , x)}\\ & & \times \mathbb{e}_{t , x , y}[\mathbb{e}_{t , x , y}[e^{\int_{t+r}^{t}h_\theta(s , x_s)ds}|x^l_{t+r},\ell(t)=l,\tau^l = r]|\ell(t)=l,\tau^l = r]f_{\tau^l|l}(r|x , y)dr\\ & = & [ 1-f_{\tau^l|l}(t - t|x , y)]e^{h_\theta(t , t , x)}\\ & & + \int_{0}^{t - t}e^{h_\theta(t , t+r , x)}\times \sum_{j\neq x^l}p^l_{x^lj}(y^l+r)\psi(t+r , r^l_jx , r^l_0(y+r\mathbf{1}))f_{\tau^l|l}(r|x , y)dr . \end{aligned}\ ] ] from and the above expression , the desired result follows . the result follows from proposition [ iesolnrs ] , proposition [ ivpuniqrs ] , and proposition [ classicalrs ] . now we are in a position to derive the expression of optimal portfolio value under risk sensitive criterion . the optimal value is given by @xmath383 where the existence and uniqueness of the classical solution to - follows from theorem [ theo3 ] . we note that the study of - becomes much simpler if the coefficients @xmath384 are independent of time @xmath90 . for time homogeneous case , proposition _ [ hcontinuityrs ] _ is immediate . furthermore , the proof of theorem _ [ theo3 ] _ does not need the results given in proposition _ [ iesolnrs ] _ , proposition _ [ ivpuniqrs ] _ , and proposition _ [ classicalrs]_. indeed theorem _ [ theo3 ] _ can directly be proved by noting the smoothness of terminal condition . we conclude this section with a proof of the verification theorem for optimal control problem . the main result is given in theorem [ markov&admissible ] . [ verificationth ] let @xmath256 be as in theorem _ [ theo2 ] _ , then * @xmath385 for every admissible markov feedback control @xmath386 . * let @xmath323 be as in proposition _ [ hcontinuityrs]_(iv ) , then @xmath387 \(i ) consider an admissible markov feedback control @xmath388 , where @xmath389 and @xmath256 , the classical solution to - as in . now by it s formula @xmath390 dr \nonumber\\ & = \displaystyle\sum_{j=1}^{m_1}\int_t^s \frac{\partial}{\partial v}{\varphi}_m(r , x_r , y_r , v^{\bar{u}}_r ) v^{\bar{u}}_r[\tilde{u}(r , x_r , y_r , v_r)^*{\sigma}(r , x_r)]_j dw^j_r\nonumber \\ & + \displaystyle\sum_{j=1}^{m_2}\int_t^s \int_\mathbb{r } \bigg[{\varphi}_m(r , x_r , y_r , v^{\bar{u}}_{r-}(1+[\tilde{u}(r , x_{r-},y_{r-},v_{r-})^*\eta(z)]_j))- { \varphi}_m(r , x_r , y_r , v^{\bar{u}}_{r-})\bigg ] \tilde{n}_j(dr , dz_j)\nonumber\\ & + \sum_{l=0}^{n}\int_t^s \int_\mathbb{r } \bigg[{\varphi}_m(r , r^l_{x^l_{r- } + h^l(x^l_{r-},y^l_{r-},z_0)}(x_{r-}),r^l_{y^l_{r-}-g^l(x^l_{r-},y^l_{r-},z_0)}(y_{r-}),v^{\bar{u}}_{r- } ) \nonumber\\ & -{\varphi}_m(r , x_{r-},y_{r-},v^{\bar{u}}_{r-})\bigg ] \tilde{\wp}^l(dr , dz_0).\end{aligned}\ ] ] we would first show that the right hand side is an @xmath391 martingale . since @xmath386 is admissible , using definition [ defi1](iii ) , it is sufficient to show , the following square integrability condition @xmath392 ^ 2dr < \infty , \end{aligned}\ ] ] to prove that the first term is a martingale . again since @xmath393 , @xmath394 . thus using the boundedness of @xmath264 the above would follow if @xmath395^{-\theta}dr<\infty\end{aligned}\ ] ] holds . now we consider the second integral . rewriting that term , we obtain @xmath396_j)^{-\frac{\theta}{2}}-1\right]\tilde{n}_j(dr , dz_j).\end{aligned}\ ] ] we first observe that @xmath397_j)>\delta$ ] , and this implies @xmath398_j)^{-\frac{\theta}{2}}<\delta^{-\frac{\theta}{2}}.\ ] ] thus the integrand of is a product of a bounded function and @xmath399 . since @xmath31 , the lvy measure of @xmath400 is a finite measure for each @xmath46 , to show is an @xmath391 martingale , it is enough to verify . similarly the third integral can be rewritten as @xmath401\tilde{\wp}^l(dr , dz_0).\end{aligned}\ ] ] in the integrand is a product of a bounded function with compact support and @xmath399 . since , the compensator of @xmath402 is @xmath403 , is also an @xmath391 martingale if holds . thus is the sufficient condition for the right side of to be a martingale . however readily follows from the lemma [ solstate](ii ) and an application of tonelli s theorem . taking conditional expectation on both sides of given @xmath404 and letting @xmath405 , we obtain @xmath406-{\varphi}_m(t , x , y , v)\nonumber\\ & = \mathbb{e}\int_t^t\bigg[\mathscr{a}^{\tilde{u}}{\varphi}_m(r , x_r , y_r , v_r^{\bar{u}})\bigg|x_t = x , y_t = y , v^{\bar{u}}_t = v\bigg]dr\geq 0.\end{aligned}\ ] ] the above non - negativity follows , since @xmath256 is the classical solution to - and @xmath407 for all @xmath0 . and implies result ( i ) . \(ii ) the right hand side of becomes zero by considering @xmath408 and this completes the proof of ( ii ) . finally we show in the following theorem that @xmath256 as in theorem [ theo2 ] indeed gives the optimal performance under all admissible controls . [ markov&admissible ] let @xmath256 be as in theorem _ [ theo2 ] _ and @xmath409 . then @xmath410 . we first note that in the proof of proposition [ verificationth](i ) , we have only used the properties ( ii ) and ( iii ) of definition [ defi1 ] of the markov control . since these two properties are true for a generic admissible control @xmath124 , we can get as in proposition [ verificationth](i ) . @xmath411 for every admissible control @xmath124 . by taking infimum , we get @xmath412 . the other side of inequality is rather straight forward . using proposition [ verificationth](ii ) and theorem [ hcontinuityrs](iv ) , @xmath323 is admissible , and @xmath413 . thus @xmath414 . hence the result is proved . now we establish a characterisation of @xmath256 using the hjb equation in the following proposition . [ comparison ] let @xmath224 be any classical solutions to - . let @xmath256 be as in theorem _ [ theo2]_. then @xmath415 , for all @xmath247 . thus the unique solution @xmath256 obtained in theorem [ theo2 ] is maximal among all classical solution to - . note that in the proof of proposition [ verificationth](i ) , to show that the right hand side of is a martingale , we have only effectively used the fact that @xmath256 satisfies conditions ( i),(ii ) and ( iii ) of definition [ defig ] . hence for any @xmath220 and @xmath323 as in proposition [ hcontinuityrs](iv ) , @xmath416 dr,\end{aligned}\ ] ] is an @xmath417 martingale . taking conditional expectation in , given @xmath418 and letting @xmath405 , we have @xmath419-{\varphi}(t , x , y , v)\nonumber\\ & = \mathbb{e}\int_t^t\bigg[\mathscr{a}^{u^\star}{\varphi}(r , x_r , y_r , v_r^u)\bigg|x_t = x , y_t = y , v^{\bar{u}^\star}_t = v\bigg]dr , \end{aligned}\ ] ] using @xmath420 . now using rhs is nonnegative and theorem [ verificationth](ii ) , we obtain @xmath421 . we have seen that the optimal portfolio value with risk sensitive criterion is given by ( [ iefinalrs ] ) and - . for illustration purpose , we are considering a simple model in which all the parameters for all assets are governed by a single semi - markov process . then @xmath422 if @xmath423 , and we denote that value as @xmath424 where @xmath425 and @xmath426 are the first components of @xmath93 , and @xmath427 respectively . hence implies @xmath428 provided @xmath429 and @xmath430 . in other words @xmath431 depends only on @xmath432 . in view of this , we may introduce a new function @xmath433 to denote @xmath434 . therefore gets reduced to @xmath435+\bar{h}_\theta(t , x)\psi(t , x , y)=0 , \end{aligned}\ ] ] for every @xmath436 , @xmath437 , @xmath438 . we further assume that @xmath439 , i.e. , the portfolio includes a single stock and a money market instrument . we also specify the state space @xmath440 , i.e. , the semi - markov process has three regimes . the drift coefficient , volatility and instantaneous interest rate at each regime are chosen as follows : @xmath441 the transition rates for @xmath169 are assumed to be given by @xmath442 where @xmath443 hence the holding time of the first component in each regime has the conditional probability density function @xmath444 and the conditional c.d.f @xmath445 . we also assumed @xmath446 and @xmath447}(z)}{b - a } dz$ ] . it is shown separately in @xcite that the classical solution to with @xmath448 , satisfies the following integral equation @xmath449 + { \displaystyle\int\limits}_{0}^{t - t}\exp\left[{\displaystyle\int\limits}_{t}^{t+r}\bar h_\theta(s , x)\,ds\right]\times \nonumber \\ & \quad\sum\nolimits_{j \neq x } p_{xj}(y+r)\bar{\psi}(t+r , j,0)\frac{f(y+r \mid x)}{1-f(y \mid x)}dr,\end{aligned}\ ] ] which also follows from . here we compute @xmath450 by discretization of above integral equation using an implicit step - by - step quadrature method as developed in @xcite . we take @xmath451 , @xmath452 so @xmath453 . the discretization is given by @xmath454 therefore from ( [ eq : iers ] ) we get @xmath455 + \delta t \sum_{l=0}^{m}w_m(l)\nonumber \\ & \frac{f(y+l\delta t \mid i)}{1-f(y \mid i ) } \left(\exp\left[h_\theta^{m - l}(i ) - h_\theta^m(i)\right ] \sum_{j \in\mathcal{x } , j \ne i}p_{ij}\psi^{m - l}(j,0)\right),\end{aligned}\ ] ] where @xmath456 are weights , chosen as below @xmath457 and @xmath458 + \frac{1}{2 } \left(-\frac{\theta}{2}\right ) \left(-\frac{\theta}{2}-1\right)u^2\sigma^2(t , i ) \right.\nonumber \\ & \quad \quad \quad \left . -1 + \frac{(1+bu)^{1-\frac{\theta}{2 } } - ( 1+au)^{1-\frac{\theta}{2}}}{u(1-\frac{\theta}{2})(b - a ) } \right].\end{aligned}\ ] ] for a given initial portfolio value @xmath225 , from and we get @xmath459 thus the numerical approximation of risk sensitive optimal wealth is given by - . in proposition [ hcontinuityrs ] we have seen that there exists a unique @xmath244 which gives @xmath460 and that we can find by using any convex optimization technique . here we have used the interior - point method to find the optimal @xmath124 . .49 .49 we use above mentioned numerical scheme to compute the risk sensitive optimal wealth function given in ( [ eq : phitildeiter ] ) . figure [ fig1 ] describes the behaviour of risk sensitive optimal wealth for different values of initial portfolio wealth and maturity . the left side plot in figure [ fig1 ] show that the optimal wealth is monotonically increasing with the value of initial investment . on the other hand the right side plot shows monotonically increasing behaviour of the optimal wealth with respect to the the maturity of investment . figure [ fig : fig2rs ] shows the movement in risk sensitive optimal wealth for different values of risk aversion parameter . the plot shows the strict diminishing behaviour of risk sensitive optimal wealth for increasing risk aversion parameter value . in this paper a portfolio optimization problem , without any consumption and transaction cost , where stock prices are modelled by multi dimensional geometric jump diffusion market model with semi - markov modulated coefficients is studied . we find the expression of optimal wealth for expected terminal utility method with risk sensitive criterion on finite time horizon . we have studied the existence of classical solution of hjb equation using a probabilistic approach . we have obtained the implicit expression of optimal portfolio . it is important to note that , the control is robust in the sense that the optimal control does not depend on the transition function of the regime . we have also implemented a numerical scheme to see the behaviour of solutions with respect to initial portfolio value , maturity and risk of aversion parameter . the results of the numerical scheme are in agreement with the theory of financial market . the corresponding problem in infinite horizon is needs further investigation . this would require appropriate results on large deviation principle for semi - markov processes which need to be carried out . first we show the uniqueness by assuming that the sde ( [ eq : sde ] ) admits a solution , @xmath461 , say , the stopping time @xmath462 . using it^ o lemma ( theorem 1.16 of @xcite ) for @xmath463 we get , @xmath464 -\frac{1}{2}(s_{s-}^l)^{-2}(s_{s-}^l)^2 a_{ll}(s , x_{s-})ds \\ & \quad + \sum\nolimits_{j=1}^{m_2}{\displaystyle\int\limits}_{\mathbb{r}}\ ! \left[\ln(s_{s-}^l + \eta_{lj}(z_{j})s_{s-}^l ) -\ln(s_{s-}^l)\ , \right]\ , n_j(ds , dz_j ) . \end{aligned}\ ] ] integrating both sides from 0 to @xmath465 yields , @xmath466 where all the integrals have finite expectations almost surely by using * ( a2)*. @xmath467 \end{aligned}\ ] ] thus any solution to ( [ eq : sde ] ) has the above expression . under * ( a2 ) * , @xmath468 has finite expectation for any finite stopping time @xmath469 . let @xmath470 . now if possible , assume @xmath471 . by letting @xmath472 in the above expression , we obtain that @xmath473 is exponential of a random variable which is finite for almost every @xmath474 . thus @xmath475 . but for almost every @xmath476 @xmath477 . hence non - positivity occurred only by jump . in other words @xmath478 for some @xmath479 . but that is contrary to the assumption on @xmath145 . hence @xmath480 . therefore , @xmath481 a.s . for all @xmath482 and is given by @xmath483 . \end{aligned}\ ] ] thus by equation ( [ eq : sdesol ] ) , @xmath484 is an adapted and rcll process and is uniquely determined with the initial condition @xmath485 . hence the solution is unique . \(i ) one can compute the conditional c.d.f @xmath183 in the following way @xmath487 we also denote the derivative of @xmath488 by @xmath489 , given by @xmath490 from the definition of @xmath491 we have , @xmath492 we also introduce a new variable @xmath493 . we denote the conditional c.d.f of @xmath494 given @xmath192 and @xmath193 as @xmath495 which is equal to @xmath496 . it is easy to see that @xmath497 . to compute this probability we use a conditioning on @xmath180 . thus @xmath498\\ { \nonumber } & = & { \displaystyle\int\limits}_0^v p_{t , x , y}(\tau^{-l}(t)>\tau^l(t)|\tau^l(t)=s ) f_{\tau^l}(s|x^l , y^l)ds\\ { \nonumber } & = & { \displaystyle\int\limits}_0^v(1-p_{t , x , y}(\tau^{-l}(t)\leq s))f_{\tau^l}(s|x^l , y^l)ds\\ & = & { \displaystyle\int\limits}_0^v\prod_{m\neq l}(1-f_{\tau^m}(s|x^m , y^m))f_{\tau^l}(s|x^l , y^l)ds.\end{aligned}\ ] ] again , @xmath499 and from , we have ( i ) . \(iv ) in order to show that @xmath191 and @xmath209 belong to @xmath196 we introduce a new function @xmath504 and @xmath505 . consider another function @xmath506 we note that @xmath507 is the derivative of @xmath508 with respect to @xmath225 and it is continuous . now we show that @xmath508 is @xmath23 in @xmath427 . to this end we first show the existence of the following limit @xmath509.\end{aligned}\ ] ] by a suitable substitution of variable , the expression in the above limit is @xmath510.\end{aligned}\ ] ] using the above expression converges to @xmath511 as @xmath355 and the limit is continuous in @xmath427 . thus @xmath512 if @xmath225 is a differentiable function of @xmath90 , then @xmath513 hence @xmath514 since @xmath515 it follows from lemma [ theo1 ] ( i ) , ( ii ) and the above notations @xmath516 and @xmath517 . hence @xmath191 and @xmath209 are in the dom(@xmath196 ) . now operating @xmath196 on @xmath191 and using , we have @xmath518 operating @xmath196 on @xmath209 @xmath519 this completes the proof of ( iv ) . * acknowledgement : * the authors are grateful to mrinal k. ghosh and anup biswas for very useful discussions . the authors also do acknowledge the referee and the associate editor for their insightful suggestions which have helped to improve the quality of this work . we are also thankful to the editor for encouraging us to revise the initial manuscript . ghosh m. k. , goswami a. , and kumar , s. , _ portfolio optimization in a semi - markov modulated market _ , appl math optim , 60(2009 ) , 275 - 296 . ghosh m. k. , goswami , a. , and kumar , s. , _ portfolio optimization in a markov modulated market _ , modern trends in controlled stochastic processes , 181 - 195 , 2010 . ghosh m. k. and saha , s. , stochastic processes with age - dependent transition rates , stoch . 29(2011 ) , 511 - 522 . goswami , a. , patel , j. and shevgaonkar , p. , a system of non - parabolic pde and application to option pricing , stoch . 34(2016 ) , 893 - 905 . hunt j. and devolder p. , semi - markov regime switching interest rate models and minimal entropy measure , physica a : statistical mechanics and its applications 390 , 15(2011 ) , 3767 - 3781 . kallsen jan , optimal portfolios for exponential lvy processes . math . methods oper . 51 ( 2000 ) , 357 - 374 .
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this article studies a portfolio optimization problem , where the market consisting of several stocks is modeled by a multi - dimensional jump diffusion process with age - dependent semi - markov modulated coefficients .
we study risk sensitive portfolio optimization on the finite time horizon .
we study the problem by using a probabilistic approach to establish the existence and uniqueness of the classical solution to the corresponding hamilton - jacobi - bellman ( hjb ) equation .
we also implement a numerical scheme to investigate the behavior of solutions for different values of the initial portfolio wealth , the maturity and the risk of aversion parameter . 1 true cm * key words * portfolio optimization , jump diffusion market model , semi - markov switching , risk sensitive criterion .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
cooperation is vital for the maintenance of public goods in human societies @xcite . but according to darwin s theory of evolution , competition rather than cooperation ought to drive our actions . the reconciliation of this theory with the fact that cooperation is widespread in human societies , as well as with the fact that it is much more common in nature as one might expect , is one of the most persistent challenges in evolutionary biology and social sciences @xcite . past decades have seen the paradigm of punishment rise as one of the more successful strategies by means of which cooperation might be promoted @xcite . indeed , punishment is also the principle tool of institutions in human societies for maintaining cooperation and otherwise orderly behavior @xcite . however , punishment is costly , and as such it reduces the payoffs of both the defectors as well as of those that exercise the punishment , hence yielding an overall lower income and acting as a drain on social welfare . thus , understanding the emergence of costly punishment is crucial for the evolution of cooperation @xcite . while recent research confirms that punishment is often motivated by negative personal emotions such as anger or disgust @xcite , raihani and mcauliffe have shown also that the decision to punish is often motivated with the aversion of inequity in mind , rather than by the desire for reciprocity @xcite . although prosocial punishment is widespread in nature @xcite , it is unlikely that cooperators are willing to commit permanently to punishing wrongdoers . for that , the action is simply to costly , and hence some form of abstinence is likely , also to avoid unwanted retaliation . several research groups have recently investigated these and related up and down sides of punishment @xcite . for example , it was shown that cooperators punish defectors selectively depending on their current personal emotions , even if the number of defectors is large @xcite . more often than not , however , whether or not to punish depends on the whiff of the moment and is thus a fairly random event . motivated by these observations , we have recently shown that sharing the effort of punishment in a probabilistic manner can significantly lower the vulnerability of costly punishment and in fact help stabilize costly altruistic strategies @xcite . here we drop the assumption that cooperators who do punish defectors do so uniformly at random . instead , we account for the diversity in punishment , taking into account the fact that some individuals are more likely to punish , while others punish only rarely . more specifically , we introduce different threshold levels for punishment , which ultimately introduces different classes of cooperators that punish defectors . the assumption of diverse players is not just a realistic hypothesis , but in general it is firmly established that it also has a decisive impact on the evolution of public cooperation @xcite . motivated by this fact , we therefore study a spatial public goods game with defectors and different types of punishing cooperators . while previously we have demonstrated the importance of randomly shared punishment @xcite , we here approach a more realistic scenario by assuming that each type of cooperators will punish with a different probability . our goal is to determine whether a specific class of punishing cooperators will be favored by natural selection , or whether despite the competition among them synergistic effects will emerge . as we will show , the evolution is governed by a counterintuitive selection mechanism , depending further on the synergistic effects of cooperative behavior . however , before presenting the main results in detail , we proceed by a more accurate description of the studied spatial public goods game with different punishing strategies . we consider a population of individuals who play the public goods game on a square lattice of size @xmath0 with periodic boundary conditions . we assume that the game is contested between @xmath1 classes of cooperators ( @xmath2 , @xmath3 , @xmath4 , @xmath5 ) and defectors ( @xmath6 ) . independently of the class a cooperator belongs to , it contributes an amount @xmath7 to the common pool , while defectors contribute nothing . after the sum of all contributions in the group is multiplied by the enhancement factor @xmath8 , the resulting amount is shared equally among all group members . moreover , cooperators with strategy @xmath9 ( @xmath10 ) choose to punish defectors with a probability @xmath11 if the latter are present . as a result , each defector in the group is punished with a fine @xmath12 , while all the cooperators who participated in the punishment equally shared the associated costs . in particular , each punishing cooperator bears the cost @xmath13 , where @xmath14 and @xmath15 are the number of cooperators and punishers in the group , respectively . we emphasize that a cooperator who decides to punish bears the same cost independently of the class it belongs to . thus , here the strategy @xmath16 only determines how frequently a cooperator is willing to punish defectors . nevertheless , it is worth pointing out that @xmath2 never punish and thus correspond to traditional second - order free - riders because they enjoy the benefits of punishment without contributing to it @xcite . on the other extreme , cooperators belonging to the @xmath5 class punish always when defectors are present in the group . since each player on site @xmath17 with von neumann neighborhood is a member of five overlapping groups of size @xmath18 , in each generation it participates in five public goods games and obtains its total payoff @xmath19 , where @xmath20 is the payoff gained from group @xmath21 . subsequently , a player @xmath17 , having strategy @xmath22 , adopts the strategy @xmath23 of a randomly chosen neighbor @xmath24 with the probability @xmath25},\ ] ] where @xmath26 denotes the amplitude of noise @xcite . without loosing generality and to ensure continuity of this line of research @xcite we set @xmath27 , meaning that it is very likely that the better performing players will pass their strategy to their neighbors , yet it is also possible that players will occasionally learn from a less successful neighbor . to conclude the description of this public good game , we would like to emphasize that different @xmath9 classes represent different strategies , as our goal is to explore how the willingness to punish evolves at specific parameter values . the model is studied by means of monte carlo simulations . initially , defectors randomly occupy half of the square lattice , and each type of cooperators randomly @xmath28 of the rest of the lattice . during one full monte carlo step ( mcs ) , all individuals in the population receive a chance once on average to adopt another strategy . depending on the proximity to phase transition points and the typical size of emerging spatial patterns , the linear system size was varied from @xmath29 to @xmath30 and the relaxation time was varied from @xmath31 to @xmath32 mcs to ensure proper statistical accuracy . the reported fractions of competing strategies were determined in the stationary state when their average values became time - independent . alternatively , we have averaged the outcomes over @xmath33 independent runs when the system terminated into a uniform absorbing state . and the probability to punish @xmath34 , as obtained for a low multiplication factor @xmath35 in the original model proposed in @xcite , where a uniform probability to punish was assumed for all cooperators . note that both @xmath12 and @xmath34 have a non - monotonous impact on the fraction of cooperators.,width=264 ] for the sake of comparison , we first present the fraction of cooperators in dependence on the punishment fine @xmath12 and the probability to punish @xmath34 at a low @xmath36 value , as obtained in the original probabilistic punishment model , where cooperators punish uniformly at random @xcite . figure [ fig1 ] illustrates that the fraction of cooperators first increases , reaches its maximum , but then again decreases , as the values of @xmath12 and @xmath34 increase along the diagonal on the @xmath37 plane . increasing one of these parameters , while the other is kept constant , returns to the same observation . both @xmath12 and @xmath34 thus have a non - monotonous impact on the fraction of cooperators , which is closely related with the fact that @xmath12 characterizes not only the level of punishment but also its cost . accordingly , too high values of @xmath12 involve too high costs stemming from the act of punishing . it is worth pointing out that @xmath35 , which is used in fig . [ fig1 ] , is a relatively low value of the multiplication factor at which the non - monotonous dependence can still be observed . in comparison with the results obtained for larger values of @xmath36 as used in ref . @xcite , however , the current plot features a significantly narrower @xmath34 region where full cooperation is possible when @xmath12 is sufficiently large . similarly , there is a limited region of intermediate @xmath12 values where cooperators that punish severely can beat defectors . based on these observations , in the present model we thus explore if there is an evolutionary selection among different punishing strategies as they compete against the defectors simultaneously , or if there is indeed cooperation in the common goal to deter defectors . when they start fighting with defectors simultaneously . panel ( c ) shows an enlarged part of panel ( a ) at low @xmath12 values , when cooperation becomes dominant over defection . to present the overall level of cooperation in the population , the cumulative fraction of @xmath9 strategies is also shown ( denoted by @xmath38 ) . for comparison , in panel ( b ) we have also plotted the resulting fraction of cooperator classes when they fight against defectors individually . as in panel ( c ) , panel ( d ) shows an enlarged part of panel ( b ) at a specific interval of @xmath12 . the multiplication factor in all panels is @xmath35.,width=321 ] for an intuitive overview , we set @xmath39 and investigate how the six types of punishing strategies compete and potentially cooperate with each other in the presence of defectors . the general conclusion , however , is robust and remains valid if we use other values of @xmath1 . using the same @xmath35 as in fig . [ fig1 ] , the panels of fig . [ fig2 ] summarize our main findings . the first panel shows the fractions of strategies in the final state in dependence of the punishment fine @xmath12 when different punishing strategies fight against defectors simultaneously . for clarity , we have also plotted the accumulated fraction of punishing strategies . in contrast to the uniform punishing model , we can see that the total fraction of cooperators should increase monotonously with increasing @xmath12 . as fig . [ fig2](a ) illustrates , cooperators can survive when @xmath40 , and become dominant over @xmath41 ( see also the enlarged part in fig . [ fig2](c ) ) . we should stress , however , that not all types of cooperators can survive at equilibrium , even if cooperators take over the whole population . it turned out that there are some `` weak '' classes of cooperators who go extinct before defectors die out , while other classes of cooperators survive . for a more in - depth explanation , the vitality of punishing classes can be estimated if we let them fight against defectors individually . the outcomes of this scenario are summarized in fig . [ fig2](b ) . results presented in this panel suggest that there are punishing classes who can dominate for all high @xmath12 values , while others become vulnerable as we increase @xmath12 . more interestingly , however , there are mildly punishing strategies who can survive only due to the support of the more successful punishing strategies . for example , for @xmath42 classes @xmath43 and @xmath44 can outperform defectors , while @xmath45 disappear when they fight against defectors individually [ fig . [ fig2](b ) and ( c ) ] . but when all punishing strategies are on the stage then @xmath45 players can survive as well . this effect is more spectacular for the second - order free riding @xmath2 class , who would die out immediately at such a low synergy factor @xmath36 if they face defectors alone . but now , especially at high @xmath12 values , their ratio becomes considerable . this indicates that some less viable classes of cooperators can survive because of the support of more viable punishing strategies via an evolutionary selection mechanism which has a biased impact on the evolution of otherwise competing strategies . to demonstrate the underlying mechanism behind the above observations , we present a series of snapshots of strategy evolutions starting from different prepared initial states the comparative analysis is plotted in fig . [ fig3 ] , where all runs were obtained for @xmath42 and @xmath35 . in the first row , we demonstrate how the class of @xmath43 punishing strategy can prevail over defectors . initially , only a tiny portion of @xmath43 cooperators is launched in the sea of defectors [ the fraction of @xmath43 is @xmath46 , see panel ( a ) ] . still , @xmath43 cooperators can expand gradually and invade the whole available territory [ shown in panels from ( a2 ) to ( a4 ) ] . the second row , which was taken at the same parameter values , demonstrates clearly the vulnerability of the @xmath45 class against defectors . despite of the fact that they occupy the majority of the available room at the beginning , shown in panel ( b1 ) , still , they will be gradually crowded out by defector players . the final state , shown in panel ( b4 ) , highlights that such a rare punishment activity represented by @xmath45 class is ineffective against defectors at the applied synergy factor @xmath36 . the third row , where all previously mentioned strategies are present at the beginning , illustrates a completely different scenario . here we start from a balanced initial state where half of the lattice sites is occupied by @xmath45 and @xmath43 strategies , while the other half is filled by defectors . as panels ( c1 ) to ( c4 ) illustrate , defectors will gradually go extinct while `` weak '' @xmath45 cooperators survive and occupy almost half of the available territory in the final state . we note that there is a neutral drift between punishing strategies in the absence of defectors , which will result in a homogeneous state where the probability to arrive to one of the possible final destinations is proportional to the initial portion of a specific class at the time defectors die out @xcite . this evolutionary outcome indicates that although @xmath45 players are , as an isolated strategy , weak against defector players , they can nevertheless survive because of the assistance of the strong @xmath43 strategy even if the initial fraction of the later is modest . in the fourth row , however , when we arrange a similar setup but replaced weak @xmath45 players with also weak @xmath47 players , the final state will always be the full @xmath43 state . here , the presence of strong @xmath43 players does not yield a relevant support to @xmath47 players who therefore die out , and subsequently the system returns to the scenario illustrated in panels ( a1 ) to ( a4 ) . and @xmath35 . the first row shows the case when just a few @xmath43 cooperators are initially present among defectors . it can be observed that even under such unfavorable initial conditions the @xmath43 strategy can successfully outperform defectors . the second row feature a similar experiment with the @xmath45 strategy , which fails to survive among defectors even though the latter are initially in minority . the third row illustrates cooperation among strategies @xmath45 and @xmath43 , which together dominate the whole population even though @xmath45 alone would fail under the same conditions ( see second row ) . we note that a neutral drift starts when defectors die out , as explained in the main text . the fourth row demonstrates , however , that the cooperation among different punishing strategies illustrated in the third row is rather fragile . if initially the strategy @xmath45 is replaced by strategy @xmath47 , then the later simply die out and subsequently the whole evolution becomes identical to the one shown in the first row , where strategy @xmath43 alone outperforms all defectors . for clarity , here the employed system size is small with just @xmath48 players.,width=321 ] the key point , which explains the significantly different trajectories for mildly punishing strategies is based on the difference of invasion velocities between the competing strategies . to demonstrate the importance of invasion velocities , we monitor how the fraction of strategies evolves in time when we launch the system from a two - strategy state where both strategies form compact domains . following the previously applied approach illustrated in fig . [ fig3 ] , we compare the strategy invasions between @xmath49 , @xmath50 , and between @xmath51 strategies . the comparison of these different cases is plotted in fig . as expected , both @xmath47 and @xmath45 loose the lonely fight against defectors , while @xmath43 will eventually crowd out defectors . note that there is only a very slight increase during the early stages of the evolutionary process that can be observed for all cases , independently of the final outcome . this is because straight initial interfaces can provide a strong temporary phalanx for every punishing strategy . nevertheless , when this interface becomes irregular due to invasions the individual weakness of @xmath47 and @xmath45 strategies reveals itself . still , there is a significant difference between their trajectories . namely , strategy @xmath45 is able to resist for a comparatively long time , which gives strategy @xmath43 enough time to crowd out defectors . on the other hand , strategy @xmath47 is a too easy prey for defectors , which is why they die out faster than the strategy @xmath43 is able to eliminate all defectors . ultimately thus , strategy @xmath45 can benefit from cooperation with strategy @xmath43 , while strategy @xmath47 is unable to do the same . , @xmath45 and @xmath43 , against defectors in dependence on time . note that initially only one cooperative strategy and defectors are present , using the same initial conditions as illustrated in fig . [ fig3 ] . positive value of @xmath52 indicates the invasion of cooperator strategy while its negative value suggests invasion to the reversed direction . note that while both @xmath47 and @xmath45 strategies ultimately loose their battle , the latter is able to prevail significantly longer . this enables an effective help of strategy @xmath43 when they compete against defectors together , as illustrated in panels ( c1 ) to ( c4 ) in fig . [ fig3].,width=321 ] in the remainder of this work , we focus on the parameter region where cooperators are able to coexist with defectors without applying punishment . namely , if the synergy factor exceeds @xmath53 , then pure cooperators ( cooperators that do not punish ) can survive permanently alongside defectors due to network reciprocity @xcite . evidently , the presence of punishers can of course still elevate the overall cooperation level and defectors can be effectively crowded out from the population @xcite . here the main question is thus how the different punishing strategies will share the available space . the results are summarized in the left panel of fig . [ r4 ] , as obtained for the representative value of @xmath54 . it can be observed that , when all the different types of punishing strategies fight against defectors simultaneously , then cooperators can dominate the whole population above a threshold value @xmath55 . however , to evaluate these final outcomes adequately , we need to know the individual relations between each particular cooperative strategy and defectors on a strategy - versus - strategy basis . therefore , as for the previously presented low @xmath36 case in fig . [ fig2 ] , in the right panel of fig . [ r4 ] we also show the stationary fractions of different cooperators classes when they compete against defectors individually . results presented in panel ( b ) highlight that too large @xmath12 values could be detrimental for the @xmath45 , @xmath44 and the @xmath43 strategy . this is the so - called `` punish , but not too hard '' effect , where too large costs of sanctioning do more damage to those that execute punishment than the imposed fines do damage to the defectors @xcite . a direct comparison with the results presented in panel ( a ) demonstrates clearly that we can observe a similar cooperation among punishing strategies as we have reported before for the low @xmath36 case , in particular because all the mentioned mildly punishing strategies can survive even at a high @xmath12 value . on the other hand , a conceptually different mechanism can be observed in the small @xmath12 region , which is reminiscent of what one would actually expect from a selection process . more specifically , panel ( a ) of fig . [ r4 ] shows that at @xmath56 only strategy @xmath45 survives and coexists with @xmath6 while all the other punishing strategies die out . the latter players are those , who could survive individually with defectors but should die out because of the presence of a more effective ( @xmath45 ) strategy . interestingly , the mentioned selection mechanism can work most efficiently when the leading strategy is less efficient against defectors . right panel of fig . [ r4 ] shows that @xmath45 would be unable to crowd out strategy @xmath6 at these @xmath12 values , while a @xmath6-free state could be obtained at higher @xmath12 value . in the latter case , when @xmath45 is too powerful , then this strategy beats defectors too fast which allows other punishing strategies to survive : this is similar to what we have observed in the third row of fig . but when @xmath45 is less effective at smaller @xmath12 values then the presence of surviving @xmath6 players enables @xmath45 players to play out their superior efficiency if comparing to other punishing strategies . thus , depending on the key parameter values , most prominently the multiplication factor @xmath36 and the punishment fine @xmath12 , the different punishing strategies can either cooperate with each other or compete against each other in the spatial public goods game . we have introduced and studied multiple types of punishing strategies that sanction defectors with different probabilities . the fundamental question that we have addressed is whether there exists a selection mechanism which would result in an unambiguous victor when these strategies compete against defectors . we have shown that the answer to this question depends sensitively on the external conditions , in particular on the value of the multiplication parameter @xmath36 . if the public goods game is demanding due to a low value of @xmath36 , then the pure payoff - driven individual selection provides a helping hand to those punishing strategies that would be unable to survive in an individual competition against defectors . in particular , we have demonstrated that the failure or success of a specific punishing strategy could depend sensitively on the relation of invasion velocities between specific punishing strategies and the defectors . accordingly , if the loosing punishing strategy can delay the complete victory of defectors sufficiently long , then a more successful punishing strategy has a chance to wipe out defectors first . this is an example of the cooperation between different punishing strategies . on the other hand , in a less demanding environment , characterized by a higher multiplication factor , a different kind of relation can emerge . while the previously summarized cooperation between punishing strategies is still possible , there also exist parameter regions where competition is the dominant mode , and indeed there is always a single and unambiguous victor among the different classes of punishers . interestingly , we have shown that this happens when the fittest punishing strategy is not effective enough to beat defectors completely . instead , by carefully taming the defectors , they help to reveal the advantages of other punishing strategies . as we have shown , the key point here is again the relation between the invasion velocities . namely , a too intensive invasion will decimate defectors too fast and the advantage of specific punishing classes will remain forever hidden . therefore , in contrast to intuitive expectation , the social diversity of cooperators in terms of their relations with defectors could be the result of an effective selection mechanism . we hope that this research will contribute relevantly to our understanding of the emergence of diversity among competing strategies , as well to their role in determining the ultimate fate of the population . this work was supported by the fundamental research funds of the central universities of china , the hungarian national research fund ( grant k-101490 ) , the slovenian research agency ( grant p5 - 0027 ) , and by the deanship of scientific research , king abdulaziz university ( grant 76 - 130 - 35-hici ) .
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inspired by the fact that people have diverse propensities to punish wrongdoers , we study a spatial public goods game with defectors and different types of punishing cooperators . during the game , cooperators punish defectors with class - specific probabilities and subsequently share the associated costs of sanctioning .
we show that in the presence of different punishing cooperators the highest level of public cooperation is always attainable through a selection mechanism .
interestingly , the selection not necessarily favors the evolution of punishers who would be able to prevail on their own against the defectors , nor does it always hinder the evolution of punishers who would be unable to prevail on their own .
instead , the evolutionary success of punishing strategies depends sensitively on their invasion velocities , which in turn reveals fascinating examples of both competition and cooperation among them .
furthermore , we show that under favorable conditions , when punishment is not strictly necessary for the maintenance of public cooperation , the less aggressive , mild form of sanctioning is the sole victor of selection process .
our work reveals that natural strategy selection can not only promote , but sometimes also hinder competition among prosocial strategies .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the 1-band hubbard model on a two - dimensional square lattice has the hamiltonian @xmath2 here , @xmath3 ( @xmath4 ) creates ( annihilates ) an electron with spin @xmath5 in a wannier orbital centered at lattice site @xmath6 . the particle density at each site is given by @xmath7 . the first sum for the kinetic energy is restricted to include only the hopping matrix element @xmath8 between next - nearest neighbor sites @xmath9 . periodic boundary conditions are used throughout the following work . the second sum describes for @xmath10 an on - site coulomb repulsion between particles of opposite spin that share the same lattice site . in the present work we restrict ourselves to @xmath11 . the chemical potential @xmath12 in the third sum controls the occupation of the finite lattice in the finite - temperature grand canonical quantum - monte - carlo ( qmc ) simulation we preformed . at half - filling , particle - hole symmetry of the kinetic and @xmath13-term implies @xmath14 . the analytic continuation of the dynamic imaginary - times qmc data to the real frequency axis is performed with state - of - the - art maximum - entropy ( me ) techniques . for exhausting discussions concerning the qmc and me methods we refer the reader to @xcite . + the 1-band hubbard model exhibits several energy scales : in the repulsive case the high - energy scale @xmath13 is important in determining the insulating gap at half - filling , @xmath15 . an important low - energy scale is set by the exchange interaction @xmath16 : in second order perturbation theory two particles with different spins on neighboring lattice sites can exchange via a virtual double occupation . this process is the source for the strong antiferromagnetic ( af ) correlations found near and at half - filling . + with the exception of some known symmetry properties like invariance under global spin - rotation , whose generators form the @xmath17algebra , and invariance under @xmath18-transformation , i.e. , charge - conservation , as well as the particle - hole transformation , no rigorous results are known for the 1-band hubbard model in two - dimensions @xcite . the mermin - wagner theorem @xcite prevents a long - range ordered state in a two - dimensional system for finite temperatures , but it is commonly believed that the ground state of the spin-@xmath19 heisenberg antiferromagnet , i.e. the large-@xmath13 limit of the repulsive half - filled hubbard model @xcite , shows long - range nel order in two dimensions . nel order results also in the weak - coupling limit @xcite , where the gap @xmath20 is due to a spin - density - wave ( sdw ) instability related to perfect nesting . + it might seem that due to the mermin - wagner theorem the physics of the ordered phase is out of reach for our numerical technique , which is limited to finite temperatures . however , one may assume that the system ` effectively orders ' as soon as the spin - correlation length becomes comparable to the system size . due to the periodic boundary conditions the spin - correlation function @xmath21 is a periodic function of @xmath22 and if the value of this function at the maximum value @xmath23 ( with @xmath24 @xmath25 cluster size ) is still appreciable , we may expect that the system is ` effectively ordered ' . a rough measure would be the spin - correlation length @xmath26 , obtained by fitting @xmath27 to the form @xmath28 . since the infinite system has the nel temperature @xmath29 , the spin - correlation length diverges as @xmath30 and we may expect that in a _ system @xmath26 becomes comparable to the cluster size @xmath24 at a _ finite _ temperature which depends on the lattice size . below this temperature we then expect that the system resembles the ordered phase , although the spin - rotation symmetry persists even in this case . in that sense , the finite size of the system creates an artificial ` nel temperature ' which , however , depends on @xmath24 and @xmath31 etc and has no real counterpart in the infinite system . this has to be kept in mind when discussing the results . + we proceed by discussing some known approximations to the hubbard model . the ` classical ' approximation to the hubbard model is the so - called hubbard - i approximation@xcite . its essence is the splitting of the electron annihilation operator into the two ` eigenoperators ' of the interaction part : @xmath32 whence @xmath33 the physical content of the hubbard - i approximation , which neglects any spin - correlations , becomes clear by realizing @xcite that the equations of motion in this approximation are completely equivalent to an ` effective hamiltonian ' for double occupancy - like particles @xmath34 and hole - like particles @xmath35 : @xmath36 this hamiltonian contains terms which describe the pair creation of a hole and a double occupancy on nearest neighbors @xmath9 , terms which describe the propagation of these effective particles , and an additional energy of formation of @xmath13 for the double occupancy . the matrix elements for the propagation are reduced by a factor of @xmath19 , because in an uncorrelated spin - background there is a probability of @xmath19 for the spin on a nearest neighbor to have the proper direction to allow for the hopping of an electron . solving ( [ hubeff ] ) by fourier and bogoliubov transform : @xmath37 yields the standard dispersion relation , which consists of the upper and lower hubbard bands : @xmath38 here @xmath39 denotes the free tight - binding dispersion @xmath40 . using ( [ interactioneigenoperators ] ) we also obtain the correct spectral weights of the two hubbard bands : @xmath41 as noted above , the key assumption in the hubbard i approximation is the neglect of spin - correlations . therefore , we expect that this approximation will become inaccurate as soon as spin - correlations become sufficiently strong so as to appreciably influence the propagation of holes and double occupancies . this effect will be strongest in the nel ordered phase believed to be realized in the ground state . if we choose the ` spin - background ' , in which the holes and double occupancies propagate , to be the nel state , the double occupancy @xmath42 with spin @xmath43 can exist only on the @xmath44 sublattice and vice versa . similarly , a hole @xmath45 can be created only on the @xmath44 sublattice and vice versa . we , thus , expect that we have to modifiy the hubbard - i hamiltonian ( [ hubeff ] ) into : @xmath46 where @xmath47 ( @xmath48 ) denotes the @xmath44 ( @xmath43 ) sublattice and @xmath49 denotes the set of nearest neighbors of site @xmath6 . note that the @xmath50 and @xmath51 propagation terms drop out ( because of the @xmath47 and @xmath48 sublattices ) , and the matrix element for pair creation are @xmath8 rather than @xmath52 - this takes into account the fact that the spins on neighboring sites are antiparallel with probability @xmath53 rather than @xmath19 as was the case in the paramagnetic state . fourier and bogoliubov transformation of ( [ swham ] ) yields the dispersion @xmath54 and using @xmath55 ( where @xmath56 is within the af brillouin zone ) , we find the spectral weight of these bands : @xmath57 this is precisely what is obtained from the sdw mean - field treatment of the hubbard model by setting the staggered magnetization @xmath58 to a value of @xmath53 ( which is a good approximation in the limit of large @xmath13 ) . in general , the sdw mean - field approximation models the af nel state by assuming @xmath59 with af nesting vector @xmath60 and staggered magnetization @xmath61@xcite . this results in the two - band dispersion @xmath62 and the spectral weight @xmath63 the gap parameter is @xmath64 and the staggered magnetization @xmath58 is determined self - consistently from the following equation : @xmath65 the solution of this self - consistency equation yields the value @xmath66 at @xmath11 , the value used in the present work . if we set @xmath67 on the other hand , as would be apropriate for @xmath68 , we obviously recover the results from our hubbard - i - like hamiltonian ( [ swham ] ) . + in the two limiting cases of no spin - correlations and of perfect nel order we can thus treat the hubbard model in a quite analogous fashion and , as will be seen below , the hubbard - i results are indeed a good approximation to the actual spectral function in the limit of high temperature . the main problem then is how to describe the effect of spin - fluctuations and how to manage the crossover from the completely disordered to the nel ordered phase . below , we will adress this crossover by qmc simulations . + thereby we want to take advantage of the possibility to calculate the spectra of specifically designed ` diagnostic operators ' . the first one of these is the _ ` shadow ' operator _ : @xmath69 we note first of all , that the fourier transform of this operator , @xmath70 has precisely the same quantum numbers as the ordinary electron operator @xmath71 : momentum @xmath72 , total spin @xmath19 , @xmath73-spin @xmath5 , and identical point group symmetry at high symmetry momenta . it follows that the poles in its dynamical correlation functions @xmath74 originate from exactly the same final states @xmath75 as those of the photoemission spectrum . it can happen only accidentally that a given state @xmath76 has an exactly vanishing weight in one of the correlation functions , but not the others . in this case , however , any arbitrarily small perturbation will remove the accidental vanishing of the peak . the only thing that can and will be different in the spectra of the diagnostic operator , are the _ weights _ of the peaks , @xmath77 . in fact , comparison of equations ( [ crep ] ) and ( [ hubihafwght ] ) shows immediately that both for the hubbard i approximation and for the sdw - approximation use of the operator @xmath70 instead of @xmath71 exchanges the weights of the two hubbard / sdw bands . it follows that band portions which have a small spectral weight in the spectra of the ordinary electron operator will aquire a large spectral weight in the spectrum of @xmath70 and vice versa . therefore , this diagnostic operator is a useful tool to map out the ` shadowy ' parts of the spectra in the outer parts of the brillouin zone . an additional benefit is that since the me technique resolves peaks with large spectral weight more reliably than those with small weight , we can get more precise information about the dispersion of these bands with weak intensity . the second diagnostic operator that we will be using is the _ ` spin-@xmath19 string ' operator _ @xmath78 where the sum on the r.h.s . runs over the nearest neighbors @xmath79 of site @xmath6 . this operator is the clebsch - gordan contraction of the adjoint rank-@xmath53 spinor @xmath80 and the spin-@xmath53 vector operator @xmath81 into yet another adjoint rank-@xmath53 spinor . it describes the annihilation of a ` dressed ' electron , i.e. , an electron with a spin - excitation on a nearest neighbor . again , since the fourier transforms of the diagnostic operator , @xmath82 , agrees with the electron annihilation operator @xmath71 in all conceivable quantum numbers ( momentum , total spin , @xmath73-component of the spin , point group operations in the case of high symmetry momenta ) , it obeys precisely the same selection rules , whence it is just as good as the @xmath83-operator itself to map out the band structure . + one further point , which is important from the technical point of view , is the following : under the particle - hole transformation @xmath84 we have @xmath85 and @xmath86 . this implies that at half - filling the spectra of the shadow and spin-@xmath19 string operator obey the same particle - hole - symmetry as those of the ordinary electron operator , i.e. @xmath87 in our maxent program particle - hole - symmetry is not implemented as an additional constraint , in other words : the maxent procedure does not ` know ' about this additional symmetry . the degree to which ( [ phsym ] ) is obeyed in the final spectra thus gives a good check for the accuracy of the reconstructed spectra . this is of particular importance in the case of the spin-@xmath19 string operator because the wick - contraction of this operator on any given time slice produces a total of @xmath88 products of noninteracting green s functions . the computation is , therefore , much more prone to inaccuracies so that an additional check is desirable . + the present work is organized as follows : first , we compare the temperature dependent dynamic single - particle properties of the hubbard model with the predicitions of the mean - field sdw and hubbard - i approximations . in addition , we consider the temperature dependent two - particle excitations . then , we will use our first diagnostic operator , the shadow operator @xmath70 , to show the existence of a total of four bands in the photoemission spectrum and to shed light onto the temperature dependent crossover from the sdw to the hubbard - i regime . then we will investigate the 4-band structure in more detail : after a phenomenological fit , which is able to produce a total of four bands , we will consider the string picture which naturally leads to our second diagnostic operator , the spin-@xmath19 string operator @xmath82 . this operator will be used to ultimately reveal the underlying mechanisms behind the generation of the 4-band structure , namely the dressing of the photoholes by clouds of af spin - excitations . to resolve the ` af mirror image ' of the narrow quasiparticle spectral weight features between @xmath89 and @xmath90 around momentum @xmath91 we introduce also a spin-@xmath92 string operator . finally , we will concentrate on the doped regime , thereby showing the violation of the luttinger theorem near half - filling , and discuss the hole concentration range in which these dressing effects dominante the low - energy physics . we start in figure [ fig1 ] with the discussion of the angle - resolved single - particle spectral function @xmath93 for @xmath11 and various temperatures in the range from @xmath94 to @xmath95 . in the left column of the figure the spectral functions are shown as ` grey - scale ' plots versus momentum @xmath96 and energy @xmath97 with dark ( light ) areas corresponding to large ( small ) spectral weight . the same spectra are shown also in the right column , but now as line - plots at each momentum @xmath96 . the qmc data in the figure are compared to the renormalized results of the hubbard - i approximation at high and medium temperatures and with the renormalized results of the mean - field sdw approximation at the lowest temperature , @xmath95 . ` renormalized ' means here that we have readjusted the parameters @xmath13 and @xmath8 in ( [ hubihafdisp ] ) and ( [ sdwdisp])so as to obtain an optimal fit to the ` bands ' of high spectral weight in the spectra . these approximate dispersions are plotted as solid lines in the left column , while their spectra are shown in the right column as line - plots at each momentum @xmath96 . thereby we have assumed a lorentzian lineshape with a suitably chosen temperature dependent width ( the hubbard - i approximation does not provide any information about linewidths ) . + starting at the highest calculated temperature , @xmath94 , we find that the hubbard - i approximation with practically unrenormalized parameters ( @xmath98 and @xmath99 ) fits the qmc spectral functions almost perfectly regarding both the general dispersion and the distribution of spectral weight ( quadratic deviation per degree of freedom @xmath100 ) . this is not surprising since the hubbard - i approximation is derived for the paramagnetic state thereby neglecting all effects of spin - correlations , i.e. , the hubbard - i approximation essentially describes the interplay between itinerant electrons and strong on - site repulsion . this should be a reasonable assumption at this high temperature since all relevant spin - degrees of freedom are thermally excited . + lowering the temperature to @xmath101 and further to @xmath102 , the data show that the hubbard - i approximation increasingly fails to reproduce the entire spectrum and is only able to fit the peaks with maximal spectral weight reasonably well . even then , in order to achieve these fits , one already has to renormalize the free parameters strongly . the values we found are @xmath103 and @xmath104 with @xmath105 at @xmath101 and @xmath106 and @xmath107 with @xmath108 at @xmath102 . the peaks that are missed by the hubbard - i approximation are the states that form the first ionization / affinity states around momentum @xmath109 on the photoemission / inverse photoemission side and two rather dispersionless bands at higher energies of @xmath110 . the former states were previously resolved by preuss and co - workers @xcite . alltogether one can distinguish a total of four ` bands ' in the single particle spectral density . as will be seen below , the temperature / doping regime where this 4-band structure is seen coincides with the regime where a collective low - energy mode with momentum @xmath111 in the spin - response exists . inspite of this , however , we stress that the 4-band structure can not be explained by a backfolding of the band structure due to ordering effects since the spin - correlation length is @xmath112 lattice spacings at temperatures @xmath113 . + for the lowest temperature , @xmath95 , the qmc data are compared with the results from the af sdw approximation . as in the case of the hubbard - i approximation at medium temperatures , the lowermost spectra of figure [ fig1 ] show that the sdw approximation is only able to fit the peaks with large spectral weight . again one has to renormalize the free parameters heavily to values of @xmath114 and @xmath115 with @xmath116 . moreover , as was the case for the hubbard - i approximation at higher temperatures , the sdw approximation neither explains the states that form the first ionization / affinity states around momentum @xmath109 on the photoemission / inverse photoemission side , nor the two dispersionless bands at higher excitation energies which can be seen rather clearly in the spectra . + all in all , the overall distribution of spectral weight is roughly reproduced by the hubbard - i and sdw approximations as long as one forgets about the 4-band structure . in fact it is well known that the _ integrated _ photoemission or inverse photoemission weight ( that means the electron momentum distribution ) at each @xmath96point is reproduced quite well by the hubbard - i approximation and the related @xmath117-pole approximation@xcite . + as already mentioned , the emerging of the 4-band structure in the photoemission somewhere in between @xmath101 and @xmath102 is closely related to a change in the spin - response : to illustrate this we consider figure [ fig2 ] , which shows the spin - correlation function , @xmath118 ( left column ) , and the charge - correlation function , @xmath119 ( right column ) , for different temperatures . whereas the spin - response is entirely incoherent at @xmath101 , with decreasing temperature it can be fitted increasingly well by the spin - wave dispersion @xmath120 this result is known from previous calculations @xcite , which demonstrated that the two - particle correlation functions like the spin - response can be described within the sdw approximation even for large values of the interaction @xmath13 . the energy scale @xmath121 directly manifests itself in the spin - response since the spin - wave dispersion takes the value of @xmath122 at momentum @xmath90 . the fit parameters are @xmath123 with @xmath124 at @xmath102 and @xmath125 with @xmath126 at @xmath127 . the latter is already quite close to the strong coupling estimate @xmath128 . furthermore , the figure shows that with decreasing temperature the spin - response concentrates its weight more and more at the af momentum @xmath60 ( as it is the case in af spin - wave theory ) and at a characteristic energy @xmath129 . the latter decreases with decreasing temperature , i.e. , the spin - response comes closer and closer to the predictions of af spin - wave theory ( [ spinwave ] ) . the spin - correlation length @xmath130 can be derived from a real - space fit of the qmc equal - imaginary - times spin - correlation function @xmath27 to the form @xmath131 thereby incorporating the effects of the periodic boundary conditions . while this is the best one can do on a finite lattice with periodic boundary conditions , the fit will only lead to roughly correct values due to the relative small system size of @xmath132 . the values obtained for the spin - correlation length then are @xmath133 , @xmath134 , @xmath135 , @xmath136 and @xmath137 for @xmath101 , @xmath102 , @xmath138 , @xmath139 and @xmath95 , respectively . we actually believe that the spin - correlation length reaches the system size already at a temperature of @xmath140 , because at this temperature the fit results in values between @xmath141 and @xmath142 ( with the exponent @xmath143 set to zero or not ) but always with error bars of roughly the system size . + the charge - response @xmath119 , on the other hand , is rather broad in both momentum @xmath96 and energy @xmath144 for all temperatures studied . furthermore , the charge - response is gapped for temperatures below @xmath145 and , therefore , can certainly not be responsible for any low - energy features of the single particle spectrum . + it is then quite obvious that at roughly the same temperature where the two narrow dispersive quasiparticle - like bands ( that can not be interpreted within the framework of the hubbard - i or sdw approximations ) appear in the single particle spectrum , the spin - response develops a sharp collective low - energy mode . we conclude that the underlying mechanism behind the occurrence of the 4-band structure consists in dynamical magnetic correlation effects , which are beyond the scope of the hubbard - i and sdw approximations . + in the following , we want to explore the single particle spectrum by means of our _ diagnostic operators_. the first of them is the shadow operator @xmath146 of equation ( [ shadoweqt ] ) which will be used to transfer spectral weight from the inner parts of the brillouin zone to the outer ones in case of normal photoemission @xmath147 . as already discussed above , this also improves the resolution of the me method in this region , since its resolution strongly depends on the spectral weight at a certain position . nevertheless the spectrum of the shadow operator has to exhibit exactly the same peak positions as the normal photoemission spectrum . + figure [ fig3 ] shows the angle - resolved spectral function of the shadow operator for moderate and low temperatures . as expected , the shadow operator has its main spectral weight near @xmath91 on the photoemission side and near @xmath89 on the inverse photoemission side . furthermore , it s spectrum supports the existence of a total of four bands , because it resolves a group of peaks forming dispersionless bands at energies of @xmath148 , a region where the normal photoemission spectrum exhibits only some weak and smeared - out spectral weight . these two dispersionless bands at energies of @xmath110 are inconsistent with the dispersions of the hubbard - i and sdw approximations of figure [ fig1 ] . we will further address this topic later in this work . + next , we turn in more detail to the temperature dependence of the photoemission spectrum . figure [ fig4 ] shows some close - ups of the normal photoemission spectrum and of the spectrum of the shadow operator at momentum @xmath91 and different temperatures . for the normal photoemission operator these close - ups show a peak at @xmath149 , which would be consistent with hubbard - i ( see figure [ fig2 ] ) . in the spectrum of the shadow operator this feature is visible as a single resolved peak only at the highest temperature , @xmath101 whereas for temperatures down to @xmath139 there is only some diffuse weight at this position . in the ordinary photoemisson spectrum the peak looses spectral weight with decreasing temperature . it disappears completely at @xmath150 where the spin - correlation length @xmath130 reaches the system size ( see above ) . thus , the temperature @xmath140 where we lose the hubbard - i - like peak at @xmath149 and @xmath91 coincides quite accurately with the temperature where ` effective ' long - range order sets in . moreover , we find that as the normal photoemission spectrum loses the peak at @xmath149 and @xmath91 , the spectrum of the shadow operator gains weight at @xmath151 . thus , we expect that both features are closely related to the temperature development of the spin - correlation length @xmath130 . we note , however , that the crossover in the shape of the dispersion from hubbard - i - like to sdw - like occurs in a quite unexpected way : the topmost band at @xmath111 does _ not _ deform into the sdw form in any continuous way , but simply ` fades away ' and eventually vanishes at the transition . + a further surprising result is the following : at @xmath152 neither the ordinary electron operator nor the shadow operator pick up the ` af umklapp band ' corresponding to the narrow dispersive band seen for example at @xmath153 for @xmath154 , i.e. there is no corresponding band at @xmath153 and @xmath155 . note that in the framework if the sdw - approximation the shadow operator _ must _ reproduce this umklapp - band at @xmath111 with the _ same weight _ as the original band at @xmath156 in the ordinary photoemission spectrum - see the discussion in the first section . that this is not the case shows that even at this lowest temperature , a simple sdw - like description of the band structure is invalid , in that the band structure can not be understood by simple backfolding of the spectrum obtained without broken symmetry . as we will see in the following the af sdw state provides only the ` background ' for the dressing of the photoholes with af spin - excitations which dynamically generate a total of four bands . we return to the discrepancy between the 2-band dispersions of the hubbard - i and sdw approximations and the 4-band structure actually observed for example in the spectrum of the shadow operator of figure [ fig3 ] . + in order to generate a ` 4-band structure ' out of the two bands of the hubbard - i and sdw approximations we try as a phenomenological ansatz to mix the dispersions of the hubbard - i / sdw approximation with two dispersionless bands at energies of @xmath157 . in other words , for both the photoemission and inverse photoemission spectrum we diagonalize an ` effective ' @xmath158 hamilton matrix : @xmath159 and plot in figure [ fig5 ] the four bands obtained in this way on top of the spectral density obtained from qmc at @xmath160 and @xmath95 . for comparison the figure also shows the original ( i.e. unhybridized ) hubbard - i bands plus the two phenomenological dispersionless bands at energies of @xmath161 . the figure shows that the overall agreement between the qmc peak positions and the four bands generated by diagonalizing @xmath162 of equation ( [ mixeqt1 ] ) is surprisingly good , particularly so in view of the fact that for both , hubbard - i and sdw approximation , only _ unrenormalized _ parameters were used . in particular , the _ self - consistently _ determined value for the sdw gap @xmath20 of @xmath163 was used at @xmath95 . the only ` external ' parameter in this figure is the mixing matrix element @xmath164 , which was set to a value of @xmath165 . + thus , we find in contrast to previous works @xcite , that the introduction of the dispersionless bands reproduces the sinlge - particle gap and the width of the quasiparticle band correctly without any renormalization of parameters . rather , the narrowing of the quasiparticle band and the reduction of the hubbard gap as compared to the unrenormalized parameters is brought about by introduction of the dispersionless bands . this naturally raises the question as to their physical origin . in the present paper we restrict ourselves to a more phenomenological and ` numerics based ' approach . a complementary and more mathematical discussion is given in ref . @xcite , where an equation of motion approach similar to hubbard s original work is pursued . + we consider the commutator of the creation operator for hole - like particles , @xmath166 , which annihilates a particle only on a singly occupied site , with the kinetic energy of the hubbard model and find@xcite : @xmath167 keeping only the first term in the square brackets of equation ( [ commute ] ) reproduces the hubbard - i approximation @xcite . the second term in the square brackets describes the dressing of the created hole by a spin - excitation and is closely related to the spin-@xmath19 string operator of equation ( [ stringoperator ] ) . the third term describes in an analogous fashion the coupling of the hole to a density fluctuations , whereas the fourth term describes the coupling to the @xmath168-excitation@xcite . the two latter types of excitation are not important for a large positive @xmath13 near half - filling , @xmath15 , and will be neglected . therefore , the operator @xmath169 is in this case the most important correction over the hubbard - i approximation . as already stated , it describes a hole dressed by a spin - excitation : this operator not only creates a hole on site @xmath79 but dresses this hole with a spin - excitation on a neighboring site , which is exactly the idea behind the spin - bag@xcite or spin - polaron@xcite pictures known in the literature . + splitting this operator into eigenoperators of @xmath170 : @xmath171 we find @xmath172 = \frac{u}{2 } \hat{d}_{i , j,\sigma}$ ] and @xmath173 = - \frac{u}{2 } \hat{c}_{i , j,\sigma}$ ] . assuming moreover that the mobility of these composite excitations is determined by the ` heavy ' spin - excitation , it seems quite resonable to assume that these ` particles ' are the source of the ( more or less ) dispersionless bands at @xmath174 . + finally , the commutation relation ( [ commute ] ) shows that the mixing matrix element between the @xmath175 and the new composite particles should be @xmath176 . based on these rough considerations we might thus expect that the two string-@xmath53 ` effective particles ' ( [ eigy ] ) are excellent candidates for explaining the two dispersionless bands at @xmath177 required to upgrade the hubbard - i or sdw approximation so as to match the qmc data . however , so far the above considerations are pure speculation and in the following we will turn to qmc - results to back up this hypothesis by numerical evidence . + before doing so , however , we want to illustrate the action of the ` string - operator ' in two extreme cases : an ideal nel state and a resonating valence - bond ( rvb ) state , i.e. a compact singlet covering of the plane@xcite ( see figure [ fig6 ] ) . in the nel state , a hole created initially on site @xmath6 can travel one place to a neighboring site @xmath79 thereby leaving behind a misaligned spin on the original site @xmath6 . exactly this process is described by the second term , @xmath178 : it creates a hole of opposite spin on a neighboring site @xmath79 and flips the spin on the original site @xmath6 . therefore , this process corresponds to the creation of a string of length @xmath53 . in fact one might think about more sophisticated diagnostic operators incorporating the effects of longer - ranged strings @xcite . indeed , dagotto and schrieffer@xcite and eder and ohta@xcite already measured the angle - resolved spectrum of a diagnostic operator containing strings with up to three lattice sites range by means of exact diagonalizations of the @xmath179 model . as already mentioned in the introduction , in the qmc method each observable has to be expressed in terms of free single - particle greens functions on each time slice by the application of wicks theorem @xcite . this results already in a quite large expression for the spin-@xmath19 string operator of equation ( [ stringoperator ] ) containing approximately @xmath180 contributions . the implementation of even longer - ranged string operators therefore was not possible . + returning to the spin-@xmath19 string operator of equation ( [ stringoperator ] ) we note that the first term , @xmath181 will always annihilate a nel state . this reflects the simple fact that spin - rotation symmetry is broken in the nel state . this is not the case , however , in the fully rotationally invariant rvb state : again , creating a hole on site @xmath6 and allowing it to hop to site @xmath79 will produce a spin - excitation . however , in the case of the rvb state , it produces the superposition of two states : in one case the dotted ellipse stands for the @xmath182 component of the triplet , and this state would again be created by the term @xmath178 . there is , however , also a second state where the dotted ellipse corresponds to the superposition of a singlet and the @xmath183 component of the triplet . this second state then would be created by the term @xmath181 , whereby the relative sign of the two terms in the string-1 operator makes sure that the two configurations are always produced with the proper phase . in both extreme cases , nel state and ` singlet soup ' , the string-@xmath53 operator thus creates a hole dressed with the proper spin - excitation : this scan be a spin wave ( i.e. a single inverted spin ) in the case of a nel state , or a singlet - triplet excitation in the case of an rvb state . + as a technical remark we still note that the excessive numerical effort which would have been necessary to compute spectra for the @xmath184 and @xmath185 ( which are products of @xmath186 fermion operators ) has made it impossible to compute the spectra of these operators - instead we have been using the ( fourier transform of ) the operator @xmath187 , defined in ( [ stringoperator ] ) . concerning the difference between this diagnostic operator and the operator @xmath188 obtained by commuting @xmath189 with the kinetic energy ( see the second term on the r.h.s . of equation ( [ commute ] ) ) , we note that their fourier transforms differ only by phase factors of the form @xmath190 . both operators are indeed identical at momentum @xmath89 and differ at momentum @xmath91 only by a factor of @xmath191 since @xmath192 and @xmath193 are next - nearest neighbor lattice sites . at all other momenta both operators have exactly the same peaks but will differ somewhat in their spectral weights . + after these remarks we discuss the angle - resolved spectral function of the spin-@xmath19 string operator @xmath194 , shown in figure [ fig7 ] . the spectrum of the spin-@xmath19 string operator indeed picks up those band portions which , according to the rough hybridization scenario in figure [ fig5 ] , should have strong ` flat - band character ' , i.e. the part of the narrow low energy band between @xmath156 and @xmath195 and between @xmath156 and @xmath196 at energies @xmath153 . we note that these are precisely the positions where the two quasiparticle - like dispersive narrow bands occurred in the normal photoemission spectrum at @xmath102 . in addition , the band portion at @xmath197 for momentum @xmath111 is also enhanced in the string-1 spectrum . + this , however , still leaves an important part of the band structure unexplained . namely , the ` af umklapp band ' of the narrow quasiparticle band dispersing upward between @xmath156 and @xmath195 at @xmath153 still is not seen in any of the spectra , not even at the lowest temperature studied . on the other hand , in a state with true antiferromagnetically broken symmetry we know that this mirror image must exist due to the backfolding of the brillouin zone . to finally resolve this part , we now introduce the last diagnostic operator of this work , which we call the spin-@xmath92 string operator : @xmath198 this describes again a composite object of a hole and a spin - excitation , but this time the two constituents are coupled to the total spin of @xmath92 . we stress that this operator will detect states which can never be seen in an actual angle - resolved photoemission spectrum on a singlet ground state , because this is forbidden by the angular - momentum selection rule . the angle - resolved spectral function @xmath199 of the spin-@xmath92 string operator is plotted in figure [ fig8 ] again for @xmath102 ( top ) and @xmath95 ( bottom ) . it is then immediately obvious that it is this operator which resolves the ` missing piece ' of the af dispersion , i.e. the ` af mirror image ' of the narrow quasiparticle band . it should also be noted that the spectrum of @xmath200 is remarkably independent of temperature , i.e. the states belonging to this ` spin-@xmath92 band ' persist irrespectively of whether there is long - range order or not . + combining the information obtained so far , suggests the following scenario for the crossover between the paramagnetic band structure at high temperature and the af band structure at low temperature : in the paramagnetic state at high temperatures ( such as @xmath102 ) , the spin is a good quantum number and the spin-@xmath92 ` band ' does exist but can not mix with any spin-@xmath19 band due to spin - conservation . the band of spin-@xmath92 quasiparticles thus plays no role whatsoever in the actual photoemission spectrum , which is presumably the reason why the outer part of the spectrum is so remarkably ` invisible ' in actual arpes experiments , leading to the idea of a ` remnant fermi surface ' in the insulator@xcite . reaching this state by photoemission would only be possible if the photohole is created in a thermally admixed state of at least spin @xmath53 . in an infinite system this situation changes discontinuously at the transition to the true broken symmetry state : there the total spin ceases to be a good quantum number , and the spin-@xmath19 band in the interior of the af zone and the spin-@xmath92 band in the exterior now suddenly can mix with each other , thus leading to the familiar sdw dispersion . we note that another way to generate a coupling between these two bands would be application of a magnetic field - this also would break spin - rotation invariance and hence enable the hybridization of a spin-@xmath19 and a spin-@xmath92 band . based on our results we thus believe that a magnetic field could enhance the spectral weight of the ` shadow part ' of the band structure as seen in arpes . summarizing our results so far , we may say that the hubbard - i approximation , slightly improved by the introduction of new quasiparticles corresponding to dressed holes provides a very good description of the spectral function for the case of half - filling , @xmath201 . in this section we want to proceed to the doped case @xmath202 , which is of prime interest for cuprate superconductors . here , an essential drawback of the qmc procedure is that reliable qmc simulations for lower temperatures are much more difficult or even impossible , since the absence of particle - hole symmetry away from half - filling introduces the notorious minus - sign problem into the algorithm . truely low temperatures like @xmath203 , which in principle correspond to the physical temperature range , are therefore out of reach . on the other hand in the study of the half - filled case we have seen that a major change takes place as the spin correlation length reaches the system size , whence ` effective long range order ' sets in . in the doped case the spin correlation length is expected to be short at any temperature , whence we may expect that the change of @xmath93 from high to small temperature is more smooth than at half - filling . in that sense , even @xmath93 data for the relatively high temperature @xmath102 are interesting to study . moreover , we can at least try to elucidate trends with decreasing temperature and thus construct a reasonably plausible scenario . + at half - filling , we have seen that the ` approximation of choice ' for the paramagnetic case was the hubbard - i approximation . this naturally poses the question as to how relevant the half - filled case is for the description of the doped case , i.e. how much of the hubbard - i physics remains valid for finite doping . at half - filling the two ` effective particles ' @xmath204 and @xmath205 , form the two separate hubbard bands . the effect of doping would now consist in the chemical potential cutting progressively into the top of the lower hubbard band , in much the same fashion as in a doped band insulator . on the other hand , for finite @xmath31 the spectral weight along this band deviates from the free - particle value of @xmath53 per momentum and spin so that the fermi surface volume ( obtained from the requirement that the integrated spectral weight up to the fermi energy be equal to the total number of electrons ) is not in any ` simple ' relationship to the number of electrons - the luttinger theorem must be violated . this is the major reason why the hubbard - i approximation has been dismissed by many authors as being unphysical . + we now wish to address the question as to what really happens if a paramagnetic ( i.e. not magnetically ordered ) insulator is doped away from half - filling , by qmc simulation . we therefore choose @xmath206 and @xmath11 . + figure [ fig9 ] then shows the development of @xmath207 with doping . it is quite obvious from this figure that initially the @xmath117 bands seen at half - filling in the photoemission spectrum ( i.e. @xmath208 ) persist with an essentially unchanged dispersion . the chemical potential gradually cuts deeper and deeper into the topmost band , forming a hole - like fermi surface centered on @xmath111 , the top of the lower hubbard band . the only deviation from a rather simple rigid - band behavior is an additional transfer of spectral weight : the part of the topmost band near @xmath111 gains in spectral weight , whereas the band with higher binding energy looses weight . in addition , there is a transfer of weight from the upper hubbard band to the inverse photoemission part below the hubbard gap . this effect is actually quite well understood@xcite . the band structure above the hubbard gap becomes more diffuse upon hole doping in that the rather clear two - band structure visible near @xmath111 at half - filling rapidly gives way to one broad ` hump ' of weight . apart from the spectral weight transfer , however , the band structure on the photoemission side is almost unaffected by the hole doping - the _ dispersion _ of the quasiparticle band becomes somewhat wider but does not change appreciably . in that sense we see at least qualitatively the behavior predicted by the hubbard i approximation . + next , we focus on the fermi surface volume . some care is necessary here : first , we can not actually be sure that at the high temperature we are using there is still a well - defined fermi surface . second , the criterion we will be using is the crossing of the quasiparticle band through the chemical potential . it has to be kept in mind that this may be quite misleading , because band portions with tiny spectral weight are ignored in this approach ( see for example ref . @xcite for a discussion ) . when thinking of a fermi surface as the constant energy contour of the chemical potential , we have to keep in mind that portions with low spectral weight may be overlooked . on the other hand the fact that a peak with appreciable weight crosses from photoemission to inverse photoemission at a certain momentum is independent of whether we call this a ` fermi surface ' in the usual sense , and should be reproduced by any theory which claims to describe the system . it therefore has to be kept in mind that in the following we are basically studying a ` spectral weight fermi surface ' , i.e. the locus in @xmath96 space where an apparent quasiparticle band with high spectral weight crosses the chemical potential . with these _ caveats _ in mind , figures [ fig10 ] and [ fig11 ] show the low - energy peak structure of @xmath207 for all allowed momenta of the @xmath209 cluster in the irreducible wedge of the brillouin zone , and for different hole concentrations . in all of these spectra there is a pronounced peak , whose position shows a smooth dispersion with momentum . around @xmath111 the peak is above @xmath12 , whereas in the center of the brillouin zone it is below . the locus in @xmath56-space where the peak crosses @xmath12 forms a closed curve around @xmath111 and it is obvious from the figure that the ` hole pocket ' around @xmath111 increases very rapidly with @xmath210 . to estimate the fermi surface volume @xmath211 we assign a weight @xmath212 of @xmath53 to momenta @xmath96 where the peak is below @xmath12 , @xmath213 if the peak is right at @xmath12 and @xmath214 if the peak is above @xmath12 . our assignments of these weights are given in figures [ fig10 ] and [ fig11 ] . the fractional fermi surface volume then is @xmath215 , where @xmath216 is the number of momenta in the @xmath209 cluster . of course , the assignment of the @xmath212 involves a certain degree of arbitrariness . it can be seen from figures [ fig10 ] and [ fig11 ] , however , that our @xmath212 would in any way tend to underestimate the fermi surface volume , so that the obtained @xmath211 data points rather have the character of a lower bound to the true @xmath211 . even if we take into account some small variations of @xmath211 due to different assignments of the weight factors , however , the resulting @xmath211 versus @xmath210 curve never can be made consistent with the luttinger volume , see figure [ fig12 ] . the deviation from the luttinger volume is quite pronounced at low doping . @xmath211 approaches the luttinger volume for dopings @xmath217% , but due to our somewhat crude way of determining @xmath211 we can not really decide when precisely the luttinger theorem is obeyed . the hubbard i approximation approaches the luttinger volume for hole concentrations of @xmath218% , i.e. the steepness of the drop of @xmath211 is not reproduced quantitatively . the latter is somewhat improved in the so - called @xmath117-pole approximation@xcite . for example the fermi surface given by beenen and edwards@xcite for @xmath219 obviously is very consistent with the spectrum in figure [ fig11 ] for @xmath220 . + we return to figure [ fig9 ] and discuss the entire width of the spectra , in particular the question of the fate of the 4-band structure in the doped system . for @xmath221 the different features that are seen at @xmath15 are still rather clearly visible , but for @xmath222 the low energy quasiparticle band at @xmath223 starts to disappear , and at @xmath224 the dominant ` band ' in the spectrum between @xmath225 and @xmath226 can be fitted by a sligtly renormalized free - electron band . as we have seen above , the luttinger theorem also is valid in this case . this suggests to classify the doping as ` underdoped ' for @xmath227 , where the luttinger theorem is invalid and the @xmath228-band structure known from half - filling persists , and ` overdoped ' where the luttinger theorem is valid and a renormalized free - electron band can be seen in the spectral function . following the convention for cuprate superconductors , we call the doping where the crossover between the two regimes occurs the ` optimal ' doping . + next , the four plots of figure [ fig13 ] show the spectra at selected @xmath56-points . the system size is only @xmath229 in this case because this allows for smaller error bars . closer inspection , especially , of the peaks at momentum @xmath89 on the photoemission side ( plot ( a ) ) and at momentum @xmath91 on the inverse photoemission side ( plot ( d ) ) confirms , that with increasing hole concentration we are losing parts of the 4-band structure seen at half - filling . + to check the physics of the band structur in more detail , we again employ our diagnostic operators . figure [ fig14 ] shows the angle - resolved spectral functions @xmath199 of the spin-@xmath19 and spin-@xmath92 string operators , @xmath187 and @xmath200 . as was the case at half - filling , the spectrum of the spin-@xmath19 string operator highlights exactly those peaks that we associate with the dipsersionless ` dressed hole ' bands in figure [ fig5 ] . the spectrum of the spin-@xmath92 string operator on the other hand has its peaks with maximal spectral weight around momentum @xmath91 , indicating that also in the doped case there is an ` antiferromagnetic mirror image ' of the quasiparticle band ( which , however , consists of spin-@xmath92 states ) . again , coupling of photoholes to thermally excited spin excitations may make these states visible in arpes spectra , thus explaining the ` shadow bands ' seen in photoemission experiments by aebi _ et al._@xcite . similarly as for half - filling one might speculate that a magnetic field , which would break spin symmetry and thus allow for a coupling of ` bands ' with different total spin , would enhance the spectral weight of these shadow bands . + all in all we have ssen that the ` band structure ' ( @xmath228-band structure , dispersion of regions of large spectral weight , ` character ' of the bands as measured by the diagnostic operators ) stays pretty much unchanged as long as we are in the underdoped regime . at half - filling the @xmath228-band structure is closely related to the sharp low - energy mode in the dynamical spin correlation function , which naturally suggests to study the spin reponse also as a function of doping . figure [ fig15 ] shows the spin - correlation function , @xmath230 ( left column ) , and the charge - correlation function , @xmath119 ( right column ) , for @xmath102 and densities @xmath221 ( underdoped ) , @xmath231 ( nearly optimally doped ) and @xmath224 ( overdoped ) . the spin - response is sharply confined in both momentum @xmath91 and energy @xmath232 only in the underdoped region i.e. the regime where we also observe the features associated with spin excitations in the single - particle spectra . as was the case at half - filling for temperatures below @xmath233 , the spin - response can be fitted by the af spin - wave dispersion ( [ spinwave ] ) in the underdoped regime . on the other hand , as soon as the system enters the overdoped regime the spin - response is no longer sharply peaked at momentum @xmath91 and energy @xmath232 : it broadens in momentum and spreads in energy by an order of magnitude with the scale changing from @xmath16 to @xmath234 accompanied by a similar change in the bandwidth of the single particle excitations . this result is already well known from previous qmc calculations @xcite and consistent with similar behaviour in the t - j model@xcite . the charge - response , @xmath119 , is always broad in both momentum @xmath96 and energy @xmath144 for all densities studied . it merely decreases its width from @xmath235 at @xmath221 to @xmath236 at @xmath224 . + although the minus - sign problem of the qmc algorithm prevents reliable simulations of large systems at low temperatures in the doped regime , we nevertheless studied the temperature evolution of the angle - resolved spectral function @xmath93 at density @xmath237 . this was possible due to the relative small system size of @xmath229 , which alleviates the minus - sign problem as compared to @xmath132 at the @xmath138 . figure [ fig16 ] shows the results from this analysis : the uppermost plot ( a ) compares the angle - resolved spectral functions @xmath93 at density @xmath238 for @xmath239 , @xmath102 and @xmath240 . we stress that the simulation at @xmath240 suffers from minus - sign problems with a drastically reduced resolution . in the center plot ( b ) , the quasiparticle peak weights around momentum @xmath91 of the @xmath238 simulation are compared with the quasiparticle peak weight at momentum @xmath91 of a half - filled , @xmath241 , simulation for different temperatures . at half - filling , the hubbard - i - like quasiparticle peak @xmath91 and @xmath149 decreases in spectral weight with decreasing @xmath242 and disappears as the spin - correlation length ( which increases with decreasing temperature ) reaches the lattice size ( at @xmath140 ) . in the underdoped case for density @xmath238 the weights of the corresponding peaks around momentum @xmath91 located also decrease with decreasing @xmath242 . closer inspection of this peak ( see the inset of the center plot ( b ) ) reveals , that this peak even raises slightly in binding energy with decreasing temperature , very similar to the peak in the half - filled case . in a real photoemission experiment this peak would have dropped below the typical resolution of roughly @xmath243 spectral weight @xcite at a temperature of @xmath244 . the spin - correlation length ( again derived by a fit of the equal - imaginary - times spin - correlation function to a form @xmath245 ) also shows similar behavior in the underdoped and half - filled cases : the values for the spin - correlation length are @xmath246 , @xmath247 and @xmath134 in the underdoped case and @xmath248 , @xmath134 and @xmath249 at half - filling for @xmath239 , @xmath102 and @xmath240 , respectively . the spin - susceptibility ( shown in plot ( c ) of figure [ fig16 ] ) also behaves very similar , but with changed magnitudes in the underdoped and in the half - filled cases . these data suggest a similar temperature evolution of the band - structure of the hubbard model in the underdoped and half - filled cases driven by the temperature dependent spin - correlation length @xmath130 . especially , we expect the hubbard - i - like quasiparticle peaks at energies of @xmath250 around momentum @xmath91 to vanish with decreasing @xmath242 in the underdoped case as the peaks at energies of @xmath149 around momentum @xmath91 do in the case of half - filling . + the latter observation suggests a profound change of the fermi surface with temperature : as seen above , it is precisely the hubbard - i - like band near @xmath111 which crosses the chemical potential and thus forms the fermi surface in the doped case . it is then quite clear that the ` disappearance ' of this band with decreasing temperature must affect the fermi surface in some dramatic way . studies at zero temperature are possible only by means of exact diagonalization . analysis of the single particle spectrum shows the same ` rigid - band ' behaviour as at high temperatures@xcite and analysis of the momentum distribution @xmath251 suggest@xcite that the doped holes accumulate at the surface of the magnetic zone ( i.e. the line @xmath252 ) rather than around @xmath111 . in the present work we have systemactically studied the temperature- and doping - dependent dynamics of the two - dimensional hubbard model by finite - temperature qmc simulations . comparing the qmc single particle spectral function , the dynamical spin response and the spectral functions of suitably chosen diagnostic operators , different physical regimes could be identified . in simplest terms there are two quantities , which basically determine the single particle spectrum : the hole concentration and the the spin - response function , whereby there is a certain relationship between the two . + at half - filling and high temeratures ( @xmath253 ) , the combined photoemission and and inverse photoemission spectrum @xmath93 displays two dispersive features , the upper and lower hubbard band , roughly separated by @xmath13 ( @xmath254 , in our work ) . at these very high temperatures the system is in a spin disordered state . we have demonstrated here that the well - known hubbard - i approximation gives an excellent description of the single particle spectrum in this state , reproducing quantitatively both the single - particle dispersion and the distribution of spectral weight . this is by no means trivial , since the hubbard - i approximation is dynamically equivalent to a simplified effective hamiltonian , which just contains hole - like ( @xmath255 ) and double occupancy - like particles in a simple bi - quadratic form . + at lower temperatures ( @xmath256 ) , the hubbard - i approximation needs to be improved ; this is to be expected , because it neglects all effects of spin - correlations . in fact , the temperature where deviations from hubbard - i become strong , coincides fairly well with the transition from a spin response @xmath118 which is diffuse both in momentum and energy ( with a spread of order @xmath8 ) , to a more ` spin - wave - like ' response . in this regime @xmath118 displays the characteristic energy scale @xmath257 , with its spectral weight being concentrated at the af wave vector @xmath60 . it should be noted that this ` spin - wave - like ' regime develops despite the fact that at @xmath102 the spin - correlations length @xmath258 is still short - ranged ( @xmath259 lattice spacings ) . only at the lower temperature @xmath95 nel order spreads over the entire qmc block , creating an effective ( finite - size ) nel state . + it is well established by previous , in particular also qmc work , that in this temperature - regime ( @xmath256 ) new spectral features appear . they have often been interpreted as four ` bands ' , two ` coherent ' bands forming the topmost valence and the lowest conduction band in the insulator plus two ` incoherent ' bands , i.e. the remaining upper and lower hubbard band features ( see for example @xcite ) . our present work not only definitively identifies these four bands but also clarifies their physical origin and their connection to the spin - excitations . in simplest terms the emerging spin waves at lower temperatures provide the excitations that can ` dress ' the hubbard quasiparticles , whence new bands corresponding to dressed holes / double occupancies appear in the single particle spectrum @xmath93 . it has been shown in ref . @xcite that the @xmath228-band structure which appears in @xmath93 at lower temperatures can be explained in this way , and our present numerical check by directly calculating the spectra of ` dressed electrons ' supports this interpretation . + this physical picture at half - filling can be extended into the underdoped regime . this is most obvious in the single particle spectral function , which stays almost unchanged in the doped case ( i.e. the @xmath228-band structure and the ` character ' of the bands as measured by the diagnostic operators ) . the main change in fact consists in the chemical potential cutting gradually into the ( top of the ) lower hubbard band , precisely as predicted by the hubbard - i approximation . contrary to widespread belief the ` fermi surface ' , if determined by the fermi surface crossings of the dominant band through the chemical potential , does not satisfy the luttinger theorem . rather , for small hole concentrations the fermi surface volume is considerable larger than that for a slightly less than half - filled free - electron band . very similar conclusions have in fact been reached by a calculation of the electron momentum distribution in the 2d t - j model by puttika _ et al._@xcite . their calculation actually was a high temperature series expansion plus a pad extrapolation to lower temperature , and it is encouraging that this method gives similar results as our qmc results which are performed at relatively high temperatures . in its range of applicability , i.e. in the absence of strong magnetic correlations and close to half - filling , the hubbard - i approximation thus works remarkably well , both at half - filling and in the doped case . we stress that this has profound implications for the theoretical treatment of the model : perturbation expansions in @xmath13 or partial and self - consistent resummations thereof , may not be expected to give any meaningful results in this strong - coupling / low doping regime . + an interesting question is the possibility to verify our results experimentally . as already mentioned above , a scan of the temperature development of @xmath207 shows that the part of the quasiparticle band near @xmath111 ( where the fermi surface is located ) is loosing weight with decreasing temperature . in fact in arpes experiments on underdoped cuprate superconductors the ` hole - pocket ' around @xmath111 seen in our simulations ( and the expansion of puttika _ et al . _ ) is not observed , but rather a small ` fermi arc ' near @xmath195 , terminated by the ` pseudo gap ' around @xmath196 although our simulations do not allow to make statements about the truely low temperatures in the experiments , we believe that this suggests a strong temperature dependence of the single particle spectrum , with the temperature scale being set by the exchange constant @xmath121 ( which controls the degree of spin disorder ) . the latter is rather large in copper oxides , so that the temperature regime studied in our simulations probably can not be accessed experimentally in these materials . we note , however , that an arpes study for the 1d material @xmath260 which has a smaller exchange constant , has indeed provided evidence for a strong @xmath242-dependence of @xmath207@xcite . clearly , it would be interesting to study the fermi surface evolution in a 2d material with lower exchange constant . + as was the case at half - filling , the dynamical spin - response plays an important role : throughout the hubbard - i phase at low doping , the spin response shows the sharp low - energy mode at @xmath111 . the simultaneous disappearance of the @xmath228-band structure in @xmath207 and the low energy spin response with scale @xmath121 in the overdoped regime then show again the close relationship between the two . the dressing of holes by spin excitations apparently remains the most important correction over hubbard - i . in the overdoped regime the spin response is spread out over an energy range of @xmath261 and thus becomes more similar to the charge response . the single - particle spectral function is most consistent with a slightly renormalized free electron dispersion , and the luttinger theorem appears to be satisfied even at the relatively high temperature @xmath206 . this is quite consistent with earlier results on the t - j model , which show that for hole concentrations @xmath262 % the spin and charge response can be approximated well by the self - convolution of the single particle spectral function@xcite . this is essentially what is to be expected for a system of weakly interacting fermions , so that we conclude that in the overdoped regime we enter a new phase which most probably extends to the low - concentration limit where the nagaoka @xmath242-matrix approximation becomes exact . finally we note that exact diagonalization studies at finite temperatures@xcite also show some evidence for a ` crossover ' between different physical regimes at a hole concentration of @xmath263% . + this work was supported by dfn contract no . tk 598-va / d03 and by bmbf ( 05sb8wwa1 ) . computations were performed at hlrs stuttgart and hlrz jlich . one of us ( w.h . ) acknowledges hospitality of the physics department in santa barbara and many useful discussions with d. j. scalapino . e. loh and j. gubernatis , _ electronic phase transitions _ , w. hanke and y. v. kopaev , north holland , amsterdam ( 1992 ) . j. e. hirsch , phys . b * 31 * , 4403 ( 1985 ) . m. jarrel , j. e. gubernatis , phys . rep . * 269 * , 133 ( 1996 ) . e. fradkin , _ field theories of condensed matter systems _ , frontiers in physics , addison - 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we present results of a systematic quantum - monte - carlo study for the single - band hubbard model .
thereby we evaluated single - particle spectra ( pes & ipes ) , two - particle spectra ( spin & density correlation functions ) , and the dynamical correlation function of suitably defined diagnostic operators , all as a function of temperature and hole doping .
the results allow to identify different physical regimes .
near half - filling we find an anomalous ` hubbard - i phase ' , where the band structure is , up to some minor modifications , consistent with the hubbard - i predictions . at lower temperatures , where the spin response becomes sharp , additional dispersionless ` bands ' emerge due to the dressing of electrons / holes with spin excitatons .
we present a simple phenomenological fit which reproduces the band structure of the insulator quantitatively .
the fermi surface volume in the low doping phase , as derived from the single - particle spectral function , is not consistent with the luttinger theorem , but qualitatively in agreement with the predictions of the hubbard - i approximation .
the anomalous phase extends up to a hole concentration of @xmath0% , i.e. the underdoped region in the phase diagram of high - t@xmath1 superconductors .
we also investigate the nature of the magnetic ordering transition in the single particle spectra .
we show that the transition to an sdw - like band structure is not accomplished by the formation of any resolvable ` precursor bands ' , but rather by a ( spectroscopically invisible ) band of spin 3/2 quasiparticles .
we discuss implications for the ` remnant fermi surface ' in insulating cuprate compounds and the shadow bands in the doped materials . 2
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in the late 80 s , inspired by witten s ideas @xcite , reshetikhin and turaev @xcite came up with a construction of new 3manifold invariants , known as wrt quantum invariants . few years later , turaev @xcite formalized this construction by introducing the notion of _ modular category_. his main result is that any modular category gives rise to a quantum @xmath0manifold invariant . a modular category is a special kind of ribbon category which has a finite set of simple objects @xmath1 , including the unit object @xmath2 , satisfying duality , domination and non - degeneracy axioms . a ribbon category is a monoidal category with braiding , twist and compatible duality . ribbon categories are universal receivers for invariants of ribbon graphs @xcite . examples of modular categories arise from representation theory of quantum groups , when the quantum parameter @xmath3 is a root of unity , or can be constructed skein theoretically @xcite . many authors @xcite , @xcite , @xcite observed independently that for some special values of @xmath3 , the @xmath4 wrt invariants admit spin and cohomological refinements . cohomological refinements give rise to homotopy quantum field theories ( hqft s ) , constructed by turaev in @xcite . however , spin refinements do not fit in the framework of hqft s . the main aim of this paper is to provide an algebraic setting for spin type refinements of quantum 3manifold invariants . before explaining our results let us recall few definitions . given a group @xmath5 , a @xmath5category was defined in ( * ? ? ? * section vi ) as an additive monoidal category @xmath6 with left duality and unit object @xmath7 that splits as a disjoint union of full subcategories @xmath8 such that * @xmath9 if @xmath10 ; * for @xmath11 and @xmath12 , @xmath13 ; * @xmath14 , and for @xmath11 , @xmath15 . we call an object @xmath16 of a modular category _ invertible _ if there exists an object @xmath17 , such that @xmath18 . it is easy to see that isomorphism classes of invertible objects form a finite abelian group under tensor multiplication . let us denote by @xmath19 the group of isomorphism classes of invertible objects in the modular category @xmath20 . in section 5 we show that the braiding ( or monodromy ) coefficients of @xmath21 with @xmath22 define a map @xmath23 . this map induces on @xmath6 the structure of a @xmath24category . note the braiding matrix defines a bilinear form on @xmath5 , and the twist coefficients extend it to a quadratic form . a special role in our approach will play a subgroup @xmath25 of @xmath5 , such that the bilinear form restricted to @xmath25 is trivial while the quadratic extension is not . let us state our main definition . [ def1 ] let @xmath20 be a modular category with a group @xmath5 of invertible objects . for any subgroup @xmath26 , we call @xmath20 @xmath25refinable if @xmath27 . moreover , an @xmath25refinable modular category @xmath6 is called _ @xmath25spin _ if the twist quadratic form restricted to @xmath25 is non - trivial , or equivalently if @xmath25 has at least one element with twist coefficient @xmath28 . when @xmath25 is cyclic of order @xmath29 , we will use shorthand @xmath29spin and @xmath29refinable . for example , the @xmath4 modular category at the @xmath30th root of unity @xmath3 is @xmath31refinable for @xmath32 and it is @xmath31spin if @xmath33 . the group @xmath34 is generated by the @xmath35dimensional representation . we say that @xmath22 has order @xmath36 if @xmath36 is the minimal integer such that @xmath37 . we will see that the order of an element with twist @xmath28 has to be even . @xmath25refinable modular categories which are non spin give rise to invariants of pairs @xmath38 for any compact orientable 3manifold @xmath39 and a cohomology class @xmath40 . they also fit in the setting of modular group - categories introduced in @xcite . in the spin case , the formalism of @xcite does not strictly apply since the subcategory in trivial degree is not modular . let us concentrate on the spin case . if we have a @xmath25spin modular category , then the twist coefficients define an order @xmath31 element @xmath41 , which we will call the spin character . we will extend the definition of generalized spin structures with modulo @xmath29 coefficients given in @xcite to this situation and define @xmath42 generalized spin structures . one of the results of this paper is the following . [ main ] any @xmath25spin modular category @xmath6 with associated spin character @xmath43 provides a topological invariant @xmath44 of a pair @xmath45 , where @xmath46 is a @xmath42 generalized spin structure on @xmath39 . moreover , @xmath47 we expect that theorem [ main ] extends naturally to a spin type tqft . in section 2 we define @xmath29_complex _ spin structures . let us denote the set of such structures on @xmath39 by @xmath48 . generalizing results of @xcite , we identify @xmath48 with the set of modulo @xmath29 chern vectors which are further used for constructing extensions of wrt invariants . [ main_c ] suppose @xmath29 is an even positive integer . for any @xmath49spin modular category @xmath6 , there exists a topological invariant @xmath44 of a pair @xmath45 , where @xmath50 is a @xmath29complex spin structure on @xmath39 . we call a modular category @xmath6 _ reduced _ if it is @xmath51refinable . assume the group of invertible objects @xmath52 of our modular category @xmath6 is cyclic , but @xmath53 with @xmath54 . if @xmath55 , then there is a positive integer @xmath56 , such that @xmath57 . clearly , @xmath6 is not reduced , but @xmath25refinable , where the subgroup @xmath25 of order @xmath58 is generated by @xmath59 . given these data , one way to construct refined invariants is by using the @xmath58refinable structure on @xmath6 . in section 7 we show that there is a more efficient way to compute this invariant . namely , there always exists a smaller reduced category @xmath60 , which leads to the same invariant up to a correction term fully determined by the linking matrix . if @xmath61 , @xmath62 is particularly simple and coincides with @xmath63 where @xmath64 . [ decomp ] let @xmath6 be a modular category with a cyclic group of invertible objects @xmath52 . assume @xmath65 , @xmath66 and @xmath57 . then , there exists a reduced @xmath58refinable category @xmath67 , a positive integer @xmath68 and a root of unity @xmath69 such that for any closed oriented 3manifold @xmath39 we have @xmath70 where @xmath71 is the refined murakami ohtsuki okada invariant . we have either @xmath72 or @xmath73 , and @xmath74 is the first betti number . in the particular case , when @xmath75 , we have @xmath76 , @xmath69 is a root of unity of order @xmath56 and @xmath77 does not depend on @xmath46 so that we have @xmath78 the murakami ohtsuki okada invariant defined in @xcite depends only on the homological information which can be obtained from the linking matrix of the surgery link . theorem 3 generalizes the well - known decompositon results for quantum invariants stated in @xcite . we expect that these decomposition results extend to refined tqfts . * organization of the paper . * after recalling the basic definitions of @xmath79structures , we give homotopy theoretical and also combinatorial descriptions of their reduction modulo @xmath29 . further we recall the definitions of ribbon and modular categories and of quantum 3manifold invariants . section [ graded ] deals with modular categories containing invertible objects . the refined invariants are studied in sections [ refined ] and 7 . the last section is devoted to the proof of theorem [ decomp ] . in this section , given a compact orientable 3manifold @xmath39 , we define the set @xmath48 . throughout this paper all manifolds are assumed to be compact and oriented ; all ( co)homology groups are computed with integer coefficients , unless otherwise is specified ; @xmath80 denotes the cyclic group of integers modulo @xmath81 . complex spin structures are additional structures some manifolds can be endowed with and just like the more common spin structures , they can be seen as a generalization of orientations . we recall basic facts and equivalent ways to define @xmath79structures following the lines of @xcite and @xcite . let @xmath82 be an integer . the group @xmath83 is defined as the non - trivial double cover of the special orthogonal group @xmath84 : @xmath85 @xmath86 , @xmath87 and @xmath88 the complex spin group is defined as the quotient @xmath89 where @xmath90 is generated by @xmath91 . it follows that the map @xmath92 , defined as @xmath93)=\lambda(a)$ ] is a principal @xmath94fibration . @xmath95 . let @xmath96 be an @xmath36dimensional riemannian manifold and denote by @xmath97 the principal bundle of oriented orthonormal frames of @xmath96 . [ definition ] a @xmath79structure on the manifold @xmath96 is a principal @xmath98bundle @xmath99 , together with a map @xmath100 that restricted to the fibers is @xmath101 , i.e , makes the following diagram commute , i.e , @xmath102^{(\pi , \rho ) } \ar[rr ] & & p_{{{\operatorname{spin}}}^c(n ) } \ar[dd]^{\pi}\ar[dr]^{s } \\ & & & x\\ p_{\so(n ) } \times \so(n ) \ar[rr ] & & p_{\so(n ) } \ar[ur]\\ } \ ] ] since @xmath103 is the @xmath94bundle of the unique non trivial line bundle over @xmath84 ( @xcite ) and isomorphism classes of principal @xmath94bundles over @xmath96 are in one - to - one correspondence with elements of @xmath104 , complex spin structures can be defined as cohomology classes : a @xmath79structure on the manifold @xmath96 is an element @xmath105 whose restriction to every fiber of @xmath106 is the unique non trivial element of @xmath107 , i.e , @xmath108 let @xmath109 denote the grassman manifold of oriented @xmath36-planes in @xmath110 and let @xmath111 be the universal @xmath36-dimensional oriented vector bundle over @xmath109 . note that for discrete topological groups @xmath5 , the classifying space @xmath112 is eilenberg - maclane of type @xmath113 . let @xmath114 be the unique ( up to homotopy ) non - homotopically trivial map from @xmath109 to the eilenberg maclane space @xmath115 . we fix @xmath114 in its homotopy class and we define the fibration @xmath116 as the pull - back under @xmath114 of the path - space fibration over @xmath115 . if we set @xmath117 , then : + a @xmath79structure on the manifold @xmath96 is a homotopy class of bundle maps between the ( stable ) tangent bundle @xmath118 of @xmath96 and @xmath119 . let us denote by @xmath120 the homomorphism @xmath121)=z^2 $ ] . then , to any @xmath79 structure @xmath46 on an @xmath36dimensional manifold @xmath96 ( in the sense of definition [ definition ] ) , one can associate a complex line bundle as follows : using the map @xmath68 , the action of the group @xmath98 on the space @xmath122 extends to an action on the product @xmath123 and we consider its orbit space @xmath124 called the _ determinant line bundle _ of @xmath46 . the chern map @xmath125 is defined as @xmath126 where @xmath127 is the first chern class of the bundle @xmath128 , and it is affine over the doubling map @xmath129 see @xcite for more details . in this subsection we define @xmath29complex spin structures ( short @xmath130structures ) on @xmath36dimensional manifolds and we describe some of their properties . then , we focus on dimension three and we present a set of refined kirby s moves for @xmath130manifolds obtained by surgery along links in @xmath131 . let @xmath132 be the bockstein homomorphism associated to the exact sequence of groups : @xmath133 the following lemma will help justify our construction of @xmath130structures . the group @xmath134 is cyclic of order @xmath31 generated by @xmath135 , where @xmath136 is the second stiefel whitney class . since the elements of the group @xmath137 are in one - to - one correspondence with homotopy classes of maps in @xmath138 $ ] , there is a unique ( up to homotopy ) non - homotopically trivial map @xmath139 we fix @xmath140 in its homotopy class and define the fibration @xmath141 as the pull - back of the path space fibration of @xmath142 under the map @xmath140 : @xmath143^{\pi_d } \ar[r ] & \pk(\z_d,3 ) \ar[d ] \\ \bso(n)\ar[r]^{g } & \k(\z_d,3 ) \\ } \ ] ] we set @xmath144 . note that another choice of the map @xmath140 in its homotopy class yields a different , but homotopy equivalent , space @xmath145 . let @xmath96 be an @xmath36dimensional riemannian manifold . [ bundlemap ] a @xmath29complex spin structure on @xmath96 is a homotopy class of a bundle map between the ( stable ) tangent bundle @xmath118 of @xmath96 and @xmath146 . [ lift ] a @xmath29complex spin structure on @xmath96 is a homotopy class of a lift @xmath147 of @xmath148 to @xmath145 , where @xmath149 is a classifying map for the bundle @xmath118 . @xmath150^{\pi_d } \\ x \ar[r]^{f } \ar@{.>}[ru]^{\overline{f } } & \bso(n ) } \\\ ] ] since the fiber of @xmath151 is the eilenberg - maclane space @xmath152 , there is a unique obstruction @xmath153 to the existence of lifts @xmath147 and this obstruction lies in the group @xmath154 . note that the universal obstruction @xmath155 ( obtained from @xmath153 by setting @xmath156 and @xmath157 ) is non - zero , therefore not all manifolds can admit @xmath130structures . to see this , assume the contrary . then , the fibration @xmath141 has a section @xmath158 and the map @xmath159 lifts @xmath140 to the contractible space @xmath160 . therefore @xmath140 must be null - homotopic which contradicts the choice we made for @xmath140 . @xmath161^{k}\ar[d]^{\pi_d } \ar[r ] & \pk(\z_d,3 ) \ar[d]^{p } \\ \bso(n)\ar@/^/[u]^{s}\ar[r]^{g } & \k(\z_d,3 ) } \\\ ] ] as a consequence , the universal obstruction @xmath162 is the generator @xmath135 of @xmath134 . the set @xmath163 of @xmath29complex spin structures on a manifold @xmath96 is non - empty , if and only if , @xmath164 . if non - empty , the set @xmath163 is affine over @xmath165 ( @xmath166 is the second stiefel whitney class of @xmath96 ) . @xmath167^{\pi_d } \\ x \ar[r]^{f } \ar@{.>}[ru]^{\overline{f } } & \bso(n ) } \\\ ] ] let @xmath148 be the classifying map of the ( stable ) tangent bundle of @xmath96 . then , the lifts @xmath147 of @xmath148 to @xmath168 are in one - to - one correspondence with the sections of the bundle @xmath169 . the obstruction @xmath153 to the existence of such sections is a characteristic cohomology class so by its naturality property we have @xmath170 the above relation together with the naturality of bockstein homomorphisms imply the result . the second part of the theorem follows by standard arguments of obstruction theory . let us consider a manifold @xmath96 with boundary . then , any section of @xmath171 transverse to @xmath172 and oriented outwards gives rise to a homotopy class of isomorphisms between the oriented vector bundles @xmath173 and @xmath171 . therefore , there is a well - defined restriction map @xmath174 affine over the map @xmath175 induced by the inclusion @xmath176 . we have seen in section [ equiv ] that the fibration @xmath116 is defined as the pull - back under @xmath114 of the path - space fibration over @xmath115 . the modulo @xmath29 restriction morphism @xmath177 induces a map @xmath178 on the level of eilenberg - maclane spaces and further , via @xmath179 , a natural map @xmath180 . as a result , there exists a well - defined natural map @xmath181 affine over @xmath182 induced by @xmath69 . @xmath183\ar[rr]\ar[dd]^{\pi } & & \pk(\z,3)\ar @{->}[dd]|!{[d];[d]}\hole \ar[dr ] & \\ & & & & \bspin^c_d(n)\ar[dd]^(.65){\pi_d}\ar[rr ] & & \pk(\z_d,3)\ar[dd]\\ x\ar[rrr]^{f } & & & \bso(n ) \ar @{=}[dr ] \ar @{->}[rr]^(.35){h}|!{[r];[r]}\hole & & \k(\z,3)\ar @{->}[dr]^{\xi _ * } & \\ & & & & \bso(n ) \ar[rr]^{g } & & \k(\z_d,3)}\ ] ] let @xmath184 be an oriented framed link in @xmath131 with linking matrix @xmath185 and denote by @xmath186 the @xmath0manifold obtained by surgery . the manifold @xmath186 is the boundary of a @xmath187manifold @xmath188 , constructed from a @xmath187ball @xmath189 by attaching @xmath36 @xmath31handles @xmath190 along embeddings of @xmath191 in concordance with the orientation and framing of each component @xmath192 . the @xmath187manifold @xmath188 is sometimes called _ trace of surgery_. let us define the set @xmath193 whose elements will be called _ modulo _ @xmath29 _ chern vectors_. [ scdk ] there is a canonical bijection @xmath194 we have seen in the previous section that @xmath79structures induce @xmath29complex spin structures . in particular , the map @xmath195 is surjective since it is affine over the surjective map @xmath196 induced by restriction modulo @xmath29 of coefficients and similarly , the restriction map @xmath197 is surjective since it is affine over the surjective map @xmath198 induced by inclusion . with the help of @xmath30 and @xmath199 , we define the map @xmath200 as follows : any @xmath201 is the image of an element @xmath202 under the composition @xmath203 . the value @xmath204 is characteristic for @xmath205 ( see @xcite ) , therefore @xmath206 with @xmath207 . we set @xmath208 to be the image of @xmath209 @xmath210 in @xmath211 . * @xmath208 is well - defined : let us assume that @xmath212 contains two different elements @xmath213 and @xmath214 . then , they differ by @xmath215 whose modulo @xmath29 reduction belongs to @xmath216 in @xmath217 . since @xmath218 @xmath219 belongs to @xmath220 in @xmath221 , we get that @xmath222=[c(\tilde{\sigma_1})+2y]=[c(\tilde{\sigma_1})]$ ] ( the first equality follows from the fact that the chern map is affine over the doubling map ) . * @xmath223 is injective : let us assume that @xmath224 . then , the preimages @xmath213 and @xmath214 ( of @xmath225 and @xmath226 , respectively ) under @xmath203 differ by @xmath215 such that @xmath218 @xmath227 belongs to @xmath220 in @xmath221 . this implies that @xmath228 @xmath229 belongs to @xmath216 in @xmath217 and therefore @xmath230 . the sets @xmath231 and @xmath211 have the same cardinality , hence @xmath223 is bijective . from now on , we will reffer to the set @xmath211 as the combinatorial description of @xmath29complex spin structures on the surgered manifold @xmath186 . a celebrated theorem of kirby @xcite states that two ( oriented ) framed links in @xmath131 produce the same manifold by surgery , if and only if , they are related by a finite sequence of local geometric transformations called _ kirby moves_. in what follows , we present a refined version of the original kirby theorem for manifolds equipped with @xmath130structures . kirby moves ( a ) stabilization ( b ) handle slide ( c ) orientation reversal . note that we use the blackboard framing and the labels refer to modulo @xmath29 chern vectors . ] [ kirby ] let @xmath232 and @xmath233 be two oriented framed links with chern vectors @xmath46 , @xmath234 . then , the manifolds @xmath235 and @xmath236 are @xmath130homeomorphic , if and only if , the pairs @xmath232 and @xmath233 are related by a finite sequence of the moves in figure [ k1 ] or their inverses . we must check that for any of the kirby moves ( stabilization , handle slide and orientation reversal ) @xmath237 , the chern vectors change under the homeomorphism @xmath238 in the way described by figure [ k1 ] . to do this , note that the map @xmath239 defined in the proof of theorem [ scdk ] is surjective and the following diagram commutes : @xmath240^{r\circ r_d } \ar[rr]^{k } & & { { \operatorname{spin}}}^c(w_{l ' } ) \ar[d]^ { r'\circ r'_d}\\ { { \operatorname{spin}}}^c_d(s^3(l))\ar[rr]^{k } & & { { \operatorname{spin}}}^c_d(s^3(l ' ) ) . \\ } \ ] ] since @xmath79structures on @xmath188 are combinatorially described as the elements of the set @xmath241 and their change under kirby moves is known ( @xcite ) , the conclusion follows . in @xcite the second author introduced spin structures modulo an even integer @xmath29 ( short @xmath242structures ) . in dimension three , he gave a combinatorial description of such structures as well as a refined set of kirby moves . in this section we recall his results using the notations of section [ combinatorial ] and extend the definition to a possibly non cyclic group of coefficient @xmath243 with distinguished order @xmath31 element @xmath244 . consider the set @xmath245 the elements of @xmath246 are called _ modulo _ @xmath29 _ characteristic solutions of _ @xmath205 . there is a canonical bijection @xmath247 given a @xmath242structure @xmath46 on @xmath186 it can be extended to @xmath188 if and only if a certain cohomology class in @xmath248 vanishes . we denote this class by @xmath249 and we call it relative obstruction . to any @xmath250 there is associated a relative obstruction @xmath249 in @xmath251 . since the group @xmath251 is free of rank @xmath36 , taking the coefficients of the relative obstruction we obtain a map @xmath252 . we will show that this map is injective and its image coincides with @xmath246 . let us consider the embedding @xmath253 . then , the relative obstruction @xmath254 , where @xmath255 is the induced map on the level of cohomology . given an integral @xmath31cycle @xmath81 , we denote by @xmath256 its self - intersection number and by @xmath257_m$ ] its modulo @xmath56 restriction . the second stiefel whitney class @xmath258 is defined by the following equation : @xmath259_2\ra = x\cdot x \pmod 2 , \forall x\ ] ] therefore , the relative obstruction is defined by @xmath260_d\ra=\frac{d}{2 } x\cdot x \pmod d , \forall x.\ ] ] using functoriality and writing @xmath249 in the preferred basis of @xmath261 , the result follows . the set @xmath246 will be referred to as a combinatorial description of @xmath242structures on the surgered manifold @xmath186 . let @xmath243 be a finite abelian group and @xmath263 a non trivial element in the @xmath31-torsion subgroup @xmath264 , then we can define @xmath262 generalized spin structures . for @xmath265 , an @xmath262 generalized spin structure on an @xmath266 principal bundle @xmath267 is a cohomology class @xmath268 whose restriction to the fiber is equal to @xmath263 in @xmath269 . a @xmath262 generalized spin structure on an oriented manifold of dimension @xmath270 is a @xmath262 spin structure on its oriented framed bundle . as usual this definition can be extended to dimensions less than 3 by using stabilization . if @xmath243 is decomposed as a product of cyclic groups , @xmath271 , then a @xmath262 generalized spin structure is a sequence @xmath272 where @xmath273 is either a @xmath274spin structure or a mod @xmath274 cohomology class , depending on the twist coefficient of the corresponding generator . refined @xmath242kirby moves ( a ) stabilization ( b ) handle slide ( c ) orientation reversal . note that we use the blackboard framing and the labels refer to modulo @xmath29 characteristic solutions of @xmath205 . ] the second author proved the following result : @xcite let @xmath275 and @xmath276 be two oriented framed links with characteristic solutions @xmath277 . then , the manifolds @xmath278 and @xmath279 are @xmath242homeomorphic , if and only if , the pairs @xmath275 and @xmath276 are related by a finite sequence of the moves in figure [ k2 ] or their inverses . we introduce basic notions about ribbon categories following @xcite . without loss of generality , we work with monoidal categories that are strict ( according to mac lane s coherence theorem @xcite , every monoidal category is equivalent to a strict one ) , i.e , categories with tensor product and unit object @xmath7 satisfying @xmath280 for any objects @xmath281 and morphisms @xmath282 of the category . braiding _ in a monoidal category @xmath6 is a family of natural isomorphisms @xmath283 where @xmath284 run over all the objects of @xmath6 , such that @xmath285 and @xmath286 . let us consider the category @xmath287 of finite dimensional vector spaces over a field @xmath288 . the family of maps @xmath289 for all @xmath290 defines a braiding . a _ twist _ in a monoidal category @xmath6 with braiding @xmath291 is a family of natural isomorphisms @xmath292 where @xmath293 runs over all the objects of @xmath6 , such that for any two objects @xmath284 of @xmath6 , we have @xmath294 in the category @xmath295 , the family @xmath296 of maps @xmath297 defines a twist . let @xmath6 be a monoidal category . assume that to each object @xmath293 of @xmath6 there are associated an object @xmath298 and two morphisms @xmath299 . the rule @xmath300 is called _ duality _ in @xmath6 if @xmath301 and @xmath302 for all objects @xmath293 of @xmath6 . the duality is called _ compatible _ with the braiding @xmath291 and the twist @xmath303 if in addition , for any object @xmath293 of @xmath6 we have @xmath304 for example , in @xmath287 , if we set @xmath305 , fix a basis @xmath306 of @xmath293 with @xmath307 , and denote by @xmath308 its dual , we can define the map @xmath309 by @xmath310 and @xmath311 by @xmath312 then , the rule @xmath300 defines a duality in @xmath313 , compatible with the braiding @xmath291 and twist @xmath303 defined above . a monoidal category @xmath6 equipped with braiding @xmath291 , twist @xmath303 and a compatible duality @xmath314 is called _ ribbon_. 1 . the category @xmath315 defined above is ribbon ; 2 . let @xmath316 be the quantum group corresponding to a semi - simple lie algebra @xmath317 , then the category of its finite dimensional representations @xmath318 is ribbon @xcite . in ribbon categories we can define traces of morphisms and dimensions of objects as follows . [ qtrace ] 1 . the _ quantum trace _ of an endomorphism @xmath319 is the composition @xmath320 2 . the _ quantum dimension _ of an object @xmath293 is @xmath321 in any ribbon category @xmath6 a. @xmath322 for any morphisms @xmath323 and @xmath324 ; b. @xmath325 for any morphisms @xmath323 and @xmath326 ; c. @xmath327 for any two objects @xmath284 ; d. @xmath328 for any object @xmath293 . a ribbon graph is a compact , oriented surface in @xmath329 $ ] decomposed into bands , annuli and coupons . bands are homeomorphic images of the square @xmath330\times [ 0,1]$ ] , annuli are homeomorphic images of the cylinder @xmath331 $ ] and coupons are bands with a distinguished base , called bottom ( the oposite base is called top ) . isotopy of ribbon graphs means isotopy in the strip @xmath329 $ ] , constant on the boundary intervals and preserving the decomposition into annuli , bands and coupons as well as preserving orientations . given a ribbon category @xmath6 , a ribbon graph is @xmath6_colored _ if each band and each annulus is equipped with an object of @xmath6 , called _ color _ , and each coupon is colored by a morphism in @xmath6 . isotopy of colored ribbon graphs means color - preserving isotopy . starting with a ribbon category @xmath6 , one can construct the category @xmath332 of @xmath6colored ribbon graphs as follows : the objects are finite sequences of end points of ribbon graphs colored with objects of @xmath6 or their duals , according to the orientation of strands , and the morphisms of @xmath332 are isotopy classes of @xmath6colored ribbon graphs ( see figure [ mf ] ) . the category @xmath332 can be given a tensor product , a natural braiding , twist and compatible duality and it becomes in this way ribbon . [ mf ] ] ribbon categories turn out to play an important role in the theory of link invariants . the following theorem , due to turaev , describes a way to associate to any ribbon category , invariants of colored ribbon graphs and in particular , of colored ribbon tangles ( i.e. graphs without coupons ) . [ functor ] let @xmath6 be a ribbon category with a braiding @xmath291 , a twist @xmath303 and a compatible duality @xmath314 . then there exists a unique covariant functor @xmath333 preserving the tensor product such that ( see figure [ mf ] ) : 1 . @xmath334 , @xmath335 ; 2 . for any objects @xmath284 of @xmath6 , we have @xmath336 3 . for any elementary @xmath6colored ribbon graph ( ribbon graph with one coupon ) @xmath337 , we have @xmath338 where @xmath148 is the color of the only coupon of @xmath337 . the functor @xmath339 has the following properties : @xmath340 @xmath341 @xmath342 in what follows we will use pictures to illustrate identities in a ribbon category . when drawing a pictorial identity , we always mean the corresponding morphisms in the ribbon category ( see figure [ formula ] for an example ) . [ formula ] ] [ or ] let @xmath343 be a @xmath6-colored ribbon graph that contains an annulus @xmath344 . if @xmath345 is the ribbon graph obtained from @xmath343 by reversing the orientation of @xmath344 and replacing the color of @xmath344 with its dual object , then @xmath346 let @xmath288 be a field of zero characteristic . a ribbon category is called @xmath288linear or pre - additive if the hom sets are @xmath288vector spaces , composition and tensor product are bilinear and @xmath347 . an object @xmath17 of the category is called _ simple _ if the map @xmath348 from @xmath349 to @xmath350 is an isomorphism . we will denote by @xmath351 the additive closure of a pre - additive ribbon category @xmath6 , which admits direct sums of objects and compositions of morphisms are modelled on the matrix multiplication ( @xcite ) . a modular category over the field @xmath288 is a @xmath288linear ribbon category @xmath6 with a finite set of simple objects @xmath1 that satifies the following axioms : 1 . _ normalization _ : the trivial object @xmath7 is in @xmath1 ; 2 . _ duality _ : for any object @xmath352 , its dual @xmath353 is isomorphic to an object in @xmath1 ; 3 . _ domination _ : for any object @xmath293 of the category , there exists a finite decomposition @xmath354 with @xmath355 ; 4 . _ non - degeneracy _ : the matrix @xmath356 is invertible over @xmath288 , where @xmath357 is the endomorphism of the trivial object associated with the @xmath358colored , @xmath359framed hopf link with linking @xmath360 . and colors @xmath361 and @xmath362 the category is called pre - modular if we remove the last axiom . replacing @xmath6 with its additive closure , we can reformulate the domination axiom as follows : for any @xmath363 there exist positive integers @xmath364 called _ structure constants _ , such that @xmath365 . the domination axiom says that any object decomposes into a direct sum of simple ones . let @xmath366 and @xmath3 be an @xmath367th root of unity . then , there exists an associated modular category @xmath368 ( @xmath369 ) with simple objects given by partitions @xmath17 from @xmath370 see e.g. @xcite for more details . [ mainex1 ] in what follows , we describe basic properties of pre - modular categories that will be relevant to the rest of the paper . we follow the lines of @xcite . unless otherwise stated , @xmath6 is a pre - modular category over a field @xmath288 with zero characteristic and @xmath337 is the set of representatives of simple objects . a. an object @xmath17 of @xmath6 is called _ transparent _ if for any object @xmath371 the following morphisms in @xmath6 are equal + b. a morphism @xmath372 is called _ negligible _ if @xmath373 for any @xmath374 . c. the _ braiding coefficients _ between two objects @xmath375 are defined as a collection @xmath376 for all @xmath377 such that + d. for any object @xmath352 the _ twist coefficient _ @xmath378 is defined by the equality : + [ twist ] + we discuss in detail a formula that will be used extensively throughout the paper . let us start by fixing some notations . given objects @xmath379 , we denote by @xmath380 the @xmath288module of morphisms @xmath381 the modules @xmath382 , @xmath383 , @xmath384 , @xmath385 , @xmath386 and @xmath387 are mutually isomorphic , as well as the modules @xmath388 , @xmath389 and all obtained from them by a cyclic permutation of colors . identifying these modules along the isomorphisms , we get a symmetrized multiplicity module @xmath390 for which only the cyclic order of colors is important . the elements of @xmath390 are represented by a round coupon with one incoming line ( colored with @xmath391 ) and two outgoing ones ( colored by @xmath17 and @xmath371 ) . the pairing @xmath392 defined as @xmath393 is non - degenerate since the category @xmath6 can be assumed without negligible morphisms ( if any , they can be quotiened out ) . the symmetrized modules @xmath390 and @xmath394 are dual to each other , therefore we can choose bases @xmath395 for @xmath394 and @xmath396 for the module @xmath390 that are dual with respect to @xmath397 note that the composition @xmath398 is an endomorphism of the simple object @xmath391 , so it is of the form @xmath399 , @xmath400 . comparing the traces gives @xmath401 so @xmath402 . for any simple objects @xmath403 , the domination axiom applied to @xmath404 yields the following relation , known as the _ fusion formula _ : @xmath405 for any pre - modular category @xmath6 , let us define the _ kirby color _ @xmath406 as an element of @xmath407 , where @xmath408 denotes the grothendieck ring of the category @xmath409 . [ slide ] in any pre - modular category the following handle sliding identity holds @xmath410 let @xmath6 be a pre - modular category with @xmath411 and @xmath412 . then , the morphism is non - trivial if and only if @xmath17 is transparent . if @xmath17 is transparent , then the map @xmath413 conversely , if we assume that for some @xmath414 , @xmath415 we have @xmath416 then , by sliding any @xmath371colored strand along the @xmath417colored one , we obtain the following equalities of morphisms in @xmath6 @xmath418 and therefore , @xmath361 is transparent . [ twisted ] a pre - modular category with @xmath411 and with no non - trivial transparent objects is modular . let @xmath419 be a matrix with entries given by the relation @xmath420 we want to prove that the product @xmath421 , where @xmath422 is the identity matrix of size @xmath423 . we have @xmath424 where @xmath425 is the kronecker index . the second equality holds by isotopy , the third equality holds by the _ fusion formula _ , while the fourth equality is a consequence of the _ killing property_. the last equality can be proved using the structure of the modules @xmath426 and @xmath427 . if @xmath6 is a modular category , then @xmath428 is invertible in @xmath288 , and hence @xmath429 . given a modular category @xmath6 and a closed @xmath0manifold @xmath430 obtained by surgery on @xmath131 along a framed link @xmath205 , whose linking matrix has @xmath431 positive and @xmath432 negative eigenvalues , we define @xmath433 where @xmath434 denotes the @xmath435framed unknot and @xmath436 is the natural ribbon functor . [ t ] for any modular category @xmath6 , @xmath437 defines a topological invariant of @xmath39 , independent on the choice of the link @xmath205 . we need to show that @xmath437 is well - defined and does not change under kirby moves . the fact that @xmath438 are non - zero follows from the properties of the ribbon functor @xmath339 and the non degeneracy axiom for @xmath6 . the first kirby move is easy to establish . the invariance under the second kirby move is provided by proposition [ slide ] . in order to prove invariance under orientation reversal , let us consider the link @xmath439 obtained from @xmath205 by reversing the orientation of a component @xmath440 . without loss of generality , we may assume @xmath441 . thanks to corollary [ or ] we have @xmath442 the linking matrix @xmath443 with @xmath444 hence the matrices @xmath445 and @xmath446 have the same eigenvalues . in particular , @xmath447 so @xmath448 . this concludes the proof . in this section , we show that for any modular category @xmath6 , its group of invertible objects @xmath5 defines the structure of @xmath449category on @xmath6 . we identify an object with its isomorphism class and hence equality between objects means an isomorphism . an object @xmath17 in @xmath1 is called _ invertible _ if there exists another object @xmath450 such that @xmath451 . isomorphism classes of invertible simple objects form a finite abelian group under tensor multiplication . let us denote this group by @xmath452 and let @xmath453 be the group of its characters . we say that the _ tensor order _ of @xmath17 is @xmath29 if @xmath454 with @xmath29 minimal . for any @xmath352 and any invertible @xmath455 , @xmath456 . by the domination axiom we have @xmath457 . multiplying this identity by the inverse of @xmath140 , we get @xmath458 . thus , the right hand is simple , and so @xmath459 has to be simple too . in example [ mainex1 ] the object @xmath243 ( the longest row ) is invertible of order @xmath460 . [ mainex2 ] assume @xmath6 is a modular category with @xmath5 the group of invertible simple objects . we can define a @xmath449structure on @xmath6 as follows : given @xmath352 , the braiding coefficient of @xmath17 with elements of @xmath5 defines a map @xmath461 which associates to each @xmath17 a character @xmath462 defined by the equality : @xmath463 indeed , the previous lemma implies that the braiding operator acts on @xmath17 and @xmath140 as a multiplication by @xmath464 , i.e. only one braiding coefficient is non - zero . using the fact that @xmath140 is of finite order , we deduce that this coefficient is a root of unity of that order . observe that @xmath465 defines a group multiplication on @xmath24 . by taking a logarithm of the character , we can identify @xmath5 with its dual @xmath24 and write the group operation on @xmath5 additively . clearly , @xmath6 splits into a disjoint union of subcategories @xmath466 for @xmath467 . moreover , for any @xmath468 , @xmath469 is either zero if @xmath470 or @xmath288 otherwise ( by the assumption these objects are simple ) . hence , we just proved the following . [ gcategory ] a modular category @xmath6 with a group @xmath5 of invertible objects splits as a disjoint union of subcategories @xmath471 such that * each @xmath472 is a full subcategory of @xmath6 ; * each object of @xmath6 belongs to @xmath472 for a unique @xmath473 ; * @xmath9 if @xmath10 ; * for @xmath11 and @xmath12 , @xmath474 ; * @xmath475 , and for @xmath11 , @xmath476 . note that this is a special case of group - categories defined by turaev in @xcite . if @xmath477 , then we will call @xmath473 _ degree _ of @xmath17 and denote by by @xmath478 . if @xmath5 is cyclic of order @xmath29 , we call @xmath6 a _ modular @xmath29category_. fixing a generator @xmath16 of @xmath5 and a primitive @xmath29th root of unity @xmath479 , we have the decomposition @xmath480 where @xmath481 . let us assume that @xmath482 in examples [ mainex1 ] and [ mainex2 ] . then @xmath483 is generated by @xmath243 . hence , the category @xmath368 is a modular @xmath460-category with @xmath484 . we will need the following lemma . [ lem21 ] let @xmath6 be a modular @xmath29category such that the group @xmath485 . then @xmath486 if @xmath29 is odd , and @xmath487 if @xmath29 is even . let @xmath488 be the braiding coefficient , such that using we have @xmath489 . on the other hand , from definition [ qtrace ] we get @xmath490 , where @xmath491 , so @xmath492 . since @xmath493 , it follows that @xmath487 . if @xmath29 i d odd , then @xmath494 . let us recall the following definition which will play a central role in our exposition . [ hspin ] let @xmath26 be a subgroup . a modular category with a group @xmath5 of invertible objects is called _ @xmath25refinable _ if @xmath495 . if , in addition , @xmath25 has at least one element with twist coefficient @xmath28 , we call the category _ @xmath25spin_. in the case @xmath25 is cyclic of order @xmath29 , we will use the shorthand @xmath29refinable and @xmath29spin . given an even integer @xmath496 , such that @xmath497 for @xmath498 and @xmath499 , the second author constructed in @xcite a category @xmath500 with @xmath501 and @xmath502 generated by @xmath503 . here @xmath504 denotes the transpose partition . the second author proved that the category @xmath500 is @xmath29spin modular . assume @xmath6 is a pre - modular @xmath29category . for any @xmath505 , let us define the refined kirby colors @xmath506 as objects in the additive closure of @xmath6 . in any pre - modular @xmath29category @xmath6 we have the following equality of morphisms : @xmath507 with @xmath508 . the proof is the same as in the non - graded case , using the fact that @xmath509 is zero unless @xmath510 . hence , the sum over @xmath391 can be restricted to @xmath511 . @xmath512 in the second and the fourth equalities we use the _ fusion formula _ , the third equality holds by isotopy . for any @xmath513 , and a primitive @xmath29th root of unity @xmath514 , we define @xmath515 the dual refined kirby color . note that @xmath516 , however other dual kirby colors depend on the choice of @xmath514 . we use the graded sliding identity to prove the following lemma . [ dual sliding property ] in any pre - modular @xmath29category @xmath6 we have the following equality of morphisms : a. @xmath517 b. @xmath518 \(a ) we have the following equalities : @xmath519 the second and fourth equalities follow from the _ fusion formula _ , the third equality holds by isotopy . + ( b ) according to corollary [ or ] , given a @xmath6colored ribbon graph @xmath343 with an annulus component @xmath344 colored with the dual kirby color @xmath520 , if we consider the ribbon graph @xmath345 obtained from @xmath343 by reversing the orientation of @xmath344 and changing its color to @xmath521 , then @xmath522 . in particular , we have @xmath523 where the second equality is a straightforward application of ( a ) . this section is devoted to the proof of theorem 1 . motivated by the known examples we will first give the proof in the cyclic case . for @xmath29spin modular categories we obtain invariants of @xmath0manifolds equipped with @xmath242 structure . in cohomological case we get refined invariants for @xmath0manifolds with modulo @xmath29 @xmath524dimensional cohomology classes . for the rest of this section @xmath430 is a closed @xmath0manifold obtained by surgery on @xmath131 along a framed link @xmath184 whose linking matrix @xmath446 has @xmath525 positive and @xmath526 negative eigenvalues . let @xmath29 be an even , positive integer and @xmath6 be a @xmath29spin modular category . for any @xmath527 let us define @xmath528 [ main - spin ] for any @xmath29spin modular category @xmath6 , @xmath529 is a topological invariant of the pair @xmath530 . moreover , @xmath531 to prove the first statement we need to show that @xmath532 is well - defined and does not change under refined kirby moves . invariance under the first two @xmath242 kirby moves follows immediately from the _ graded sliding property _ and the next lemma which implies that @xmath533 is invertible for @xmath534 . in order to prove invariance under orientation reversal , let us consider the link @xmath439 obtained from @xmath205 by reversing the orientation of a component @xmath440 . without loss of generality , we may assume that @xmath441 . we have seen in the proof of theorem [ t ] that the linking matrices @xmath445 and @xmath446 have the same eigenvalues so @xmath535 , @xmath536 . + since @xmath537 applying corollary [ or ] gives @xmath538 and therefore @xmath539 . it remains to prove that @xmath540 if @xmath541 do not solve @xmath542 . the proof is in three steps . assume that the first component @xmath543 of our link is the @xmath435framed unknot . then it can be unlinked from the rest of @xmath205 by applying the fenn - rourke move . the graded sliding identity and lemma [ cancel ] tell us that @xmath544 should solve the above equations . assume @xmath543 is the @xmath344framed unknot . then we add a @xmath435framed unknot to our link ( with an invertible invariant ) and slide it along @xmath543 ( perform the inverse fenn - rourke move ) . this changes the framing on @xmath543 by @xmath545 and allows to reduce this case to the previous one . finally , assume @xmath543 is arbitrary . then we can unknot @xmath543 by adding @xmath435framed unknots to our link in such a way , that their linking number with @xmath543 is zero . this again reduces the situation to the previous case . [ cancel ] for any @xmath29spin modular category @xmath6 , @xmath546 is zero unless @xmath547 . recall that invertible objects form an abelian group under tensor multiplication , which acts on @xmath1 . in particular , its cyclic subgroup @xmath548 acts on each @xmath549 . let us denote by @xmath550 the set of orbits under this action and by @xmath551 the corresponding reduced kirby color . note that @xmath552 . let @xmath553 be the @xmath554framed hopf link with linking matrix @xmath28 . . ] after sliding the second component of @xmath555 along the first one we get @xmath556 which is zero unless @xmath557 . here we used that @xmath558 and @xmath559 , by lemma [ lem21 ] . since @xmath560 and @xmath561 are complex conjugate to each other , the result follows . in this subsection we assume that @xmath6 is a non - spin @xmath29refinable modular category . the elements @xmath562 are combinatorially given by solutions of @xmath563 let @xmath205 be an oriented framed link and @xmath564 . the usual kirby moves admit refinements for manifolds equipped with such structures as follows : * stabilization : @xmath565 ; * handle slide : @xmath566 where @xmath439 is obtained from @xmath205 by sliding component @xmath192 along @xmath567 and @xmath568 if @xmath569 and @xmath570 . here the sign depends on whether the orientations of @xmath192 and @xmath567 match or not , respectively ; * orientation reversal : @xmath566 where @xmath439 is obtained from @xmath205 by reversing the orientation of component @xmath192 and @xmath571 if @xmath572 and @xmath573 . for any @xmath574 let us define @xmath575 [ main - coho ] for any non - spin @xmath29refinable modular category , @xmath576 is a topological invariant of the pair @xmath38 . moreover , @xmath577 the proof is based on the following lemma . [ vanish ] for any non - spin @xmath29refinable modular category @xmath6 , @xmath546 is zero unless @xmath578 . just as in the proof of lemma [ cancel ] , we consider the hopf link @xmath579 with linking number @xmath28 and we slide the second component along the first one . we get @xmath580 which is zero , unless @xmath578 . the proof of theorem 1 follows from the two previous cases . an @xmath42 generalized spin structure @xmath46 on @xmath430 is described by a sequence of coefficients @xmath581 satisfying a characteristic equation : @xmath582 indeed , the kirby element decomposes using @xmath583-grading , and the formula for the refined invariant given in still holds . the condition for non vanishing in lemma [ cancel ] is @xmath584 . the solution of these equations is a sequence of @xmath585 where the index @xmath586 runs over cyclic components @xmath587 of @xmath25 . moreover , either @xmath588 or @xmath589 , depending on the twist coefficient of the corresponding generator . combining the two previous theorems we get @xmath590 this section is devoted to the proof of theorem 2 . again , according to the twist coefficients , we will either get an extension of wrt invariants for @xmath0manifolds equipped with modulo @xmath29 complex spin structures or with @xmath31dimensional cohomology classes . throughout this section @xmath6 is a @xmath49spin modular category with @xmath29 even . for any @xmath591 , let us define @xmath592 where the shorthand @xmath593 means that the summation is taken over all elements of @xmath221 in the equivalence class of @xmath46 . [ main - spinc ] let @xmath29 be an even integer . for any @xmath594spin modular category @xmath6 , @xmath595 is a topological invariant of the pair @xmath596 . in order to prove that @xmath595 is a topological invariant of the @xmath130manifold @xmath45 , we have to check invariance under the @xmath597 kirby moves of theorem [ kirby ] . we start by checking invariance under the first kirby move . let @xmath233 be obtained from @xmath598 by a positive stabilization . we have that @xmath599 to compute the sum @xmath600 , we write the dual kirby color @xmath601 in terms of the refined ( graded ) kirby colors @xmath602 as follows : @xmath603 lemma [ cancel ] together with the identity @xmath604 gives @xmath605 so @xmath606 . analogously , @xmath607 is invariant under a negative stabilization . the invariance under the second kirby move is provided by the _ dual sliding property_. finally , we must check invariance under orientation reversal . for that , let @xmath608 be obtained from @xmath232 by changing the orientation of a component @xmath440 . without any loss of generality , we may assume that @xmath441 and , just like in the proof of theorem [ t ] , we get @xmath535 , @xmath536 . we have that @xmath609 the first equality above is a consequence of the fact that @xmath443 , for @xmath444 while the third equality is an immediate application of corollary [ or ] . this concludes the proof . let @xmath29 be a positive integer and @xmath20 be a non spin @xmath29refinable modular category . the group @xmath610 is described combinatorially as the set @xmath611 . the kirby moves for the pair @xmath38 where @xmath39 is obtained by surgery on a link @xmath205 and @xmath612 can be described as follows : * stabilization : @xmath613 ; * handle slide : @xmath566 where @xmath439 is obtained from @xmath205 by sliding component @xmath192 along @xmath567 and @xmath568 if @xmath614 and @xmath615 . here the sign depends on whether the orientations of @xmath192 and @xmath567 match or not , respectively ; * orientation reversal : @xmath566 where @xmath439 is obtained from @xmath205 by changing the orientation of component @xmath192 and @xmath571 if @xmath572 and @xmath573 . for any @xmath616 let us define @xmath617 where the shorthand @xmath618 means that the summation is taken over all elements of @xmath217 in the equivalence class of @xmath114 . [ main - cohom2 ] for any non spin @xmath29refinable modular category @xmath6 , @xmath576 is a topological invariant of the pair @xmath38 . in order to prove that @xmath576 is a topological invariant of the manifold @xmath619 , we have to check invariance under the kirby moves listed above . we start by checking invariance under the first kirby move . let @xmath620 be obtained from @xmath621 by a positive stabilization . @xmath622 we compute @xmath623 since @xmath624 unless @xmath625 . using lemma [ vanish ] we get that @xmath626 and therefore @xmath627 . analogously , @xmath628 is invariant under a negative stabilization . the invariance under the second kirby move is provided by the _ dual sliding property_. finally , we must check invariance under orientation reversal . for that , let @xmath620 be obtained from @xmath621 by changing the orientation of a component @xmath440 . without any loss of generality , we may assume that @xmath441 and , just like in the proof of theorem [ t ] , we get @xmath535 , @xmath536 . since @xmath629 , it follows that @xmath630 the first equality above is a consequence of the fact that @xmath443 , for @xmath444 while the third equality is an immediate application of corollary [ or ] . this concludes the proof . this section is devoted to the proof of theorem 3 . reader interested in the case @xmath631 only can skip this section and consult an easy direct argument in appendix . throughout this section @xmath29 is any positive integer , it needs not to be even . let us recall the setting . we assume that @xmath6 be a modular category with cyclic group of invertible objects @xmath52 . let @xmath632 and @xmath633 . moreover , let us split @xmath634 , such that @xmath635 , @xmath636 @xmath637 . for any @xmath638 we can choose @xmath639 where @xmath640 is a primitive @xmath29th root of unity . let us fix the generator @xmath16 , so that @xmath641 . the twist coefficient for @xmath16 is @xmath642 and satisfies @xmath643 . we consider the subgroup of invertible objects @xmath644 . clearly , @xmath495 , so the modular category @xmath20 is @xmath58refinable . the twist coefficient for the generator @xmath645 is @xmath646 . it is equal to @xmath28 if @xmath58 is even , @xmath56 is odd and either @xmath486 and @xmath647 , or @xmath648 and @xmath649 ; it is equal to @xmath524 in all other cases . the modular category @xmath20 is @xmath58spin if @xmath58 is even , @xmath56 is odd and @xmath650 has order @xmath651 , and @xmath58cohomological in all other cases . we now present the idea of the proof of the decomposition statement . we define a tensor category @xmath652 with simple objects represented by @xmath653 . the tensor product in @xmath652 mimics central extension of groups using @xmath31-cocycles . we lift the map @xmath654 into a map @xmath655 , which plays the role of a section . further , we extend @xmath148 into a map @xmath656 , such that @xmath657 given two elements @xmath658 and @xmath659 of @xmath660 we define their tensor product as @xmath661 . we allow in @xmath662 direct sums of objects with homogenous @xmath148 value . for @xmath663 and @xmath664 we set @xmath665 the category @xmath652 is a tensor category over @xmath288 with unit object @xmath666 and compatible duality . note that @xmath667 is a left and right dual for @xmath658 if @xmath668 is choosen so that @xmath669 . the category @xmath652 is semisimple with @xmath670 as representative set of simple objects . the group of invertible objects is @xmath671 , generated by @xmath672 and @xmath673 . the object @xmath674 is invertible in @xmath675 if and only if @xmath293 is invertible in @xmath20 . we deduce the last statement . to prove semisimplicity , it is enough to decompose the tensor product of two objects objects @xmath674 , @xmath676 in @xmath677 . we have in the category @xmath20 a decomposition @xmath678 we set @xmath679 . then we have in the category @xmath675 the following decomposition @xmath680 further , let us give @xmath662 a ribbon structure which twists the one given on @xmath20 . the braiding is given by a formula @xmath681 with appropriate choice of a root of unity @xmath69 whose order @xmath682 if @xmath29 is even and @xmath683 if @xmath29 is odd . using duality , the twist is then given by @xmath684 the corresponding colored link invariants @xmath685 and @xmath686 are equal up to a power of @xmath69 which is computed from map @xmath148 and linking numbers . note that @xmath69 is choosen such that @xmath687 elements of @xmath688 become transparent . the @xmath688category @xmath689 is premodular and can be modularized as described in @xcite . simple objects in the modularization @xmath690 are obtained from those of @xmath689 quotienting by a free action . the set @xmath691 of simple objects in @xmath60 has cardinality @xmath692 . below we give a detailed proof of the decomposition formula in the spin case , the cohomological cases can be proven similarly . we consider here the spin case , which means that @xmath29 is even , @xmath56 is odd and the twist @xmath650 has order @xmath651 . the generator @xmath16 can be choosen so that @xmath693 . let @xmath694 with @xmath695 mod @xmath696 . note that @xmath697 so that the modified braiding is well defined . the braiding coefficients for the generators of @xmath688 are : @xmath698 @xmath699 the twist coefficients are @xmath700 @xmath701 it follows that the group of transparent objects is generated by @xmath702 which has trivial twist and quantum dimension @xmath524 . applying the results @xcite we see that the category @xmath675 is modularizable , i.e. that there exists a modular category @xmath690 and a dominant ribbon functor @xmath703 . here the group of transparent objects acts freely on the set @xmath670 of simple objects in @xmath675 . this is proved using the map @xmath148 and the fact that @xmath704 has order @xmath687 which is the order of @xmath702 . hence , the simple objects @xmath705 in @xmath690 are represented by cosets in @xmath660 under this free action . in the category @xmath690 the group of invertible objects is @xmath706 . the twist coefficient for @xmath672 is equal to @xmath28 , so the category @xmath690 is @xmath58spin . denote by @xmath707 the kirby element in @xmath675 which represents @xmath687 times the kirby element @xmath708 in @xmath690 . we write the graded decomposition @xmath709 so that @xmath710 moreover , for any @xmath711 mod @xmath58 , the kirby color @xmath712 in @xmath67 is represented by @xmath713 , i.e. @xmath714 indeed , the set @xmath713 consists of all @xmath715 such that @xmath716 mod @xmath717 . there are @xmath718 such elements . acting with @xmath719 we can shift the degree of solutions by @xmath58 . in this way we obtain all @xmath715 with @xmath720 mod @xmath58 . taking the quotient by this action we get @xmath721 . it makes sense to evaluate both reshetikhin - turaev ribbon functors @xmath722 , and @xmath723 on @xmath675 colored links . let @xmath430 be a @xmath0manifold given by surgery on the @xmath36-component link @xmath205 with signature @xmath724 , and @xmath73 represented by coefficients @xmath725 , @xmath726 . for objects @xmath727 , @xmath728 , we have @xmath729 where @xmath730 . note that the left hand side is invariant under action of @xmath702 on objects and can be used for the evaluation of the reduced invariant @xmath731 which we want to compare with @xmath732 . we have @xmath733 @xmath734 after normalization we get @xmath735 and @xmath736 is the complex conjugate . one can check , following the graded construction in section [ refined ] that the formula @xmath737 defines an invariant of @xmath45 . we conclude @xmath738 in the case @xmath739 , we have @xmath76 and @xmath740 mod @xmath58 . furthermore , we can assume that our surgery presentation has even linking matrix ( the obstruction given by the spin cobordism group vanishes ) , so that @xmath741 mod @xmath742 . decomposing @xmath743 with @xmath744 and @xmath745 we see that @xmath746 does not depend on @xmath747 . summing over @xmath46 we get @xmath748 @xmath749 + here we give a simple direct proof of theorem 3 in the case when @xmath631 . for readers convenience , we repeat the statement . [ decomposition ] let @xmath6 be a modular @xmath29category with the group @xmath52 of invertible objects , such that @xmath750 , @xmath751 . then there are exists a subcategory @xmath752 and a root of unity @xmath69 , such that for any closed orientable @xmath0manifold @xmath39 @xmath753 let @xmath56 be such that @xmath754 and we set @xmath755 let @xmath67 be the full ribbon subcategory of @xmath6 generated by @xmath756 and @xmath708 be the corresponding kirby color . let @xmath757 in this situation @xmath243 has order @xmath58 and @xmath758 moreover , consider @xmath759 such that clearly , @xmath767 since @xmath768 . duality axiom holds since for @xmath769 , @xmath770 and hence @xmath771 . domination follows trivially from the same property of @xmath6 . it remains to prove the non - degeneracy . observe that the kirby color @xmath417 decomposes as the sum @xmath772 since @xmath773 . decomposing @xmath417 as above and using proposition [ crossing ] ( a ) we get for any @xmath774 the following equalities @xmath775 the _ killing property _ combined with proposition [ twisted ] implies non - degeneracy . let @xmath39 be presented by an oriented framed link @xmath184 with all components @xmath192 unknotted ( such a link always exists , see @xcite ) . then , the invariant @xmath776 if we replace @xmath777 in @xmath778 and apply proposition ( a ) we obtain : @xmath779 in particular , for @xmath534 , we have @xmath780 given the link @xmath205 with components colored by @xmath781 , @xmath782 can be computed as follows : first we make each component of @xmath205 zero framed , then we unlink the components ( using proposition ( c ) and ( d ) as many times as necessary ) . finally , we obtain @xmath36 disjoint and unlinked copies of the zero framed unknot with colors @xmath781 and the relation : @xmath783 where @xmath446 is the linking matrix of @xmath205 . similarly @xmath784 and the reshetikin - turaev invariant decomposes as @xmath785 note that for @xmath29 even , @xmath69 is an @xmath56th root of unity if @xmath56 is odd and a @xmath651th root of unity if @xmath56 is even and according to @xcite , @xmath786 is a topological invariant of @xmath39 independent on the choice of @xmath205 , known as the murakami ohtsuki okada invariant . @xmath749 c. blanchet . _ hecke algebras , modular categories and @xmath0 manifolds quantum invariants _ , ( 2000 ) , 193223 . c. blanchet . _ a spin decomposition of the verlinde formulas for type a modular categories _ , , ( 2005 ) , 1 28 .
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modular categories are a well - known source of quantum 3manifold invariants . in this paper
we study structures on modular categories which allow to define refinements of quantum 3manifold invariants involving cohomology classes or generalized spin and complex spin structures . a crucial role in our construction is played by objects which are invertible under tensor product .
all known examples of cohomological or spin type refinements of the witten - reshetikhin - turaev 3manifold invariants are special cases of our construction . in addition
, we establish a splitting formula for the refined invariants , generalizing the well - known product decomposition of quantum invariants into projective ones and those determined by the linking matrix .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the purpose of this contribution is twofold : to review how an su(n ) transformation can be experimentally realized using optical elements , and to show how such an experimental realization can be used to investigate the cyclic evolution of a state over the manifold su(n)/u(n-1 ) . the bulk of the results will be presented explicitly for su(3 ) ( see @xcite for further details ) and su(4 ) , although it will become clear that the method can be applied to any su(n ) . recall that cyclic evolution of a wave function yields the original state plus a phase shift , and this phase shift is a sum of a dynamical phase @xmath0 and a geometric ( or topological , or quantal , or berry ) phase @xmath1 shift@xcite . the geometric phase shift is important not just for quantum systems but also for all of wave physics . thus far , controlled geometric phase experiments , both realized and proposed , have been exclusively concerned with the abelian geometric phase arising in the evolution of u(1)invariant states @xcite here , we generalize the above results to an abelian geometric phase which arises from geodesic transformations of u(n-1)invariant states in su(n)/u(n-1 ) space . the scheme employs a sequence of optical element , henceforth called su(n ) elements because they perform transformations described by an su(n ) matrix , arranged so that the net result of the sequence cyclically evolves an initial state back to itself up to a phase . it will be seen that the decomposition of an su(n ) transformation into a product of appropriate su(2 ) subgroup transformations is the prescription to construct each su(n ) element as a sequence of su(2 ) elements . it is important to distinguish the evolution of states in the geometric space su(n)/u(n-1 ) from the transformations of the optical beam as it progresses through the interferometer . it is possible to set up the experiment so as to eliminate the dynamical phase associated with these optical transformations , thus making the dynamical phase irrelevant for our purpose . the cyclic evolution described here occurs in the geometric space , and the geometric phase of interest is related to this evolution . consider an optical element which mixes two input beams . it is , formally , a black box which performs some transformation , as the output is not the same as the input . we are here interested by optical elements which mix the input beams in a linear way , _ i.e. _ the output is a linear combination of the inputs . furthermore , we will assume that the optical element is passive , _ i.e. _ it does not globally create or annihilate photons . the optical elements the enter in the construction of su(n ) device are beam splitters , mirrors and phase shifters . a phase shifter is essentially a slab of material which increases the optical path lenght of one beam relative to the other . a beam splitter is a partially silvered mirror which lets photons through with some probability . provided that losses can be ignored , each of these optical elements can be associated with an su(2 ) unitary transformation@xcite . it is therefore advantageous to factorize each su(n ) transformation into a product of su(2)@xmath2 subgroup transformations mixing fields @xmath3 and @xmath4 . an optical element mixing two fields is associated with an su(2 ) transformation in the following way . suppose that _ one _ photon enters the black box . we may assume that it will enter the optical system either via beam one or beam two . thus the hilbert space of input states is two dimensional . as the transformation is linear , the set of all possible output states will also be a two dimensional space . clearly this conclusion does not change if the input state is a general state @xmath5 , where the photon has probability @xmath6 of being in beam 1 and probability @xmath7 of being in beam 2 ( @xmath8 , @xmath9 are complex numbers ) . suppose now that _ two _ photons enter the black box . then , we can have one of three possibilities : two photons enter in beam 1 , one photon enters in each beam , or both photons enter in beam 2 . in this case , the hilbert space of states is three dimensional . continuing in this way , and using the fact that the input photons are indistinguishable , one rapidly works out that , in a system containing @xmath10 photons , the relevant hilbert space is of dimension @xmath11 . the conservation of photon number leads to the following constraint on the possible form of the linear transformation . consider first the case of a single photon . the optical transformation @xmath12 with @xmath13 complex numbers , transforms a general state @xmath14 into the output state @xmath15 such that @xmath16 taking the transpose complex conjugate of that to find @xmath17 , multiplying from the right by @xmath15 , we find that , if the number of photon ( = 1 ) is to be conserved for any input state , @xmath18 implies that @xmath19 , the unit matrix . thus , @xmath20 is a @xmath21 unitary transformation . because we are only interested in the relative phase between the beams , the determinant of @xmath20 can chosen without loss generality to be + 1 , so that @xmath20 is an su(2 ) matrix . if the black box performs a transformation @xmath20 that is an su(2 ) transformation when there is a single photon in the system , it must also perform an su(2 ) transformation when there are @xmath22 photons in the system : the transformtation effected by the black box can not depend on the number of photons in the system ( at least not in the regimes that we are considering ) . thus , in a system of two photons , where state space is three dimensional , @xmath20 will be @xmath23 representation of the relevant su(2 ) transformation . in a system containing @xmath10 photon , @xmath20 will be an su(2 ) matrix of dimension @xmath24@xcite . it is well known that an su(2 ) transformation can be factored into a product of three subtransformations : @xmath25 this factorization is a prescription on how to construct the su(2 ) device : a slab of material is inserted in one beamline so as to create a relative phase shift of @xmath26 , a partially silvered mirror which lets @xmath27 photons from beam 1 through is then inserted , and another phase shifter completes the design . in an su(3 ) interferometer , an general su(3 ) matrix is decomposed into a product of three su(2 ) matrices @xcite : @xmath28 where @xmath29 . this factorization , symbolically written @xmath30 , is a _ de facto _ prescription on how to build the su(3 ) device : fields @xmath31 and @xmath32 are mixed followed by a mixing of the output field @xmath31 with the field in channel @xmath33 , and , finally , the output field @xmath31 is mixed with field @xmath32 . as it is possible to factorize an su(n ) matrix in terms of su(2 ) submatrices @xcite , the process of constructing a general su(n ) device is perfectly obvious and follows the lines illustrated explicitly for the su(3 ) device . for instance , the appropriate factorization of an su(4 ) matrix is a product of the type @xmath34 where @xmath35 is an su(2 ) matrix mixing fields @xmath36 and @xmath37 . the factorization of an su(n ) matrix into su(2 ) subgroup matrices is not unique , and the number of su(2 ) elements required to construct an su(n ) device can vary according to the parametrization : an estimate of the number of su(2 ) devices required to construct an su(n ) element was given in @xcite . finally , we mention that su(3 ) and su(4 ) devices have been constructed but with an aim to study non - classical statistics @xcite . the total phase @xmath38 acquired by a state during a generic cyclic evolution is the sum of @xmath39 . a special type of evolution is the geodesic evolution@xcite ; by transforming the output state along geodesic paths in the geometric space , the geometric phase shift along each path is zero . an essential property of geodesic evolutions is that they are not transitive : the product of two such evolutions is not necessary another geodesic evolution . this is most easily illustrated by drawing three points on a plane at random . it is well known that the geodesic on a plane is a straight line . let @xmath40 and @xmath41 denote the three points . then it is clear that , even if @xmath42 is the straight ( geodesic ) line that connects @xmath43 and @xmath44 , and even if @xmath45 is the straight line that connect @xmath46 and @xmath41 , the combined segment @xmath47 is _ not _ a geodesic between @xmath48 and @xmath41 . this property makes it possible to construct a cyclic evolution from a sequence of geodesic legs : the geometric phase acquired during the circuit is then a global property of the entire circuit . for definiteness , let us consider su(3 ) . there , the evolution of the state @xmath49 to the state @xmath50 via 3 geodesic paths in the geometric space can be described by 3 one parameter su(3 ) group elements @xmath51 , with @xmath52 an evolution parameter . these transformations satisfy the conditions that @xmath53 is the identity element and @xmath54 for some fixed end values @xmath55 of the evolution parameters . we consider evolutions @xmath56 of the form @xmath57 with @xmath58 an element of su(3 ) satisfying @xmath59 the form of the one parameter subgroup @xmath60 with real entries was guided by the definition of a geodesic curve between two states @xmath61 and @xmath62 , which can be written in the form @xcite @xmath63 with @xmath64 . as it is always possible to choose unit vectors @xmath61 such that @xmath65 is real and positive , it is straightforward to show that any @xmath66 of the form given by eq . ( [ eq : decomposeu ] ) satisfying @xmath67 real and positive gives evolution along a geodesic curve in su(3)/u(2 ) . the form of the geodesic evolution makes it easy to obtain its physical interpretation . the transformation @xmath60 is a transformation of appropriate length along some reference geodesic ( some generalized greenwich meridian on su(3)/u(2 ) ) . the transformation @xmath58 is a principal axis transformation which correctly orients the reference geodesic so that it passes through @xmath68 and @xmath69 . @xmath58 therefore depends on the initial and final states . the three states in su(3)/u(2 ) must be chosen in a sufficiently general way to ensure that they can represent any triangle in su(3)/u(2 ) @xcite . since the latter is of dimension @xmath70 , there are @xmath70 free parameters to be chosen . the first state can be chosen , wlog , to be the `` north pole '' state . again wlog , the second state can always be chosen to lie along the reference geodesic some distance away from the initial state . the last state must therefore contain the remaining @xmath32 parameters . in short , the vertices of a geodesic triangle in su(3)/u(2 ) can , in general , be chosen as @xmath71 with @xmath72 , @xmath73 , @xmath8 and @xmath9 arbitrary . since @xmath74 , the geometric phase for the complete circuit is extracted from the overlap real positive overlap @xmath75 . this works out immediately to @xmath76 the generalization to su(4 ) is immediate : the form of eq.([eq : decomposeu ] ) remains the same , but the matrix of eq.([eq : rt ] ) is augmented to a @xmath77 matrix : @xmath78 the condition of eq.([eq : geocurve ] ) remains . as we have argued , the first two vertices of the geodesic triangle remain unchanged , but the last vertex now depends on the @xmath79 parameters of su(4)/u(3 ) : @xmath80 a form which obviously reduces to the su(3 ) case if @xmath81 . for su(4)/u(3 ) , the berry phase is again related to the inner product of @xmath82 through @xmath83 and can be seen to depend on the required number of parameters . an optical su(n ) transformation can be realized by a n channel optical interferometer@xcite . the su(2)@xmath84 matrix @xmath85 in eq . ( [ eq : rt ] ) is a special case of the generalized lossless beam splitter transformation for mixing channels @xmath33 and @xmath31 . more generally a beam splitter can be described by a unitary transformation between two channels@xcite . for example , a general su(2)@xmath86 beam splitter transformation for mixing channels @xmath31 and @xmath32 is of the form @xmath87 with @xmath88 and @xmath89 the transmitted and reflected phase shift parameters , respectively , and @xmath90 the beam splitter transmission . a generalized beam splitter can be realized as a combination of phase shifters and 50/50 beam splitters in a mach zehnder interferometer configuration . the goal of the following is to construct an su(3 ) optical transformations in terms of su(2 ) elements which realize the geodesic evolution in the geometric space by appropriately adjusting parameters of the interferometer . it will be convenient to write @xmath91 in eq.([vertices ] ) as @xmath92 , where @xmath93 , @xmath94 , @xmath95 and @xmath96 are functions of @xmath72 , @xmath73 , @xmath8 and @xmath9 , the parameters of @xmath91 in eq . ( [ vertices ] ) . following our factorization scheme , the geodesic evolution operators @xmath66 , connecting @xmath61 to @xmath62 , can be expressed as @xmath97 with @xmath85 given by eq . ( [ eq : rt ] ) , the parameters @xmath52 ranging from @xmath98 , and @xmath99 . note that @xmath100 and , in fact , all the parameters of @xmath101 are fixed by the requirement that @xmath102 also note that , for each @xmath36 , @xmath53 is the identity in su(3 ) and @xmath103 as required . once it is observed that @xmath104 , it is trivial to verify that each evolution satisfies eq . ( [ eq : geocurve ] ) and is therefore geodesic . the geometric phase for the cyclic evolution @xmath105 is given explicitly by eq.([berryphase ] ) . this phase depends on four free parameters in the experimental scheme : @xmath72 , @xmath73 , @xmath8 and @xmath9 , which describe a general geodesic triangle in su(3)/u(2 ) . the interferometer configuration for realizing the necessary evolution about the geodesic triangle is depicted in fig . [ fig : config ] . this configuration consists of a sequence of su(2)@xmath2 transformations , and we use the shorthand notation @xmath106 to designate the three parameters associated with the generalized beam splitter . the three channel interferometer consists of a sequence of nine su(2)@xmath2 transformations . the field enters port @xmath107 , and the vacuum state enters ports @xmath108 and @xmath109 . = 6.0 cm for su(4 ) , the first evolution is the same , but the second will depend on more parameters as the dimensionality of su(4)/u(3 ) is larger than that of su(3)/u(2 ) . briefly , we have : @xmath110 where @xmath111 and where @xmath112 is an su(3 ) matrix of the form @xmath113 whose details are unimportant for our purposes . although su(3 ) and su(4 ) interferometry have been considered in some details , the methods employed here can be extended to su(n ) , or n channel , interferometry@xcite . the schemes discussed above employing such a device would produce and enable observation of the geometric phase shift for geodesic transformations of states invariant under subgroups of states in the dimensional coset space . this contribution is a summary of recent theoretical work on the possibility of measuring berry phases using optical elements . the scheme depends on the optical realization of su(n ) transformations in the optical domain ; this is possible because the lie algebra su(n ) can be realized in terms of boson creation and destruction operators which have immediate interpretation as photon field operators . there also exists the possibility of realizing sp(2n,@xmath114 ) transformation using optical elements @xcite : the lie algebra sp(2n,@xmath114 ) also has a realization in terms of boson operators . the setup to measure berry phase in an optical experiment is interesting because it provides a very practical realization of otherwise abstract ideas and allows one to do `` experimental differential geometry '' over su(n)/u(n-1 ) . this contribution has dealt exclusively with the optical realization of abelian berry phase : even if states are invariant under u(n ) , two states are equivalent if they differ by a u(1)phase . it is possible to enlarge the equivalence class to obtain the so called non - abelian berry phase @xcite , which has been studied in the context of degenerate states . it is possible to study the non - abelian version of the results presented here by using polarization : two states of different polarization are declared equivalent . the larger equivalence class comes about because a rotation of the polarization plane is an su(2 ) transformation . the experimental aspects of this remain , at the moment , unclear . an optical experiment to measure an su(2 ) phase would require optical devices which perform `` tunable '' polarization dependent transformation . the theoretical aspects of this questions are currently under investigation . this work has been supported by two macquarie university research grants and by an australian research council large grant . bcs appreciates valuable discussions with j. m. dawes and a. zeilinger , and hdg acknowledges the support of fonds f.c.a.r . of the qubec government . m. v. berry , proc . ( lond . ) * 392 * , 45 ( 1984 ) . b. simon , phys.rev.lett * 51 * , 2167 ( 1983 ) ; f. wilczek and a. shapere , _ geometric phases in physics _ , advanced series in mathematical physics - vol . * 5 * ( world scientific , singapore , 1989 ) . a. tomita and r. y. chiao , phys.rev.lett * 57 * , 937 ( 1986 ) ; r. y. chiao , a. antaramian , k. m. ganga , h. jiao , s. r. wilkinson , and h. nathel , phys.rev.lett * 60 * , 1214 ( 1988 ) ; d. suter , k. t. mueller , and a. pines , phys.rev.lett * 60 * , 1218 ( 1988 ) . g. kwiat and r. y. chiao , phys.rev.lett * 66 * , 588 ( 1991 ) . r. simon and n. mukunda , phys.rev.lett * 70 * , 880 ( 1993 ) . b. yurke , s. l. mccall , and j. r. klauder , phys.rev.a * 33 * , 4033 ( 1986 ) . r. a. campos , b. e. a. saleh and m. c. teich , phys.rev.a * 40 * , 1371 ( 1989 ) . d. j. rowe , b. c. sanders and h. de guise , j. math . phys . * 40 * , 3604 ( 1999 ) . m. reck , a. zeilinger , h. j. bernstein , and p. bertani , phys.rev.lett * 73 * , 58 ( 1994 ) ; b. c. sanders , h. de guise , d. j. rowe and a. mann , j. phys . a : math . gen . * 32 * , 7791 ( 1999 ) . t. h. chyba , l. j. wang , l. mandel , and r. simon , opt . * 13 * , 562 ( 1988 ) ; r. simon , h. j. kimble and e. c. g. sudarshan , phys.rev.lett * 61 * , 19 ( 1988 ) . arvind , k. s. mallesh , and n. mukunda , j. phys . a : math . gen . * 30 * , 2417 ( 1997 ) j. anandan , phys . a * 133 * , 171 ( 1988 ) , f. wilczek and a. zee , phys . lett . * 52 * , 2111 ( 1984 ) .
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an interferometric scheme to study abelian geometric phase shift over the manifold su(n)/su(n-1 ) is presented . #
1/#2.1em .5ex-.1em /-.15em.25ex # 1 # 1
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You are an expert at summarizing long articles. Proceed to summarize the following text:
conformal field theories in two dimensions can be used to describe string theory from the worldsheet perspective and statistical systems at a second order phase transition . the study of these field theories is often manageable because the infinite dimensional symmetry algebras which exist can reduce the field content to a finite number of representations . in such minimal cases , there are ` null vectors ' which should decouple from all correlation functions . as a result the non - zero correlation functions satisfy differential equations which enable one to solve the theory completely . each symmetry algebra has its own series of minimal models for which this procedure works . the simplest case is that of minimal models of the virasoro algebra . in @xcite , belavin et al showed how to relate correlation functions of arbitrary fields in a virasoro minimal model to those of a particular class of fields , called primary fields , and in turn showed how differential equations for four - point functions of primary fields could be found from the singular vectors that are present in these models . the virasoro algebra can be extended by extra generators to include supersymmetry , and the simplest of these extensions is the @xmath0 superconformal algebra . although the @xmath0 superconformal algebra includes the virasoro algebra , there is an infinite set of minimal models of the @xmath0 superconformal algebra which only includes a few minimal models of the virasoro algebra , and the methods of @xcite need to be generalised to find the differential equations satisfied by the correlation functions of primary fields of the @xmath0 superconformal algebra . the first thing to note is that the superconformal algebra comes in two forms , called the neveu - schwarz ( ns ) and ramond ( r ) algebras , and accordingly has two classes of representations and fields associated to these representations . the differential equations for correlations functions of four ns type fields were worked out in @xcite , but the extension to correlation functions of four ramond fields has problems @xcite and the solution to these problems presented here requires an extension of ideas in @xcite . in this paper we perform this extension and find differential equations for correlation functions of four r type fields . the structure of the paper is as follows : we first introduce the ns and r algebras and describe their representations and singular vectors . we discuss their three point functions and describe the chiral blocks and how the singular vectors allow them to be found level - by - level . next we describe the toy model of the ising model and show how one can obtain a differential equation for the four - spin field correlation function , as in @xcite . next we apply these ideas to the ramond correlation functions . this lets us write down a matrix - differential equation for the correlation functions . we check these by comparison with known results and present the exact solution for a variety of cases . the algebra of chiral superconformal transformations in the plane is generated by two fields , @xmath1 and @xmath2 . according to the choice of boundary conditions on @xmath3 , one may choose the labels @xmath4 on @xmath5 to be integer or half - integer ; the algebra in these two sectors are called the ramond and neveu schwarz algebras respectively . in both sectors the generators obey @xmath6 & = & \frac c{12 } m(m^2 - 1)\delta_{m+n,0 } + ( m - n)l_{m+n } \;,\\ \{g_m , g_n\ } & = & \frac c3 ( m^2 - \frac 14 ) \delta_{m+n,0 } + 2 l_{m+n } \;,\\ { } ~[l_m , g_n ] & = & ( \frac m2 - n)g_{m+n}\;.\end{aligned}\ ] ] it is usual to adjoin the operator @xmath7 to the superconformal algebra , where @xmath8 is the fermion number operator which anti - commutes with @xmath5 and commutes with @xmath9 . this allows one to consider states of definite parity and is essential in many constructions . in a superconformal field theory the physically relevant representations of the superconformal algebra are irreducible highest weight representations . these are graded by @xmath10 eigenvalue , or level . see @xcite for more details . highest weight representations of the neveu - schwarz algebra have a state of least @xmath10 eigenvalue @xmath11 such that @xmath12 if one considers the extended algebra then it is usual to take the highest weight state bosonic ( @xmath13 ) or fermionic ( @xmath14 ) in which case the addition of the fermion number operator does not alter the representation theory . it is useful to parametrise @xmath15 and @xmath16 as @xmath17 since there is a singular vector of @xmath10 eigenvalue @xmath18 whenever @xmath19 and @xmath20 and @xmath21 , @xmath22 ( see refs . note that the highest weight state can be either bosonic or fermionic , for example , the ns vacuum state @xmath23 is usually regarded as bosonic and so the ns highest weight state @xmath24 is fermionic . the highest weight representation @xmath25 is spanned by states of the form @xmath26 and the level of such as state is @xmath27 . states with even numbers of modes of @xmath28 have integer level and states with odd numbers of modes have half - integer level and hence @xmath29 where @xmath30 is spanned by integer level states and @xmath31 by half - integer level states . highest weight representations of the extended ramond algebra have , in general , a two - dimensional highest weight space spanned by states @xmath32 of definite fermion parity @xmath33 satisfying @xmath34 where @xmath35 and @xmath36 and @xmath37 can be chosen freely ; representations with different @xmath37 are equivalent . such a representation has two singular vectors at level @xmath38 whenever @xmath39 , @xmath40 an odd integer , @xmath20 and @xmath41 , @xmath42 we shall mostly be concerned with the constraints arising from the vanishing of the singular vectors at level 1 in the representation @xmath43 which take the form @xmath44 we would like to find the chiral blocks of four ramond fields @xmath45 where the intermediate states lie in a particular ns representation . we can do this step - by - step by first calculating the operator product expansions ( opes ) @xmath46 and then forming the chiral block by summing over intermediate states . to do this we need to define the vertex operators @xmath47 of fixed fermion parity and we also need to specify whether the ns representation in the intermediate channel is bosonic or fermionic . for the field operators , we shall use the definition in @xcite : @xmath48 where the combinations @xmath49 and @xmath50 are @xmath51 @xmath52 , @xmath53 and @xmath4 can be integral or half - integral as circumstances dictate ( here @xmath54 play the same role as @xmath55 in and @xmath56 as @xmath37 ) . it turns out that the four opes in depend on only two quantities . it is convenient to form the following four linear combinations : @xmath57 we shall first consider the case where the ns highest weight state is bosonic and the opes depend on the two constants @xmath58 . using and the standard result for the virasoro algebra ( see ) , we can calculate the overlaps of @xmath59 and @xmath60 . we list here the first few , taking @xmath61 for convenience : @xmath62 if we take the ns representation to be fermionic rather than bosonic , then @xmath63 and @xmath64 just swap roles in these equations , so without loss of generality we consider henceforth the intermediate channel to be bosonic . given these combinations @xmath65 , it is possible to define the chiral blocks and calculate them to be @xmath66 where ` even ' denotes projection of the intermediate states onto @xmath67 and ` odd ' , projection onto @xmath68 , where we again note that we have assumed that @xmath11 is bosonic . the vanishing of the state imposes constraints on the allowed fusions of the fields @xmath69 and in fact allows one to determine the operator products @xmath70 and @xmath71 recursively . we take @xmath72 as this simplifies the constraints on @xmath15 , and we shall denote @xmath73 by @xmath63 and @xmath74 by @xmath75 for simplicity . if we use the equation ( valid for any virasoro primary field of weight @xmath15 ) @xmath76 and the relations , we find that @xmath63 and @xmath75 satisfy @xmath77 where we assume the representation @xmath15 to be bosonic . taking @xmath15 fermionic simply swaps @xmath63 with @xmath75 . considering the contribution of the highest weight state @xmath11 to @xmath63 , we see that @xmath78 so that @xmath63 is only non - zero if @xmath79 . the two allowed fusions and the operator products are thus @xmath80 & & \hbox { $ -$ channel:}\qquad ( 1,2 ) \times ( 1,s ) \longrightarrow ( 1,s{-}1 ) \label{eq :- channel } \\ & { { \ensuremath{\cal f}}}^- & = \vec h + \tfrac{st{+}t-2}{st{-}2 } l_{-1}\vec h + \ldots \nonumber\\ & { { \ensuremath{\cal g}}}^- & = -\tfrac{\sqrt{2t}}{2t{-}2 } g_{-1/2}\vec h + \sqrt{\tfrac t2}\tfrac 1{st{-}4}\left ( g_{-3/2 } - \tfrac{2(st{+}t{-}4)}{st{-}2}l_{-1}g_{-1/2}\right)\vec h + \ldots \nonumber\end{aligned}\ ] ] using these expressions it is then possible in principle to calculate the chiral blocks order by order and turn the recursion relations for the operator product expansions into differential equations on the chiral blocks . rather than carry this out in detail we will instead turn to a method to derive differential equations directly . we would like to find differential equations for a chiral block of the form @xmath81 coming from the vanishing of , where @xmath82 , @xmath83 and @xmath84 are possibly matrix - valued ramond fields . as a first step , we consider the contribution coming from the mode @xmath85 . this is easy to calculate using the relations for a virasoro primary field of weight @xmath15 @xmath86 & = & z^{m+1}\phi'(z ) + z^m h(m+1)\phi(z ) \;. \label{eq.l.c4}\end{aligned}\ ] ] the only complication is the need to remove the term in @xmath87 that would normally arise . this can be done by considering @xmath88 to find @xmath89 f(z ) \;. \label{eq : l-1}\ ] ] the difficulty now is to treat @xmath90 . we shall adapt some ideas used in the construction of the co - invariant spaces that are used to classify the fusion algebra of virasoro and superconformal algebra representations . in this method , one finds linear combinations of modes which annihilate the state @xmath91 . the simplest way to find these is to find polynomials @xmath92 such that @xmath93 has at most a simple pole at @xmath94 , @xmath95 and @xmath96 . the modes of @xmath97 then take simple values when acting on @xmath91 . for example , for @xmath98 @xmath99 satisfies @xmath100 as a toy example , we shall first apply this idea to the ising model to reproduce the result of @xcite before applying it to the superconformal algebra and ramond fields . the ising model can be formulated as the theory of a single free fermion field @xmath101 . as with the superconformal algebra , this field can have half - integer ( ns ) or integer ( r ) modes @xmath102 satisfying @xmath103 the ramond algebra has a zero mode @xmath104 satisfying @xmath105 and so there are two inequivalent irreducible highest weight representations of the ramond algebra , with highest weights @xmath106 satisfying @xmath107 a unitary highest weight representation of the ramond algebra will have a highest weight space on which @xmath104 is represented by a matrix with eigenvalues @xmath108 . we shall denote the ( vector - valued ) highest weight of such a general representation by @xmath109 and the chiral fields corresponding to such a state by @xmath110 . the energy - momentum tensor can be written in terms of @xmath111 as @xmath112 so that @xmath113 this last equation can be viewed as a null - vector equation , the analogue of and we can use it to find a differential equation for the correlation functions of the form @xmath114 here , each @xmath115 is some vector - value representation as is the function @xmath8 . we will not actually need to specify the exact form of these representations , as we will see shortly . we know that the space of virasoro conformal blocks with the correct properties is two dimensional so we have to consider @xmath116 as a vector in some space of solutions and the differential equation we obtain will be a matrix differential equation . according to the idea outlined above , we want to find combinations of modes of the ramond algebra @xmath117 and @xmath118 so that @xmath119 we will require that these operators square to @xmath120 and mutually anti - commute , that is satisfy the algebra @xmath121 we start by defining operators @xmath122 as integrals , @xmath123 where the contour encloses the origin but not the points @xmath94 or @xmath95 and @xmath124 are functions which remove the unwanted singularities in @xmath125 from the state @xmath126 since the operator product of the field @xmath111 with a ramond field @xmath115 is of the form @xmath127 suitable combinations are @xmath128 it is easy to calculate the anti - commutators of these operators either as contour integrals or directly in terms of the modes to find @xmath129@xmath130 consequently we can define new combinations @xmath131 which satisfy as @xmath132 we can now use the combinations @xmath131 to replace the mode @xmath133 in the singular vector . there remains a large degree of choice in how to do this as we can replace @xmath133 by any of the @xmath131 as follows : @xmath134 without loss of generality , we will now just consider the correlation functions @xmath135 . the general expression for the null vector relation we can obtain in terms of @xmath131 using is of the form @xmath136 where @xmath137 and @xmath138 are functions of @xmath95 . acting on this equation on the left by @xmath139 leads to a matrix differential equations for the correlation functions of the form @xmath140 where @xmath141 are matrices representing the action of the zero modes on the fields at @xmath95 , 1 and @xmath96 satisfying the clifford algebra @xmath142 we must now choose a representation of this algebra . the smallest representations of this algebra are two dimensional and there are two inequivalent choices for which @xmath143 . it is essential for constructing the correct differential equation for the correlation functions of the spin field that the correct representation is chosen . to fix the equivalence class of the representation , we note that the operators @xmath131 satisfy further relations , for example @xmath144 either of these is sufficient to fix the class of the representation . since we are considering @xmath135 , we note that @xmath145 implies we must choose the representation of the clifford algebra for which @xmath146 , ie for which @xmath147 . returning to equation , the simplest choices are for two of the functions @xmath148 to be zero , the other non - zero , so that only a single matrix @xmath149 appears in the matrix differential equation . in this case we can take the matrix to be diagonal ( or alternatively consider the one - dimensional representations of the algebra @xmath150 ) and we obtain a set of first order differential equations for the correlation functions as follows which give the two components of the function @xmath151 as the chiral blocks associated to a particular channel . we illustrate this below . setting @xmath153 , we find @xmath154 and @xmath155 , so that the differential equation becomes @xmath156 at this stage the only requirement on @xmath157 is that it squares to @xmath94 , so we can consider one - dimensional subspaces of the space of correlation functions on which @xmath157 takes values @xmath33 , leading to the two solutions @xmath158 which are the well known chiral blocks of the virasoro algebra associated to the following choice of channel : @xmath159 the differential equation becomes @xmath161 taking @xmath162 diagonal with values @xmath33 leads to the two solutions @xmath163 which are the chiral blocks of the virasoro algebra associated to the following channel : @xmath164 the differential equation becomes @xmath165 taking @xmath166 diagonal with values @xmath33 leads to the two solutions @xmath167 which are the well - known chiral blocks of the virasoro algebra associated to the channel : @xmath168 taking inspiration from the case of the ising model , we will try to find differential equations for the chiral blocks @xmath169 the expectation is that we can express the singular vector in terms of combinations of the modes @xmath5 which ( inside the correlation function ) lead to a matrix representation of the algebra of the zero modes acting on the primary fields inserted at @xmath94 , @xmath95 and @xmath96 . that is , we will try to find combinations @xmath170 of the modes @xmath5 ( for @xmath171 ) which , when taken inside a four - point function lead to a matrix representation @xmath172 of the zero - mode algebra @xmath173 and which will lead to a matrix differential equation for in the form @xmath174 the first step in repeating the analysis of the ising model for the superconformal algebra is to identify analogues of the combinations @xmath131 . from , the operator product of @xmath3 with a ramond field @xmath175 takes the form @xmath176 where @xmath177 is a matrix representation of the zero mode satisfying @xmath178 , @xmath179 and @xmath180 . furthermore , @xmath181 consequently , we are motivated to consider the three combinations which remove all the singularities at two of the points @xmath182 and 1 , and turn the leading singularity at the remaining point into a simple pole : @xmath183 these operators , however , do not square to constants as in , nor do they simply anti - commute . their algebra is more complicated , and only simplifies inside the four - point functions . we can express their algebra in terms of suitable combinations of modes of the virasoro algebra : @xmath184 these combinations have the following properties when acting on the state @xmath185 @xmath186 @xmath187 in terms of these operators , the @xmath131 satisfy @xmath188 we thus define the combinations @xmath170 as @xmath189 inside the four - point function , the terms in @xmath190 and @xmath191 vanish and @xmath192 , so that the action of the operators @xmath170 inside is given by matrices @xmath172 satisfying the algebra . to use these , we have to express the singular vector in terms of the @xmath131 . if we act with any of the @xmath131 in the highest weight state @xmath193 , the leading term is a multiple of @xmath194 , so that it is not possible to express the singular vector in terms of just one of the @xmath131 . instead , it is necessary to use all three and one finds that @xmath195 combining with leads to the following matrix differential equation for , the main result of this article : @xmath196 & & ( z-1 ) f ' + \textstyle ( h_{12 } + h_1 + h_z - h_{\infty } ) f(z ) \nonumber \\[2 mm ] & + & \textstyle \sqrt{\frac t2}\left [ \frac{1+z}{2z } \lambda_{12 } - i \left(\frac{1-z}z\right)^{1/2 } \!\!\!\ ! \!\!g_1 - \frac{(1-z)^{1/2}}z g_z + z^{-1/2 } g_\infty \right ] f(z ) = 0 \;. \\[3 mm ] \end{array } } \label{eq : gde}\ ] ] we now turn to the analysis of this equation and its solutions . it will also be convenient to parametrise @xmath197 according to as @xmath198 the first thing we can do is to check the indices of the solutions around the points @xmath199 and @xmath200 , and then we can find the explicit solutions in various cases . finally we compare these to solutions known by other methods . these are the equations for the leading behaviour of the solution around a singular point . the equation has singular points @xmath201 and @xmath96 . we first consider the point @xmath200 around which the solution will have an expansion of the form @xmath202 where @xmath203 are vectors . note that the expansion will be in half - integer powers of @xmath95 as the differential equation explicitly contains @xmath204 . around @xmath205 , the leading behaviour is determined by substituting in and examining the coefficient of @xmath206 : @xmath207 ) a_0 = 0 \;. \label{eq : gde2 } \end{aligned}\ ] ] since this equation only involves @xmath208 , we can take @xmath208 to be diagonal with eigenvalues @xmath209 . this leads to two solutions for @xmath210 , @xmath211 \;.\ ] ] if we use the parametrisation @xmath212 , we find @xmath213 these values are exactly the expected exponents for the chiral blocks shown below : @xmath214 it is easy to check that similar results hold for the other two channels , corresponding to expanding the solution @xmath8 around @xmath96 in powers of @xmath215 and around @xmath94 in powers of @xmath216 . to solve the full equation we must choose a matrix representation for @xmath172 . up to now we have not had to specify the action of the zero modes on the primary fields , and as in the ising model , we do not need to do it now . there are only two inequivalent representations for which @xmath217 , and the choice of representation is invariant under monodromy around @xmath205 and @xmath61 . we shall take @xmath172 to be given in terms of the pauli matrices as @xmath218 where @xmath219 . this choice turns the equation for @xmath8 into a real matrix differential equation and taking @xmath208 diagonal also means that the two components of @xmath8 have expansions in @xmath95 ( and not @xmath204 ) . the two components of @xmath8 can be identified by comparison with and and are the odd and even chiral blocks ( up to a sign ) @xmath220 in some cases it is possible to find exact solutions to . the simplest case to consider is where all the @xmath197 are equal to @xmath221 . in this case @xmath222 and the fusion rules force @xmath223 . writing the components of @xmath8 as @xmath224 , with the representation , the differential equation becomes @xmath225 the coefficients in the equation as presented have branch cuts at @xmath205 and @xmath61 , and it is convenient to remove these by changing variables to @xmath226 with this change and a redefinition @xmath227 to remove the leading singularity in @xmath95 , the equations simplify dramatically to @xmath228 with solutions @xmath229 g^- = \left(1{-}u^4\right)^{\frac{3}{4}-\frac{3 t}{2 } } \pmatrix { \frac{2t{-}1}{t{-}2 } \cdot u^3 \cdot { f\ ! } \left(\frac{3}{2}{-}t,\frac{3}{2}{-}\frac{3t}{2};2{-}\frac{t}{2};u^4\right ) \cr { f\ ! } \left(\frac{1}{2}{-}t,\frac{3}{2}{-}\frac{3t}{2};1{-}\frac{t}{2};u^4\right ) } \end{aligned}\ ] ] where @xmath230 is the standard hypergeometric function . for special values of @xmath231 , the blocks with identity intermediate channel simplify further . for example at @xmath232 , that is for four - point blocks of the field with @xmath233 in the tri - critical ising model , @xmath234 simplifies to @xmath235 as can be seen , the two components are related by @xmath236 which corresponds to @xmath95 encircling the branch point at @xmath61 once . the blocks can also be found exactly in the more general case @xmath237 , that is @xmath238 in the parametrisation . for example , the solutions for @xmath8 with @xmath223 are @xmath239 f^- & = & u^{\frac{3}{4}{-}\frac{1}{2 } ( s{+}1 ) t } \left(1{-}u^2\right)^{\frac{1}{4 } \left(2 s{-}1{-}2 ( r{-}1 ) t{-}s^2 t\right ) } \left(1{+}u^2\right)^{\frac{1}{4 } ( 4{-}2 r t{+}s ( s t{-}2 ) ) } \\ & & \pmatrix { \frac{2rt{-}2}{st{-}4 } \cdot u^3 \cdot { f\ ! } \left(\frac{3{-}rt}{2},\frac{6{-}2rt{-}st}{2};\frac{8{-}st}{4};u^4\right ) \cr { f\ ! } \left(\frac{1{-}rt}{2},\frac{6{-}2rt{-}st}{2};\frac{4{-}st}{4};u^4\right ) } \end{aligned}\ ] ] we can check the differential equation and its solutions against solutions known by other methods . principally , there are three values of the central charge in the superconformal minimal series which also appear in list of virasoro minimal models , so that a - series of the superconformal minimal models can be identified with the following invariants of the virasoro minimal models , @xmath240 the representations in the superconformal models can be found as sums of representations in the virasoro minimal models , and the chiral blocks of the superconformal models must be sums of virasoro chiral blocks . power series expansions of the virasoro chiral blocks can be found easily , either by using one of the recursion relations of zamolodchikov @xcite or by solving the differential equation from the singular vector . we present two examples here to show how this works . in this model , the tricritical ising model , the ramond representations of the superconformal algebra are @xmath43 and @xmath242 with conformal dimensions @xmath243 and @xmath244 , and consequently all correlation functions of four ramond fields can be found using the method in this paper . if we consider just one case , the following two blocks can be found by series solution of the differential equation @xmath245 @xmath246 since the ramond representations of the unextended superconformal algebra and the even and odd sectors of the intermediate channel are each irreducible representations of the virasoro algebra , these blocks can also be found using the representation theory of the virasoro algebra . in this case , the irreducible virasoro representation of weight @xmath247 has virasoro kac labels @xmath248 and so the odd and even chiral blocks are two solutions of a fourth order differential equation . this differential equation can also be solved for a series solution and two of the solutions are exactly those given in and . this model is related to the @xmath250 invariant of the virasoro minimal model @xmath251 . the irreducible super virasoro representations of interest split into direct sums of irreducible virasoro representations as follows @xmath252 most of the superconformal chiral blocks are given by sums of virasoro chiral blocks , but in some cases there is only a single virasoro representation contributing to the intermediate channel and so the results of solving and the virasoro null vector equations can be compared directly . if we denote virasoro representations by @xmath253 then two such cases are @xmath254 @xmath255 this time the blocks and can be calculated either as series solutions of or as the series solutions of eighth order differential equations corresponding to the level eight null vector in the virasoro representation @xmath256 , giving the same answers shown . they can also be compared with the general series solution for virasoro chiral blocks found by zamolodchikov @xcite . we have found a matrix differential equation for ramond four - point chiral blocks . this can solved completely in some cases but as yet the full general solution is now known . these solutions were known already exactly in some other cases and as series solutions in a few more cases based on relations with virasoro minimal models . the differential equations presented here reproduce these known results . exact integral formulae based on free - field constructions are also known @xcite and it remains to check that these satisfy the equations we have found . recently , recursive formulae generalising zamolodchikov s elliptic recursion formulae in @xcite have been found @xcite and again it remains to check that these give series expansions which satisfy our equations . for the future , these equations and their solutions should enable one to extend calculations that have only fully been worked out in the virasoro case to the superconformal case , such as the construction of the full set of boundary structure constants in @xcite gmtw is very grateful to f. wagner and i. runkel for discussions on many aspects of fusion and superconformal field theory and to ir for detailed comments on the manuscript . this work was supported by pparc and stfc through rolling grants pp / c507145/1 and st / g000395/1 and by the studentship ppa / s / s/2005/4103 . 99 z. a. qiu , , nucl . b * 270 * ( 1986 ) 205234 . g. mussardo , g. sotkov and m. stanishkov , _ ramond sector of the supersymmetric minimal models _ , phys . b * 195 * , 397 ( 1987 ) . g. mussardo , g. sotkov and h. stanishkov , _ fine structure of the supersymmetric operator product expansion algebras , _ , nucl . b * 305 * , 69 ( 1988 ) . v. g. kac , , proc . int . congress math . 1978 , helsinki . b. l. feigin and d. b. fuchs , , repts . stockholm univ . ( 1986 ) , published in ` _ representations of infinite - dimensional lie groups and lie algebras _ ' , eds . a. vershik and d. zhelobenko , gordon and breach ( 1989 ) . a. b. zamolodchikov , _ conformal symmetry in two - dimensions : an explicit recurrence formula for the conformal partial wave amplitude , _ commun . * 96 * ( 1984 ) 419 . g mussardo , g m sotkov , and m s stanishkov , _ fine structure of the supersymmetric operator product expansion algebras , _ nucl . , * b305 * [ fs23 ] ( 1988 ) 69 .
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we consider chiral blocks of four ramond fields of the @xmath0 super virasoro algebra where one of the fields is in the ( 1,2 ) representation . we show how the null vector in the ( 1,2 ) representation determines the chiral blocks as series expansions .
we then turn to the ising model to find an algebraic method to determine differential equations for the blocks of four spin fields . extending these ideas to the super virasoro case
, we find a first order differential equation for blocks of four ramond fields .
we are able to find exact solutions in many cases .
we compare our blocks with results known from other methods .
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superfluids are peculiar states of matter in which , at the cost of losing part of their individuality , particles gain the ability of cooperative lossless motion @xcite . the occurrence of superfluidity can be attributed to the formation of certain non - local correlations within the system . for two - dimensional ( 2d ) systems , mermin and wagner showed that heisenberg models can be neither ferromagnetic or anti - ferromagnetic at finite temperatures @xcite . hohenberg further ruled out the existence of long - range ordering in 2d bose and fermi systems @xcite . these rigorous results imply that bose - einstein condensation ( bec ) does not exist at any finite temperature in uniform , interacting 2d systems in the thermodynamic limit , since spontaneous long - range ordering is prevented by long - wavelength fluctuations . hence one might expect that conventional superfluidity would not occur in 2d systems . nevertheless , a different path to superfluid behavior is possible in 2d systems . at low temperatures quasi - long - range correlations may form with an associated power - law decay that eventually reaches zero instead of extending throughout the system . it was theoretically shown by berezinskii @xcite and by kosterlitz and thouless @xcite ( bkt ) that in 2d a transition to a superfluid state may occur at finite temperatures . qualitatively the physics of such low - temperature 2d system is conveniently described using the notion of vortex - antivortex pairs ( vaps ) . at temperatures below the bkt transition , long - wavelength fluctuations destroy true long - range order and yield spontaneous creation and annihilation of bound vaps at the boundaries of the local domains of the resulting `` quasicondensate '' @xcite . these locally coherent blocks contribute to the power - law decay of the two - point correlation function resulting in a superfluid response of the system . on increasing the temperature , fluctuations increase and vaps unbind at the bkt transition . the breaking of vaps results in the proliferation of free vortices and an exponentially decaying correlation length , and hence the system loses its superfluid properties . this was further quantified by nelson and kosterlitz who predicted a universal jump in the superfluid density at the critical point , which may be used to empirically detect the bkt transition @xcite . in a trapped ultra - cold bose gas the situation is rather complicated due to the inhomogeneity arising from the confining potential . bagnato and kleppner showed that a trapped , _ ideal _ bose gas undergoes bec at finite temperatures @xcite , implying that in principle a trapped , _ interacting _ bose gas may exist in a coherent bec phase and/or in a bkt - type phase . the low temperature structure of a real 2d bose gas in a trap has therefore attracted a fair amount of theoretical discussion in the recent literature and it has been debated whether the superfluid transition in such systems is of bec - type or bkt - type @xcite . experimentally , the superfluid bkt transition in a bulk was first realized in liquid helium thin films by bishop and reppy @xcite . et al . _ reported an observation of the transition in superconducting josephson junction arrays @xcite , followed by safonov _ et al . _ who measured a kink in the three - body loss rate in spin - polarized hydrogen @xcite . these experiments relied on indirect methods of observation where as in the trapped atomic gases vaps and their dynamics can be directly imaged . the quasi-2d regime in trapped quantum degenerate gases has been approached experimentally by using tight axial confinement with the aid of optical potentials and through centrifugal expansion in rapidly rotated condensates @xcite . however , probing the details of the quasicondensation transition has only recently become experimentally accessible @xcite . indeed , dalibard and co - workers observed phase defects in the interference patterns of multiple quasi-2d gases trapped in the valleys of an optical lattice @xcite . these phase defects were evidently caused by unbound free vortices @xcite . further observations on spatial phase correlations in the system provided evidence for the cross - over between the bkt quasicondensate and normal state @xcite . while the observed correlations were shown to be consistent with the bkt quasicondensation picture , the question whether the system is superfluid or not still remained . recently , an observation of the bkt cross - over has been achieved in a 2d lattice of josephson - coupled becs @xcite and in a single 2d dipole trap @xcite . the interplay between interactions and inhomogeneous effects arising from the trapping potential have made theoretical predictions for the low temperature phases of dilute , atomic bose gas difficult . the main point we address in this paper is the question of superfluidity in this system at low temperatures . there is not a single observable that categorically defines superfluidity and we present results of microscopic calculations for a variety of observables , including off - diagonal long - range order ( condensate ) , fluctuations , scissor mode dynamics and presence of vortices . from the combined analysis of these quantities we are able to infer a cross - over temperature , @xmath0 , below which the system exhibits superfluidity . the paper is organized as follows : in sec . ii we discuss various measurables useful in examining the superfluid properties of our system . our computational approach is explained in sec . iii , and the results are presented in sec . iv followed by the concluding remarks in sec . v. a substance which has the ability to flow without dissipation is superfluid . although the difference between a superfluid and a classical fluid may seem intuitively clear , it is difficult , if not impossible , to find a single universal definition for superfluidity against which any material could be tested . indeed , the complete description of superfluidity is not a single feature but a complex of phenomena @xcite . in an interacting 3d atomic gas , the formation of bec is essentially equivalent to the emergence of a macroscopic wave function , which inherently exhibits long - range order throughout the system . furthermore , the system attains finite superfluid fraction at the bec transition . the situation is more subtle in 2d where the condensation process is plagued by long - wavelength phase fluctuations . in the following , we introduce measurables relevant for providing evidence of superfluidity in a quasi - two - dimensional sample of trapped ultra - cold atoms and apply these definitions to discuss the superfluidity of quasicondensates . quantized vortices are the hallmark of superfluids . the flow @xmath1 of a superfluid described by a macroscopic wave function with a phase @xmath2 obeys the condition of irrotationality , @xmath3 , and the onsager - feynman quantization of circulation , @xmath4 where @xmath5 is an integer and @xmath6 is the mass of an atom . therefore rotation ( @xmath7 ) is only possible around the phase singularity at the core of a quantized vortex where the superfluid density vanishes . if normal fluid is present , it occupies the volume in the vortex core . the quantized vortices can therefore be seen to play a two - fold role in superfluid systems . on the one hand their coherent role is vital in enabling superfluids to rotate , while on the other hand they can be viewed as ( topological ) defects that are a source of incoherence causing a reduction in the superfluid fraction . it is worth noting that the observation of a vortex alone is not a sufficient criterion from which superfluidity may be deduced . for instance , while a persistent vortex in a zero temperature condensate is readily accepted as _ proof of superfluidity _ , in a 2d system near bkt cross - over an observation of a transient isolated vortex would be more likely to signify unbinding of vaps and hence _ loss of superfluidity_. therefore , special care must be taken in the treatment of a situation where both spontaneously ( thermally activated ) and actively ( by external rotation ) created vortices may exist simultaneously . a particularly vivid example of a classical vortex which has many of the characteristics of a superfluid vortex has been realized in a recent experiment in which the @xmath8 vortex phase winding was imprinted into a non - superfluid cloud of atoms and was observed to persist for extended times due to the cancellation of spins in the diffusion process @xcite . consider a system described by the usual one - body reduced density matrix @xmath9 where @xmath10 is the second quantized bosonic field operator and the brackets denote quantum mechanical ensemble averaging . if @xmath11 has a macroscopic eigenvalue @xmath12 where @xmath13 is the number of particles in the system , it is said to possess bose - einstein condensation in the state determined by the corresponding eigenvector , also known as the macroscopic condensate wave function . the concept of long - range order is often encountered in the context of superfluidity . in a homogeneous bose - einstein condensed system , true long - range order exists in the sense that @xmath14 where as in the normal state the off - diagonal correlations decay exponentially with the spatial separation @xmath15 where @xmath16 characterizes the length scale over which the correlations decay . although the existence of bose - einstein condensation does not _ a priori _ imply the system to be superfluid , the existence of off - diagonal order in the system can be considered as a prerequisite for superfluidity . in the case of finite systems , such as trapped atomic gases , eq . ( [ eq3 ] ) generalises by understanding that the boundary of the system is mapped to infinity . two - dimensional systems lacking true long - range order may attain superfluidity through the berezinskii - kosterlitz - thouless mechanism . in this case the correlations decay algebraically with distance @xmath17 and are characterised by the exponent , @xmath18 , where @xmath19 is the 2d superfluid density , and @xmath20 is the thermal de broglie wavelength . for temperatures below the bkt transition temperature ( @xmath21 ) the first order approximation to the critical exponent is @xmath22 , corresponding to the universal jump at @xmath23 in the superfluid density @xcite . further insight into the state of the system is obtained by studying the higher - order coherence properties of the system @xcite . the second - order coherence function @xmath24 yields information about the local coherence of the bose field , @xmath25 , and in general for @xmath26th order coherence @xmath27 for @xmath28 , while for the corresponding thermal state @xmath29 . by decomposing the field into ` coherent ' ( @xmath30 ) and ` incoherent ' ( @xmath31 ) parts , @xmath32 , eq . ( [ eq6 ] ) becomes @xmath33 where @xmath34 . in a similar fashion @xmath35 which essentially measures the probability of three - particle coincidences . the total three - body recombination rate @xmath36 where @xmath37 is the rate constant and @xmath38 is the total density , is an experimentally measurable quantity yielding overall information of the third order coherence properties of the system . while it has been used experimentally to infer the quasicondensation transition point in helium thin films @xcite , obtaining similar information in trapped 2d gases is more problematic since the local gas density increases as the transition is crossed from normal to quasicondensed state and this partially compensates for the corresponding decrease in @xmath39 . the moment of inertia , @xmath40 , of a superfluid about a chosen axis is reduced from its classical value , @xmath41 , due to the irrotational motion of superfluid matter . the temperature dependent superfluid fraction @xmath42 where @xmath43 is the total particle number , is a quantity of special interest . a finite value of this macroscopic measurable may be used as an evidence of superfluidity . microscopically , this information about superfluidity is encoded in the elementary excitation spectrum of the system . the collective scissors mode oscillation has been employed to prove that the occurrence of bec in 3d implies superfluidity @xcite . essentially , the scissors mode may be viewed as an oscillation of an ellipsoidal cloud of atoms about its semi - axis in the plane . in the collisionless regime , a gas in a normal state has two prominent undamped scissors mode eigenfrequencies @xmath44 where @xmath45 and @xmath46 are the planar trapping frequencies . in eq . ( [ eq11 ] ) @xmath47 corresponds to an irrotational quadrupole oscillation and @xmath48 is related to a classical rotational motion . if superfluid component is present , it oscillates at an additional characteristic frequency @xmath49 whose existence thus provides a clear sign for superfluidity of the system in the collisionless regime . it is to be noted , however , that in the hydrodynamic limit @xmath48 becomes over - damped and both the remaining thermal mode , @xmath47 , and the superfluid scissors mode , @xmath50 , attain the same value . in such situation , the damping rate of this mode may in principle be used to reveal superfluid response , although this may prove to be difficult to achieve in practice . the scissors mode excitations of the system are directly related to the reduced moment of inertia @xcite @xmath51 where @xmath52 is the fourier transform of the time - dependent quadrupole moment @xmath53 . substitution of eq . ( [ eq13 ] ) in eq . ( [ eq10 ] ) , yields a formula for the superfluid fraction in terms of the scissors mode excitations . we emphasize that the presence of a superfluid scissors mode , @xmath50 , implies non - classical rotational inertia , @xmath54 , finite superfluid fraction , @xmath55 , and hence superfluidity . we employ the method of classical fields as detailed in refs . essentially , this amounts to propagating the projected gross - pitaevskii equation @xmath56 in time for the field , @xmath57 , restricted in the subspace determined by an energy cut - off in the harmonic oscillator basis states . the projector , @xmath58 , serves to constrain the evolution of the field within the subspace of highly occupied states . each simulation corresponds to the evolution of a single trajectory through the phase space and therefore , in order to construct thermodynamic observables , one should ensemble average over many different but equivalent trajectories . however , when considering an equilibrium quantities , we may assume the system to be ergodic and replace such ensemble averages by time - averages over the instantaneous field configurations taken from a single trajectory . the field , @xmath57 , is normalized to the number of particles , @xmath59 , described by the restricted basis . the total number of particles , @xmath60 , is obtained by using the semiclassical hartree - fock approximation for the @xmath61 above cut - off particles as in refs . @xcite and as described below . the in - plane phase function , @xmath62 , of the complex field , @xmath57 , allows for an explicit detection of the locations of vortices and antivortices . the classical field is completely described by the conserved total energy , @xmath63 , an energy cut - off for the restricted basis , @xmath64 , the dimensionless nonlinearity constant , @xmath65 , and the harmonic trap frequencies , @xmath66 and @xmath67 . here the spatial length scale is @xmath68 . from these simulations we can also compute the equilibrium temperature , @xmath69 , and chemical potential , @xmath70 , as described in refs . @xcite . an inherent feature in our numerical method requires that the total particle number , @xmath43 , must be computed _ a posteriori_. this is done within the self - consistent hartree - fock approximation by computing the particles not included in the simulated field , @xmath30 , from the semiclassical density . the form of the semiclassical integral reflects the quasi-2d nature of the trap . for the temperatures considered here , @xmath71 and therefore several of the lowest axial oscillator states , @xmath72 , contribute significantly to the total number of particles . however , the temperature is too low for the equipartition theorem to apply and therefore these axial levels need to be treated discretely in the semiclassical integral . the hartree - fock energy for this system is given by @xmath73 with @xmath74 the kinetic energy . the thermal densities are given by @xmath75 where @xmath76 and @xmath70 is the chemical potential . the interaction term in eq . ( [ hfegy ] ) contains the multilevel coupling constant @xmath77 which accounts for the interactions between the particles in different axial levels . we have denoted @xmath78 and @xmath79 are the normalized harmonic oscillator eigenstates . finally , the number of above cut - off particles @xmath80 is obtained by integrating over the 2d densities and summing the contributions from different axial energy levels . to obtain data points for a fixed number of particles @xmath43 for a range of temperatures , we estimate the cutoff energy @xmath64 , total number of particles @xmath59 and energy @xmath63 of the classical region of the system according to the prescription of ref . we then simulate this within the pgpe and calculate the temperature and total number of thermal particles using the procedure described above . we use this knowledge to make any necessary adjustment to the initial guesses to end up with a target @xmath43 . in principle we could construct an approximation to the full green s function from our time - dependent classical field simulations allowing us to extract the collective excitation frequencies and their damping rates for the system . it would be , however , a formidable task . furthermore , an identification of near - degenerate modes would become cumbersome . instead , we concentrate on a specific class of excitations the so - called scissors modes which may be selectively excited and consequently their oscillation frequency can be individually measured from our dynamical simulations . in order to accurately compute the scissors mode collective oscillation frequencies , an ensemble averaging over many equivalent trajectories is required . therefore we first prepare a large set of initial field configurations by time - sampling a single equilibrium simulation . these instantaneous field configurations are then rotated 11 degrees in the @xmath81 -plane with respect to the semi - axis of the anisotropic trapping potential . to facilitate lossless rotation of the state , we must first project the classical field into a larger eigenbasis in order to account for the increase in the energy after rotation due to the anisotropic trapping geometry . subsequently the seed states thus prepared are propagated in time and the quadrupole moment , @xmath82 , is measured . the above procedure is repeated for @xmath83 microstates for each temperature point . the scissors mode frequencies are then obtained from the fourier transformation of the ensemble averaged quadrupole signal @xmath84 , from which also the damping rates of those modes can be estimated . in order to allow practical comparison with future experiments , we choose experimentally realistic system parameters which allow us to study the scissors - mode collective oscillation frequencies . the strength of particle interactions is determined by a constant , @xmath85 , and the harmonic confining potential , @xmath86 , is characterized by the cartesian frequencies @xmath87 hz . the trap is chosen to be anisotropic in the @xmath81 -plane in order to lift the degeneracy between the quadrupole and scissors modes . however , the planar anisotropy is kept moderate , in contrast to ref . @xcite , in order to separate the frequency of the superfluid scissors - mode from both of the two classical modes . we consider @xmath88 @xmath89rb atoms interacting with the @xmath90-wave scattering length @xmath91 nm . several results we present are generated by performing an average along elliptical trajectories in the 2d plane about the trap centre . this is done in order to utilize the full information contained in the simulated fields . since our system does not possess cylindrical symmetry ( in order to facilitate computation of scissors modes ) , we perform this averaging by considering elliptical shells of constant ( 2d ) trap potential energy and average over all spatial points falling on such strips . in what follows , @xmath92 , denotes the distance from origin to such ellipse along the weakly trapped @xmath93-axis of the trap . in this section we present our numerical results and analysis for the superfluid indicators described above . taking the compendium of results described below , we claim that the quasicondensate studied here is superfluid . all measurables extracted from our simulations point to a cross - over temperature @xmath0 at which our system attains superfluidity . in order to lay down the qualitative features of the system we have plotted instantaneous 2d classical field densities , @xmath94 , in fig . [ f1](a ) and ( b ) , and the corresponding phases @xmath95 , fig . [ f1](c ) and ( d ) . the temperatures are @xmath96k for fig . [ f1](a ) and ( c ) and @xmath97k for fig . [ f1](b ) and ( d ) . at low temperatures the density and phase are relatively uniform , while at high temperatures both exhibit strong fluctuation and and vortices and antivortices are nucleated . this observation is in striking contrast to the usual situation in 3d and highlights the main qualitative difference between 2d and 3d systems . to further quantify the emergence of vortices and antivortices due to the phase fluctuations , we have measured at each temperature point the probablility , @xmath98 , of finding a vortex or antivortex at radius @xmath92 this is done by locating all phase singularities in an instantaneous classical field configuration and averaging over 1000 different microstates . the classical field area is then divided into ellipsoidal strips of equal width and the vortex occupation probability is obtained by counting the number of phase singularities detected within each strip divided by the number of microstates sampled ( this is the _ radial _ averaging discussed earlier ) . thus obtained probabilities for a set of temperatures , @xmath99 nk , are plotted as functions of radial distance , @xmath92 , in fig . the bullets denote the coherence length , measured as @xmath100 radius of @xmath101 ( defined in sec . c below ) and the red curve is for @xmath102k . there is a sudden jump in the vortex occupation probability at the cross - over . in the superfluid phase there is a vortex - free region at small radii . vortex pairs are observed in a narrow band near the edge of the coherent region of the system , i.e. the spatial region that has a flat phase in fig [ f1](c ) . thus the system can be divided in three concentric regions : central coherent and vortex free bec - like region , coherent bkt - like region where vortices are bound , and an incoherent thermal outmost region where vortices are free . we compute the one - body density matrix , eq . ( [ eq1 ] ) , for the classical field by assuming ergodicity , which allows us to replace the ensemble average by a time average . the number of condensed particles , @xmath103 , is obtained by computing the largest eigenvalue of the density matrix . figure [ f3 ] displays the condensate fraction as a function of temperature . the curve is plotted to provide comparison with the pure 2d ideal gas relation @xmath104 where @xmath105 is the critical temperature calculated for a quasi-2d ideal - gas in our trap geometry containing @xmath106 particles . the vertical line is the temperature @xmath107 nk , below which this system is superfluid and the markers are the simulation data . unlike in the recent experiment of krger _ et . @xcite we find the interactions to cause only a minor shift in the critical temperature from the non - interacting boson result . in terms of the definition based on the eigenvalues of the density matrix , the system may be claimed to show bose - einstein condensation at all temperatures below @xmath0 . nevertheless it turns out that the system is best described in terms of phase fluctuating quasicondensate apart from the very lowest temperatures . coherence is an essential feature of superfluidity . the density matrix , eq . ( [ eq1 ] ) , provides an useful probe for the global coherence between two spatially separated points in the system . in particular it conveys the knowledge of the correlation length and the information on the possible presence of long - range order . figure [ f4a ] shows the two - point function , @xmath108 for different temperatures . for low temperatures , @xmath109 , and small radii we witness power - law decay of @xmath101 in accordance with eq . ( [ eq5 ] ) , where as for temperatures , @xmath110 , and/or near the edge of the coherent region , exponential decay is observed . the qualitative behavior changes at the cross - over temperature , @xmath0 , denoted by the red line . the second - order coherence function , eq . ( [ eq6 ] ) , measures local coherence in the gas . particularly , for a purely thermal sample @xmath111 and for completely coherent state @xmath112 . in our inhomogeneous system @xmath113 interpolates between these two values as shown in fig . [ f4b ] where @xmath114 is displayed for different temperatures as functions of the radial distance from the trap centre . at the lowest temperatures , @xmath114 , shows a flat part in accordance with the presence of a nearly phase coherent bec . at large enough radii and for high enough temperatures @xmath114 approaches its thermal value . the plateau disappears near the temperature where the macroscopic wave function vanishes . the thicker red line indicates @xmath114 for a result with @xmath115 . in fig . [ f5 ] , we have plotted the coherence length as function of temperature . the values are obtained by measuring the @xmath100 width of the two - point correlation function . we have also plotted a function , @xmath116 , where , @xmath117 , ( solid curve ) for @xmath118 and @xmath109 . the value for the exponent , @xmath26 , may be crudely explained in terms of the thomas - fermi radius , @xmath119 of an isotropic 2d condensate , since @xmath120 varies linearly with the temperature in the vicinity of the cross - over point , @xmath0 . we have computed the scissors mode frequencies according to the description in the methods section . the obtained oscillation frequencies and their relative intensities are displayed in fig . the blue bullets are the mean oscillation frequencies obtained by fitting double gaussian functions to the normalised fourier spectrum at each temperature , which is indicated in gray in the background . the horizontal dashed red lines are the analytical predictions eqs . ( [ eq11 ] ) and ( [ eq12 ] ) . above the cross - over temperature , @xmath0 , indicated by the red vertical line , we obtain signal for two thermal modes whose frequencies are found to agree with the predictions of eq . ( [ eq11 ] ) . at temperatures well below the cross - over , only one scissors mode persists with a frequency corresponding to that of a superfluid , see eq . ( [ eq12 ] ) . this is the key feature verifying the quasicondensate to be superfluid . it is interesting to notice that we only observe two different scissors modes at all temperatures . the upper ` irrotational ' mode simply experiences frequency shift across the cross - over associated with the change in superfluid density of the system . this is to be contrasted with 3d systems where in general three different scissors modes exist and the superfluid scissors mode experiences downward ( as opposed to the upward shift seen in fig . [ f7 ] ) frequency - shift on increasing temperature across the cross - over @xcite . we have also computed the scissors modes for different systems of varying particle interaction strengths which could be experimentally realized using feshbach resonances . we have verified that in the strongly interacting systems only a single scissors mode survives at all temperatures , making it difficult to distinguish the superfluid and thermal response of the system from one another . in conclusion , we have studied the problem of superfluid - normal cross - over in a real , trapped quasi-2d bose gas . in such systems the formation of pure bose - einstein condensation is challenged by the long - wavelength phase fluctuations and this fact has made the characterization of such systems difficult both theoretically and experimentally . by performing classical field simulations for these systems , we have shown that such quasicondensates are superfluid below the cross - over temperature @xmath0 . this conclusion is based on observations of the global coherence properties and scissors mode excitations of the system , which constitute the two major results of this article proving superfluidity of quasicondensates . in particular , the emergence of condensate scissors - mode below the cross - over temperature provides an unequivocal microscopic evidence of non - classical rotational inertia and thus the superfluidity of the system . we have not found signs of fragmentation in terms of the eigenvalues of the density matrix below the cross - over temperature . a similar conclusion is obtained from the fact that only a single condensate peak is observed in momentum space . this indicates that despite of the prevailing phase fluctuations , the superfluid state of these systems resembles more closely that of a single - mode trapped bose - einstein condensate than the bulk berezinskii - kosterlitz - thoules superfluid phase . quasi-2d quantum gases are currently under active experimental investigation . while experiments have verified aspects of first - order coherence in this system consistent with the berezinskii - kosterlitz - thouless transition , there has yet to be any direct evidence of superfluidity . in this paper we have reported a series of tests that can be realized in an experiment and should cast light on the formation of superfluidity in these systems . this research was supported by the marsden fund of new zealand , the university of otago , and the new zealand foundation for research science and technology under the contract nerf - uoox0703 : quantum technologies . mjd acknowledges the financial support of the australian research council and the queensland state government . 90 a. j. leggett , _ quantum liquids _ , ( oxford university press , oxford , 2006 ) n. d mermin , h. wagner , phys . * 17 * , 1133 ( 1966 ) . p. c. hohenberg , phys . 158 * , 383 ( 1967 ) . v. l. berezinskii , sov . jetp * 32 * , 493 ( 1971 ) ; 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we investigate the harmonically trapped interacting bose gas in a quasi-2d geometry using the classical field method .
the system exhibits quasi - long - range order and non - classical rotational inertia at temperatures below the berezinskii - kosterlitz - thouless cross - over to the superfluid state .
in particular , we compute the scissors - mode oscillation frequencies and find that the irrotational mode changes its frequency as the temperature is sweeped across the cross - over thus providing microscopic evidence for the emergence of superfluidity .
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the discovery of giant magnetoteresistance in 1998 by the groups of fert and grnberg led to new reading heads for hard disks @xcite . moreover for the first time , a device based on magnetic phenomena replaced a conventional electronics device based on the movement of the electrons charge and thus opened the way to the field of spintronics or magnetoelectronics . the aim is to replace conventional electronics with new devices where magnetism plays a central role leading to smaller energy consumption . several architectures have been proposed @xcite but only in 2009 dash and collaborators managed to inject spin - polarized current from a metallic electrode into si , which is a key issue in current research in this field . showing that spintronic devices can be incorporated into conventional electronics @xcite . in order to maximize the efficiency of spintronic devices , the injected current should have as high spin - polarization as possible @xcite . to this respect half - metallic compounds have attracted a lot of interest ( for a review see reference @xcite ) . these alloys are ferromagnets where the majority spin channel is metallic while the minority - spin band structure is that of a semiconductor leading to 100%spin - polarization of the electrons at the fermi level and thus to possibly 100% spin - polarized current into a semiconductor when half metals are employed as the metallic electrode . the term half - metal was initially used by de groot et al in the case of the nimnsb heusler alloy @xcite . ab - initio ( also known as first - principles ) calculations have been widely used to explain the properties of these alloys and to predict new half - metallic compounds . an interesting case is the transition - metal pnictides like cras and mnas . akinaga and collaborators found in 2000 that when a cras thin film is grown on top of a zinc - blende semiconductor like gaas , the metallic film adopts the lattice of the substrate and it crystallizes in a meta - stable half - metallic zinc - blende phase @xcite structure . later cras was successfully synthesized in the zinc - blence structure in the form of multilayers with gaas @xcite and other successful experiments include the growth of zinc - blende mnas in the form of dots @xcite and crsb in the form of films @xcite . experiments agree with predictions of ab - initio calculations performed by several groups @xcite . in the case of the half - metallic ferromagnets like cras or crse , the gap in the minority - spin band arises from the hybridization between the @xmath0-states of the @xmath1 atom and the triple - degenerated @xmath2 states of the transition - metal and as a result the total spin - moment , @xmath3 , follows the slater - pauling ( sp ) behavior being equal in @xmath4 to @xmath5 where @xmath6 the total number of valence electrons in the unit cell @xcite . recently theoretical works have appeared attacking also some crucial aspects of these alloys like the exchange bias in ferro-/antiferromagnetic interfaces @xcite , the stability of the zinc - blende structure @xcite , the dynamical correlations @xcite , the interfaces with semiconductors @xcite , the exchange interaction @xcite , the emergence of half - metallic ferrimagnetism @xcite and the temperature effects @xcite . an extended overview on the properties of these alloys can be found in reference @xcite . of the lattice constant . note that in the case of the cras / cdse we have two non - equivalent interfaces : ( i ) when the sequence of the atoms is ... -cr - as - cd- ... denoted as cras / cdse-1 and ( ii ) when the sequence is ... -cr - se - cd- ... denoted as cras / cdse-2 . finally we should note that we have assumed the lattice constant of the two semiconductors ( 0.606 nm ) . [ fig1 ] ] [ cols="<,^,^,^,^,^ " , ] [ table6 ] finally , in the last section we will present our results concerning the case of in , for the cras / inas interface , and cd , for both cras / cdse interfaces , impurities at various sites . all three interfaces show similar behavior and thus in figure [ fig9 ] we present the dos for all possible in impurities for the cras / inas multilayer . we should note that with respect to the conservation of the half - metallicity this is the most interesting case since for the other two cras / cdse interfaces the half - metallic character is conserved for all cases under study . in table [ table6 ] we have gathered the atom - resolved spin moments for all cases under study and as it can be easily deduced from the table the variation of the spin moments for the same position of the in(cd ) impurity is similar for all three interfaces and thus we will restrict our discussion to the cras / inas case . . [ fig9 ] ] we expect that the most frequent case to occur would be the in impurity at the cr site since such an impurity does not disrupt the zinc - blende structure . in atoms have only two valence electrons occupying the deep - energy - lying @xmath7-states and thus for the energy window which we examine the @xmath0-states , which we observe , have their origin at the nearest as neighbors whose @xmath0-states penetrate in the in sites ( cd has only one valence @xmath7-electron ) . thus the in impurity acts similarly to a void , although it does not lead to such large reorganization of the charge of the neighboring atoms , leading to slightly larger spin moment of the neighboring atoms with respect to the perfect interfaces as shown in table [ table6 ] . due to the small weight of the in @xmath0-states we have multiplied the corresponding dos with a factor 5 or 10 in figure [ fig9 ] to make it visible . with respect to the case of void impurity at the cr site , here the shift of the bands of the nearest - neighboring as atoms is smaller keeping the half - metallic character of the interface although the gap is considerably shrinking . when the in impurity is located at the void1 site , the disturbance of the lattice is smaller with respect to the case just presented , although both cr and void1 sites have the same nearest - neighbors and as shown in figure [ fig9 ] the width of the gap remains unchanged . due to the negligible weight of the in @xmath0 states also the occurrence of in impurities at void2 and void3 sites leads to a slight variation of the spin moments but the half - metallicity is preserved and the gap retains a large width . to conclude we should discuss also the case of in impurities at as sites . as atoms at the interface have two cr atoms as nearest neighbors and the hybridization between the as @xmath0- and cr @xmath2-orbitals is strong . the substitution of an as atom by an in one leads to reduced hybridization for the cr orbitals and cr atoms at the interface regain the charge participating at the bonds with the missing as atom . this extra charge is accommodated at the cr spin - up states leading to larger spin magnetic moments of the cr atoms at the interface which now are about 3.59 @xmath4 instead of 3.25 @xmath4 in the case of the perfect cras / inas interface presented in table [ table1 ] . the in impurity atom and its nearest - neighboring in atoms have states within the gap , as shown in figure [ fig9 ] but if we take into account that we have multiplied the in dos by 10 their real weight at the fermi level is negligible with respect to the cr majority - spin dos . thus we can safely consider that the half - metallicity is conserved although as shown by the cr dos , the gap in the minority - spin band seriously shrinks and the fermi level is near the right edge of the gap . we have studied using the korringa - kohn - rostoker method the appearance of single impurities at interfaces between the half - metallic ferromagnet cras and the binary inas and cdse semiconductors . in the case of bulk cras studied in reference @xcite we had shown that most impurities affect the half - metallic character of cras inducing states either at the edges of the gap or in the middle of the gap . but multilayers are very thin as experiments show @xcite and thus we can not use the impurity calculations for the bulk to discuss the case of interfaces . we have studied al possible single impurities at the interfaces . contrary to the bulk systems almost all defects were not affecting the half - metallic character of the perfect interfaces . the only exception were void impurities at cr or in(cd ) sites . the missing cr or in(cd ) atom leads to a reorganization of the charge of the surrounding atoms and as a result the @xmath0 bands of the nearest neighboring as(se ) atom shift to higher energies crossing the fermi level and leading to loss of the half - metallicity . this is the opposite behavior that the one exhibited by the void impurities at cr site in the bulk cras alloy where the dos remained almost unchanged @xcite . the different behavior of the void impurities should be attributed to the lower dimensionality of the interfaces with respect to the bulk . but void impurities are schottky - type and we expect them to exhibit high formation energies and thus their number should be small in the experimental systems . thus contrary to the bulk cras and eventually thick films showing a bulk - like behavior , impurities are expected to affect only marginally the half - metallic character of the interfaces in the case of thin multilayers and the latter ones are promising for spintronic devices . zhao and a. zunger , phys . rev . b 71 ( 2005 ) 132403 ; m.s . miao and w.r.l . lambrecht , phys . b 72 ( 2005 ) 064409 ; l.j . shi b.g . liu , j. phys . : condens . matter 17 ( 2005 ) 1209 ; b. g. liu , phys . b 67 ( 2003 ) 172411 . c.y . fong , m.c . qian , j.e . pask , l.h . yang , s. dag , appl . 84 ( 2004 ) 239 ; m. moradi and z. soltani , j. appl . phys . 105 ( 2009 ) 023701 ; f. ahmadian , m.r . abolhassani , m. ghoranneviss , m. elahi , physica b : condens . matter 404 ( 2009 ) 3684 . e. aioglu , i. galanakis , l.m . sandratskii , p. bruno , j. phys . : condens . matter 17 ( 2005 ) 3915 . i. galanakis , k. zdogan , e. aioglu , b. akta , phys . b 74 ( 2006 ) 140408(r ) ; k. zdogan , i. galanakis , b. akta , e. aioglu , j. magn . 320 ( 2008 ) 197 . m. leai , ph . mavropoulos , j. enkovaara , g. bihlmayer , s. blgel , phys . 97 ( 2006 ) 026404 ; l. chioncel , m.i . katsnelson , g.a . de wijs , r.a . de groot , a.i . lichtenstein , phys . b 71 ( 2005 ) 085111 ; r. skomski and p. a. dowben , europhys . lett . 58 ( 2002 ) 544 .
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we present an extended study of single impurity atoms at the interface between the half - metallic ferromagnetic zinc - blende cras compound and the zinc - blende binary inas and cdse semiconductors in the form of very thin multilayers .
contrary to the case of impurities in the perfect bulk cras studied in [ i. galanakis and s.g .
pouliasis , j. magn .
magn . mat . 321 ( 2009 )
1084 ] defects at the interfaces do not alter in general the half - metallic character of the perfect systems .
the only exception are void impurities at cr or in(cd ) sites which lead , due to the lower - dimensionality of the interfaces with respect to the bulk cras , to a shift of the @xmath0 bands of the nearest neighboring as(se ) atom to higher energies and thus to the loss of the half - metallicity .
but void impurities are schottky - type and should exhibit high formation energies and thus we expect the interfaces in the case of thin multilayers to exhibit a robust half - metallic character . electronic structure , half - metals , cras 75.47.np , 75.50.cc , 75.30.et
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You are an expert at summarizing long articles. Proceed to summarize the following text:
there is currently interest in exploiting electron spin for nano - spintronic@xcite and quantum information processing@xcite . this is partly motivated by electron spin long coherence times @xcite and availability of scalable semiconductor technology . in the simplest approach , a physical qubit is identified with the two states of an electron spin , which can be manipulated by applying local magnetic fields . much progress has been achieved using micro - magnet@xcite technology with electron spin qubits . an alternative approach is to encode a logical qubit in a two level system constructed with spin complexes@xcite . this includes singlet - triplet two electron qubit@xcite and a logical qubit encoded in a degenerate ground state of a three spin complex@xcite . as suggested by divincenzo et al.@xcite , in the framework of a heisenberg model for a logical qubit encoded in a linear chain of three spins , such a logical qubit can be manipulated by the control of the exchange interaction between pairs of spins . the ability to manipulate spin state with a voltage is related to the relation between the orbital and spin part of the many - electron wavefunction . a detailed microscopic model of a three electron complex in a single semiconductor quantum dot ( qd ) , exploring the orbital and spin relation , has been investigated and compared with experiment by one of us@xcite . a similar microscopic model of three electrons localized in a lateral gated triangular triple quantum dot molecule in the plane of a gaas / gaalas heterojunction was proposed and explored by one of us in ref . . the coded qubit was identified with chirality of three electron spin complex , or equivalently , two possible directions of a minority spin motion . other proposals to implement coded qubits with both triple quantum dots@xcite ( tqds ) and atom traps@xcitealso exist . the advantages of working with a tqd - based coded qubit are two - fold . first , every quantum gate can be implemented electrically . in such a scheme , magnetic field will only be used for initialization and ( or ) measurement of a coded qubit . second , the coded qubit involves decoherence free subspace@xcite ( dfs ) and is immune to channels of collective decoherence . this reduces decoherence of a tqd - based coded qubit due to charge fluctuations as discussed in ref . . other interesting phenomena involving tqd molecules include non - fermi liquid behavior when coupled to leads@xcite , potential for generating maximally entangled three - partite ghz and w states@xciteand manipulation of total spin@xcite . in this work we develop a microscopic theory of quantum circuits based on coded qubits encoded in chirality of electron spin complexes in tqd . we use a combination of linear combination of harmonic orbitals - configuration interactions ( lcho - ci)@xcite , hubbard and heisenberg models to determine a set of optimal conditions for single qubit operations and describe the two qubit gate . we show that there exists an in - plane magnetic field direction and magnitude optimal for single qubit operations . however , the magnetic field will rotate qubit states in an undesirable way ; and we show how this rotation can be controlled by tuning voltages on the gates . once the optimal sets of magnetic fields and voltages have been obtained , the exact diagonalization techniques are used to establish and verify an effective two - qubit hamiltonian . it is shown that the effective two qubit interaction is ising - like , leading to a two - qubit phase gate . the present work establishes both single qubit and two - qubit operations necessary for performing quantum computation using tqd - based coded qubits . the paper is organized as follows . [ sec : intro ] contains introduction [ sec : sys ] describes the quantum circuit based on chirality of 3 electron spin complexes and computational methodology . in sec . [ sec : cqbt ] , we present a definition of a tqd - based coded qubit , discuss initialization , single qubit operations using voltages , and measurement of the state of the coded qubit . in sec . [ sec : dqop ] , we present an effective hamiltonian of 2 coupled coded qubits and show that it can be canonically transformed to an ising interaction . [ sec : con ] contains summary . fig.([fig : layout]a ) shows schematically layout of quantum circuits based on coded qubits encoded in chirality of electron spin complexes in triangular tqds . the circles denote individual lateral quantum dots formed in the 2d electron gas ( 2deg ) at the heterojunctions of algaas / gaas by metallic gates on the algaas surface . the gates are set to confine a single electron in each dot denoted by an arrow . additional gates ( not shown here ) are used to control tunneling between dots in the same tqd molecule , shown schematically as solid lines . dashed lines indicate tunneling between neighboring tqds . the tunneling between any pair of qds is responsible for exchange interaction of electron spins localized on each qd . the brackets indicate two tqds isolated from the rest of the circuit . it is assumed that any number of tqds can be isolated from the rest of the circuit by turning off the boundary exchange interactions with gate voltages . for comparison , fig.([fig : layout]b ) shows the circuit composed of linear , instead of triangular , tqds , similar to the linear chain of spins proposed by divincenzo et al . , in which the coded qubits in the chain are implemented with only two exchange interactions . fig.([fig : layout]c ) is another possible architecture studied by weinstein@xcite et . al . for spin system however , in this design , bringing tqds close together induces interactions between quantum dots beyond the ones indicated by dashed line . in this work we focus mainly on triangular tqds - based quantum circuit shown in fig.([fig : layout]a ) . since electrons are well localized in each qd , a system of two tqds in a chain can be very well described by an extended hubbard model@xcite . with @xmath0 electron creation ( annihilation ) operator for the electron with spin @xmath1 on the @xmath2 qd , the hubbard hamiltonian reads : @xmath3 where @xmath4 labels the two tqd molecules from left to right , indices @xmath5 range from @xmath6 to @xmath7 label each qd from left to right in the @xmath8 tqd molecule . the intra - tqd hubbard parameters @xmath9 , @xmath10 , @xmath11 and @xmath12 are the tunneling matrix element between the @xmath2 and the @xmath13 qds , the off - site coulomb interaction parameter between any two qds in the same tqd molecule , the on - site coulomb interaction strength for any qd , and the on - site energy for the @xmath2 qd respectively . the on - site energy depends on spin and magnetic field which is applied in the plane of a tqd molecule . the hubbard parameters @xmath14 and @xmath15 represent the inter - molecular tunneling matrix element and inter - molecular coulomb interactions . @xmath16 is the number operator for the @xmath17 electron on the @xmath2 qd . @xmath18 is the electron charge density operator on the @xmath2 qd . when we discuss triangular resonant tqds , we drop the subscripts on all the parameters . in our model , we consider the parameters corresponding to regime of strong correlations : @xmath19 . the intra - tqd hubbard parameters , @xmath20 , @xmath21 , @xmath10 , and @xmath11 , in eq.([eq : nhubbard ] ) are obtained from a microscopic calculation for single tqd based on lcho - ci method as explained in ref . . the inter - tqd hubbard parameter @xmath15 is taken to be the direct coulomb interaction between 2 charges localized on adjacent edge dots of two neighboring tqd molecules . the energy spectrum and eigenstates of the hubbard hamiltonian for one and two tqd molecules are obtained using configuration interaction technique . for a given number of electons @xmath22 we construct all possible configurations @xmath23 , build hamiltonian matrix in the space of configurations , and diagonalize it numerically . at half - filling , the low - energy spectrum of the hubbard model can be approximated by a spectrum of a heisenberg model@xcite describing electron spins localized in each dot : @xmath24 the exchange interactions , @xmath25 , for the tqds can be expressed in terms of tunneling matrix elements and quantum dot energies : @xmath26 the exchange interaction can be controlled by either tuning the tunneling matrix element @xmath9 by , for example , additional gates controlling the height of the tunneling barrier , or by biasing the dots and changing their on - site energy @xmath27 . in this section , we discuss a single coded qubit shown in fig.([fig : bfield]a ) : its preparation , initialization , operation and measurement in the presence of a lateral magnetic field in the @xmath28 direction . the qubit is encoded in quantum states of a three electron spin complex in a fully symmetric and half - filled tqd . an example of one of the three possible configurations @xmath29 with @xmath30 is shown schematically in fig.([fig : bfield]a ) . the three configurations with the minority spin on qd 1 , 2 or 3 form a doubly degenerate ground state with total spin @xmath31 and an excited state with total spin @xmath32 separated from the ground state by @xmath33 , as shown in fig.([fig : bfield]b ) . the doubly degenerate ground state with fixed @xmath30 forms an effective two level system . we identify the coded qubit levels @xmath34 and @xmath35 with @xmath36 where @xmath37 and @xmath38 . the three spin complex is characterized by chirality @xmath39 which measures the degree of collinearity of the three spins . the two coded qubit levels , eq.([eq : chiralstates ] ) , are the eigenstates of chirality operator with eigenvalues @xmath40 . these two states also can be more intuitively characterized by minority spin moving either to the left or to the right as in resonant valence bond ( rvb ) plaquette @xcite . we note that in the absence of the magnetic field the coded qubit states in @xmath41 subspace are degenerate with coded qubit states in @xmath30 subspace . if we are to work in the computational space corresponding to @xmath30 , any process which flips the electron spin will remove the coded qubit from its computational space . in order to separate the computational hilbert space @xmath30 from @xmath42 subspace , a magnetic field @xmath43 is applied along the @xmath28 direction as shown in fig([fig : bfield]a ) . we avoid applying magnetic field applied in the @xmath44 direction , because it activates undesirable , higher order spin - spin interactions@xcite , whereas the magnetic field applied in the plane only modifies the on - site energies of qds : for @xmath43 field , qds 1 and 3 energy level as @xmath45 , where @xmath46 is the spatial separation of qds 1 and 3 and @xmath47 is the cyclotron frequency , while the qd 2 energy level only acquires the zeeman term . hence magnetic field in the @xmath28 direction effectively lowers the energy of the qd 2 by @xmath48 with respect to qds 1 and 3 . fig.([fig : bfield]b ) shows the energy spectrum of the coded qubit as a function of @xmath49 obtained in lcho - ci method . in our lcho - ci calculation , we use the following parameters for a specific resonant tqd . the inter - dot distance is 10 @xmath50 , where @xmath51 is the effective bohr radius for @xmath52 . the confining gaussian potential on the @xmath2 qd is of the form , @xmath53 , where @xmath54 and @xmath55 . @xmath56 is the effective rydberg for @xmath52 . in fig.([fig : bfield]b ) , the optimal cyclotron frequency @xmath57 corresponds to @xmath58 . at @xmath59 , the four - fold degenerate ground state is separated from the four - fold degenerate spin polarized excited state by @xmath33 . the two computational hilbert spaces corresponding to @xmath60 separate energetically with increasing magnetic field while the spin polarized states decrease in energy . at magnetic field @xmath61 , such that @xmath62 , the energy gap is maximized and should correspond to the working point that can maintain the longest coherence time of the coded qubit . the coded qubit should operate at this value of magnetic field . however , as discussed above , the magnetic field effectively biases the qd 2 . this removes the degeneracy of the two qubit levels and rotates them from their zero magnetic field states . the energy splitting of the two coded qubit levels as a function of applied magnetic field is shown in fig.([fig : gate]a ) . the splitting is a fraction of the large energy scale @xmath33 . for the largest gap , i.e. @xmath63 , the splitting is @xmath64 . in order to restore the degeneracy of the two coded qubit levels , one can apply voltage to the qd 2 . in fig.([fig : gate]b ) , microscopic calculations done in lcho - ci show that a positive voltage bias on the qd 2 can indeed bring the tqd back on resonance in the presence of @xmath65 field . we have now established the coded qubit and the best conditions for its operation . in order to operate the coded qubit , we need to be able to initialize it . we propose to initialize the coded qubit by turning off both interaction between qds 1 and 3 and interaction between qds 1 and 2 . the only remaining interaction is between qds 2 and 3 . the ground state of a tqd , @xmath66 , becomes a product of a spin down state of an electron on the qd 1 and a singlet state of electrons across qds 2 and 3 . this intuitively is a ground state in magnetic field . the singlet state of two electrons in a pair of qds 2 and 3 can be generated in real time starting from two electrons in a biased qd 3 . this procedure does not generate directly the coded qubit levels @xmath67 and @xmath35 , but the state @xmath68 is a linear superposition of the two qubit levels : @xmath69 . once interactions are turned on , any state can be obtained from the initial state . this can be seen by writing the tqd heisenberg hamiltonian in the basis of the two coded qubit levels @xmath70 , @xmath71 if we take @xmath72 , and let @xmath73 , then the heisenberg hamiltonian corresponds to @xmath74 operation . if we take @xmath75 and @xmath76 , then the coded qubit hamiltonian eq.([eq : singlecodedqubit ] ) corresponds to @xmath77 operation . the capability to rotate a qubit with respect to two different axes on a bloch sphere allowes us to generate arbitrary single qubit operations . in practice , we tune the exchange interaction @xmath25 through biasing qds and changing their energies . as long as the biasing @xmath78 is satisfied , we expect the quantum state to remain in the qubit subspace during the process of tuning the exchange interactions as discussed in ref . . next , we discuss the measurement of coded qubits . several proposals@xcite discusses methods of detecting chirality of a triangular ( three - body ) antiferromagnetic cluster . for our specific tqd - based coded qubit , we propose to apply a @xmath79 field to split the two coded qubit levels with different chirality . as discussed earlier , the application of @xmath79 field will give rise to additional terms proportional to chirality operator on top of the heisenberg model , eq.([eq : heis1 ] ) , used for qubit modeling . since the additional term to the heisenberg model is strictly proportional to the chirality , it splits the two coded qubit levels with the gap given by @xmath80 where @xmath81 is the magnetic flux through the device and @xmath82 is the flux quanta . this energy gap can be interpreted as the energy difference between two magnetic dipole moments@xcite oriented in opposite directions under @xmath79 . in ref . , scarola et . al . also showed that the unintended qubit rotation due to the additional chirality term can be significantly reduced for a range of specific weak @xmath79 fields that satisfy @xmath83 and @xmath84 , where @xmath85 is the modified magnetic length , and @xmath86 is the confining frequency used to approximate the potential of a qd . thus , initializing the system under @xmath79 might not be a good idea as the field has to stay on for the entire period of quantum computation and eventually modify the quantum state of the coded qubit due to magnetic moment coupling with @xmath79 . however , a measurement done with advanced spectroscopy only requires @xmath79 to be turned on for a relatively short time and significantly limit the extent to which the coded qubit will be modified . we now turn to discuss double coded qubit operations using both hubbard and heisenberg models . starting with the hubbard hamiltonian , eq.([eq : nhubbard ] ) , we derive perturbatively a heisenberg hamiltonian for a complex of six electron spins with a set of exchange interactions @xmath25 @xmath87 with all other exchange interactions set to zero . for all the hubbard parameters considered in this study ( i.e. weak inter - tqd tunneling and strong coulomb interaction ) , @xmath88 is usually about two order of magnitudes smaller than @xmath89 and @xmath90 , mainly due to the fact that @xmath91 in our study . we treat the inter - tqd exchange interaction , scaled by @xmath88 , perturbatively to derive an effective interaction in the coupled coded qubits subspace , @xmath92 , where the superscript @xmath93 and @xmath46 stand for left and right tqd respectively . the effective qubit - qubit interacting hamiltonian reads @xmath94 where @xmath95 and @xmath96 and the direction vectors in eq.([eq : codedqubit4 ] ) are defined as follows : @xmath97 . thus , the effective interaction for 2 tqd - based code qubits is equivalent to an @xmath98 hamiltonian under a uniform in - plane magnetic field for 2 spins . if we rotate the effective qubit - qubit interacting hamiltonian , eq.([eq : codedqubit4 ] ) , from the coded qubit basis to the jacobian basis , @xmath99 , the rotated hamiltonian reads @xmath100 where @xmath101 , and @xmath102 . the jacobian state @xmath103 is defined in sec.[sec : cqbt ] when we discuss the initialization of the coded qubit . the jacobian state @xmath104 , where @xmath105 is a @xmath106 triplet state of electrons on qds 2 and 3 and @xmath107 is a @xmath108 triplet state of electrons on qds 2 and 3 . similarly , @xmath109 can also be written as a linear combination of the coded qubit levels . therefore this rotation of basis only requires single qubit operations applied to each coded qubit ; the two hamiltonians , eq.([eq : codedqubit4 ] ) and eq.([eq:2qubitl ] ) , are locally equivalent@xcite . as ising interaction can be used to generate cnot gate@xcite , we show how to generate non - local two qubit interactions with 2 tqd - based coded qubits . fig.([fig : ising ] ) shows results of exact diagonalization of the hubbard hamiltonian for two coupled tqds as a function of increasing coupling @xmath14 between the two molecules . the inset shows the entire energy spectrum at @xmath110 . the low energy spectrum consists of four levels , characterized mostly by the two lowest levels of each tqd . these 4 low - lying energy levels are well separated from the rest of the spectrum and constitute the two coupled coded qubit subspace . the low - lying spectrum contains a doubly degenerate level , which is a signature of the ising model in an external field . in practice , the tuning of @xmath88 , the inter - tqd exchange interaction , should be done via tuning the tunneling parameter @xmath14 . ( [ fig : ising ] ) shows that the ising model features of the energy spectrum are maintained over a wide range of values of @xmath14 , and hence the ising model hamiltonian , derived above , describes the coupling of two coded qubits very well . in summary , we present a theory of quantum circuits based on coded qubits encoded in chirality of electron spin complexes in lateral gated semiconductor triple quantum dot molecules with one electron spin in each dot . using microscopic hamiltonian and exact diagonalization techniques we show how to initialize , coherently control and measure the quantum state of a chirality based coded qubit using static in - plane magnetic field and voltage tuning of individual qds . the microscopic model of two interacting coded qubits is established and mapped to an ising hamiltonian . hence both conditional two - qubit phase gate and voltage controlled single qubit operations are demonstrated . the authors thank nserc , quantumworks , cifar , nrc - cnrc crp and ogs for support . hsieh would like to thank y .- shim and a. sharma for useful discussions . subspace under a @xmath43 field . the @xmath43 field breaks the discrete rotational symmetry , and it affects qd 2 differently from qds 1 and 3 . the energy spectrum of a triangular tqd as a function of @xmath111 , where @xmath112 is the cyclotron frequency used in lcho - ci calculation . @xmath57 corresponds to a magnetic field @xmath61 such that the gap between computational subspace ( the lowest energy level in the plot ) and the rest of spectrum is maximized as indicated by the black arrow . , scaledwidth=90.0% ] . the @xmath43 field splits the doubly degenerate ground state , and @xmath64 corresponds to the gap between the two levels at @xmath57 or equivalently at @xmath113 . the energy spectrum of the qubit levels under @xmath113 as a function of voltage @xmath114 on qd2 , and @xmath115 is the unbiased voltage at resonant condition when there is no external @xmath65 field . the splitting of levels under @xmath65 field can be restored via gate voltage tuning.,scaledwidth=70.0% ] . the energy spectrum resembles that of the ising model with an external field : a doubly degenerate levels corresponding to state @xmath116 and @xmath117 and two unique levels @xmath118 and @xmath119 . `` @xmath120 '' denotes a doubly degenerate level . inset : the entire energy spectrum at @xmath121 calculated with lcho - ci method.,scaledwidth=90.0% ]
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we present a theory of quantum circuits based on logical qubits encoded in chirality of electron spin complexes in lateral gated semiconductor triple quantum dot molecules with one electron spin in each dot . using microscopic hamiltonian
we show how to initialize , coherently control and measure the quantum state of a chirality based coded qubit using static in - plane magnetic field and voltage tuning of individual dots .
the microscopic model of two interacting coded qubits is established and mapped to an ising hamiltonian , resulting in conditional two - qubit phase gate .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
we are interested in numerical solutions of the fully nonlinear elliptic monge - ampre equation @xmath1 on a convex bounded domain @xmath2 of @xmath3 with boundary @xmath4 . the unknown @xmath5 is a real valued function and @xmath6 are given functions with @xmath7 in the non degenerate case and @xmath8 in the degenerate case . we will also assume that @xmath9 and @xmath10 in @xmath11 . starting with @xcite , interest has grown for finite element methods which are able to capture non smooth solutions of second order fully nonlinear equations . for smooth solutions , the problem was studied in the context of @xmath0 conforming approximations by b@xmath12hmer @xcite and in the context of lagrange elements by brenner and al @xcite . non smooth solutions can be handled in the context of the vanishing moment methodology @xcite which is a singular perturbation of . in this paper we give numerical evidence that @xmath0 conforming approximations of a natural variational formulation of converge for non smooth solutions of the two dimensional problem . this is achieved by discretizing new iterative methods we introduce . we establish the convergence of the iterative methods under the assumption that the discrete problem has a convex solution . we prove that such a solution exists when has a smooth convex solution . we do not assume that has a smooth solution for our iterative methods to converge . the existence of a convex solution to the discrete problem in the general case and the convergence of the discretization will be addressed in a subsequent paper . even with such an existence result , it is still a non trivial task to solve the discrete nonlinear systems in situations where has a non smooth solution . this paper addresses this issue . we refer to @xcite for a review of numerical methods for fully nonlinear equations of monge - ampre type . the main technical innovation of this paper is the proof that in the context of @xmath0 conforming approximations , discrete functions near a strictly convex solution are strictly convex . this explains why convexity did not need to be imposed explicitly in some previous studies . newton s method remains the most appropriate iterative method for solving the discrete nonlinear equations when has a smooth convex solution . we give a new proof of convergence of newton s method in the context of @xmath0 conforming approximations . the results of this paper extend easily to finite dimensional spaces of piecewise smooth @xmath0 functions provided that the approximation property and inverse estimates below hold . our results can be described in the general context of discretizations by @xmath0 elements of iterative methods for a general nonlinear elliptic equation @xmath13 . in the case of the monge - ampre equation , @xmath14 . we first describe the iterative methods at the continuous level . however , we will not address convergence at the continuous level . as an initial guess we take the solution of the poisson equation @xmath15 in @xmath16 on @xmath4 . we assume that @xmath17 differentiable and consider the sequence of problems @xmath18 where @xmath19 is a linear operator which can be taken as @xmath20 where @xmath21 is the identity operator or @xmath22 where @xmath23 is the laplace operator and @xmath24 is a parameter . pseudo transient continuation methods @xcite form a general class of methods for solving nonlinear singular equations . in the case of the monge - ampre equation , the method consists in solving the sequence of approximate problems @xmath25 here @xmath26 denotes the matrix of cofactors of the matrix @xmath27 . given @xmath24 , we consider the sequence of iterates @xmath28 this can be interpreted as an euler discretization of the pseudo time dependent equation @xmath29 or as a laplacian preconditioner of a simple pseudo time marching algorithm , @xcite @xmath30 see also a remark in @xcite . to the author s best knowledge , this is the first time the pseudo transient continuation method and the time marching method are used to indicate numerically convergence to viscosity solutions of finite element type methods for the monge - ampre equation . the methods we propose can be used in the context of different types of discretizations allowing us in particular to treat more easily non - rectangular domains . the methods can be accelerated with fast poisson solvers and multigrid methods . this latter property is even more striking for the time marching method as its implementation requires only having access to a multigrid poisson solver . although the theory of the monge - ampre equation has concentrated on convex solutions , one can equally focus on concave solutions . we found out that is better able to capture concave solutions . it is easy to implement , requiring only a poisson solver . for example one can capture weak solutions of the monge - ampere equation by simply discretizing with the standard lagrange finite elements . the time marching method can also be applied to fully nonlinear equations such as the pucci equation where @xmath17 is not differentiable . in summary the pseudo transient continuation methods are better for smooth solutions and singular solutions on a coarse mesh . otherwise the method of choice is the time marching method . we organize the paper as follows : in the second section we introduce some notation and prove the key result that discrete functions near a strictly convex solution are strictly convex . we introduce the natural variational formulation of and state an existence and uniqueness result for the discrete problem . the result can be inferred from the ones in @xcite . we give new proofs in section [ existence ] . as a corollary the discrete variational problem has a convex solution when has a smooth convex solution . we study the pseudo transient continuation methods in section [ continuation ] . a special case is newton s method for which we prove a quadratic convergence rate . the time marching methods are studied in section [ time ] . the last section is devoted to numerical results . we give a brief description of the spline element method which is used for the computations and offer heuristics about why our methods appear to preserve convexity . we use the standard notation for the sobolev spaces @xmath31 with norms @xmath32 and semi - norm @xmath33 . in particular , @xmath34 and in this case , the norm and semi - norms will be denoted respectively by @xmath35 and @xmath36 . for a vector field @xmath37 with values in @xmath38 , we set @xmath39 and a similar notation for @xmath40 . in the case @xmath41 , we set @xmath42 with a similar notation for @xmath43 . for matrix valued fields , the above notation is extended canonically . we make the usual convention of denoting constants by @xmath44 but will occasionally index some constants . for constants which depend on the mesh size , we may use @xmath45 or @xmath46 . we make the assumption that the boundary of @xmath2 is polygonal and that the triangulation @xmath47 is shape regular in the sense that there is a constant @xmath48 such that for any triangle @xmath49 , @xmath50 , where @xmath51 denotes the diameter of @xmath49 and @xmath52 the radius of the largest ball contained in @xmath49 . we also require the triangulation to be quasi - uniform in the sense that @xmath53 is bounded where @xmath54 and @xmath55 are the maximum and minimum respectively of @xmath56 . we define @xmath57 where @xmath58 denotes the space of polynomials of degree less than or equal to @xmath59 . in _ two dimensions _ , it is known that , @xcite , for @xmath60 and @xmath61 , there exists a linear quasi - interpolation operator @xmath62 mapping @xmath63 into the spline space @xmath64 and a constant @xmath44 such that if @xmath65 is in the sobolev space @xmath66 @xmath67 for @xmath68 . if @xmath2 is convex , the constant @xmath44 depends only on @xmath69 and on the smallest angle @xmath70 in @xmath47 . in the nonconvex case , @xmath44 depends only on the lipschitz constant associated with the boundary of @xmath2 . it is also known c.f . @xcite that the full approximation property for spline spaces holds on special triangulations for certain values of @xmath59 . in _ three dimensions , holds in general for @xmath71 , c.f . @xcite . note that , by , @xmath72 for all @xmath73 . we assume that the following inverse inequality which holds for @xmath0 finite element spaces , c.f . theorem 4.5.11 of @xcite , also holds for the spline spaces @xmath74 and for @xmath75 . the local estimates may be viewed as a consequence of the assumption of uniform triangulation and of markov inequality , @xcite p. 2 . passing from local estimates to global estimates can be done as in @xcite . we first recall the divergence form of the determinant and the expression of its frchet derivative . for two @xmath76 matrices @xmath77 , we recall the frobenius product @xmath78 . [ det - lem ] we have @xmath79 and for @xmath80 we have @xmath81 for @xmath82 sufficiently smooth . note that for any @xmath76 matrix @xmath27 , @xmath83 , where @xmath26 is the matrix of cofactors of @xmath27 . this follows from the row expansion definition of the determinant . for any sufficiently smooth matrix field @xmath27 and vector field @xmath65 , @xmath84 . here the divergence of a matrix field is the divergence operator applied row - wise . if we put @xmath85 , then @xmath86 and @xmath87 but @xmath88 , c.f . for example @xcite p. 440 . hence since @xmath89 and @xmath90 are symmetric matrices follows . the assertion about the frchet derivative of @xmath17 follows from the definition of the determinant as a multilinear map ( e.g ) and the definition of matrix of cofactors . see also @xcite p. 440 . using the divergence form of the determinant and integration by parts , one obtains the variational formulation of given by : find @xmath91 , @xmath92 on @xmath4 such that @xmath93 we show that for @xmath91 , is well defined . _ case @xmath94_. for @xmath94 , each entry of @xmath90 consists of a second derivative @xmath95 . by h@xmath12lder s inequality , @xmath96 next for @xmath97 and by sobolev embedding , i.e. the embedding of @xmath98 in @xmath99 for @xmath100 when @xmath94 , the right hand side above is bounded by @xmath101 @xmath102 . _ case @xmath103_. for @xmath103 , each entry of @xmath90 involves the product of two second order derivatives . we have by h@xmath12lder s inequality and sobolev embedding , i.e. the embedding of @xmath98 in @xmath99 for @xmath104 when @xmath103 , @xmath105 we conclude that for @xmath103 , @xmath106 in summary for @xmath107 , we may write @xmath108 put @xmath109 and @xmath110 . note that @xmath111 given by satisfies @xmath112 . let @xmath113 and furthermore let @xmath114 be the interpolant in @xmath111 of a smooth extension of @xmath10 . we have the following conforming discretization of : find @xmath115 , @xmath116 on @xmath4 such that @xmath117 we now present a number of preliminary results . we first state a result on the well posedness of . the @xmath118 error estimate can be inferred from @xcite . we give a new proof in section [ existence ] . [ errorest ] let @xmath119 and assume that @xmath120 is a strictly convex function , that @xmath2 is convex with a polygonal boundary and that the spaces @xmath111 have the optimal approximation property and satisfy the inverse estimates . then the problem has a unique solution @xmath121 for @xmath54 sufficiently small and we have the error estimates latexmath:[\ ] ] the ellipticity assures uniqueness of the component @xmath446 and the saddle point problems are solved by a version of the augmented lagrangian algorithm @xmath447 the convergence properties of the iterative method were given in @xcite . extensive implementation details can be found in @xcite . for @xmath94 , the computational domain is the unit square @xmath448 ^ 2 $ ] which is first divided into squares of side length @xmath54 . then each square is divided into two triangles by the diagonal with negative slope . for @xmath103 , the initial tetrahedral partition @xmath449 consists in six tetrahedra . each tetrahedron is then uniformly refined into 8 subtetrahedra forming @xmath450 . in the tables , @xmath451 denotes the number of iterations . we refer to @xcite for implementation details of the method . all numerical experiments are with the versions of the iterative methods with laplacian preconditioner . in general we did not try to choose the value of @xmath325 that would give the smallest number of iterations except in tables [ tab - comp1 ] and [ tab - comp2 ] where we compare the performance of the two methods . we use some standard test cases for numerical evidence for convergence to non smooth solutions of the elliptic monge - ampere equation . test 1 : @xmath452 so that @xmath453 and @xmath454 on @xmath4 . test 2 : @xmath455 so that @xmath456 and @xmath457 on @xmath4 . barring roundoff errors , the methods introduced in this paper capture smooth solutions . for the two dimensional test function , test 1 , we give numerical results for successive refinements and for the three dimensional test function , we give numerical results for increasing values of the degree @xmath59 on two successive refinements . .time marching method for test 1 , @xmath458 , @xmath459 [ cols="^,^,^,^,^,^,^,^",options="header " , ] the time listed is in seconds and obtained on an imac running mac os 10.6.8 with a 2.4 ghz intel core 2 duo and 4 gb of sdram memory . while for small values of @xmath54 the time marching method appears to take significantly more time , it is also significantly more accurate . for @xmath460 the time took by the two methods is almost the same with the time marching method giving a more accurate solution . next we consider a non square domain . test 5 : we consider the unit circle discretized with a delanauy triangulation with 824 triangles and @xmath461 which vanishes on the boundary , figure [ fig11 ] . on a non square domain with pseudo transient @xmath462 , height=170 ] we conclude this section with a test problem for a degenerate monge - ampre equation test 6 : @xmath463 and @xmath464 . the graph of the function , figure [ fig2 ] is singular along the line @xmath465 . the approximations been @xmath0 do appear to capture the singularity but not the convexity of the solution . somewhat better results are obtained with another iterative method discussed in @xcite . when the time marching method is discretized by the standard finite difference method the singularity is captured correctly . we wish to discuss these results in separate works . and @xmath463 with time marching @xmath466 , title="fig:",height=170 ] and @xmath463 with time marching @xmath466 , title="fig:",height=170 ] when has a smooth strictly convex solution , theorem [ errorest ] establishes that the approximate solution is automatically convex . the numerical experiments indicate that in the non smooth case , discrete solutions are also convex . the result can be easily explained at the continuous level ( for a smooth solution ) . assume that @xmath467 and that the sequence @xmath250 defined by @xmath468 has been shown to converge to @xmath5 in the h@xmath12lder space @xmath469 for some @xmath470 in @xmath471 . from the arithmetic - geometric inequality , we have @xmath472 by the continuity of the eigenvalues , , @xmath473 is bounded in a neighborhood of @xmath5 in which all @xmath250 belong for @xmath474 large enough . choose @xmath325 such that @xmath475 for all @xmath474 and note that the right hand of is equal to @xmath476 . by the assumption on @xmath325 , we get @xmath477 . in the limit , we obtain @xmath478 . since @xmath479 by assumption , we get @xmath480 . as for the time marching method @xmath481 assume now again that @xmath482 and that the sequence @xmath250 has been shown to converge to @xmath5 in @xmath469 for some @xmath470 in @xmath471 . choose @xmath325 such that @xmath483 . we have @xmath484 . it follows from the arithmetic - geometric inequality that @xmath485 and so @xmath486 and it follows that the time marching method also preserves the positivity of the laplacian . in two dimensions @xmath480 and @xmath487 imply that @xmath89 is positive definite . the author acknowledges discussions with f. celiker , b. cockburn , w. gangbo , r. glowinski , m.j . lai , r. nochetto , a. oberman and a. regev . the author was supported in part by nsf grant dms-0811052 and the sloan foundation . this research was supported in part by the institute for mathematics and its applications with funds provided by the national science foundation . awanou , g. , lai , m.j . , wenston , p. : the multivariate spline method for scattered data fitting and numerical solution of partial differential equations . in : wavelets and splines : athens 2005 , mod . methods math . , pp . nashboro press , brentwood , tn ( 2006 ) davydov , o. , saeed , a. : stable splitting of bivariate splines spaces by bernstein - bzier methods . in : curves and surfaces , _ lecture notes in comput . . 6920 , pp . 220235 . springer , heidelberg ( 2012 ) dyer , b.w . , hong , d. : algorithm for optimal triangulations in scattered data representation and implementation . j. comput . appl . * 5*(1 ) , 2543 ( 2003 ) . approximation theory and wavelets ( austin , tx , 1999 ) farag , i. , kartson , j. : numerical solution of nonlinear elliptic problems via preconditioning operators : theory and applications , _ advances in computation : theory and practice _ , vol . nova science publishers inc . , hauppauge , ny ( 2002 )
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we present two numerical methods for the fully nonlinear elliptic monge - ampre equation .
the first is a pseudo transient continuation method and the second is a pure pseudo time marching method .
the methods are proved to converge to a convex solution of a natural discrete variational formulation with @xmath0 conforming approximations .
the assumption of existence of a convex solution to the discrete problem is proven for smooth solutions of the continuous problem and supported by numerical evidence for non smooth solutions .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the theory of hyperplane arrangements has deep connections with many areas of mathematics , such as combinatorics , representation theory , algebraic geometry , algebraic topology , singularity theory , and the theory of hypergeometric functions . one beautiful application of the use of hyperplane arrangements in algebraic geometry is the proof of brieskorn s conjecture by deligne @xcite in 1972 . real hyperplane arrangements serve as the motivation for the axioms that define oriented matroids . see @xcite for detailed reviews of the theory of hyperplane arrangements . real hyperplane arrangements have been used recently in the physics literature to analyze the fiber structure and the network of flop transitions between different resolutions of elliptically fibered calabi - yau threefolds @xcite . compactifications of m - theory on calabi - yau threefolds form an elegant bridge between the study of supersymmetric gauge theories in five dimensional spacetime and certain types of highly structured hyperplane arrangements defined by a lie algebra @xmath2 and a representation @xmath3 of @xmath2 . these hyperplane arrangements have the peculiarity of being defined not in the full affine space as they are restricted to the dual fundamental weyl chamber of the lie algebra @xmath2 . [ def : incgeogeneral ] let @xmath2 be a reductive lie algebra over @xmath4 , let @xmath5 be a split , real form of a cartan subalgebra of @xmath6 and let @xmath7 be a representation of @xmath2 . we denote by @xmath8 the real hyperplane arrangement consisting of the kernels of the weights of @xmath7 restricted to a dual fundamental weyl chamber in @xmath5 . because all cartan subalgebras are conjugate and all fundamental weyl chambers canonically related by the weyl group action , the incidence geometry @xmath8 is independent of the choice of @xmath5 and of a dual fundamental weyl chamber in @xmath5 . in the rest of the paper , we call the dual fundamental weyl chamber " the _ weyl chamber_. an arrangement of hyperplanes stratifies the ambient space into open polyhedral cones called _ faces_. a non - empty intersection of a finite number of hyperplanes of the arrangement is called a _ flat_. a @xmath9-face ( resp . @xmath9-flat ) is a face ( resp . a flat ) that generates a @xmath9-dimensional linear space . faces of maximum dimension are called _ a classical combinatorial problem asks how many @xmath9-flats and @xmath9-faces a given hyperplane arrangement contains . zaslasky @xcite enumerated chambers and bounded chambers of an affine hyperplane arrangement using the mbius function of its semi - lattice of flats . the purpose of this paper is to count the numbers of @xmath9-faces and @xmath9-flats of hyperplane arrangements @xmath10 where @xmath11 is the vector representation of @xmath0 ( the first fundamental representation ) and @xmath12 is its second exterior power ( the second fundamental representation ) . since our arrangements are restricted to the weyl chamber , we can not apply zaslasky s result directly . however , the highly symmetric nature of the hyperplane arrangements under consideration allows us to study them via their extreme rays . we also analyze the geometries @xmath13 when @xmath2 is not simply laced . the case of @xmath14 is discussed in @xcite . the counting problem solved in this paper is formulated purely in terms of representation theory . however , its motivation comes from the geometry of string theory . in this subsection , we review the motivation . the reader uninterested in the interface of string theory and algebraic geometry can safely ignore this subsection . hyperplane arrangements of the form @xmath15 arise naturally in the study _ coulomb branch _ of a gauge theory in five - dimensional supersymmetric theory with eight supercharges @xcite . such theories contain different types of particles organized into representations of the supersymmetric algebra called _ supermultiplets_. we will consider only vector multiplets and hypermultiplets . when a gauge theory is defined in a five dimensional supersymmetric theory , the vector multiplets transform in the adjoint representation while the hypermultiplets transform in some representation @xmath3 of @xmath2 . the hypermultiplets can become massive through the higgs mechanism along hyperplanes passing through the origin of the weyl chamber of @xmath2 . these hyperplanes partition the weyl chamber into different connected regions called _ coulomb phases _ of the gauge theory , which are characterized by a cubic prepotential depending on real coordinates @xmath16 of the weight space . intriligator - morrision - seiberg computed the prepotential of a five dimensional gauge theory for a theory in which hypermultiplets transform in the representation @xmath17 of a lie algebra @xmath2 . the result is the real function @xcite : @xmath18 depending on the real vector @xmath19 in the weight space . we only wrote the terms of @xmath20 that are relevant for our purposes . the variable @xmath21 runs through the simple roots of @xmath2 , and @xmath22 through the weights of the representation @xmath7 under which the hypermultiplets transform . the cubic absolute values generate singularities , which correspond to hyperplanes along which new massless particles are generated . a well - defined phase of the coulomb branch of the theory is a connected region in which the quantities @xmath23 and @xmath24 each take fixed signs . we fix the sign of @xmath23 by requiring @xmath19 to be in the fundamental weyl chamber . the condition that @xmath25 for all @xmath26 restricts us inside a specific chamber of the arrangement @xmath27 . the counting of flats is connected to the enumeration of mixed _ coulomb - higgs branches _ of the gauge theory . a five dimensional gauge theory with 8 supercharges can be geometrically engineered by a compactification of m - theory on a calabi - yau threefold @xcite . when the calabi - yau threefold is elliptically fibered , its singular fibers determine the lie algebra @xmath2 and the matter representation @xmath3 of the gauge theory . a smooth elliptic fibration with a rational section is a resolution of a singular weierstrass model , and flop transitions join pairs of crepant resolutions of the weierstrass model . it is conjectured that the network of ( partial ) crepant resolutions of the weierstrass model is isomorphic to the adjacency graph of the faces of the hyperplane arrangement @xmath15 . more precisely , it is conjectured that each crepant resolution corresponds to a unique chamber of @xmath15 @xcite , and that two crepant resolutions are connected by a flop if and only if the intersection of the corresponding chambers is a face of codimension one . their common face is then conjectured to correspond to a unique partial resolution of the weierstrass model . see @xcite for examples of explicit matchings between ( partial ) resolutions and faces of a hyperplane arrangements @xmath15 . there are subtleties in presence of non - trivial @xmath28-factorial terminal singularities , see for example @xcite . a proof of this conjecture will likely rely heavily on ideas from mori s program along the lines of @xcite . the intriligator - morrision - seiberg prepotential @xmath29 can also be obtained geometrically @xcite . in the case of @xmath30 , the gauge group is @xmath1 , and therefore the relevant elliptic fibrations each have a mordell - well group of rank one and a singular fiber of type i@xmath31 over a divisor of the discriminant locus . models for elliptic fibrations with a mordell - weil group of rank one are discussed in @xcite and @xcite . elliptic fibrations for geometric engineering of @xmath1 gauge theories have been extensively studied recently in the physics literature ( see for example @xcite and references therein ) . we are now ready to discuss the enumerations performed in this paper . we denote the number of @xmath9-faces of @xmath32 by @xmath33 and collect these numbers in the following formal power series ( generating function ) : @xmath34 the following theorem enumerates faces of @xmath30 . it is proved in sec : genfuncfaces . [ thm : facecount ] the generating function counting faces of @xmath35 by dimension is given by the following rational function : @xmath36 thm : facecount implies a simple recurrence relation for @xmath33 . [ cor : facerecurse ] the sequence @xmath33 obeys the recurrence relation : @xmath37 for @xmath38 . we can also obtain formulas for the generating polynomials @xmath39 g(s , t ) = \sum_{k=0}^n g(n , k)t^k.\ ] ] [ cor : facegenpoly ] the generating polynomials @xmath40 are given by @xmath41 cor : facegenpoly yields the following values of @xmath40 : .3 cm from the above formulae , we can read off the value of @xmath33 for @xmath42 as the coefficient of @xmath43 in @xmath40 : @xmath44 we can also extract formulae for @xmath33 for @xmath9 small and @xmath45 small from thm : facecount . [ cor : facesmalllarge ] we have @xmath46 and @xmath47 cor : facesmalllarge yields the following formulae for @xmath33 and @xmath48 for @xmath49 : @xmath50 thm : facecount and corollaries [ cor : facerecurse ] , [ cor : facegenpoly ] , and [ cor : facesmalllarge ] are proved in sec : genfuncfaces . we denote the number of @xmath9-flats of @xmath32 by @xmath51 and collect these numbers in the following formal power series ( generating function ) : @xmath52 the following theorem enumerates flats of @xmath30 . [ thm : flatcount ] the generating function counting flats of @xmath35 by dimension is given by the following rational function : @xmath53 thm : flatcount is proven sec : genfuncflats . it implies a simple recurrence relation for @xmath51 , the analogue of cor : facerecurse for flats . [ cor : flatrecurse ] the sequence @xmath51 obeys the recurrence relation @xmath54 for @xmath55 . the taylor expansion of thm : flatcount to order 10 in @xmath56 is : from the above formula , we can read off the value of @xmath51 for @xmath42 as the coefficient of @xmath57 in @xmath58 : @xmath59 the form of the series @xmath58 does not allow simple explicit formulae for @xmath60 or for @xmath51 as in corollaries [ cor : facegenpoly ] and [ cor : facesmalllarge ] , respectively . thm : flatcount is proved in sec : genfuncflats . ( 1,1 ) & ( 1,2 ) & ( 1,3 ) & & ( 1,n ) + & ( 2,2 ) & ( 2,3 ) & & ( 2,n ) + & & ( 3,3 ) & & ( 3,n ) + & & & & + & & & & ( n , n ) a face is determined by assigning signs to @xmath61 for all values of @xmath62 such that @xmath63 . these weights can be organized in a right - justified young tableau as in figure [ fig : signtableau ] . it is important to realize that not all sign patterns are allowed . since we are in the weyl chamber , we have @xmath64 using the identities @xmath65 one can prove the following simple sign rules called _ sign flows _ in the physics literature @xcite : the signs in a tableau corresponding to a chamber satisfy the following conditions . * all the boxes above or on the left of a box with positive entries are also positive . * all the boxes below or on the right of a box with negative entries are also negative . a tableau satisfying these two rules corresponds to a unique chamber of @xmath66 . however , a different notation for chambers will be more convenient for dealing with extreme rays . a different notation for chambers will be more convenient for dealing with extreme rays . the sign rules implies that the positive entries of each row are next to each other and start on the left border of the table . it is therefore efficient to denote a given chamber by the numbers of positive entries on each row . the sign rules is then automatically satisfied if these numbers form a decreasing sequence @xmath67 , where we denote by @xmath68 the number of positive entries on the @xmath69th row of the tableau . we do nt count the rows that do not have any positive entries . an entry of the tableau located on the @xmath69th row and the @xmath70th column ( with @xmath71 ) is positive if and only if @xmath72 and @xmath73 . it is negative otherwise . this justifies the following definition . [ def : subsettochamber ] for @xmath74 $ ] , define a face @xmath75 of @xmath35 as the subset of @xmath5 on which @xmath76 we can equivalently write a subset @xmath77 $ ] as a _ characteristic vector _ @xmath78 , where @xmath79 and @xmath80 . a characteristic vector @xmath79 defines a subset @xmath81 $ ] such that @xmath82 ( resp . @xmath83 ) if and only if @xmath84 ( resp . @xmath85 ) . the notation of def : subsettochamber allows us to state the following classification of chambers and extreme rays . [ thm : chamberstructure ] the chambers and extreme rays of @xmath30 satisfy the following properties . a. [ assert : chamberbiject ] the map @xmath86 defines a bijection from @xmath87}$ ] to the set of chambers of @xmath35 . in particular , @xmath35 has @xmath88 chambers . b. [ assert : chamberraysprelim ] the extreme rays of @xmath75 are generated by the vectors @xmath89 where @xmath90 where @xmath91 where @xmath91 counts the elements of @xmath92 that are greater or equal to @xmath93 , and the vectors @xmath94 non - negatively span @xmath75 . in particular , the geometry @xmath30 is simplicial . the enumeration of chambers of @xmath30 announced in thm : chamberstructureassert : chamberbiject recovers the count of phases of @xmath1 with fundamental and antisymmetric matter from the physics literature @xcite . theorem [ thm : chamberstructure ] is proven in sec : extremerayschambers . suppose that @xmath95 with @xmath96 . a direct calculation shows that the box diagram associated to @xmath75 has @xmath97 plusses in the @xmath69th row for @xmath98 and all minus signs in the @xmath69th row for @xmath99 . see tables [ tab : example.gl2 ] and [ tab : example.gl3 ] for examples . c | l|c| l sign tableau & subset of @xmath100 $ ] & characteristic vector & interior points + ' '' '' & @xmath101 & @xmath102 & @xmath103 + ' '' '' & @xmath104 & @xmath105 & @xmath106 + ' '' '' & @xmath107 & @xmath108 & @xmath109 + ' '' '' & @xmath110 & @xmath111 & @xmath112 l | c| c | l sign tableau & subset of @xmath100 $ ] & characteristic vector & interior points of the chamber + ' '' '' - & - & - + & - & - + & & - & @xmath110 & @xmath113 & @xmath114 + ' '' '' + & - & - + & - & - + & & - & @xmath107 & @xmath115 & @xmath116 + ' '' '' + & + & - + & - & - + & & - & @xmath104 & @xmath117 & @xmath118 + ' '' '' + & + & + + & - & - + & & - & @xmath119 & @xmath120 & @xmath121 + ' '' '' + & + & + + & + & - + & & - & @xmath122 & @xmath123 & @xmath124 + ' '' '' + & + & - + & + & - + & & - & @xmath101 & @xmath125 & @xmath126 + ' '' '' + & + & + + & + & + + & & - & @xmath127 & @xmath128 & @xmath129 + ' '' '' + & + & + + & + & + + & & + & @xmath130 & @xmath131 & @xmath132 in light of thm : chamberstructure , we can introduce additional structure on the set of extreme rays of @xmath30 . we will equip the set of extreme rays with a partial order and use the combinatorics of the resulting partially ordered set ( poset ) to study faces and flats in @xmath30 . the following definitions describe the posets that will be relevant to us . [ def : quarterplaneposet ] the _ discrete quarter plane poset _ is the set @xmath133 endowed with the cartesian order induced by the usual order of the set @xmath134 of non - negative integers : @xmath135 this is a graded poset with grading function @xmath136 . we call @xmath137 the _ level _ of @xmath138 . [ def : posetforextrays ] we denote by @xmath139 the subset of the discrete quarter plane that consists of points at level less or equal to @xmath140 . @xmath139 is a poset with the order induced by the cartesian order defined above . we denote by @xmath141 the poset @xmath142 where @xmath143 is greater than all the elements of @xmath139 . we denote by @xmath144 the poset @xmath139 with the origin removed : @xmath145 we are now ready to relate @xmath144 to the set of extreme rays of @xmath30 . [ def : vfn ] define a function @xmath146 by @xmath147 the following theorem follows directly from thm : chamberstructure . [ thm : raystruture ] the extreme rays of @xmath35 satisfy the following properties . a. the function @xmath148 defines a bijection from @xmath144 to the set of extreme rays of @xmath35 . b. the set of extreme rays that lie in a chamber @xmath75 are the extreme rays @xmath149 where @xmath150 . we will characterize faces and flats by the extreme rays they contain . for that reason we define a function which returns the extreme rays lying in a given subset of the weyl chamber . [ def : raysoperator ] define a function @xmath151 from @xmath152 to the power set of the set of extreme rays of @xmath30 as follows . for @xmath153 let @xmath154 be the set of extreme rays of @xmath30 that lie in @xmath155 . the following theorem , proven in sec : extremeraysfaces relates faces in @xmath30 to the combinatorics of @xmath144 . [ thm : characterization.faces ] for all @xmath9 , the function @xmath156 induces a bijection from the set of @xmath9-faces of @xmath30 to the set of @xmath9-chains in @xmath144 . here , we say that @xmath157 is a @xmath9-chain if @xmath92 is a chain and @xmath158 . to state an analogue of thm : characterization.faces for flats , we will need to define structures called _ ensembles _ that will play the role of chains . [ def : ensemble ] an _ ensemble _ is the restriction to @xmath144 of a union @xmath159 \right)\cap { \mathbb{e}^\ast}_n\ ] ] of intervals @xmath160 $ ] of @xmath141 satisfying the following four conditions : 1 . @xmath161 2 . @xmath162 for @xmath163 ; 3 . @xmath164 for @xmath165 ; and 4 . @xmath166 or @xmath167 . we say that @xmath168 is a _ @xmath9-ensemble _ if @xmath169 , so that @xmath9 counts the number of distinct levels of elements of @xmath168 . figure [ fig : ensembleoneeg ] . shows one example of an ensemble . 2 -ensemble in @xmath170 . * figure ( a ) represents the @xmath171-ensemble @xmath92 which is the restriction to @xmath172 of the union of intervals @xmath173 \cup [ ( 3,3),(4,4 ) ] \cup [ ( 5,5),\infty]$ ] . theorem [ thm : characterization.flats ] guarantees that @xmath174 is the set of extreme rays of a unique 10-flat in @xmath175 . figure ( b ) illustrates a maximal chain of @xmath92 , which corresponds to a @xmath171-face of @xmath175 . ] -ensemble in @xmath170 . * figure ( a ) represents the @xmath171-ensemble @xmath92 which is the restriction to @xmath172 of the union of intervals @xmath173 \cup [ ( 3,3),(4,4 ) ] \cup [ ( 5,5),\infty]$ ] . theorem [ thm : characterization.flats ] guarantees that @xmath174 is the set of extreme rays of a unique 10-flat in @xmath175 . figure ( b ) illustrates a maximal chain of @xmath92 , which corresponds to a @xmath171-face of @xmath175 . ] [ thm : characterization.flats ] for all @xmath9 , the function @xmath156 induces a bijection from the set of @xmath9-flats of @xmath30 to the set of @xmath9-ensembles of @xmath144 . theorem [ thm : characterization.flats ] is proven in section [ sec : extremeraysflats ] on page . denote by @xmath11 the vector representation of @xmath176 and of @xmath177 . recall that the representation @xmath178 of @xmath177 factors as @xmath179 , where @xmath180 is irreducible . let @xmath181 ( resp . @xmath182 ) denote the unique 26-dimensional ( resp . 7-dimensional ) irreducible representation of @xmath183 ( resp . @xmath184 ) . we are now ready to discuss the geometry @xmath185 when @xmath2 is non - simply laced simple lie algebra . [ thm : nonsimplylaced ] suppose that @xmath186 and suppose that @xmath2 has rank @xmath140 . the geometry @xmath185 is a cone over a simplex @xmath187 . in particular , there is 1 chamber and the number of @xmath9-faces ( resp . @xmath9-flats ) is @xmath188 . there is only one other geometry @xmath13 when @xmath2 is simple and non - simply laced and @xmath7 is a direct sum of minuscule and quasi - minuscule representation . this is the geometry @xmath189 it is mentioned in ( * ? ? ? * section 6.3 ) that @xmath190 and @xmath191 have no chambers . this is in contradiction with the conclusion of thm : nonsimplylaced . a direct calculation shows that all the weights of @xmath7 are scalar multiples of the roots of @xmath2 in the cases discussed in the theorem . therefore , the hyperplanes @xmath192 for @xmath22 a weight of @xmath7 coincide with vanishing loci of roots of @xmath2 . the theorem then follows from the fact that the weyl chamber is simplicial . . * the geometry contains 4 chambers , 5 half - lines and the origin . there is one 2-flat ( the half - plane below the line @xmath193 ) , three 1-flats ( the interior walls @xmath194 , @xmath195 , @xmath196 ) , and one 0-flat ( the origin ) . ] in sec : prelims , we recall basic facts regarding the lie algebra @xmath0 and from the theory of hyperplane arrangements . in sec : extrayschmfaces , we prove theorems [ thm : chamberstructure ] , [ thm : raystruture ] , and [ thm : characterization.faces ] . in sec : facecount , we prove thm : facecount , and in sec : flatcount , we prove theorems [ thm : flatcount ] and [ thm : characterization.flats ] . m.e . is grateful to shu - heng shao and shing - tung yau for helpful discussions . m.e . and a.n . thank william massey and all the organizers of the 20th conference for african american researchers in mathematical sciences ( caarms 20 ) where this started . is supported in part by the national science foundation ( nsf ) grant dms-1406925 elliptic fibrations and string theory " . is supported by the harvard college research program . in sec : prelimliealg , we fix notation for the weyl chamber of @xmath0 . in sec : hyparrange , we recall some facts from the theory of hyperplane arrangements . a detailed list of symbols is presented in tab : notations . let @xmath197 denote the cartan subalgebra of @xmath0 consisting of the diagonal matrices . we identify @xmath5 with @xmath198 via the isomorphism @xmath199 the positive roots of @xmath0 are @xmath200 . we define the open weyl chamber of @xmath0 as the set of vectors @xmath201 forming a non - increasing sequence @xmath202 and denote by @xmath203 the closure of @xmath204 . define @xmath205 . the weights of the vector representation are @xmath206 for @xmath207 and the weights of the antisymmetric representation are @xmath61 for @xmath208 . when @xmath209 , the hyperplane @xmath210 is the vanishing locus of the weight @xmath206 of the vector representation of @xmath0 . when @xmath211 , the hyperplane @xmath212 is the vanishing locus of the weight @xmath61 of the antisymmetric representation of @xmath0 . by definition , each chamber is a convex rational polyhedral cone ( finite intersection of rational half spaces ) . in the following we will refer to a convex rational polyhedral cone simply as `` cone '' . face _ is either a chamber or the intersection of a chamber with a supporting hyperplane . there are finitely many faces , and each face is a lower dimensional cone . the chambers cover @xmath155 and form a _ fan_. a fan is a finite collection of cones such that for each cone all its faces are in the fan and the intersection of two cones is a face of each @xcite . by classical theorems of weyl and minkowski , a cone is finitely generated if and only if it is a finite intersection of closed linear half spaces ( see ( * ? ? ? * theorem 1.3 ) ) . it follows that each chamber of @xmath32 admits a system of generators . a cone of dimension @xmath140 is said to be _ simplical _ if it is non - negatively spanned by @xmath140 linearly independent vectors . a face of dimension one which is a half - line is called an _ extreme ray_. a _ facet _ is a face of co - dimension one . the roots of the lie algebra @xmath0 define a hyperplane arrangement known as the _ braid arrangement _ @xcite , which has @xmath213 chambers , namely the weyl chambers of @xmath214 . the braid arrangement is also called the coxeter arrangement of type @xmath214 . the weights of the fundamental representation define an arrangement of hyperplanes known as the _ boolean arrangement _ @xcite . the weights of the antisymmetric representation @xmath215 define an arrangement of hyperplanes known as the _ threshold arrangement _ @xcite . altogether , the roots of @xmath0 and the weights of the first two fundamental representations define the coxeter arrangement of type @xmath216 @xcite . in sec : extremerayschambers , we prove theorems [ thm : chamberstructure ] and [ thm : raystruture ] , and in sec : extremeraysfaces , we prove thm : characterization.faces . given a vector @xmath217 , we denote by @xmath218 ( resp . @xmath219 ) the vector obtained from @xmath220 by rearranging its components in non - increasing ( resp . non - decreasing ) order . the idea of the proof is to derive for each @xmath78 an explicit linear isomorphism @xmath221 from @xmath198 to @xmath5 such @xmath222 . the elementary basis vectors of @xmath198 then map under @xmath221 to the set of extreme rays of @xmath223 . define @xmath224 the absolute values of the coordinates @xmath225 of a point @xmath226 are pairwise distinct and non - zero , because @xmath227 and @xmath228 for all @xmath208 . after reordering , we can write the set @xmath229 as a strictly increasing vector @xmath230 so that @xmath231 for all @xmath69 because @xmath232 there exists a unique sequence @xmath233 of signs such that @xmath234 for all @xmath235 . we see immediately that @xmath236 defines a continuous function from @xmath237 to @xmath238 . a direct calculation shows that @xmath239 and that @xmath240 . we now prove part assert : chamberbiject of the theorem . suppose that @xmath241 . consider the vector @xmath242 note that the absolute values of the components of @xmath243 are @xmath244 . we see immediately that @xmath245 for all @xmath78 , so that @xmath246 . because @xmath247 and @xmath246 , the face @xmath223 contains a point of @xmath204 not lying in any hyperplane of @xmath30 . it follows that @xmath223 is a chamber of @xmath30 for all @xmath248 . it remains to prove that @xmath249 defines a bijection from @xmath238 to the set of chambers of @xmath30 . consider a chamber @xmath250 of @xmath30 and let @xmath251 be an interior point . note that @xmath240 . we see immediately that @xmath252 because @xmath253 is a chamber of @xmath30 . suppose that @xmath254 . it follows that @xmath243 and @xmath255 lie in the same connected component of @xmath237 . because @xmath236 is continuous , we obtain that @xmath256 so that @xmath249 is injective . part assert : chamberbiject follows . we now prove part assert : chamberraysprelim . consider a chamber @xmath223 of @xmath30 . let @xmath257 denote the subset of @xmath100 $ ] corresponding to @xmath258 . note that @xmath259 with @xmath260 . hence , @xmath223 is non - negatively spanned by @xmath261 . it follows that @xmath94 are the extreme rays of @xmath223 , as desired . recall the definition of a chain in a poset . a _ * chain * _ in a poset is a subset of the poset in which any two elements are comparable . the _ length _ of a chain is its size , and a _ @xmath9-chain _ is a chain of length @xmath9 . it follows from thm : chamberstructure that @xmath262 induces a bijection from the set of chambers of @xmath30 to the set of @xmath140-chains in @xmath144 . to prove thm : characterization.faces , we will exploit the simpliciality of the chambers of @xmath30 , which was also proved in thm : chamberstructure . suppose that @xmath250 is a @xmath9-face of @xmath30 . consider a chamber @xmath249 containing @xmath250 . because @xmath249 is simplicial ( by thm : chamberstructure ) , @xmath250 must be the non - negative span of @xmath9 extreme rays of @xmath249 . in particular , we obtain that @xmath263 is a subset of @xmath264 of size @xmath9 . because @xmath265 is an @xmath140-chain in @xmath144 and @xmath266 is a subset of @xmath265 of size @xmath9 , it follows that @xmath266 is a @xmath9-chain in @xmath144 . therefore , @xmath262 is a function from the set of @xmath9-faces of @xmath30 to the set of @xmath9-chains of @xmath144 . the fact that every face is non - negatively spanned by the extreme rays that it contains ensures that @xmath262 induces an injection from the set of @xmath9-faces of @xmath30 to the set of @xmath9-chains in @xmath144 . it remains to prove that @xmath262 is surjective . suppose that @xmath267 is a @xmath9-chain . let @xmath250 denote the non - negative span of @xmath268 . consider an @xmath140-chain @xmath269 containing @xmath270 which clearly exists , and the corresponding chamber @xmath249 ( which satisfies the property that @xmath271 ) . it follows from the simpliciality of @xmath249 ( thm : chamberstructure ) that @xmath250 is a sub - face of @xmath249 of dimension @xmath9 and that @xmath272 . therefore , every @xmath9-chain in @xmath144 can be expressed in the form @xmath273 for some @xmath9-face @xmath250 of @xmath30 . the theorem follows . in sec : prelimonposet , we describe some basic properties of the posets @xmath139 and @xmath144 . in sec : genfuncfaces , we prove thm : facecount . given two points @xmath274 and @xmath275 in @xmath144 , such that @xmath276 , consider the rectangle with vertical and horizontal edges whose ne - sw diagonal is the segment @xmath277 . the lattice points of this rectangle are exactly the points of the interval @xmath278 $ ] in @xmath144 . see figures [ fig : posetsn ] on page for examples of intervals in @xmath144 . the discrete quarter - plane @xmath133 is equipped with a grading given by the level function . the origin is the only point of level zero . denote by @xmath279 the set of points of level @xmath69 of @xmath133 . the level sets @xmath279 are all anti - chains and form a partition of @xmath133 . see fig : levelset for an example . note that @xmath280 for all @xmath281 , the interval @xmath282 in @xmath139 ( or in @xmath144 ) is isomorphic to @xmath144 as a graded poset . the isomorphism is induced by the translation by @xmath274 in @xmath133 : the map @xmath283 defines an isomorphism from @xmath284 to @xmath282 . 3 and @xmath144 . * ] and @xmath144 . * ] and @xmath144 . * ] in order to compute the generating function @xmath285 we first derive a recurrence relation for @xmath33 . [ prop : facerecursion ] the integers @xmath33 satisfy the recurrence relation @xmath286 for @xmath287 , where @xmath288 for @xmath289 and for @xmath290 $ ] . we prove prop : facerecursion by tracking the minimum of each @xmath9-chain and interpreting the @xmath291 largest elements of the chain as @xmath292-chain in @xmath293 with @xmath294 . the proposition is clear for @xmath295 . assume for the remainder of the proof that @xmath296 . denote by @xmath297 the set of @xmath9-chains in @xmath144 with minimum @xmath274 . suppose that @xmath298 . recall that translation by @xmath299 in @xmath133 defines an isomorphism from @xmath282 to @xmath300 . we see immediately that the process of removing @xmath274 from a @xmath9-chain and translating by @xmath299 ( i.e. , the function defined by @xmath301 ) defines a bijection from @xmath297 to the set of @xmath292-chains in @xmath302 . it follows that @xmath303 . note in particular that @xmath304 for @xmath305 . recall that @xmath144 contains @xmath306 elements of level @xmath307 for @xmath308 . grouping by level , we obtain that @xmath309 as desired . we need to introduce the correction term @xmath310 in order to deal with the case of @xmath295 . after some algebra , we can derive the expression for @xmath311 announced in thm : facecount . translating prop : facerecursion in to the language generating functions yields the linear equation @xmath312 where we used the identity @xmath313 the theorem follows by solving the linear equation for @xmath311 . corollaries [ cor : facerecurse ] , [ cor : facegenpoly ] and [ cor : facesmalllarge ] follow from thm : facecount by elementary algebraic manipulations . in particular , cor : facerecurse can be derived by noting that the coefficient of @xmath57 in @xmath314 vanishes for @xmath315 due to thm : facecount . cor : facegenpoly can be derived by computing a partial fraction decomposition for the expression for @xmath311 given in thm : facecount . on the other hand , cor : facesmalllarge is proved by regarding @xmath311 as the sum of geometric series with ratio @xmath316 . in sec : extremeraysflats , we prove thm : characterization.flats , and in sec : genfuncflats , we prove thm : flatcount . thm : characterization.flats will be derived from the following two lemmas . the first asserts that @xmath317 is a flat whenever @xmath318 is an ensemble , and the second asserts that every ensemble is of the form @xmath317 for some flat @xmath318 . to prove lem : flatyieldsensemble , we exploit the the fact that the set of extreme rays of a flat is the intersection of the sets of extreme rays of the hyperplanes containing it . recall that the set of extreme rays of @xmath319 @xmath320 corresponds in the poset @xmath144 to the union of intervals @xmath321\cup [ \varphi(i , j)+(1,1),\infty],\ ] ] where @xmath322 notice that @xmath323 with equality if and only if @xmath324 when @xmath209 , the second interval is empty and @xmath325 is a weight of the fundamental representation @xmath11 . the first interval is empty when @xmath326 and completely contained in the @xmath327-axis ( reps . @xmath220-axis ) when @xmath328 ( resp . @xmath329 ) . pedagogical examples are presented in fig : flats.hyperplane . * . the set of extreme rays of a hyperplane @xmath330 ( @xmath71 ) corresponding to a union of intervals @xmath331\cup [ \varphi(i , j)+(1,1),\infty]$ ] in the poset @xmath139 with @xmath332 . for extreme values ( @xmath328 , @xmath333 , @xmath329 , @xmath334 , @xmath335 ) one of the two intervals can be contained in one of the axis , shrink to a point , or disappear altogether . the points @xmath336 and @xmath337 are the only points of @xmath144 not correspond to any extreme ray of any hyperplane @xmath330 . [ fig : flats.hyperplane ] ] * . the set of extreme rays of a hyperplane @xmath330 ( @xmath71 ) corresponding to a union of intervals @xmath331\cup [ \varphi(i , j)+(1,1),\infty]$ ] in the poset @xmath139 with @xmath332 . for extreme values ( @xmath328 , @xmath333 , @xmath329 , @xmath334 , @xmath335 ) one of the two intervals can be contained in one of the axis , shrink to a point , or disappear altogether . the points @xmath336 and @xmath337 are the only points of @xmath144 not correspond to any extreme ray of any hyperplane @xmath330 . [ fig : flats.hyperplane ] ] * . the set of extreme rays of a hyperplane @xmath330 ( @xmath71 ) corresponding to a union of intervals @xmath331\cup [ \varphi(i , j)+(1,1),\infty]$ ] in the poset @xmath139 with @xmath332 . for extreme values ( @xmath328 , @xmath333 , @xmath329 , @xmath334 , @xmath335 ) one of the two intervals can be contained in one of the axis , shrink to a point , or disappear altogether . the points @xmath336 and @xmath337 are the only points of @xmath144 not correspond to any extreme ray of any hyperplane @xmath330 . [ fig : flats.hyperplane ] ] * . the set of extreme rays of a hyperplane @xmath330 ( @xmath71 ) corresponding to a union of intervals @xmath331\cup [ \varphi(i , j)+(1,1),\infty]$ ] in the poset @xmath139 with @xmath332 . for extreme values ( @xmath328 , @xmath333 , @xmath329 , @xmath334 , @xmath335 ) one of the two intervals can be contained in one of the axis , shrink to a point , or disappear altogether . the points @xmath336 and @xmath337 are the only points of @xmath144 not correspond to any extreme ray of any hyperplane @xmath330 . [ fig : flats.hyperplane ] ] * . the set of extreme rays of a hyperplane @xmath330 ( @xmath71 ) corresponding to a union of intervals @xmath331\cup [ \varphi(i , j)+(1,1),\infty]$ ] in the poset @xmath139 with @xmath332 . for extreme values ( @xmath328 , @xmath333 , @xmath329 , @xmath334 , @xmath335 ) one of the two intervals can be contained in one of the axis , shrink to a point , or disappear altogether . the points @xmath336 and @xmath337 are the only points of @xmath144 not correspond to any extreme ray of any hyperplane @xmath330 . [ fig : flats.hyperplane ] ] * . the set of extreme rays of a hyperplane @xmath330 ( @xmath71 ) corresponding to a union of intervals @xmath331\cup [ \varphi(i , j)+(1,1),\infty]$ ] in the poset @xmath139 with @xmath332 . for extreme values ( @xmath328 , @xmath333 , @xmath329 , @xmath334 , @xmath335 ) one of the two intervals can be contained in one of the axis , shrink to a point , or disappear altogether . the points @xmath336 and @xmath337 are the only points of @xmath144 not correspond to any extreme ray of any hyperplane @xmath330 . [ fig : flats.hyperplane ] ] * . the set of extreme rays of a hyperplane @xmath330 ( @xmath71 ) corresponding to a union of intervals @xmath331\cup [ \varphi(i , j)+(1,1),\infty]$ ] in the poset @xmath139 with @xmath332 . for extreme values ( @xmath328 , @xmath333 , @xmath329 , @xmath334 , @xmath335 ) one of the two intervals can be contained in one of the axis , shrink to a point , or disappear altogether . the points @xmath336 and @xmath337 are the only points of @xmath144 not correspond to any extreme ray of any hyperplane @xmath330 . [ fig : flats.hyperplane ] ] * . the set of extreme rays of a hyperplane @xmath330 ( @xmath71 ) corresponding to a union of intervals @xmath331\cup [ \varphi(i , j)+(1,1),\infty]$ ] in the poset @xmath139 with @xmath332 . for extreme values ( @xmath328 , @xmath333 , @xmath329 , @xmath334 , @xmath335 ) one of the two intervals can be contained in one of the axis , shrink to a point , or disappear altogether . the points @xmath336 and @xmath337 are the only points of @xmath144 not correspond to any extreme ray of any hyperplane @xmath330 . [ fig : flats.hyperplane ] ] * . the set of extreme rays of a hyperplane @xmath330 ( @xmath71 ) corresponding to a union of intervals @xmath331\cup [ \varphi(i , j)+(1,1),\infty]$ ] in the poset @xmath139 with @xmath332 . for extreme values ( @xmath328 , @xmath333 , @xmath329 , @xmath334 , @xmath335 ) one of the two intervals can be contained in one of the axis , shrink to a point , or disappear altogether . the points @xmath336 and @xmath337 are the only points of @xmath144 not correspond to any extreme ray of any hyperplane @xmath330 . [ fig : flats.hyperplane ] ] * . the set of extreme rays of a hyperplane @xmath330 ( @xmath71 ) corresponding to a union of intervals @xmath331\cup [ \varphi(i , j)+(1,1),\infty]$ ] in the poset @xmath139 with @xmath332 . for extreme values ( @xmath328 , @xmath333 , @xmath329 , @xmath334 , @xmath335 ) one of the two intervals can be contained in one of the axis , shrink to a point , or disappear altogether . the points @xmath336 and @xmath337 are the only points of @xmath144 not correspond to any extreme ray of any hyperplane @xmath330 . [ fig : flats.hyperplane ] ] consider a @xmath9-flat @xmath318 and denote by @xmath168 the set of extreme rays of @xmath318 . we will prove that @xmath168 is an ensemble . suppose that @xmath338 note that @xmath339 using the structure of extreme rays of hyperplanes @xmath319 and the identity @xmath340\cap[c , d]=[a\vee c , b\wedge d],\ ] ] we see immediately that @xmath341 can be minimally expressed in the form @xmath342 \cup [ a_m , b_m].\ ] ] we claim that this expression exhibits @xmath168 as an ensemble . condition ( 1 ) is satisfied because @xmath343 condition ( 2 ) follows from the minimality of the expression for @xmath168 . note that for all @xmath344 there exists @xmath345 such that @xmath346 and @xmath347 . in particular , we have @xmath348 for all @xmath349 , which is condition ( 3 ) . note that either @xmath350 for some @xmath345 for @xmath351 . condition ( 4 ) follows from the fact that @xmath323 for all @xmath352 . therefore , @xmath168 is an ensemble . it remains to prove that @xmath168 is a @xmath9-ensemble . note that because @xmath168 is a union of intervals , the number of distinct levels of elements of @xmath168 is the same as the length of any maximal chain in @xmath168 . suppose that @xmath168 is an @xmath353-ensemble and that @xmath354 is an @xmath353-chain . thm : characterization.faces ensures that there exists an @xmath353-face @xmath250 whose set of extreme rays is @xmath355 . because @xmath250 is a non - negatively spanned by @xmath355 and @xmath318 is non - negatively spanned by @xmath268 , we obtain that @xmath356 . it follows that @xmath357 . consider a @xmath9-face @xmath356 , which exists because @xmath318 is a @xmath9-flat and every flat is the union of the ( finitely many ) faces it contains . we see immediately that @xmath358 . thm : characterization.faces guarantees that @xmath266 is a @xmath9-chain in @xmath144 , from which it follows that @xmath359 . hence , we have @xmath360 , so that @xmath168 is a @xmath9-ensemble . the lemma follows . [ lem : getspecialflats ] given two points of @xmath361 and @xmath362 of @xmath141 such that @xmath363 , there exists a flat @xmath318 of @xmath30 such that @xmath364\cup [ a , \infty]\right ) \cap { \mathbb{e}^\ast}_n = \operatorname{rays}(l).\ ] ] \cup [ a,\infty]\cap { \mathbb{e}^\ast}_n$ ] * with @xmath365 in @xmath141 , @xmath366 , and @xmath367 . the three figures illustrate how to construct a flat @xmath318 whose set of extreme rays @xmath317 corresponds to the ensemble @xmath368 \cup [ a,\infty]\cap { \mathbb{e}^\ast}_n$ ] . in every case , @xmath318 is the intersection of two hyperplanes . there are three cases to consider : ( case 1 ) @xmath369 ( case 2.a ) @xmath369 and @xmath370 , ( case 2.b ) @xmath371 and @xmath372 . ] \cup [ a,\infty]\cap { \mathbb{e}^\ast}_n$ ] * with @xmath365 in @xmath141 , @xmath366 , and @xmath367 . the three figures illustrate how to construct a flat @xmath318 whose set of extreme rays @xmath317 corresponds to the ensemble @xmath368 \cup [ a,\infty]\cap { \mathbb{e}^\ast}_n$ ] . in every case , @xmath318 is the intersection of two hyperplanes . there are three cases to consider : ( case 1 ) @xmath369 ( case 2.a ) @xmath369 and @xmath370 , ( case 2.b ) @xmath371 and @xmath372 . ] \cup [ a,\infty]\cap { \mathbb{e}^\ast}_n$ ] * with @xmath365 in @xmath141 , @xmath366 , and @xmath367 . the three figures illustrate how to construct a flat @xmath318 whose set of extreme rays @xmath317 corresponds to the ensemble @xmath368 \cup [ a,\infty]\cap { \mathbb{e}^\ast}_n$ ] . in every case , @xmath318 is the intersection of two hyperplanes . there are three cases to consider : ( case 1 ) @xmath369 ( case 2.a ) @xmath369 and @xmath370 , ( case 2.b ) @xmath371 and @xmath372 . ] * case 1 : @xmath369 . consider the flat @xmath375 of @xmath30 . a direct calculation shows that @xmath364 \cup [ a , \infty]\right ) \cap { \mathbb{e}^\ast}_n = \operatorname{rays}\left(l\right).\ ] ] * case 2 : @xmath376 . write @xmath377 . consider the flat @xmath378 of @xmath30 . a direct calculation shows that @xmath364 \cup [ a , \infty]\right ) \cap { \mathbb{e}^\ast}_n = \operatorname{rays}\left(l\right).\ ] ] consider an ensemble given by @xmath379\cup [ a_1 , b_1]\cup \cdots \cup [ a_k , b_k]\right).\ ] ] define the intervals @xmath380\cup [ a_{i+1 } , \infty ) \text { for } 0 \le i \le k-1\\ d_k&= ( 0 , b_k].\end{aligned}\ ] ] lem : getspecialflats guarantees that each @xmath381 is the set of extreme rays of a flat @xmath382 . consider the flat @xmath383 a direct calculation shows that @xmath384 as desired . lem : flatyieldsensemble ensures that @xmath151 induces a function from the set of @xmath9-flats of @xmath30 to the set of @xmath9-ensembles in @xmath144 . because every flat is non - negatively spanned by its extreme rays , the induced function is injective . it remains to prove that the induced function is surjective . let @xmath168 be a @xmath9-ensemble . lem : ensembleyieldsflat guarantees the existence of a flat @xmath250 whose set of extreme rays is @xmath168 . the fact that @xmath250 is a @xmath9-flat follows from lem : flatyieldsensemble and the fact that @xmath168 is a @xmath9-ensemble . therefore , the induced function is surjective , which implies the theorem . a subset @xmath267 is a @xmath9-ensemble if and only if @xmath387 is a @xmath9-pseudo - ensemble ( necessarily with starting point 0 ) . in particular , the function defined by @xmath388 induces a bijection from the set of @xmath9-ensembles in @xmath144 to the set of @xmath9-pseudo - ensembles in @xmath139 with starting point @xmath389 . for all @xmath390 the set @xmath391 is the unique @xmath392-pseudo - ensemble in @xmath139 . denote by @xmath393 the number of @xmath9-pseudo - ensembles in @xmath139 for @xmath287 . define @xmath394 for @xmath289 and note that @xmath394 for @xmath395 and for @xmath396 . the following two lemmas illustrate the role of @xmath393 in counting ensembles : thm : flatcount will follow from the recurrence relations presented in the lemmas by routine algebraic manipulations involving generating functions . the idea of lem : rhonkrecursion is to divide into cases based on the starting point of a pseudo - ensemble . after translating the starting point to @xmath389 and removing @xmath389 , we obtain an ensemble . the lemma is clear for @xmath399 . assume for the remainder of the proof that @xmath400 . denote by @xmath401 the set of @xmath9-pseudo - ensembles in @xmath139 with starting point @xmath274 . recall that translation by @xmath299 induces an isomorphism from @xmath282 to @xmath302 . we see immediately that the composite of translation by @xmath299 and the removal of @xmath389 ( i.e. , the function @xmath402 ) induces a bijection from @xmath401 to the set of @xmath9-ensembles in @xmath302 . hence , we have @xmath403 recall that @xmath144 contains @xmath306 elements of level @xmath307 for @xmath308 . grouping by level , we obtain that @xmath404 as desired . we need to introduce the correction term @xmath405 in order to deal with the case of @xmath399 . the proof of lem : hnkrecursion is similar . we instead divide into cases based on the level of @xmath406 , the maximum of the first interval in the definition of an ensemble . translating @xmath407 to @xmath408 and intersecting with @xmath139 , we obtain a pseudo - ensemble . the lemma is clear for @xmath409 and for @xmath410 . assume for the remainder of the proof that @xmath411 . denote by @xmath412 the set of @xmath9-ensembles in @xmath144 whose first interval is @xmath413 $ ] . the hypothesis that @xmath414 ensures that @xmath415 , while the definition of an ensemble guarantees that @xmath415 for @xmath416 . recall that translation by @xmath299 induces an isomorphism from @xmath282 to @xmath302 . recall that translation by @xmath417 induces an isomorphism from @xmath418 to @xmath419 . we see immediately that the composite of translation by @xmath417 and intersection with @xmath133 ( i.e. , the function @xmath420 ) induces a bijection from @xmath297 to the set of @xmath421-pseudo - ensembles in @xmath422 . hence , we have @xmath423 the proof of cor : flatrecurse is similar to that of cor : facerecurse . in particular , cor : flatrecurse can be derived by noting that the coefficient of @xmath57 in @xmath428 vanishes for @xmath55 due to thm : flatcount . p. orlik and h. terao , http://dx.doi.org/10.1007/978-3-662-02772-1 [ _ arrangements of hyperplanes _ ] , vol . 300 of _ grundlehren der mathematischen wissenschaften [ fundamental principles of mathematical sciences]_. springer - 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we study the central hyperplane arrangement whose hyperplanes are the vanishing loci of the weights of the first and the second fundamental representations of @xmath0 restricted to the dual fundamental weyl chamber .
we obtain generating functions that count flats and faces of a given dimension .
this counting is interpreted in physics as the enumeration of the phases of the coulomb and mixed coulomb - higgs branches of a five dimensional gauge theory with 8 supercharges in presence of hypermultiplets transforming in the fundamental and antisymmetric representation of a @xmath1 gauge group as described by the intriligator - morrison - seiberg superpotential .
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the paper deals with the estimation problem in the heteroscedastic nonparametic regression model @xmath0 where the design points @xmath1 , @xmath2 is an unknown function to be estimated , @xmath3 is a sequence of centered independent random variables with unit variance and @xmath4 are unknown scale functionals depending on the design points and the regression function @xmath5 . typically , the notion of asymptotic optimality is associated with the optimal convergence rate of the minimax risk ( see e.g. , ibragimov , hasminskii,1981 ; stone,1982 ) . an important question in optimality results is to study the exact asymptotic behavior of the minimax risk . such results have been obtained only in a limited number of investigations . as to the nonparametric estimation problem for heteroscedastic regression models we should mention the papers by efromovich , 2007 , efromovich , pinsker , 1996 , and galtchouk , pergamenshchikov , 2005 , concerning the exact asymptotic behavior of the @xmath6-risk and the paper by brua , 2007 , devoted to the efficient pointwise estimation for heteroscedastic regressions . heteroscedastic regression models are largely used in financial mathematics , in particular , in problem of calibrating ( see e.g. , belomestny , reiss , 2006 ) . an example of heteroscedastic regression models is given by econometrics ( see , for example , goldfeld , quandt , 1972 , p. 83 ) , where for consumer budget problems one uses some parametric version of model with the scale coefficients defined as @xmath7 where @xmath8 , @xmath9 and @xmath10 are some unknown positive constants . the purpose of the article is to study asymptotic properties of the adaptive estimation procedure proposed in galtchouk , pergamenshchikov , 2007 , for which a non - asymptotic oracle inequality was proved for quadratic risks . we will prove that this oracle inequality is asymptotically sharp , i.e. the asymptotic quadratic risk is minimal . it means the adaptive estimation procedure is efficient under some the conditions on the scales @xmath4 which are satisfied in the case . note that in efromovich , 2007 , efromovich , pinsker , 1996 , an efficient adaptive procedure is constructed for heteroscedastic regression when the scale coefficient is independent of @xmath5 , i.e. @xmath11 . in galtchouk , pergamenshchikov , 2005 , for the model the asymptotic efficiency was proved under strong the conditions on the scales which are not satisfied in the case . moreover in the cited papers the efficiency was proved for the gaussian random variables @xmath3 that is very restrictive for applications of proposed methods to practical problems . in the paper we modify the risk . we take a additional supremum over the family of unknown noise distributions like to galtchouk , pergamenshchikov , 2006 . this modification allows us to eliminate from the risk dependence on the noise distribution . moreover for this risk a efficient procedure is robust with respect to changing the noise distribution . it is well known to prove the asymptotic efficiency one has to show that the asymptotic quadratic risk coincides with the lower bound which is equal to the pinsker constant . in the paper two problems are resolved : in the first one a upper bound for the risk is obtained by making use of the non - asymptotic oracle inequality from galtchouk , pergamenshchikov , 2007 , in the second one we prove that this upper bound coincides with the pinsker constant . let us remember that the adaptive procedure proposed in galtchouk , pergamenshchikov , 2007 , is based on weighted least - squares estimates , where the weights are proper modifications of the pinsker weights for the homogeneous case ( when @xmath12 ) relative to a certain smoothness of the function @xmath5 and this procedure chooses a best estimator for the quadratic risk among these estimators . to obtain the pinsker constant for the model one has to prove a sharp asymptotic lower bound for the quadratic risk in the case when the noise variance depends on the unknown regression function . in this case , as usually , we minorize the minimax risk by a bayesian one for a respective parametric family . then for the bayesian risk we make use of a lower bound ( see theorem 6.1 ) which is a modification of the van trees inequality ( see , gill , levit , 1995 ) . the paper is organized as follows . in section [ sec : ad ] we construct an adaptive estimation procedure . in section [ sec : co ] we formulate principal the conditions . the main results are presented in section [ sec : ma ] . the upper bound for the quadratic risk is given in section [ sec : up ] . in section [ sec : lo ] we give all main steps of proving the lower bound . in subsection [ subsec : tr ] we find the lower bound for the bayesian risk which minorizes the minimax risk . in subsection [ subsec : fa ] we study a special parametric functions family used to define the bayesian risk . in subsection [ subsec : br ] we choose a prior distribution for bayesian risk to maximize the lower bound . section [ sec : np ] is devoted to explain how to use the given procedure in the case when the unknown regression function is non periodic . in section [ sec : cn ] we discuss the main results and their practical importance . the proofs are given in section [ sec : pr ] . the appendix contains some technical results . in this section we describe the adaptive procedure proposed in galtchouk , pergamenshchikov , 2006 . we make use of the standard trigonometric basis @xmath13 in @xmath14 $ ] , i.e. @xmath15x)\,,\ j\ge 2\,,\ ] ] where the function @xmath16 for even @xmath17 and @xmath18 for odd @xmath17 ; @xmath19 $ ] denotes the integer part of @xmath20 . to evaluate the error of estimation in the model we will make use of the empiric norm in the hilbert space @xmath21 $ ] , generated by the design points @xmath22 of model . to this end , for any functions @xmath23 and @xmath24 from @xmath21 $ ] , we define the empiric inner product @xmath25 moreover , we will use this inner product for vectors in @xmath26 as well , i.e. if + @xmath27 and @xmath28 , then @xmath29 the prime denotes the transposition . notice that if @xmath30 is odd , then the functions @xmath31 are orthonormal with respect to this inner product , i.e. for any @xmath32 , @xmath33 where @xmath34 is kronecker s symbol , @xmath35 if @xmath36 and @xmath37 for @xmath38 . [ re.ad.1 ] note that in the case of even @xmath30 , the basis is orthogonal and it is orthonormal except the @xmath30th function for which the normalizing constant should be changed . the corresponding modifications of the formulas for even @xmath30 one can see in galtchouk , pergamenshchikov,2005 . to avoid these complications of formulas related to even @xmath30 , we suppose @xmath30 to be odd . thanks to this basis we pass to the discrete fourier transformation of model : @xmath39 where @xmath40 , @xmath41 , @xmath42 and @xmath43 we estimate the function @xmath5 by the weighted least squares estimator @xmath44 where the weight vector @xmath45 belongs to some finite set @xmath46 from @xmath47^n$ ] with @xmath48 . here we make use of the weight family @xmath46 introduced in galtchouk , pergamenshchikov , 2008 , i.e. @xmath49 where @xmath50 and @xmath51 $ ] . we suppose that the parameters @xmath52 and @xmath53 are functions of @xmath30 , i.e. @xmath54 and @xmath55 , such that , @xmath56 & \lim_{{\mathchoice{n\to\infty}{n\to\infty}{\lower.25ex\hbox{$\scriptstylen\to\infty$ } } { \lower0.25ex\hbox{$\scriptscriptstylen\to\infty$}}}}\,{\varepsilon}_{{\mathchoice{n}{n}{\lower.25ex\hbox{$\scriptstylen$ } } { \lower0.25ex\hbox{$\scriptscriptstylen$}}}}\,=\,0 \quad\mbox{and}\quad \lim_{{\mathchoice{n\to\infty}{n\to\infty}{\lower.25ex\hbox{$\scriptstylen\to\infty$ } } { \lower0.25ex\hbox{$\scriptscriptstylen\to\infty$}}}}\,n^{\nu}\,{\varepsilon}_{{\mathchoice{n}{n}{\lower.25ex\hbox{$\scriptstylen$ } } { \lower0.25ex\hbox{$\scriptscriptstylen$}}}}\,=+\infty\ , , \end{array } \right.\ ] ] for any @xmath57 . for example , one can take for @xmath48 @xmath58 where @xmath59 is any nonnegative constant . for each @xmath60 we define the weight vector @xmath61 as @xmath62 here @xmath63 $ ] with @xmath64 where @xmath65 is any nonnegative constant and @xmath66 [ re.ad.2 ] note that the weighted least squares estimators have been introduced by pinsker , 1981 , for continuous time optimal signal filtering in the gaussian noise . he proved that the mean - square asymptotic risk is minimized by weighted least squares estimators with weights of type . moreover he has found the sharp minimal value of the mean - square asymptotic risk , which was called later as the pinsker constant . nussbaum , 1985 , used the same method with proper modification for efficient estimation of the function @xmath5 of known smoothness in the homogeneous gaussian model , i.e. when @xmath12 and @xmath3 is i.i.d . @xmath67 sequence . to choose weights from the set we minimize the special cost function introduced by galtchouk , pergamenshchikov , 2007 this cost function is as follows @xmath68 where @xmath69 and @xmath70 $ ] . the penalty term we define as @xmath71 where @xmath72 is any slowly increasing sequence , i.e. @xmath73 for any @xmath57 . finally , we set @xmath74 the goal of this paper is to study asymptotic ( as @xmath75 ) properties of this estimation procedure . [ re.ad.3 ] now we explain why does one choose the cost function in the form . developing the empiric quadratic risk for estimate , one obtains @xmath76 it s natural to choose the weight vector @xmath77 for which this function reaches the minimum . since the last term on the right - hand part is independent of @xmath77 , it can be droped and one has to minimize with respect to @xmath77 the function equals the difference of two first terms on the right - hand part . it s clear that the minimization problem cannt be solved directly because the fourier coefficients @xmath78 are unknown.to overcome this difficulty , we replace the product @xmath79 by its asymptotically unbiased estimator @xmath80 ( see , galtchouk , pergamenshchikov , 2007 , 2008 ) . moreover , to pay this substitution , we introduce into the cost function the penalty term @xmath81 with a small coefficient @xmath82 . the form of the penalty term is provided by the principal term of the quadratic risk for weighted least - squares estimator , see galtchouk , pergamenshchikov , 2007 , 2008.the coefficient @xmath82 means , that the penalty is small , because the estimator @xmath80 approximates in mean the quantity @xmath79 asymptotically , as @xmath75 . note that the principal difference between the procedure and the adaptive procedure proposed by golubev , nussbaum , 1993 , for a homogeneous gaussian regression , consists in presence of the penalty term in the cost function . [ re.ad.4 ] as it was noted at remark [ re.ad.2 ] , nussbaum , 1985 , has shown that the weight coefficients of type provide the asymptotic minimum of the mean - squared risk at the regression function estimation problem for the homogeneous gaussian model , when the smoothness of the function @xmath5 is known . in fact , to obtain an efficient estimator one needs to take a weighted least squares estimator with the weight vector @xmath83 , where the index @xmath84 depends on smoothness of function @xmath5 and on coefficients @xmath4 , ( see below ) , which are unknown in our case . for this reason , galtchouk , pergamenshchikov , 2007 , 2008 , have proposed to make use of the family of coefficients , which contains the weight vector providing the minimum of the mean - squared risk . moreover , they proposed the adaptive procedure for which a non - asymptotic oracle inequality ( see , theorem [ th.m.1 ] below ) was proved under some weak conditions on the coefficients @xmath4 . it is important to note that due the properties of the parametric family , the secondary term in the oracle inequality is slowly increasing ( slower than any degree of @xmath30 ) . first we impose some conditions on unknown function @xmath5 in the model . let @xmath85 be the set of @xmath86-periodic @xmath87 times differentiable @xmath88 functions . we assume that @xmath5 belongs to the following set @xmath89 where @xmath90 denotes the norm in @xmath21 $ ] , i.e. @xmath91 moreover , we suppose that @xmath92 and @xmath93 are unknown parameters . note that , we can represent the set @xmath94 as an ellipse in @xmath21 $ ] , i.e. @xmath95\,:\ , \sum_{{\mathchoice{j=1}{j=1}{\lower.25ex\hbox{$\scriptstylej=1 $ } } { \lower0.25ex\hbox{$\scriptscriptstylej=1$}}}}^\infty\,a_{{\mathchoice{j}{j}{\lower.25ex\hbox{$\scriptstylej$ } } { \lower0.25ex\hbox{$\scriptscriptstylej$}}}}\theta^2_{{\mathchoice{j}{j}{\lower.25ex\hbox{$\scriptstylej$ } } { \lower0.25ex\hbox{$\scriptscriptstylej$}}}}\le r\}\,,\ ] ] where @xmath96 and @xmath97)^{2i}\,.\ ] ] here @xmath13 is the trigonometric basis defined in . now we describe the conditions on the scale coefficients @xmath98 . * _ @xmath99 for some unknown function @xmath100\times { { \cal l}}_{{\mathchoice{1}{1}{\lower.25ex\hbox{$\scriptstyle1 $ } } { \lower0.25ex\hbox{$\scriptscriptstyle1$}}}}[0,1 ] \to \bbr_+$ ] , which is square integrable with respect to @xmath20 such that @xmath101 where @xmath102 . moreover , @xmath103 and @xmath104 _ * _ for any @xmath105 $ ] , the operator @xmath106\to \bbr$ ] is differentiable in the frchet sense for any fixed function @xmath107 from @xmath108 $ ] , i.e. for any @xmath109 from some vicinity of @xmath107 in @xmath108 $ ] , @xmath110 where the frchet derivative @xmath111\to \bbr$ ] is a bounded linear operator and the residual term @xmath112 , for each @xmath105 $ ] , satisfies the following property : @xmath113 where @xmath114 . _ * _ there exists some positive constant @xmath115 such that for any function @xmath5 from @xmath108 $ ] the operator @xmath116 defined in the condition @xmath117 satisfies the following inequality for any function @xmath109 from @xmath108 $ ] : @xmath118 where @xmath119 . _ * _ the function @xmath120 corresponding to @xmath121 is continuous on the interval @xmath47 $ ] . moreover , @xmath122 _ [ re.co.1 ] let us explain the conditions @xmath123@xmath124 . in fact , this is the regularity conditions of the function @xmath125 generating the scale coefficients @xmath4 . condition @xmath123 means that the function @xmath126 should be uniformly integrable with respect to the first argument in the sens of convergence . moreover , this function should be separated from zero ( see inequality ) and bounded on the class ( see inequality ) . boundedness away from zero provides that the distribution of observations @xmath127 is nt degenerate in @xmath26 , and the boundedness means that the intensity of the noise vector should be finite , otherwise the estimation problem hasnt any sens . conditions @xmath117 and @xmath128 mean that the function @xmath129 is regular , at any fixed @xmath130 , with respect to @xmath5 in the sens , that it is differentiable in the frchet sens ( see e.g. , kolmogorov , fomin , 1989 ) and moreover the frchet derivative satisfies the growth condition given by the inequality which permits to consider the example . last the condition @xmath124 is the usual uniform continuity the condition of the function @xmath131 at the function @xmath132 . now we give some examples of functions satisfying the conditions @xmath123-@xmath124 . we set @xmath133 with some coefficients @xmath134 , @xmath135 , @xmath136 . in this case @xmath137 the frchet derivative is given by @xmath138 it is easy to see that the function satisfies the conditions @xmath123@xmath124 . moreover , the conditions @xmath123@xmath124 are satisfied by any function of type @xmath139 where the functions @xmath140 and @xmath141 satisfy the following the conditions : * @xmath140 is a @xmath47\times\bbr\to [ c_{{\mathchoice{0}{0}{\lower.25ex\hbox{$\scriptstyle0 $ } } { \lower0.25ex\hbox{$\scriptscriptstyle0$}}}}\,,\,+\infty)$ ] function ( with @xmath134 ) such that @xmath142 * @xmath141 is a continuously differentiable @xmath143 function such that @xmath144 where @xmath145 is the derivative of @xmath141 . in this case @xmath146 and @xmath147 where @xmath148 . now to estimate the last term in this inequality note that @xmath149 therefore , from the condition we get @xmath150 and through the bounyakovskii - cauchy - schwarz inequality , for any @xmath151 , @xmath152 now , the condition implies @xmath123 . moreover , the frchet derivative in this case is given by @xmath153 one can check directly that this operator satisfies the inequality with @xmath154 . denote by @xmath155 the family of distributions @xmath156 in @xmath26 of the vectors @xmath157 in the model such that the components @xmath158 are jointly independent , centered with unit variance and @xmath159 where @xmath160 is slowly increasing sequence , that is it satisfies the property . it is easy to see that , for any @xmath161 , the centered gaussian distribution in @xmath26 with unit covariation matrix belongs to the family @xmath155 . we will denote by @xmath162 this gaussian distribution . for any estimator @xmath163 we define the following quadratic risk @xmath164 where @xmath165 is the expectation with respect to the distribution @xmath166 of the observations @xmath167 with the fixed function @xmath5 and the fixed distribution @xmath168 of random variables @xmath3 in the model . moreover , to make the risk independent of the design points , in this paper we will make use of the risk with respect to the usual norm in @xmath21 $ ] also , i.e. @xmath169 if an estimator @xmath163 is defined only at the design points @xmath22 , then we extend it as step function onto the interval @xmath47 $ ] by setting @xmath170 , for all @xmath130 , where @xmath171}{[0,x_{1}]}{\lower.25ex\hbox{$\scriptstyle[0,x_{1}]$ } } { \lower0.25ex\hbox{$\scriptscriptstyle[0,x_{1}]$}}}}(x ) + \sum_{k=2}^n\,f(x_k)\chi_{{\mathchoice{(x_{k-1},x_k]}{(x_{k-1},x_k]}{\lower.25ex\hbox{$\scriptstyle(x_{k-1},x_k]$ } } { \lower0.25ex\hbox{$\scriptscriptstyle(x_{k-1},x_k]$}}}}(x)\,.\ ] ] in galtchouk , pergamenshchikov , 2007 , 2008 the following non - asymptotic oracle inequality has been shown for the procedure . [ th.m.1 ] assume that in the model the function @xmath5 belongs to @xmath172 . then , for any odd @xmath48 and @xmath92 , the estimator @xmath173 satisfies the following oracle inequality @xmath174 where the function @xmath175 is such that , for any @xmath57 , @xmath176 [ re.m.1 ] note that in galtchouk , pergamenshchikov , 2007 , 2008 , the oracle inequality is proved for the model , where the random variables @xmath3 are independent identically distributed . in fact , the result and the proof are true for independent random variables which are not identically distributed , i.e. for any distribution of the random vector @xmath157 from @xmath155 . now we formulate the main asymptotic results . to this end , for any function @xmath151 , we set @xmath177 where @xmath178 it is well known ( see e.g. , nussbaum , 1985 ) that the optimal rate of convergence is @xmath179 when the risk is taken uniformly over @xmath180 . [ th.m.2 ] assume that in the model the sequence @xmath181 fulfills the condition @xmath182 . then the estimator @xmath173 from satisfies the inequalities @xmath183 and @xmath184 the following result gives the sharp lower bound for risk and show that @xmath185 is the . [ th.m.3 ] assume that in the model the sequence @xmath181 satisfies the conditions @xmath186 @xmath187 . then the risks and admit the following asymptotic lower bounds @xmath188 and @xmath189 [ re.m.2 ] note that in galtchouk , pergamenshchikov , 2005 , an asymptotically efficient estimator has been constructed and results similar to theorems [ th.m.2 ] and [ th.m.3 ] were claimed for the model . in fact the upper bound is true there under some additional condition on the smoothness of the function @xmath5 , i.e. on the parameter @xmath87 . in the cited paper this additional condition is not formulated since erroneous inequality @xmath190 . to avoid using this inequality we modify the estimating procedure by introducing the penalty term @xmath191 in the cost function . by this way we remove all additional conditions on the smoothness parameter @xmath87 . [ re.m.3 ] in fact to obtain the non - asymptotic oracle inequality , it is nt necessary to make use of equidistant design points and the trigonometric basis . one may take any design points ( deterministic or random ) and any orthonormal basis satisfying . but to obtain the property one needs to impose some technical conditions ( see galtchouk , pergamenshchikov , 2008 ) . note that the results of theorem [ th.m.2 ] and theorem [ th.m.3 ] are based on equidistant design points and the trigonometric basis . in this section we prove theorem [ th.m.2 ] . to this end we will make use of the oracle inequality . we have to find an estimator from the family - for which we can show the upper bound . we start with the construction of such an estimator . first we put @xmath192 where @xmath193 . then we choose an index from the set @xmath194 as @xmath195 where @xmath87 is the parameter of the set @xmath180 and @xmath196 . finally , we set @xmath197 now we show the upper bound for this estimator . [ th.u.1 ] assume that the condition @xmath182 holds . then @xmath198 [ re.u.1 ] note that the estimator @xmath199 belongs to the family - , but we ca nt use directly this estimator because the parameters @xmath87 , @xmath200 and @xmath201 are unknown . we can use this upper bound only through the oracle inequality proved for procedure . now theorem [ th.m.1 ] and theorem [ th.u.1 ] imply the upper bound . to obtain the upper bound we need the following auxiliary result . [ sec : le.u.1 ] for any @xmath202 and any estimate @xmath203 of @xmath151 , @xmath204 where the function @xmath205 is defined in . proof of this lemma is given in appendix [ subsec : a.1 ] . now inequality and this lemma imply the upper bound . hence theorem [ th.m.2 ] . in this section we give the main steps of proving the lower bounds and . in common , we follow the same scheme as nussbaum , 1985 . we begin with minorizing the minimax risk by a bayesian one constructed on a parametric functional family introduced in section [ subsec : fa ] ( see ) and using the prior distribution . further , a special modification of the van trees inequality ( see , theorem [ th.tr.1 ] ) yields a lower bound for the bayesian risk depending on the chosen prior distribution , of course . finally , in section [ subsec : br ] , we choose parameters of the prior distribution ( see ) providing the maximal value of the lower bound for the bayesian risk . this value coincides with the pinsker constant as it is shown in section [ subsec : th.m.3 ] . let @xmath206 be a statistical model relative to the observations @xmath127 governed by the regression equation @xmath207 where @xmath208 are i.i.d . @xmath67 random variables , @xmath209 is a unknown parameter vector , @xmath210 is a unknown ( or known ) function and @xmath211 , with the function @xmath125 defined in the condition @xmath123 . assume that a prior distribution @xmath212 of the parameter @xmath213 in @xmath214 is defined by the density @xmath215 of the following form @xmath216 where @xmath217 is a continuously differentiable bounded density on @xmath218 with @xmath219 let @xmath220 be a continuously differentiable @xmath221 function such that , for any @xmath222 , @xmath223 where @xmath224 let @xmath225 be an estimator of @xmath226 based on observations @xmath127 . for any @xmath227 - measurable integrable function @xmath228 , we set @xmath229 where @xmath230 is the expectation with respect to the distribution @xmath231 of the vector @xmath232 . note that in this case @xmath233 where @xmath234 we prove the following result . [ th.tr.1 ] assume that the conditions @xmath235 hold . moreover , assume that the function @xmath236 with @xmath237 is uniformly over @xmath130 differentiable with respect to @xmath238 , i.e. for any @xmath222 there exists a function @xmath239 $ ] such that @xmath240 where @xmath241 , all coordinates are @xmath242 , except the i - th equals to @xmath86 . then for any square integrable estimator @xmath225 of @xmath226 and any @xmath222 , @xmath243 where @xmath244 , @xmath245 and @xmath246 @xmath247 , the operator @xmath248 is defined in the condition @xmath186 . * proof * is given in appendix [ subsec : a.2 ] . [ re.tr.1 ] note that the inequality is some modification of the van trees inequality ( see , gill , levit , 1995 ) adapted to the model . in this section we define and study some special parametric family of kernel function which will be used to prove the sharp lower bound . let us begin by kernel functions . we fix @xmath249 and we set @xmath250 where @xmath251 is the indicator of a set @xmath252 , the kernel @xmath253 is such that @xmath254 it is easy to see that the function @xmath255 possesses the properties : @xmath256 moreover , for any @xmath257 and @xmath258 @xmath259 we divide the interval @xmath47 $ ] into @xmath260 equal subintervals of length @xmath261 and on each of them we construct a kernel - type function which equals to zero at the boundary of the subinterval together with all derivatives . it provides that the fourier partial sums with respect to the trigonometric basis in @xmath262 $ ] give a natural parametric approximation to the function on each subinterval . let @xmath263 be the trigonometric basis in @xmath264 $ ] , i.e. @xmath265 x\right)\,,\ j\ge 2\,,\ ] ] where the functions @xmath266 are defined in . now , for any array @xmath267 we define the following function @xmath268 where @xmath269 , @xmath270 - 1\,.\ ] ] we assume that the sequences @xmath271 and @xmath272 , satisfy the following conditions . @xmath273 _ the sequence @xmath274 as @xmath75 and for any @xmath57 @xmath275 moreover , there exist @xmath276 and @xmath277 such that @xmath278 _ to define a prior distribution on the family of arrays , we choose the following random array @xmath279 with @xmath280 where @xmath281 are i.i.d . @xmath67 random variables and @xmath282 are some nonrandom positive coefficients . we make use of gaussian variables since they possess the minimal fisher information and therefore maximize the lower bound . we set @xmath283 we assume that the coefficients @xmath284 satisfy the following conditions . @xmath285 _ there exists a sequence of positive numbers @xmath286 such that @xmath287 moreover , for any @xmath288 , @xmath289 _ @xmath290 _ for some @xmath291 @xmath292 _ @xmath293 _ there exists @xmath294 such that @xmath295 _ [ sec : pr.fa.1 ] let the conditions @xmath273@xmath285 . then , for any @xmath57 and for any @xmath296 , @xmath297 [ sec : pr.fa.2 ] let the conditions @xmath273@xmath293 . then , for any @xmath57 , @xmath298 [ sec : pr.fa.3 ] let the conditions @xmath273@xmath293 . then , for any @xmath57 , @xmath299 [ sec : pr.fa.4 ] let the conditions @xmath273@xmath293 . then for any function @xmath300 satisfying the conditions and @xmath124 @xmath301 proofs of propositions [ sec : pr.fa.1][sec : pr.fa.4 ] are given in appendix . now we will obtain the lower bound for the bayesian risk that yields the lower bound for the minimax risk . we make use of the sequence of random functions @xmath302 defined in - with the coefficients @xmath282 satisfying the conditions @xmath273@xmath293 which will be chosen later . for any estimator @xmath203 we introduce now the corresponding bayes risk @xmath303 where the kernel family @xmath304 is defined in , @xmath212 denotes the distribution of the random array @xmath213 defined by in @xmath214 with @xmath305 . we remember that @xmath162 is a centered gaussian distribution in @xmath26 with unit covariation matrix . first of all , we replace the functions @xmath203 and @xmath5 by their fourier series with respect to the basis @xmath306 by making use of this basis we can estimate the norm @xmath307 from below as @xmath308 where @xmath309 moreover , from the definition one gets @xmath310 it is easy to see that the functions @xmath311 satisfy the condition for gaussian prior densities . in this case ( see the definition in ) we have @xmath312 where @xmath313 now to obtain a lower bound for the bayes risk @xmath314 we make use of theorem [ th.tr.1 ] which implies that @xmath315 where @xmath316 and @xmath317 with @xmath318 . in the appendix we show that @xmath319 and @xmath320 this means that , for any @xmath57 and for sufficiently large @xmath30 , @xmath321 where @xmath322 is defined in . therefore , if we denote in @xmath323 we obtain , for sufficiently large @xmath30 , @xmath324 in the appendix we show that @xmath325 where @xmath326 therefore we can write that , for sufficiently large @xmath30 , @xmath327 obviously , to obtain a `` good '' lower bound for the risk @xmath314 one needs to maximize the right - hand side of the inequality . hence we choose the coefficients @xmath328 by maximization the function @xmath329 , i.e. @xmath330 the parameter @xmath331 will be chosen later to satisfy the condition @xmath290 . by the lagrange multipliers method it is easy to find that the solution of this problem is given by @xmath332 with @xmath333 to obtain a positive solution in we need to impose the following condition @xmath334 moreover , from the condition @xmath290 we obtain that @xmath335 where @xmath336 note that by the condition @xmath124 the function @xmath120 is continuous on the interval @xmath47 $ ] , therefore @xmath337 now we have to choose the sequence @xmath338 . note that if we put in @xmath339 we can rewrite the inequality as @xmath340 where @xmath341 it is clear that @xmath342 therefore to obtain a positive finite asymptotic lower bound in we have to take the parameter @xmath343 as @xmath344 with some positive coefficient @xmath345 . moreover , the conditions - imply that , for sufficiently large @xmath30 , @xmath346 moreover , taking into account that for sufficiently large @xmath30 @xmath347 we obtain the following condition on @xmath345 @xmath348 where @xmath349 to maximize the function @xmath350 on the right - hand side of the inequality we take @xmath351 defined in . therefore we obtain that @xmath352 where @xmath353 furthermore , taking into account that @xmath354 we get @xmath355 where @xmath356 this means that to obtain in the maximal lower bound one has to take in @xmath357 it is important to note that if one defines the prior distribution @xmath212 in the bayesian risk by formulas , , and , then the bayesian risk would depend on a parameter @xmath358 , i.e. @xmath359 . therefore , the inequality implies that , for any @xmath358 , @xmath360 where the function @xmath361 is defined in for @xmath121 . now to end the definition of the sequence of the random functions @xmath362 defined by and one has to define the sequence @xmath363 . let us remember that we make use of the sequence @xmath362 with the coefficients @xmath282 constructed in for @xmath351 given in and for the sequence @xmath343 given by and for some fixed arbitrary @xmath358 . we will choose the sequence @xmath363 to satisfy the conditions @xmath273@xmath293 . one can take , for example , @xmath364 + 1 $ ] . then the condition @xmath273 is trivial . moreover , taking into account that in this case @xmath365 we find thanks to the convergence @xmath366 therefore , the solution , for sufficiently large @xmath30 , satisfies the following inequality @xmath367 now it is easy to see that the condition @xmath285 holds with @xmath368 and the condition @xmath293 holds for arbitrary @xmath369 . as to the condition @xmath290 , note that in view of the definition of @xmath370 in we get @xmath371 hence the condition @xmath290 . now we consider the estimation problem of the non periodic regression function @xmath5 in the model . in this case we will estimate the function @xmath5 on any interior interval @xmath372 $ ] of @xmath47 $ ] , i.e. for @xmath373 . it should be pointed out that at the boundary points @xmath374 and @xmath375 , one must to make use of kernel estimators ( see brua , 2007 ) . let now @xmath376 be a infinitely differentiable @xmath47\to\bbr_{{\mathchoice{+}{+}{\lower.25ex\hbox{$\scriptstyle+$ } } { \lower0.25ex\hbox{$\scriptscriptstyle+$}}}}$ ] function such that @xmath377 for @xmath378 and @xmath379 for all @xmath380 , for example , @xmath381}{[a',b']}{\lower.25ex\hbox{$\scriptstyle[a',b']$ } } { \lower0.25ex\hbox{$\scriptscriptstyle[a',b']$}}}}(z)\ , \d z\,,\ ] ] where @xmath141 is some kernel function introduced in , @xmath382 multiplying the equation by the function @xmath383 and simulating the i.i.d . @xmath67 sequence @xmath384 one comes to the estimation problem of the periodic regression function @xmath385 , i.e. @xmath386 where @xmath387 , @xmath388 and @xmath389 is some sufficiently small parameter . it is easy to see that if the sequence @xmath4 satisfies the conditions @xmath390 , then the sequence @xmath391 satisfies these conditions as well with @xmath392 in conclusion , it should be noted that this paper completes the investigation of the estimation problem of the nonparametric regression function for the heteroscedastic regression model in the case of quadratic risk . it is proved that the adaptive procedure satisfies the non asymptotic oracle inequality and it is asymptotically efficient for estimating a periodic regression function . moreover , in section [ sec : np ] we have explained how to apply the procedure to the case of non periodic function . as far as we know , the procedure is unique for estimating the regression function at the model . let us remember once more the main steps of this investigation . the procedure combines the both principal aspects of nonparametric estimation : non asymptotic and asymptotic . non - asymptotic aspect is based on the selection model procedure with penalization ( see e.g. , barron , birg and massart , 1999 , or fourdrinier , pergamenshchikov , 2007 ) . our selection model procedure differs from the commonly used one by a small coefficient in the penalty term going to zero that provides the sharp non - asymptotic oracle inequality . moreover , the commonly used selection model procedure is based on the least - squares estimators whereas our procedure uses weighted least - squares estimators with the weights minimizing the asymptotic quadratic risk that provides the asymptotic efficiency , as the final result . from practical point of view , the procedure gives an acceptable accuracy even for small samples as it is shown via simulations by galtchouk , pergamenshchikov , 2008 . to prove the theorem we will adapt to the heteroscedastic case the corresponding proof from nussbaum , 1985 . first , from we obtain that , for any @xmath168 , @xmath393 where @xmath394 setting now @xmath395 with the function @xmath396 defined in , the index @xmath397 defined in , @xmath398 $ ] , @xmath399 $ ] and @xmath400 we rewrite as follows @xmath401 with @xmath402 note that we have decomposed the first term on the right - hand of into the sum @xmath403 this decomposition allows us to show that @xmath404 is negligible and further to approximate the first term by a similar term in which the coefficients @xmath405 will be replaced by the fourier coefficients @xmath406 of the function @xmath5 . taking into account the definition of @xmath396 in we can bound @xmath407 as @xmath408 therefore , by lemma [ sec : le.a.1 ] we obtain @xmath409 let us consider now the next term @xmath410 . we have @xmath411 now by lemma [ sec : le.a.2 ] and the definition we obtain directly the same property for @xmath410 , i.e. @xmath412 setting @xmath413 and applying the well - known inequality @xmath414 to the first term on the right - hand side of the inequality we obtain that , for any @xmath296 and for any @xmath168 , @xmath415 \label{sec : up.1 - 3 } & + { \widetilde}{\delta}_{{\mathchoice{1,n}{1,n}{\lower.25ex\hbox{$\scriptstyle1,n$ } } { \lower0.25ex\hbox{$\scriptscriptstyle1,n$}}}}+ { \widetilde}{\delta}_{{\mathchoice{2,n}{2,n}{\lower.25ex\hbox{$\scriptstyle2,n$ } } { \lower0.25ex\hbox{$\scriptscriptstyle2,n$}}}}+(1 + 1/\delta)\ , { \widetilde}{\delta}_{{\mathchoice{3,n}{3,n}{\lower.25ex\hbox{$\scriptstyle3,n$ } } { \lower0.25ex\hbox{$\scriptscriptstyle3,n$}}}}\,,\end{aligned}\ ] ] where @xmath416 taking into account that @xmath93 and that @xmath417 we can show through lemma [ sec : le.a.3 ] that @xmath418 therefore , the inequality yields @xmath419 and to prove it suffices to show that @xmath420 first , it should be noted that the definition and the inequalities - imply directly @xmath421 moreover , by the definition of @xmath422 in , for sufficiently large @xmath30 , for which @xmath423 we find @xmath424 therefore , by the definition of the coefficients @xmath425 in @xmath426 furthermore , in view of the definition we calculate directly @xmath427 now , the definition of @xmath180 in and the condition imply the inequality . hence theorem [ th.u.1 ] . in this section we prove theorem [ th.m.3 ] . lemma [ sec : le.u.1 ] implies that to prove the lower bounds and , it suffices to show @xmath428 where @xmath429 for any estimator @xmath203 , we denote by @xmath430 its projection onto @xmath431 , i.e. + @xmath432 . since @xmath180 is a convex set , we get @xmath433 now we introduce the following set @xmath434 where @xmath281 are i.i.d . @xmath67 random variables from and the sequence @xmath286 is given in the condition @xmath285 . therefore , we can write that @xmath435 here the kernel function family @xmath304 is given in in which @xmath436 + 1 $ ] and the parameter @xmath437 is defined in and ; the measure @xmath212 is defined in . moreover , note that on the set @xmath438 the random function @xmath439 is uniformly bounded , i.e. @xmath440 where the coefficient @xmath441 is defined in . thus , we estimate the risk @xmath442 from below as @xmath443 with @xmath444 by making use of the bayes risk with the prior distribution given by formulae , , and for any fixed parameter @xmath358 we rewrite the lower bound for @xmath442 as @xmath445 with @xmath446 in section [ subsec : br ] we proved that the parameters in chosen prior distribution satisfy the conditions @xmath273@xmath293 . therefore propositions [ sec : pr.fa.2][sec : pr.fa.3 ] and the limit imply that , for any @xmath57 , @xmath447 moreover , by the condition @xmath124 the sequence @xmath448 goes to @xmath361 as @xmath75 . therefore , from this , and we get , for any @xmath358 , @xmath449 where @xmath450 is defined in . limiting here @xmath451 implies inequality . hence theorem [ th.m.3 ] . for any @xmath202 , by making use of the elementary inequality @xmath456 one gets @xmath457 moreover , for any @xmath452 with @xmath93 , by the bounyakovskii - cauchy - schwarz inequality we obtain that @xmath458 hence lemma [ sec : le.u.1 ] . for any @xmath459 we set @xmath460 note that due to the condition , the density is bounded , i.e. @xmath461 so through we obtain that @xmath462 therefore , integrating by parts yields @xmath463 now the bounyakovskii - cauchy - schwarz inequality gives the following lower bound @xmath464 to estimate the denominator in the last ratio , note that @xmath465 > from it follows that @xmath466 moreover , the conditions @xmath117 and imply @xmath467 from which it follows @xmath468 this implies inequality . hence theorem [ th.tr.1 ] . first note that , for @xmath130 , we can represent the @xmath469th derivative as @xmath470 where @xmath471 therefore @xmath472 and by the bounyakovskii - cauchy - schwarz inequality we obtain that @xmath473 with @xmath474 and @xmath475 now we show that , for any @xmath476 and @xmath296 , @xmath477 to this end note that @xmath478 therefore , taking into account the definition of the set @xmath479 in , the functions @xmath480 with @xmath481 can be estimated on this set as @xmath482 and by we get , for any @xmath296 and sufficiently large @xmath30 , @xmath483 moreover , for sufficiently large @xmath30 , @xmath484 therefore , the condition @xmath273 implies @xmath485 for any @xmath57 . hence proposition [ sec : pr.fa.1 ] . first of all we prove that for @xmath486 from the condition @xmath290 @xmath487 indeed , putting in @xmath488 we can represent the @xmath87th derivative of @xmath489 as follows @xmath490 with @xmath491 and @xmath492 first , note that , we can estimate the norm of @xmath493 by the same way as in the inequality , i.e. @xmath494 by making use of we obtain that , for any @xmath495 and for any @xmath296 , @xmath496 let us consider now the last term in . taking into account that @xmath497 we get @xmath498 therefore from the condition @xmath290 we get for sufficiently large @xmath30 @xmath499 with @xmath500 we show that for any @xmath57 and for any @xmath296 @xmath501 indeed , by the chebychev inequality for any @xmath502 @xmath503 note now that according to the burkholder - davis - gundy inequality for any @xmath504 there exists a constant @xmath505 such that @xmath506 moreover , by putting @xmath507 we can estimate the random variable @xmath508 as @xmath509 therefore , by the condition @xmath293 , for sufficiently large @xmath30 , @xmath510 where @xmath511 . now the condition @xmath273 implies , for sufficiently large @xmath30 , @xmath512 thus , choosing in @xmath513 we obtain the limiting equality which together with - implies . now it is easy to deduce that proposition [ sec : pr.fa.1 ] yields proposition [ sec : pr.fa.2 ] . first of all , we recall that , due to the condition @xmath285 , @xmath514 therefore , taking into account that @xmath515 we obtain , for sufficiently large @xmath30 , @xmath516 moreover , for any @xmath517 and @xmath518 , we estimate the last term as @xmath519 & + n\,\p(\xi^c_{{\mathchoice{n}{n}{\lower.25ex\hbox{$\scriptstylen$ } } { \lower0.25ex\hbox{$\scriptscriptstylen$ } } } } ) + 2\e\,\zeta^2\,\chi_{{\mathchoice{\{\zeta^2\ge n\}}{\{\zeta^2\ge n\}}{\lower.25ex\hbox{$\scriptstyle\{\zeta^2\ge n\}$ } } { \lower0.25ex\hbox{$\scriptscriptstyle\{\zeta^2\ge n\}$}}}}\,,\end{aligned}\ ] ] where @xmath520 . by applying now proposition [ sec : pr.fa.2 ] and the limit we come to proposition [ sec : pr.fa.3 ] . [ sec : le.a.2 ] for any @xmath524 , @xmath525}n^{-m}\left|\sum_{l=2}^{n}\,l^m \ , \left ( \phi^2_{{\mathchoice{l}{l}{\lower.25ex\hbox{$\scriptstylel$ } } { \lower0.25ex\hbox{$\scriptscriptstylel$}}}}(x)-1 \right)\ , \right|\le 2^m\,.\ ] ] first of all , note that proposition [ sec : pr.fa.4 ] , the condition and the condition @xmath124 imply that @xmath532 let us show now that for any continuously differentiable function @xmath109 on @xmath533 $ ] @xmath534 indeed , setting @xmath535 we deduce @xmath536 where @xmath537 + 1 $ ] , @xmath538 $ ] , @xmath539 + 1-n{\widetilde}{x}_{{\mathchoice{m}{m}{\lower.25ex\hbox{$\scriptstylem$ } } { \lower0.25ex\hbox{$\scriptscriptstylem$}}}})/(nh ) \quad \mbox{and } \quad v^*=([n{\widetilde}{x}_{{\mathchoice{m}{m}{\lower.25ex\hbox{$\scriptstylem$ } } { \lower0.25ex\hbox{$\scriptscriptstylem$}}}}+nh]-n{\widetilde}{x}_{{\mathchoice{m}{m}{\lower.25ex\hbox{$\scriptstylem$ } } { \lower0.25ex\hbox{$\scriptscriptstylem$}}}})/(nh)\,.\ ] ] therefore , taking into account that the derivative of the function @xmath109 is bounded on the interval @xmath533 $ ] we obtain that @xmath540 taking into account the conditions on the sequence @xmath272 given in @xmath273 we obtain limiting equality which together with implies . now we study the behavior of @xmath541 . due to the inequality we estimate the frchet derivative as @xmath542 consider now the fisrt term on the right - hand side of this inequality . we have @xmath543 we recall that the sequence @xmath441 is defined in . therefore , property implies @xmath544 as to the second term on the right - hand side of , we get @xmath545 similarly , @xmath546 and , by @xmath547 therefore , @xmath548 and the condition @xmath273 implies . indeed , by the direct calculation it easy to see that , for any @xmath549 and for any vector @xmath550 , @xmath551 where the operator @xmath552 is defined in . moreover , we remember that @xmath553 . therefore , taking into account the property we obtain . brua , j .- . asymptotic efficient estimators for non - parametric heteroscedastic model . _ statistical metodologie _ , accepted to publication , available at _ http://hal.archives-ouvertes.fr/hal/-00178536/fr/ _ galtchouk , l.,@xmath554 pergamenshchikov , s.(2005 ) . efficient adaptive nonparametric estimation in heteroscedastic regression models . preprint of the strasbourg louis pasteur university , irma , 2005/020 available at _ http://www.univ-rouen.fr/lmrs/persopage/pergamenchtchikov _ galtchouk , l.,@xmath554 pergamenshchikov , s.(2007 ) . adaptive nonparametric estimation in heteroscedastic regression models . sharp non - asymptotic oracle inequalities . preprint of the strasbourg louis pasteur university , irma , 2007/09 , available at _ http://hal.archives-ouvertes.fr/hal/-00179856/fr/ _ galtchouk , l.,@xmath554 pergamenshchikov , s.(2008 ) . sharp non - asymptotic oracle inequalities for nonparametric heteroscedastic regression models . _ journal of nonparametric statistics _ , accepted to publication
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the paper deals with asymptotic properties of the adaptive procedure proposed in the author paper , 2007 , for estimating a unknown nonparametric regression . we prove that this procedure is asymptotically efficient for a quadratic risk , i.e. the asymptotic quadratic risk for this procedure coincides with the pinsker constant which gives a sharp lower bound for the quadratic risk over all possible estimates .
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the euclidean tsp with neighborhoods ( tspn ) is the following problem : given a set @xmath0 of @xmath1 regions ( subsets of @xmath2 ) , find a shortest tour that visits at least one point from each region . even for disjoint or connected regions , the tspn does not admit a ptas unless @xmath6 @xcite . aiming for a ptas under additional restrictions on the input , @xcite and @xcite require connected and disjoint regions , and both introduce a notion of @xmath3-fatness . [ def : afat - mit ] a region @xmath4 of points in the plane is * @xmath7-fat * , if it contains a disk of diameter @xmath8 . [ def : afat - elb ] a region @xmath4 in the plane is * @xmath7-fat@xmath9 * , if for every disk @xmath10 , such that the center of @xmath10 is contained in @xmath4 but @xmath10 does not fully contain @xmath4 , the area of the intersection @xmath11 is at least @xmath12 times the area of @xmath10 . for @xmath3-fat@xmath13 regions , chan and elbassioni @xcite developed a quasi - polynomial time approximation scheme ( even for a more general notion of fatness and in more general metric spaces ) . mitchell @xcite was the first to consider @xmath3-fat regions . bodlaender et al . @xcite introduced the notion of geographic clustering , where each region contains a square of size @xmath14 and has diameter at most @xmath15 for a fixed constant @xmath16 , which is a special case of @xmath3-fat regions . they showed that the tspn with geographic clustering admits a ptas based on arora s framework for the euclidean tsp . in all cases , @xmath3-fatness provides a lower bound ( in terms of their diameters ) on the length of a tour visiting disjoint regions , but in the following , the second definition will turn out to be more useful . throughout this paper , @xmath17 and @xmath18 will be constants . the core of mitchell s algorithm is dynamic programming , which requires certain restrictions on the space of solutions . to this end , mitchell claims the following : there is an almost optimal tour ( up to a factor of @xmath19 ) such that : a. the tour visits the minimum - diameter axis - aligned rectangle @xmath20 intersecting all regions , and therefore has to be located within a window @xmath21 of diameter @xmath22 intersecting @xmath20 . we distinguish internal regions @xmath23 that are entirely contained in @xmath21 , and external regions . b. we can require the vertices of the tour to lie on a polynomial - size grid ( in @xmath1 and @xmath24 ) within this rectangle . c. the tour is a connected eulerian graph fulfilling the @xmath25-guillotine property " ( which roughly states that there is a recursive decomposition of the bounding box of the tour by cutting it into subwindows such that the structure of internal regions and tour segments on the cut is of bounded complexity in @xmath26 and @xmath27 ) , again at a loss of only @xmath28 for appropriately chosen @xmath26 and @xmath27 . d. the tour obeys ( b ) and ( c ) simultaneously . e. the external regions can be dealt with efficiently as there is only a polynomial number of ways for them to be visited by an @xmath25-guillotine tour ( i. e. for every cut , there is a polynomial number of options for which regions will be visited on which side of it ) . under these assumptions , mitchell states a dynamic programming algorithm . starting with a window ( axis - parallel rectangle ) @xmath21 , which is assumed to contain all edges of the tour , every subwindow @xmath29 defines several subproblems ( see figure [ pic : sub ] ) . the subproblems also enumerate all possible configurations of edge segments intersecting its boundary , connection patterns of these segments , internal ( contained in @xmath21 ) and external ( intersecting @xmath30 ) regions to be visited inside and outside of the window , cuts ( horizontal or vertical lines dividing @xmath29 into two subwindows ) and configurations on the cut . for each cut , the subproblems to both sides will already have been solved through a bottom - up recursion , therefore we can select an optimal solution with compatible configurations . the optimum ( shortest ) solution for the subproblem ( among all possible cuts ) is stored and can be used for the next recursion level . assumption a is false , and will be rectified in section [ sec : loc ] , lemma [ lem : loc ] . statement b is correct . for the third statement , a stronger assumption on the regions can be used to mend the upper bound for the additional length incurred in mitchell s construction ; see section [ sec : guillotine ] , theorem [ thm : charge ] . preserving connectivity in a graph with guillotine property is difficult , not accounted for in @xcite and for mitchell s line of argument not clear . we present a counterexample in section [ sec : cnn ] , figure [ pic : region - span ] . while not proven by mitchell , statement d is still correct ( if assumption c holds for the given tour ) , a technical argument will be sketched in section [ sec : grid ] . the last statement is again false , but can be fixed using a different notion of @xmath3-fatness , which we will show in section [ sec : ext ] . in mitchell s algorithm , the search for a ( nearly ) optimal tour among a set @xmath0 of connected regions is restricted to a small neighborhood of the minimum - diameter axis - aligned rectangle @xmath20 that intersects all regions . there exists an optimal tour @xmath31 of the regions in @xmath0 that lies within the ball @xmath32 of radius @xmath33 around the center point @xmath34 of @xmath20 . however , figure [ pic : loc ] shows that in general , this is false : a nearly optimal tour need not be within @xmath35 distance of @xmath20 , even if the regions are @xmath3-fat , disjoint and connected as in figure [ pic : loc ] . the vicinity of @xmath20 only contains a @xmath36-approximation of the optimum , which is instead found within @xmath37 . we now show how this problem can be resolved : if an optimal tour intersected @xmath20 , mitchell s lemma would be correct . he argues that , if some regions were to be visited far away from @xmath20 , the path leading to them could be replaced by @xmath38 , which due to connectivity must visit those regions . otherwise , no region can be fully contained in @xmath20 , so the same argument yields that every region must intersect @xmath38 , making @xmath39 an upper bound for the length @xmath40 of an optimal solution . combining this with the fact that @xmath41 , @xmath40 is now known up to a constant factor . now , there are two cases to consider : if there is a small region ( of diameter @xmath42 ) , an area of diameter @xmath42 around this region must contain an optimal tour . otherwise , all regions are of diameter @xmath43 . if the regions are required to be polygons , it is possible to limit the number of possible ( approximate ) locations of an optimal tour by adapting an approach by j. gudmundsson and c. levcopoulos ( * ? ? ? * section 5.1 ) , who show that in that case a tour must be the boundary of a convex polygon . this additional structural information then allows them to deduce the existence of an optimal tour within @xmath42 of a vertex of one of the polygonal regions . considering rectangles of the right size ( since @xmath40 is known up to a constant factor ) yields the following lemma : [ lem : loc ] for a set @xmath0 of disjoint , connected polygons in the plane with a total of @xmath44 vertices , @xmath45 rectangles of size @xmath42 can be found in polynomial time , such that an optimal tour of length @xmath40 is contained in at least one of them . if there were no further problems , a ptas could be obtained by applying mitchell s algorithm to all rectangles from lemma [ lem : loc ] . the main idea of the algorithm is to find a nearly optimal tour that satisfies the @xmath46-guillotine property , which will be defined in the following . consider a polygonal planar embedding @xmath47 of a graph @xmath48 with edge set @xmath49 an a total length of @xmath50 . without loss of generality , let @xmath49 be a subset of the interior of the unit square @xmath51 . let @xmath0 be a set of regions and @xmath21 the axis - aligned bounding box of an optimal tour ( we can afford to enumerate all possibilities on a grid and get a @xmath52-approximation of @xmath21 ) ; let @xmath23 be the subset of regions that lie in the interior of @xmath21 . a * window * is an axis - aligned rectangle @xmath53 . let @xmath54 be a horizontal or vertical line through the interior of @xmath29 , then @xmath54 is called a * cut * for @xmath29 . the intersection @xmath55 consists of a set of subsegments of the restriction of @xmath49 to @xmath29 . let @xmath56 be the endpoints of these segments ordered along @xmath54 . then the * @xmath57-span * @xmath58 of @xmath54 ( with respect to @xmath29 ) is empty , if @xmath59 , and consists of the line segment @xmath60 otherwise ( see figure [ pic : cuts ] ) . a cut @xmath54 is * @xmath57-good * with respect to @xmath29 and @xmath49 , if @xmath61 . and its @xmath26-span for @xmath62 . the cut is @xmath63-good , but not @xmath64-good.,width=453 ] mitchell defines the @xmath27-region - span analogously : [ def : m - region - span ] the intersection @xmath65 of a cut @xmath54 with the regions @xmath23 restricted to @xmath29 consists of a set of subsegments of @xmath54 . the * @xmath66-region - span * @xmath67 of @xmath54 is the line segment @xmath68 along @xmath54 from the @xmath27th entry point @xmath69 , where @xmath54 enters the @xmath27th region of @xmath23 , to the @xmath27th - from - the - last exit point @xmath70 , assuming that the number of intersected regions is @xmath71 . otherwise , the @xmath27-region - span is empty . this definition is ambiguous if the regions are not required to be convex , because the order of the regions is unclear and there might be a number of points at which @xmath54 enters or exits the same region . for example , on the right , many of the line segments connecting two red dots could be the @xmath72-region - span according to this definition . mitchell s @xmath27-region - span does not `` behave well '' in the corresponding charging scheme . we propose the following alternative definition . its benefits will become apparent in the proof of theorem [ thm : charge ] and in figure [ pic : charging - problems ] : [ def : region - span ] the intersection @xmath73 of a cut @xmath54 with the internal regions @xmath23 restricted to @xmath29 consists of a ( possibly empty ) set of subsegments of @xmath54 . let @xmath74 be the endpoints of these segments which are in @xmath75 , ordered along @xmath54 . then the * @xmath66-region - span * @xmath67 of @xmath54 ( with respect to @xmath23 and @xmath29 ) is empty , if @xmath76 and consists of the line segment @xmath68 otherwise ( see figure [ pic : rcuts ] ) . a cut @xmath54 is * @xmath66-region - good * with respect to @xmath29 , @xmath23 and @xmath49 , if @xmath77 . and its @xmath27-region - span ( according to definition [ def : region - span ] ) for @xmath78.,width=453 ] with either definition of the @xmath27-region - span , we can define a corresponding version of the @xmath46-guillotine property as follows : an edge set @xmath49 of a polygonal planar embedded graph satisfies the * @xmath79-guillotine property * with respect to a window @xmath29 and regions @xmath23 , if one of the following conditions holds : * no edge of @xmath49 lies completely in the interior of @xmath29 , _ or _ * there is a cut @xmath54 of @xmath29 that is @xmath26-good with respect to @xmath29 and @xmath49 and @xmath27-region - good with respect to @xmath29 , @xmath23 and @xmath49 , such that @xmath54 splits @xmath29 into two windows @xmath80 and @xmath81 , for which @xmath49 recursively satisfies the @xmath46-guillotine property with respect to @xmath80 resp . @xmath81 and @xmath23 . it is clear from this definition that transforming a tour into an edge set with this property will induce an additional length that depends both on the edges and the regions present . the crucial property of a tour connecting disjoint , @xmath3-fat regions is that their number and diameter provide a lower bound on its length . it is worth noting that the following lemma holds for a tour among @xmath3-fat regions in either mitchell s ( definition [ def : afat - mit ] ) or elbassioni s and fishkin s ( definition [ def : afat - elb ] ) sense : [ lem : afat - mit ] let @xmath82 , then there is a constant @xmath83 ( that depends on @xmath28 and @xmath3 ) , such that for every tspn - tour @xmath31 of length @xmath40 , connecting @xmath1 disjoint , connected , @xmath3-fat ( @xmath3-fat@xmath13 ) regions in the plane , @xmath84 , where @xmath85 is the sum of the diameters of the regions that are completely contained in the axis - aligned bounding box @xmath21 of @xmath31 . @xcite provides a proof for this lemma with respect to @xmath3-fat regions in the sense of definition [ def : afat - mit ] , which can easily be adapted for @xmath3-fat@xmath13 regions as in definition [ def : afat - elb ] ( even without requiring connected regions ) . in the dynamic programming algorithm , @xmath27 can be chosen as @xmath86 ; therefore we can `` afford '' to construct additional edges of length @xmath87 for every @xmath88 and still obtain a @xmath89-approximation algorithm . the following definitions were not explicitly given in @xcite and are therefore adapted from the corresponding definitions in @xcite for the standard tsp : let @xmath54 be a cut through window @xmath29 , and @xmath90 a point on @xmath54 , then @xmath90 is called * @xmath57-dark * with respect to @xmath29 and an edge set @xmath49 , if @xmath90 is contained in the @xmath26-span of the cut through @xmath90 that is orthogonal to @xmath54 . similarly , a point @xmath90 on a cut @xmath54 is said to be * @xmath66-region - dark * , if it is contained in the @xmath27-region - span of a cut through @xmath90 that is orthogonal to @xmath54 . a segment on a cut @xmath54 is called * @xmath57-dark * and * @xmath66-region - dark * , respectively , if every point of it is . a cut @xmath54 is called * favorable * if the sum of the lengths of its @xmath26-dark and @xmath27-region - dark portions is at least as big as the sum of the lengths of its @xmath26-span and @xmath27-region - span . while our definition of the @xmath27-region - span removes the ambiguity and ensures the correctness of the proof techniques used by mitchell , it yields a weaker ( but correct ) overall statement : [ thm : charge ] let @xmath48 be an planar embedded connected graph , with edge set @xmath49 consisting of line segments of total length @xmath50 . let @xmath0 be a set of disjoint , polygonal , @xmath3-fat regions and assume that @xmath91 for every @xmath92 . let @xmath21 be the axis - aligned bounding box of @xmath49 . then , for any positive integers @xmath26 and @xmath27 , there exists an edge set @xmath93 that obeys the @xmath46-guillotine property with respect to @xmath21 and regions @xmath23 , and for which the length of @xmath94 is at most @xmath95 , where @xmath96 is the sum of the perimeters of the regions in @xmath23 . the only deviation from mitchell s version is that the length of @xmath94 is bounded using @xmath96 instead of @xmath85 as defined in lemma [ lem : afat - mit ] . to apply the lower bound on the optimum obtained from @xmath3-fatness as in the original paper , a further restriction can be imposed on the regions that the ratio between the perimeter and diameter of regions is bounded by a constant , which is true , for example , for convex regions . since polygonal regions only ensure that this ratio is bounded by @xmath45 , this is a very restrictive assumption . note further that this theorem , as well as ( * ? ? ? * theorem 3.1 ) , does not establish the existence of a _ connected _ edge set with the properties of @xmath94 ; see section [ sec : cnn ] . the proof relies on the following key lemma by mitchell : [ lem : fav - cut ] for any planar embedded graph @xmath48 with edge set @xmath49 , any set of regions @xmath23 and any window @xmath29 , there is a favorable cut . now , given an edge set @xmath49 as in the theorem , we recursively find a favorable cut @xmath54 , add the @xmath26-span and @xmath27-region - span to @xmath49 and proceed with the two subwindows , into which @xmath54 splits the current window . this procedure terminates , because the proof of lemma [ lem : wfav - cut ] yields that the cut can always be chosen to be at one of finitely many candidate coordinates since we assume that all vertices of the tour lie on a grid . as for the additional length induced by the @xmath26-spans and @xmath27-region - spans , we know that it can be bounded by the length of the respective dark portions of the cuts in question . since @xcite omits some of the details , we will give them here . the charging scheme works as follows : every edge and the boundary ( in mitchell s version , diameter ) of every region is split up into finitely many pieces , to each of which we assign a `` charge '' that specifies which multiple of the length of that segment was added to @xmath49 as part of @xmath26-spans and @xmath27-region - spans . if we can establish that the charge for every edge segment is at most @xmath97 , the charge for every region boundary is at most @xmath98 , and the @xmath26-span and @xmath27-region - span never get charged during the recursive process , we obtain the statement of theorem [ thm : charge ] . let @xmath54 be a favorable cut . the charging scheme for the edge set is described in @xcite : for each @xmath26-dark portion of @xmath54 , the @xmath99 inner edge segments ( the @xmath26 segments closest to the cut on each side ) are each charged with @xmath100 . in the recursive procedure , each segment @xmath101 can be charged no more than once from each of the four sides of its axis - parallel bounding box , since in order for it to be charged , there have to be at least @xmath26 edges to the corresponding side of it , but there are less than @xmath26 edges between @xmath101 and any cut that charges it . therefore , after placing a cut and charging @xmath101 from one side , there will be less than @xmath26 edges to the respective side of @xmath101 in the new subwindow , preventing it from being charged from that direction again . thus , each side of the axis - parallel bounding box of the segment gets charged @xmath102 times , and since the perimeter of the bounding box is at most @xmath103 times the length of the edge segment @xmath101 , it gets charged no more than @xmath97 times in total . the @xmath26-span and @xmath27-region span never get charged , because after they are inserted , they are not in the interior of any of the windows which are considered afterwards . with definition [ def : region - span ] of the @xmath27-region - span , it is possible to replace the regions by their boundaries ( which form a polygonal edge set of total length @xmath96 ) and to treat them the same way as the edge set @xmath49 ( in particular , the @xmath27-region - span and @xmath27-region - dark parts become @xmath27-span and @xmath27-dark ) . for mitchell s original definition ( definition [ def : m - region - span ] ) of the @xmath27-region - span , a scenario as in figure [ pic : charging - problems ] can become a problem . every black line pictured is a favorable cut , every red line segment is @xmath72-region - dark on the respective cut ( even if the window in question has been cut by the black line directly below and above already ) . the total length of the red line segments is however proportional to the perimeter , not the diameter , of the blue and green regions . within one window , mitchell s statement holds ; the problem with his definition is its lack of a monotone additive behavior : when cutting a window @xmath29 into two parts , the sum of the diameters of all relevant regions is @xmath29 might be less than the sums of the diameters of the relevant regions for each subproblem combined , and while no part of the diameter of a region is charged more than @xmath98 times in each subproblem , the combined charge might still be greater than @xmath98 . in order for the dynamic programming algorithm to work , the number of possible endpoints for an edge has to be restricted ( for example , to a grid ) . in @xcite , an optimal solution will thus first be moved to a fine grid through slight perturbation , and subsequently transformed into an @xmath46-guillotine subdivision . mitchell claims that there is always a favorable cut that has grid coordinates , arguing that in the charging lemma ( lemma [ lem : wfav - cut ] ) , the functions considered are piecewise linear between grid points , therefore the maximum of such a function must be attained at a grid point . the proof fails to take into account that the function might be discontinuous ( and not even semi - continuous ) at grid points . even if this were true , it is not in general true in the euclidean case ( unlike the rectilinear case ) that the @xmath26-span ends at a grid coordinate on the cut ( for example , it could instead end at an interior point of an edge ) . a ( slightly technical ) solution to this uses a weaker version of this claim , i. e. that a favorable cut has to be at a grid coordinate or the mean value of two consecutive grid points , which follows from the simple observation that when integrating affine functions over an interval , the sign of the integral is the same as the sign of the function value at the midpoint of the interval . this can be used in the proof of lemma [ lem : fav - cut ] as follows : the existence of a favorable cut is shown via changing the order of integration then the integral of the length of the dark portions along the @xmath104-axis is the same as the integral of the length of the spans along the @xmath105-axis , and vice versa . therefore , there is one axis , such that there is more dark than spanned area in that direction , i. e. the total area of all dark points with respect to some horizontal ( resp . vertical ) cut is greater than the area of all points that are contained in some horizontal ( resp . vertical ) cut . thus , there has to be a single cut with that property as well : a favorable cut . if all previous edges and regions are restricted to the grid , the aforementioned observations imply that in particular , there is a favorable cut at a grid coordinate or in the center between two consecutive ones . it can then be shown that a non - empty @xmath26-span in an optimal solution always has a certain minimum length , or it contains a grid point . this observation allows us to slightly modify the edge set , so that a cut becomes @xmath26-good , while all edges still have grid endpoints . the @xmath27-region - span can be dealt with in a similar way . moving it to the grid requires a slight change in the definition of the @xmath106-guillotine property , which will preserve its algorithmic properties . however , there are further problems with the @xmath27-region - span , which will be explained in the following section . to transform an optimal tour into an @xmath46-guillotine subdivision , the @xmath26-span and @xmath27-region - span of a favorable cut are inserted into the edge set through a recursive procedure . the @xmath26-span is always connected to the original edge set @xmath49 , since its endpoints are intersection points of the cut with @xmath49 . this is not true for the @xmath27-region - span , and in fact , it can be `` far away '' from @xmath49 , as seen in figure [ pic : region - span ] . the optimal tour ( green ) and the @xmath72-region - span @xmath107 of the favorable cut @xmath54 ( which is favorable , because the two grey squares at the top make a portion of @xmath54 with the same length as @xmath107 @xmath72-region - dark ) are far away from each other . connecting it to the tour does not preserve the approximation ratio of @xmath108 . -region - span , width=113 ] note that , in the dynamic programming algorithm , we can not afford to decide whether to connect the @xmath27-region - span of a cut to the edge set : if we choose not to connect it , we have to decide which subproblems is responsible for each region on the @xmath27-region - span , but this is exactly what was to be avoided by introducing it in the first place . on the other hand , if we do connect the @xmath27-region - span to the edge set , both its length and its possible interference with other subproblems have to be taken care of . with the second definition of @xmath3-fatness ( definition [ def : afat - elb ] ) , which implies the lower bounds mentioned in section [ sec : ext ] , the length of a segment connecting the @xmath27-region - span to @xmath49 could be charged off to the length of the @xmath27-region - span itself , whereas mitchell s definition of @xmath3-fatness does not even guarantee this ( because in the proof of the lower bound , we relied on the regions being contained in the bounding box of the tour ) . it is not clear whether the connection of tour and region - span intersects another subproblem , possibly violating the @xmath46-guillotine property there , therefore even for @xmath3-fat@xmath13 regions , this problem remains open . in addition to dealing with the internal regions @xmath23 , the dynamic programming algorithm has to determine how to visit external regions . mitchell s strategy is to enumerate all possible options , restricting the complexity with the following argument : given a situation as in figure [ pic : sub ] , with some external regions protruding from the outside into a subproblem @xmath29 , we know that since there are only @xmath109 edges on each side of @xmath29 , they can be split into @xmath109 intervals of regions , such that along the corresponding side of @xmath29 , the regions are consecutive with no edges passing through @xmath110 between them ( e. g. the red , green and blue region in figure [ pic : sub ] ) . it seems clear now that , in order for the green region to be visited by an edge outside of @xmath29 , either the red or the blue region would have to be crossed as well . if this were true , it could be deduced that the set of regions in one of the intervals in question that are not visited outside @xmath29 , and that thus @xmath29 is responsible for , is a connected subinterval , leading to @xmath111 possibilities for each interval and @xmath112 possibilities overall for each window @xmath29 . this argument fails if the green region has a disconnected intersection with @xmath21 . an example is given on the left in figure [ pic : ext ] : an @xmath46-guillotine tour can visit any subset of the regions outside of @xmath29 , thus there is no polynomial upper bound on the number of possibilities anymore . mitchell s argument holds for convex regions , but as seen left in figure [ pic : ext ] , in general it does not apply to disjoint , connected , @xmath3-fat regions . the number of regions such that their intersection with @xmath21 ( or even a slightly extended rectangle ) is disconnected could be @xmath113 , for example if the construction in figure [ pic : ext ] is extended beyond the yellow region , which is possible , because the regions here actually become `` more fat '' as their size increases , i.e. @xmath3 decreases and eventually converges to 2 . it can be shown that in order for this to be a problem , the size of the regions has to increase exponentially due to the logarithmic lower bound in the packing lemma , and the fact that the boundary of @xmath21 is a tour of the external regions . one solution is therefore to restrict the diameter of the regions ; many of the approximation algorithms for similar problems do in fact require the regions to have comparable diameter ( @xcite , @xcite ) . alternatively , requiring convexity solves the issue , but is quite a strong condition . another option is using the notion of @xmath3-fatness@xmath114 from definition [ def : afat - elb ] as established by k. elbassioni , a. fishkin , n. mustafa and r. sitters @xcite . this definition implies that a path connecting @xmath1 regions has a length of at least @xmath115 , where @xmath116 is the diameter of the smallest region @xcite ; adding a variant of this up by diameter types yields the same lower bound as for mitchell s definition of @xmath3-fatness , up to a constant factor . unlike mitchell s version , this definition however estimates the length of a tour in terms of the minimum diameter of the regions involved and can therefore be used to give a constant upper bound on the number of large external regions ( see figure [ pic : ext ] , on the right ) : since @xmath30 is a path connecting them , the number of external regions with diameter @xmath117 is at most @xmath118 . in both cases , the small external regions can be added to @xmath23 , since @xmath30 is a tour of them , which is at least @xmath119 times the length of an optimal tour , and thus these regions provide a lower bound of the length of @xmath120 , which in turn provides a lower bound on @xmath40 . this way , the statement of lemma [ lem : afat - mit ] ( for which the fact that @xmath21 is the bounding box of the tour was exploited in the proof ) remains intact with modified constants . for the large regions , we can afford to explicitly enumerate which subwindow should visit them . overall , we have the following result : [ thm : main ] let @xmath18 be fixed . given a set of @xmath1 disjoint , connected , polygonal regions that are @xmath3-fat@xmath13 , convex or @xmath3-fat and of bounded diameter , with a total of @xmath44 vertices in the plane , we can find a connected , @xmath46-guillotine , eulerian grid - rounded graph visiting all regions in polynomial time in the size of the grid , @xmath44 , @xmath1 , @xmath121 and @xmath122 . among all such graphs , it will be shortest possible up to a factor of @xmath19 . here , grid - rounded means that all edge endpoints are on a grid of polynomial size , and that there is only a polynomial number of possible positions for every cut . mitchell claims that for @xmath123 and @xmath124 and @xmath3-fat regions , a connected @xmath25-guillotine graph is a @xmath52-approximation of a tour ; his proofs only apply to not necessarily connected graphs and regions with bounded perimeter - to - diameter ratio . in general , because of the connectivity problem in section [ sec : cnn ] and some technical difficulties choosing an appropriate grid , it is not clear whether there is a grid of polynomial size , such that a graph with the properties of the theorem is a @xmath52-approximation of an optimal tspn tour . for unit disks and with a slightly modified definition of the guillotine property , there are @xmath26 and @xmath27 such that the guillotine subdivision is only by a factor of @xmath125 longer than a tour ; this will be shown in theorem [ thm : grid ] . the criticism put forward in section [ sec : cnn ] extends to a joint paper of mitchell and dumitrescu @xcite . despite being published earlier than the ptas candidate , the approach chosen there actually takes into account that the @xmath27-region - span ( or @xmath26-disk - span in the notation of the paper ) has to be connected to the edge set , and this is done at a sufficiently low cost . however , no proof is given that the edges added during this process preserve the @xmath106-guillotine property . in particular , even if a subproblem contains disks that do not intersect its boundary , the @xmath27-region - span might not visit one of them ; if it always did , then we could add the connecting edge and guarantee that it remains within the same subproblem . with some additional effort , this problem can be avoided , as we will show now . definition [ def : afat - elb ] yields a useful lower bound : [ lem : afat ] a shortest path connecting @xmath1 disjoint , @xmath3-fat@xmath13 regions of diameter @xmath126 has length at least @xmath115 . all results apply not only to unit disks ( for which @xmath127 ) , but to disk - like regions : a set of regions is * disk - like * , if all regions are disjoint and connected , and have comparable size ( their diameters range between @xmath128 and @xmath129 , which are constant ) , are @xmath3-fat@xmath13 or @xmath3-fat for constant @xmath3 , and their perimeter - to - diameter ratio is bounded by a constant @xmath130 . let @xmath131 denote the input set of @xmath1 disjoint unit disks . throughout the rest of this paper , we will use a slightly modified version of the @xmath46-guillotine property : an edge set @xmath49 of a polygonal planar embedded graph satisfies the * @xmath79-guillotine property * with respect to a window @xmath29 and regions @xmath23 , if one of the following conditions holds : 1 . there is no edge in @xmath49 with its interior ( i. e. the edge without its endpoints ) completely contained in the interior of @xmath29 . 2 . there is a cut @xmath54 of @xmath29 that is @xmath26-good with respect to @xmath29 and @xmath49 and @xmath27-region - good with respect to @xmath29 , @xmath23 and @xmath49 , such that @xmath54 splits @xmath29 into two windows @xmath80 and @xmath81 , for which @xmath49 recursively satisfies the @xmath46-guillotine property with respect to @xmath80 resp . @xmath81 and @xmath23 . the first case differs from mitchell s definition , which only requires that no entire edge lies completely in the interior of @xmath29 . however , with that definition , adding the @xmath26-span to the edge set does not reduce the complexity of the subproblem ( possibilities for edge configurations on the boundary of the window ) , because then we would still have to know the positions of the edges that intersect the @xmath26-span . let @xmath26 and @xmath27 be fixed . then a cut is called * @xmath132-favorable * , if the sum of the lengths of its @xmath26-span and @xmath27-region - span is at most @xmath16 times the sum of the lengths of its @xmath26-dark and @xmath27-region - dark portions . a cut is * weakly @xmath132-favorable * , if the sum of the lengths of its @xmath26-span and @xmath27-region - span is at most @xmath16 times the sum of the lengths of its @xmath26-dark and @xmath133-region - dark portions . for a cut @xmath54 , define the following notation : * @xmath58 length of the @xmath26-span * @xmath67 length of the @xmath27-region - span * @xmath134 length of the @xmath26-dark segments * @xmath135 length of the @xmath27-region - dark segments let @xmath26 and @xmath27 be fixed . a cut is called * weakly central * , if it is horizontal and has distance at least @xmath136 from the top and bottom edge of the window or it is vertical and has distance at least @xmath137 from the left and right edge of the window , where @xmath138 and @xmath139 denote the width and height of the window , respectively . it is * perfect * , if it is weakly @xmath140-favorable and at least one of the following holds : * it is * central * , i. e. if it is a horizontal cut , it has distance at least 2 from its top and bottom edge ; if it is a vertical cut , it has distance at least 2 from the left and right edge , _ or _ * it is weakly central and its @xmath27-region - span is empty . any central cut is weakly central , any @xmath16-favorable cut is weakly @xmath16-favorable . [ lem : wfav - cut ] let @xmath26 and @xmath141 be fixed . every window has a perfect cut . lemma 3.1 in @xcite states that every window has a favorable cut . we can use the same techniques to show that almost every window has a central @xmath64-favorable cut : by definition , any point @xmath90 is in the @xmath26-span of a vertical cut , if and only if it is @xmath26-dark in a horizontal cut ; analogously for regions . therefore , @xmath142 , where @xmath143 is the vertical cut with @xmath104-coordinate @xmath104 and @xmath144 is the horizontal cut with @xmath105-coordinate @xmath105 ; without loss of generality @xmath145 . then there is a @xmath72-favorable vertical cut , i. e. an @xmath104 such that @xmath146 . using markov s inequality , we can also conclude that at least half of the vertical cuts are @xmath64-favorable . therefore , if the window has width @xmath147 , we can choose a central @xmath64-favorable cut . if the window has width @xmath148 , then the same argument yields that there is still a @xmath64-favorable cut @xmath143 with distance at least @xmath149 from the left and right edge of the window . if its @xmath27-region - span is empty , it is a perfect cut . otherwise , there are at least @xmath150 intersection points with disks , i. e. at least @xmath27 disks , each of which has to intersect this cut and thus have at least @xmath149 of its width within the window . consider the interval @xmath151 $ ] of the window ( and note that it has width @xmath152 ) . let @xmath153 be the set of disks on @xmath143 , then each of them must intersect @xmath154 or @xmath155 ( or both ) . therefore , at least @xmath133 disks of @xmath153 intersect one of them , without loss of generality , @xmath154 . this means that every cut with @xmath104-coordinate in @xmath156 has a total of @xmath27 intersection points with these disks . let @xmath105 be the @xmath105-coordinate of the horizontal cut such that half of these intersection points are below and half of them above @xmath144 . then , the segment from @xmath157 to @xmath104 on @xmath144 is @xmath133-region - dark . on the other hand , since @xmath148 , any horizontal cut can intersect at most @xmath158 disks ( because lemma [ lem : afat ] implies @xmath159 ) . therefore , there are at most @xmath160 total intersection points with disks on the cut . as @xmath161 , @xmath144 has an empty @xmath27-region - span . for @xmath144 , we now have @xmath162 , @xmath163 , @xmath164 , and @xmath165 . this implies that @xmath144 is weakly @xmath140-favorable . finally , since there are at least @xmath166 disks above and below @xmath144 on @xmath143 , @xmath144 has distance @xmath167 from the top and bottom of the window , so it is weakly central and therefore a perfect cut . [ thm : guillotines ] let @xmath168 and @xmath161 be fixed , and let @xmath0 be a set of @xmath169 unit disks . let @xmath40 be the length of a shortest tour with edge set @xmath170 connecting them and @xmath21 its axis - parallel bounding box , then there exists a connected eulerian @xmath171-guillotine subdivision with edge set @xmath172 of length @xmath173 connecting all regions . using different constants for @xmath26 and @xmath27 is not necessary here . note , however , that the algorithm has polynomial running time if @xmath174 and @xmath175 , so choosing @xmath27 differently might help with different applications . we recursively partition @xmath21 using perfect cuts . these always exist by the previous lemma . for each such cut , we add its @xmath26-span and @xmath176-region - span to the edge set ( and not the @xmath27-region - span , because if a segment is added , we need a lower bound on the length of the @xmath27-region - span ) as well as possibly an additional segment for connectivity ( see figure [ pic : guill ] ) . -region - span @xmath107 and the connecting segment ( twice ) makes @xmath54 @xmath72-region - good , width=264 ] we refine mitchell s charging scheme and assign to each point @xmath104 of an edge or the boundary of a disk a charge " @xmath177 , such that the additional length incurred throughout the construction equals @xmath178 . this charge will be piecewise constant . we will show that the charge on an edge segment is bounded by @xmath179 , and the charge of a disk boundary segment is bounded by @xmath180 , for constants @xmath181 and @xmath83 . this proves the theorem , because @xmath182 , but @xmath183 , hence @xmath184 for the disks , and for the edges , @xmath185 . from now on , a segment will refer to a disk boundary or edge segment . in the beginning , every segment has charge 0 . every charge that is applied to a segment gets added to its previous charge . we will distinguish direct and indirect charge , and show that each segment is directly charged at most 4 times , once from each axis - parallel direction . indirect charge will be charge that is added to a segment in @xmath94 that was constructed during the proof . since we can not charge these segments , we pass their charge on : the new segments at some point were charged to a segment in @xmath170 or @xmath186 , to which we add the new charge recursively . for example , in figure [ pic : guill ] , the blue region span can not be charged by any cut , since it is on the boundary of a window . on the other hand , the connecting segment might be charged by a different cut during the construction . if the direct charge for inserting the blue edges was applied to the disks making ( different ) parts of the cut 1-region - dark , then if the connecting segment is charged , we will instead pass the charge on to the disks ( each of them receives half the charge ) . now , let @xmath29 be a window with perfect cut @xmath54 . then , we add its @xmath26-span and @xmath176-region - span to @xmath94 . furthermore , we have that if the @xmath176-region - span is non - empty , it is not necessarily connected to the tour . but in this case the cut is central . therefore , no disk intersected by the @xmath176-region - span can intersect the boundary of the window ( without loss of generality let @xmath54 be vertical : then no disk can intersect the left and right boundary , because the cut is central , and the parts of the cut above and below the region - span intersect at least 12 disks each , therefore their length is at least 1 ) . connect the @xmath176-region - span to the closest point of the tour within the same window ( which will have distance @xmath187 , because all the disks are visited ) . note the the @xmath26-span , by definition , is connected to the edge set , hence the connecting segments for the @xmath27-span are sufficient for connectivity . this makes the cut @xmath26-good and @xmath176-region - good and leaves the edge set connected , therefore recursive application will make the entire window @xmath188-guillotine . we have to show that this procedure terminates . but this follows from the fact that all cuts are weakly central : each recursion step reduces one coordinate of the window by at least 1 or @xmath189 of its width / height . we only add new edges in the interior of a subwindow when the @xmath190-region - span is non - empty , therefore , for small enough windows , we do not add edges to the interior of their subwindows . at that point , the minimum length , width and height of an edge inside of the window remains fixed , so at some point , all edges that lie completely in the interior of @xmath29 will be axis - parallel , and as soon as one coordinate gets small enough , all of them are parallel . but then we can cut between them , so each window only contains one such edge . and lastly , we can cut that edge in half . ( there is probably a simpler argument . ) there are separate arguments for the length charged to disks and edges . for each cut @xmath54 , we have added a length of @xmath191 . we can charge it off as follows : there are @xmath99 edge segments making a segment of length @xmath134 @xmath26-dark . charge each of them with @xmath192 . ( more precisely , these are not necessarily the same edges or even connected , but there are edges of total length at least @xmath193 within the same window such that their orthogonal projection onto @xmath54 intersects at most @xmath194 other edges in @xmath172 . we can charge all of those with @xmath195 . ) similarly , there are disk boundary segments of total length @xmath196 making parts of the cut @xmath133-region - dark . we can charge each of them with @xmath197 . in total , the charged length is @xmath198 , because the cut is @xmath140-favorable . we also know that @xmath199 , because the additional segments in @xmath67 both visit 12 disks . therefore , @xmath200 , which is an upper bound on the actual length of what we insert @xmath26-span , @xmath176-region - span , and possibly a connecting segment of length at most 2 . so the charge indeed will be an upper bound on the additional length as described in the beginning of the proof . it remains to show that each segment is charged directly only a constant number of times , and that the total indirect charge is sufficiently small . for edge segments , there are at most @xmath194 other segments between a charged segment @xmath101 and the cut @xmath54 . but the cut then becomes the boundary of the next subwindow . therefore , this edge will not make a cut @xmath201 between @xmath101 and @xmath54 @xmath26-dark , and not be charged again from this direction . since all cuts are axis - parallel , each edge is indeed only charged at most 4 times . for disks , the same argument works : only the @xmath133 disks closest to a cut ( and making it @xmath133-dark or even @xmath27-dark ) in a given direction are charged , and each disk can only be among those and make the cut @xmath133-dark once for each direction . finally , we have to take care of indirect charge . this applies to both disks and edges , because while our analysis shows that we can upper bound the additional length for the connecting segments by the @xmath27-region - span , the length of this span may be accounted for by @xmath26-dark and not by @xmath133-region - dark segments . but for each edge segment , the direct charge is at most @xmath202 . the segment of that length might be charged with @xmath202 again , adding a charge of @xmath203 to the original segment , yielding a geometric series . therefore , the total charge is at most @xmath204 for @xmath205 . for disks , the same analysis works : the direct charge is at most @xmath197 , and since @xmath205 , the indirect charge is bounded by @xmath206 . finally , we duplicate each new edge segment to make the resulting graph eulerian , increasing the additional length by a factor of 2 . this concludes the proof . we did not use the fact that the regions are unit disks : it it sufficient to assume they are disk - like and modify the constants accordingly . by computing a @xmath52-approximation of a tour visiting the centers of the disks , we obtain a tspn tour which is at most an additive @xmath207 from the optimum ( for @xmath1 large , this is a constant - factor approximation algorithm and was analyzed in @xcite ) . if the tour has length at least @xmath208 , this is a sufficiently good solution ; otherwise , the disks are within a square of size @xmath209 , if @xmath210 and @xmath211 . such a square can be found ( or shown that none exists ) in polynomial time . we then equip it with a regular rectilinear grid with edge length @xmath212 . [ def : grid ] an edge set is * grid - rounded * w. r. t. a grid @xmath48 , if all edge endpoints are on the grid . a polygon is grid - rounded , if its boundary is grid - rounded . a set of regions is grid - rounded , if all regions are grid - rounded polygons . a coordinate ( or axis - parallel line segment ) is said to be a * half - grid coordinate * of @xmath48 , if it is on the grid or in the middle between two consecutive grid points . at a cost of a factor @xmath52 , the instance can be grid - rounded , such that every disk center is on a grid point . the same can be done for an optimum solution , i. e. every edge should begin and end in a grid point . such a solution is still feasible , if we replace each disk @xmath213 by the convex hull of the set of grid points @xmath214 it contains . as this convex hull contains at least the diamond inscribed in the disk ( a square of area 2 ) , and it has rotational symmetry , it will still be @xmath3-fat@xmath13 , for slightly smaller @xmath3 . the previous theorem still holds for these regions ( with modified constants ) . the regions @xmath215 are polygonal . therefore , in lemma [ lem : wfav - cut ] , the functions @xmath216 and @xmath217 are piecewise linear , with discontinuities at grid points . this implies : [ lem : grid - cut ] given grid - rounded disk - like regions and a window @xmath29 with half - grid coordinates , there is a perfect cut with half - grid coordinates . this follows from lemma [ lem : wfav - cut ] , because the integrals involved can be replaced by ( @xmath116 times ) the sums of the function values at all half - grid points ( that are not grid points ) . for piecewise linear functions , this sum is equal to the integral . if the @xmath26-span and @xmath27-region - span are inserted at half - grid cuts ( with their endpoints not even at half - grid points ) , the solution does not remain on the grid . in particular , the connecting segment for the @xmath27-region - span could lead to discontinuities in @xmath58 and @xmath134 at non - grid points , thus preventing the recursive application of this lemma . therefore , it will be moved to the grid in the following . we can assume that every disk is visited by the endpoint of an edge that lies on the grid . this costs a factor of @xmath52 and means that when restructuring the edge set , as long as the resulting graph remains eulerian , connected and has the same ( or more ) edge endpoints , we can still extract at tspn tour from it . therefore , the following lemma can be applied without taking the regions into account : [ lem : span ] consider a regular rectilinear grid with edge length @xmath116 and a grid - rounded set @xmath49 of edges that is an optimum tour of the set of its edge endpoints , and a window @xmath29 . let @xmath54 be a cut in @xmath29 , such that its @xmath26-span is non - empty and @xmath54 has distance at least @xmath116 from the boundary edges of the window that are parallel to @xmath54 . if the @xmath104- or @xmath105-coordinate of @xmath54 ( depending on its orientation ) is a grid coordinate , and @xmath26 intersects at least 15 different edges in their interior ( or has 16 intersection points with edges ) , then @xmath218 . furthermore , if @xmath54 is in the center between two consecutive grid coordinates , then @xmath218 , if its @xmath26-span intersects at least 19 different edges ( necessarily in their interior , because their endpoints can only be on the grid ) . this is a grid - rounded version of arora s patching lemma ( * ? ? ? * lemma 3 ) . the proof uses similar ideas and exploits the grid structure to show that in some cases , the patching construction actually decreases the tour length . first , let @xmath54 be on the grid and without loss of generality vertical . if @xmath54 has 16 intersection points with edges , then either two of them are ( different ) grid points , and @xmath219 , or at @xmath54 intersects at least 15 different edges in their interior , therefore it is sufficient to consider this case . if @xmath220 , this configuration can never be optimal . this follows form the construction in figure [ pic : patching ] : expand the @xmath26-span to the nearest grid coordinates above and below , and then consider the box that has as left and right edge this expanded @xmath26-span translated by @xmath221 . this box will have height @xmath116 or @xmath222 and width @xmath222 . if it has height @xmath222 , an additional length of @xmath223 is used to connect to the grid point in the center of the box . for every edge intersecting the box , split it into different parts at the intersection points . the inner part ( inside of the box ) has length at least @xmath116 , but it does not visit any new endpoints , and can therefore be removed . for the other parts , if we consider their second endpoint ( not on the boundary of the box ) fixed and choose their first endpoint among the points of the boundary of the box , they become shortest possible when connected in such a way , that the endpoint is a vertex of the box or the segment is orthogonal to the boundary edge of the box . in both cases , the first endpoint is a grid point . therefore , for these parts there is a grid point on the boundary of the box , such that the intersection point can be moved there without increasing the length of the edge set . this preserves connectivity and parity except possibly on the boundary of the box . since it has perimeter @xmath224 , edges of length @xmath225 can be used to correct parity . overall , we have added edges of total length @xmath226 and removed edges of length @xmath227 for every edge intersecting @xmath58 in its interior . hence , for an optimal tour , there can be at most 14 such edges . if @xmath54 is not on the grid , but at a half - grid coordinate , we can apply a similar argument and construction , see figure [ pic : patching2 ] . the box has perimeter @xmath228 and for every edge intersecting it , the inner segment of length at least @xmath229 can be removed . there are no interior points to be visited , and correcting parity costs at most @xmath230 . therefore , there can be at most @xmath231 edges intersecting the @xmath26-span , if its length is less than @xmath116 . for the @xmath27-region - span , no such lemma is needed , because if we insert it , we can also afford a connecting segment of length @xmath64 . therefore , it can be extended to the grid without increasing the length we used for the entire construction by more than a factor of @xmath64 ( and actually , @xmath232 ) provided the cut is on the grid . choosing only cuts on the grid is not sufficient , as the following example shows : even without regions , there is no @xmath72-good cut with grid coordinates . the recursive construction in theorem [ thm : guillotines ] makes cuts @xmath26-good and @xmath27-region - good by inserting edges on the cut , which is not possible for a cut that is not at a grid coordinate . since we can not change the position of the regions , no modification of the edge set ( preserving grid - roundedness ) can make the cut @xmath27-region - good . moving the cut to the grid is also not an option , since figure [ pic : grid - cut ] shows that this is not always possible . therefore , we can not hope to find an @xmath25-guillotine subdivision with this construction . however , for algorithmic purposes , the main aim of the guillotine property was avoiding the enumeration of which subproblem is responsible for visiting which regions . what happens , if we make them _ both _ responsible for all regions in the @xmath27-region - span ? first , note , that `` smaller '' subproblems never rely on the fact that their containing windows are guillotine , because they do not yet know the respective cuts with the exception of the four cuts defining their boundaries . for these cuts , we can easily enumerate the possibilities `` visit all regions in the @xmath27-region - span '' and `` visit none of them '' . intuitively , it seems that this construction would significantly increase the length of the subdivision . but we know that those regions can be visited by a segment with at most the length of the @xmath27-region - span . more importantly , we know that they can be visited on each side of the cut by a segment such that their combined length is at most @xmath233 , which we can afford by the charging scheme . not both of these can necessarily be connected to @xmath49 within their containing window ( but at least one ) , therefore we should add ( or at least `` reserve '' ) an edge across the cut , i. e. only obtain an @xmath234-guillotine subdivision . to accommodate these changes , we redefine @xmath27-region - good and thus obtain a new @xmath25-guillotine property : a cut @xmath54 is * @xmath66-region - good * with respect to @xmath29 , @xmath23 and @xmath49 , if there are no two regions @xmath235 that intersect the @xmath27-region - span of @xmath54 , but @xmath49 visits @xmath236 and @xmath237 only on different sides of the cut . in other words , if @xmath54 does not visit @xmath236 on one side of the cut , it must visit all other regions that intersect the @xmath27-region - span on the other side of the cut . here , `` side of the cut '' denotes a closed half - space ; in particular , a cut that was @xmath27-region - good w. r. t. the previous definition remains @xmath27-region - good , since the @xmath27-region - span is in @xmath49 and thus @xmath49 visits all regions in the @xmath27-region - span on both sides of the cut . the following two lemmas show that this construction works for both edges and regions , we can replace the operations `` insert the span '' by one of the following in the charging scheme , and get the statement of the charging scheme ( with modified constants ) for grid - rounded subdivisions . [ lem : m - good ] given a perfect half - grid cut @xmath54 in a window @xmath29 of width @xmath126 , and a connected eulerian grid - rounded edge set @xmath49 , then there is a grid - rounded edge set @xmath94 that differs from @xmath49 only at edges intersecting @xmath29 , has the @xmath238-guillotine property outside of @xmath29 if @xmath49 does , is by at most an additive @xmath239 longer than @xmath49 , visits the same ( or a superset ) of the grid points @xmath94 visits , and is connected and eulerian , such that @xmath54 is @xmath240-good w. r. t. @xmath94 and @xmath29 . if the window has width @xmath241 , then the constructions in the proof might not be inside of the window . on the other hand , such a window is @xmath25-guillotine by definition , since it can not contain ( the entire interior of ) grid edges in its interior . without loss of generality , let @xmath54 be a vertical cut . if @xmath54 is @xmath240-good , there is nothing to show . otherwise , there are at least @xmath242 edges on the @xmath26-span , so it has length at least @xmath116 by lemma [ lem : span ] , or @xmath49 can be made shorter by applying the construction there , thereby making the cut @xmath243-good . if @xmath218 , there are two cases : if @xmath54 is not on the grid , as in figure [ pic : half - cut ] , we insert an `` h '' shape , which has length @xmath244 . for all edges intersecting this h , the intersection point should be moved to a grid point without increasing the length or violating guillotine property . to see that the length does not increase , let @xmath245 be the endpoints of an edge intersecting the h , then replacing the edge by segments from @xmath90 to the first intersection point @xmath246 and from the last intersection point @xmath247 to @xmath14 preserves connectivity ( and parity can be correcting using edges of the h ) . let @xmath90 be to the left of @xmath54 , then @xmath246 is either the left vertical edge or the bar . in the latter case , moving the intersection point to the left endpoint of the bar only decreases the length of the segment . in the former case , @xmath246 is either a grid point or can be moved vertically on the h. in that case , either the edge from @xmath90 to @xmath246 is horizontal ( so @xmath246 is a grid point because @xmath90 is ) , or there is a direction such that the angle at @xmath246 gets less acute when moving @xmath246 , thereby making @xmath248 shorter . whenever @xmath248 intersects a grid point , we subdivide it and continue with the segment containing @xmath246 , thus ensuring that the edge set remains planar an @xmath238-guillotine , if @xmath49 is . the resulting graph is grid - rounded and @xmath54 is @xmath26-good , since the @xmath26-span only contains one point , and this point is part of the edge set . it might not be eulerian , but the only points whose parity might have changed are on the h , hence duplicating some of its edges will make the edge set eulerian again . if the cut is at a grid coordinate , we increase the length of the @xmath26-span by at most @xmath249 as shown in figure [ pic : span - again ] ( so that it becomes grid - rounded ) and then proceed analogously to the first case for all edges intersecting it . [ lem : region - good ] given a perfect half - grid @xmath240-good cut @xmath54 in a window @xmath29 , and a connected eulerian grid - rounded edge set @xmath49 and grid - rounded disk - like regions @xmath0 , then there is a grid - rounded edge set @xmath94 that differs from @xmath49 only in edges intersecting of @xmath29 , is @xmath250-guillotine outside @xmath29 if @xmath49 is , is by at most an additive @xmath251 longer than @xmath49 , visits the same ( or a superset ) of the grid points @xmath94 visits , and is connected and eulerian , such that @xmath54 is @xmath252-good and @xmath253-good w. r. t. @xmath94 , @xmath0 and @xmath29 . the constant @xmath83 here depends on the constants for the disk - like regions ( definition [ def : grid ] ) and can be chosen as 24 for unit disks . without loss of generality , let @xmath54 be vertical . if @xmath54 is at a grid coordinate , extend @xmath254 to the grid and move all intersection points with edges to the grid as in previous lemmas . correct parity on the extended @xmath253-region - span . otherwise , since the regions are polygonal , the set of regions in the @xmath253-region - span can be visited by a vertical segment of at most the same length either at the grid coordinate directly to the left or to the right of @xmath54 . insert this segment and possibly a connecting segment to the edge set ( which might cross the cut ) inside of @xmath29 . for all edges intersecting the new vertical segment , proceed as before . for the connecting segment , note that it is not necessarily rectilinear . therefore , if it intersects an edge , we can subdivide this edge and move the intersection point to a grid point . this costs at most @xmath230 , because there are 3 incident edges , and connects to the edge set hence it is sufficient to do this at most once . since @xmath255 by choice of @xmath83 , this is not too expensive , and can be done inside @xmath29 . again , correcting parity is only necessary on new segments . applying these lemmas to the charging scheme yields that there is an @xmath25-guillotine grid - rounded subdivision that approximates a tour well , more precisely : [ thm : grid ] let @xmath18 . for every set @xmath131 of @xmath256 disjoint unit disks within a square of size @xmath257 , let @xmath258 and @xmath259 . then , there is an edge set @xmath49 with the following properties : 1 . it satisfies the ( new ) @xmath260-guillotine property . 2 . the endpoints of every edge are on a regular rectilinear grid with edge length @xmath261 . it visits at least one point from each of @xmath262 , the grid points of the slightly perturbed , polygonal approximations of the disks . it is eulerian and connected . the total length of all its segments is @xmath263 , where @xmath40 is the length of an optimum tour visiting @xmath131 ( or @xmath262 , as both lengths only differ by a factor of at most @xmath264 ) . this theorem implies an approximation ratio of @xmath265 for mitchell s dynamic programming algorithm for the tspn with disjoint unit disks , if the grid and @xmath26 and @xmath27 are chosen as above , and for these parameters , such a subdivision can be found by mitchell s algorithm ( together with the refinements to preserve grid - roundedness ) in polynomial time ( theorem [ thm : main ] ) . using the two previous lemmas , one can construct @xmath94 as follows : starting with an optimal tour on the grid , recursively find a perfect half - grid cut , insert edges so that it becomes @xmath240-good and @xmath176-region - good , and continue with the new subwindows . the edge set remains a tour , and becomes @xmath266-guillotine . the increase in length can be bounded using the same charging scheme as in theorem [ thm : guillotines ] . the guillotine subdivision method of mitchell @xcite can be used to derive a ptas for the tsp with unit disk neighborhoods . all arguments carry over to disk - like regions , for which mitchell s framework can be used to derive a ptas as well . this includes geographic clustering as the special case when the regions are @xmath3-fat ( they could be @xmath3-fat@xmath13 instead ) . however , the approach of bodlaender et al . @xcite based on curved dissection in arora s ptas for tsp @xcite achieves a faster theoretical running time for disjoint connected regions with geographic clustering . their algorithm , like arora s , can be generalized to more than two dimensions . for @xmath3-fat regions in mitchell s sense , the existence of a ptas remains open . the problem with external regions of section [ sec : ext ] can be avoided by using @xmath3-fat@xmath13 or convex regions instead , the charging scheme ( section [ sec : charge ] ) can be fixed by bounding the ratio of perimeter and diameter , the grid can be handled as in the unit disk case , and localization does not require any additional assumptions . however , even for those stronger conditions on the regions , it is unclear how to handle connectivity ( section [ sec : cnn ] ) for neighborhoods of varying size . the length of the connecting segment can be bounded for @xmath3-fat@xmath13 regions as in the unit disk case , but it might still destroy the guillotine property of other windows . to our knowledge , no ptas for any form of the tsp with neighborhoods of varying size exists . mitchell s constant factor approximation algorithm for disjoint connected regions @xcite relies on the ptas for @xmath3-fat regions , but only applies it to disjoint balls , which are @xmath3-fat@xmath13 . therefore , a constant factor approximation algorithm by elbassioni et al . @xcite can be used instead , so that the overall algorithm in @xcite still works and yields a constant factor approximation for the tsp with general disjoint connected , and in particular @xmath3-fat , regions . h. l. bodlaender , c. feremans , a. grigoriev , e. penninkx , r. sitters , t. wolle . _ on the minimum corridor connection problem and other generalized geometric problems _ , 2009 . computational geometry , volume 42 , issue 9 , november 2009 , pages 939 - 951 . h. chan , k. elbassioni : _ a qptas for tsp with fat weakly disjoint neighborhoods in doubling metrics _ , 2010 . proceedings of the twenty - first annual acm - siam symposium on discrete algorithms ( soda 10 ) . society for industrial and applied mathematics , philadelphia , pa , usa , 256 - 267 . k. elbassioni , a. fishkin , n. mustafa , r. sitters : _ approximation algorithms for euclidean group tsp _ , 2005 . proceedings of the 32nd international conference on automata , languages and programming ( icalp05 ) . springer - verlag , berlin , heidelberg , 1115 - 1126 . k. elbassioni , a. fishkin , r. sitters : _ approximation algorithms for the euclidean traveling salesman problem with discrete and continuous neighborhoods _ , 2009 . j. comput . 19 , 173 ( 2009 ) . j. s. b. mitchell . : _ guillotine subdivisions approximate polygonal subdivisions : a simple new method for the geometric k - mst problem _ , 1996 . proceedings of the seventh annual acm - siam symposium on discrete algorithms ( soda 96 ) . society for industrial and applied mathematics , philadelphia , pa , usa , 402 - 408 . j. s. b. mitchell : _ guillotine subdivisions approximate polygonal subdivisions : a simple polynomial time approximation scheme for geometric tsp , k - mst , and related problems _ , 1999 . siam journal on computing , volume 28 , 4 ( march 1999 ) , 1298 - 1309 . j. s. b. mitchell : _ a ptas for tsp with neighborhoods among fat regions in the plane _ proceedings of the eighteenth annual acm - siam symposium on discrete algorithms ( soda 07 ) . society for industrial and applied mathematics , philadelphia , pa , usa , 11 - 18 . j. s. b. mitchell : _ a constant - factor approximation algorithm for tsp with pairwise - disjoint connected neighborhoods in the plane _ , 2010 . proceedings of the twenty - sixth annual symposium on computational geometry ( socg 10 ) . acm , new york , ny , usa , 183 - 191 . s. safra , o. schwartz : _ on the complexity of approximating tsp with neighborhoods and related problems _ , 2005 . proceedings 11th annual european symposium on algorithms ( esa ) , volume 2832 of lecture notes in computer science , pages 446 - 458 . springer .
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the euclidean tsp with neighborhoods ( tspn ) is the following problem : given a set @xmath0 of @xmath1 regions ( subsets of @xmath2 ) , find a shortest tour that visits at least one point from each region .
we study the special cases of disjoint , connected , @xmath3-fat regions ( i.e. , every region @xmath4 contains a disk of diameter @xmath5 ) and disjoint unit disks . for the latter , dumitrescu and mitchell @xcite proposed an algorithm based on mitchell s guillotine subdivision approach for the euclidean tsp @xcite , and
claimed it to be a ptas .
however , their proof contains a severe gap , which we will close in the following .
bodlaender et al .
@xcite remark that their techniques for the minimum corridor connection problem based on arora s ptas for tsp @xcite carry over to the tspn and yield an alternative ptas for this problem .
for disjoint connected @xmath3-fat regions of varying size , mitchell @xcite proposed a slightly different ptas candidate
. we will expose several further problems and gaps in this approach .
some of them we can close , but overall , for @xmath3-fat regions , the existence of a ptas for the tspn remains open .
_ keywords : _ tsp with neighbourhoods , approximation scheme , guillotine subdivision , travelling salesman problem
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the tw hya association , a loose grouping of very young stars that does not appear to be related to any nearby molecular cloud material , has been the subject of considerable attention in recent years . the prototype of the association , tw hya itself , was the first recognized example of an isolated t tauri star . the first few members of the association were discovered more or less by accident , and were only later recognized to be related . aside from these , the systematic search for additional stars belonging to the group has relied on their x - ray properties ( such as their detection as a rosat source ) , or their similar kinematics ( proper motions ) . follow - up studies have been made in many cases to establish the youth of the candidate members , for example by the presence of strong lithium @xmath06708 absorption , h@xmath1 emission , infrared excess , etc . to date there are 19 objects recognized as true members in the published literature ( see , e.g. , kastner et al . 1997 ; webb et al.1999 ; sterzik et al . 1999 ; zuckerman et al . 2001 ) , along with many other potential members that need to be studied further . one of the goals of the present work is to investigate the kinematics of the candidate members , but in the direction to previous studies that have focussed only on proper motions . we present measurements along the line of sight ( radial velocities ) , which are complementary to the proper motions and help define the space motion of these objects . another goal of this project is to detect any spectroscopic binaries that might be present through repeated measurements of the radial velocities of our targets , and to determine their orbits . our observations were obtained using nearly identical echelle spectrographs on the mmt prior to conversion ( mt . hopkins , az ) , the 1.5 m tillinghast reflector at the f. l. whipple observatory ( also on mt . hopkins , az ) , and on the 1.5 m wyeth reflector at the oak ridge observatory ( harvard , ma ) . a single echelle order was recorded with a photon - counting reticon diode array , at a resolving power of @xmath2 and a central wavelength of 5187 . the spectral coverage is 45 . we report here on the results based on some 300 spectra , and an additional 150 archival spectra from the cfa echelle database . table 1 presents the sample of stars studied in this program , including for completeness hd 98800 ( twa-4 ) , which was observed several years ago with the same instrumental setup and was discovered to be a quadruple system ( torres et al . the second column in the table gives the official " designation within the association , and the last column collects the available measurements of the li @xmath06708 equivalent width for these stars . -5pt rllcccc@ c & & & r.a . & dec . & @xmath3 & & li @xmath06708 eq . + & twa # & other name & ( j2000 ) & ( j2000 ) & ( mag ) & spt & width ( ) + 1 & & hip 48273 & 09:50:30.1 & @xmath404:20:37 & 6.24 & f6 & ... + 2 & & tyc 6604 - 0118 - 1 & 09:59:08.4 & @xmath522:39:35 & 10.09 & k2 & 0.143 + 3 & & hip 50796 & 10:22:18.0 & @xmath510:32:15 & 10.80 & ... & ... + 4 & & hip 53486 & 10:56:31.0 & @xmath407:23:19 & 7.37 & k0 & ... + 5 & & rxj1100.0@xmath53813 & 11:00:02.4 & @xmath538:13:20 & 12.29 & ... & ... + 6 & & hd 95490 & 11:00:51.2 & @xmath535:33:38 & 8.79 & f7 & 0.15 + 7 & twa-1 & tw hya & 11:01:52.0 & @xmath534:42:16 & 10.92 & k8e & 0.43 + 8 & twa-2a & cd@xmath68887a & 11:09:13.9 & @xmath530:01:39 & 11.07 & m2e & 0.52 + 9 & & rxj1109.7@xmath53907 & 11:09:40.1 & @xmath539:06:48 & 10.58 & g3 & 0.190 + 10 & twa-3a & hen 3 - 600a & 11:10:28.0 & @xmath537:31:53 & 12.04 & m4e & 0.57 + 11 & & hd 97131 & 11:10:34.2 & @xmath530:27:19 & 9.01 & f2 & 0.03 + 12 & & cd@xmath77097 & 11:12:42.7 & @xmath538:31:04 & 10.24 & f5 & ... + 13 & & rxj1115.1@xmath53233 & 11:15:06.9 & @xmath532:32:46 & 12.36 & ... & ... + 14 & twa-12 & rxj1121.1@xmath53845 & 11:21:05.5 & @xmath538:45:17 & 12.85 & m2 & 0.530 + 15 & twa-13a & rxj1121.3@xmath53447n & 11:21:17.3 & @xmath534:46:47 & 11.46 & m2e & 0.650 + 16 & twa-13b & rxj1121.3@xmath53447s & 11:21:17.5 & @xmath534:46:51 & 12.00 & m1e & 0.51 + 17 & twa-4a & hd 98800a & 11:22:05.3 & @xmath524:46:40 & 9.41 & k5 & 0.425 + 18 & twa-4b & hd 98800b & 11:22:05.3 & @xmath524:46:39 & 9.94 & ... & [email protected] + 19 & twa-5a & cd@xmath87795a & 11:31:55.4 & @xmath534:36:27 & 11.37 & m3 & 0.55 + 20 & twa-9a & cd@xmath97429a & 11:48:24.3 & @xmath537:28:49 & 11.26 & k5 & 0.46 + all recognized members are seen to have strong li in absorption , while some of the other objects ( proposed by a number of authors as candidate members ) have much weaker li . not all of the potential members have estimates of this crucial youth indicator , so their status is unclear . radial velocities were obtained using standard cross - correlation techniques ( the xcsao task running under iraf ) , as described , e.g. , by latham ( 1992 ) . the templates were taken from an extensive library of synthetic spectra based on the latest model atmospheres by kurucz ( morse & kurucz , in preparation ) . in addition , we derived effective temperatures ( @xmath10 ) and projected rotational velocities ( @xmath11 ) for all our stars by comparison with the synthetic spectra . a number of objects have turned out to be double - lined . for these we used the two - dimensional cross - correlation algorithm known as todcor ( zucker & mazeh 1994 ) , and in most cases we were able to derive the @xmath10 and @xmath12 for both components . table 2 lists our results for the temperatures ( k ) and rotational velocities ( km s@xmath13 ) . a comparison with @xmath10 values derived from the spectral types ( adopting the calibration by de jager & nieuwenhuijzen 1987 ) and other sources shows generally good agreement . the agreement for the cooler stars is poorer due to limitations in the model atmospheres used to compute the synthetic spectra , which do not include several key molecular opacity sources . our measures of @xmath12 are also fairly consistent with other determinations . rll@ c@ c@ c@c@c@c & & & & t@xmath14 & t@xmath14 & t@xmath14 & @xmath12 & @xmath12 + & twa # & other names & spt & ( cfa ) & ( spt ) & ( other ) & ( cfa ) & ( other ) + 1 & & hip 48273 & f6 & 6300/6150 & 6530 & & 22/22 & + 2 & & tyc 6604 - 0118 - 1 & k2 & 5050/4850 & 4840 & & 19/15 & 19 + 3 & & hip 50796 & ... & 4750 & ... & & 8 & + 4 & & hip 53486 & k0 & 5050 & 5150 & & 1 : & + 5 & & rxj1100.0@xmath53813 & ... & 5750 & ... & & 35 & + 6 & & hd 95490 & f7 & 6500 & 6430 & & 13 & + 7 & twa-1 & tw hya & k8e & 4150 & 4150 & 4150 & 4 & 10 , 15 , 14 + 8 & twa-2a & cd@xmath68887a & m2e & 4050 : & 3690 & & 13 & 15 + 9 & & rxj1109.7@xmath53907 & g3 & 5900 & 5710 & 5800 & 27 & 23 + 10 & twa-3a & hen 3 - 600a & m4e & 4750 : & 3350 & & 20 : & 15 + 11 & & hd 97131 & f2 & 6750/6750 & 7180 & & 12/16 & + 12 & & cd@xmath77097 & f5 & 6250 & 6650 & & 14 & + 13 & & rxj1115.1@xmath53233 & ... & 5750/5250 : & ... & & 25/0 : & + 14 & twa-12 & rxj1121.1@xmath53845 & m2 & 4000 : & 3520 & 3600 & 15 & 21 + 15 & twa-13a & rxj1121.3@xmath53447n & m2e & 4150 : & 3520 & 3600 & 12 & 16 , 10 + 16 & twa-13b & rxj1121.3@xmath53447s & m1e & 4100 : & 3660 & 3800 & 12 & 16 , 12 + 17 & twa-4a & hd 98800a & k5 & & 4410 & 4350 & & 5 + 18 & twa-4b & hd 98800b & ... & & ... & 4250/3700 & & 3/2 + 19 & twa-5a & cd@xmath87795a & m3 & 4050 : & 3490 & & 36 : & 58 + 20 & twa-9a & cd@xmath97429a & k5 & 4350 & 4410 & & 11 & + in addition to the well known quadruple system hd 98800 , multiple measurements of the radial velocity of our other targets have revealed several binaries ( some double - lined ) with short orbital periods ( rxj1115.1@xmath53233 , tyc 6604 - 0118 - 1 , hip 48273 , rxj1100.0@xmath53813 ) and one double - lined triple system ( hd 97131 ) . none of these appear to be true members of the tw hya association , based on their li abundance or kinematics ( radial velocity of the center of mass ) . details on these orbital solutions will be reported in a forthcoming paper . in table 3 we list the mean heliocentric radial velocities of all our targets . for the multiple systems we give the center - of - mass velocity . our measurements are consistent with other determinations from the literature , but much more precise . except for the case of hen 3 - 600a ( twa-3a ) and cd@xmath5337795a ( twa-5a ) , which are double - lined or possibly triple - lined but have not yet had their orbits determined ( see also webb et al . 1999 ) , the other recognized members of the association have similar radial velocities . in the case of hd 98800 we have assumed that the average velocity of components a and b corresponds to the center of mass of the quadruple system . similarly for rxj1121.3@xmath53447 ( twa-13 ) , which is a visual binary . the mean velocity of twa-1 , twa-2 , twa-4 , twa-9 , twa-12 , and twa-13 is @xmath15 km s@xmath13 . rllcc@c@l & & & & mean @xmath16 & mean @xmath16 & + & twa # & other names & n@xmath17 & ( cfa ) & ( other ) & remarks + 1 & & hip 48273 & 112 & @xmath416.249 @xmath18 0.071 & & binary ( orbit ) + 2 & & tyc 6604 - 0118 - 1 & 34 & @xmath426.96 @xmath18 0.24 & & binary ( orbit ) + 3 & & hip 50796 & 2 & @xmath413.1 @xmath18 1.0 & & + 4 & & hip 53486 & 6 & @xmath45.54 @xmath18 0.29 & & + 5 & & rxj1100.0@xmath53813 & 18 & @xmath42.03 @xmath18 0.92 & & binary ( orbit ) + 6 & & hd 95490 & 4 & @xmath57.51 @xmath18 0.17 & & + 7 & twa-1 & tw hya & 5 & @xmath412.92 @xmath18 0.23 & @xmath413.5 @xmath18 1.5 & + 8 & twa-2a & cd@xmath529@xmath198887a & 6 & @xmath411.20 @xmath18 0.32 & @xmath20 & + 9 & & rxj1109.7@xmath53907 & 7 & @xmath51.2 @xmath18 1.1 : & @xmath52.0 @xmath18 1.1 & var ? + 10 & twa-3a & hen 3 - 600a & 14 & @xmath46.9 @xmath18 1.7 : & @xmath21 & double - lined ? + 11 & & hd 97131 & 37 & @xmath527.10 @xmath18 0.26 & & triple ( orbit ) + 12 & & cd@xmath77097 & 3 & @xmath58.19 @xmath18 0.26 & & + 13 & & rxj1115.1@xmath53233 & 11 & @xmath510.0 @xmath18 1.1 & & binary ( orbit ) + 14 & twa-12 & rxj1121.1@xmath53845 & 2 & @xmath412.23 @xmath18 0.60 & @xmath410.9 @xmath18 1.1 & + 15 & twa-13a & rxj1121.3@xmath53447n & 4 & @xmath411.72 @xmath18 0.61 & @xmath410.5 @xmath18 1.2 & + 16 & twa-13b & rxj1121.3@xmath53447s & 4 & @xmath412.41 @xmath18 0.48 & @xmath412.0 @xmath18 1.2 & + 17 & twa-4a & hd 98800a & 152 & @xmath412.75 @xmath18 0.10 & & binary ( orbit ) + 18 & twa-4b & hd 98800b & 152 & @xmath45.73 @xmath18 0.14 & & binary ( orbit ) + 19 & twa-5a & cd@xmath533@xmath197795a & 26 & @xmath46.9 @xmath18 2.0 : & @xmath21 & double - lined ? + 20 & twa-9a & cd@xmath536@xmath197429a & 10 & @xmath410.17 @xmath18 0.36 & & + though it is tempting to assign this mean radial velocity to the group as a whole , and to then use it as a criterion to accept or reject other candidate members , reality is more complex and various kinematical studies have shown that there is probably a gradient in the radial velocity across the large sky area covered by this association ( several tens of degrees ) . one of such studies , by makarov & fabricius ( 2001 ) , has modeled the kinematics of the association as a moving group using a variant of the convergent - point method . the authors used proper motions from the tycho-2 catalogue along with hipparcos parallaxes for the few members which have them , and searched an area of more than 3000 deg@xmath22 for additional members with the same motion that are also x - ray sources from the rosat bright source catalog . for each star they computed a kinematical " distance as well as the predicted radial velocity . comparing the latter with the few measurements available to the authors from the literature for the classical members , they found it necessary to include an expansion term in their model , which not only is not unexpected for this group , but also gives a dynamical age of 8.3 myr , in good agreement with other estimates from pre - main sequence evolutionary tracks ( @xmath23 myr ) . the internal velocity dispersion they derived is 0.8 km s@xmath13 , and the depth of the group is significant ( tens of parsecs ) . makarov & fabricius ( 2001 ) produced a list of 23 additional candidates , of which we have observed several ( see table 4 ) . our radial velocity measurements allow us to test these objects for membership . the known members of the association ( designation in column 2 ) have radial velocities ( @xmath24 ) very close to the predicted values ( @xmath25 ) in most cases , but two of the other stars ( hip 48273 and tyc 6604 - 0118 - 1 ) , which happen to be double - lined binaries , do not . on the other hand , the velocity for hip 50796 agrees perfectly with the prediction . hip 53486 is an especially interesting case because it the nearest candidate member ( 17 pc ) , and its predicted distance ( @xmath26 ) agrees with the value measured by hipparcos ( @xmath27 ) . it has the lowest expected radial velocity ( @xmath28 km s@xmath13 ) , and we do indeed measure a low value ( @xmath29 km s@xmath13 ) . further evidence that some of these new candidates may be true members is given by the fact that , unbeknownst to makarov & fabricius , one of their stars ( hip 57524 , too faint for us and not shown in table 4 ) is actually twa-19 , which is li - rich . follow - up observations of all these candidate are necessary to establish their youth , and our colleagues at this workshop are encouraged to do so . l@ c@ c@ c@ c@ c@ cc@ c & & r.a . & dec . & @xmath30 & @xmath27 & @xmath26 & @xmath25 & @xmath24 + name & twa # & ( 2000 ) & ( 2000 ) & ( mas / yr ) & ( pc ) & ( pc ) & ( km s@xmath13 ) & ( km s@xmath13 ) + hip 53911 & 1 & 11:01:51.9 & @xmath534:42:17 & 75.4 & 56.4 & 57.1 & @xmath412.7 & @xmath412.9 + tyc 7201 - 0027 - 1 & 2 & 11:09:13.8 & @xmath530:01:40 & 92.6 & & 47.1 & @xmath410.6 & @xmath411.2 + hip 55505 & 4 & 11:22:05.3 & @xmath524:46:40 & 96.8 & 46.7 & 45.7 & @xmath49.1 & @xmath49.2 + tyc 7223 - 0275 - 1 & 5 & 11:31:55.3 & @xmath534:36:28 & 86.7 & & 50.9 & @xmath410.0 & @xmath46.9 : + hip 57589 & 9 & 11:48:24.2 & @xmath537:28:49 & 158.1 & 50.3 & 76.3 & @xmath412.6 & @xmath410.2 + hip 48273 & & 09:50:30.1 & @xmath404:20:37 & 157.9 & 45.9 & 26.6 & @xmath410.7 & @xmath416.2 + tyc 6604 - 0118 - 1 & & 09:59:08.4 & @xmath522:39:35 & 163.7 & & 62.8 & @xmath416.5 & @xmath427.0 + hip 50796 & & 10:22:18.0 & @xmath510:32:16 & 179.1 & 34.0 & 53.8 & @xmath413.1 & @xmath413.1 + hip 53486 & & 10:56:30.8 & @xmath407:23:18 & 268.1 & 17.6 & 16.7 & @xmath43.7 & @xmath45.5 +
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we present our spectroscopic measurements of the radial velocity , effective temperature , and projected rotational velocity of several of the known members of the tw hya association , as well as measurements for candidate members selected on the basis of their x - ray or kinematic properties .
a number of our targets turn out to be binaries , but most are non - members . the radial velocities for some of the other candidates support the conclusion that they are kinematically associated with the group , although further observations are required to show that they are indeed pre - main sequence objects .
# 1_#1 _ # 1_#1 _ = # 1 1.25 in .125 in .25 in
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the prompt emission detected from gamma - ray bursts ( grbs ) is believed to originate at large distances from the central engine , from within an ultrarelativistic outflow @xcite . this ultra - relativistic motion is necessary to avoid strong @xmath0 annihilation , a signature that is not observed ( see , e.g. , * ? ? ? thermal emission is naturally expected in such a scenario . indeed , since the densities at the base of the relativistic flow are very large , the medium is optically thick to radiation owing to thomson scattering by entrained electrons . the optical depth decreases during the relativistic expansion and the outflow eventually becomes transparent for its own radiation , at the photospheric radius . any internal energy that is still carried out by the flow can be radiated at the photosphere and will be observed as a thermal component in the prompt spectrum . this expected photospheric emission in grb spectra was early suggested on such theoretical grounds by @xcite , @xcite , and @xcite , among others . the non - thermal component observed in the spectrum has to be produced by another mechanism in the optically thin region , i.e. , well above the photosphere . due to the ultra - relativistic motion , this difference in the radius of the emission implies a delay between the observation of the two components that is usually small compared to the typical duration of a long grb , and is also small compared to the typical duration of time intervals used for time - dependent spectroscopic analysis . the thermal and non - thermal components should then appear superimposed for the observer ( e.g. * ? ? ? @xcite pointed out that in the standard fireball model , the photospheric component can easily be dominant in the spectrum if the efficiency @xmath1 of the mechanism responsible for the non - thermal emission is only moderate ( @xmath2 ) . observationally , @xcite , @xcite , and @xcite argued that a photospheric component is present in cgro batse data . the limited energy range provided by batse ( 20 - 2000 kev ) , however , hampered the possibility of unambiguously identifying the emission process . since the launch of fermi in 2008 , the combination of the gamma ray burst monitor ( gbm ) and the large area telescope ( lat ) provides an unprecedented energy range for grb spectroscopy , and the identification of the emission processes responsible for the gamma - ray prompt emission may become a reality . gbm alone covers a wider energy range than its predecessor batse , and the design of its data enables finer resolution spectroscopy . this allows better constraints on spectral fits , with increasingly complex models . grb energy spectra in the kev - mev energy range are usually well represented by the band function @xcite , two power - laws , smoothly joined and parameterized by e@xmath3 which represents the energy at which peak power is radiated @xcite . the value of the low - energy power - law index , @xmath4 , is higher than the value of the high - energy power - law index , @xmath5 , and the parameter e@xmath3 of the band function for grbs generally appears to follow predictable trends with time and flux level @xcite . it is , however , an empirical function rather than a physically - motivated model . the meaning of the parameters in the context of emission and transport mechanisms taking place in grbs is not well understood , but is generally believed to represent the non - thermal emission from accelerated charged particles . in section 1 , we describe our observational results consisting of a grb prompt - emission spectrum best fit with the combination of a thermal component and a standard band function . in section 2 , we use these results to constrain the origin of the energy released in the grb jet . gbm is composed of 12 sodium iodide ( nai ) detectors covering an energy range from 8 kev to 1 mev and two bismuth germanate ( bgo ) detectors sensitive between 200 kev and 40 mev @xcite . the instrument triggered on 24 july 2010 , at t@xmath6=00:42:05.992 ut on the very bright grb _ 100724_b @xcite . the event was also seen at higher energies in the fermi large area telescope ( lat ) @xcite . the most precise position for the direction of the burst is the intersection of the interplanetary network annulus obtained using gbm , konus - wind @xcite , and messenger data with the 90% lat confidence level location error box , and is a strip of sky centered on ra = 118.8@xmath7 and dec = 75.8@xmath7 which is 1.2@xmath7 long and 0.2@xmath7 wide ( k.hurley and v.palshin , private communication ) . figure [ fig : figure1 ] ( top two panels ) shows the gbm light curve of grb _ 100724_b in two energy bands . multiple peaks of varying intensity are superimposed on a pre - trigger plateau , with a decaying tail that is detected over 200 s from t@xmath8 . ( in blue ) and the bb temperature kt ( in red ) over the duration of the burst . the vertical dashed lines indicate the period used in the time - integrated analysis . [ fig : figure1 ] ] we simultaneously fit the spectral data of the nai detectors with a source angle less than 60@xmath7 ( nais 0 , 1 , 2 , 3 , and 5 ) and the data of the brightest bgo detector ( bgo 0 ) using the analysis package rmfit 3.3rc8 . an effective area correction is applied between each of the nais and bgo 0 during the fit process . this correction is used to handle possible discrepancies between the flux in the detectors due to the choice of the model to generate the instrument responses for instance . we performed a time - integrated spectral analysis over the main part of the burst ( t@xmath6 - 1.024s to t@xmath6 + 83.969s ) using the band function . the band parameters are in part fairly typical of the ensemble of grbs , with @xmath9 and e@xmath3 = @xmath10 kev @xcite . however , with an index @xmath11 , the high - energy power law systematically overshoots the observed flux above 1 mev in bgo , as can be seen by the fit residuals in figure [ fig : figure2 ] ( top two panels ) , which also indicate systematic patterns at low energy . this suggests a simple band function does not adequately represent the spectrum of this burst . we identify the best shape to fit the above - mentioned spectral deviation by fitting the same data simultaneously with a band function combined with each of the following models : single power- law ( pl ) , black body ( bb ) , band function , power law with exponential cut off ( comp " for comptonized model ) , and gaussian . we select the best model by choosing the fit with the lowest castor c - stat value ( later c - stat ) . c - stat differs from the poisson likelihood statistic by an offset which is a constant for a particular dataset . table [ table : spectraldeviationfit ] shows the results of these fits . the effective area correction described above is on the order of a few percent and does not change the c - stat for each fit more than a few units , nor does it change the value of the parameters resulting from the fit . [ cols= " < , < , < , < , < , < , < , < , < , < , < , < , < , < " , ] while spectral deviations from the standard band function were previously identified in the form of an additional pl to the band function sometimes extending from the lower energy in the gbm to the higher energy in the lat @xcite , in the case of grb _ 100724_b a pl spectral component does not improve on the band - only fit and an additional bb component to the band function is the best model to fit the spectral deviation . an equal c - stat is obtained for band+band and band+compt , but with @xmath4 close to + 1 for the additional band and compt functions , and a very low value for @xmath5 ( only constrained as an upper limit , below -5 ) for the extra band function , the band and compt functions can be interpreted as a planck function . even with more parameters , the additional band and compt functions resemble a bb component , reinforcing band+bb as the best combination . additional models were tried , such as a log - parabola function @xcite , but the results were highly disfavored , and we exclude them from table 1 . figure [ fig : figure2 ] ( bottom two panels ) shows the bb contribution below e@xmath3 . compared to the band - only fit , e@xmath3 is shifted towards higher energy to @xmath12 kev and @xmath5 is lower with a value of @xmath13 . this index is consistent with the flux detected above 1 mev , and the spectrum seen in the lat @xcite at higher energies . @xmath4 is also significantly lowered to @xmath14 . while the simultaneous fit of all the selected detectors provides the best constraints on the two spectral components , fits with band+bb to combinations of individual nai detectors with bgo 0 result in similar parameter values and offer significant improvement over the band - only fit . this provides a check that the bb component is real and not introduced by effects such as detector deadtime or spectral distortions that would affect each detector in a different way depending on the angle of the detector to the source . to verify that the improvement in the fit obtained by adding a bb component to the band function is not a statistical fluctuation , we generated 20,000 synthetic spectra for each selected detector . for the simulations we used the parameters from the fit performed with the band - only function , which we take as the null hypothesis . to create the simulated spectra , for each detector the real background is added to the source spectrum model and poissonian fluctuations are applied to the sum all the detectors are then fit simultaneously with both band and band+bb , and their c - stat are compared . none of the 20,000 simulated spectra give a difference larger than 45 units of c - stat ( [ band][band+bb ] ) , while in the real data , this difference is 95 units , corresponding to a probability lower than @xmath15 that the bb excess is due to statistical fluctuations . to study the evolution of the spectral components , 22 time intervals were devised by requiring that each interval produce a band+bb spectral fit with well - constrained band function parameters , while attempting to separate the peaks and valleys of the light curve so that the spectral fit parameters can be tracked with burst flux as well as with time . the bottom panel of figure [ fig : figure1 ] exhibits the evolution of e@xmath3 and bb temperature , kt , through these intervals . we notice that e@xmath3 tracks the lightcurve and globally decreases over time . the bb component is detected throughout the burst , and its temperature shows weak correlation with e@xmath3 . the significance of this correlation is difficult to assess , mostly because the variation in temperature is small . overall , it appears the temperature is quite stable , with figure [ fig : figure3 ] showing more clearly the small scatter in kt . can be fit by a gaussian distribution with a mean of @xmath16 , and a @xmath17 standard deviation of @xmath18 . [ fig : figure3 ] ] with a fluence of @xmath19 erg @xmath20 measured in 85 sec from [email protected] between 8 kev and 40 mev , grb _ 100724_b is the most intense grb detected by gbm over this energy range through 2010 september . combined with the broad energy range of the gbm , this allows for accurate modeling of its energy spectrum even with this complex model . previous observational results regarding thermal components in grbs were ambiguous and some were limited to individual fine time - slices rather than a spectral fit over the entire emission period . some studies showing bb fits did not demonstrate that the bb fit was statistically preferred to a simple non - thermal component @xcite . other analyses found bb+pl spectra for isolated portions of selected grbs , raising the possibility that these spectra are actually adequately fit with a standard band function but that due to a weak signal in small time slices and extreme parameters for the band function , a bb shape is competitive with the band function @xcite . the non - thermal component fit with a single power law suggested a break well beyond the common e@xmath3 values , and the bb temperature and its variations intriguingly matched those of a typical e@xmath3 . despite the broader energy range of rhessi grb observations , one analysis found difficulties in fitting combined thermal plus non - thermal models @xcite . we find here that the joint bb plus non - thermal ( band ) fit is highly statistically preferred so that in simultaneously detecting both components we are confident in their correct identification . time - resolved spectroscopy of grb _ 100724_b reveals that this bb component is seen throughout the burst and does nt evolve much over time , while the non - thermal component follows the typical variations @xcite . the consistency of the mean kt value with the temperature obtained in the time - integrated spectral fit , combined with the detection of the bb component throughout the burst , strengthen the case for an underlying thermal component in the gamma - ray emission seen from grb _ 100724_b and show that the presence of the bb in the time - integrated spectrum can not be attributed to spectral evolution of the band function during the burst . e@xmath3 varies substantially , from @xmath21 to @xmath22 kev . at the same time , the thermal component remains relatively steady with the temperature varying only modestly between 30 and 50 kev as suggested by a @xmath23 dependence expected from a bb component . time - averaged values of the temperature and the flux of the thermal component , and of the ratio of this flux over the total gamma - ray flux are @xmath24 kev , @xmath25 erg / s/@xmath26 and @xmath27 . in the standard fireball model these observables allow determination of the physical properties of the outflow and its photosphere . owing to the imprecise and delayed localization of grb _ 100724_b , optical follow up to determine the distance to the source was impossible . for this reason the temperature of the bb can be translated into a real source temperature only as a function of source distance . we assume in the following argument a typical redshift @xmath28 . we find that the lorentz factor is @xmath29 , the photospheric radius is @xmath30 and the radius at the base of the flow is @xmath31 @xcite . here @xmath32 is a geometrical factor of order unity and @xmath1 is the efficiency of the mechanism responsible for the non - thermal emission . with an extreme efficiency @xmath33 , these estimates are in good agreement with the typical values expected in the fireball model . the dependence on redshift is not strong : at z=3 ( resp . 8) , @xmath34 ( resp . 1290 ) , @xmath35 ( resp . @xmath36 ) , and @xmath37 ( resp . @xmath38 ) . using more realistic values for the efficiency , the radius @xmath39 is the most altered , with @xmath40 for @xmath41 . such small values are puzzling . if the central engine is a rotating black hole , as in the popular collapsar model for long grbs @xcite , with a minimal mass in the range 5 10 @xmath42 , such radii are smaller than the typical value expected for the innermost stable orbit , from 44 - 89 km for a non - rotating black hole to 22 - 43 km for a highly - rotating black hole having a spin @xmath43 . these results for the time - integrated spectrum imply a small @xmath44 or a very large efficiency and the constraint is even stronger in some time bins . we conclude that observations of grb _ 100724_b require either a very high efficiency for the non - thermal process , or a very small size of the region at the base of the flow , both of which are quite challenging for the standard fireball model , if not excluding it . a simple solution to this discrepancy between the standard fireball model and the observations is to assume that the initial energy release by the central engine is not purely thermal , but that the flow is highly magnetized close to the source @xcite . the magnetization @xmath45 is the ratio of the poynting flux over the power ( thermal + kinetic ) carried by the baryons . if no magnetic dissipation occurs below the photosphere , the efficiency @xmath1 in the estimates of @xmath46 , @xmath47 , and @xmath39 above should be replaced by @xmath48 . a magnetization @xmath49 will therefore reconcile the observations with physically acceptable values for the radius at the base of the flow and the efficiency of the mechanism responsible for the non - thermal emission . a similar conclusion is reached for scenarios where magnetic dissipation occurs early and contributes efficiently to the acceleration of the jet . however , the appearance of a low intensity thermal component in the spectrum probably excludes the most extreme version of the magnetized outflow scenario , where the energy is released by the central engine as a pure poynting flux ( @xmath50 ) . we have shown that the simultaneous presence of thermal and non - thermal components to the spectra of grb _ 100724_b is statistically preferred . although the non - thermal component is dominant , the black body flux is well within the gbm sensitivity . deviations from the band function may be measurable in less fluent bursts or in bursts where the thermal component is less prominent , providing that the black body component lies in the band pass of the instrument and its peak in energy is distinguishable from @xmath51 . if the presence of an unresolved thermal component in other bursts modifies the band function parameters in the same sense as the band - only fit for grb _ 100724_b , then we might expect a systematic bias yielding values of @xmath4 and @xmath5 that are higher ( harder ) than in the true non - thermal component . two important consequences of this bias are that the perceived violation of the synchrotron limit that disallows values @xmath52 ( for slow - cooling electrons ) and @xmath53 ( for fast - cooling electrons ) may not be as common as suggested by @xcite and @xcite , and that the relatively low rate of bursts detected by the lat compared to the predictions of @xcite and the observations in abdo et al . ( in preparation ) based on extrapolations of @xmath5 from lower energies might be explained by this bias in @xmath5 , a possibility suggested also by @xcite . our observations provide strong evidence for the presence of a photospheric spectral component , long suspected to exist in the standard fireball model . in addition , our results require implausible parameters for the standard baryonic fireball model and therefore favor a substantial magnetic component to the outflow . the gbm project is supported by the german bundesministerium fr wirtschaft und technologie ( bmwi ) via the deutsches zentrum fr luft- und raumfahrt ( dlr ) under the contract numbers 50 qv 0301 and 50 og 0502 . , f. , axelsson , m. , zhang , b. b. , mcglynn , s. , peer , a. , lundman , c. , larsson , s. , battelino , m. , zhang , b. , bissaldi , e. , bregeon , j. , briggs , m. s. , chiang , j. , de palma , f. , guiriec , s. , larsson , j. , longo , f. , mcbreen , s. , omodei , n. , petrosian , v. , preece , r. , and van der horst , a. j. : 2010 , , l172
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observations of grb _ 100724b _ with the fermi gamma - ray burst monitor ( gbm ) find that the spectrum is dominated by the typical band functional form , which is usually taken to represent a non - thermal emission component , but also includes a statistically highly significant thermal spectral contribution . the simultaneous observation of the thermal and non - thermal components allows us to confidently identify the two emission components .
the fact that these seem to vary independently favors the idea that the thermal component is of photospheric origin while the dominant non - thermal emission occurs at larger radii .
our results imply either a very high efficiency for the non - thermal process , or a very small size of the region at the base of the flow , both quite challenging for the standard fireball model .
these problems are resolved if the jet is initially highly magnetized and has a substantial poynting flux .
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since the initial observation of exciton - polaritons in a strongly coupled microcavity in 1992 @xcite , a wide range of quantum many - body effects have been observed in polariton fluids such as bose - einstein condensation @xcite , and superfluidity exhibiting quantized vortices @xcite and solitons@xcite . most of these results have been interpreted in terms of nonequilibrium bose gas theory , because the thermalization of the polaritons has been limited by their short cavity lifetime , on the order of 10 ps , compared to a thermalization time of the order of 1 ps . our recent results @xcite have indicated that we can now produce structures which allow much longer lifetime , of the order of 200 ps . here we report on accurate measurements of this lifetime using a unique method in which we inject polariton pulses at finite momentum into a microcavity and track their motion in time and space . this allows us to observe `` slow reflection , '' in which renormalized light slows down to zero velocity , turns around , and goes back the other way . in addition to providing a measure of the lifetime , the long - distance propagation seen here allows the possibility of beam - like polariton - interaction experiments and all - optical switching methods over long distances . as the technology of microcavity polaritons is now well established , much attention has turned to increasing the lifetime of the polaritons , to allow better thermalization and to allow propagation over longer distances . the lifetime of polaritons is a function of the intrinsic photon lifetime of the cavity and the fraction of photon in the polariton states . as amply discussed elsewhere@xcite , a polariton state @xmath2 is a superposition of an exciton state @xmath3 and a photon state @xmath4 , @xmath5 where @xmath6 and @xmath7 are the @xmath8-dependent hopfield coefficients . the @xmath0 signs indicate that there are two superpositions , known as the upper and lower polaritons ; in the experiments reported here we focus entirely on the lower polariton branch . at resonance , @xmath9 , while far from resonance the polariton can be nearly fully photon - like or exciton - like . this implies that the @xmath8-dependent lifetime @xmath10 of the polaritons is given by @xmath11 for polaritons in our gaas - based samples , the rate of nonradiative recombination @xmath12 is negligible , so the lifetime is essentially entirely determined by the photon fraction and the cavity lifetime . in early polariton experiments@xcite , the cavity lifetime was on the order of 1 ps while the polariton lifetime was at most 10 - 15 ps , even well into excitonic detunings . this implied that polaritons would only scatter a few times on average before decaying . in recent experiments@xcite , the polariton lifetime has been extended to about 30 ps . our previous work @xcite has given estimates of the polariton lifetime in new samples of the order of 100 - 200 ps , allowing polaritons to propagate hundreds of microns to millimeters within the cavity and to show a sharp transition to a superfluid state . because of the propagation of the polaritons to long distances away from the excitation spot , the configuration of those experiments made it difficult to get an accurate measure of the lifetime . a measurement spatially restricted to the laser excitation spot would give a severe underestimate of the lifetime , because the polaritons do not stay put they feel a force due to the cavity thickness gradient that pulls them to one side , leading them to travel hundreds of microns from the excitation spot . therefore , to accurately measure the lifetime , a measurement must track the polaritons in space as they move . the measurements reported here do just that . these measurements confirm the earlier estimates of the lifetime but considerably reduce the uncertainty . the sample was arranged such that the gradient was aligned with the streak camera time slit , and then polaritons were injected at a large angle such that they moved directly against the gradient . the experimental setup is shown in figure [ fig1 ] . we used an objective with a wide field of view in addition to a large numerical aperture . a resonantly injected picosecond pulse of polaritons was tracked as it entered the field of view , turned around and traveled away . this occurs because the sample has a cavity thickness variation that leads to an energy gradient of the polariton . in simple terms , one can think of the motion of the polaritons as governed by energy conservation with the following hamiltonian , which is just the same as that of a massive object moving in a potential gradient : @xmath13 here @xmath14 is the effective mass of the lower polariton branch that we observe , which depends weakly on @xmath8 , and is equal approximately to @xmath15 times the vacuum electron mass in these experiments . the force @xmath16 is given by the gradient in space of the @xmath17 cavity resonance energy , and is approximately equal to @xmath18 mev / mm for the section of the microcavity studied here . we will refer to `` uphill '' as moving toward higher cavity resonance energy ( narrower cavity width ) and `` downhill '' as moving to lower energy ( wider cavity width ) . this experimental setup utilizes the fact that the polaritons in these high-@xmath1 samples flow over a great spatial distance and change in - plane momentum rapidly . the lifetime of shorter - lived polaritons is more difficult to directly observe by streak camera measurements due to the overlap of any emission with the injecting laser . upon resonantly injecting polaritons , the created population is in the same state as the exciting laser . the initial polariton population therefore will have the same characteristics as the exciting laser and can not be separated from it . observing any other state ( for example by looking at cross - polarized emission ) will inherently measure the scattering time of the polaritons to enter that state . in this experiment , we rely on the fact that polaritons will flow ballistically from the point of injection to the point of detection in order to separate the observed luminescence from the reflected laser . to the extent that this motion is ballistic , integrating the population over the observed spatial region will directly yield the population decay of the polaritons . unlike the case of observing luminescence from a different energy or polarization state than the initial population , this method directly follows the decay of a single population rather than relying on an average over many @xmath8-states . the momentum of the injected polaritons is controlled by the angle of the laser which generates them . the angle of incidence used here was @xmath19 , corresponding to an initial wavevector of @xmath20 @xmath21 . after propagating uphill for over two millimeters , the polaritons enter our spatial field of view and optical collection angle . observing the polaritons far from the injection point reduces collection of scattered light from the laser excitation , and injecting polaritons at a large angle ensures that the reflected laser is outside the collection angle of the lens . as the polaritons flow against the cavity gradient they lose momentum , effectively exchanging in - plane kinetic energy for confinement energy , similar to a ball rolling uphill exchanging kinetic energy for gravitational potential energy . since polaritons have a one - to - one relationship of in - plane momentum to external angle of emission for emitted photons , watching the luminescence change emission angle while the gas of polaritons propagates gives us a direct observation of their slowing . because the entire process is energy conserving , the injection laser , the ballistic polaritons and the emission all have the same wavelength . once the polaritons reach a turn - around point , they flow back downhill and the emission angle increases to the negative direction . figure [ fig2 ] shows time - integrated real space emission intensity from the microcavity near the turn - around point of the polariton gas . the coordinates in this image are such that the injection point is at roughly ( 0,0 ) , and the force due to the cavity gradient is nearly directly toward @xmath22 . while polaritons were injected primarily in the @xmath23-direction , the initial narrow spread of momenta in the y - direction led to a spread in real space after propagation over a long distance . at roughly @xmath24 mm , polaritons are seen entering the field of view , which also corresponds to the acceptance angle of the optics . the brightest streak , directly horizontal at y=0 mm , is the trajectory of the most intense part of the injected population which was peaked at zero momentum in the @xmath25-direction . other bright streaks can be seen arcing to @xmath26 , and the entire range of states reach their respective turn - around points at @xmath27 mm . the fact that there are bright streaks in this image rather than a smooth cloud suggests that the injection of the polaritons into the cavity occurs unevenly in momentum space . the asymmetry of the cloud between @xmath28 and @xmath29 may be due to a slight misalignment between the cavity gradient and the injection direction . , @xmath25)=(0,0 ) , and the gradient is approximately toward @xmath22 . polaritons approach this field of view from the left and turn around at @xmath30 mm before flowing back to @xmath22 . the sharp cutoff at @xmath311.7 mm is due to clipping in the spectrometer.,width=326 ] to measure the lifetime , the bright jet of polaritons was time - resolved using a hamamatsu streak camera . to facilitate this , the sample was initially installed such that the gradient was aligned with the horizontal time slit on the streak camera . this enabled us to track a single jet of polaritons while they propagate against the gradient , turn around , and travel backwards , as shown in figure [ fig3](a ) . the vertical distance axis in this figure corresponds to the horizontal @xmath32-axis in figure [ fig2 ] . the trajectory of the polaritons is easily seen in the data , which in this region is well described by a parabolic fit , as expected for the hamiltonian ( [ ham ] ) , which is equivalent to that of a ball moving with a constant force due to gravity . indeed , these data directly demonstrate the in - plane velocity and acceleration of the polaritons during their trajectory . one should note that this region of observation is already more than a millimeter and nearly 200 ps from the injection point , indicating that these polaritons are propagating farther and persisting longer than those in earlier samples , even without confinement in 1d structures , such as used in ref . . 150 @xmath33 m @xmath34 150 @xmath33 m of fig . ( a ) intensity vs @xmath32-distance vs time of the propagating polaritons . the dashed red line is a fit to the polariton motion as they feel a constant acceleration of @xmath35 mm / ns@xmath36 . this acceleration is in good agreement with the expected value based on the known cavity gradient and the effective mass . ( b ) the polariton intensity of ( a ) summed in the @xmath32-dimension to highlight the exponential decay of the population . the data are well fit by a single exponential decay with lifetime of @xmath37 ps.,width=326 ] a simple analysis of this data yields the polariton lifetime after summing in the spatial dimension , as shown in figure [ fig3](b ) . the data are well fit by a single exponential with a lifetime of @xmath38 ps . for the region of the sample observed in figure [ fig3 ] , the detuning of the polariton corresponds to the lower polariton approximately 75% photonic . ( although the polaritons move long distances , their detuning does not change much because they stay at the same energy . ) from this we estimate that the cavity photon lifetime is approximately 135 ps , which corresponds to a @xmath1-factor of over 320,000 . it should be noted that this lifetime measurement may still be an underestimation of the lifetime . close inspection of figure [ fig2 ] reveals that individual jets of polaritons are still spreading out from the central jet . a population with some spread in initial momenta perpendicular to the cavity gradient must spread out horizontally while propagating uphill . the fraction of polaritons that move out of our field of view will lead to an underestimation of the lifetime . this error can be compensated for by using a narrower time slit to cut out adjacent jets at early times ; however , narrow slit widths can result in errors that will either underestimate or overestimate the polariton lifetime if the entirety of the main jet is not aligned with the time slit . in this experiment , data was collected over a range of slit widths from 50 to 300 @xmath33 m with consistent results . in another set of experiments , we generated polaritons via a 200-fs pump pulse tuned to one - half the energy of the polaritons . such experiments will be presented in more depth elsewhere , but we can see here that the propagation of long - lived polaritons can be used to probe the dynamics of two - photon generated polaritons . we conducted two experiments . first , we recreated geometry of the one - photon experiments discussed above by injecting polaritons uphill and observing the turn - around point , as shown in figure [ fig4](a ) . second , we injected polaritons at normal incidence , directly at the point of observation , as shown in figure [ fig4](b ) . figure [ fig4](a ) clearly shows characteristics similar to the one - photon resonant injection data presented in figure [ fig3 ] , with polaritons entering the field of view at @xmath39 200 ps after the injecting laser pulse and following a roughly parabolic trajectory . however , the trajectory is very broad with a poorly defined turn - around point . this can be understood by recognizing that the initial population is created in a range of energy states due to the spectral width of the exciting femtosecond pulse . since there is a broad range of photon energy and momenta in the pump , a range of polariton states can satisfy energy and momentum conservation in the two photon absorption . this method of observing polariton propagation can therefore enable us to characterize the initial polariton population . in fig . [ fig4](b ) we present data from a different setup employing two - photon generation of polaritons . in this case , we pump at normal incidence directly through the microscope objective used to image the luminescence . since the pump laser has wavelength far from the polariton wavelength , there is no difficulty with scattered laser light . since the entire na of the objective was used , the angle of incidence of the pump light ranged over @xmath0 20@xmath40 even though the intensity was maximum at 0@xmath40 and the laser was spectrally tuned to the @xmath17 state . the data indicate that polaritons created directly via two - photon absorption are peaked at an initial wavevector uphill , with no @xmath17 polaritons created initially . the long - lived population stationary at @xmath41 is substrate luminescence excited by two - photon absorption . figure [ fig4](c ) shows the same data as figure [ fig4](b ) , except that the image is centered to eliminate the clipping at @xmath42 in ( b ) . the majority of the injected polaritons have finite @xmath8 , even though the pump light was centered at @xmath17 . this supports the view that that two - photon absorption of the polaritons , which should be forbidden at @xmath17 due to the selection rules , becomes allowed at finite @xmath8 , due to valence - band mixing with the higher - lying light - hole states . these results show clearly that two - photon resonant generation of polaritons is possible . one can expect very strong nonlinear effects from microcavity polaritons due to the strong interaction of the cavity mode with the quantum well exciton@xcite . we point out that there is no comparable one - photon excitation experiment such an experiment will not work because the exciting laser will be reflected directly back into the imaging system . polaritons can be viewed as `` renormalized photons , '' especially in the region of the cavity where the the detuning makes the polaritons mostly photon - like . as mentioned above , the behavior we have seen here can thus be viewed as a type of `` slow light , '' or `` slow reflection , '' in which the photons decelerate from @xmath39 3.5% of the speed of light to a full stop and then go back the other way . this behavior is expected for light in a wedge - shaped cavity , without any need for the excitonic part of the polaritons . however , it has been hard to directly observe , because one must have very high @xmath1 and fast time resolution to track the motion of the photons . these measurements show that the photons can truly be viewed as having effective mass and feeling a force . with such long distance propagation and long lifetime , it is now possible to construct experiments in which two or more beams are used and caused to interact . this could be used to directly measure the polariton - polariton interactions and also for schemes of optical gating using polaritons , as presented e.g. in ref . @xcite . additionally , we have shown that two - photon injection of polaritons is a very relevant phenomena in gaas microcavities , but there is a noticeable dependence of the absorption strength on the angle of incidence . by temporally- and spatially - resolving the propagation of polaritons after generation we can probe the mechanisms by which different states are occupied by this nonlinear absorption . the work at the university of pittsburgh was supported by the national science foundation under grants phy-1205762 and eccs-1243778 . the work at princeton university was partially funded by the gordon and betty moore foundation as well as the national science foundation mrsec program through the princeton center for complex materials ( dmr-0819860 ) . 10 c. weisbuch , m. nishioka , a. ishikawa , and y. arakawa , `` observation of the coupled exciton - photon mode splitting in a semiconductor quantum microcavity , '' physical review letters * 69 * , 33143317 ( 1992 ) . j. kasprzak , m. richard , s. kundermann , a. baas , p. jeambrun , j. m. j. keeling , f. m. marchetti , m. h. szymaska , r. andr , j. l. staehli , v. savona , p. b. littlewood , b. deveaud , and l. s. dang , `` bose einstein condensation of exciton polaritons , '' nature * 443 * , 40914 ( 2006 ) . r. balili , v. hartwell , d. snoke , l. pfeiffer , and k. west , `` bose - einstein condensation of microcavity polaritons in a trap , '' science * 316 * , 10071010 ( 2007 ) . k. g. lagoudakis , f. manni , b. pietka , m. wouters , t. c. h. liew , v. savona , a. v. kavokin , r. andr , and b. deveaud - pldran , `` probing the dynamics of spontaneous quantum vortices in polariton superfluids , '' physical review letters * 106 * , 115301 ( 2011 ) . a. amo , s. pigeon , d. sanvitto , v. g. sala , r. hivet , i. carusotto , f. pisanello , g. lemenager , r. houdre , e. giacobino , c. ciuti , and a. bramati , `` polariton superfluids reveal quantum hydrodynamic solitons , '' science * 332 * , 11671170 ( 2011 ) . b. nelsen , g. liu , m. steger , d. w. snoke , r. balili , k. west , and l. pfeiffer , `` dissipationless flow and sharp threshold of a polariton condensate with long lifetime , '' physical review x * 3 * , 041015 ( 2013 ) . m. steger , g. liu , b. nelsen , c. gautham , d. w. snoke , r. balili , l. pfeiffer , and k. west , `` long - range ballistic motion and coherent flow of long - lifetime polaritons , '' physical review b * 88 * , 235314 ( 2013 ) . a. v. kavokin , j. j. baumberg , g. malpuech , and f. p. laussy , _ microcavities _ ( oxford university press , new york , 2007 ) . f. tassone and y. yamamoto , `` lasing and squeezing of composite bosons in a semiconductor microcavity , '' physical review a * 62 * , 063809 ( 2000 ) . h. deng , g. weihs , d. snoke , j. bloch , and y. yamamoto , `` polariton lasing vs. photon lasing in a semiconductor microcavity , '' proceedings of the national academy of sciences of the united states of america * 100 * , 1531815323 ( 2003 ) . g. tosi , g. christmann , n. g. berloff , p. tsotsis , t. gao , z. hatzopoulos , p. g. savvidis , and j. j. baumberg , `` sculpting oscillators with light within a nonlinear quantum fluid , '' nature physics * 8 * , 190194 ( 2012 ) . e. wertz , l. ferrier , d. solnyshkov , r. johne , d. sanvitto , a. lematre , i. sagnes , r. grousson , a. v. kavokin , p. senellart , g. malpuech , and j. bloch , `` spontaneous formation and optical manipulation of extended polariton condensates , '' nature physics * 6 * , 860864 ( 2010 ) . v. pellegrini , r. colombelli , i. carusotto , f. beltram , s. rubini , r. lantier , a. franciosi , c. vinegoni , and l. pavesi , `` resonant second harmonic generation in znse bulk microcavity , '' applied physics letters * 74 * , 19451947 ( 1999 ) . s. lei , y. yao , z. li , t. yu , and z. zou , `` design and theoretical analysis of resonant cavity for second - harmonic generation with high efficiency , '' applied physics letters * 98 * , 13 ( 2011 ) . d. ballarini , m. de giorgi , e. cancellieri , r. houdr , e. giacobino , r. cingolani , a. bramati , g. gigli , and d. sanvitto , `` all - optical polariton transistor . '' nature communications * 4 * , 1778 ( 2013 ) . * supplemental information this sample is the same as used in refs @xcite . a @xmath43 microcavity contains three sets of four gaas quantum wells located at the antinodes of the cavity mode . the qws are 70 pure gaas embedded in pure alas barriers at least 30 thick . the optical mode is confined between distributed bragg reflectors made of alas / al@xmath44ga@xmath45as with 32 pairs on the top surface and 40 pairs on the bottom surface . molecular beam epitaxial growth of the sample leads to an inherent wedge to the cavity thickness , resulting in a gradient of the cavity as well as exciton energies . the polariton exhibits a rabi coupling of 6 mev at 5 k ; the cavity mode gradient is 13 mev / mm and the exciton gradient is 1.5 mev / mm . the sample was held in a cold - finger cryostat at 5 k for all experiments . emission was collected using a n.a.=0.42 microscope objective . a preliminary imaging lens permitted spatial filtering of the real space image data , and a subsequent iris in the fourier image plane permitted filtering of the emission angle . secondary lenses could be exchanged to image either the real - space or angle - resolved emission . luminescence was imaged through a spectrometer onto either a standard ccd or onto a hamamatsu streak camera . in the resonant injection experiment , polaritons were resonantly injected at @xmath46 nm with a picosecond laser far on the photonic side of the sample . the injected state had a detuning of approximately -2.6 mev and corresponded to an external angle of roughly 42@xmath47 ( @xmath48 @xmath21 ) . the sample and pump laser were arranged such that the polaritons were moving anti - parallel to the cavity gradient , which was aligned with the time slit . the angle of incidence was larger than the collection angle of the optics , so the reflected beam was not collected . additionally , the pump spot was spatially outside the field of view such that scattered light was not collected . at a distance of approximately 2 mm from the injection point , emission entered the collection range of the optics . at the turn - around point , the polaritons are more photonic with a detuning of -7.4 mev which corresponds to a photon fraction of 75% . for the two - photon injection experiment , a 200-fs pulse generated by a coherent opa system was tuned to one - half the energy of the desired polariton transition . for the 40@xmath40 injection case , the laser was tuned to excite at @xmath46 nm . for the 0@xmath40 injection case , the laser was tuned to excite at @xmath49 nm . the x - distance from the injection point for the 40@xmath40 injection cases was estimated as follows : the @xmath32-distance in figs . [ fig2],[fig3](a ) , and [ fig4](a ) was determined from the fit of the polariton @xmath32 vs @xmath50 trajectory presented in fig . [ fig3](a ) . extrapolation of this fit back to time @xmath51 as determined by locating scattered laser light determines the initial position of excitation . this initial excitation position is consistent with the sample parameters and injection conditions . this method assumes that the acceleration of the polaritons is strictly constant from creation to turn around . variation in the acceleration due to a non - constant energy gradient in addition to the changing mass of the polariton implies uncertainty on the overall offset of this axis , but the spatial magnification was measured directly .
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we resonantly inject polaritons into a microcavity and track them in time and space as they feel a force due to the cavity gradient .
this is an example of `` slow reflection , '' as the polaritons , which can be viewed as renormalized photons , slow down to zero velocity and then move back in the opposite direction .
these measurements accurately measure the lifetime of the polaritons in our samples , which is 180 @xmath0 10 ps , corresponding to a cavity leakage time of 135 ps and a cavity @xmath1 of 320,000 .
such long - lived polaritons propagate millimeters in these wedge - shaped microcavities .
additionally , we generate polaritons by two - photon excitation directly into the polariton states , allowing the possibility of modulation of the two - photon absorption by a polariton condensate .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
for the stellar populations of our milky way galaxy ( mwg ) - the halo , the bulge , and the disk ( thick plus thin disk ) - the fundamental questions that have to be addressed are : when , how and on what timescales did the galactic components form , and was there any connection between them ? if yes , simultaneously or sequentially ? one possible approach to disentangle the evolutionary scenario is to look for evolutionary signatures in age , dynamics , and chemistry of long - lived stars , in the stellar populations within our mwg . at present , two major and basically different strategies for modelling galaxy evolution can be followed : dynamical investigations which include hydrodynamical simulations of isolated galaxy evolution and of protogalactic interactions reaching from cosmological perturbation scales to direct mergers , and , on the other hand , studies which neglect any dynamical effects but consider either the whole galaxy or particular regions and describe the temporal evolution of mass fractions and element abundances in detail . for the case of a closed box a linear relation between the time - dependent metallicity @xmath0 and the initial - to - temporal gas ratio @xmath1 $ ] follows analytically , where the slope is determined by the yield @xmath2 , i.e. the metallicity release per stellar population . deviations from this simple relation are explained by lower `` effective '' yields @xmath3 due to outflow of metal - rich gas from the ( now open ) volume or infall of low - metallicity ( presumably primordial ) gas . such dynamical effects can only be properly treated if simulations can account for the energetics , the composition , and the dynamical state of the galactic gas , as well as the relevant interchange processes , in a self - consistent manner . this includes the pollution of the different gas phases with characteristic elements by means of various stellar mass - loss mechanisms , the gas phase transitions , and the self - consistent large - scale streaming motions of hot intercloud gas ( icm ) or cool gas ( cm ) that falls in from a reservoir of protogalactic gas . it follows that galactic regions and components experience mutual dynamical interactions and their evolutions are not decoupled . let us consider a number of chemical peculiarities of different kinds of galaxies which justify the use of sophisticated multi - phase dynamical descriptions of the interstellar medium ( ism ) in studies of the global scenario of galaxy evolution : \1 ) in the solar vicinity at least three severe problems arise by considering the metallicity distribution of f or g dwarfs , stars that live sufficiently long to trace the evolution of a galaxy : the age - metallicity relation , @xmath2 of the metallicity distribution and , finally , the lack of metal - poor g dwarfs ( the well - known g - dwarf problem ) . various influences on the evolution of the solar neighbourhood have therefore been invoked by different authors and are applied by artificial parametrizations ranging from time - dependent accretion of pristine halo gas to temporal variations of the stellar initial mass function ( imf ) . although they lack self - consistency , the results can often provide very helpful basic insights into influences of distinct effects . \2 ) it is also important to explain the observed differences of @xmath2 in bulge , disk and halo of the mwg ( pagel 1987 ) . \3 ) there is now much evidence that metal - absorption line systems ( als ) in qso spectra arise from the gaseous halos of forming galaxies . in spite of the lack of knowledge of their dynamical properties and their origin , as a working hypothesis it is assumed that the early hot halo gas is produced by supernovae typeii ( sneii ) , because the observed abundance ratios agree well with snii yields ( reimers et al . the metallicity and radial extent of als should therefore appear automatically in galaxy models as the result of chemical evolution . \4 ) dwarf galaxies ( dgs ) present a variety of morphological types . their structural and chemical properties differ from those of giant galaxies . the dwarf irregular galaxies ( dirrs ) appear with lower @xmath4 at the same gas fraction as gss . this implies that the metal - enriched gas from sneii was lost from the galaxy rather than astrated ( larson 1974 , dekel & silk 1986 ) . yet many dwarf spheroidals ( dsphs ) which represent the low - mass end of dgs show not only a significant intermediate - age stellar population , but also more recent sf events ( smecker - hane et al . 1994 , han et al . 1997 ) with increasing metallicity , indicating that gas was kept in the system . \5 ) gss and dgs differ significantly in their n / o ratios . a fundamental explanation is needed to explain why gss like the mwg reach higher ratios at larger o abundances than dgs , while n / o is almost restricted to around -1.5 for dgs over a wide range in o ( see fig.3 ) . for systems and sites of low potential energy we know from empirical studies and theoretical investigations that the ism is on average held in balance by counteracting processes like heating and cooling , turbulence and dissipation ( burkert & hensler 1989 , hensler et al . since these processes are non - linearly coupled , the effect of neglecting one of them will alter the evolution completely . a number of studies of self - regulated sf exist with particular attention to the influences of stellar radiation , supernova explosions , and the evaporation / condensation balance between the two chemically and dynamically distinct gas phases , the cloudy medium ( cm ) and the hot intercloud medium ( icm ) ( franco & cox 1983 , ikeuchi et al . 1984 , mckee 1989 , bertoldi & mckee 1995 , kppen et al . 1995,1998 ) . the evolution of dgs is self - regulated and determined by large - scale outflows ( hensler et al . 1993,1998a ) . external effects like extended dm halos , the igm pressure , etc . ( vilchez 1995 ) could cause the morphological differences of dgs by regulating the otherwise unbound hot gas that can be held in the galactic halo . the gas could then either be stripped off , or it cools and recollapses , igniting subsequent sf . self - regulation with snii energy deposition can also characterise the structure and evolution of galactic disks which reach a lower effective gravitational potential in rotational equilibrium ( firmani & tutukov 1992 ; burkert et al . 1992 , rosen & bregman 1995 ) . to approach global models of galaxy evolution which yield the structural differences and details , adequate treatment of the dynamics of stellar and gaseous components is essential . at least the following processes have to be taken into account : sn , sf , heating , cooling , stellar mass loss , condensation and evaporation . this includes the treatment of the multi - phase character of the ism as well as the star - gas interactions and phase transitions . since gas and stars evolve dynamically , and because several processes both depend on their metallicities and also influence the element abundances in each component , these models are called * chemodynamical * ( _ cd _ ) . it must be emphasized , however , that the number of free parameters in the _ cd _ scheme is not large but actually smaller than in multi - zone models . the allowed ranges of parameter values are strongly constrained , either because they are theoretically evaluated ( like e.g. evaporation and condensation ) , empirically determined ( like e.g. stellar winds ) , or because they force self - regulation in a way that is independent of the parameterization . because of limited space we refer the interested reader to more comprehensive descriptions of the _ cd _ treatment and to different applications ( non - rotating galaxies : theis , burkert & hensler 1992 , hensler , burkert & theis 1993 , hensler , gallagher & theis 1998a ; vertical settling of the galactic disk : burkert , truran & hensler 1992 ; disk galaxies : samland & hensler 1996 ; the mwg : samland , hensler & theis 1997 ( sht97 ) , samland 1998 ; dwarf galaxies : hensler & rieschick 1998 , also section 5 ) . as a striking success of the _ cd _ treatment we will first briefly present some of the results from a published model ( sht97 ) which can be compared to the observational features of the mwg . the model starts from an isolated spheroidal , rotating but purely gaseous cloud with a mass of @xmath5 , a radius of 50 kpc , and an angular momentum of about @xmath6 pc@xmath7 myrs@xmath8 , corresponding to a spin parameter @xmath9=0.05 . we assume that the protogalaxy consists initially of cm and icm with a density distribution of plummer - kuzmin - type ( satoh 1980 ) with 10 kpc scalelength . the initial cm / icm mass division ( 99%/1% ) does not affect the later collapse , because the onset of sf determines the physical state within less than 10@xmath10 years . while the evolutionary phases are described in detail in sht97 , here we wish to emphasize the striking agreement of the _ cd _ model after 15 gyrs with first the metallicity distributions of the halo , the bulge , and the solar vicinity ( fig.1 ) , and secondly the radial oxygen gradient within the disk ( fig.2 ) . convincingly one single _ cd _ model reproduces the different ( otherwise implausible ) @xmath3 in the different regions and demonstrates that they result simply from large - scale streaming effects of the hot gas . the icm is produced by sneii in overpressure to its surrounding ism and expands until its cooling leads to condensation , i.e. the phase transition to the cm . in addition , the model shows agreement with the mwg structure , i.e. gas - star content , baryonic mass distributions , velocity distributions , pn and sn rates . from this agreement it may be safe to assume that the temporal evolution of the model is reliable , e.g. the formation epochs of the components and the temporal variation of the radial metallicity gradient ( see sht97 ) . the enormous energy release by massive stars from their combined wind , radiation and snii explosion leads to violently expanding hot gas bubbles . they act dynamically on the ambient ism by sweeping it up and squeezing it into dense shells , which break up due to dynamical instabilities . since massive stars have peeled off their unprocessed shell material during the h - main sequence lifetime , they expel their nucleosynthesized products before and during their wolf - rayet phase and even more intensely by snii explosion . due to its rapid expansion the hot gas engulfs the denser cool clumps and clouds . the effect is twofold : first , as it passes the clouds the icm significantly perturbs their shape and surface ( elmegreen , this volume ) . secondly , the contact interface between cm and icm is blown up by heat conduction . if the cloud is able to get rid of the diffused energy , hot gas can condense onto its surface ; if not , the cloudy material evaporates from the surface and immigrates into the icm . reasonably , this mass exchange by means of evaporation / condensation is a self - regulated process in static media ( kppen et al . streaming motions , however , can alter this picture and can lead to runaway behaviour in either direction . certainly this mechanism causes a highly efficient small - scale mixing between the gas phases and by this homogenizes abundances on the local scales of massive star associations . indeed , kobulnicki ( this volume ) reports the non - detection of any sizable o , n , and he anomalies from hii regions in the vicinity of young super starclusters in starburst dgs with one exception , ngc 5253 , which reveals a central n overabundance . while c and n are mainly contributed to the cm by pne from intermediate - mass stars , o and fe are the dominant tracers of snii and snia ejecta , respectively . since the mixture of e.g. n and o can only result from phase transitions between cm and icm , the n / o ratio allows to make qualitative deductions about the mixing direction and its temporal efficiency . additionally , its radial distribution provides an insight into dynamical effects of the ism . as mentioned in the introduction ( point 5 ) the n / o ratio is smaller in dgs than in gss by almost 0.7 dex , while o is lower by one order of magnitude . the averaged n / o value for dgs ( see fig.3 ) at -1.46 ( garnett 1990 ) lies only 0.2 dex below the ratio determined from metal - dependent yields ( woosley & weaver 1995 ) for @xmath11and integrated over a salpeter imf . the almost linear dependence of the nitrogen production on @xmath4 allows for even smaller n / o in dgs because of the generally lower @xmath4 , which explains the observed scatter to even smaller ratios . in order to study and compare abundances and structural signatures in _ cd _ models of dgs , we have performed 2d simulations of rotating gaseous protogalactic clouds for a large range of initial masses with and without dark matter halos . the density distribution is again of the plummer - kuzmin - type with the same spin parameter as the above - mentioned mwg model . here we discuss 10@xmath12 dg models starting with a 2 kpc scalelength . fig.3 presents a diagram for the n / o vs. o / h ratios of dgs and gss ( also the solar value ) compared with evolutionary tracks by different authors . in contrast to the other models shown ( see garnett 1990 ) which begin and partly remain at too large n / o ranges , both our _ cd _ models commence at very low values due to the delayed nitrogen release by pne and the lower n production in massive stars at low metallicities . the most evolved track ( 10@xmath12 with dm ) rises rapidly and reaches n / o of -1.8 after 1 gyr and -1.6 after 2 gyrs , respectively . this small value results numerically as the yield ratio , but as n and o enrich different gas phases , an almost ideal mixing of cm and icm is required . since the hot material can not fully condense onto the clouds , an almost total evaporation of the cm in the vicinity of the sf and snii explosion sites must be invoked . reasonably , the n mixes perfectly in this case with o in the icm to an almost constant abundance ratio and expands over larger distances within the dg . due to its cooling the icm forms new condensations of cm where the abundances are observed in hii regions at a constant value ( kobulnicki , this volume ) . in the case of phase transition by means of condensation , only parts of the icm ( and therefore of the o content ) is incorporated into the cm , which leads to higher n / o ratios , but also reveals inhomogeneous n / o distributions . larger gravitational potentials and resulting mass densities in more massive galaxies lead to a faster cooling of the icm and significantly higher condensation rates which produces a larger n / o . with the same differential mixing processes the observed c / o tendency of dgs ( garnett et al . 1995 ) can also be explained ( rieschick & hensler , in preparation ) . because of limited space here , we could only briefly demonstrate that various structural and chemical agreement with observations can be achieved self - consistently by global evolutionary _ cd _ models . if the _ cd _ prescription is applied , important physical processes like large - scale coupling of different galactic regions by dynamical interactions as well as small - scale mixing effects between the gas phases are adequately taken into account , and this substantially fixes the element abundances . abundances can serve as reliable diagnostic tools of galaxy evolution and provide a chance to deconvolve it in detail , if studies couple the above - mentioned gas processes with the dynamics of gas and stars as well as with their mutual interactions . is very grateful to the organizers of the conference for their invitation and their kindest hospitality . the authors achnowledge gratefully cooperations and discussions with j.kppen , m.samland and ch.theis and their contributions to the field of chemodynamics . we also thank the referee mike edmunds for the careful reading of the text and his comments for clarification . this work is supported by the _ deutsche forschungsgemeinschaft _ ( dfg ) under grant no . he 1487/5 - 3 ( a.r . ) , the participation at this meeting by one of us ( g.h . ) under grant no . he 1487/21 - 1 . the numerical calculations are partly performed at the computer centers rz kiel , zib berlin , and hlrz jlich . + [ -0.8 cm ] bertoldi f. , mckee c.f . , 1995 , _ amazing light _ chiao ( new york : springer ) burkert a. , hensler g. , 1989 , in _ evolutionary phenomena in galaxies _ , eds . beckmann & b.e.j . pagel , cambridge university press , p. 230 burkert a. , truran j.s.w . , hensler g. 1992 , _ apj _ , * 391 * , 651 dekel a. , silk j. , 1986 , _ apj _ , * 303 * , 39 firmani c. , & tutukov a. v. , 1992 , _ a&a _ , * 264 * , 37 franco j. , cox d.p . , 1983 , _ apj _ , * 273 * , 243 garnett d.r . , 1990 , _ apj _ , * 360 * , 142 garnett d.r . , skillman e.d . , dufour r.j . , et al.,1995 , _ apj _ , * 443 * , 64 han m. , hoessel j.g . , gallagher j.s . , et al . , 1997 , _ aj _ , * 113 * , 1001 hensler g. , theis ch . , burkert a. , 1993 , in proc . 3@xmath13 daec meeting _ the feedback of chemical evolution on the stellar content of galaxies _ , eds . d. alloin & g. stasinska , observatoire de paris , p. 239 hensler g. , gallagher j.s . , , 1998a , _ apj _ , * * , submitted hensler g. , kppen j. , samland m. , theis ch . , 1998b , in proc . of the german - japanese workshop _ galaxy evolution _ , eds . n. arimoto & w. duschl , springer , in press hensler g. , rieschick a. , 1998 , iau highlights in astronomy , in press ikeuchi s. , habe a. , tanaka y. d. 1984 , _ mnras _ , * 207 * , 909 kppen j. , theis ch . , hensler g. , 1995 , _ a&a _ , * 296 * , 99 kppen j. , theis ch . , hensler g. , 1998 , _ a&a _ , * 328 * , 121 larson r.b . , 1974 , _ mnras _ , * 169 * , 229 matteucci f. , tosi m. , 1985 , _ mnras _ , * 217 * , 391 mckee , c.f . 1989 , apj , 345 , 782 pagel b.e.j . , 1987 , in : _ the galaxy _ , eds . g. gilmore & b. carswell , reidel , p. 341 reimers d. , vogel s. , hagen h .- j . , et al . , 1992 , nature , 360 , 561 rosen a. , bregman j.n . , 1995 , _ apj _ , * 440 * , 634 samland m. , 1994 , phd . thesis , university of kiel samland m. , 1998 , _ apj _ , * * , in press samland , m. , & hensler , g. 1996 , rev . , 9 , 277 samland m. , hensler g. , theis ch . , 1997 , _ apj _ , * 476 * , 544 ( sht97 ) satoh c. , 1980 , _ pasj _ , * 32 * , 41 smecker - hane t.a . , stetson p.b . , hesser j.e . , et al . , 1994 , _ aj _ , * 108 * , 507 theis c. , burkert a. , hensler g. , 1992 , _ a&a _ , * 265 * , 465 vilchez j.m . , 1995 , _ aj _ , * 110 * , 1090 woosley s.e . , weaver t.a . , 1995 , _ apjs _ , * 101 * , 181
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since stellar populations enhance particular element abundances according to the yields and lifetimes of the stellar progenitors , the chemical evolution of galaxies serves as one of the key tools that allows the tracing of galaxy evolution . in order to deduce the evolution of separate galactic regions one has to account for the dynamics of the interstellar medium , because distant regions can interact by means of large - scale dynamics . to be able to interpret the distributions and ratios of the characteristic elements and their relation to e.g.the galactic gas content ,
an understanding of the dynamical effects combined with small - scale transitions between the gas phases by evaporation and condensation is essential . in this paper , we address various complex signatures of chemical evolution and present in particular two problems of abundance distributions in different types of galaxies : the discrepancies of metallicity distributions and effective yields in the different regions of our milky way and the n / o abundance ratio in dwarf galaxies .
these can be solved properly , if the chemodynamical prescription is applied to simulations of galaxy evolution .
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there are quantum mechanical problems in which the hamiltonian depends explicitly on time , for example , the interaction of the system with an external time - dependent field where @xmath0 . in such cases , the system does not remain in any stationary state , and the behavior of it is governed by the time - dependent schrdinger equation @xmath1 the problem will now be an initial - value ( cauchy ) problem of solving this partial differential equation for a prescribed initial condition at @xmath2 . when problems of this short are discussed formally , it is common to speak of the perturbation ( @xmath3 ) as causing _ transitions _ between eigenstates of @xmath4 . if this statement is interpreted to mean that the state has changed from its inital value to a final value , than it is incorrect . the effect of a time - dependent perturbation is to produce a nonstationary state , rather than to cause a jump from one stationary state to another which were determined by considering boundary conditions solely . experiments open the possibility to investigate dynamical properties of confined systems . in that field , of particular interest is the response of the system to time - dependent variations of the confining field . then , even assuming complete isolation of the many - body confined system from the environment , there is the question of how the correlation properties change . furthermore , one can consider an atom in its ground state . if at times @xmath5 it becomes subject to a time - varying potential energy caused by a charged heavy projectile passing by , its electron cloud will be shaken up so that its energy increases . the energy change , related to the stopping power , is a measurable quantity . however , when we have shake - up processes , the application of one - particle auxiliary pictures is not necessarily useful _ a priori_. motivated by such problems , in this work we consider a simplified interacting model system introduced by heisenberg @xcite in the early days of quantum mechanics @xmath6 where @xmath7}$ ] measures the strength of repulsive interparticle interaction . we will use ideas and methods @xcite for solving the quantum motion of a particle in a harmonic oscillator with time - dependent frequency , by adding a quadrupolar ( @xmath8 ) time - dependent perturbation , of @xmath9 character , to the above ground - state hamiltonian @xmath10 in the field of heavy - particle interaction with atoms , such term could mimic the shake - up process due to close encounters . there , as was stressed @xcite by fermi , quantum mechanics is needed since bohr s classical treatment is valid @xcite only for dipolar perturbation . one might think that a two - particle model is a bit trivial to test time - dependent many - body methods . but , this is not the case . indeed , it is the few - body correlated dynamics that serve @xcite as benchmark for methods beyond , for instance , time dependent density functional theory , td - dft . furthermore , to the theory of breathing modes of many - body systems in harmonic confinement , it was utilized @xcite to replace a given ( say , with coulombic interparticle interaction ) hamiltonian with a quadratic hamiltonian for which an analytic solution exist . a mapping at the hamiltonian level could be a practical version of the isospectral deformation discussed earlier @xcite in the light of the , unfortunately , formal character of pure existence theorems and mapping lemmas behind td - dft . we believe that experience about relevant details in interacting systems is best governed by studying time - dependent quantities based on exactly solvable instructive examples . in particular , the simultaneous treatment of the notorious kinetic and interparticle energy components could be more accurate than in td - dft where both are approximated . besides , our result allows a transparent implementation of the mapping - formalism @xcite designed to construct independent - particle potentials ( and to discuss @xcite an alarming paradox arising there ) by using exact probabilistic quantities of the correlated model as inputs . as a last motivation , we note that there have been efforts to clarify the basic relations between interparticle correlation and information - theoretic measures for entanglement . for instance , the stationary ground - state of the correlated model applied in the present work has already been investigated in this respect . a surprising duality between rnyi s entropies characterizing entangled systems , with attractive ( @xmath11 ) and repulsive interparticle interactions was found @xcite and explained @xcite recently . the question of how such a remarkable duality will change when the correlated system is perturbed by a time - dependent external field is an exciting one of broad relevance . one of our goals here is to provide an answer . we stress that we restrict ourselves to the linear - response limit , by taking our @xmath12 sufficiently small or sudden , since the main goal is on the time - dependent spectral aspects of correlation - dynamics and not on comparison with data . however , we should note that similarly to the born series of stationary scattering theory , an order - by - order expansion could result in only an asymptotic series . indeed , higher - order response to external field is an important topic in the energy change during shake - up dynamics of a nucleus @xcite or an atom @xcite . the idea that by renormalization of the kinetic energy one could @xcite extend the validity of linear - response formalism is also a challenging one . such questions need accurate solutions in linear and nonlinear responses . the nonlinear version of the exact determination of energy changes in our correlated model system will be published separately . the rest of the paper is organized as follows . section ii contains our theoretical results . section iii is devoted to a short summary and few relevant comments . we will use natural , rather than atomic , units in this work , except where the opposite is explicitly stated . by introducing the normal coordinates @xmath13 and @xmath14 , one can easily rewrite @xcite the unperturbed hamiltonian in the form @xmath15 where @xmath16 and @xmath17 denote the resulting independent - mode frequencies . it is this separated form which shows that a time - dependent perturbation of dipolar character , @xmath18\ , x_1 $ ] , will couple only to one normal mode characterized by the unperturbed angular frequency @xmath19 . therefore , such perturbation @xcite produces an energy change independent of the correlated aspect of our model . in the case investigated here , we have @xmath20 , i.e. , there is time - dependent quadrupolar perturbation in both independent normal modes , which will evolve in time independently . thus , we proceed by one oscillator [ @xmath21 in a time - dependent harmonic confinement , following the established theoretical path , where one has to solve a time - dependent schrdinger equation , of the form given by eq . ( 1 ) , with @xmath22 the solution rests on making proper changes of the time and distance scales @xcite in order to consider frequency variations in @xmath23 as it changes from @xmath24 during time - evolution . the exact mode , a nonstationary evolving state @xmath25 $ ] , contains these scales as @xmath26^{1/4}\ , \exp\left[-\frac{x^2}{2}\ , \frac{m\ , \omega_0}{\hbar \ , r^2(t ) } \left(1-i\ , \frac{r(t)\dot{r}(t)}{\omega_0}\right)\right]\ , e^{-i\ , \gamma(t)}.\ ] ] this exact quantum mechanical , time - dependent solution is obtained @xcite by considering a classical equation of motion @xmath27 with complex @xmath28 $ ] substitution , where the real @xmath29 gives the length scale at the instant @xmath30 . the nonlinear differential equation , determining this scale becomes @xmath31 ^ 3},\ ] ] after taking @xmath32 . the initial conditions are @xmath33 and @xmath34 . since we are dealing with a weak external perturbation in this work on spectral dynamics , we solve eq . ( 7 ) via the substitutions @xmath35 and @xmath36 . with @xmath37 to eq . ( 7 ) , we get a forced - oscillator - like differential equation @xmath38 which is treated by going to the complex notation @xmath39 $ ] in order to derive a first - order differential equation for @xmath40 . remarkably , eq . ( 8) shows transparently that the breathing frequency @xcite is precisely @xmath41 , i.e. , the double of the confinement frequency characterizing the ground - state mode of the unperturbed @xmath42 . once an explicit form for @xmath43 is prescribed to be used in both ( @xmath44 ) differential equations for @xmath45 [ considering the two independent _ modes _ , with @xmath46 and @xmath47 and the solutions for @xmath48 are found , the time - dependent wave function becomes @xmath49 which is valid for @xmath50 , and to which the form for @xmath51 is given by eq . ( 5 ) . by using this normal - coordinate - based representation for the exact wave - function we can easily calculate with it and the _ unperturbed _ hamiltonian , the expectation values of the kinetic and potential energy components , @xmath52 and @xmath53 , respectively . we get for these quantities @xmath54\ ] ] @xmath55 in the linear - response limit , the total energy _ change _ @xmath56 becomes @xmath57}.\ ] ] to arrive at the consistent r.h.s . above , which is valid for weak external fields , we linearized eqs . ( 10 - 11 ) . in this case [ with eq . ( 8) ] one gets @xmath58 $ ] for the rate . next , we derive for the overlap @xmath59 the following expression @xmath60 in terms of @xmath61 . this quantity is a measure of correlated ( when @xmath62 dynamics . its product form reflects the independence of two normal modes during propagation . when we are close to the stability limit at @xmath63 , even a weak external field could result in an almost perfect orthogonality at certain , @xmath43-dependent , time @xmath5 . now , we turn to quantities which show explicitly the entangled nature of the correlated two - particle system in the time domain . by rewriting the wave function in terms of original coordinates , we determine the reduced single - particle density matrix [ @xmath64 from @xmath65 after a long , but straightforward , calculation we obtain @xmath66 where , for further clarifications below , we introduced the following abbreviations @xmath67^{1/4}\ , e^{-\frac{m}{2\hbar}\omega_s(t)\ , x^2[1-i\ , \alpha(t)/\omega_s(t)]},\ ] ] @xmath68\ ] ] @xmath69 ^ 2 + [ \dot{r}_1(\omega_1,t)/r_1(\omega_1,t ) - \dot{r}_2(\omega_2,t)/r_2(\omega_2,t)]^2 } { \omega_1(t)+\omega_2(t)}\ ] ] with @xmath70 $ ] , where @xmath71 ^ 2 $ ] . the one - matrix is hermitian , as it must be , since @xmath72 . later we will give , as in the stationary case @xcite earlier , a direct spectral decomposition of this important hermitian matrix in eq.(15 ) , without considering , as more usual , an eigenvalue problem . the diagonal of the reduced single - particle density matrix gives the basic variable of td - dft , i.e. , the one - particle probability density @xmath73^{1/2}\ , e^{-\frac{m}{\hbar}\omega_s(t)\ , x^2}.\ ] ] the single - particle probability current , @xmath74 , is calculated as usual @xcite from @xmath75 which , in terms of the functions introduced above , becomes @xmath76 these probability densities satisfy the continuity equation : @xmath77 . by taking the @xmath78 substitution in eq.(15 ) , i.e. , neglecting the important role of the interparticle coordinate , we can define an auxiliary independent - particle model , where the particles move in a _ certain _ external field . the wave function of this modeling becomes @xmath79 this approximate state will still result in the exact probability density and exact probability current , by construction . so , one may consider it as an _ output _ of td - dft . indeed , using eqs . ( 6.50 - 6.52 ) of @xcite or eqs . ( 21 - 22 ) of @xcite , an effective potential @xmath80 is given by @xmath81\ , = \ , \frac{1}{2}\omega_{s}^2(t)\ , x^2\ , -\ , \frac{1}{2}\left[\sqrt{\omega_{s}(t)}\ , \frac{d^2}{d t^2}\left(\frac{1 } { \sqrt{\omega_{s}(t)}}\right)\right ] x^2,\ ] ] at least upto a time - dependent constant @xcite . in the so - called adiabatic treatment in td - dft , one uses @xcite only the first term as a time - dependent external field . but , with such a single - particle potential , which produces from a time - dependent wave equation the @xmath82 state , we also arrive at a dilemma . we know the exact schrdinger hamiltonian . especially , we know how the unperturbed hamiltonian is modified by adding a time - dependent quadrupolar perturbation to it . furthermore , since the perturbation acts only over a limited time in our problem , in the schrdinger picture we can follow the standard quantum mechanical prescription to determine the energy change by calculating expectation values with the exact wave function @xmath83 and unperturbed hamiltonian @xmath84 . in the light of @xmath85 $ ] , the situation in the orbital version of td - dft is not so clear . for instance , it will contain a @xmath86-proportional higher - order derivative @xcite . quite unfortunately , in td - dft , which is based @xcite on a density - potential mapping lemma , we do not @xcite know universal functionals to determine an energy change via them . after the above remarks on open questions in applied dt - dft , and motivated partly by dreizler s early suggestion @xcite on using an isospectral deformation ( @xmath87 ) and a constrained search , we turn to the decomposition of our reduced one - particle density matrix , and , in addition , put forward an idea on the possibility of using an _ approximate _ nonidempotent one - matrix , @xmath88 , designed below . following our experience @xcite on the decomposition of the stationary @xmath89 , we will use mehler s formula @xcite now to @xmath90 . thus , via the @xmath91=\bar{\omega}(t)[1+z^2(t)]/[1-z^2(t)]$ ] and @xmath92 $ ] correspondences , we introduce two new variable @xmath93 and @xmath94 . the solutions are @xmath95 @xmath96 - \sqrt{\left[\frac{\omega_s(t)}{a(t)}+1\right]^2 - 1}.\ ] ] now , in term of these we get our point - wise decomposition directly @xmath97[z(t)]^l\ , \phi_l(x_1,t)\ , \phi_l^{*}(x_2,t),\ ] ] where the @xmath98 elements of the orthonormal basis set ( natural orbitals ) are @xmath99^{1/4}\ , e^{i\ , \frac{m}{2\hbar}\alpha(t)x^2}\ , \frac{1}{\sqrt{2^l\ , l!}}\ , e^{-\frac{m}{2\hbar}\bar{\omega}(t)x^2}\ , h_m(\sqrt{m\bar{\omega}(t)/\hbar}x).\ ] ] of course , extracted quantities , like the exact one - matrix @xmath90 and the associated exact probability density @xmath100 , contain less and less information then the wave function @xmath101 and the pair - function @xmath102 . however , even by these extracted quantities one can calculate the exact kinetic , and the external - potential energies , respectively , while in td - dft only the last energy is , in principle , exact . thus , to formulate an idea , which is based on inversion from _ known _ probability density @xmath100 and probability current @xmath74 ( i.e. , from the crucial @xcite @xmath103 phase ) , we make a rewriting @xmath104[z(t)]^l [ 1-z_d(t)][z_d(t)]^l|\phi_l[x , t,\omega_d(t)]|^{2},\ ] ] which is valid if and only if @xmath105/[1-z_d(t)]$ ] . so , with one - particle probabilistic functions as input , we can prescribe , from the r.h.s . of the above equation , a trace - conserving @xmath88 which has the form of @xmath90 . such nonidempotent form contains two ( interconnected ) parameters @xmath106 and @xmath107 , instead of the uniquely derived @xmath93 and @xmath94 . whether this deformed one - matrix could result in , because of its _ tunable _ flexibility , an essentially better approximation for the kinetic energy than the td - dft [ @xmath108 where the one - matrix is idempotent , requires future investigations within the framework of a constrained search @xcite . the information - theoretic aspects of iterparticle interaction , i.e. , of a crucial term in the hamiltonian , may be considered via entropic measures which do not use expectation values with physical dimensions but use pure probabilities encoded in the occupation numbers @xmath109\ , [ z(t)]^l},\ ] ] which are , in the present case , time - dependent quantities as well . first , we calculate rnyi s entropy @xcite , with @xmath110 and @xmath111 , from @xmath112^q}{1-[z(t)]^q}}.\ ] ] the von neumann entropy is obtained as a limiting case when @xmath113 . it is given by @xmath114\ , -\frac{z(t)}{1-z(t)}\ , \ln[z(t)]}.\ ] ] in the knowledge of @xmath94 , these entropies characterize the deviation from the stationary case described by @xmath115 , i.e. , without external perturbation on the correlated two - body system . by considering repulsive [ @xmath116 and attractive [ @xmath11 ] versions , we can investigate the duality @xcite problem with respect to entropies , now in the time domain . notice , that the knowledge of rnyi entropies for arbitrary values of @xmath118 provides , mainly from information - theoretic point of view , more information than the von neumann entropy alone , since one could extract @xcite from them the full spectrum of the one - matrix . but , as eq . ( 26 ) shows transparently , to physical expectation values one also needs the proper basis set . armed the above , strongly interrelated , theoretical details on the dynamics of interparticle correlation we come to their implementation . we use a simple model for illustration , with @xmath119 . by tuning the value of the effective rate of change ( denoted by @xmath120 ) we can easily investigate the sudden and adiabatic limits . we get @xmath121\ ] ] @xmath122\ ] ] @xmath123\ ] ] the sum ( @xmath44 ) of @xmath124 measures the total deviation from the unperturbed ground - state energy @xmath125 . in the kick - limit , where @xmath126 , we have @xmath127}$ ] . as expected on physical grounds , the change at long time , i.e. , taking @xmath128 first , becomes a constant which tends to zero in the @xmath129 sudden limit . in the long - time limit the overlap parameter of eq . ( 13 ) , which measures the time - evolution of orthogonality between exact states , also tends to a constant value . simple independent - particle modelings of the correlated model system could be based on effective ( @xmath130 ) , prefixed external fields @xmath131 before time - dependent perturbation . in such cases one gets @xmath132}}$ ] for the total energy change , following the standard quantum mechanical averaging procedure discussed at eq . two options are analyzed here , and a remarkable result is found , as figure 1 signals . namely , with @xmath133 , i.e. , with the density - optimal modeling , the corresponding energy _ change _ tends to the exact one at high enough values of @xmath120 , since @xmath134 . somewhat surprisingly , by using the so - called energy - optimal hartree - fock @xcite modeling , where @xmath135 , one can not get agreement with the exact result at any value of @xmath120 . for illustration , a convenient ratio - function is defined as @xmath136 and plotted in figure 1 as a function of the @xmath120 with fixed , density - optimal , @xmath137 . we have checked , using both sides of eq . ( 12 ) , that with a @xmath138 value we are in the linear - response limit , where the energy changes are proportional to @xmath139 . notice , that we use standard hartree atomic units , where @xmath140 , to both figures of this work . [ 0.3 ] , as a function of @xmath30 which is measured in atomic units . the solid and dashed curves refer to interparticle repulsion ( @xmath141 ) and attraction ( @xmath142 ) , respectively . we used @xmath138 and @xmath143 . the top , middle , and bottom panels refer , respectively , to the @xmath144 , @xmath145 , and @xmath146 values . [ figure1],title="fig : " ] as we can see from the figure , far from the sudden limit , i.e. , at a small @xmath120 value , the density - optimal modeling underestimates the true energy change in the correlated case . we speculate that such sensitivity close to the adiabatic limit could be , at least partially , behind yet unclarified controversies in the applied td - dft @xcite to energy transfer by slow ions in wide - gap insulators , where @xmath147 is positive of course . in section iii , we will return to the informations given in figure 1 , considering there the energy changes for negative @xmath147 . at the end of this section , we turn to illustrations of the spectral aspect of dynamic interparticle correlation . we apply von neumann entropy to get a precise insight into the time - dependence of an information - theoretic measure . in the light of facts on hamiltonian - based measures , the energy - change and the wave - function overlap at @xmath128 , such analysis is desirable . based on it , one can discuss , and maybe understood , challenging interrelations between measures which are based on different emphasizes within quantum mechanics . we stress here that the exact energy - changes depend , as expected , on the sign of @xmath147 . [ 0.3 ] , as a function of the time @xmath30 , which is measured in atomic units . the solid and dashed curves refer to a moderate interparticle repulsion @xmath148 , and the corresponding ( see the text ) interparticle attraction @xmath149 , respectively . three values for the rate parameter @xmath120 , measured in inverse time - units , are employed to illustration . top panel : @xmath150 . middle panel : @xmath151 . bottom panel : @xmath152 . [ figure2],title="fig : " ] in figure 2 we exhibit @xmath153 , by taking @xmath143 , @xmath138 , @xmath154 and @xmath142 . these couplings for repulsion and attraction , respectively , are fixed to yield equal entropies at @xmath155 , in order to shed further light on duality @xcite , now in the time - domain as well . notice , that in the unperturbed case to _ any _ allowed [ @xmath156 $ ] repulsive coupling there exists a corresponding attractive ( @xmath11 ) one for which the calculated entropies are equal . for instance , when we are close to the stability limit in the repulsive ( @xmath157 ) case , i.e. , when @xmath158 , the corresponding attractive ( @xmath159 ) coupling becomes @xmath160 . the solid and dashed curves refer to repulsion and attraction , as we mentioned . they are plotted as a function of the time measured in atomic units . the rate - parameter of the time - dependent external perturbation is described by @xmath120 and measured in inverse units of time . in order to discuss tendencies , we used three values for this parameter . as expected , at very short times the entropies , similarly to the energy changes , grow in time . after about 1 - 2 atomic units ( i.e. , about 25 - 50 attoseconds ) they start to reflect the oscillating character builded into the time - dependent occupation numbers @xmath161 , the primary root of which is in the time - evolving correlated wave function . the entropies will keep , in absence of an environment - made dissipative @xcite coupling , such oscillating behavior without limitation . from this point of view , they signal the time - evolvement of the closed interacting system . more importantly , one can see from the illustration that the initial dual character of entropies , fixed at @xmath155 , will _ disappear _ due to a new physical variable , i.e. , the time . finally , there are reductions in the oscillation - amplitudes by increasing @xmath120 . clearly , at the so - called sudden limit ( @xmath129 ) for a time - dependent perturbation , they will diminish as expected , since the system s reaction - rapidity is limited by its normal mode frequencies . in order to justify the above argument more directly , we take now a somewhat less realistic modeling by using an abrupt quench of _ all _ interactions at @xmath162 in the hamiltonian . in that case , instead of eqs . ( 33 - 34 ) , one gets @xcite the following exact expressions @xmath163 @xmath164 there is no change in the system s kinetic energy since @xmath165 ^ 2+[\dot{r}_i(\omega_i , t)/\omega_i]^2=1 $ ] . by using time - dependent @xmath166 and @xmath167 , which are needed to eq . ( 25 ) , we get @xmath168 for all @xmath169 since @xmath170\equiv{2[2\sqrt{\omega_1\omega_2}/(\omega_1-\omega_2)]^2}$ ] . thus , there is no entropy-_change _ in the total - quench case . this @xmath171 character could be a useful constraint in practical attempts , with a good starting @xmath100 , using the interconnected @xmath107 and @xmath106 variables of eq . one can take @xmath172 , where @xmath173 . based on the time - dependent schrdinger equation , an exact calculation is performed for the time - evolving energy change , and orthogonality of initial and final states , in a correlated two - particle model system driven by a weak external field of quadrupolar character . besides , considering the structure of the exact time - dependent one - matrix , an independent - particle model is defined from it which contains exact information on the single - particle probability density and probability current . the resulting noninteracting auxiliary state is used to construct an effective potential and discuss its applicability . we analyzed the energy changes in a comparative manner , and pointed out ( see , below as well ) the limited capability of an optimized independent - particle modeling for the underlying time - dependent process . a point - wise decomposition of the reduced one - particle density matrix is given in terms of time - dependent occupation numbers and time - dependent orthonormal orbitals . based on such exact spectral decomposition on the time domain , an entropic measure of inseparable correlation is also investigated . it is found that the duality behavior characterizing the stationary ground - state may disappear depending on the rapidity of the external time - dependent perturbation . at realistic rapidity , the undamped time - dependent oscillations , found in an entropic measure , reflect the fact that the correlated system evolves in time . in this respect , we have a signal on the excited state , similarly as one can see in the time - evolving probability density and probability current . our first comment is based on results obtained for the energy changes , and plotted in figure 1 , by considering the sign of @xmath147 . in the case of an attractive interparticle interaction , the case of a nucleus @xcite , the exact result for @xmath174 becomes bigger at moderate @xmath120 values than the result obtained within the density - optimal framework @xmath175 . so , we speculate that a recent result @xcite on the excitation of a deformed nucleus within td - dft could have the same , underestimating character in its quadrupole channel , as the atomistic case has . thus , the true enhancement in excitations over the result based on the modeling of teller @xcite , might also need further theoretical refinement for that channel . in the second comment , we focus on the role of the switching rate @xmath120 in the case of energy changes . as a preliminary step to future realistic calculations , here we would like to estimate a characteristic interaction - time ( @xmath176 ) in classical atom - atom collision . to do that , we model the three - dimensional screened interaction via a finite - range ( @xmath177 ) potential @xmath178 solving the classical problem @xcite for the collision time ( @xmath179 ) we get @xmath180\ ] ] where @xmath181 is the energy of a heavy charged ( @xmath182 ) projectile moving with velocity @xmath183 and colliding at impact parameter @xmath184 with a fixed center of _ effective _ charge @xmath185 . the dimensionless parameter introduced is @xmath186 $ ] , i.e. , it is related to the ratio of potential and kinetic energies . a closer inspection of this compact ( and reasonable at short - range solid - state conditions ) form shows that practically one may use @xmath187 to get a very acceptable estimation as a function of the heavy projectile velocity @xmath183 , with which @xmath188 is _ decreasing _ from 4 ( at @xmath189 ) to 2 ( at @xmath190 ) . if we take @xmath191 and consider the energy change determined above , we get a @xmath192^{-2}}\sim{[\alpha(v)v]^2}$ ] character at very low velocities . for the opposite , high - velocity , limit @xmath192^{-2}}\sim{[\alpha(v)v]^{-2}}$ ] is the scaling , which also looks quite reasonable physically . however , the detailed work on the exactly determined non - linear energy changes is left for a dedicated publication . there , and also in bohr s pioneering modeling @xcite with time - dependent dipole fields of a passing charge , the time - scale derived above using collision theory , could play a more quantitative role to understand fine details behind experimental predictions @xcite on energy losses of slow ions in insulators . furthermore , in the renewed field of time - dependent energy losses , a proper time - scale also could help to analyze further a remarkable theoretical prediction @xcite on the strong interplay of nuclear and electronic stopping components of the observable total energy transfer . indeed , few - electron shake - up processes can be dominant at close encounters in condensed matter . in such dynamical cases , one - electron approximations , with double - occupancy for an auxiliary spatial orbital , could result in inaccuracies . we thank professor p. m. echenique for the very warm hospitality at the dipc . one of us ( in ) is grateful to professor p. bauer and professor d. snchez - portal for useful discussions on energy transfer processes in insulators . this work was supported in part by the spanish ministry of economy and competitiveness mineco ( project no . fis2013 - 48286-c2 - 1-p ) . 00 w. heisenberg , z. phys . * 38 * , 411 ( 1926 ) . m. moshinsky , am . j. phys . * 36 * , 52 ( 1968 ) . l. e. ballentine , _ quantum mechanics _ ( world scientific , singapore , 1998 ) . v. s. popov and a. m. perelomov , sov . jetp * 30 * , 910 ( 1970 ) . yu . kagan , e. l. surkov , and g. v. shlyapnikov , phys . a * 54 * , r1753 ( 1996 ) . a. del campo , phys . * 111 * , 100502 ( 2013 ) . e. fermi , phys . rev . * 57 * , 485 ( 1940 ) . n. bohr , phil . mag . * 25 * , 10 ( 1913 ) . j. f. dobson , phys . rev . lett . * 73 * , 2244 ( 1994 ) . m. brics and d. bauer , phys . rev . a * 88 * , 052514 ( 2013 ) . c. r. mcdonald , g. orlando , j. w. abraham , d. hochstuhl , m. bonitz , and t. brabec , phys . lett . * 111 * , 256801 ( 2013 ) . h. kohl and r. m. dreizler , phys . . lett . * 56 * , 1993 ( 1986 ) . c. a. ullrich , _ time - dependent density - functional theory _ ( oxford university press , oxford , 2012 ) , and references therein . m. ruggenthaler , m. penz , and r. van leeuwen , arxiv : 1412.7052v1 [ cond-mat.other ] . j. schirmer and a. dreuw , phys . rev . a * 75 * , 022513 ( 2007 ) . j. pipek and i. nagy , phys . a * 79 * , 052501 ( 2009 ) . schilling , d. gross , and m. christandl , phys . lett . * 110 * , 040404 ( 2013 ) . m. l. glasser and i. nagy , phys . a * 377 * , 2317 ( 2013 ) . schilling and r. schilling , j. phys . a : math . theor . a * 47 * , 415305 ( 2014 ) . d. w. robinson , nucl . phys . * 25 * , 459 ( 1961 ) . k. w. hill and e. merzbacher , phys . rev . a * 9 * , 156 ( 1974 ) . m. mierzejewski and p. prelovsek , phys . . lett . * 105 * , 186405 ( 2010 ) . m. lein and s. kmmel , phys . lett . * 94 * , 143003 ( 2005 ) . i. nagy , phys . a * 87 * , 052512 ( 2013 ) . a. erdlyi , _ higher transcendental functions _ ( mcgraw - hill , new york , 1953 ) , p. 194 . p. koscik , phys . a * 379 * , 293 ( 2015 ) . j. rapp , m. brics , and d. bauer , phys . a * 90 * , 012518 ( 2014 ) . c. l. benavides - riveros and i. nagy , arxiv : 1406.2809v1 [ quant - ph ] . a. rnyi , _ probability theory _ ( north - holland , amsterdam , 1970 ) . p. calabrese , p. le doussal , and s. n. majumdar , phys . rev . a * 91 * , 012303 ( 2015 ) . m. ashan zeb , j. kohanoff , d. snchez - portal , and e. artacho , nucl . methods phys . b * 303 * , 59 ( 2013 ) . f. mao , ch . zhang , j. dai , and r .- s . zhang , phys . rev . a * 89 * , 022707 ( 2014 ) . k. albrecht , phys . 56b * , 127 ( 1975 ) , and references therein . i. v. tokatly , phys . 110 * , 233001 ( 2013 ) . i. stetcu , c. a. bertulani , a. bulgac , p. magierski , and k. j. roche , phys . lett . * 114 * , 012701 ( 2015 ) . m. goldhaber and e. teller , phys . rev . * 74 * , 1046 ( 1948 ) . i. nagy , nucl . methods phys . b * 94 * , 377 ( 1994 ) . s. p. moller , a. csete , t. ichioka , h. knudsen , u. i. uggerhoj , and h. h. anderson , phys . lett . * 93 * , 042502 ( 2004 ) , and references therein . s. n. markin , d. primetzhofer , and p. bauer , phys . . lett . * 103 * , 113201 ( 2009 ) . a. a. correa , j. kohanoff , e. artacho , d. snchez - portal , and a. caro , phys . * 108 * , 213201 ( 2012 ) , and references therein .
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time - dependent quantities are calculated in the linear response limit for a correlated one dimensional model atom driven by an external quadrupolar time - dependent field . besides the analysis of the time - evolving energy change in the correlated two - particle system , and orthogonality of initial and final states , mehler s formula is applied in order to derive a point - wise decomposition of the time - dependent one - matrix in terms of time - dependent occupation numbers and time - dependent orthonormal , natural orbitals .
based on such exact spectral decomposition on the time domain , rnyi s entropy is also investigated .
considering the structure of the exact time - dependent one - matrix , an independent - particle model is defined from it which contains exact information on the single - particle probability density and probability current .
the resulting noninteracting auxiliary state is used to construct an effective potential and discuss its applicability .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the problem of the crossover from superfluidity to superconductivity with charge carrier density or coupling constant changing has a long history @xcite . the interest to this phenomena has arisen again after the discovery of high - temperature superconductors ( htscs ) in 1986 @xcite . it was already known upon that time that the superfluidity of composite bosons transforms into the superconductivity of overlapped cooper pairs with chemical doping in the case of the s - wave pairing . now , in the s - wave pairing case the problem is quite well explored for the 3d systems @xcite and , particularly for the quasi-2d case @xcite . for the 2d case this problem was studied at @xmath6 ( when a long - range superconducting is still possible in a 2d system @xcite ) for the case of local attraction ( see , for example @xcite ) and for the phonon - exchange model @xcite . most of these problems are reviewed in @xcite . due to a layered structure and the anisotropic symmetry of the order parameter in htscs , the crossover in the 2d system in the @xmath1-wave pairing channel is of a special interest . however , because if its complexity , this case is not so well understood at present . the d - wave pairing for the case of the extended hubbard model with the nearest neighbor attraction case was analyzed in @xcite . the crossover from superfluidity to superconductivity in the @xmath0-wave and @xmath1-wave pairing channels for a 2d continuum model was studied in paper @xcite , where also thermodynamic properties of the system in the crossover particle density region were considered . the authors proposed an interaction potential , which is attractive at distances between particles shorter of some value @xmath7 and longer of @xmath8 , and repulsive due to electron correlations at short distances , @xmath9 . it was found , in particular , that there exists a minimal value of the attractive coupling constant in the @xmath1-wave pairing channel , which gives the crossover from superfluidity to superconductivity at small carrier densities , i.e. the fermion chemical potential changes its sign and becomes positive with carrier density growth . the correlation length @xmath7 was assumed in @xcite to be a parameter , which does not depend on the carrier density @xmath10 per cell . however , concerning htscs it would be interesting to consider a more realistic case , when at small carrier densities @xmath3 and the proportionality coefficient is of order of a lattice constant . such a dependence was experimentally observed in htsc cuprates for the length of spin - spin correlations , which are believed to be responsible for the hole attraction in these materials . in particular , it was found for @xmath4 in underdoped regime , that the magnetic correlation length decreases with carrier density per cell according to the dependence @xmath11@xmath12 @xcite . in what follows we analyze the possibility of low carrier density crossover for a model analogous to @xcite with particle repulsion at distances @xmath13 and attraction at @xmath14 , where , however , @xmath15 ( @xmath16 is parameter of order of the lattice constant , and its possible value is discussed in the next section ) . obviously , at large carrier densities such that @xmath17 the superconductivity in this system should disappear . however , since we are interested in the small carrier concentrations , it is assumed that this relation does not take place . as it will be shown below such a dependence of @xmath18 leads to a qualitatively different behavior of the system with respect to the case with @xmath19 . in particular , there exists a minimal value for the coupling constant when the two - particle bound states exist at low carrier densities in the @xmath0-wave pairing channel . the existence of such a threshold value of the coupling constant is typical for the @xmath1-wave pairing case ( see , for example @xcite ) . another interesting property is : for any coupling constant in both channels there exists a corresponding carrier density value , below which the system is in the superconducting state . the hamiltonian of the system which describes the non - retarded fermion interaction can be written in a standard form @xmath20 where @xmath21 is the effective fermion mass , and @xmath22 is the chemical potential ; fermi - operators @xmath23 depend on the space - time coordinate @xmath24 . the instantaneous interaction potential is chosen in the next form @xmath25 with @xmath26 which corresponds to potential used in @xcite . here @xmath27 is an inter - particle distance . positive parameters @xmath28 and @xmath29 correspond to particle repulsion at @xmath9 and particle attraction at @xmath14 . the charge carrier density dependence of the correlation radius @xmath7 is given by ( [ r0 ] ) with the parameter @xmath30 , @xmath31 is the square lattice constant . this relation can be easily estimated from the equality @xmath32 , where on the left side the volume of the 2d system is expressed as a volume ( circle of the radius @xmath33 ) occupied by one particle , multiplied by the full number of particles @xmath34 , @xmath35 is an elementary cell number in the system . the free fermion bandwidth @xmath36 is connected with @xmath37 as @xmath38 . it should be noted , that the relation ( [ r0 ] ) at @xmath39 is in a good agreement with the experimental data for @xmath4 @xcite , where the plane magnetically ordered lattice parameters are equal to @xmath40 and @xmath41 , and the corresponding parameter @xmath16 is @xmath42 . in order to study the superconducting properties of the model in the channels with different pair angular momentum @xmath43 , it is convenient , similarly to @xcite , to approximate the fourier transform of ( [ magnoninteraction ] ) by a separable potential : @xmath44 where @xmath45 is an effective coupling constant , and functions @xmath46 with coefficients @xmath47 @xmath48 is the momentum modulus @xmath49 , and @xmath50 is its angle in polar coordinates @xmath51 . parameters @xmath52 and @xmath53 put the momentum range in the proper region . they are connected with the potential ( [ magnoninteraction ] ) parameters as @xmath54 and @xmath55 . below we put @xmath56 and @xmath57 . obviously , the expression ( [ potentialk ] ) is not the exact fourier transform of ( [ magnoninteraction ] ) , but it sets the interaction in right momentum range and has the correct asymptotic behavior at small and large momenta : @xmath58 and @xmath59 , correspondingly . since we are mainly interested in the low carrier density region , where the crossover can take place , the correct behavior of the interaction potential at small momenta is the most important . in the case of low carrier concentrations the small momenta give the main contribution in the integral of the gap equation ( see eq . ( [ gapequation ] ) below ) . we shall study @xmath0- and @xmath1-wave channels with @xmath60 and @xmath61 separately , so we assume that the parameters @xmath45 for both channels are independent . it will be assumed also that the pairing takes place at zero pair momentum at @xmath6 . the minimization of the ground state energy with respect to the superconducting order parameter @xmath62 and the chemical potential gives in the case of the approximation ( [ potentialk ] ) the standard pair of equation for the crossover problem ( see , for example @xcite ) : @xmath63 @xmath64 where @xmath65 is a free particle dispersion law . the form of the equation ( [ gapequation ] ) allows to search the solution for the superconducting order parameter in the next form @xmath66 where @xmath67 does not depend on momentum @xmath68 and @xmath69 is defined by ( [ w ] ) and ( [ h ] ) . in this case the solution of the system ( [ gapequation]),([numberequation ] ) gives the dependence of the gap parameter @xmath67 ( @xmath70 for @xmath60 and @xmath71 for @xmath72 ) and the corresponding chemical potential @xmath22 on the particle density @xmath2 and the coupling @xmath73 . the solution of the system ( [ gapequation]),([numberequation ] ) at @xmath74 gives the crossover line @xmath75 , which separates the parameters regions , where the local ( @xmath76 ) and cooper pairing ( @xmath77 ) take place . note , that in the @xmath0-wave pairing case the coulomb repulsion parameter @xmath53 does not enter in the equations . the numerical solution for the crossover line for the @xmath0-wave pairing case is presented in fig.1 , where we have put fermi energy @xmath78 instead of @xmath10 , since in the 2d case they are connected by a linear relation @xmath79 , and the dimensionless coupling constant @xmath80 is introduced . the solution for the @xmath19 case is also presented . as it follows from the numerical calculations , there exists a minimal value of coupling @xmath81 , which corresponds to carrier density @xmath82 on the crossover line , necessary to generate the crossover from superfluidity to superconductivity with doping , otherwise the cooper pairing regime takes place at any charge carrier density . in other words , there is a minimal value of coupling constant which leads to the two - particle bound states in the @xmath0-wave pairing channel at small @xmath83 . it is important to note , that there is no such a minimal coupling in the @xmath0-wave pairing channel when the correlation length @xmath19 @xcite . moreover , the crossover with charge carrier density increasing in the @xmath0-wave channel takes place for any known doping independent interaction potential @xcite . another interesting property , which follows from the fig.1b ) , is the `` inverse '' crossover from superfluidity to superconductivity with charge carrier density decreasing at small values of @xmath78 ( @xmath84 ) . this is a consequence of the competition between two opposite processes which occur with @xmath85 decreasing . the lowering of @xmath78 tends the system to become a superfluid , but at the same time it leads to growing of @xmath18 and makes the pair size larger , i.e. the pairs become bounded weaker . on the other hand , the carrier localization on dopants at very small carrier densities also makes a possible density region of superfluidity more narrow @xcite . however , the region of extremely small particle densities @xmath86 is not very interesting from the point of view of the connection with htsc , since in this region the relation @xmath87 does not hold , and the model is not correct . as it will be shown in the next subsection , in the @xmath1-wave case this crossover from from superfluidity to superconductivity with doping decreasing takes place at rather large fermi energy values . the doping dependencies of the gap @xmath70 and the chemical potential @xmath22 at different values of the coupling constant @xmath88 are presented in figs . 2 , and 3 . the gap is increasing with the doping almost linearly , except the region of extremely low concentrations . it is interesting to note , that the chemical potential in the superfluidity region ( @xmath76 ) has a minimum at a finite value of @xmath83 and it is equal to zero at @xmath89 , since the effective coupling constant at low doping @xmath90 is zero at @xmath89 . in other words , the local pairs are the most strongly coupled at some finite carrier density value . this is also a consequence of the competition between the fermi surface formation and correlation radius decreasing with charge carrier density growth . this situation is qualitatively different from the @xmath19 case , where @xmath91 does not depend on particle density and renormalized coupling constant is @xmath83-independent in the @xmath0-channel . in this case @xmath92 at @xmath89 , where @xmath93 is a two - particle bound state energy . in this case the coulomb repulsion parameter @xmath94 is present in the equations ( [ gapequation ] ) and ( [ numberequation ] ) . however , the presence of this parameter leads just to renormalization of the dimensionless coupling constant @xmath95 and the energy gap parameter : @xmath96 , @xmath97 , where @xmath98 is characteristic energy of coulomb repulsion . we shall consider the case , when the coulomb repulsion is much smaller than @xmath36 , i.e. this is the case of a large free - fermion bandwidth therefore , the coupling constant is assumed to be large : @xmath99 ( see figs . the crossover line @xmath100 for the @xmath1-wave pairing case is presented in fig.4 . qualitatively , the behavior of the system with doping and coupling changing in the @xmath1-wave pairing channel is similar to the @xmath0-wave case . the important difference is that the low carrier density superconducting state exists at rather high values of @xmath83 . also in this case there is a minimal value of the coupling constant for two - particle bound states @xmath101 at @xmath102 . it should be noted that the existence of the large threshold value for the coupling constant in both channels can be a possible answer on the question why the crossover has been not observed in cuprates . the charge carrier density dependence of @xmath103 and @xmath22 at different coupling parameters @xmath104 are presented in figs.5 and 6 . the superconducting gap @xmath71 at low charge carrier densities is much smaller than in the @xmath0-case , due to stronger @xmath83-dependence of effective coupling constant at low carrier densities ( [ renormg ] ) . the magnitude of the order parameter @xmath103 starts to grow almost linearly with @xmath83 increasing when the fermi energy is larger of some minimal value ( fig.5 ) . this behavior is qualitatively similar to the doping dependence of the gap of cuprates in the underdoped regime . however , because of its simplicity , the model ca nt describe the decreasing of the gap with charge carrier density increasing at large values of @xmath2 htscs . for this other properties of the charge carrier interaction in cuprates have to be taken into account ( see the last section ) . the small-@xmath83 superconducting region with @xmath105 as well as region of superfluidity are rather large in the @xmath1-wave pairing channel . also in this case the chemical potential is equal to zero at @xmath89 , i.e. there are no two - particle bound states at very low charge carrier densities . it should be noted that the decreasing of the chemical potential with increasing charge carrier density at small @xmath83 both in @xmath0- and @xmath1-wave pairing cases indicates a negative electronic compressibility . this can be related to increasing of antiferromagnetic correlations at low carrier densities . theoretical description of the behavior of superconductor with carrier density changing is an interesting and important problem , in particular , because of its possible association with htscs . the microscopic mechanism of the superconductivity in cuprates is not known so far , and the solution of phenomenological models , which take into account some of the properties of htscs , can help to clarify the nature of their unusual behavior , and maybe even help to understand the microscopic mechanism of the htsc phenomenon . in this paper the possibility of the crossover from superfluidity to superconductivity with charge carrier density and coupling constant changing in different pairing channels at @xmath6 was studied for a model , which qualitatively takes into account one of the properties of htscs , namely , the doping dependence of correlation length @xmath7 at low carrier densities . it has been shown , that even this simple model results in interesting and unusual properties , which are rather different from a more standard case with @xmath5 . in particular , the two - particle bound states in the @xmath0-channel exist only if the coupling constant is larger of the threshold value , similar to the @xmath1-wave pairing case . at any value of coupling constant larger of the threshold one , the `` inverse '' crossover from superfluidity to superconductivity takes place with doping decreasing in both @xmath0- and @xmath1-wave channels . of course , such a simple model can not pretend to describe doping dependence of the gap and chemical potential of htscs . the momentum dependence of the interaction potential has to be taken into account more carefully , especially in the overdoped regime , where @xmath78 is rather large and the separable potential may be not correct . in general , also the effect of the retardation of interaction can not be neglected . these and some other questions are planed to be studied in a future work . v.t . thanks the cfif members , especially prof . p.d . sacramento and prof . vieira for kind hospitality . is partly supported by scopes - project 7ukpj062150.00/1 of the swiss national science fundation . eagles , phys . 186 ( 1969 ) 456 ; a.j . leggett , in : a. pekalski and j. przystawa ( eds . ) , modern trends in the theory of condensed matter , springer - verlag , berlin , 1980 , p.13 ; p. nozieres and s. schmitt - rink , j. low temp . phys . 59 ( 1985 ) 195 .
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zero temperature crossover from superfluidity to superconductivity with carrier density increasing is studied for a two - dimensional system in the @xmath0-wave and @xmath1-wave pairing channels .
it was assumed that the particle attraction correlation length depends on carrier density @xmath2 as @xmath3 .
such a dependence was found experimentally for the radius of magnetic correlations in @xmath4 .
the short range coulomb repulsion was also taken into account .
it is shown that the behavior of the system with doping is fundamentally different from the case with @xmath5 .
in particular , similarly to the @xmath1-wave case , the crossover in the @xmath0-channel takes place only if the coupling is larger of some minimal value , otherwise the cooper pairing scenario takes place at any small carrier density .
the relevance of the model to the high - temperature superconductors is discussed . : 74.20.-z , 74.62.dh , 74.72.-h _ keywords _ : symmetry , @xmath0-wave , @xmath1-wave , effects of doping
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topological open strings and topological d - branes have recently been enjoying the attention of both physicists and mathematicians . the most obvious physical motivation for studying topological string theory is that it is a toy - model for `` physical '' string theory . thus a better understanding of topological d - branes could shed light on the general definition of a boundary condition for a two - dimensional conformal field theory ( 2d cft ) , something which is not known at present . further , if a 2d topological field theory ( 2d tft ) is obtained by twisting a 2d supersymmetric field theory , then it is possible to regard topological d - branes as a special class of `` physical '' d - branes ( bps d - branes ) . in fact , much of recent progress in string theory has resulted from studying bps d - branes . from the mathematical viewpoint , topological string theory is an alternative way of describing certain important geometric categories , such as the category of coherent sheaves on a calabi - yau manifold , and can serve as a powerful source of intuition . an outstanding example of such intuition is the homological mirror symmetry conjecture @xcite . most works on topological string theory considered the case of topologically twisted @xmath0 sigma - models @xcite with a calabi - yau target space . this is the case when the world - sheet theory is conformal , and topological correlators can also be interpreted in terms of a physical string theory @xcite . however , one can also consider more general topologically twisted @xmath0 field theories and the corresponding d - branes . one class of such theories is given by sigma - models whose target is a fano variety ( say , a complex projective space , or a complex grassmannian ) . such qfts , although conformally - invariant on the classical level , have non - trivial renormalization - group flow once quantum effects are taken into account . another set of examples is provided by @xmath0 landau - ginzburg models ( lg models ) @xcite . in fact , in many cases these two classes of @xmath0 theories are related by mirror symmetry @xcite . for example , the sigma - model with target @xmath1 is mirror to a landau - ginzburg model with @xmath2 fields @xmath3 taking values in @xmath4 , and a superpotential @xmath5 thus if one wants to extend the homological mirror symmetry conjecture to the non - calabi - yau case , one needs to understand d - branes in topologically twisted lg models . note that all critical points of this superpotential are isolated and non - degenerate ; this means that all the vacua have a mass gap , and the infrared limit of this lg model is trivial . in what follows we will call such lg models _ massive_. despite the triviality of the infrared limit , the homological mirror symmetry conjecture remains meaningful and non - trivial in this case . very recently it has been proposed that massive @xmath0 @xmath6 qfts can be used to describe certain non - standard superstring backgrounds with ramond - ramond flux @xcite . thus a study of d - branes in massive qfts could be useful for understanding open strings in such ramond - ramond backgrounds . in order to formulate our problem more concretely , let us first summarize the situation in the calabi - yau case , where the @xmath0 field theory is conformal . @xmath0 superconformal field theories have two topologically twisted versions : a - model and b - model @xcite . the corresponding d - branes are called a - branes and b - branes . mirror symmetry exchanges a - branes and b - branes . tree - level topological correlators give the set of either a - branes or b - branes the structure of an @xmath7-category ; gauge - invariant information is encoded by the corresponding derived categories . it has been argued that the derived category of b - branes is equivalent to the derived category of coherent sheaves @xcite . a detailed check of this proposal has been performed in ref . @xcite . for a - branes on calabi - yau manifolds , it has been proposed that the relevant @xmath7-category is the so - called fukaya category , whose objects are ( roughly ) lagrangian submanifolds carrying vector bundles with flat connections @xcite . recently it has been shown that the derived fukaya category is too small and does not accommodate certain physically acceptable a - branes @xcite . in particular , if we want the homological mirror symmetry conjecture to be true for tori , then the fukaya category must be enlarged with non - lagrangian ( more specifically , coisotropic ) branes . in the case of fano varieties , the sigma - model is not conformal . what is more important , the axial @xmath8 r - current is anomalous , and therefore one can not define the b - twist @xcite . one _ can _ consider d - branes which preserve b - type supersymmetry , but the relation with the derived category of coherent sheaves is less straightforward @xcite . mirror symmetry relates b - branes on fano varieties with a - branes in lg models . the latter have been studied from a variety of viewpoints in refs . @xcite . in the case when the fano variety is @xmath1 , the prediction of mirror symmetry has been tested in ref . in particular , the mirrors of `` exceptional '' bundles on @xmath1 have been identified , and in the case of @xmath9 and @xmath10 it has been checked that morphisms between these bundles in the derived category of coherent sheaves on @xmath1 agree with the floer homology between their mirror a - branes . one can also consider the category of a - branes on a fano manifold . since the vector @xmath8 r - current is not anomalous , the a - twist is well - defined , and a - branes can be regarded as topological boundary conditions for the a - model . presumably , the category of a - branes contains the derived fukaya category as a subcategory , but other than that little is known about it , even in the case of @xmath1 . if we assume mirror symmetry , we can learn about the category of a - branes on @xmath1 by studying b - branes in the mirror lg model . the b - twist is well - defined for any lg model whose target has a trivial canonical bundle , thus b - branes in such a lg model can be regarded as topological boundary conditions for the b - model . an obvious question is how the introduction of the superpotential deforms the relation between the category of b - branes and the derived category of coherent sheaves . important steps towards understanding b - branes in lg models have been taken in refs . @xcite ( see also refs . @xcite for a related work ) . in these papers general properties of b - branes have been studied , and several concrete examples have been discussed . a somewhat surprising lesson from these works is that the category of b - branes remains non - trivial even in a massive lg model , where the bulk 2d tft is trivial . for example , if we take the superpotential @xmath11 to be a non - degenerate quadratic function on @xmath12 , the graded algebra of endomorphisms of a d0-brane sitting at the critical point of @xmath11 is isomorphic to a clifford algebra with @xmath2 generators @xcite . this raises the question if one can determine the category of b - branes in any massive lg model . a proposal which accomplishes this has been put forward by m. kontsevich . roughly speaking , the proposal is that the superpotential @xmath11 deforms the derived category of coherent sheaves by replacing complexes of locally free sheaves with `` twisted '' complexes . here `` twisted '' means that compositions of successive morphisms in a complex are equal to @xmath11 , instead of zero . one also needs to switch from @xmath13-graded complexes to @xmath14-graded ones . kontsevich s proposal is supposed to describe b - branes in any lg model such that the critical set of @xmath11 is compact ; in particular , it does not require the critical points of @xmath11 to be non - degenerate . the main goal of this paper is to provide evidence for kontsevich s proposal . our evidence is of two kinds . first , we argue on physical grounds that twisted complexes arise as a consequence of brst - invariance . more precisely , while in the presence of the superpotential a holomorphic vector bundle or a complex of vector bundles does not correspond to a b - type boundary condition , we show that any twisted complex of holomorphic vector bundles is a valid b - brane . second , we test the proposal in some specific cases where morphisms between branes ( i.e. spectra of topological open strings ) can be easily computed . we focus on the massive case , where the proposal simplifies considerably . namely , the category of @xmath14-graded twisted complexes can be related to the sum of several copies of the category of finite - dimensional @xmath14-graded modules over a clifford algebra . we will denote the latter category @xmath15 in what follows . this reformulation is helpful , because the functor from the category of b - branes in a massive lg model to @xmath15 is very simple to describe . in this paper we perform some checks that this functor is an embedding of graded categories . in view of the above - mentioned `` duality '' between @xmath14-graded twisted complexes and @xmath15 , this provides a test of kontsevich s proposal . assuming the validity of the proposal , we infer that the category of b - branes in an arbitrary massive lg model is a full sub - category of @xmath15 . the latter has a very simple and explicit description . since for many fano varieties the mirror lg model is known , our results allow one to effectively compute the category of a - branes for such varieties . from the mathematical viewpoint , it is an interesting challenge to reproduce such results using methods of symplectic geometry . axiomatic definitions of topological d - branes for 2d topological field theories ( 2d tfts ) have been recently proposed by g. moore and g. segal @xcite and c. i. lazaroiu @xcite . one of the main unresolved problems in the axiomatic approach is whether these axioms determine unambiguously the category of topological d - branes associated to a given tft . we show that the category of b - branes for a massive lg model with a quadratic superpotential provides a counter - example to uniqueness . in fact , this example shows that uniqueness , if understood naively , fails also for ordinary ( i.e. non - topological ) d - branes in any closed superstring background . however , this particular failure is rather mild , i.e. it does not seem to have serious physical consequences . now let us describe the content of the paper in more detail . in section [ sec : a ] we recall some basic facts about mirror symmetry between fano varieties and lg models . in section [ sec : b ] we review general properties of b - branes in lg models . we argue that if the superpotential has only isolated critical points , then it is sufficient to study b - branes in the infinitesimal neighborhood of each critical point . for example , if all critical points of @xmath11 are non - degenerate , one does not lose anything if one replaces the superpotential by its quadratic approximation near each critical point . the material in this section is not new and has been previously discussed in refs . @xcite . in sections [ sec : c ] and [ sec : cp ] we study b - branes in the lg model with the superpotential @xmath16 this is the simplest lg model where b - branes of dimension larger than @xmath17 are present . in section [ sec : d ] we discuss b - branes in more general lg models with the superpotential @xmath18 in section [ sec : e ] we explain kontsevich s proposal and show that our results are consistent with it . we also explain why brst invariance of boundary conditions requires twisted complexes of vector bundles instead of ordinary complexes . this provides a physical explanation of kontsevich s proposal . we also relate b - branes in massive lg models to @xmath14-graded clifford modules . in section [ sec : f ] we use homological mirror symmetry to compute the category of a - branes for @xmath10 and @xmath19 . section [ sec : g ] contains concluding remarks . a fano variety is a compact complex manifold whose anti - canonical line bundle is ample . this is equivalent to saying that the first chern class of the canonical line bundle is negative - definite . an @xmath0 sigma - model whose target space is a fano variety describes an @xmath0 @xmath6 field theory which is free in the ultraviolet . in the infrared , it can either flow to a massive vacuum , or to a non - trivial @xmath0 scft . note that classically @xmath0 sigma - models have both vector and axial @xmath8 r - symmetries , but for fano varieties quantum anomalies break the axial r - symmetry down to a discrete subgroup . generically , this subgroup is @xmath14 , but in special cases it can be larger . in the calabi - yau case the full axial r - symmetry is non - anomalous , and it is this fact that makes calabi - yau target spaces so special . the simplest examples of fano varieties are complex projective spaces @xmath1 . the corresponding @xmath0 field theories are well studied ; in fact , these models are integrable , in the sense that the exact s - matrix is known @xcite . these theories have only massive vacua . a more general set of examples is given by grassmann varieties @xmath20 , which are defined as spaces of complex @xmath21-planes in an @xmath2-dimensional complex vector space . the corresponding @xmath0 field theories are also integrable @xcite . an @xmath0 field theory which has a conserved vector ( resp . axial ) r - current admits a topological a - twist ( resp . b - twist ) , which yields a 2d topological field theory called the a - model ( resp . b - model ) . @xmath0 superconformal field theories have both axial and vector r - symmetries , and therefore admit both kinds of twisting . a - branes and b - branes are `` defined '' as boundary conditions which are consistent with a - twist and b - twist , respectively . these two sets of branes have the structure of a category . the space of morphisms is defined as the state space of topological open strings stretched between pairs of branes . composition of morphisms is defined by means of 3-point correlators in topological open string theory . since state spaces of open strings are graded vector spaces , brane categories are graded categories . in the case of a - branes , spaces of morphisms are graded by the axial r - charge ; in the case of b - branes , by the vector r - charge . for fano varieties , the axial r - symmetry is generically @xmath14 , so spaces of morphisms in the category of a - branes are @xmath14-graded vector spaces . in the calabi - yau case , the full axial @xmath8 r - symmetry is non - anomalous , and therefore the category of a - branes is @xmath13-graded . this is the reason one has to work with @xmath13-graded lagrangian submanifolds in the calabi - yau case @xcite . in the fano case , one only needs to require that lagrangian submanifolds be oriented . in the calabi - yau case , we can also consider the category of b - branes . since the vector r - current is non - anomalous , this category is @xmath13-graded . in the fano case , the b - twist is not defined , and there is no obvious way to define the category of b - branes . given a graded category , one can enlarge it by adding for any object @xmath22 its shifts @xmath23 $ ] , where @xmath24 or @xmath25 , and defining morphisms as follows : @xmath26,y_2[j]\right)=mor^{k+j - i}(y_1,y_2).\ ] ] in string theory , the r - charge of strings connecting two different branes is defined only up to an integer constant ; changing this constant by @xmath21 shifts the degree of all morphisms by @xmath21 . the effect of this arbitrariness is that for any brane @xmath22 its shifts @xmath23 $ ] are automatically included . this implies that no information is lost if we replace groups of morphisms with their degree-0 components . this is what one usually does when working with categories of complexes , such as the derived category . nevertheless , in this paper we will keep morphisms of all degrees , since this conforms better to physical conventions . from this viewpoint , the mathematical counterpart of the category of b - branes on a calabi - yau @xmath27 is not @xmath28 , but a @xmath13-graded category which is called the completion of @xmath28 with respect to the shift functor . for a sigma - model on a calabi - yau manifold which is a complete intersection in a toric variety , the mirror theory is again a sigma - model of the same kind . for fano varieties which are complete intersections in a toric variety , the mirror theory is a lg model whose target is a non - compact calabi - yau @xcite . a general definition of a lg model involves , besides a choice of a target manifold , a choice of a holomorphic function @xmath11 on this manifold ( the superpotential ) . thus non - trivial lg models require non - compact target spaces . this non - compactness usually does not cause trouble : the important thing is for the critical set of @xmath11 to be compact . in general , superpotential breaks vector r - symmetry down to @xmath14 . thus the a - twist is not defined , in general . on the other hand , since the canonical bundle of the target manifold is trivial , the axial r - symmetry is not anomalous , and the b - twist is well - defined . we expect that the category of a - branes on a fano variety is equivalent to the category of b - branes on the mirror lg model . for example , the mirror of @xmath1 is a lg model whose target is @xmath29 with the superpotential eq . ( [ wcpn ] ) . this superpotential has @xmath30 non - degenerate critical points given by @xmath31 the physical interpretation is that the theory has @xmath30 massive vacua . this agrees with the count of vacua in the @xmath1 model . furthermore , the superpotential breaks vector @xmath32 symmetry down to @xmath14 given by @xmath33 where @xmath34 are the usual odd coordinates on the chiral @xmath35 superspace . as a consequence , spaces of morphisms in the category of b - branes in this lg model are @xmath14-graded . this is mirror to the fact that morphisms in the category of a - branes on @xmath1 are @xmath14-graded . r - symmetry ; the @xmath14 symmetry discussed in the text is its subgroup . this is mirror to the fact that the @xmath1 sigma - model has non - anomalous axial @xmath36 r - symmetry . in this paper we will only keep track of @xmath14-gradings . ] as a rule , it is easier to understand b - branes , rather than a - branes . therefore , we now turn to the study of b - branes in lg models , in the hope that it will illuminate the properties of a - branes on fano varieties . the classical geometry of b - branes was described in refs . @xcite ( see also ref . @xcite ) . in this section we summarize the results of refs . @xcite which are relevant for us and discuss some simple consequences . let @xmath27 be the target space of a lg model . on general grounds , it must be khler manifold ( possibly non - compact ) . let the @xmath11 be a fixed holomorphic function on @xmath27 ( the superpotential ) . let @xmath22 be a submanifold of @xmath27 , and let @xmath37 be a hermitian vector bundle over @xmath22 with a unitary connection @xmath38 . the rank of @xmath37 will be called the multiplicity of the corresponding d - brane . it is shown in ref . @xcite that the triple @xmath39 defines a classical b - type boundary condition if and only if @xmath22 is a complex submanifold of @xmath27 , @xmath11 is constant on @xmath22 , and the pair @xmath40 , where @xmath41 is the anti - holomorphic part of @xmath38 , is a holomorphic vector bundle . for example , a point on @xmath27 together with a choice of multiplicity @xmath42 defines a b - type boundary condition . the class of b - branes described in the previous paragraph does not exhaust all possible b - branes . but it appears plausible that all b - branes can be obtained as bound states of the branes described above . it was noticed in ref . @xcite ( see also ref . @xcite ) that most of the `` classical '' b - branes should be regarded as zero objects in the category of b - branes . a classical b - brane is isomorphic to the zero object if and only if the space of its endomorphisms is zero - dimensional , i.e. when there are no supersymmetric open string states connecting the brane with itself . in this case one says that world - sheet supersymmetry is spontaneously broken . for example , it is explained in refs . @xcite that if @xmath22 is a point on @xmath27 which is not a critical point of @xmath11 , then b - type supersymmetry is spontaneously broken , and therefore @xmath22 is isomorphic to the zero object in the category of b - branes . this phenomenon reduces enormously the number of b - branes that one needs to consider , and makes it plausible that the whole category can be described combinatorially , using only the number and type of critical points of @xmath11 . to substantiate this claim , we first notice that to any b - brane in the class described above one can assign a complex number , the value of @xmath11 on this brane . further , there are no non - zero morphisms between branes with different values of @xmath11 , because any string connecting such branes will have non - zero energy and will not be supersymmetric @xcite . ( unlike in the case of a - branes , there is no central charge in the supersymmetry algebra , and supersymmetric states must have zero energy . ) thus the category of b - branes can be regarded as a family of categories parametrized by @xmath43 , and the categories at different points in @xmath43 do not `` talk '' to each other . second , zero - energy classical configurations of an open string must be constant maps from the interval to @xmath22 , such that the potential energy @xmath44 vanishes . this implies that unless a b - brane passes through a critical point of @xmath11 , there are no supersymmetric states for strings connecting this b - brane to any other b - brane ( including itself ) . it follows that categories corresponding to non - critical values of @xmath11 are trivial ( contain only the zero object ) . now let us assume that all critical points of @xmath11 are isolated . by scaling up the khler form , we can make the semi - classical approximation arbitrarily good . this means that wave - functions of all string states will be arbitrarily well localized near a particular critical point of @xmath11 , and the overlap between wave - functions associated to different critical points will be arbitrarily small . since topological correlators do not depend on the khler form , it is clear that morphisms between b - branes can be computed using only the leading terms in the taylor expansion of @xmath11 around the critical points .. ] to be more precise , we can attach a category to each isolated critical point of @xmath11 as follows : we replace @xmath11 by a polynomial which has the same singularity , and consider the category of b - branes on an affine space with this polynomial superpotential . now let us form the direct sum of such categories over all critical points of @xmath11 and call it @xmath45 . there is an obvious map which associates to any b - brane an object of @xmath15 . invariance of topological correlators under variations of the khler form means that this map extends to a functor , and this functor is full and faithful . in other words , the category of b - branes is a full sub - category of @xmath15 . in particular , when all critical points of @xmath11 are non - degenerate ( i.e. when all vacua are massive ) , the problem reduces to understanding b - branes in the lg model with target @xmath12 and superpotential @xmath46 the corresponding bulk theory is free , but since the boundary conditions need not be linear , the problem of determining all b - branes is far from trivial . in this paper we will study b - branes which correspond to linear boundary conditions . some such branes have been considered in refs . we will see below that if kontsevich s conjecture is true , then these branes generate the whole category of b - branes . note that the lg superpotential eq . ( [ wcpn ] ) satisfies the conditions stated above . thus , assuming mirror symmetry , we can gain information about a - branes on @xmath1 by studying b - branes in the free lg model with the superpotential eq . ( [ wfreen ] ) . in the case @xmath47 this has been done in ref . @xcite ; in that case the category of a - branes is independently known , and one can see that the mirror conjecture holds true . in section [ sec : f ] we discuss the less trivial case @xmath48 . before continuing , let us make some further comments on the relation between @xmath15 and the category of b - branes . we do not claim that the two categories are equivalent , only that the latter is a full sub - category of the former . this means that each b - brane can be regarded as a direct sum of `` local '' b - branes attached to critical points , but not every direct sum is a valid b - brane . we will see in section [ sec : f ] some examples where the category of b - branes is strictly smaller than @xmath15 . nevertheless , since each b - brane behaves as a composite of `` local '' b - branes , it is reasonable to enlarge the category of b - branes by allowing arbitrary sums of `` local '' b - branes . then the category of b - branes becomes equivalent to @xmath15 . from a purely algebraic standpoint , this is a very natural procedure ( c.f . a discussion in ref . @xcite concerning reducible and irreducible branes ) , but the drawback is that the new branes lack a clear geometric interpretation . section [ sec : f ] contains a further discussion of this issue . we begin by recalling the results of ref . @xcite concerning b - branes in the simplest lg model with the superpotential @xmath50 . in this case , the only allowed b - branes are d0-branes located at @xmath51 . it has been shown in ref . @xcite that the space of endomorphisms of a single d0-brane is two - dimensional , with one - dimensional even subspace and one - dimensional odd subspace . as a graded algebra , it is generated over @xmath43 by the identity and an odd element @xmath52 with the relation @xmath53 this is a clifford algebra @xmath54 . if we take @xmath55 d0-branes , then the algebra of endomorphisms becomes @xmath56 where @xmath57 is the algebra of @xmath58 complex matrices . we are interested in the next simplest lg model with the superpotential @xmath49 . again , d0-branes must be localized at @xmath59 , and it has been shown in ref . @xcite that the algebra of endomorphisms of a single d0-brane is generated by two odd elements @xmath60 with the relations @xmath61 this is a clifford algebra @xmath62 . more invariantly , if we denote by @xmath63 the complex vector space which is the target space of our lg model , we can say that fermion zero modes for the open string take values in @xmath63 . the hessian of the superpotential @xmath64 defines a non - degenerate symmetric bilinear form on @xmath63 , and the endomorphism algebra of the d0-brane is the clifford algebra associated to the pair @xmath65 . ( some standard facts about clifford algebras and their modules are described in the appendix . we will freely use these facts in what follows . ) but in this case there can also be b - branes of higher dimension , namely d2-branes . ( this is briefly discussed in ref . irreducible d2-branes are irreducible components of the critical level set @xmath66 , which is a singular quadric @xmath67 thus there are two candidate irreducible d2-branes , given by @xmath68 and @xmath69 , respectively . our immediate goal is to compute their endomorphisms , as well as morphisms between d2-branes and d0-branes . in physical terms , we will compute the spectrum and disk correlators of the topological open string with appropriate boundary conditions . we consider the landau - ginzburg model on @xmath70 $ ] with two chiral superfields @xmath71 and @xmath72 . the superpotential assumes the following form @xmath73 we include a positive factor @xmath74 in the superpotential in order to keep track of dimensions of topological correlators later . in physical terms , @xmath75 is a measure of the mass gap in the landau - ginzburg model . assuming the standard khler potential @xmath76 the world - sheet action reads @xmath77}\!\!{{\rm d}}^2x { \left}\{\sum_{{{\alpha}}=1}^{2}{\left}(|\partial_t\phi^{{{\alpha}}}|^2- & & { \left}.-\,|m\phi^1|^2 -|m\phi^2|^2- m{\left}(\psi_+^1\psi_-^2+\psi_+^2\psi_-^1{\right } ) - \bar{m}{\left}(\psi_-^{\bar1}\psi_+^{\bar2}+\psi_-^{\bar2}\psi_+^{\bar1}{\right}){\right}\},{\nonumber}\end{aligned}\ ] ] where @xmath78 are the bosonic components of @xmath79 , and @xmath80 are their fermionic partners . the world - sheet parametrization @xmath81 is such that @xmath82 is the world - sheet time . from the bosonic lagrangian density @xmath83 one readily obtains the eom s for @xmath84 @xmath85 similarly , in terms of new variables @xmath86 the fermionic lagrangian density can be written as @xmath87 the fermionic eom s are given by @xmath88 b - type supersymmetry transformations are well known ( see , for example , ref . @xcite ) and look as follows : @xmath89 now we will perform canonical quantization of this system with various boundary conditions which correspond to d0-d0 strings , d2-d2 strings , and d0-d2 strings . we would like to find supersymmetric states of d0-d0 strings , since these correspond to endomorphisms of the d0-brane . the relevant boundary conditions are @xmath90 first consider the bosonic degrees of freedom . the boundary conditions give the following mode expansions for @xmath84 and its conjugate momentum @xmath91 : @xmath92 where @xmath93 to quantize , we impose the following commutation relations : @xmath94=\big[{{\tilde{a}}}_{{{\alpha}},n } , { { \tilde{a}}}_{{{\beta}},m}^\dagger\big]=\delta_{{{\alpha}}{{\beta}}}\delta_{mn},\ ] ] with all other commutators vanishing . it is easy to check that ( [ eq : comm_a ] ) is compatible with the following canonical commutation relations @xmath95 = i\delta_{{{\alpha}}{{\beta}}}\cdot \frac2{\pi}\sum_{n=1}^{\infty } \sin n{\sigma}\sin n{\sigma } ' = i\,\delta_{{{\alpha}}{{\beta}}}\,\delta({\sigma}-{\sigma } ' ) \label{eq : can_phi}\ ] ] the bosonic hamiltonian is given by @xmath96 where the additive constant `` @xmath97 '' in the sum is the bosonic zero point energy . we shall see later that it is exactly canceled by the fermionic zero point energy . next we consider the fermionic degrees of freedom . it will turn out convenient to use the following combinations as new dynamical variables : @xmath98 the main advantage of using @xmath99 and @xmath100 is that the eom s for the unbarred quantities are decoupled from those for the barred quantities . the mode expansions for these fields have the following form : @xmath101 to fix the commutation rules for the oscillators we impose the canonical commutation relations for @xmath102 : @xmath103 a convenient choice of compatible commutation rules for oscillators is @xmath104 with all others vanishing . one can easily check that the canonical commutation relations for @xmath105 fields are also respected , provided that the following relations are imposed : @xmath106 where @xmath107 and @xmath108 are defined by @xmath109 the fermionic hamiltonian is @xmath110{\nonumber}\\ & = \int_0^\pi\!{{\rm d}}{\sigma}\ , { \left}[\ , i(c_+'-mc_+)\bar{b}_+ + i(c_-'+mc_-)\bar{b}_- + { \rm h.c . } { \right}]{\nonumber}\\ & = \sum_{n=1}^\infty \ ; { \omega}_n { \left}({{\alpha}}^\dagger_{1,n}{{\alpha}}_{1,n}+ { { \alpha}}^\dagger_{2,n}{{\alpha}}_{2,n}+{{\tilde{{{\alpha}}}}}^\dagger_{1,n}{{\tilde{{{\alpha}}}}}_{1,n}+{{\tilde{{{\alpha}}}}}^\dagger_{2,n}{{\tilde{{{\alpha}}}}}_{2,n}-2{\right}).{\nonumber}\end{aligned}\ ] ] note that the additive constant @xmath111 cancels the bosonic zero point energy . the hamiltonian is diagonalized in the fock basis , and the zero - energy states are @xmath112 the supercharge @xmath113 can also be expanded in terms of oscillators . it can be shown that each term in the expansion contains an annihilation operator for non - zero modes , and that @xmath113 does not depend on zero - mode oscillators @xmath108 and @xmath107 . therefore @xmath113 annihilates all four ground states . since we have two different d2-branes related by a symmetry , there are two inequivalent possibilities : either our string begins and ends on the same d2-brane , or it begins on one d2-brane , and ends on the other d2-brane . the first situation corresponds to endomorphisms of a d2-brane , while the second one corresponds to morphisms from one d2-brane to the other one . first we consider the case when both boundaries end on the same brane , say , the one given by the equation @xmath114 . the relevant boundary conditions are @xmath115 first let us look at the bosons . the mode expansions for @xmath116 and its conjugate momentum are the same as before , while for @xmath117 and its conjugate momentum they are given by @xmath118 canonical commutation relations for bosonic fields imply the commutation relations eq . ( [ eq : comm_a ] ) for the oscillators . in terms of oscillators the bosonic hamiltonian is @xmath119 for the fermions , the mode expansions are given by @xmath120\sin n{\sigma},{\nonumber}\\ b_2 & = & \frac{-1}{\sqrt{2\pi } } { \left } ( { { \tilde{{{\alpha}}}}}_{0 } - { { \alpha}}_0^\dagger{\right}){\nonumber}\\ & + & \frac{-i}{\sqrt{2\pi}}\sum_{n=1}^\infty { \left}[-\frac{n+im}{{\omega}_n}{\left}({{\tilde{{{\alpha}}}}}_{1,n}-{{\alpha}}_{1,n}^\dagger{\right } ) + \frac{n - im}{{\omega}_n}{\left}({{\tilde{{{\alpha}}}}}_{2,n}-{{\alpha}}_{2,n}^\dagger{\right}){\right}]\cos n{\sigma},{\nonumber}\\ c_1 & = & \frac{1}{\sqrt{2\pi}}{\left}({{\alpha}}_{0}+{{\tilde{{{\alpha}}}}}_{0}^\dagger{\right } ) + \frac{1}{\sqrt{2\pi}}\sum_{n=1}^\infty { \left}({{\alpha}}_{1,n}+ { { \alpha}}_{2,n}+{{\tilde{{{\alpha}}}}}_{1,n}^\dagger+{{\tilde{{{\alpha}}}}}_{2,n}^\dagger{\right})\cos n{\sigma},{\nonumber}\\ c_2 & = & \frac{1}{\sqrt{2\pi}}\sum_{n=1}^\infty { \left}({{\tilde{{{\alpha}}}}}_{2,n}-{{\tilde{{{\alpha}}}}}_{1,n}+{{\alpha}}_{2,n}^\dagger-{{\alpha}}_{1,n}^\dagger{\right})\sin n{\sigma}.{\nonumber}\end{aligned}\ ] ] the canonical commutation relations for the fields @xmath121 and @xmath122 are equivalent to the following commutation relations for the oscillators : @xmath123 with all others vanishing . the fermionic hamiltonian can be shown to be @xmath124{\nonumber}\\ & = & \sum_{n=1}^\infty \ ; { \omega}_n { \left}({{\alpha}}^\dagger_{1,n}{{\alpha}}_{1,n}+ { { \alpha}}^\dagger_{2,n}{{\alpha}}_{2,n}+{{\tilde{{{\alpha}}}}}^\dagger_{1,n}{{\tilde{{{\alpha}}}}}_{1,n}+ { { \tilde{{{\alpha}}}}}^\dagger_{2,n}{{\tilde{{{\alpha}}}}}_{2,n}-2{\right})\nonumber\\ & & + \ ; m{\left}({{\alpha}}_0^\dagger{{\alpha}}_0 + { { \tilde{{{\alpha}}}}}_0^\dagger{{\tilde{{{\alpha}}}}}_0 - 1{\right}).{\nonumber}\end{aligned}\ ] ] the fermionic zero point energy cancels the bosonic zero point energy , and we see that there is a unique state with zero energy : the fock vacuum . for the same reason as in the d0-d0 case , this state is supersymmetric ( is annihilated by the supercharge ) . now consider the case when one end of the string ( @xmath125 ) is attached to @xmath114 , and the other one ( @xmath126 ) is attached to @xmath127 . the boundary conditions are @xmath128 the mode expansions for the bosons are @xmath129 while for the fermions they are @xmath130\sin k_n{\sigma},{\nonumber}\\ b_2 & = & \frac{1}{\sqrt{2\pi}}\sum_{n=1}^\infty { \left}[\frac{k_n+im}{{\omega}_n}{\left}({{\tilde{{{\alpha}}}}}_{1,n}-{{\alpha}}_{1,n}^\dagger{\right } ) - \frac{k_n - im}{{\omega}_n}{\left}({{\tilde{{{\alpha}}}}}_{2,n}-{{\alpha}}_{2,n}^\dagger{\right}){\right}]\cos k_n{\sigma},{\nonumber}\\ c_1 & = & \frac{1}{\sqrt{2\pi}}\sum_{n=1}^\infty { \left}({{\alpha}}_{1,n}+{{\alpha}}_{2,n}+{{\tilde{{{\alpha}}}}}_{1,n}^\dagger+{{\tilde{{{\alpha}}}}}_{2,n}^\dagger{\right})\cos k_n{\sigma},{\nonumber}\\ c_2 & = & \frac{i}{\sqrt{2\pi}}\sum_{n=1}^\infty { \left}({{\tilde{{{\alpha}}}}}_{1,n}-{{\tilde{{{\alpha}}}}}_{2,n}+{{\alpha}}_{1,n}^\dagger-{{\alpha}}_{2,n}^\dagger{\right})\sin k_n{\sigma},{\nonumber}\end{aligned}\ ] ] where @xmath131 one can show as before that commutation relations ( [ eq : comm_a ] ) and ( [ eq : comm_alpha ] ) yield all the canonical commutation relations , and the total hamiltonian is diagonalized in the fock basis as follows : @xmath132 again there is a single ground state which is annihilated by the supercharge . the boundary conditions for the bosons are @xmath133 the corresponding mode expansions are @xmath134 where @xmath135 imposing ( [ eq : can_phi ] ) , we infer that the bosonic oscillators obey ( [ eq : comm_a ] ) . the bosonic hamiltonian is given by @xmath136 now let us consider fermions . the boundary conditions imposed by supersymmetry are @xmath137 which , when combined with the eom s , give the following boundary conditions for @xmath102 and @xmath105 fields separately : @xmath138 @xmath139 the mode expansions are @xmath140 where @xmath141 imposing canonical commutation relations on the fields implies the following commutation relations for the oscillators : @xmath142 @xmath143 all other anti - commutators vanish . in view of the commutation relations for @xmath144 and @xmath145 , we set @xmath146 so that @xmath147 the fermionic hamiltonian has the form @xmath148{\nonumber}\\ & = & \sum_{n=1}^\infty { \omega}_{1,n } { \left}({{\alpha}}^\dagger_{n}{{\alpha}}_{n}+{{\tilde{{{\alpha}}}}}^\dagger_{n}{{\tilde{{{\alpha}}}}}_{n}-1{\right } ) + \sum_{n=1}^\infty { \omega}_{2,n } { \left } ( { { \beta}}^\dagger_{n}{{\beta}}_{n } + \tbb^\dagger_{n}\tbb_{n}-1{\right}).{\nonumber}\end{aligned}\ ] ] as before the bosonic zero - point energy is canceled by the fermionic zero - point energy . there are two zero - energy states : @xmath149 again it can be shown that they are annihilated by the supercharge . therefore both ground states are supersymmetric . the landau - ginzburg model admits a topological twist to yield the so - called b - model . this topological twist turns the world - sheet spinor fields @xmath150 and @xmath151 of the original lg model into a pair of sections of the pullback bundle @xmath152 , which we denote by @xmath153 , and a world - sheet one - form @xmath154 with values in @xmath155 . the brst transformations of the twisted fields are @xmath156 to make connection with the fields in the original landau - ginzburg theory , we note that @xmath107 and @xmath52 are the twisted versions of @xmath157 and @xmath158 respectively , while @xmath154 comes from @xmath102 and @xmath105 . we also adopt the common notation @xmath159 . the local physical observables are in one - to - one correspondence with the brst - cohomology , i.e. local quantities which are brst invariant but not brst exact . it is easy to see from the above brst transformations that in the bulk the physical observables correspond to holomorphic functions of @xmath84 modulo @xmath160 , where @xmath161 is an arbitrary holomorphic vector field . there are no additional local observables from the fermionic fields , as long as @xmath11 is nontrivial . in particular , when @xmath162 , the space of bulk observables is @xmath163/i$ ] , where @xmath164 is the ideal generated by the first partial derivatives of @xmath11 . in the boundary sector , where @xmath11 is constrained to be constant , additional observables will arise from the @xmath52 fields . we now specialize to the d0 and d2 branes studied above . we consider the boundary component which is mapped to the d0 brane located at @xmath165 . the boundary conditions require , among other things , that @xmath166 therefore boundary observables can only come from the @xmath52 fields . from the brst transformation @xmath167 one sees immediately that both @xmath168 and @xmath169 are brst invariant on the boundary . let us denote the restriction of @xmath170 to the boundary by the same letter @xmath170 . thus the ring of boundary observables is generated by @xmath168 and @xmath169 . without loss of generality , we may assume that the d2 brane sits at the locus @xmath171 . the relevant boundary conditions read @xmath172 from this one sees that @xmath169 no longer gives rise to a boundary degree of freedom . also , since @xmath117 is not constrained to vanish on the boundary , @xmath168 is no longer brst invariant . thus there are no boundary observables ( except for the identity operator ) associated with the d2 brane . first let us compute topological correlators for strings connecting a brane with itself . since in the d2-d2 case there is only the vacuum state , the problem is non - trivial only in the d0-d0 case . in order to compute disk correlators with products of @xmath168 and @xmath169 inserted on the circumference , we proceed as in ref . we start with a world - sheet diagram which has the topology of a cylinder , with the d0 and d2 boundary conditions imposed on the two boundary circles ( c.f . [ fig : cyl ] ) . on the d0 boundary there can be operator insertions . viewed in the open - string channel , this world - sheet diagram computes the one - loop amplitude @xmath173\ ] ] where the @xmath174 s are operators inserted at the d0 boundary . the trace on the rhs can be reduced to the hilbert space of open - string zero modes by the standard argument . in the specific case at hand , the zero mode space is spanned by @xmath175 and @xmath176 as described in sec . [ sec : d0d2 ] , and one easily obtains @xmath177 more generally , one can compute @xmath178 from ( [ eq : cyl_1 ] ) and ( [ eq : cyl_2 ] ) one deduces the relations in the boundary operator product algebra for the d0-brane : @xmath179 this is the clifford algebra with two generators corresponding to the quadratic form @xmath180 up to a numerical factor , this matrix is the hessian of @xmath11 in the basis @xmath60 . thus one can state the result more invariantly by saying that the boundary operator product algebra for the d0-brane is the clifford algebra @xmath181 , where @xmath63 is target space of our lg model , and @xmath113 is the quadratic form given by the hessian of @xmath11 . topological correlators on the disk can be inferred from the computation on the cylinder using factorization in the closed string channel @xcite . namely , we insert a complete set of states in the closed string channel ( cf . fig . [ fig : cyl]b ) and rewrite the cylinder amplitude as @xmath182 where @xmath183 s form a complete set of bulk operators , and @xmath107 is the ( inverse ) metric on the space of bulk operators defined via topological correlators on the sphere . the relative normalization is fixed by demanding the following relation for the d2-d2 cylinder amplitude @xmath184 in our case , the only bulk operator is the identity , therefore all disk correlators for the d0-brane simply coincide with the cylinder correlators . besides the algebra structure , another important datum is a non - degenerate inner product on the space of endomorphisms . this inner product is determined by the two - point disk correlator and makes the endomorphism algebra into a ( non - commutative ) graded frobenius algebra . in our case the only non - vanishing inner products are @xmath185 note that the bilinear form corresponding to this product is even . in contrast , in the model @xmath50 the bilinear form is odd and given by @xmath186 so far we have determined the endomorphism algebra of the d0-brane ( it is isomorphic to @xmath62 ) and the d2-brane ( it is isomorphic to @xmath43 ) . now we turn to the computation of compositions of morphisms between different branes . we begin with the case when both branes are d2-branes . let us denote the d2-brane given by the equation @xmath114 ( resp . @xmath127 ) by @xmath187 ( resp . @xmath188 ) . it was shown in the previous section that the vector space @xmath189 is one - dimensional for all @xmath190 and @xmath191 . when @xmath192 , this space is even , but for @xmath193 there is no canonical choice for the r - charge . in other words , for @xmath193 the `` vacuum '' vector spanning @xmath189 can equally well be regarded as even or odd . for reasons which will become clear later , we define @xmath194 to be purely odd ; since @xmath195 is dual to @xmath194 , it is also purely odd , while @xmath196)$ ] is purely even . here @xmath197 $ ] denotes the shift of @xmath187 . let @xmath198 and @xmath199 be generators of @xmath194 and @xmath200 , respectively . since the endomorphism algebra of a d2-brane is spanned by the identity morphism , we only need to determine if @xmath201 is zero or not . this product is evaluated by the disk amplitude with two insertions of boundary - changing operators . by conformal invariance , this is the same as the vacuum - vacuum transition amplitude for open strings stretched between @xmath188 and @xmath187 . since there are no fermionic zero modes in this case , this amplitude is non - zero . this means that @xmath202 with @xmath203 . this trivial computation implies that the even generator of @xmath196)$ ] is an isomorphism . in physical language , @xmath188 is isomorphic to the anti - brane of @xmath187 . there remain compositions of morphisms involving both d2-branes and the d0-brane . in view of the previous paragraph , it is sufficient to consider morphisms between @xmath188 and the d0-brane . no new computations are actually required , the result being fixed by general properties of topological string theory @xcite . first of all , we note that by cyclic symmetry of topological correlators computing compositions of morphisms from d2 to d0 and back ( or the other way around ) is equivalent to computing how the endomorphism algebra of the d0-brane acts on the space of morphisms from d2 to d0 . in more detail , we have a non - degenerate pairing @xmath204 given by the path - integral on an infinite strip . ( this paring is odd in our case , because there is a single fermionic zero mode . ) similarly , we have an even non - degenerate pairing @xmath205 thus computing the product map @xmath206 is the same as computing the map @xmath207 furthermore , in our case @xmath208 is isomorphic to @xmath62 , and we know from the previous section that @xmath209 is two - dimensional . the @xmath14 graded algebra @xmath62 has a unique representation on @xmath210 , up to a flip of parity ( up to isomorphism , it is given by any two pauli matrices ) . since in string theory the parity of morphisms is not canonically fixed anyway , we conclude that the module structure of @xmath209 is completely determined , up to the unavoidable ambiguity in the overall parity . this in turn determines the composition of morphisms going from d0 to d2 and back . we now turn to massive landau - ginzburg models which involve more than two fields . without loss of generality , we may assume that the superpotential on @xmath12 is given by @xmath211 we can construct examples of b - branes in this lg model for any @xmath2 , using the results of the previous section . for @xmath212 , @xmath213 , we consider an equivalent superpotential @xmath214 since it is a sum of @xmath21 copies of the superpotential @xmath49 , we can construct a b - type boundary condition by picking @xmath21 arbitrary b - type boundary conditions for the latter model , and tensoring them . for example , if we take all boundary conditions to be d0-branes , the tensor product state will also be a d0-brane , and its endomorphism algebra will be @xmath215 . if we take all boundary conditions to be d2-branes , the tensor product boundary state will be a d(@xmath216)-brane , and its endomorphism algebra will be @xmath43 . similarly , for @xmath217 we consider an equivalent superpotential @xmath218 clearly , b - branes for this lg model can be constructed by taking tensor product of @xmath21 boundary states for the lg model with @xmath49 and a boundary state for the lg model with @xmath50 . in this way one obtains b - branes of dimension up to @xmath216 . it is easy to see that the endomorphism algebra of the d0-brane will be isomorphic to @xmath219 , while the endomorphism algebra of the d(@xmath216)-brane will be isomorphic to @xmath54 . more generally , one can explicitly construct all b - branes which correspond to linear subspaces of the critical level set @xmath66 . since @xmath11 is quadratic , these are the same as linear subspaces isotropic with respect to the bilinear form @xmath113 . classification of such isotropic subspaces is well known @xcite . the maximal dimension of an isotropic subspace is @xmath220.$ ] for @xmath2 odd , there is a single irreducible family of isotropic subspaces of maximal dimension parametrized by @xmath221 parameters . for @xmath2 even , there are two irreducible families of isotropic subspaces of maximal dimension parametrized by @xmath222 parameters . any isotropic subspace lies in one of the maximal isotropic subspaces . it is straightforward to compute morphisms and their compositions ( i.e the spectrum and topological correlators ) between all linear b - branes . in the next subsection we discuss in some detail the results for the case @xmath223 , when the lg superpotential has the form @xmath224 . then we will describe the general case . note that in principle there could also be b - branes corresponding to non - linear boundary conditions ( e.g. non - linear submanifolds of the quadric @xmath66 ) . such b - branes are hard to study directly . in what follows we shall focus on linear boundary conditions . maximal isotropic linear subspaces on the quadric surface @xmath66 are complex lines , and there is a single irreducible family of them . this family is parametrized by @xmath9 as follows : @xmath225 where @xmath226 $ ] are homogeneous coordinates on @xmath9 . any two distinct lines in the family intersect at a single point ( @xmath227 ) . using a linear change of basis in the target space which preserves @xmath11 , one can always map any line in the above family to the line @xmath228 . for the brane @xmath228 we already know that the endomorphism algebra is isomorphic to @xmath54 , and since linear changes of variables preserving the superpotential are invariances of the topological lg model , we conclude that the same is true for any d2-brane in the above family . next we consider morphisms between different lines in the family . clearly , there are no bosonic zero modes , so the space of morphisms will be spanned by the `` vacuum '' state and its fermionic excitations with zero energy . as remarked in the previous section , only some components of @xmath52 have a chance to be brst - non - trivial boundary observables . thus all brst - invariant states can be obtained by acting by some components of @xmath52 on the vacuum state . let @xmath229 be the target space of our lg model , and let @xmath230 and @xmath231 be two distinct lines in @xmath63 isotropic with respect to the quadratic form @xmath11 . the corresponding b - branes will be denoted @xmath188 and @xmath187 . let us look at the @xmath52-field restricted to the boundary of the world - sheet which is mapped to @xmath230 . we can regard @xmath232 as basis elements of @xmath63 . brst transformations are @xmath233 on the boundary the vector with components @xmath234 can be an arbitrary element of @xmath230 . it follows that brst - invariant components of @xmath52 must be orthogonal to @xmath230 with respect to the form @xmath113 . we denote the orthogonal subspace by @xmath235 . of course , since @xmath230 is isotropic , we have an inclusion @xmath236 . similarly , brst - invariant fermionic fields on the @xmath231-boundary are parametrized by elements of @xmath237 . the total space of brst - invariant fermionic fields is @xmath238 however , not all of these are non - zero . neumann boundary conditions plus supersymmetry imply that the components of @xmath52 along @xmath239 vanish . thus non - trivial brst invariant fermionic zero modes are parametrized by elements of the quotient space @xmath240 this space is one - dimensional . thus there is a single fermionic zero mode , and the space of morphisms between two different lines is isomorphic to its exterior algebra ( as a @xmath14-graded vector space ) . that is , @xmath194 has one - dimensional even subspace , and one - dimensional odd subspace . composition of morphisms between two distinct lines is fixed by consistency considerations . if @xmath188 and @xmath187 are any two lines , then @xmath194 must be a left module over @xmath241 and right module over @xmath242 . there is only one such module of dimension two : the clifford algebra itself , regarded as a bi - module over itself . together with various parings given by the 2-point correlators , this fixes the structure of correlators involving any two d2-branes . in particular , it is easy to see that the element in @xmath194 corresponding to the identity element in @xmath54 is invertible . this means that any two lines give isomorphic objects in the category of b - branes . similar arguments can be used to determine boundary correlators involving both d2 and d0 . as explained above , the endomorphism algebra of the d0-brane is isomorphic to @xmath243 . as for the space of morphisms between a d2-brane and d0-brane , it is 4-dimensional , with two - dimensional even subspace and two - dimensional odd subspace . indeed , since all d2-branes are isomorphic , it is sufficient to consider morphisms from the d2-brane @xmath228 . this d2-brane is the tensor product of the d2-brane in the lg model @xmath49 and the d0-brane in the lg model @xmath50 . hence the computation of the space of morphisms and their compositions is reduced to the one we have performed in the previous section . the above arguments can be easily generalized to arbitrary @xmath2 . we shall consider only linear boundary conditions . let the target space be @xmath245 , let @xmath11 be a non - degenerate quadratic function on @xmath63 , and let @xmath246 be its hessian . a b - brane is a linear subspace @xmath247 which is isotropic with respect to @xmath113 . as mentioned above , @xmath248 is less or equal to @xmath220 $ ] . using linear changes of variables , we can bring @xmath11 to the standard form , and @xmath249 to the subspace given by @xmath250 . such a d(2k)-brane is a tensor product of @xmath21 copies of d2-branes in the model @xmath49 , @xmath220-k$ ] copies of the d0-brane in the model @xmath49 , and , for @xmath2 odd , one copy of a d0-brane in the model @xmath50 . it follows that the space of endomorphisms has dimension @xmath251 and is isomorphic as a @xmath14-graded algebra to the clifford algebra with @xmath252 generators . in particular , the algebra of endomorphisms of a d - brane of maximal possible dimension is isomorphic to @xmath43 or @xmath54 depending on whether @xmath2 is even or odd , while the the endomorphism algebra of the d0-brane is isomorphic to @xmath253 . next let us discuss morphisms between two different b - branes . let @xmath230 and @xmath231 be isotropic linear subspaces corresponding to b - branes @xmath188 and @xmath187 . the same arguments as in the previous subsection tell us that the space of fermionic zero modes can be identified with @xmath254 the space of morphisms is isomorphic as a graded vector space to the exterior algebra of this vector space ( up to an overall flip of parity ) . it is easy to see that the dimension of the space of zero modes is given by @xmath255 , where @xmath256 . therefore the dimension of the space of morphisms is given by @xmath257 in particular , in the case @xmath258 we recover the result eq . ( [ dimend ] ) obtained by other means . let us give a few examples . first , let @xmath2 be even , and @xmath230 and @xmath231 be distinct maximal isotropic subspaces . then @xmath259 for @xmath260 , and there are no zero modes . this means that the space of morphisms between any two maximal isotropic subspaces is one - dimensional . as usual , the r - charge assignment is ambiguous , but it is natural to require the r - charge to vary continuously as one varies @xmath261 . since in the case @xmath258 the space of endomorphisms is even and isomorphic to @xmath43 , this implies that for any two maximal isotropic subspaces in the same irreducible family the space of morphisms is even and isomorphic to @xmath43 . we will fix the remaining ambiguity by saying that the space of morphisms between two maximal isotropic subspaces in _ different _ irreducible families is odd . the reason for such a convention will be explained in the next section . the fact that for even @xmath2 there are no fermionic zero modes for open strings connecting two maximal isotropic subspaces implies that the vacuum - vacuum transition amplitude is non - zero in this sector . this is equivalent to saying that the composition of non - zero morphisms between two maximal b - branes is a non - zero multiple of the identity endomorphism . if these two b - branes are in the same irreducible family , this means that they represent isomorphic objects in the category ; if they are in different irreducible families , then the interpretation is that they are isomorphic up to a shift . if @xmath2 is odd , and @xmath230 and @xmath231 are maximal linear subspaces , then there is a single fermionic zero mode . thus @xmath194 is two - dimensional , with one - dimensional even and one - dimensional odd subspaces . the composition of morphisms going between two maximal b - branes is fixed by consistency requirements . namely , the space of morphisms must be a graded bi - module over @xmath54 ( the endomorphism algebra of a single b - brane ) , and there is only one such graded bi - module of dimension two : @xmath54 itself . furthermore , there is an odd non - degenerate pairing @xmath262 which is invariant with respect to both actions of @xmath54 in an obvious sense . up to isomorphism , there is only one such pairing , namely @xmath263 where @xmath264 is defined by @xmath265 together with the module structure of @xmath194 , this pairing determines the composition of morphisms going between any two lines . as in the case @xmath224 , it is easy to see that the morphism corresponding to the identity element of @xmath54 is invertible , and therefore any two maximal b - branes are isomorphic . our third example is the case @xmath266 , that is , the case of the d0-brane . the space of zero modes coincides with @xmath63 , and the space of endomorphism is isomorphic to @xmath267 as a @xmath14-graded vector space ( @xmath63 is regarded as odd ) . this agrees with an independent argument of subsection [ secdgen ] . there we also showed that the algebra of endomorphisms is isomorphic to @xmath181 . our fourth and final example is the case when @xmath230 is an arbitrary isotropic subspace of dimension @xmath268 $ ] , and @xmath269 . in other words , the second brane is the d0-brane . then the space of zero modes is @xmath270 . its dimension is @xmath271 , and therefore the space of open strings stretched between a maximal linear subspace and the d0-brane has dimension @xmath272 the space of morphisms has the structure of a graded module over the endomorphism algebra of the d0-brane , which is isomorphic to @xmath181 . if we neglect the grading , there is a unique such module , which is a sum of @xmath273-k}$ ] irreducible ( spinor ) modules . let us recall how to define the derived category of coherent sheaves on a smooth affine variety @xmath27 following ref . let @xmath274 be the category of coherent sheaves on @xmath27 , or equivalently the category of finite modules over the coordinate ring @xmath275 of @xmath27 . we define @xmath276 to be a category whose objects are bounded @xmath13-graded complexes of projective objects of @xmath274 . equivalently , we can think about the coordinate ring of @xmath27 as a differential graded algebra ( dg - algebra ) which is concentrated in degree zero and has a trivial differential ; then objects of @xmath276 are differential graded modules ( dg - modules ) over this dg - algebra such that all homogeneous components are projective @xmath275-modules , and all but a finite number of homogeneous components are trivial . morphisms in @xmath276 are morphisms of these dg - modules regarded simply as @xmath275-modules ( i.e. morphisms do not necessarily preserve the grading or respect the differentials ) . groups of morphisms in the category @xmath276 are naturally @xmath13-graded and have a natural differential of degree @xmath277 . for example , closed morphisms of degree 0 in the category @xmath276 are ordinary morphisms of complexes ( the ones which preserve the grading and commute with the differentials ) , while exact morphisms of degree 0 are morphisms of complexes which are homotopic to zero . thus @xmath276 is a dg - category . there is a general way to make a triangulated category out of any dg - category @xcite . one takes the category of `` twisted objects '' of the dg - category , which is again a dg - category , and then passes to degree-0 homology , i.e. replaces groups of morphisms with their degree-0 homology . in the present case , since we are working with complexes of projective modules , it is not necessary to consider twisted objects , and one can simply apply the functor @xmath278 to @xmath276 . the resulting triangulated category is simply the homotopy category of bounded complexes of projective @xmath275 modules , and it is well known that it is equivalent to the bounded derived category of @xmath274 ( see e.g. ref . alternatively , one can apply to @xmath276 the functor @xmath279 . this gives a graded category which is the completion of @xmath28 with respect to the shift functor . as discussed in section [ sec : a ] , the latter alternative conforms better to physical conventions . now we can formulate kontsevich s proposal rather simply . let @xmath27 be a smooth affine variety , and @xmath11 be a holomorphic function on @xmath27 ( the superpotential ) , whose critical set is compact . let @xmath280 be a critical value of @xmath11 . first , since in the presence of the superpotential morphisms between b - branes are @xmath14-graded , we will have to use @xmath14-graded complexes in order to construct the analogue of @xmath276 . second , we deform our @xmath14-graded complexes of projective modules by asking that the composition of two successive morphisms be equal to @xmath281 , instead of zero . thus objects of the deformed category @xmath282 are pairs of finitely generated projective @xmath275-modules @xmath283 and morphisms @xmath284 and @xmath285 such that @xmath286 we can regard the pair @xmath287 as a @xmath14-graded @xmath275-module , and @xmath288 as an odd endomorphism @xmath289 of this module whose square is @xmath281 ( `` twisted differential '' ) . morphisms in this category are defined as ( ungraded ) morphisms of the corresponding @xmath275-modules . they have a natural @xmath14-grading , and a natural differential . the differential on @xmath290 is defined as @xmath291 here @xmath292 acts as @xmath277 on the even component and as @xmath293 on the odd component . it is easy to see that @xmath294 is an odd operator whose square is zero . thus @xmath282 is a differential @xmath14-graded category . in what follows the term `` dg - category '' ( resp . `` graded category '' ) will refer to a differential @xmath14-graded category ( resp . @xmath14-graded category ) , unless specified otherwise . applying to @xmath282 the functor @xmath279 , we obtain a graded category , which is proposed to be equivalent to the category of b - branes corresponding to the critical value @xmath295 . one can show that all spaces of morphisms in this category are finite - dimensional , provided the critical set of @xmath11 is compact . an unsatisfactory feature of this construction is that one needs to use complexes of projective modules , instead of general complexes . this causes problems if one tries to extend the definition from affine varieties to algebraic ones . there is a way to repair this defect @xcite , but we will not try to explain this more complicated definition in this paper . in this subsection we give a physical argument supporting the identification of b - branes with objects of the category @xmath282 . our argument is modelled on those in refs . @xcite , where it was explained why complexes of locally free sheaves on a calabi - yau manifold can be thought of as b - branes . for our purposes , it is sufficient to consider @xmath14-graded complexes . then the argument of refs . @xcite can be summarized as follows . consider a pair of locally free sheaves ( i.e. holomorphic vector bundles ) @xmath296 and @xmath297 . we already know that @xmath296 and @xmath297 can be thought of as b - branes , i.e. as topological boundary conditions for a topological sigma - model . the same goes for @xmath298 and @xmath299 $ ] . ( in the physical setting , one has additional data , such as hermitian metrics on @xmath296 and @xmath297 and compatible connections . ) now we can deform the boundary condition corresponding to @xmath299 $ ] by adding a boundary term to the action which depends on a pair of sections @xmath300 and @xmath301 ( the `` tachyons '' ) . in order to preserve brst invariance , one has to require that @xmath302 and @xmath303 be holomorphic , and @xmath304 , @xmath305 . this deformed boundary condition corresponds to a @xmath14-graded complex @xmath306 one expects ( although there is no iron - clad argument ) that any b - brane is isomorphic to a b - brane of this kind . this provides a physical explanation for the relation between complexes of locally free sheaves on a calabi - yau and b - branes . in the case of a lg model , locally free coherent sheaves on @xmath27 are not valid b - branes , because their support is the whole @xmath27 , and @xmath11 is not constant on @xmath27 . technically , the problem occurs because the brst variation of the bulk action contains a non - vanishing @xmath11-dependent term which is a total derivative on the world - sheet . this is the so - called warner problem @xcite . the sum @xmath299 $ ] is not a b - brane either . however , we can try to add a boundary term to the action of the sigma - model so that brst invariance is restored . we take the same term as in the case @xmath66 . as in refs . @xcite , we have two holomorphic sections @xmath302 and @xmath303 to play with . as we show below , the condition of brst - invariance is modified to @xmath307 , @xmath308 , where @xmath295 is a constant . this shows that any object of @xmath282 corresponds to a b - brane . now let us work out the brst - invariance conditions and show that @xmath302 and @xmath303 must satisfy the constraints stated above . for simplicity , let us assume at first that both @xmath296 and @xmath297 are line bundles . the boundary lagrangian is taken to be @xmath309 here @xmath310 is a complex fermion living on the boundary , @xmath311 is the restriction of a bulk fermionic field to the boundary , and @xmath312 are holomorphic sections of @xmath313 and @xmath314 , respectively . they depend on the fields @xmath315 restricted to the boundary . the fermion @xmath310 takes values in @xmath314 , and the covariant derivative @xmath316 along the boundary makes use of the unitary connections on @xmath296 and @xmath297 . the boundary lagrangian is manifestly gauge - invariant . if we set @xmath317 , we get the usual path - integral representation of the parallel transport operator in the bundle @xmath318 @xcite . for non - zero @xmath302 or @xmath303 we get a deformation of the usual boundary condition . in the special case @xmath319 we get the boundary lagrangian used in ref . @xcite . we postulate the following supersymmetry transformations for @xmath310 : @xmath320 here @xmath321 and @xmath322 are regarded as independent complex grassmann variables . we also note that @xmath302 and @xmath303 transform as follows : @xmath323 brst transformations are obtained by setting @xmath324 . one can check that the brst variation of the boundary lagrangian is given , up to a total derivative , by @xmath325 on the other hand , the brst variation of the bulk action is a boundary term given by @xmath326.{\nonumber}\end{aligned}\ ] ] the first two terms in the bulk variation are standard and vanish when the standard neumann boundary conditions are imposed on @xmath315 and @xmath327 . the last term is the warner term @xcite . obviously , in order for the variation of the boundary lagrangian to cancel the warner term , we need to require @xmath328 one can get rid of the factor @xmath190 by redefining @xmath329 . this implies that instead of an ordinary complex of holomorphic vector bundles we are dealing with an object of @xmath330 it is straightforward to generalize the construction to the higher - rank case . the fermion @xmath310 still takes values in @xmath314 , which means that it is a matrix of size @xmath331 . in order for the path - integral over @xmath332 to reproduce a path - ordered exponential in the representation of the gauge group of dimension @xmath333 , one needs to insert a projector onto the sector where the total fermion number ( including the boundary contribution from @xmath310 ) is equal to @xmath17 or @xmath277 @xcite . the rest of the argument is unchanged . the conditions of brst - invariance now read @xmath334 the two conditions arise by requiring that the brst variation of the bulk term be cancelled on both boundaries of the world - sheet . by taking the trace of these two equations and comparing them , one infers that the ranks of @xmath296 and @xmath297 are in fact the same , and the constant terms in the equations are also the same . it follows that any object of @xmath282 is a b - brane . one can also check that the total brst charge is nilpotent . indeed , it is easy to see that the square of the bulk contribution to the brst charge is equal to @xmath335 on the other hand , the boundary supercharge coming from one of the two boundaries is given by @xmath336 canonical quantization yields @xmath337 and therefore @xmath338 it is also easy to check that @xmath339 and @xmath340 anti - commute ( the holomorphicity of @xmath302 and @xmath303 is important here . ) hence the sum of @xmath340 and the two boundary supercharges is nilpotent . we start with the case @xmath341 , @xmath50 , where there is only a d0-brane to worry about . in the absence of the superpotential , d0-brane on @xmath43 is associated with the structure sheaf of a point , which has a two - term projective resolution @xmath342 if we pass to @xmath14-graded complexes , we obtain the following dg - module : @xmath343 where @xmath344 and it is understood that the module @xmath345 has the obvious grading . if we turn on the superpotential , we need to deform the differential so that its square be equal to @xmath50 . it is clear how to do this : simply consider the object @xmath346 where @xmath347 this is our candidate object for the d0-brane at @xmath51 . as a check , let us compute its endomorphism algebra , following kontsevich s prescription . in the category @xmath282 , the algebra of endomorphisms is @xmath348).\ ] ] the @xmath14-grading is the natural grading on @xmath349 matrices ( diagonal elements are even , off - diagonal elements are odd ) . the differential acts on this graded vector space as follows : @xmath350 here @xmath351 are elements of @xmath352 i.e. simply polynomials in @xmath353 . computing @xmath279 , we find that this abelian group is isomorphic to the group of complex matrices of the form @xmath354 multiplicative structure is given by matrix multiplication . clearly , this algebra is generated over @xmath43 by the identity and an odd matrix @xmath355 which squares to identity . this agrees with the endomorphism algebra of the d0-brane in the lg model @xmath50 @xcite . now let us discuss b - branes in the lg model @xmath49 . using the same reasoning as above , it is easy to guess that the d2-brane given by the equation @xmath68 should correspond to the object @xmath356 where the map is defined as @xmath357 similarly , the d2-brane given by the equation @xmath69 should correspond to @xmath358 with the twisted differential @xmath359 the group of endomorphisms of the former object in the category @xmath282 is @xmath360),\ ] ] with the differential which acts as follows : @xmath361 the homology is readily computed ; the result is that it is spanned by the identity matrix . thus the algebra of endomorphisms in the derived category is isomorphic to @xmath43 . this agrees with the computation in section [ sec : cp ] . of course , for the other d2-brane we get the same result . finally , in order to compute morphisms between the two d2-branes , we note that one is a shift of the other . is isomorphic to the anti - brane for the d2-brane with the equation @xmath69 . ] thus the space of morphisms is the space of endomorphisms with gradings reversed , i.e. it is spanned by the identity matrix regarded as odd . composing two such odd morphisms going in the opposite directions we get the identity endomorphism . this agrees with the computations in section [ sec : cp ] and explains why we declared the space of morphisms between two different d2-branes to be odd . now let us discuss the d0-brane . consider the direct sum of objects corresponding to d2-branes with equations @xmath68 and @xmath69 . it is easy to see that its algebra of endomorphism is the clifford algebra @xmath62 , so we propose that this object corresponds to the d0-brane . it is easy to check that morphisms to and from other objects agree with our computations in section [ sec : cp ] . finally , we propose an object of @xmath362 corresponding to the d0-brane in the free massive lg model with @xmath2 fields . if we bring the superpotential to the standard form eq . ( [ generalw ] ) , then we can simply tensor @xmath2 copies of the object eq . ( [ d0object ] ) . consequently , the endomorphism algebra will also be the graded tensor product of @xmath2 copies of @xmath54 , which is isomorphic to @xmath363 . more invariantly , let @xmath364 be the complex vector space whose coordinates we denoted by @xmath365 , let @xmath366 be the corresponding basis in @xmath63 , let @xmath367 be the dual basis in @xmath368 , and let @xmath246 be the hessian of @xmath11 . we start with the @xmath14-graded version of the koszul resolution of the point at the origin : @xmath369 where @xmath370 , and the differential is induced by the wedge product with @xmath371 ( we use einstein s convention of summing over repeating indices ) . now we modify the differential so that its square be @xmath11 instead of zero . the obvious guess is @xmath372 where @xmath373 , denotes the interior product with an element of @xmath368 . using the identity @xmath374 one can easily check that @xmath375 . note that @xmath376 is essentially the fourier transform of the dirac operator , if we identify @xmath377 and @xmath378 with spinor bundles . we regard the above computations as a convincing check of kontsevich s proposal for massive lg models . in this subsection , we would like to address the following three questions . first , how do we match b - branes with objects in the category @xmath362 if @xmath379 ? ( as explained in section [ sec : b ] , it is sufficient to consider the case @xmath380 , @xmath11 quadratic non - degenerate , and @xmath381 . ) second , is there an efficient method to compute morphisms in the category @xmath362 ? third , assuming the validity of kontsevich s proposal , what do we learn about b - branes in massive lg models ? that is , is there a simpler way to describe @xmath362 ? to answer these questions , we will define a functor from the category of b - branes to the category of finite - dimensional @xmath14-graded modules over the clifford algebra @xmath363 . let us denote this category @xmath382 . ( as usual , we allow both even and odd morphisms ; thus @xmath382 is a @xmath14-graded category . ) since we set @xmath380 , @xmath381 , and @xmath11 is quadratic and non - degenerate , the category @xmath362 really depends only on @xmath2 ; we will denote this category @xmath383 for short . there is a further functor from @xmath382 to @xmath383 . composing these two functors gives a way to associate objects of @xmath383 to b - branes . in fact , as explained below , the second functor is an equivalence of categories which implies that we can calculate morphisms in @xmath382 instead of @xmath383 . this equivalence is a cousin of the much - studied koszul duality for quadratic algebras ( see below ) . the structure of @xmath382 is quite simple : any object is a direct sum of irreducible objects ( spinor modules ) , and there is one or two non - isomorphic irreducible objects , depending on whether @xmath2 is odd or even . thus we have a completely explicit description of @xmath382 , and therefore , by koszul duality , of @xmath383 . assuming the validity of kontsevich s conjecture , this amounts to a solution of topological open string theory for any massive lg model . one can associate a @xmath14-graded clifford module to a b - brane as follows . for any b - brane @xmath22 , consider the graded vector space @xmath384 . since @xmath208 is isomorphic to @xmath363 as a graded algebra , @xmath385 is a functor from the category of b - branes to the category of left @xmath14-graded modules over @xmath363 . since spaces of open strings are expected to be finite - dimensional , @xmath386 is expected to be a finite - dimensional vector space . thus @xmath385 is a graded functor from the graded category of b - branes to @xmath382 . the results of section [ sec : d ] ( see also the appendix ) imply that the functor @xmath385 maps the maximal linear b - brane to an irreducible clifford module ; for even @xmath2 maximal isotropic subspaces which belong to different irreducible families are mapped to non - isomorphic clifford modules related by parity reversal . the d0-brane is mapped to the free module of rank one . a linear b - brane of complex dimension @xmath387 $ ] is mapped to a module which is a direct sum of @xmath273-\ell}$ ] irreducible modules . next we would like to explain why @xmath382 is equivalent to @xmath383 . the relation between these two rather different - looking categories is a generalization of the so - called koszul duality for quadratic algebras @xcite . any serious attempt to discuss koszul duality would take us out of our depth , so we will just make a few remarks which may help to orient the reader who would like to study these questions deeper . classical koszul duality applies to quadratic algebras , i.e. @xmath13-graded algebras generated by degree-1 elements , such that all relations between generators are homogeneous quadratic . the basic example of a dual pair is the pair @xmath388 , where @xmath389 is the symmetric algebra of a finite - dimensional vector space @xmath368 , and @xmath267 is the exterior algebra of the dual vector space . the statement of koszul duality is that their derived categories of finitely - generated @xmath13-graded modules are equivalent . there is a generalization of koszul duality to the case where the relations are non - homogeneous quadratic @xcite . but the dual object in this case is not a graded algebra , but a quadratic cdg algebra . a cdg algebra is a triple @xmath390 , where @xmath391 is a graded algebra , @xmath376 is a degree-1 derivation , and @xmath392 is a degree-2 element @xmath392 such that @xmath393 $ ] for any @xmath394 . cdg means `` curved differential graded '' ; another name for a cdg algebra is a `` q - algebra '' @xcite . a module over a cdg algebra @xmath390 is a graded module @xmath395 over @xmath391 equipped with a degree-1 derivation @xmath396 such that @xmath397 for any @xmath398 . what we need is a @xmath14-graded version of non - homogeneous koszul duality . indeed , on one hand , the clifford algebra is a @xmath14-graded quadratic algebra , while on the other hand , the category @xmath282 can be regarded as a category of modules over a certain @xmath14-graded cdg algebra . this cdg algebra is purely even and isomorphic to @xmath275 as an algebra . the derivation @xmath376 is identically zero , but the even element @xmath392 is not : it is given by @xmath11 . the category @xmath383 can be regarded as the derived category of the category of finitely generated cdg modules over the cdg algebra @xmath399 . this cdg algebra is koszul - dual to the clifford algebra in the sense of refs . @xcite , and we expect that the corresponding derived categories of modules are equivalent . more precisely , we expect that the derived category of finite - dimensional @xmath14-graded clifford modules is equivalent to the derived category of finitely - generated modules over the cdg algebra @xmath399 . since the clifford algebra can be regarded as a deformation of the exterior algebra , and the cdg algebra @xmath399 is a deformation of the polynomial algebra , this claim looks like a generalization of the classic result of ref . in fact , the deformed duality is in some sense simpler than the classic one , since the category @xmath382 is semi - simple and `` deriving '' it is a trivial operation ( gives us back the same category ) . it is also more useful : while the classic duality of ref . @xcite reduced the problem of classifying coherent sheaves on @xmath1 to a _ very difficult _ problem in linear algebra , the deformed duality reduces the problem of classifying b - branes in the free massive lg model to a _ very simple _ problem in linear algebra ( classification of finite - dimensional graded modules over a clifford algebra . ) let us describe the functors which establish the equivalence of @xmath383 and @xmath382 . the first one , from @xmath383 to @xmath382 , is obvious : it takes an object @xmath22 of @xmath383 to @xmath400 , where @xmath401 is the object of @xmath383 described in the last paragraph of subsection [ sec : konts ] . the mapping of morphisms is the obvious one . to define the functor acting in the opposite direction , let us consider for any object @xmath395 of @xmath382 the vector space @xmath402 where @xmath275 is simply the algebra of polynomial functions on @xmath12 . since @xmath395 is @xmath14-graded , this vector space is also @xmath14-graded . it is also an @xmath275 module , for obvious reasons . it remains to define the twisted differential @xmath403 , i.e. an odd endomorphism of @xmath55 which squares to @xmath11 . let @xmath63 be the vector space which appears in the definition of the clifford algebra ; we will also identify the target space @xmath27 of the lg model with @xmath63 . the twisted differential will be @xmath404 where @xmath405 is a basis in @xmath63 , @xmath365 are the corresponding linear coordinates , and the dot denotes the clifford algebra action . it is easy to check that @xmath403 is odd , and that @xmath406 . thus we defined a map which sends an object of @xmath382 to an object of @xmath383 . the mapping of morphisms is the obvious one : if @xmath407 is a morphism of clifford modules @xmath395 and @xmath408 , then the corresponding element of @xmath409 is @xmath410 . it is easy to check that @xmath410 is closed , and thus is a well - defined morphism in the category @xmath383 . the claim is that compositions of these two functors in any order are isomorphic to identity functors . we will not try to prove this claim here , but to make it more plausible note that the mapping of objects is given by essentially the same formulas as in the classic case @xcite . the mirror of the the nonlinear sigma model with target @xmath10 is the affine @xmath411 toda model @xcite . the affine @xmath411 toda model is an @xmath412 landau - ginzburg theory of two chiral superfields @xmath413 and @xmath414 taking values in @xmath4 and a rational superpotential @xmath415 we can test the homological mirror symmetry conjecture by comparing the fukaya category of @xmath10 with the category of b - branes in the toda model . as discussed above , every b - brane in the toda model lies on some holomorphic curve @xmath416 . in addition , in order for open strings to have a supersymmetric ground state , we require this curve to pass through a critical point of @xmath11 . in the @xmath411 theory there are three distinct critical points : @xmath417 the values of @xmath11 corresponding to these critical points are pairwise distinct : @xmath418 . there is an obvious @xmath419 symmetry which permutes the critical points . this implies that the categories @xmath420 are all equivalent . from now on we will focus on one of them , say , the one corresponding to @xmath421 . all b - branes associated to this critical point must be complex submanifolds of the holomorphic curve in @xmath422 given by @xmath423 this curve is a singular cubic with a single node ( see fig . [ fig : cubic ] ) . thus the category of b - branes is a full sub - category of the category of b - branes in the lg model @xmath49 . we have seen that the latter is equivalent to the category @xmath424 . it remains to understand which objects in the latter category correspond to b - branes . clearly , the d0-brane sitting at the critical point @xmath425 is a valid b - brane . as for d2-branes , they must be ( desingularizations of the ) irreducible components of the curve eq . ( [ curve ] ) . but it is easy to see that the singular cubic is irreducible . thus there is only one d2-brane of type b : the one which corresponds to the structure sheaf of the desingularized cubic . the corresponding object in the `` local '' category associated to the critical point is the direct sum of the d2-brane @xmath68 and the d2-brane @xmath69 . this direct sum is isomorphic to the d0-brane ( see section ( [ sec : cp ] ) ) . we conclude that the basic b - brane in the toda model is the d0-brane , all other branes being direct sums of several copies of the d0-brane . the endomorphism algebra of the d0-brane is isomorphic to @xmath62 . we see that the category of b - branes in this case is strictly smaller than the `` local '' category @xmath426 which is equivalent to @xmath424 . as discussed in section [ sec : b ] , since the d0-brane looks like a composite of two d2-branes , one can formally add these missing d2-branes to the category of b - branes for the toda model . the enlarged category is equivalent to the category @xmath424 . now let us interpret these results from the point of view of homological mirror symmetry . the mirror of the d0-brane has been identified in ref . @xcite using the dualization argument of ref . the mirror is a certain lagrangian 2-torus in @xmath10 equipped with a rank one trivial vector bundle and a certain flat connection . let us be more specific . consider the unit 5-sphere in @xmath427 , i.e. a hypersurface defined by the equation @xmath428 the quotient of this 5-sphere by a free @xmath429 action @xmath430 is diffeomorphic to @xmath10 . in fact , the standard symplectic form on @xmath10 is obtained by restricting to @xmath431 the standard khler form on @xmath427 and then pushing it down to the quotient . now consider a 3-torus in @xmath427 defined by the equations latexmath:[\[\label{3torus } and invariant with respect to the @xmath429 action . hence by passing to the quotient , we obtain a 2-torus embedded in @xmath10 . it is trivial to check that this 2-torus is lagrangian with respect to the standard symplectic form on @xmath10 . the flat connection can be specified by its monodromy representation . let @xmath433 and @xmath434 be the loops on the 3-torus ( [ 3torus ] ) defined by @xmath435 their images under the quotient map generate the fundamental group of our lagrangian 2-torus . according to ref . @xcite , the mirror of the d0-brane sitting at the point @xmath436 @xmath437 corresponds to the monodromy representation which maps both generators to @xmath438 . in particular , the d0-brane which sits at the point @xmath425 is mirror to the trivial flat connection on the lagrangian 2-torus . as a simple check of this claim , note that the algebra of endomorphisms of a d0-brane in the model @xmath49 has euler characteristic zero . in the mirror picture , the corresponding object is the euler characteristic of the floer complex , which coincides with the classical euler characteristic of the 2-torus . thus the euler characteristics match . it would be nice to compute the floer homology groups as well and to check that they agree with the predictions of mirror symmetry . namely , we expect that 1 . the floer homology of the lagrangian 2-torus equipped with a rank - one flat connection is non - vanishing only for the three special flat connections defined above ; 2 . for these choices of the flat connection , the floer homology is isomorphic to the classical cohomology of the torus as a @xmath14-graded vector space ; 3 . as a @xmath14-graded algebra , the floer homology is isomorphic to the clifford algebra with two generators , i.e. it is a quantum deformation of the classical cohomology ring ; 4 . floer homology groups which compute morphisms between different flat connections of rank one vanish . it was argued above that we can formally add d2-branes to the category of b - branes . it is reasonable to ask if this procedure is consistent with or perhaps even forced on us by homological mirror symmetry . to answer this question we need to identify the mirrors of the added d2-branes . there are two such d2-branes for each critical level set . for each of them the euler characteristic of the endomorphism algebra is @xmath277 . if we assume that the mirror of a d2-brane is a lagrangian submanifold , then it must be homeomorphic to a real projective plane @xmath439 . but since @xmath439 is not orientable , it is not an admissible object of the fukaya category ( one needs orientability in order to define @xmath14-graded maslov index and @xmath14-grading on the floer complex ) . we conclude that the mirrors of the added d2-brane can not be lagrangian submanifolds , and therefore homological mirror symmetry does not force us to include them on the b - side . on the other other hand , if we added d2-branes on the b - side , we can maintain homological mirror symmetry by adding certain objects on the a - side . in other words , we would like to regard the lagrangian 2-torus with a trivial flat connection , which is mirror to the d0-brane , as a direct sum of two irreducible objects , which are mirror to the d2-branes . but since there are no such objects in the fukaya category , we simply add these direct summands `` by hand . '' let us clarify what we mean by adding direct summands `` by hand . '' let @xmath37 be an object of an additive category @xmath440 . a projector is an element of @xmath441 which satisfies @xmath442 . given any projector , we would like to have the corresponding direct summand , i.e. an object @xmath443 and a pair of morphisms @xmath444 and @xmath445 such that @xmath446 and @xmath447 . if @xmath443 does not exist for all projectors and for all @xmath37 , then we look for the smallest additive category which contains @xmath440 as a full subcategory and in which every projector has a direct summand . to summarize , to maintain homological mirror symmetry , we must either add formal direct summands on both a and b sides , or on neither side . the mirror in this case is the lg model with target @xmath448 and the superpotential @xmath449 here @xmath450 and @xmath451 are nonzero complex numbers whose logarithms are mirror to the periods of the complexified khler form on the two @xmath9 s . this superpotential has four non - degenerate critical points . for generic @xmath452 there are four critical level sets all of which look like a cubic with a node . thus we are in exactly the same situation as in the previous subsection , and the only b - branes are d0-branes sitting at the critical points . another way to see these d0-branes is to note that the lg model is a product of two lg models with the superpotential @xmath453 . this model is mirror to @xmath9 and has been studied in ref . its only b - branes are d0-branes sitting at the two critical points of the superpotential . taking tensor products of pairs of such b - branes gives us four d0-branes discussed above . the mirror of each d0-brane is a lagrangian 2-torus with some flat connection . indeed , the mirror of a d0-brane in the model @xmath453 is the equatorial circle on @xmath9 @xcite , therefore the mirror of a d0-brane in the product model is the product of two equatorial circles . the monodromy around the two generators of the fundamental group is @xmath454 . for @xmath455 something special happens both on the a and b sides . on the a side , we get a new lagrangian submanifold which is homeomorphic to a 2-sphere . to see this , let @xmath353 and @xmath456 be coordinates on the standard affine patches on the two @xmath9 s . consider the `` anti - diagonal '' 2-sphere given by @xmath457 . let @xmath458 be the fubini - study form on @xmath9 , @xmath459 be the projection maps from @xmath19 to the two factors , and @xmath460 be complex numbers . it is trivial to check the restriction of @xmath461 to the `` anti - diagonal '' 2-sphere vanishes if and only if @xmath462 . thus the 2-sphere is lagrangian if and only if @xmath455 . on the b side , the critical level set @xmath66 now contains two critical points . the equation of this critical level set @xmath463 shows that it is reducible . the irreducible components are a line and a non - singular quadric which intersect transversally at two points ; these are the two critical points mentioned above . we have two irreducible d2-branes of type b corresponding to the two irreducible components of the critical level set . it is easy to see that one is isomorphic to the shift of the other , while their sum is isomorphic to the sum of two d0-branes sitting at the two critical points . note that this is another example where the category of b - branes is strictly smaller than the sum of `` local '' categories associated to critical points . this happens because all d2-branes pass through both critical points in the set @xmath66 . thus a single d0-brane sitting at a critical point is irreducible . of course , if we only look at the infinitesimal neighborhood of one of the critical points , then we are in the same situation as in the model @xmath49 , and the d0-brane appears to be composite . if desired , we can enlarge the category of b - branes by adding all formal direct summands . then it will become equivalent to the sum of categories attached to the two critical points ( each of which is equivalent to @xmath424 ) , and each d0-brane will be the sum of two irreducible objects . now let us match the objects on a and b sides . d0-branes correspond to `` equatorial '' lagrangian tori , as before . the mirror of a d2-brane must be a lagrangian 2-sphere . indeed , each d2-brane passes through two critical points , each of which contributes @xmath277 to the euler characteristic of the endomorphism algebra . an obvious conjecture is that the two d2-branes are mirror to the lagrangian 2-sphere discussed above and its shift ( i.e. orientation - reversal ) . if this is true , then the sum of the lagrangian 2-sphere and its shift must be isomorphic ( in the fukaya category ) to the sum of two `` equatorial '' lagrangian tori with monodromies @xmath464 and @xmath465 . it would be interesting to check this by computing the floer homology between all the objects involved . in this paper we have described the category of b - branes for the free massive lg model with @xmath2 chiral fields . we also argued that this allows one to determine the category of b - branes for an arbitrary massive lg model . the most striking feature of our results is their simplicity . for example , if we consider the free massive lg model , there is a multi - parameter family of maximal isotropic subspaces of the quadric @xmath66 , but they are all isomorphic as objects of the category of b - branes ( up to a shift ) . moreover , b - branes of lower dimension , including the d0-brane , are isomorphic to direct sums of b - branes of maximal dimension . these rather counter - intuitive observations solve the problem of computing tree - level topological open string correlators in these models . it is interesting to compare our results with those of refs . @xcite , where a general framework for classifying d - branes in 2d topological field theories has been proposed . in our case , the 2d tft in the bulk is rather trivial : it is isomorphic , as a frobenius algebra , to @xmath43 with its unique frobenius structure . the theory of ref . @xcite ( generalized to the @xmath14-graded case ) tells us that the algebra of open strings connecting a brane with itself must be simple . we saw that in our case endomorphism algebras of b - branes are all isomorphic to clifford algebras , and these are indeed simple ( as @xmath14-graded algebras ) . however , unlike in the purely bosonic case , in the @xmath14-graded case not every two simple finite - dimensional algebras are morita equivalent . in fact , there are two morita - equivalence classes of such algebras , represented by @xmath43 and @xmath54 . a clifford algebra with @xmath21 generators is morita - equivalent to @xmath43 or @xmath54 depending on whether @xmath21 is even or odd . we have seen that when the number of fields @xmath2 is even ( resp . odd ) only even ( resp . odd ) values of @xmath21 occur . this suggests that it is impossible to have a topological open string theory which includes d - branes of both kinds . indeed , we have seen that all pairings between spaces of morphisms induced by 2-point correlators are either even or odd , depending on whether @xmath2 is even or odd . on the other hand , one of the basic axioms of topological open string theory is that all pairings must have the same parity @xcite . this observation provides a simple counter - example to the belief that a 2d tft determines uniquely the associated category of topological boundary conditions . in fact , we can make a stronger statement . given any 2d scft representing a superstring background , we can tensor it with the topological lg model @xmath50 . since the latter theory is trivial , this does not change the closed string sector . but the open string sector does change : one has to tensor every `` physical '' d - brane with the d0-brane of the lg model , and this results in tensoring the open string spectrum of each d - brane with @xmath54 . this is equivalent to the introduction of an odd chan - paton label . thus we have two inequivalent open string theories for a given closed string theory . this particular ambiguity is rather mild and can be easily eliminated . the difference between odd and even @xmath2 comes from the number of fermionic zero modes on a disk , or equivalently from the parity of the bilinear forms computed by the 2-point disk correlators . thus to specify completely the open string theory we are dealing with , it is sufficient to fix the parity of all bilinear forms . it would be interesting to extend the considerations of this paper to lg models which flow to non - trivial scfts in the infrared limit . for example , one could study landau - ginzburg realizations of @xmath0 minimal models . for these theories much information about b - branes is available from the boundary state formalism , and it would interesting to see if it is consistent with kontsevich s proposal . by analogy with the massive case , one expects that the category of b - branes will be describable in terms of modules over the algebra of endomorphisms of a d0-brane . from the mathematical viewpoint , this algebra must be related by a koszul - like duality to the cdg algebra which appears in kontsevich s proposal . it appears that for @xmath11 of degree higher than two koszul duality relates kontsevich s cdg algebra to a finite - dimensional @xmath7-algebra @xcite . b - branes should correspond to finite - dimensional @xmath7-modules over this @xmath7-algebra . in this way solving topological open string theory is reduced to a problem in linear algebra . hopefully , the latter problem is manageable . in the axiomatic approach of refs . @xcite , topologically twisted @xmath0 minimal models correspond to non - semi - simple frobenius algebras . it would be interesting to explore the uniqueness of the open string sector in such models . in this appendix we collect some well - known facts about complex clifford algebras and their modules . let @xmath63 be a complex vector space of dimension @xmath2 , and @xmath113 be a non - degenerate symmetric bilinear form on @xmath63 . clifford algebra @xmath181 has @xmath63 and the identity as its set of generators , and the following relations : @xmath467 as a vector space , @xmath181 is isomorphic to @xmath267 and therefore has dimension @xmath468 . we can regard @xmath181 either as an ordinary associative algebra , or as a @xmath14-graded algebra , such that all the generators are odd . in the latter case , the grading corresponds to the decomposition of @xmath267 into polyvectors of even and odd degree . since the isomorphism class of @xmath181 depends only on the dimension @xmath2 of @xmath63 , we will also use the notation @xmath363 to denote this isomorphism class . if @xmath469 and @xmath470 are complex vector spaces with non - degenerate bilinear forms @xmath471 and @xmath472 , then @xmath473 here all clifford algebras are regarded as @xmath14-graded algebras , and @xmath474 denotes their @xmath14-graded tensor product . if @xmath2 is even , then @xmath181 regarded as an ungraded algebra is isomorphic to the algebra of complex @xmath475 matrices , which we will denote @xmath476 . if @xmath2 is odd , then @xmath181 regarded as an ungraded algebra is isomorphic to @xmath477},{{\mathbb c}}){\oplus}mat(2^{[n/2]},{{\mathbb c}})$ ] . in particular , @xmath54 is isomorphic to @xmath478 . we see that @xmath181 is a simple algebra only for even @xmath2 . however , if we regard it as a @xmath14-graded algebra , then it is simple for all @xmath2 . now let us discuss finite - dimensional modules over @xmath181 . the category of clifford modules is semi - simple , i.e. every exact sequence splits . thus every clifford module is a direct sum of irreducible modules . the number and properties of irreducible modules depend on the parity of @xmath2 , as well as whether we regard @xmath181 as a @xmath14-graded algebra . if we neglect the grading , then for even @xmath2 we have a unique irreducible module @xmath479 of dimension @xmath480 . it is called the spinor module and can be constructed as follows . pick a pair of subspaces @xmath481 of @xmath63 such that both @xmath249 and @xmath11 are isotropic with respect to @xmath113 , @xmath482 , and @xmath483 . one can easily see that @xmath113 gives a non - degenerate pairing between @xmath249 and @xmath11 and thus we may identify @xmath11 with @xmath484 . set @xmath485 , and define the action of clifford algebra on @xmath479 as follows : if @xmath486 where @xmath487 and @xmath488 , then for any @xmath489 we let @xmath490 here we used the identification of @xmath11 with @xmath484 mentioned above . for odd @xmath2 clifford algebra is a sum of two matrix algebras , and therefore there are two non - isomorphic irreducible modules of dimension @xmath273}$ ] ( two spinor modules ) . for example , for @xmath47 the algebra is generated by the identity and an odd element @xmath491 with a single relation @xmath492 ; the two irreducible modules are one - dimensional , with the action of @xmath491 given by @xmath493 . for general @xmath2 one can use the property eq . ( [ tensorcl ] ) to reduce the problem to the cases already considered . if we regard @xmath181 as a @xmath14-graded algebra , then we should look for @xmath14-graded irreducible modules . for even @xmath2 there are two inequivalent choices of grading on the spinor module related by parity reversal . therefore we have two non - isomorphic irreducible spinor modules @xmath479 and @xmath494 . for odd @xmath2 the `` minimal '' @xmath14-graded module has dimension @xmath495 ; as an ungraded module , it is isomorphic to the direct sum of two inequivalent irreducible ungraded modules . furthermore , the choice of grading is unique up to isomorphism . we will denote this unique spinor module by @xmath479 . for example , for @xmath47 @xmath496 , and @xmath491 acts as any of the three pauli matrices , say @xmath497 . then the parity operator can be chosen to be @xmath498 . to summarize , for even @xmath2 any @xmath14-graded clifford module is a direct sum of several copies of two inequivalent spinor modules @xmath479 and @xmath494 . for odd @xmath2 the situation is the same , except that @xmath479 is isomorphic to @xmath494 . the dimension of the spinor module is given by @xmath499}$ ] for any @xmath500 . in particular , @xmath181 regarded as a left module over itself is a direct sum of @xmath273}$ ] copies of spinor modules . for @xmath2 even half of them are @xmath479 s , and the other half are @xmath494 s . we are deeply grateful to maxim kontsevich for sharing with us his ideas about b - branes in landau - ginzburg models , and to alexander polishchuk for pointing out the relevance of non - homogeneous koszul duality . we also thank kentaro hori and dmitri orlov for reading a preliminary draft of the paper and making a number of valuable comments . the first author would like to thank institut des hautes tudes scientifiques for hospitality during the writing of this paper . this work was supported in part by the doe grant de - fg03 - 92-er40701 .
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we study topological d - branes of type b in @xmath0 landau - ginzburg models , focusing on the case where all vacua have a mass gap . in general ,
tree - level topological string theory in the presence of topological d - branes is described mathematically in terms of a triangulated category .
for example , it has been argued that b - branes for an @xmath0 sigma - model with a calabi - yau target space are described by the derived category of coherent sheaves on this space .
m. kontsevich previously proposed a candidate category for b - branes in @xmath0 landau - ginzburg models , and our computations confirm this proposal .
we also give a heuristic physical derivation of the proposal .
assuming its validity , we can completely describe the category of b - branes in an arbitrary massive landau - ginzburg model in terms of modules over a clifford algebra .
assuming in addition homological mirror symmetry , our results enable one to compute the fukaya category for a large class of fano varieties .
we also provide a ( somewhat trivial ) counter - example to the hypothesis that given a closed string background there is a unique set of d - branes consistent with it .
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ngc6712 is a small ( tidal radius @xmath10 ) and relatively loose ( @xmath11 ) and faint ( @xmath12 ; djorgovski @xcite ) galactic globular cluster ( gc ) that has not yet received much observational attention . its main claim to fame so far is due to the presence in its core of the high luminosity x - ray burster x1850086 whose optical counterpart may be a faint uv - excess object ( anderson et al . this fact presents somewhat of a puzzle since one would expect such an x - ray source to be located in a highly concentrated cluster where the stellar density favors its formation via tidal capture of a neutron star ( hertz & grindlay @xcite ) . most other sources of this type have indeed been found in high density core collapse clusters suggesting that , perhaps , ngc6712 has already undergone such an event in the past and is now in a state of re - expansion ( grindlay et al . @xcite ) . this unusual situation may also be connected in some way to its galactic orbit as computed recently by dauphole et al.(@xcite ) that is fairly well restricted to the vicinity of the disk and penetrates very deeply in the galactic bulge . this certainly means that one would expect this cluster to have undergone severe tidal shocking during the numerous encounters with both the disk and the bulge during its lifetime and the consequences on the dynamical status of the cluster to be significant and observable . a simple single - mass approximation of these effects was computed by gnedin & ostriker ( @xcite ) for both disk and bulge shocks under differing assumptions on the galactic model with a resultant time to destruction as small as @xmath13h@xmath14 . according to these calculations , then , the cluster should have evaporated long ago and at the very least may have lost a very substantial portion of its original mass during its lifetime . clearly , this catastrophe should be well impressed on its present day distribution of stars on the main sequence ( ms ) with its lowest mass members beyond the half - mass radius particularly vulnerable to escape . this effect may well have been detected already in m4 , another cluster at significant risk of tidal disruption ( kanatas et al . @xcite ; pulone et al . @xcite ) , but until this cluster s structural parameters are pinned down more accurately this remains still speculative . there is , therefore , much interest today in determining accurately the present day mass function ( pdmf ) of ngc6712 to look for the signature of such powerful effects . currently available observations of the color magnitude diagram ( cmd ) of this cluster , however , only reach to just above the ms turn - off ( cudworth @xcite ; anderson et al . @xcite ) and are , therefore , of limited use for this task . in order to push the observations well into the relevant part of the ms below the turn - off ( to ) , the vlt was used to probe deeply into this cluster with its unprecedented sensitivity and resolution . this paper describes the first results of these observations that give clear evidence that there is indeed a distortion of the mf of ngc6712 with respect to that of other dynamically much less disturbed clusters . the observational data used in this paper were collected during the science verification ( sv ; leibundgut & renzini @xcite ) phase of the first 8m - diameter very large telescope ( vlt ) at eso , using the vlt test camera ( vlt - tc ) . readers interested in the vlt and its instruments should consult eso s world - wide web at http://www.eso.org/paranal , while the scope of the vlt science verification is described in leibundgut , de marchi , & renzini ( @xcite ) . images of the globular cluster ngc6712 were taken with the vlt - tc in the v and r bands . with a @xmath15 square pixel detector and a plate scale of @xmath16pixel@xmath17 , the vlt - tc offers a field of view of @xmath18 . sv observations , however , were obtained with an electronically enforced @xmath19 binning of the ccd , so that the actual size of each pixel in these images is of @xmath20 on a side . observations of four regions of the cluster are available , located between one and two times the half - light radius ( @xmath21 ; djorgovski @xcite ) . the coordinates ( j2000 ) of the center of each field are given in table1 along with the total exposure time in each band . fields f1 and f2 were observed during the night of 1998 aug 23 , and f3 and f4 on 1998 aug 27 . .vlt sv observations of ngc6712 [ cols= " < , < , < , > , > , > " , ] the lfs measured in this way and corrected for photometric incompleteness are shown in figure3 as a function of the r - band magnitude . the lfs have been registered through a vertical shift in the logarithmic plane by imposing a least square fit in the range @xmath22 . the error bars associated with each point reflect the poisson statistics of the counting process ( and include the correction for incompleteness ) . these lfs can be directly compared to one another as they have all been measured at the same radial distance from the center ( @xmath23 or @xmath24 times the half - light radius @xmath25 ) . they show the same overall trend , i.e. an increase with decreasing luminosity up to a peak at @xmath26 ( close to the to luminosity ) , and from there they all flatten out and possibly drop with decreasing luminosity even after the incompleteness of our photometry has been accounted for . and indeed , to ensure that our lfs are robust , we have not included in figure3 any datapoints whose associated photometric completeness is worse than @xmath27 . stars brighter than @xmath28 have already evolved off the ms and , therefore , their lf provides no information on the underlying mf without uncertain corrections for evolution ( scalo @xcite ) . moreover , because of saturation at the bright end of our cmds , the brightest portion of our lfs is uncertain . for cluster stars which are still on their ms , however , the lfs in figure3 directly reflect the pdmf of the local population and immediately indicate a relative deficiency of low mass objects with respect to the stars with the to mass ( @xmath29m@xmath7 , ) , as we discuss below . indeed , the most important conclusion that one can draw from figure3 is that the shape of the lfs completely deviates from that of any other gc for which relatively deep photometric data are available near the half - mass radius . observations carried out with the wfpc2 on board the hst over the past few years ( paresce , de marchi & romaniello @xcite ; cool , piotto , & king @xcite ; elson et al . @xcite ; de marchi & paresce @xcite ; piotto , cool , & king @xcite ; pulone et al . @xcite ; king et al . @xcite ; de marchi @xcite ) have consistently revealed lfs that , near the cluster half - mass radius , increase with decreasing luminosity from the to magnitude all the way down to about @xmath30 ( @xmath31m@xmath7 , ) where they flatten out and drop at fainter luminosities . inverted lfs such as those shown in figure3 have been observed right in the core of high density gcs ( 47tuc , ngc6397 , m15 ) but in those cases a simple isothermal model of a cluster in equilibrium can easily explain this effect as being due to mass segregation ( paresce , de marchi , & jedrzejewski @xcite ; king , sosin , & cool @xcite ; de marchi & paresce @xcite ) . more complete multi - mass king michie models show , however , that thermal relaxation is much less efficient ( if at all ) at depleting low - mass stars near the half - mass radius ( see pulone , de marchi , & paresce @xcite ) , and we can not therefore trace the origin of the lfs that we observe back to the effects of mass segregation alone . to make it easier to compare the lf of ngc6712 with that of other clusters , we display it in figure4 as a function of the absolute r - band magnitude , assuming @xmath32 and @xmath33 or @xmath34 ( djorgovski @xcite ) . rather than showing the three individual lfs , we have combined them together into one single function by averaging their values in each magnitude bin , and have taken the standard deviation as a measure of the associated uncertainty ( error bars ) . the dashed line shown in figure4 corresponds to the lf of the low - metallicity cluster ngc6397 as measured by king et al . ( 1998 ) , while the dot - dashed line reproduces the lf of the metal rich cluster 47tuc from hesser et al . ( @xcite ) . both lfs have been translated into the r - band by using the m l relationship of baraffe et al . ( @xcite ) for the appropriate metallicity , i.e. the magnitude corresponding to each observed point in the lf has been converted into a mass which has then been used to read the corresponding magnitude in the r band from the appropriate m - l relation . the size of each magnitude bin has also been rescaled to reflect the difference in the slopes of the m - l relationships for different bands . we have selected ngc6397 and 47tuc as they both have accurate lf measurements at and below the to luminosity , where we have normalized them to our observations , and because the metal content of these clusters nicely brackets that of ngc6712 ( [ fe / h]@xmath35 ; zinn & west @xcite ) . it should , nevertheless , be clear that , due the uncertainties in the theoretical m - l relations and in the observed lfs , our comparison will only provide an indication of the true differences . the difference between these two lfs and that of ngc6712 is striking . while the lfs of ngc6397 , measured at @xmath36 , shows a steep increase starting from the to , the lf of ngc6712 sampled at @xmath37 slowly drops from the to all the way to the detection limit at @xmath38 . we would like to point out that the discrepancy is so large that to bring the two lfs into agreement would require us to have underestimated the photometric incompleteness by a factor of @xmath39 . the same reasoning holds true for the lf of 47tuc , which has been measured at @xmath40 . this difference must thus be physical and reflect the properties of the local stellar population . under the simple assumption that the mf should be represented by an exponential distribution in the mass range @xmath41m@xmath7 , ( a reasonable hypothesis given the narrow mass range ) , we have used the m l relationship of baraffe et al . ( @xcite ) appropriate for the metallicity of ngc6712 to reproduce the observed lf . we obtain a fairly reasonable fit to the observations with a power - law distribution of the type @xmath42 ( salpeter s imf would be @xmath43 ) , in which the number of objects decreases with mass ( solid line in figure4 ) . richer et al . ( @xcite ) and , more recently , de marchi & paresce ( 1997 ) , vesperini & heggie ( @xcite ) , and pulone et al . ( @xcite ) have convincingly shown that near the cluster half - mass radius the lf should closely reflect the imf , as dynamical modifications should leave these regions almost untouched . in fact , the internal relaxation mechanism governed by energy equipartition through two- and three - body encounters mostly affects the region within a few core radii , while the interaction with the galactic tidal field is expected to simply speed up the evaporation of light stars near the tidal boundary , but none of these processes should , in principle , significantly alter the properties of stars located close to the much safer half - mass radius area . if this were true for ngc6712 as well , one should conclude that this cluster is the only one so far to feature an inverse imf ( increasing with mass ) that has not been observed in any other environment . while this hypothesis can not be safely ruled out , there are far better reasons to believe that ngc6712 might have experienced a much stronger interaction with the galaxy than any other of the clusters studied so far . and indeed , with a perigalactic distance smaller than 300pc this cluster ventures so frequently and so deeply into the galactic bulge ( dauphole et al . @xcite ) that it is likely to have undergone severe tidal shocking during the numerous encounters with both the disk and the bulge during its lifetime . the latest galactic plane crossing could have happened as recently as @xmath44year ago ( cudworth @xcite ) which is much smaller than its half - mass relaxation time ( @xmath45yr ) . such an event might have imparted strong modifications on the mass distribution not only of the stars in the cluster periphery but also well into its innermost regions , perhaps even reaching the core where it could have triggered a premature collapse because of tidally induced relaxation ( see kundi & ostriker @xcite and gnedin & ostriker @xcite for a detailed description of this mechanism ) . as a result of such a catastrophe , it would be surprising if the present - day mf were still to bear any memory of its parent imf anywhere in the cluster , including the half - mass radius region . vesperini & heggie ( @xcite ) have estimated that these effects would substantially decrease the slope of a simple power law mf , much in the same way as we are observing here . we , therefore , conclude that the vlt has revealed the consequences of the strong tidal stripping that the galaxy ( and particularly its bulge ) exerts on gcs orbiting close to the center , and which might have contributed to the destruction of an initially much more numerous population of gcs ( aguilar , hut , & ostriker @xcite ; vesperini @xcite ) . although kanatas et al . ( @xcite ) and piotto et al . ( @xcite ) had speculated that similar events could have happened respectively in m4 and ngc6397 , the result that we show here is the first , clear , unambiguous detection of this mechanism . to characterize the strength and extent of these phenomena more accurately would require the investigation of the ms population outside the half - mass radius in many more clusters , and possibly at larger distance from the galactic center , so as to probe the intensity of the stripping process as a function of the depth of the galactic potential well . if the z component of the space velocity of ngc6712 is indeed appropriate for a halo cluster , as suggested by cudworth ( @xcite ) , then this violent stripping process might not be restricted only to objects orbiting the innermost galactic regions . grindlay , j. , bailyn , c. , mathieu , r. , & latham , d. 1988 , in the harlow shapley symposium on globular cluster systems in galaxies , iau symp . j. grindlay & a. davis philip ( dordrecht : kluwer ) , 659
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the vlt on cerro paranal was used to observe four fields located at @xmath0 from the center of the galactic globular cluster ngc6712 in the v and r bands .
the resulting color - magnitude diagram shows a well defined main sequence reaching down to the @xmath1 detection limit at @xmath2 , @xmath3 or approximately 4 magnitudes below the main sequence turn - off , the deepest obtained so far on this cluster .
this yields a main sequence luminosity function that peaks at @xmath4 and drops down to the 50% completeness limit at @xmath5 .
transformation to a mass function via the latest mass - luminosity relation appropriate to this object indicates that the peak of the luminosity function corresponds to @xmath6m@xmath7 , a value significantly higher than the @xmath8m@xmath7 , measured for most other clusters observed so far .
since this object , in its galactic orbit , penetrates very deeply into the galactic bulge with perigalactic distance of @xmath9kpc , this result is the first strong evidence that tidal forces have stripped this cluster of a substantial portion of its lower mass star population all the way down to its half - light radius and possibly beyond .
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the last decade brought a dynamic evolution of the computing capabilities of graphics processing units ( gpus ) . in that time , the performance of a single card increased from tens of gflops in nvxx to tflops in the newest kepler / maxwell nvidia chips @xcite . this raw processing power did not go unnoticed by the engineering and science communities , which started applying gpus to accelerate a wide array of calculations in what became known as gpgpu general - purpose computing on gpus . this led to the development of special gpu variants optimized for high performance computing ( e.g. the nvidia tesla line ) , but it should be noted that even commodity graphics cards , such as those from the nvidia geforce series , still provide enormous computational power and can be a very economical ( both from the monetary and energy consumption point of view ) alternative to large cpu clusters . the spread of gpgpu techniques was further facilitated by the development of cuda and opencl parallel programming paradigms allowing efficient exploitation of the available gpu compute power without exposing the programmer to too many low - level details of the underlying hardware . gpus were used successfully to accelerate many problems , e.g. the numerical solution of stochastic differential equations @xcite , fluid simulations with the lattice boltzmann method @xcite , molecular dynamics simulations @xcite , classical @xcite and quantum monte carlo @xcite simulations , exact diagonalization of the hubbard model @xcite , _ etc_. parallel computing in general , and its realization in gpus in particular , can also be extremely useful in many fields of solid state physics . for a large number of problems , the ground state of the system and its free energy are of special interest . for instance , in order to determine the phase diagram of a model , free energy has to be calculated for a large number of points in the parameter space . in this paper , we address this very issue and illustrate it on a concrete example of a superconducting system with an oscillating order parameter ( op ) , specifically an iron - based multi - band superconductor ( fesc ) . our algorithm is not limited to systems of this type and can also be used for systems in the homogeneous superconducting state ( bcs ) . the discovery of high temperature superconductivity in fesc @xcite began a period of intense experimental and theoretical research . @xcite all fesc include a two - dimensional structure which is shown in fig . [ fig.feas].a . the fermi surfaces ( fs ) in fesc are composed of hole - like fermi pockets ( around the @xmath1 point ) and electron - like fermi pockets ( around the @xmath2 point ) fig . [ fig.feas].b . moreover , in fesc we expect the presence of @xmath3 symmetry of the superconducting op . @xcite in this case the op exhibits a sign reversal between the hole pockets and electron pockets . for one @xmath4 ion in the unit cell , the op is proportional to @xmath5 . layers in fesc are built by @xmath4 ions ( red dots ) forming a square lattice surrounded by @xmath6 ions ( green dots ) which also form a square lattice . @xmath6 ions are placed above or under the centers of the squares formed by @xmath4 . this leads to two inequivalent positions of @xmath4 atoms , so that there are two ions of @xmath4 and @xmath6 in an elementary cell . ( panel b ) true ( folded ) fermi surface in the first brillouin zone for two @xmath4 ions in unit cell . the colors blue , red and green correspond to the fs for the 1st , 2nd , and 3rd band , respectively . ] fesc systems show complex low - energy band structures , which have been extensively studied . @xcite a consequence of this is a more sensitive dependence of the fs to doping . @xcite in the superconducting state , the gap is found to be on the order of 10 mev , small relative to the breadth of the band . @xcite this increases the required accuracy of calculated physical quantities needed to determine the phase diagram of the superconducting state , such as free energy . @xcite in this paper we show how the increased computational cost of obtaining thermodynamically reliable results can be offset by parallelizing the most demanding routines using cuda , after a suitable transformation of variables to decouple the interacting degrees of freedom . in section [ sec.theory_ph ] we discuss the theoretical background of numerical calculations . in section [ sec.algorithm ] we describe the implementation of the algorithm and compare its performance when executed on the cpu and gpu . we summarize the results in section [ sec.summary ] . many theoretical models of fesc systems have been proposed , with two @xcite , three @xcite , four @xcite and five bands @xcite . most of the models mentioned describe one ` fe ` unit cell and closely approximate the band and fs structure ( fig [ fig.feas].b ) obtained by lda calculations . @xcite in every model the non - interacting tight - binding hamiltonian of fesc in momentum space can be described by : @xmath7 where @xmath8 is the creation ( annihilation ) operator for a spin @xmath9 electron of momentum @xmath10 in the orbital @xmath11 ( the set of orbitals is model dependent ) . the hopping matrix elements @xmath12 determine the model of fesc . here , @xmath13 is the chemical potential and @xmath14 is an external magnetic field parallel to the ` feas ` layers . for our analysis we have chosen the minimal two - band model proposed by raghu _ et al . _ @xcite and the three - band model proposed by daghofer _ _ @xcite ( described in [ app.twoband ] and [ app.threeband ] respectively ) . the band structure and fs of the fesc system can be reconstructed by diagonalizing the hamiltonian @xmath15 : @xmath16 where @xmath17 is the creation ( annihilation ) operator for a spin @xmath9 electron of momentum @xmath18 in the band @xmath19 . [ [ superconductivity - in - multi - band - iron - base - systems - in - high - magnetic - fields ] ] superconductivity in multi - band iron - base systems in high magnetic fields + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + fesc superconductors are layered @xcite , clean @xcite materials with a relatively high maki parameter @xmath20 . @xcite all of the features are shared with heavy fermion systems , in which strong indications exist to observe the fulde ovchinnikov ( fflo ) phase @xcite a superconducting phase with an oscillating order parameter in real space , caused by the non - zero value of the total momentum of cooper pairs . in contrast to the bcs state where cooper pairs form a singlet state @xmath21 , the fflo phase is formed by pairing states @xmath22 . these states can occur between the zeeman - split parts of the fermi surface in a high external magnetic field ( when the paramagnetic pair - breaking effects are smaller than the diamagnetic pair - breaking effects ) . @xcite in one - band materials , the fflo can be stabilized by anisotropies of the fermi - surface and of the unconventional gap function , @xcite by pair hopping interaction @xcite or , in systems with nonstandard quasiparticles , with spin - dependent mass . @xcite this phase can be also realized in inhomogeneous systems in the presence of impurities @xcite or spin density waves @xcite . in some situations , the fflo can be also stable in the absence of an external magnetic field . @xcite in multi - band systems , the experimental @xcite and theoretical @xcite works point to the existence of the fflo phase in fesc . through the analysis of the cooper pair susceptibility in the minimal two - band model of fesc , such systems are shown to support the existence of an fflo phase , regardless of the exhibited op symmetry . it should be noted that the state with nonzero cooper pair momentum , in fesc superconductors with the @xmath3 symmetry , is the ground state of the system near the pauli limit . @xcite this holds true also for the three - band model ( e.g. [ app.suscept ] and ref . @xcite ) . [ [ free - energy - for - intra - band - superconducting - phase ] ] free energy for intra - band superconducting phase + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + in absence of inter - band interactions , the bcs and the fflo phase ( with cooper pairs with total momentum @xmath23 equal zero and non - zero respectively ) can be described by the effective hamiltonian : @xmath24 where @xmath25 is the amplitude of the op for cooper pairs with total momentum @xmath26 ( in band @xmath19 with symmetry described by the form factor @xmath27 for more details see ref . @xcite ) . using the bogoliubov transformation we can find the eigenvalues of the full hamiltonian @xmath28 : @xmath29 in this case we formally describe two independent bands . the total free energy for the system is given by @xmath30 , where latexmath:[\[\begin{aligned } \label{eq.freeene } \omega_{\varepsilon } = - \frac{1}{\beta } \sum_{\alpha \in \{+ , -\ } } \sum_{\bm k } \ln \left ( 1 + \exp ( - \beta \mathcal{e}_{\varepsilon{\bm k}}^{\alpha } ) \right ) + \sum_{\bm k } \left ( e_{\varepsilon{\bm k } \downarrow } - \frac { 2 | \delta_{\varepsilon } the free energy in @xmath19-th band , where @xmath32 is the respective interaction intensity and @xmath33 . for different parameters @xmath32 in @xmath34 results for @xmath35 ( panels a , c and e ) and @xmath36 ( panels b , d and f ) . ] [ [ historical - and - technical - note ] ] historical and technical note + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the historically basic concept of the fflo phase was simultaneously proposed by two independent groups , fulde - ferrell @xcite and larkin - ovchinnikov @xcite in 1964 . the first group proposed a superconducting phase where cooper pairs have only one non - zero total momentum @xmath37 , and the superconducting order parameter in real space @xmath38 . in the second case , cooper pairs have two possible momenta : @xmath37 and the opposite @xmath39 , with an equal amplitude of the order parameter . thus in real space the superconducting order parameter is given by @xmath40 . however , the most general case of fflo is a superconducting order parameter given by a sum of plane waves , where the cooper pairs have all compatible values of the momentum @xmath41 in the system : @xmath42 where @xmath43 is the cardinality of the first brillouin zone ( in the square lattice it is equal to @xmath44 ) . for the historical reasons described above , whenever @xmath45 ( @xmath46 and @xmath47 ) we can speak about the fulde ferrell ( ff ) phase , whereas for @xmath48 ( and @xmath49 , @xmath50 ) about the larkin ovchinnikov ( lo ) phase . larger @xmath43 impose a more demanding spatial decomposition of the order parameter , both in the theoretical and computational sense . however , every time it can be reduced to the diagonalization of the ( block ) matrix representation of the hamiltonian . using the translational symmetry of the lattice , the problem for the ff phase ( @xmath51 ) in one - band systems corresponds to the independent diagonalization of @xmath52 matrices ( with eigenvalues given like in eq . [ eq.enequasiparticle ] with the number of bands @xmath53 ) for each of the @xmath44 different momentum sectors . in case of the lo phase ( @xmath54 ) , the calculation can be similarly decomposed in momentum space or using other spatial symmetries of the system ( an example of this procedure can be found in ref . @xcite ) , with a much greater computational effort due to the lower degree of symmetry , leading to @xmath55 independent diagonalization problems of size @xmath56 . in the _ full _ fflo phase ( i.e. in a system with impurities @xcite or a vortex lattice @xcite ) , the spatial decomposition is determined in real space using the self - consistent bogoliubov - de gennes equations , which require the full diagonalization of a hamiltonian of maximal rank @xmath57 @xcite at every self - consistent step . to work around these limitations , iterative methods @xcite or the kernel polynomial method @xcite can be used . these methods are based on the idea of expressing functions of the energy spectrum in an orthogonal basis , e.g. chebyshev polynomial expansion . @xcite by doing so , it becomes possible to conduct self - consistent calculations in the superconducting state without performing the diagonalization procedure . the time expense of iterative methods can also be reduced by a careful gpu implementation , which is currently a work in progress . in the present work , we describe how the calculation of the free energy can be accelerated in the ff phase , which due to its greater symmetry allows optimal parallelization on a gpu architecture . parallel programing can be realized in cpus and gpus in many different ways . in this section we compare the performance of the same algorithm implemented using openmp @xcite , pgi cuda / openacc fortran @xcite , and directly in cuda c @xcite . the first two are generic extensions of fortran / c++ that make it easy to , respectively , use multiple cpu cores , and compile a subset of existing fortran / c++ code for a gpu . they take the form of annotations which can be added to existing code , and as such , enable the use of additional computational power with very little overhead by the programmer . typically , much better efficiency can be achieved by the third option i.e. a specifically optimized implementation targeting the gpu architecture directly . this requires more work on the part of the programmer , both in adjusting the algorithms and in rewriting the code , but it makes it possible to fully utilize the available resources . the global ground state for a fixed magnetic field strength @xmath14 and temperature @xmath58 is found by minimizing the free energy over the set of @xmath59 and @xmath23 . in case of @xmath60 independent bands this corresponds to global minimization of the free energy @xmath61 in every band separately , for every @xmath26 in the first brillouin zone ( fbz ) algorithm [ alg.1 ] . for the calculation of the free energy @xmath61 , we must know the eigenvalues @xmath62 reconstructing the band structure of our systems . in the case of the two - band model , it can simply be found analytically ( see [ app.twoband ] ) . however , for models with more bands ( such as the three - band model [ app.threeband ] ) the band structure has to be determined numerically ( e.g. using a linear algebra library , such as lapack ( cpu ) or magma ( gpu ) @xcite ) . with this approach , the calculation of @xmath63 and @xmath64 becomes a computationally costly procedure , and if it were to be repeated inside the inner loop of algorithm [ alg.1 ] , it would significantly impact the execution time . for this reason , we propose to precalculate the eigenvalues for every momentum vector @xmath65 and store them in memory for models with more than two bands . the main downside of this approach is the large increase in memory usage . while algorithm [ alg.1 ] is simple to realize on a cpu , its execution time is proportional to the system size @xmath44 , and as such scales quadratically with @xmath55 for a square lattice ( @xmath55 and @xmath66 are the number of lattice sites in the @xmath67 and @xmath68 direction , respectively ) . generate matrices @xmath63 and @xmath69 for @xmath65 calculate matrices @xmath70 for @xmath65 eq . [ eq.enequasiparticle ] calculate @xmath61 find and save @xmath59 corresponding to a fixed @xmath23 and minimal value @xmath61 find and save @xmath23 and @xmath59 corresponding to minimum of @xmath61 sometimes the physical properties of the system make it possible to reduce the amount of computation for instance when it is known that the minimum of the energy is attained for values of momentum @xmath23 in specific directions fig . [ fig.minene ] . @xcite in this case , the outer loop of algorithm [ alg.1 ] can be restricted to @xmath71 , where @xmath72 is a set of @xmath73 vectors . such reductions are not unique to linear systems with translational symmetry but are also the case for systems with rotational symmetry . @xcite in the case of bcs - type superconductivity where cooper pairs have zero total momentum ( @xmath74 ) , algorithm [ alg.1 ] can be further simplified by taking into account the following property of the dispersion relation : @xmath75 in eq . [ eq.enequasiparticle ] . this can be particularly useful in determining the system energy in the presence of the bcs phase i.e. either in complete absence of external magnetic fields or when only weak fields are present . a more general approach to the reduction of the execution time of our algorithm is to exploit the large degree of parallelism inherent in the problem . in fact , algorithm [ alg.1 ] can be classified as , , embarrassingly parallel since the vast majority of computation can be carried out independently for all combinations of @xmath76 . for simplicity , in this paper we concentrate on optimizing the inner loop , as all the presented methods apply to the outer loop in a similar fashion . we present two approaches to this problem . the first is to parallelize the execution of the serial loop over @xmath59 with openmp to fully utilize all available cpu cores . this has the advantage of simplicity , as the implementation requires minimal changes to the original ( serial ) code . the second approach is to implement algorithm [ alg.1 ] on a gpu using the cuda environment . modern gpus are capable of simultaneously executing thousands of threads in simt ( same instruction , multiple threads ) mode . from a programmer s point of view , all the threads are laid out in a 1- , 2- or 3-dimensional grid and are executing a _ kernel function_. the grid is further subdivided into blocks ( groups of threads ) , which are handled by a physical computational subunit of the gpu ( the so - called streaming multiprocessor ) . threads within a block can exchange data efficiently during execution , but cross - block communication can only take place through global gpu memory , which is significantly slower . mapped to gpu hardware . ] to fully utilize the gpu hardware , we split algorithm [ alg.1 ] into three steps . in the first step , we execute the ` computefreeenergy ` kernel ( algorithm [ alg.2 ] ) on a 3d grid @xmath77 . to take advantage of the efficient intra - block communication , we also carry out partial sums within the block ( corresponding to a subset of values spanning @xmath78 ) using the parallel sum - reduction algorithm . @xcite in the second step , we execute the sum - reduction algorithm again on the partial sums that were generated by algorithm [ alg.2 ] . in the third and last step , we copy the output of step 2 from gpu memory to host memory , and look for the value of @xmath79 corresponding to the lowest free energy with a linear search . depending on the exact configuration of the kernels in step 1 and 2 , the summation might not be complete at the beginning of step 3 . if this is the case , we carry out the remaining summation within the serial loop computing @xmath79 . with block sizes of 128 and 1024 used for the kernels in steps 1 and 2 , we can sum up to @xmath80 terms in parallel on the gpu . we found that the remaining summation was not worth the overhead of carrying it out on the gpu . should this not be the case for some larger problems , further parallel execution can be trivially achieved by repeating step 2 one more time . compute @xmath81 and @xmath18 corresponding to the current thread load @xmath82 and @xmath83 from global memory ( precomputed by a separate kernel ) compute @xmath84 and @xmath85 sum @xmath85 for a range of @xmath18 corresponding to one block of threads save the partial sum from the previous step in global gpu memory to test our approach , we executed algorithms [ alg.1 ] and [ alg.2 ] on linux machines with the following hardware : * cpu : intel(r ) core(tm ) i7 - 3960x cpu @ 3.30ghz 6 cores / 12 threads , * gpu : nvidia tesla k40 ( gk180 ) with the sm clock set to 875 mhz . the programs were run for a single value of @xmath86 and 200 values of @xmath79 . calculations were done for a square lattice of size @xmath87 for various values of @xmath88 . the execution times ( including only the computation part of the code , and excluding any time spent on startup or input / output ) are presented in figure [ fig.scaling ] . and [ alg.2 ] for one vector @xmath89 . right panel : speedup factors for all configurations at @xmath90 . the last 3 case names correspond to runs of the same cuda c code in double precision ( dp ) , single precision ( sp ) , and single precision with fast intrinsic functions ( spfm ) . all versions of the fortran code used double precision calculations . ] comparing the best cpu execution time ( with openmp ) to the gpu fortran code using openacc , we find a speedup factor of @xmath91 in the limit of large lattices . the custom gpu code shows slightly better performance , with a @xmath92x speedup for the double precision version , and additional speedup factors of @xmath93 for single precision , and @xmath94 for intrinsic functions . when taken together , the fastest gpu version is @xmath95 times faster than the openmp code and @xmath96 times faster than the serial cpu code utilizing only a single core . it is remarkable that the original fortran code enhanced with openacc annotations provides performance comparable to a manual implementation in cuda c. this result shows the power of appropriately used annotations marking parallelizable regions of the code . while still requiring explicit input from the programmer and a good understanding of the structure of the code , this approach is in practice significantly faster than writing the program from scratch in cuda c and dealing with low level details of gpu programming and resource allocation . this conclusion however only applies in the limit of large lattices ( see the left panel in figure [ fig.scaling ] ) . for smaller ones , the cuda c code can be seen to be noticeably faster than openacc , which is likely caused by the automatically generated gpu code introducing unnecessary overhead . it should be noted that the last two speedup factors were achieved by trading off precision of calculations for performance e.g. intrinsic functions are faster , but less precise implementations of transcendental functions . in our tests , we obtained the same results with all three approaches . this might not be true for some other systems though , so we advise careful experimentation . with a factor of 2.8x between the most and least precise method , it might also be worthwhile to run larger parameter scans at lower precision and then selectively verify with double precision calculations . the rich phenomenology and the subtle competing and interplaying phenomena of high-@xmath97 materials such as fesc ( section [ sec.intro ] ) , require us to probe fine regimes and precisely determine possible experimental signatures of exotic phases such as fflo ( section [ sec.theory_ph ] ) . by conducting our calculations in momentum space , and by fully exploiting the symmetries of the system , we are able to increase the size of the studied system by two orders of magnitude compared to previously reported results and practically eliminate finite size effects . the cost is borne by the increased complexity of the efficient custom - tailored gpu implementation , described in section [ sec.algorithm ] . our method shown here on the example of an iron - based multi - band superconductor exhibiting a fflo phase , can also be used in calculations of the ground state in standard bcs - type superconductors . overall , we achieved a 19x speedup compared to the cpu implementation ( 119x compared a single cpu core ) . in the spectrum of gpu - accelerated results in physics , this puts us towards the higher end , with the highest speedups being @xmath98x for compute - bound problems with large inherent parallelism . d.c . is supported by the forszt phd fellowship , co - funded by the european social fund . is supported by the ncn project dec-2011/01/n / st3/02473 . the authors would like to thank nvidia for providing hardware resources for development and benchmarking . the model of fesc proposed by raghu _ et al . _ in ref . @xcite , is a minimal two - band model of iron - base pnictides describing the @xmath99 and @xmath100 orbitals with hybridization : @xmath101 where @xmath102 , @xmath103 , @xmath104 , @xmath105 . @xmath106 is the energy unit . half - filling , a configuration with two electrons per site requires @xmath107 . the model is exactly diagonalizable , with eigenvalues : @xmath108 the spectrum @xmath109 reproduces the band structure and fermi surface of fesc for @xmath110 we get the electron - like ( hole - like ) band . this model of fesc was proposed by daghofer _ et al . _ in ref . @xcite and improved in ref . @xcite . beyond the @xmath99 and @xmath100 orbitals , the model also accounts for the @xmath111 orbital : @xmath112 in ref . @xcite the hopping parameters in electron volts are given as : @xmath113 , @xmath114 , @xmath115 , @xmath116 , @xmath117 , @xmath118 , @xmath119 , @xmath120 , @xmath121 , @xmath122 , @xmath123 , @xmath124 and @xmath125 . the average number of particles in the system @xmath126 is attained for @xmath127 . the fs for this model is shown in fig . [ fig.fsdag ] . the static cooper pair susceptibility indicates the possible formation of the fflo phase : @xcite @xmath128 where @xmath129 is the retarded green s function and @xmath130 is the op in band @xmath19 . the operator @xmath131 in real space corresponds to the operator @xmath132 in momentum space . the factor @xmath133 defines the op symmetries for @xmath3 pairing , @xmath134 is equal to @xmath135 for next nearest neighbors and zero otherwise . @xcite in momentum space : @xmath136 @xmath137 where @xmath138 is the structure factor corresponding to the @xmath3-wave symmetry , and @xmath139 is the fermi function . this quantity can be calculated numerically similarly to the procedure used for free energy in section [ sec.theory_ph ] . j. a. anderson , c. d. lorenz , a. travesset , general purpose molecular dynamics simulations fully implemented on graphics processing units , http://dx.doi.org/10.1016/j.jcp.2008.01.047[j . * 227 * ( 2008 ) 5342 ] t. preis , p. virnau , p. wolfgang and j. j. schneider , gpu accelerated monte carlo simulation of the 2d and 3d ising model , http://dx.doi.org/10.1016/j.jcp.2009.03.018[j . comput . phys . * 228 * ( 2009 ) 4468 ] y. kamihara , t. watanabe , m. hirano , h. hosono , iron - 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obtaining a thermodynamically accurate phase diagram through numerical calculations is a computationally expensive problem that is crucially important to understanding the complex phenomena of solid state physics , such as superconductivity . in this work
we show how this type of analysis can be significantly accelerated through the use of modern gpus .
we illustrate this with a concrete example of free energy calculation in multi - band iron - based superconductors , known to exhibit a superconducting state with oscillating order parameter ( op ) .
our approach can also be used for classical bcs - type superconductors . with a customized algorithm and compiler tuning
we are able to achieve a 19x speedup compared to the cpu ( 119x compared to a single cpu core ) , reducing calculation time from minutes to mere seconds , enabling the analysis of larger systems and the elimination of finite size effects .
fflo , pnictides , nvidia cuda , pgi cuda fortran , superconductivity * program summary * _ manuscript title : _ gpu - based acceleration of free energy calculations in solid state physics + _ authors : _ micha januszewski , andrzej ptok , dawid crivelli , bartomiej gardas + _ journal reference : _ + _ catalogue identifier : _
+ _ licensing provisions : lgplv3 _ + _ programming language : _ fortran ,
cuda c + _ computer : _ any with a cuda - compliant gpu + _ operating system : _ no limits ( tested on linux ) + _ ram : _ typically tens of megabytes .
+ _ keywords : _ superconductivity , fflo , cuda , openmp , openacc , free energy + _ classification : _ 7 , 6.5 + _ nature of problem : _ gpu - accelerated free energy calculations in multi - band iron - based superconductor models .
+ _ solution method : _ parallel parameter space search for a global minimum of free energy .
+ _ unusual features : _
+ the same core algorithm is implemented in fortran with openmp and openacc compiler annotations , as well as in cuda c. the original fortran implementation targets the cpu architecture , while the cuda c version is hand - optimized for modern gpus . + _ running time : _ problem - dependent , up to several seconds for a single value of momentum and a linear lattice size on the order of @xmath0 .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
since the copula of two continuous random variables is scale - invariant , copulas are regarded as the functions that capture dependence structure between random variables . for many purposes , independence and monotone dependence have so far been considered two opposite extremes of dependence structure . however , monotone dependence is just a special kind of dependence between two random variables . more general complete dependence happens when functional relationship between continuous random variables are piecewise monotonic , which corresponds to their copula being a shuffle of min . see @xcite . mikusinski et al . @xcite showed that shuffles of min is dense in the class of all copulas with respect to the uniform norm . this surprising fact urged the discovery of the ( modified ) sobolev norm by siburg and stoimenov @xcite which is based on the @xmath0-operation introduced by darsow et al . they @xcite showed that continuous random variables @xmath1 and @xmath2 are mutually completely dependent , i.e. their functional relationship is any borel measurable bijection , if and only if their copula has unit sobolev - norm . darsow et al . @xcite showed that for a real stochastic processes @xmath3 , the validity of the chapman - kolmogorov equations is equivalent to the validity of the equations @xmath4 for all @xmath5 , where @xmath6 denotes the copula of @xmath7 and @xmath8 . it is then natural to investigate how dependence levels of @xmath9 and @xmath10 are related to that of @xmath11 . aside from @xmath12 , @xmath13 and @xmath14 , the easiest case is when @xmath9 and @xmath10 are mutual complete dependence copulas . in light of our result on denseness of shuffles of min in the mcd copulas , we shall show that if @xmath15 then @xmath16 . now , if @xmath17 and @xmath18 is a copula then we prove that @xmath19 coincides with a generalized shuffle of @xmath18 in the sense of durante et al . we also give similar characterizations of shuffles of @xmath18 and generalized shuffles of min . these characterizations have advantages of simplicity in calculations because it avoids using induced measures . then we use this relationship to obtain a simple proof of a characterization of copulas whose orbit is singleton ( theorem 10 in @xcite ) . note that there are many examples where shuffles of @xmath18 , i.e. @xmath19 or @xmath20 , do not have the same sobolev norm as @xmath18 . however , we show that multiplication by unit norm copulas preserves independence , complete dependence and mutual complete dependence . since left- and right - multiplying a copula @xmath21 by unit norm copulas amount to `` shuffling '' or `` permuting '' @xmath1 and @xmath2 respectively , we introduce a new norm , called the @xmath0-norm , which is invariant under multiplication by a unit norm copula . mutual complete dependence copulas still has @xmath0-norm one . this invariant property implies that complete dependence copulas also possess unit @xmath0-norm . based on the @xmath0-norm , a new measure of dependence is defined in the same spirit as the definition by siburg et al . it turns out that this new measure of dependence satisfies most of the seven postulates proposed by rnyi @xcite . the only known measure of dependence that satisfies all rnyi s postulates is the maximal correlation coefficient . this manuscript is structured as follows . we shall summarize related basic properties of copulas , the binary operator @xmath0 and the sobolev norm in section 2 . then we obtain a characterization of copulas with unit sobolev norm which implies that the @xmath0-product of mcd copulas is a mcd copula in section 3 . section 4 contains our characterizations of generalized shuffles of min and ( generalized ) shuffles of copulas in the sense of durante et al . in terms of the @xmath0-product . we then show that shuffling a copula preserves independence , complete dependence and mutual complete dependence . in section 5 , a new norm is introduced and its properties are proved . and in section 6 , we define a new measure of dependence and verify that it satisfies most of rnyi s postulates . a _ bivariate copula _ is defined to be a joint distribution function of two random variables with uniform distribution on @xmath22.$ ] since such a joint distribution is uniquely determined by its restriction on @xmath22 ^ 2 $ ] one can also define a copula as a function @xmath23 ^ 2 \to [ 0 , 1]$ ] satisfying the following properties @xmath24 @xmath25 for all @xmath26 ^ 2 $ ] and @xmath27 ^ 2 $ ] such that @xmath28 note that the two definitions are equivalent . every copula @xmath18 induces a measure @xmath29 on @xmath30 ^ 2 $ ] by @xmath31\times[y , v ] ) = c(u , v)-c(u , y)-c(x , v)+c(x , y).\ ] ] the induced measure @xmath29 is doubly stochastic in the sense that for every borel set @xmath10 , @xmath32\times b ) = m(b ) = \mu_c(b\times[0,1])$ ] where @xmath33 is lebesgue measure on @xmath34 . important copulas include the frchet - hoeffding upper and lower bounds @xmath35 and the product , or independent , copula @xmath36 . a fundamental property is that @xmath13 is a copula of @xmath1 and @xmath2 if and only if @xmath2 is almost surely an increasing bijective function of @xmath1 . if @xmath1 and @xmath2 are uniformly distributed on @xmath30 $ ] then that bijection is the identity map on @xmath30 $ ] . its graph , the main diagonal , is the support of the induced measure @xmath37 , also called the support of @xmath13 . at the other extreme , the minimum copula @xmath38 corresponds to random variables being monotone decreasing function of each other . listed below are some basic properties of any copula @xmath18 , some of which shall be used frequently in the manuscript . 1 . @xmath39 for all @xmath40.$ ] 2 . @xmath41 @xmath42 ^ 2 $ ] and hence @xmath18 is uniformly continuous . 3 . @xmath43 and @xmath44 exist almost everywhere on @xmath30 ^ 2 $ ] . 4 . for a.e . @xmath45 $ ] , @xmath46 is nondecreasing in the domain where it exists and similar statement holds for @xmath47 . perhaps , the most important property of copulas is given by the sklar s theorem which states that to every joint distribution function @xmath48 of continuous random variables @xmath1 and @xmath2 with marginal distributions @xmath49 and @xmath50 respectively , there corresponds a unique copula @xmath18 , called the _ copula of @xmath1 and @xmath2 _ for which @xmath51 for all @xmath52 this means that the copula of @xmath53 captures all dependence structure of the two random variables . @xmath1 and @xmath2 are said to be _ mutually completely dependent _ if there exists an invertible borel measurable function @xmath54 such that @xmath55 shuffles of min were introduced by mikusinski et al . @xcite as examples of copulas of mutually completely dependent random variables . by definition , a shuffle of min is constructed by shuffling ( permuting ) the support of the min copula @xmath13 on @xmath56 vertical strips subdivided by a partition @xmath57 . it is shown ( * ? ? ? * theorems 2.1 & 2.2 ) that the copula of @xmath1 and @xmath2 is a shuffle of min if and only if there exists an invertible borel measurable function @xmath54 with finitely many discontinuity points such that @xmath58 . in @xcite , such an @xmath54 is called strongly piecewise monotone function . following @xcite , the binary operation @xmath0 on the set @xmath59 of all bivariate copulas is defined as @xmath60\ ] ] and the _ sobolev norm _ of a copula @xmath18 is defined by @xmath61 it is well - known that @xmath62 is a monoid with null element @xmath12 and identity @xmath13 . so a copula @xmath18 is called _ left invertible _ ( _ right invertible _ ) if there is a copula @xmath63 for which @xmath64 ( @xmath65 ) . it was shown in ( * ? ? ? * theorem 7.6 ) and ( * ? ? ? * theorem 4.2 ) that the @xmath0-product on @xmath59 is jointly continuous with respect to the sobolev norm but not with respect to the uniform norm . moreover , they @xcite gave a statistical interpretation of the sobolev norm of a copula . [ thm : normc ] let @xmath18 be a bivariate copula of continuous random variables @xmath1 and @xmath2 . @xmath66 ; 2 . ) @xmath67 if and only if @xmath68 ; and 3 . ) the following are equivalent . 1 . @xmath69 . 2 . @xmath18 is invertible with respect to @xmath0 . [ thm : unitnorm3 ] for each @xmath70 $ ] , @xmath71 a.e . 4 . there exists a borel measurable bijection @xmath72 such that @xmath73 a.e . it follows readily that all shuffles of min have norm one . let @xmath18 be a copula with unit sobolev norm . then @xmath74 and @xmath75 take values @xmath76 or @xmath77 almost everywhere . see , for example , theorem 7.1 in @xcite and theorem 4.2 in @xcite . let us recall from ( * ? ? ? * theorem 2.2.7 ) that , for a.e . @xmath78 , \partial_1 c(x , y)$ ] is a nondecreasing function of @xmath79.$ ] similar statement holds also for @xmath80 so for a.e . @xmath78,$ ] there is @xmath81 $ ] such that for almost every @xmath82 , @xmath83 if @xmath84 and @xmath85 if @xmath86 ( @xmath87 ) denote the set of such @xmath88 s by @xmath89 so that @xmath90 . and for every @xmath91 , by redefining @xmath92 on a set of measure zero , we may assume that @xmath92 is defined and nondecreasing for all @xmath93 $ ] . to show that @xmath54 is measurable , let @xmath94,$ ] and observe that since @xmath92 is increasing in @xmath82 @xmath95 which is measurable because each @xmath96 is measurable . in exactly the same fashion , there exists a measurable function @xmath97 \to [ 0 , 1]$ ] for which @xmath98 let us recall the definition that a measurable function @xmath99\to[0,1]$ ] is said to be _ measure - preserving _ if @xmath100 for any lebesgue measurable set @xmath101 $ ] . [ thm : suppcopulanormone ] suppose @xmath18 is a copula with unit sobolev norm . then there exists a unique invertible borel measurable function @xmath102 \to [ 0 , 1]$ ] such that @xmath54 is measure - preserving and for almost every @xmath103 in @xmath30 ^ 2 $ ] @xmath104 furthermore , if @xmath54 is continuous on an interval @xmath105 then it is differentiable on @xmath105 with constant derivative equal to either @xmath77 or @xmath106 . during the preparation of this manuscript , we have come across similar results such as proposition 1 in @xcite and theorem 2.4 and corollary 2.4.1 in @xcite . we first claim that @xmath54 and @xmath107 defined above are inverses of each other in the sense that @xmath108 and @xmath109 are identity on @xmath22 $ ] a.e . , i.e. @xmath110 and @xmath111 both have measure @xmath77 . this is equivalent to saying that @xmath112 if and only if @xmath113 for a.e . @xmath114 ^ 2 $ ] . indeed , observe that for any open interval @xmath115 $ ] , @xmath116 if and only if @xmath117 . now let @xmath118 and @xmath119 be open intervals in @xmath30 $ ] for which @xmath120 does not intersect the graph @xmath112 , i.e. @xmath121 , hence @xmath92 is independent of @xmath122 for a.e . so @xmath124 for a.e . @xmath123 and all @xmath122 . then for @xmath125 in @xmath126 @xmath127 and so @xmath128 is independent of @xmath123 which implies that @xmath120 does not intersect the graph @xmath113 . the converse can be shown by a similar argument . since the graph of a borel function is a borel subset of @xmath30 ^ 2 $ ] , @xmath112 and @xmath113 give the same graph . and the claim follows . let @xmath29 denote the doubly stochastic measure associated with @xmath18 . a straightforward verification gives @xmath129 for all borel rectangles @xmath120 , which implies by a standard measure - theoretic technique that holds for all borel sets @xmath130 $ ] . so @xmath54 is measure - preserving since it is equivalent to the validity of for all borel sets @xmath9 and @xmath10 . lastly , we prove that if @xmath54 is continuous on an open interval @xmath131 then it is differentiable with @xmath132 being constant and equal to @xmath133 . since @xmath54 is continuous and one - to - one on @xmath105 , it has to be strictly monotonic on @xmath105 . let us consider the case where @xmath54 is strictly increasing on @xmath105 . this implies that @xmath134 is strictly increasing on the interval @xmath135 and that @xmath84 if and only if @xmath136 for @xmath137 . for @xmath138\subset ( a , b),$ ] @xmath139 since @xmath140 and @xmath141 are differentiable with respect to @xmath82 almost everywhere , we have for a.e . @xmath82 , @xmath142 as @xmath143 and @xmath75 and @xmath144 are equal to @xmath76 or @xmath77 , @xmath145 and hence @xmath146 for all @xmath137 . similarly , if @xmath54 is strictly decreasing on @xmath147 then @xmath148 a.e . on @xmath147 a natural question is then to investigate the set on which an invertible measure - preserving function @xmath54 is continuous . unfortunately , the support of a unit norm copula may be the graph of a function which is discontinuous on a dense subset of @xmath30 $ ] , and hence there is no interval on which it is continuous . define a sequence of shuffles of min @xmath149 by letting @xmath150 be the comonotonic copula supported on the main diagonal . @xmath151 is defined so that it shares the same support with @xmath150 in @xmath152\times[0,1]$ ] and its support in the other half @xmath153=[\frac{1}{2},1]\times[0,1]$ ] is that of @xmath150 flipped horizontally . @xmath154 is then obtained from @xmath151 by flipping the support in each stripe of the set @xmath155 $ ] where @xmath156\cup{\left([\frac{1}{2 ^ 2},\frac{1}{2}]+\frac{1}{2}\right)}$ ] . for general @xmath157 , we define @xmath158 and let the shuffle of min @xmath159 be obtained from @xmath160 by flipping the support in each stripe of @xmath161 horizontally . to sum up , each shuffle of min @xmath159 is supported on the graph @xmath162 where @xmath163 is constructed according to the above iterative procedure , starting from @xmath164 and @xmath165}$ ] . the first few @xmath159 s are illustrated in figure [ fig : s_n ] . [ fig : s_n ] @[email protected]@ , @xmath154 , and @xmath166,title="fig:",scaledwidth=26.0% ] , @xmath154 , and @xmath166,title="fig:",scaledwidth=26.0% ] & , @xmath154 , and @xmath166,title="fig:",scaledwidth=80.0% ] from construction , @xmath161 consists of @xmath167 stripes , each of width @xmath168 . on each of these stripes , the supports of @xmath159 and @xmath160 differ by a flip which implies that @xmath169 and @xmath170 are equal on the stripe except on two triangles of total area @xmath171 where @xmath172 . similarly , on each stripe of @xmath161 , @xmath173 on two triangles of total area @xmath174 and zero elsewhere . therefore , @xmath175 now , given @xmath176 , @xmath177 which converges to @xmath76 as @xmath178 . since the set of copulas is complete with respect to the sobolev norm ( see p. 424 in @xcite ) , the cauchy sequence @xmath149 converges to a copula @xmath179 . it follows that @xmath180 . it can also be shown that the support of @xmath179 contains the graph of the pointwise limit @xmath54 of @xmath163 . finally , we shall show that the mutual complete dependence copula @xmath179 has support on the graph of a function discontinuous on the set of dyadic points in @xmath30 $ ] . in fact , it is straightforward to calculate the jump of @xmath54 at a dyadic point @xmath181 where @xmath182 is indivisible by @xmath183 : @xmath184 we note here that the support of @xmath179 is self - similar with hausdorff dimension one . a surprising fact by mikusinski , sherwood and taylor ( * ? ? ? * theorem 3.1 ) is that every copula , in particular the independence copula , can be approximated arbitrarily close in the uniform norm by a shuffle of min . consequently , the uniform norm can not distinguish dependence structures among copulas . however , if @xmath149 is a sequence of shuffles of min converging in the sobolev norm to a copula @xmath18 , then it is necessary that @xmath69 , hence @xmath18 is a copula of two mutually completely dependent random variables . conversely , one might ask whether any copula @xmath18 with @xmath69 can be approximated arbitrarily close in the sobolev norm by a shuffle of min . we quote here without proof a result from @xcite which will be useful in answering the question . [ thm : chounguyen ] for every measure - preserving function @xmath54 over @xmath30 $ ] , there exists a sequence of bijective piecewise linear measure - preserving functions @xmath185 whose slopes are either @xmath186 or @xmath106 and such that @xmath163 converges to @xmath54 a.e . [ lem : norm - supp ] let @xmath187 and @xmath188 be copulas with norm one which are supported on the graphs of @xmath189 and @xmath190 , respectively . then @xmath191 by assumption , for a.e . @xmath103 , @xmath192 if and only if @xmath82 is between @xmath193 and @xmath194 . likewise , @xmath195 if and only if @xmath88 is between @xmath196 and @xmath197 for a.e . so @xmath198 [ thm : c - s ] for any copula @xmath18 with @xmath199 , there exists a sequence of shuffles of min @xmath149 such that @xmath200 . suppose @xmath18 is a copula with norm one and @xmath18 is supported on the graph of @xmath54 . it follows from theorem [ thm : suppcopulanormone ] that @xmath54 is a measure - preserving bijection from @xmath30 $ ] onto itself . by theorem [ thm : chounguyen ] , one can construct a sequence of measure - preserving functions @xmath185 for which each @xmath163 is bijective piecewise linear with slopes @xmath186 or @xmath106 and @xmath163 converges to @xmath54 a.e . a corresponding sequence of shuffles of min @xmath201 can then be chosen so that the graph of @xmath163 is the support of @xmath159 . by lemma [ lem : norm - supp ] , @xmath202 . since @xmath203 a.e . , an application of dominated convergence theorem shows that @xmath204 . consequently , @xmath205 in the sobolev norm . from the proof , it is worth noting that one can approximate a copula @xmath18 by only straight shuffles of min whose slopes on all subintervals are @xmath186 . let @xmath206 . 1 . if @xmath207 and @xmath208 then @xmath209 . 2 . if @xmath210 then @xmath207 if and only if @xmath208 . [ lem1 ] \1 . let @xmath206 be such that @xmath207 and @xmath208 . by theorem [ thm : c - s ] , there exist sequences @xmath149 , @xmath211 of shuffles of min such that @xmath212 and @xmath213 in the sobolev norm . hence , with respect to the sobolev norm , @xmath214 by the joint continuity of the @xmath0-product . since a product of shuffles of min is still a shuffle of min , @xmath215 \2 . let @xmath216 and @xmath217 be copulas of sobolev norm 1 . since @xmath218 , an application of 1 . at least as soon as shuffles of min were introduced in @xcite , the idea of simple shuffles of copulas was already apparent . see , e.g. , @xcite . in @xcite , durante , sarkoci and sempi gave a general definition of shuffles of copulas via a characterization of shuffles of min in terms of a _ shuffling _ @xmath220 ^ 2\to [ 0,1]^2 $ ] defined by @xmath221 where @xmath222\to [ 0,1]$ ] . before stating their results , let us recall the definition of push - forward measures . let @xmath54 be a measurable function from a measure space @xmath223 to a measurable space @xmath224 . push - forward of @xmath225 under @xmath54 _ is the measure @xmath226 on @xmath224 defined by @xmath227 for @xmath228 . a copula @xmath18 is a shuffle of min if and only if there exists a piecewise - continuous measure - preserving bijection @xmath229\to[0,1]$ ] such that @xmath230 . dropping piecewise continuity of @xmath231 , a _ generalized shuffle of min _ is defined as a copula @xmath18 whose induced measure is @xmath230 for some measure - preserving bijection @xmath229\to[0,1]$ ] . replacing @xmath13 by a given copula @xmath63 , a _ shuffle of @xmath63 _ is a copula @xmath18 whose induced measure is @xmath232 for some piecewise - continuous measure - preserving bijection @xmath231 . @xmath18 is also called the _ @xmath231-shuffle of @xmath63_. if the bijection @xmath231 is only required to be measure - preserving in , then @xmath18 is called a _ generalized shuffle of @xmath63_. the following lemma will be useful in our investigation . [ lem:1 ] let @xmath231 be a measure - preserving bijection on @xmath30 $ ] and @xmath18 be a copula defined by @xmath233\times[0,y]\right)}\quad\text{for } x , y\in[0,1].\ ] ] then the copula @xmath18 , or equivalently its induced measure @xmath234 , is supported on the graph of @xmath235 . moreover , the converse also holds , i.e. if @xmath18 is supported on the graph of a measure - preserving bijection @xmath231 then @xmath236 . let @xmath237\times[c , d]$ ] be a closed rectangle in @xmath238 and @xmath239 be the map on @xmath30 ^ 2 $ ] associated with a given measure - preserving bijection @xmath231 on @xmath30 $ ] , i.e. @xmath240 . so @xmath241\times[c , d]\right ) } = { \left(t^{-1}[a , b]\right)}\times[c , d]$ ] and , by definition of the push - forward measure , @xmath242\times[c , d]\right ) } & = \mu_m{\left(s_t^{-1}{\left([a , b]\times[c , d]\right)}\right)}\\ & = \mu_m{\left({\left(t^{-1}[a , b]\right)}\times[c , d]\right ) } = m{\left({\left(t^{-1}[a , b]\right)}\cap[c , d]\right)}. \end{aligned}\ ] ] thus , @xmath243\times[c , d]\right ) } = 0 $ ] if and only if the projection of @xmath244\times[c , d]\right)}$ ] onto @xmath245 $ ] has measure zero . consequently , since borel measurable subsets of @xmath30 ^ 2 $ ] are generated by rectangles , the desired result is obtained . a copula @xmath18 is a generalized shuffle of min if and only if @xmath69 . ( @xmath246 ) let @xmath18 be a generalized shuffle of min , i.e. there exists a measure preserving bijection @xmath231 on @xmath30 $ ] such that @xmath247 . by theorem [ thm : chounguyen ] , there is a sequence @xmath211 of piecewise - continuous measure - preserving bijection on @xmath30 $ ] such that @xmath248 a.e . so @xmath249\times[0,y]\right)}$ ] defines a sequence of shuffles of min . we claim that @xmath250 . in fact , by lemma [ lem:1 ] , @xmath251 and @xmath252 are supported on the graphs of @xmath235 and @xmath253 respectively . now , lemma [ lem : norm - supp ] implies that @xmath254 which converges to @xmath76 as a result of the lusin - souslin theorem ( see , e.g. , ( * ? ? ? * corollary 15.2 ) ) which states that a borel measurable injective image of a borel set is a borel set and the dominated convergence theorem . therefore , @xmath255 in the sobolev norm . ( @xmath256 ) let @xmath18 be a copula with @xmath69 . then theorem [ thm : suppcopulanormone ] gives a measure - preserving bijection @xmath54 whose graph is the support of @xmath18 . so lemma [ lem:1 ] implies that @xmath257 . [ thm : mu*nu ] if @xmath225 and @xmath258 are doubly stochastic measures on @xmath30 ^ 2 $ ] then @xmath259 induces a doubly stochastic ( borel ) measure @xmath260 on @xmath30 ^ 2 $ ] , where @xmath261)\quad\text{and}\quad \partial_1\nu(t , j ) = \frac{d}{dt}\nu([0,t]\times j).\ ] ] furthermore , if @xmath9 and @xmath10 are copulas and @xmath262 and @xmath263 denote their doubly stochastic measures then @xmath264 we shall prove only which shows that @xmath260 is a doubly stochastic measure when the measures @xmath225 and @xmath258 are doubly stochastic and inducible by copulas . let @xmath9 and @xmath10 be copulas and @xmath265 , j=[b_1,b_2 ] \subseteq [ 0,1]$ ] . then @xmath266 \,dt\\ & = \int_0 ^ 1 \partial_2{\left(a(a_2,t)-a(a_1,t)\right)}\partial_1{\left(b(t , b_2)-b(t , b_1\right)}\,dt\\ & = \int_0 ^ 1 \frac{d}{dt}\mu_a(i\times[0,t])\frac{d}{dt}\mu_b([0,t]\times j)\,dt\\ & = \mu_a*\mu_b(i\times j ) . \end{aligned}\ ] ] the usual measure - theoretic techniques allow to extend this result to the product of all borel sets . [ lem : s_tassoc ] let @xmath231 be a measure - preserving bijection on @xmath30 $ ] and @xmath225 , @xmath258 be doubly stochastic measures on @xmath30 ^ 2 $ ] . then @xmath267 let @xmath105 and @xmath268 be borel sets in @xmath30 $ ] . then @xmath269 [ thm : socchar ] let @xmath18 and @xmath63 be bivariate copulas . then 1 . @xmath18 is a shuffle of @xmath63 if and only if there exists a shuffle of min @xmath9 such that @xmath270 ; 2 . @xmath18 is a generalized shuffle of @xmath63 if and only if there exists a generalized shuffle of min @xmath9 such that @xmath270 . we shall only prove 2 . since 1 . is just a special case . ( @xmath246 ) if @xmath18 is a shuffle of @xmath63 , i.e. @xmath271 for some measure - preserving bijection @xmath231 of @xmath30 $ ] , then the copula @xmath9 defined by @xmath272 is a shuffle of min by theorem [ thm : mu*nu ] . then @xmath273 which means that @xmath274 . ( @xmath256 ) if @xmath274 for some copula @xmath9 with @xmath17 then @xmath272 for some measure - preserving bijection @xmath231 and @xmath275 note the repeated uses of theorem [ thm : mu*nu ] and lemma [ lem : s_tassoc ] in both derivations . since @xmath12 is the only null element of @xmath0 ( see @xcite ) , it follows easily from theorem [ thm : socchar ] that @xmath12 is the only copula which is invariant under shuffling by generalized shuffles of min . this is a result first proved in ( * ? ? ? * theorem 10 ) . even though all generalized shuffles of min have equal unit norm , not all shuffles of @xmath18 have the same norm . here is a class of examples . [ exam : shufflediffnorm ] for @xmath276 , let @xmath277 denote the straight shuffle of min whose support is on the main diagonals of the squares @xmath278\times[1-\alpha,1]$ ] and @xmath279\times[0,1-\alpha]$ ] . then by straightforward computations , for any copula @xmath18 , @xmath280 and @xmath281 let us now consider the farlie - gumbel - morgenstern ( fgm ) copulas @xmath282 , @xmath283 $ ] , defined by @xmath284 . then @xmath285 and @xmath286 so that @xmath287 which is equal to @xmath288 only if @xmath289 or @xmath290 or @xmath77 . for each @xmath291 , @xmath292 is maximized when @xmath293 and the maximum value is @xmath294 . if @xmath295 and @xmath2 are conditionally independent given @xmath1 , then @xmath296 [ da - con - ind ] let @xmath297 be borel measurable and @xmath298 be random variables . then @xmath299 and @xmath2 are conditionally independent given @xmath1.[special - case ] since @xmath72 is borel measurable , @xmath299 is measurable with respect to @xmath300 , the @xmath301-algebra generated by @xmath1 . hence , by properties of conditional expectations , @xmath302 for all @xmath303 . this completes the proof . [ cor : star - shuffling ] let @xmath304 be borel measurable functions . then @xmath305 for all random variables @xmath298.[decompose ] since @xmath54 and @xmath107 are borel measurable , by propositions [ da - con - ind ] and [ special - case ] , we have @xmath306 for all random variables @xmath298 . transposing both sides of , we obtain @xmath307 then , we have @xmath308 let @xmath309 , the set of invertible copulas or , equivalently , the set of copulas with unit sobolev norm . a _ shuffling map _ @xmath310 is a map on @xmath311 defined by @xmath312 . the motivation behind the word shuffling comes from the fact that a shuffling image of a copula is a two - sided generalized shuffle of the copula . note that @xmath313 let @xmath298 be continuous random variables and @xmath309 . then the following statements hold : 1 . @xmath1 and @xmath2 are independent if and only if @xmath314 . @xmath1 is completely dependent on @xmath2 or vice versa if and only if @xmath315 is a complete dependence copula . 3 . @xmath1 and @xmath2 are mutually completely dependent if and only if @xmath315 is a mutual complete dependence copula . [ dependence - invariant ] \1 . this clearly follows from the fact that @xmath12 is the zero element in @xmath316 . 2 . with out loss of generality , let us assume that @xmath2 is completely dependent on @xmath1 , i.e. there exists a borel measurable transformation @xmath72 such that @xmath73 with probability one . let @xmath54 and @xmath107 be borel measurable bijective transformations on @xmath34 such that @xmath317 and @xmath318 . by corollary [ decompose ] , we have @xmath319 . thus , it suffices to show that @xmath320 is completely dependent on @xmath321 . from @xmath73 with probability one , @xmath322 with probability one . it is left to show that @xmath323 is borel measurable . this is true because of lusin - souslin theorem ( see , e.g. , @xcite , corollary 15.2 ) which states that a borel measurable injective image of a borel set is a borel set . the converse automatically follows because the inverse of a shuffling map is still a shuffling map . the proof is completely similar to above except that the function @xmath72 is also required to be bijective . corollary [ decompose ] implies that a shuffling image of a copula @xmath324 is a copula of transformed random variables @xmath325 for some borel measurable bijective transformations @xmath54 and @xmath107 . together with the above lemma , we obtain the following theorem . let @xmath1 and @xmath2 be continuous random variables . let @xmath54 and @xmath107 be any borel measurable bijective transformations of the random variables @xmath1 and @xmath2 , respectively . then @xmath1 and @xmath2 are independent , completely dependent or mutually completely dependent if and only if @xmath321 and @xmath320 are independent , completely dependent or mutually completely dependent , respectively . the above theorem suggests that shuffling maps preserve stochastic properties of copulas . in the next section , we contruct a norm which , in some sense , also preserves stochastic properties of copulas . our main purpose is to construct a norm under which shuffling maps are isometries and then derive its properties . define a map @xmath326 , by @xmath327 by straightforward verifications , @xmath328 is a norm on @xmath311 , called the _ @xmath0-norm_. moreover , it is clear from the definition that @xmath329 for all @xmath330 . the following proposition summarizes basic properties of the @xmath0-norm . observe that properties 2.4 . are the same as those for the sobolev norm . [ prop:*norm ] let @xmath331 . then the following statements hold . 1 . @xmath332 if @xmath333 . 2 . @xmath334 if and only if @xmath68 . [ prop:*norm3 ] @xmath335 . @xmath336 for all @xmath330 . [ props ] 1 . is a consequence of the inequality @xmath337 . 2 . follows from the fact that @xmath12 is the zero of @xmath316 . to prove 3 . , we first observe that @xmath338 for all @xmath309 . the result follows by taking supremum over @xmath309 on both sides . finally , using the facts that @xmath339 for all @xmath340 , @xmath341 let @xmath342 and @xmath343 . then @xmath344 . therefore , shuffling maps are isometries with respect to the @xmath0-norm . we shall prove only one side of the equation as the other can be proved in a similar fashion . let @xmath345 and @xmath346 . then by corollary [ lem1 ] , for any @xmath331 , @xmath347 if and only if @xmath348 . hence , @xmath349 from example [ exam : shufflediffnorm ] , let @xmath350 , @xmath283\setminus{\left\{0\right\}}$ ] and @xmath351 . then @xmath352 . since @xmath353 for any @xmath354 . then @xmath355 hence , the sobolev norm and the @xmath356-norm are distinct . [ norms - distinct ] [ exam : equalnorms ] let @xmath357 $ ] and @xmath18 be a copula . recall that one can show using only the property @xmath358 of the norm @xmath359 ( see @xcite ) that @xmath360 . since the @xmath0-norm shares this same property with the sobolev norm ( see proposition [ prop:*norm]([prop:*norm3 ] ) ) , we also have @xmath361 so @xmath362 for all copulas @xmath18 satisfying @xmath363 . in particular , the sobolev norm and the @xmath356-norm coincide on the family of convex sums of an invertible copula and the product copula , where the norms are equal to @xmath364 . [ lem : shufflingcd ] let @xmath365 $ ] be a borel measurable set . define the function @xmath366 \to [ 0,1]$ ] by @xmath367\cap a ) & \text{if } x\in a,\\ m(a ) + m([0,x]\setminus a ) & \text{if } x\notin a. \end{cases}\ ] ] then @xmath368 is measure - preserving and _ essentially invertible _ in the sense that there exists a borel measurable function @xmath369 for which @xmath370 a.e . @xmath45 $ ] . such a @xmath369 is called an _ essential inverse _ of @xmath368 . clearly , @xmath368 is borel measurable . * @xmath371 @xmath368 is measure - preserving : * it suffices to prove that @xmath372)=m([0,b])$ ] for all @xmath373 $ ] . now if @xmath374 , then @xmath375 = \ { x\in a\colon s(x ) \in [ 0,b ] \ } = { \left\ { x \in a\colon m(a \cap [ 0,x ] ) \leq b \right\}}.\ ] ] by continuity of @xmath33 , there exists a largest @xmath376 such that @xmath377 ) = b$ ] . then @xmath378 = a \cap \{x\colon m(a \cap [ 0,x ] ) \leq b\ } = a \cap [ 0,x_0]$ ] . therefore , @xmath372 ) = m(a \cap [ 0,x_0 ] ) = b$ ] . the case where @xmath379 can be proved similarly . * @xmath371 @xmath368 is essentially invertible : * using continuity of @xmath33 , we shall define an auxiliary function @xmath369 on @xmath30 $ ] as follows . if @xmath380 , there exists a corresponding @xmath123 such that @xmath381\cap a\right ) } = y$ ] . if @xmath382 , there exists a corresponding @xmath383 such that @xmath384\setminus a\right ) } = y$ ] . in these two cases , we define @xmath385 . generally , @xmath369 is not unique as there might be many such @xmath88 s . we shall show that @xmath368 is injective outside a borel set of measure zero by proving that @xmath386 is the identity map on @xmath30\setminus z$ ] for some borel set @xmath295 of measure zero . if @xmath123 then @xmath387 so that @xmath388 and @xmath389\cap a ) = s_a(x ) = m([0,x]\cap a)$ ] . similarly , if @xmath383 then @xmath390 , @xmath391 and @xmath389\setminus a ) = m([0,x]\setminus a)$ ] . consider the set of all @xmath123 for which @xmath392 . if @xmath393 then @xmath394\cap a\right ) } = 0 $ ] which implies that @xmath395 wherever exists . now , a simple application of the radon - nikodym theorem ( see , e.g. , @xcite ) yields that the limit in is equal to @xmath77 for all @xmath396 where @xmath295 is a borel set of measure zero . therefore , @xmath397 has zero measure . similarly , @xmath398 is a null set and hence @xmath399 . therefore , @xmath400 except possibly on a borel set @xmath295 of measure zero . by the lusin - souslin theorem ( see , e.g. , ( * ? ? ? * corollary 15.2 ) ) , the injective borel measurable function @xmath368 , mapping a borel set @xmath30\setminus z$ ] onto a borel set @xmath401\setminus z)$ ] , has a borel measurable inverse , still denoted by @xmath369 . now , since @xmath368 is measure - preserving , its range which is the domain of @xmath369 has full measure . this guarantees that @xmath369 can be extended to a borel measurable function on @xmath30 $ ] which is an essential inverse of @xmath368 . [ thm : cdnorm1 ] let @xmath298 be random variables on a common probability space for which @xmath2 is completely dependent on @xmath1 or @xmath1 is completely dependent on @xmath2 . then @xmath402 . assume that @xmath2 is completely dependent on @xmath1 . then @xmath21 is a complete dependence copula for which @xmath403 where @xmath216 is a uniform random variable on @xmath30 $ ] and @xmath404 \rightarrow [ 0,1]$ ] is a measure - preserving borel function . note that @xmath405 is also a uniform random variable and that @xmath18 is left invertible . as the first step , we shall construct an invertible copula @xmath151 such that @xmath406 is supported in the two diagonal squares @xmath407 ^ 2 \cup [ 1/2,1]^2 $ ] . let @xmath408)$ ] , denote @xmath409 as defined in and put @xmath410 . by lemma [ lem : shufflingcd ] , @xmath411 is invertible a.e . and hence @xmath151 is invertible . by corollary [ cor : star - shuffling ] , @xmath412 where @xmath413 is still a uniform random variable on @xmath30 $ ] . it is left to verify that the support of @xmath406 lies entirely in the two diagonal squares which can be done by showing that the graph of @xmath414 is contained in the area . in fact , since @xmath415 ) \subseteq a$ ] , it follows that @xmath416 ) \subseteq f(a ) \subseteq [ 0,\frac{1}{2}]$ ] . the inclusion @xmath417 ) \subseteq [ \frac{1}{2},1]$ ] can be shown similarly . as a consequence , @xmath406 can be written as an ordinal sum of two copulas , @xmath187 and @xmath188 , with respect to the partition @xmath418,[\frac{1}{2},1]\}$ ] . since left - multiplying a copula @xmath18 by an invertible copula amounts to shuffling the first coordinate of @xmath18 , it follows that @xmath187 and @xmath188 are still supported on closures of graphs of measure - preserving functions . next , we apply the same process to @xmath187 and @xmath188 which yields invertible copulas @xmath419 and @xmath420 for which @xmath421 and @xmath422 are both supported in @xmath152 ^ 2 \cup [ \frac{1}{2},1]^2 $ ] and define @xmath154 to be the ordinal sum of @xmath419 and @xmath420 with respect to the partition @xmath418,[\frac{1}{2},1]\}$ ] . @xmath154 is again an invertible copula . then the support of @xmath423 is contained in the four diagonal squares @xmath424}^2 $ ] . therefore , @xmath423 is an ordinal sum with respect to the partition @xmath425,[\frac{1}{4},\frac{1}{2}],[\frac{1}{2},\frac{3}{4}],[\frac{3}{4},1]\}$ ] of four copulas each of which is supported on the closure of graph of a measure - preserving function . by successively applying this process , we obtain a sequence of invertible copulas @xmath426 , defined by @xmath427 , such that the support of @xmath428 is a subset of the @xmath167 diagonal squares . so @xmath429 pointwise outside the main diagonal and so are their partial derivatives . hence @xmath430 . thus @xmath431 . if @xmath1 is completely dependent on @xmath2 then @xmath21 is right invertible and similar process where suitably chosen @xmath159 s are multiplied on the right yields a sequence @xmath432 of invertible copulas such that @xmath433 as desired [ cor : l*r ] let @xmath434 and @xmath435 be left invertible and right invertible copulas , respectively . then @xmath436 . from theorem [ thm : cdnorm1 ] , there exist sequences of invertible copulas @xmath159 and @xmath437 such that @xmath438 and @xmath439 in the sobolev norm . by the joint continuity of the @xmath0-product with respect to the sobolev norm , @xmath440 . therefore , @xmath441 . let us give some examples of copulas of the form @xmath442 . consider a copula @xmath18 , @xmath443 and @xmath444 whose supports are shown in the figure below . ( -0.25,-0.25)(4.2,1.2 ) ( 0,1)(1,1 ) ( 1,0)(1,1 ) ( 0,0)(0,1 ) ( 0,0)(1,0 ) ( 1.5,1)(2.5,1 ) ( 2.5,0)(2.5,1 ) ( 1.5,0)(1.5,1 ) ( 1.5,0)(2.5,0 ) ( 3,1)(4,1 ) ( 4,0)(4,1 ) ( 3,0)(3,1 ) ( 3,0)(4,0 ) ( 0.5,0)(0.5,1 ) ( 1.5,0.5)(2.5,0.5 ) ( 3,0.5)(4,0.5 ) ( 3.5,0)(3.5,1 ) as mentioned before , the copula @xmath444 , though neither left nor right invertible , has unit @xmath0-norm . in @xcite , rnyi triggered numerous interests in finding the `` right '' sets of properties that a natural ( if any ) measure of dependence @xmath445 should possess . for reference , the seven postulates proposed by rnyi are listed below . 1 . [ renyi : def ] @xmath445 is defined for all non - constant random variables @xmath1 , @xmath2 . [ renyi : sym ] @xmath446 . [ renyi:01 ] @xmath447 $ ] . [ renyi:0 ] @xmath448 if and only if @xmath1 and @xmath2 are independent . [ renyi:1 ] @xmath449 if either @xmath450 or @xmath451 a.s . for some borel - measurable functions @xmath54 , @xmath107 [ renyi : scale ] if @xmath452 and @xmath453 are borel bijections on @xmath34 then @xmath454 . [ renyi : rho ] if @xmath1 and @xmath2 are jointly normal with correlation coefficient @xmath455 , then @xmath456 . recently , siburg and stoimenov @xcite introduced a measure of mutual complete dependence @xmath457 defined via its copula @xmath324 by @xmath458 . while @xmath459 is defined only for continuous random variables , it satisfies the next three properties [ renyi : sym].[renyi:0 ] . enjoyed by most if not all measures of dependence . however , instead of the conditions [ renyi:1 ] . and [ renyi : scale ] . , @xmath459 satisfies the following conditions which makes it suitable for capturing mutual complete dependence regardless of how the random variables are related . * @xmath460 if and only if there exist borel measurable bijections @xmath54 and @xmath107 such that @xmath450 and @xmath451 almost surely . * if @xmath452 and @xmath453 are strictly monotonic transformations on images of @xmath1 and @xmath2 , respectively , then @xmath461 . now , the property [ renyi : scale].@xmath462 means that @xmath459 is invariant under only strictly monotonic transformations of random variables . using the @xmath0-norm which is invariant under all borel measurable bijections , we define @xmath463 where the last equality follows from proposition [ props](3 ) . since the @xmath0-norm shares many properties with the sobolev norm ( see proposition [ prop:*norm ] ) , the properties of @xmath464 are for the most part analogous to those of @xmath459 s . main exceptions are that [ renyi:1].@xmath462[renyi : scale].@xmath462 are replaced back by [ renyi:1].[renyi : scale ] . 1 . 2 . @xmath467 3 . @xmath468 if and only if @xmath1 and @xmath2 are independent . [ * :1 ] @xmath469 if @xmath2 is completely dependent on @xmath1 or @xmath1 is completely dependent on @xmath2 . [ * : scale ] if @xmath54 and @xmath107 are borel measurable bijective transformations , then we have @xmath470 6 . if @xmath471 is a sequence of pairs of continuous random variables with copulas @xmath472 and if @xmath473 , then @xmath474 1 . follows from the fact that @xmath475 . see proposition [ props ] . 2 . is clear from the definition of @xmath476 and the fact that @xmath477 $ ] . the statement 3 . is a result of proposition [ props ] which says that @xmath478 if and only if @xmath479 . 4 . follows immediately from theorem [ thm : cdnorm1 ] . to prove 5 . , let @xmath480 be borel measurable bijective transformations . then , @xmath1 and @xmath321 are mutually completely dependent , and so are @xmath2 and @xmath320 . thus @xmath481 and @xmath482 . therefore , the copulas @xmath483 and @xmath484 are invertible . hence @xmath485 finally , 6 . can be proved via the inequality @xmath486 therefore , we have constructed a measure of dependence for continuous random variables which satisfies all of renyi s postulates except possibly the last condition [ renyi : rho ] . the @xmath0-norm of a convex sum of a unit @xmath0-norm copula and the independence copula is computed . corollary [ cor : l*r ] implies that there are many more copulas with unit @xmath0-norm , i.e. any copulas of the form @xmath490 where @xmath480 are borel measurable transformations . by the characterization of idempotent copulas in @xcite , all singular idempotent copulas are of this form and hence have unit @xmath0-norm . 00 brown , j.r . doubly stochastic measures and markov operators , michigan math . j. 12:367375 . chou , s.h . nguyen , t.t . on frchet theorem in the set of measure preserving functions over the unit interval . international journal of mathematics and mathematical sciences 13:373378 . chaidee , n. santiwipanont , t. sumetkijakan , s. ( 2012 ) . denseness of patched mins in the sobolev norm , preprint . darsow , w.f . nguyen , b. olsen , e.t . copulas and markov processes . illinois j. math 36:600642 . darsow , w.f . olsen , e.t . norms for copulas . international journal of mathematics and mathematical sciences 18:417436 . darsow , w.f . olsen , e.t . characterization of idempotent 2-copulas . note di matematica 30:147177 . mikusinski , p. sherwood , h. taylor , m.d . probabilistic interpretations of copulas and their convex sums . in : dallaglio , g. kotz , s. salinetti , g. ed . , _ advances in probability distributions with given marginals : beyond the copulas . _ 67:95112 . mikusinski , p. sherwood , h. taylor , m.d . shuffles of min , stochastica 13:6174 nelsen , r.b . ( 2006 ) . _ an introduction to copulas , _ 2@xmath491 ed . springer verlag . siburg , k.f . stoimenov , p.a . ( 2008 ) . a scalar product for copulas . journal of mathematical analysis and applications 344:429439 . siburg , k.f . stoimenov , p.a . ( 2009 ) . a measure of mutual complete dependence . metrika 71 : 239251 . sklar , m. ( 1959 ) . fonctions de rpartition @xmath56 dimensions et leurs marges . paris 8 : 229231 .
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using a characterization of mutual complete dependence copulas , we show that , with respect to the sobolev norm , the mcd copulas can be approximated arbitrarily closed by shuffles of min .
this result is then used to obtain a characterization of generalized shuffles of copulas introduced by durante , sarkoci and sempi in terms of mcd copulas and the @xmath0-product discovered by darsow , nguyen and olsen .
since shuffles of a copula is the copula of the corresponding shuffles of the two continuous random variables , we define a new norm which is invariant under shuffling .
this norm gives rise to a new measure of dependence which shares many properties with the maximal correlation coefficient , the only measure of dependence that satisfies all of rnyi s postulates .
copulas , shuffles of min , measure - preserving , sobolev norm @xmath0-product , shuffles of copulas , measure of dependence 28a20 , 28a35 , 46b20 , 60a10 , 60b10
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nowadays , in condensed matter physics and semiconductor microelectronics , two - dimensional ( 2d ) electron system is one of the main objects of detailed study . such a system is formed by , e.g. , surface - state electrons or electrons in semiconductor heterostructures . phenomenon that is observed in such systems and makes them of great interest , especially in context of spintronic applications , is spin - orbit interaction ( soi ) . this interaction arise from the structure inversion asymmetry of potential confining the electron system in directions perpendicular to the confinement plane ( the rashba spin - orbit interaction@xcite ) and the bulk inversion asymmetry that is present in semiconductor heterostructures based on materials with a zinc - blende structure ( the dresselhaus spin - orbit interaction@xcite ) . the dresselhaus interaction depends on semiconductor material and growth geometry , whereas the interaction strength of the rashba soi can be tuned via an externally applied electric field perpendicular to the confinement plane.@xcite as a result , one can controllably manipulate the spin in devices without recourse to an external magnetic field.@xcite . in order to efficiently exploit the mentioned phenomenon , a theoretical study of dynamics of electrons and holes in the 2d spin - orbit coupled electron systems is needed . the most discussed and studied processes concerning this problem are spin relaxation and spin dephasing.@xcite however , to our knowledge , such crucial quasiparticle property as the lifetime caused by inelastic electron - electron scattering remains still insufficiently studied . to all appearance the first attempt to analyze what effect the soi has on the quasiparticle lifetime has been made in ref . . in the work cited , a particular case of the 2d electron gas ( 2deg ) with the rashba soi was considered at the limit of @xmath1 , where @xmath2 is the fermi energy and @xmath3 with @xmath4 and @xmath5 being the interaction strength and the effective electron mass , respectively ( unless stated otherwise , atomic units are used throughout , i.e. , @xmath6 . ) . within the @xmath0 approximation , it has been shown that in a small vicinity of @xmath2 a modification of the lifetime due to the soi is insignificant and does not depend on the subband index of the spin - orbit split band . to go beyond the limits of ref . , in ref . the inelastic lifetime ( decay rate ) of quasiparticles in the 2deg with the rashba soi has been studied within a wide energy region . for material parameters typical for in@xmath7ga@xmath8as 2degs , it has been revealed that modifications induced by the soi and the dependence on the subband index become noticeable , when the decay channel due to plasmon emission appears . the first joint theoretical and experimental investigation of hole lifetimes in a 2d spin - orbit coupled electron system has been done in ref . . in addition to a demonstration of the weak influence of the soi on hole lifetimes by the case of the au(111 ) surface state , a hypothetical system , where the soi can have a profound effect , has been considered . in this work , we generalize the results on effect of the soi on the quasiparticle lifetime . within the @xmath0 approach with the screened interaction @xmath9 evaluated in the random phase approximation ( rpa ) , we study the inelastic decay rate of quasiparticles in a 2deg with the rashba and dresselhaus interactions linear in @xmath10magnitude of the electron 2d momentum @xmath11 . in our @xmath0-calculations , material parameters suitable for inas quantum wells are taken . we compare the inelastic decay rates calculated at different ratios between the interaction strengths of the mentioned spin - orbit interactions . we show that on the energy scale , for the taken material parameters , the main visible effect induced by the soi is modifications of the plasmon - emission decay channel via the extension of the landau damping region . we also consider a hypothetical small - density case , when in the 2d spin - orbit coupled electron system the fermi level is close to the band energy at @xmath12 . for such a system , we predict strong subband - index dependence and anisotropy of the inelastic decay rate for electrons and appearance of a plasmon decay channel for holes . and @xmath13 determining the spin - quantization axis with polar angles @xmath14 and @xmath15 and the rotation axis , respectively . ] we consider a 2deg described by the hamiltonian @xmath16 with @xmath17 and the spin - orbit contribution @xmath18 that includes both rashba and dresselhaus terms . the latter is written with the assumption that a quantum well grown in [ 001 ] direction is considered . in eq . ( [ hamiltonian ] ) , @xmath19 are the electron momenta along the [ 100 ] and [ 010 ] cubic axes of the crystal , respectively , @xmath20 are the pauli matrices , @xmath5 is the effective electron mass , @xmath4 and @xmath21 are the interaction strengths for the rashba and dresselhaus spin - orbit interactions . to bring the hamiltonian to a diagonal form , we perform the rotation in spin space generated by @xmath22 $ ] dependent on the momentum @xmath11 . the rotation is performed with the angle @xmath14 around the axis determined by @xmath23 . a positional relationship of the axis @xmath23 and the spin - quantization axis @xmath24 is shown in fig . [ fig1 ] . we suppose that we deal with the in - plane spin polarization , i.e. , @xmath25 . in the new , unitary transformed , spin basis the spin - orbit contribution has the form@xcite @xmath26\sigma_{z},\end{aligned}\ ] ] where the angle @xmath15 is related to the polar angle @xmath27 of the momentum @xmath11 as @xmath28 due to the diagonal form of @xmath29 , the energy bands are simply given by@xcite @xmath30\ ] ] and correspond to the wave functions @xmath31 with the subband index @xmath32 , where @xmath33 are the spin components in the new spin basis . this means that for the initial , untransformed , hamiltonian we have the following eigenstates @xmath34 . the spin orientation in @xmath11 space reads as ( see fig . [ fig2 ] ) @xmath35 note that the case with @xmath36 and @xmath37 ( pure rashba ) is characterized by the angle @xmath38 , whereas in the situation with @xmath39 and @xmath40 ( pure dresselhaus ) one has @xmath41 . in the special case of @xmath42 , the angle @xmath43 does not depend on @xmath27 . the inelastic decay rate ( inverse lifetime @xmath44 caused by inelastic electron - electron scattering ) is determined by the imaginary part of the matrix elements of the quasiparticle self - energy @xmath45 at the energy @xmath46 as @xmath47 . at the hartree - fock ( hf ) mean - field level , these elements are totally real and have the form @xmath48 where @xmath49 is the bare coulomb interaction with @xmath50 being the static dielectric constant . the factors @xmath51/2 $ ] come from @xmath52 and @xmath53 is the fermi factor . such a form ( [ hf_sigma ] ) is similar to the exchange contribution to the single - particle energies considered in refs . in the pure rashba case . however , in order to examine quasiparticle lifetimes , one has to go beyond the hf approximation . the simplest variant is the @xmath0 approximation ( for details about the approximation , we refer the reader to refs . and ) . within such an approximation , we arrive at the following expression for the imaginary part of the mentioned matrix elements @xmath54 when @xmath55 , and @xmath56\\ & \times&\mathrm{im}w^0(\mathbf{k}-\mathbf{q},\omega - e_{\mathbf{q}s'})\theta(\omega - e_{\mathbf{q}s'}),\nonumber\end{aligned}\ ] ] when @xmath57 . in these equations , @xmath58 is the step function and the screened interaction @xmath59^{-1}\ ] ] is defined by the rpa irreducible polarizability ( see also refs . and , where the retarded part of @xmath60 was examined ) @xmath61 using the example of an inas quantum well , we take the effective mass @xmath62 ( see , e.g. , ref . ) and the static dielectric constant @xmath63 ( see , e.g. , ref . ) . the interaction strength of the dresselhaus soi is chosen to be @xmath64 evm to simulate a quite narrow quantum well.@xcite at the ratio @xmath65 ( see , e.g. , ref . ) , we have the rashba interaction strength @xmath66 evm . the electron density is put at @xmath67 @xmath68 that corresponds to the fermi energy @xmath69 mev . [ fig3 ] shows our results on the inelastic decay rate @xmath70 obtained with the material parameters listed above . two main points caused by the spin - orbit splitting of the band make the considered 2deg different from that without the soi . these are an angle - dependent relative shift of @xmath71 and @xmath72 on the momentum scale and some smoothing of sharp forms of the peak caused by opening of the plasmon decay channel . the former reflects the fact that subbands of the split band reach the same energy at different momenta , while the latter originates from the extension of the landau damping region ( the region where plasmons decay into single - particle excitations@xcite ) due to appearance of inter - subband transitions ( for a detailed discussion of the screening properties of the 2deg with the soi we refer the reader to refs . and ) this extension varying with the polar angle @xmath27 leads to a nonzero plasmon linewidth , when the plasmon spectrum enters into the soi - induced damping region . in order to show what effect the soi has on the inelastic decay rate for different subbands , in the inset of fig . [ fig3 ] , by setting up a correspondence between @xmath70 and @xmath73 via the momentum @xmath11 , we plot the decay rate as a function of energy . on first glance , it may seem that we have an ordinary energy dependence of the decay rate as in a 2deg without the soi : the quadratic behavior with the logarithmic enhancement in the vicinity of the fermi energy with @xmath74 at @xmath2 and the jump above the fermi energy , which is caused by opening the plasmon decay channel for excited electrons.@xcite however , on examining the energy dependence of @xmath75 in detail , we can say that due to the finite plasmon linewidth the plasmon decay channel manifests itself at lower energies , when it occurs in a 2deg without the soi . the same reason leads to reduction in the jump . also , we can reveal distinctions between @xmath71 and @xmath72 , which become noticeable , when the plasmon - emission decay channel appears , and increase upon moving from @xmath76 to @xmath77 . an analysis of the inelastic mean - free path ( imfp ) @xmath78 as a function of energy has shown that , as a consequence of the distinctions between @xmath71 and @xmath72 , the imfp of electrons can vary with the subband index . for example , at @xmath79 for electrons this variation can reach , e.g. , @xmath80% . the obtained results can be understood by inspecting constant - energy contours shown in fig . [ fig2 ] with mental drawing of possible transitions selected by the factors @xmath81 of eqs . ( [ me_sigma_im_below_ef ] ) and ( [ me_sigma_im_above_ef ] ) at a given @xmath27 ( see also ref . ) . actually , for each subband ( @xmath82 ) one has a set of intra- and inter - subband transition momenta as arguments of @xmath83 . for the chosen material parameters and for @xmath76 , these momenta do not vary considerably with the subband index @xmath84 . for @xmath79 differences in both intra- and inter - subband transitions for @xmath85 and @xmath86 become already sensible for values of @xmath83 , especially in the vicinity of plasmon peaks of the latter . now , remaining @xmath5 , @xmath50 , @xmath87 , and @xmath2 unchanged , we consider the case of @xmath88 [ @xmath89 evm ] and the pure dresselhaus ( rashba ) case [ @xmath90 evm and @xmath91 . the case of equal interaction strengths , when the rashba and dresselhaus interactions can cancel each other , is distinguished by various significant effects reported in the literature ( see , e.g. , refs . ) . in this case , one has the 2d electron system with two uncoupled spin components ( see fig . [ fig2 ] ) , each of which demonstrates the properties peculiar to a 2deg without the soi.@xcite our results@xcite on the inelastic decay rate at @xmath88 are shown in fig . the sharp edges of the plasmon contribution are evidence of the fact that there is no modifications of the landau damping region induced by the soi . as is seen from the inset of the figure , due to the shifting property @xmath92 , where @xmath93 , the @xmath71 and @xmath72 curves coincide and have the form of that in a 2deg without the soi . in the pure rashba or pure dresselhaus cases ( see fig . [ fig4 ] ) , the resulting @xmath75 does not tell the difference between spin orientations in the momentum plane , which correspond to the rashba or dresselhaus soi ( see fig . [ fig2 ] ) . as well as before , we have the relative shift ( but angle - independent ) on the momentum scale and main modifications induced by the soi in the energy region , where a quasiparticle can decay into plasmons . all the considered cases meet the condition of @xmath94 , where @xmath95 is the measure of influence of the soi on the band structure . however , as is partly discussed in ref . , in two - dimensional electron systems with much greater @xmath96 as compared to @xmath2 the inelastic decay rate can substantially differ from that in the 2deg without the soi . a striking example of such a system is that formed by surface - state electrons in ordered surface alloys,@xcite which are very promising materials for spintronics applications . in order to predict how the inelastic decay rate can behave in a system , where @xmath97 , we consider the hypothetical case with the unchanged @xmath62 , @xmath63 , and @xmath64 evm , but with @xmath98 evm ( @xmath99 ) and @xmath100 @xmath68 , which give @xmath101 mev and @xmath102 mev . the obtained results are presented in fig . [ fig5 ] . the main feature we would like to note first is that for holes the decay rate @xmath72 as a function of @xmath10 has an `` outgrowth '' at @xmath103 , where @xmath104^{1/2}$ ] the momentum , at which @xmath105 has a minimum . a close analysis of the imaginary part of the screened interaction @xmath9 and the region of integration in eq . ( [ me_sigma_im_below_ef ] ) has shown that the outgrowth is caused by opening of the plasmon decay channel for transitions between the @xmath86 and @xmath106 subbands . it is important that in a 2deg without the soi such a channel is impossible for holes.@xcite as is evident from the figure , in this case on the energy scale we have a strong anisotropy of the inelastic decay rate . also the presented curves clearly demonstrate that the latter depends strongly on the subband index @xmath84 of the spin - orbit split band . keeping in mind that the index @xmath84 distinguishes spin components , we can say that the subband - index dependence reflects a spin asymmetry of the inelastic decay rate @xmath107 for a given direction . the asymmetry shows its worth most brightly in the @xmath79 direction . in fact , in that very direction there are significant distinctions in @xmath72 and @xmath71 as functions of the exciting energy and , as a consequence , in the corresponding imfp for electrons . for instance , the ratio @xmath107 is about 3 at @xmath108 mev and about 2 at @xmath109 mev . at further increasing of energy , the ratio continues to decrease . note that the imfp spin asymmetry makes a basis of the spin filter effect observed in hot electron transport through a ferromagnetic ( see , e.g. , refs . and ) . in the considered case of the quantum well , the spin asymmetry is not such big as in ferromagnetics ( see , e.g. , ref . ) , but , as distinct from the latters , values of the imfp spin symmetry depend strongly on direction and can be tuned by external electric field . in conclusion , we have presented a study of the inelastic decay rate of quasiparticles in a two - dimensional electron gas with the @xmath10-linear spin - orbit interaction that includes both rashba ( interaction strength @xmath4 ) and dresselhaus ( interaction strength @xmath21 ) contributions . in this study , the electron gas is characterized by material parameters suitable for [ 001]-grown inas quantum wells . we have considered the cases of @xmath110 , @xmath42 , @xmath40 ( @xmath111 ) , and @xmath37 ( @xmath112 ) . the cases meet the condition of @xmath94 , where @xmath95 is the measure of influence of the spin - orbit interaction on the band structure . as compared to a two - dimensional electron gas without the spin - orbit interaction , we have revealed a relative shift of the inelastic decay rates for different subbands of the spin - orbit split band on the momentum scale . also , except for the case of equal interaction strengths , we have found a some smoothing of sharp forms of the peak concerned with opening of the plasmon decay channel for electrons . we have shown that , on the energy scale , in this very region distinctions between the decay rates for different subbands become noticeable . these distinctions depend on the polar angle @xmath27 and cause the inelastic mean free path to be angle- and subband - dependent . as to the case of @xmath42 , due to the shifting property , the decay rate as a function of energy has the form of that in a two - dimensional electron gas without the spin - orbit interaction . in order to predict how the inelastic decay rate can behave in a system , where @xmath97 , we have considered the hypothetical case of small electron density . we have revealed that in such a system the decay rate demonstrates strong anisotropy and subband dependence within all the considered interval of momenta and exciting energies . since the subband dependence can be interpreted as a spin asymmetry of the decay rate in a given direction of @xmath11 , one can expect the spin - filter effect driven by externally applied electric field . also , we have found that in the system with @xmath97 holes can decay into plasmons , what is impossible in a two - dimensional electron gas without the spin - orbit interaction . we acknowledge partial support from the university of the basque country ( grant no . gic07it36607 ) and the spanish ministerio de ciencia y tecnologa ( grant no . fis2007 - 66711-c02 - 01 ) . calculations were partly performed on skif - cyberia supercomputer of tomsk state university . rashba , sov . solid state * 2 * , 1109 ( 1960 ) ; 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( [ energies ] ) can be cast into the form that is frequently used in the literature . actually , from @xmath117\right)^2 $ ] with the help of eq . ( [ phi_k_def ] ) we arrive at @xmath118 $ ] . the latter can be solved as @xmath119^{1/2}$ ] . however , in such an expression the subband index @xmath82 distinguishes the inner- and outer - branch and , e.g. , in the case of @xmath42 does not correspond to spin components . bernevig , j. orenstein , and s .- c . zhang , phys . lett . * 97 * , 236601 ( 2006 ) . g .- h . chen and m.e . raikh , phys . b * 60 * , 4826 ( 1999 ) . s. chesi and g.f . giuliani , phys . b * 75 * , 155305 ( 2007 ) . juri and p.i . tamborenea , phys . b * 77 * , 233310 ( 2008 ) . nechaev , i.yu . sklyadneva , v.m . silkin , p.m. echenique , and e.v . chulkov , phys . b * 78 * , 085113 ( 2008 ) . m. pletyukhov and v. gritsev , phys . b * 74 * , 045307 ( 2006 ) . badalyan , a. matos - abiague , g. vignale , and j. fabian , phys . b * 79 * , 205305 ( 2009 ) . w. knap , c. skierbiszewski , a. zduniak , e. litwin - staszewska , d. bertho , f. kobbi , j.l . robert , g.e . pikus , f.g . pikus , s.v . iordanskii , v. mosser , k. zekentes , yu . b. lyanda - geller , phys . b * 53 * , 3912 ( 1996 ) . o.g . lorimor and w.g . spitzer , j. appl . phys . * 36 * , 1841 ( 1965 ) . s. giglberger , l.e . golub , v.v . belkov , s.n . danilov , d. schuh , c. gerl , f. rohlfing , j. stahl , w. wegscheider , d. weiss , w. prettl , and s.d . ganichev , phys . b * 75 * , 035327 ( 2007 ) . h. bruus and k. flensberg , _ many - body quantum theory in condensed matter physics : an introduction _ ( oxford university press , oxford , 2004 ) . g. f. giuliani , g. vignale , _ quantum theory of the electron liquid _ ( cambridge university press , cambridge , 2005 ) . n.s . averkiev and l.e . golub , phys . b * 60 * , 15582 ( 1999 ) . j. schliemann , j.c . egues , and d. loss , phys . 90 * , 146801 ( 2003 ) . koralek , c.p . weber , j. orenstein , b.a . bernevig , s .- c . zhang , s. mack , d.d . awschalom , nature * 458 * , 610 ( 2009 ) . we have chosen the direction @xmath76 , in which we can reproduce the momentum dependence of the inelastic decay rate peculiar for a 2deg without the soi . for each spin components , such a dependence becomes isotropic if one shifts the origin in the momentum plane on the vector @xmath120 . ast , j. henk , a. ernst , l. moreschini , m.c . falub , d. pacil , p. bruno , k. kern , and m. grioni , phys . * 98 * , 186807 ( 2007 ) ; c.r . ast , d. pacil , l. moreschini , m.c . falub , m. papagno , k. kern , m. grioni , j. henk , a. ernst , s. ostanin , and p. bruno , phys . b * 77 * , 081407(r ) ( 2008 ) ; h. mirhosseini , j. henk , a. ernst , s. ostanin , c .- t . chiang , p. yu , a. winkelmann , and j. kirschner , phys . rev . b * 79 * , 245428 ( 2009 ) . v.p . zhukov and e.v . chulkov , phys . usp . * 52 * , 105 ( 2009 ) . i.a . nechaev and e.v . chulkov , phys . solid state * 51 * , 754 ( 2009 ) .
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we present a study of the inelastic decay rate of quasiparticles in a two - dimensional electron gas with spin - orbit interaction .
the study is done within the @xmath0 approximation .
the spin - orbit interaction is taken in the most general form that includes both rashba and dresselhaus contributions linear in magnitude of the electron 2d momentum .
spin - orbit interaction effect on the inelastic decay rate is examined at different parameters characterizing the electron gas and the spin - orbit interaction strength in it .
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understanding biodiversity and coevolution is a central challenge in modern evolutionary and theoretical biology @xcite . in this context , for some decades much effort has been devoted to mathematically model dynamics of competing populations through nonlinear , yet deterministic , set of rate equations like the equations devised by lotka and volterra @xcite or many of their variants @xcite . this heuristic approach is often termed as population - level description . as a common feature , these deterministic models fail to account for stochastic effects ( like fluctuations and spatial correlations ) . however , to gain some more realistic and fundamental understanding on generic features of population dynamics and mechanisms leading to biodiversity , it is highly desirable to include internal stochastic noise in the description of agents kinetics by going beyond the classical deterministic picture . one of the main reasons is to account for discrete degrees of freedom and finite - size fluctuations @xcite . in fact , the deterministic rate equations always ( tacitly ) assume the presence of infinitely many interacting agents , while in real systems there is a _ large _ , yet _ finite _ , number of individuals ( recently , this issue has been addressed in refs . @xcite ) . as a consequence , the dynamics is intrinsically stochastic and the unavoidable finite - size fluctuations may have drastic effects and even completely invalidate the deterministic predictions . interestingly , both _ in vitro _ @xcite and _ in vivo _ @xcite experiments have recently been devoted to experimentally probe the influence of stochasticity on biodiversity : the authors of refs . @xcite have investigated the mechanism necessary to ensure coexistence in a community of three populations of _ escherichia coli _ and have numerically modelled the dynamics of their experiments by the so - called ` rock - paper - scissors ' model , well - known in the field of game theory @xcite . this is a three - species cyclic generalization of the lotka - volterra model @xcite . as a result , the authors of ref . @xcite reported that in a well - mixed ( non - spatial ) environment ( i.e. when the experiments were carried out in a flask ) two species got extinct after some finite time , while coexistence of the populations was never observed . motivated by these experimental results , in this work we theoretically study the stochastic version of the cyclic lotka - volterra model and investigate in detail the effects of finite - size fluctuations on possible population extinction / coexistence . for our investigation , as suggested by the flask experiment of ref . @xcite , the stochastic dynamics of the cyclic lotka - volterra model is formulated in the natural language of urn models @xcite and by adopting the so - called individual - based description @xcite . in the latter , the explicit rules governing the interaction of a _ finite number _ of individuals with each other are embodied in a master equation . the fluctuations are then specifically accounted for by an appropriate fokker - planck equation derived from the master equation via a so - called van kampen expansion @xcite . this program allows us to quantitatively study the deviations of the stochastic dynamics of the cyclic lotka - volterra model with respect to the rate equation predictions and to address the question of the extinction probability , the computation of which is the main result of this work . from a more general perspective , we think that our findings have a broad relevance , both theoretical and practical , as they shed further light on how stochastic noise can dramatically affect the properties of the numerous nonlinear systems whose deterministic description , like in the case of the cyclic lotka - volterra model , predicts the existence of neutrally stable solutions , i.e. cycles in the phase portrait @xcite . this paper is organized as follows : the cyclic lotka - volterra model is introduced in the next section and its deterministic rate equation treatment is presented . in section [ stoch_appr ] , we develop a quantitative analytical approach that accounts for stochasticity , a fokker - planck equation is derived from the underlying master equation within a van kampen expansion . this allows us to compute the variances of the agents densities . we also study the time - dependence properties of the system by carrying out a fourier analysis from a set of langevin equations . section [ sect - ext - prob ] is devoted to the computation of the probability of having extinction of two species at a given time , which constitutes the main issue of this work . in the final section , we summarize our findings and present our conclusions . , @xmath0 , and @xmath1 . the latter may correspond to the strategies in a rock - paper - scissors game @xcite , or to different bacterial species @xcite . [ cycle ] ] the cyclic lotka volterra model under consideration here is a system where three states @xmath2 , @xmath0 , and @xmath1 cyclically dominate each other : @xmath2 invades @xmath0 , @xmath0 outperforms @xmath1 , and @xmath1 in turn dominates over @xmath2 , schematically drawn in fig . [ cycle ] . these three states @xmath2 , @xmath0 , and @xmath1 allow for various interpretations , ranging from strategies in the rock - paper - scissors game @xcite over tree , fire , ash in forest fire models @xcite or chemical reactions @xcite to different bacterial species @xcite . in the latter case , a population of poison producing bacteria was brought together with another being resistant to the poison and a third which is not resistant . as the production of poison as well as the resistance against it have some cost , these species show a cyclic dominance : the poison - producing one invades the non - resistant , which in turn reproduces faster than the resistant one , and the latter finally dominates the poison - producing . in a well - mixed environment , like a flask in the experiments @xcite , eventually only one species survives . the populations are large but finite , and the dynamics of reproduction and killing events may to a good approximation be viewed as stochastic . ) , red or medium gray ( @xmath0 ) , and blue or dark gray ( @xmath1 ) . at each time step , two random individuals are chosen ( indicated by arrows in the left picture ) and react ( right picture ) . , title="fig : " ] ) , red or medium gray ( @xmath0 ) , and blue or dark gray ( @xmath1 ) . at each time step , two random individuals are chosen ( indicated by arrows in the left picture ) and react ( right picture ) . , title="fig : " ] motivated from these biological experiments , we introduce a stochastic version of the cyclic lotka - volterra model . consider a population of size @xmath3 which is well mixed , i.e. in the absence of spatial structure . the stochastic dynamics used to describe its evolution , illustrated in fig . [ urn ] , is referred to as `` urn model '' @xcite and closely related to the moran process @xcite . at every time step , two randomly chosen individuals are selected , which may at certain probability react according to the following scheme : @xmath4 with reaction rates @xmath5 , and @xmath6 . we observe the cyclic dominance of the three species . also , the total number @xmath3 of individuals is conserved by this dynamics ; this will of course play a role in our further analysis . we now proceed with the analysis of the deterministic version of the system ( [ react ] ) . this will prove insightful for building a stochastic description of the model , which is the scope of sec . [ stoch_appr ] . the deterministic rate equations describe the time evolution of the densities @xmath7 , @xmath8 , and @xmath9 for the species @xmath2 , @xmath0 , and @xmath1 ; they read @xmath10 where the dot stands for the time derivative . these equations describe a well - mixed system , without any spatial correlations , as naturally implemented in urn models @xcite or , equivalently , infinite dimensional lattice systems or complete graphs . in the following , the eqs . ( [ re ] ) are discussed and , from their properties , we gain intuition on the effects of stochasticity . already from the basic reactions ( [ react ] ) we have noticed that the total number of individuals is conserved , which is a property correctly reproduced by the the rate equations ( [ re ] ) . setting the total density , meaning the sum of the densities @xmath11 , @xmath12 , and @xmath13 , to unity , we obtain @xmath14 for all times @xmath15 . only two out of the three densities are thus independent , we may view the time evolution of the densities in a two - dimensional phase space . ( [ re ] ) together with ( [ total_dens ] ) admit three trivial ( absorbing ) fixed points : @xmath16 ; @xmath17 ; and @xmath18 . they denote states where only one of the three species survived , the other ones died out . in addition , the rate equations ( [ re ] ) also predict the existence of a fixed point @xmath19 which corresponds to a reactive steady state , associated with the coexistence of all three species : @xmath20 to determine the nature of this fixed point , we observe that another constant of motion exists for the rate equations ( [ re ] ) , namely the quantity @xmath21 does not evolve in time . in contrast to the total density ( [ total_dens ] ) , this constant of motion is only conserved by the rate equations but does not stem from the reaction scheme ( [ react ] ) . hence , when considering the stochastic version of the cyclic model , the total density remains constant but the expression ( [ const2 ] ) will no longer be a conserved quantity . the above fixed point ( [ fp - c ] ) and constant of motion ( [ const2 ] ) have been derived and discussed also within the framework of game theory , see e.g. ref . @xcite . in fig . [ simplex_as ] , we depict the ternary phase space @xcite for the densities @xmath11 , @xmath12 , and @xmath13 : the solutions of the rate equations ( [ re ] ) are shown for different initial conditions and a given set of rates @xmath22 , @xmath23 , and @xmath6 . as the rate equations ( [ re ] ) are nonlinear in the densities , we can not solve them analytically , but use numerical methods . due to the constant of motion , the solutions yield cycles around the reactive fixed point ( thus corresponding to case 3 in durrett and levin s classification @xcite ) . the three trivial steady states , corresponding to saddle points within the linear analysis , are the edges of the simplex . the reactive stationary state , as well as the cycles , are neutrally stable , stemming from the existence of the constant of motion ( [ const2 ] ) . especially , the reactive fixed point is a _ center fixed point_. the boundary of the simplex denotes states where at least one of the three species died out ; as cyclic dominance is lost , states that have reached this boundary will evolve towards one of the edges , making the boundary _ absorbing_. . the rate equations predict cycles , which are shown in black . their linearization around the reactive fixed point is solved in proper coordinates @xmath24 ( blue or dark gray ) . the red ( or light gray ) erratic flow denotes a single trajectory in a finite system ( @xmath25 ) , obtained from stochastic simulations . it spirals out from the reactive fixed point , eventually reaching an absorbing state . [ simplex_as ] ] the nonlinearity of eqs . ( [ re ] ) induces substantial difficulties in the analytical treatment . however , much can already be inferred from the linearization around the reactive fixed point ( [ fp - c ] ) , which we will consider in the following . we therefore introduce the deviations from the reactive fixed point , denoted as @xmath26 : @xmath27 using conservation of the total density ( [ total_dens ] ) , we can eliminate @xmath28 , and the remaining linearized equations ( [ re ] ) may be put into the form @xmath29 , with the vector @xmath30 and the matrix @xmath31 the reactive fixed point is associated to the eigenvalues @xmath32 of @xmath33 , where @xmath34 is given by @xmath35 oscillations with this frequency arise in its vicinity . in proper coordinates @xmath36 , these oscillations are harmonic , being the solution of @xmath37 , with @xmath38 . for illustration , we have included these coordinates , in which the solutions take the form of circles around the origin , in fig . [ simplex_as ] . the linear transformation @xmath39 is given by @xmath40 the equations @xmath41 are easily solved : @xmath42 eventually , we obtain the solutions for the linearized rate equations : @xmath43\cos(\omega_0 t)\cr & + \omega_0\bigg\{\frac{1}{k_c}[a(0)-a^*]+\frac{k_b+k_c}{k_bk_c}[b(0)-b^*]\bigg\}\sin(\omega_0 t)\cr \label{re - lin - sol}\end{aligned}\ ] ] where @xmath8 and @xmath9 follow by cyclic permutations . to establish the validity of the linear analysis ( [ a - lin])-([re - lin - sol ] ) , we compare the ( numerical ) solution of the rate equations ( [ re ] ) with the linear approximations ( [ re - lin - sol ] ) . as shown in fig . [ dens_comp ] , when @xmath44 , the agreement between the nonlinear rate equations ( [ re ] ) and the linear approximation ( [ re - lin - sol ] ) is excellent , both curves almost coincide . on the other hand , the nonlinear terms appearing in eqs . ( [ re ] ) become important already when @xmath45 and are responsible for significant discrepancies both in the amplitudes and frequency from the predictions of eq . ( [ re - lin - sol ] ) . ( color online ) the deterministic time - evolution of the density @xmath11 for small and large amplitudes . the prediction ( [ re - lin - sol ] ) , shown in black , is compared to the numerical solution ( red or gray ) of the rate equations ( [ re ] ) . for small amplitudes ( @xmath46 ) , both coincide . however , for large amplitudes ( @xmath47 ) , they considerably differ both in amplitude and frequency . we used reaction rates @xmath48 . ] we now aim at introducing a measure of distance @xmath49 to the reactive fixed point within the phase portrait . in the next section , this quantity will help quantify effects of stochasticity . as it was recently proved useful in a related context @xcite , we aim at taking the structure of cycles predicted by the deterministic equations , see fig . [ simplex_as ] , into account by requiring that the distance should not change on a given cycle . motivated from the constant of motion ( [ const2 ] ) , we introduce @xmath50 with the normalization factor ( see below ) @xmath51 being conserved by the eqs . ( [ re ] ) , @xmath49 remains constant on every deterministic cycle . as it vanishes at the reactive fixed point and monotonically grows when departing from it , @xmath49 yields a measure of the distance to the latter . expanding the radius @xmath49 in small deviations @xmath52 from the reactive fixed point results in @xmath53\cr & + o({\bf x}^2)\quad . \label{r - x}\end{aligned}\ ] ] in the variables @xmath36 , with our choice for @xmath54 , it simplifies to @xmath55 corresponding to the radius of the deterministic circles , which emerge in the variables @xmath56 . the fact that the number @xmath3 of particles is _ finite _ induces fluctuations that are not accounted for by eqs . ( [ re ] ) . in the following , our goal is to understand the importance of fluctuations and their effects on the deterministic picture ( [ re ] ) . we show that , due to the neutrally stable character of the deterministic cycles , fluctuations have drastic consequences . intuitively , we expect that in the presence of stochasticity , each trajectory performs a random walk in the phase portrait , interpolating between the deterministic cycles ( as will be revealed by considering @xmath49 ) , eventually reaching the boundary of the phase space . there , the cyclic dominance is completely lost , as one of the species gets extinct . of the two remaining ones , one species is defeating the other , such that the latter soon gets extinct as well , leaving the other one as the only survivor ( this corresponds to one of the trivial fixed points , the edges of the ternary phase space ) . we thus observe the boundary to be absorbing , and presume the system to always end up in one of the absorbing states . a first indication of the actual emergence of this scenario can be inferred from the stochastic trajectory shown in fig . [ simplex_as ] . in this section , we set up a stochastic description for the cyclic lotka - volterra model in the urn formulation with finite number @xmath3 of individuals . starting from the master equation of the stochastic process , we obtain a fokker - planck equation for the time evolution of the probability @xmath57 of finding the system in state @xmath58 at time @xmath15 . it allows us to gain a detailed understanding of the stochastic system . in particular , we will find that , as anticipated at the end of the last section , after long enough time , the system reaches one of the absorbing states . our main result is the time - dependence of the extinction probability , being the probability that , starting at a situation corresponding to the reactive fixed point , after time @xmath15 two of the three species have died out . it is obtained through mapping onto a known first - passage problem . we compare our analytical findings to results from stochastic simulations . for the sake of clarity and without loss of generality , throughout this section , the case of equal reaction rates @xmath59 is considered . details on the unequal rates situation are relegated to appendix [ app_gen_rates ] . we carried out extensive stochastic simulations to support and corroborate our analytical results . an efficient simulation method originally due to gillespie @xcite was implemented for the reactions ( [ react ] ) . time and type of the next reaction taking place are determined by random numbers , using the poisson nature of the individual reactions . for the extinction probability , to unravel the universal time - scaling , system sizes ranging from @xmath60 to @xmath61 were considered , with sample averages over @xmath62 realizations . let us start with the master equation of the processes ( [ react ] ) . we derive it in the variables @xmath63 , which were introduced in ( [ x ] ) as the deviations of the densities from the reactive fixed point . using the conservation of the total density , @xmath64 , @xmath28 is eliminated and @xmath30 kept as independent variables . the master equation for the time - evolution of the probability @xmath65 of finding the system in state @xmath52 at time @xmath15 thus reads @xmath66 where @xmath67 denotes the transition probability from state @xmath52 to the state @xmath68 within one time step ; summation extends over all possible changes @xmath69 . we choose the unit of time so that , on average , every individual reacts once per time step . + according to the kramers - moyal expansion of the master equation to second order in @xmath69 , this results in the fokker - planck equation @xmath70+\frac{1}{2}\partial_i\partial_j[\mathcal{b}_{ij}({\bf x})p({\bf x},t ) ] ~. \label{fokker_planck}\ ] ] here , the indices @xmath71 stand for @xmath2 and @xmath0 ; in the above equation , the summation convention implies summation over them . the quantities @xmath72 and @xmath73 are , according to the kramers - moyal expansion : @xmath74 note that @xmath75 is symmetric . for the sake of clarity , we outline the calculation of @xmath76 : the relevant changes @xmath77 in the density @xmath11 result from the basic reactions ( [ react ] ) , they are @xmath78 in the first reaction , @xmath79 in the second and @xmath80 in the third . the corresponding rates read @xmath81 for the first reaction ( the prefactor of @xmath3 enters due to our choice of time scale , where @xmath3 reactions occur in one unit of time ) , and @xmath82 for the third , resulting in @xmath83 . the other quantities are calculated analogously , yielding + @xmath84 with @xmath85 . van kampen s linear noise approximation @xcite further simplifies these quantities . in this approach , the values @xmath86 are expanded around @xmath87 to the first order . as they vanish at the reactive fixed point , we obtain : @xmath88 where the matrix elements @xmath89 are given by @xmath90 the matrix @xmath33 , already given in ( [ a - lin ] ) , embodies the deterministic evolution , while the stochastic noise is encoded in @xmath75 . to take the fluctuations into account within the van kampen expansion , one approximates @xmath75 by its values at the reactive fixed point . hence , we find : @xmath91 and the corresponding fokker - planck equation reads @xmath92+\frac{1}{2}\mathcal{b}_{ij}\partial_i\partial_jp({\bf x},t ) \quad . \label{gen - f - p}\ ] ] for further convenience , we now bring eq . ( [ gen - f - p ] ) into a more suitable form by exploiting the polar symmetry unveiled by the variables * y*. as for the linearization of eqs . ( [ re ] ) , it is useful to rely on the linear mapping @xmath93 , with @xmath94 . interestingly , it turns out that this transformation diagonalizes @xmath95 . one indeed finds @xmath96 , with @xmath97 in the @xmath56 variables , the fokker - planck equation ( [ gen - f - p ] ) takes the simpler form @xmath98p({\bf y},t)\cr & + \frac{1}{12n}[\partial_{y_a}^2+\partial_{y_b}^2]p({\bf y},t)\quad . \label{f - p - y}\end{aligned}\ ] ] to capture the structure of circles predicted by the deterministic approach , we introduce polar coordinates : @xmath99 note that in the vicinity of the reactive fixed point , @xmath100 denotes the distance @xmath49 , which is now a random variable : @xmath101 . the fokker - planck equation ( [ f - p - y ] ) eventually turns into @xmath102p(r,\phi , t)~ .\cr \label{pol - f - p}\end{aligned}\ ] ] the first term on the right - hand - side of this equation describes the system s deterministic evolution , being the motion on circles around the origin at frequency @xmath34 . stochastic effects enter through the second term , which corresponds to isotropic diffusion in two dimensions with diffusion constant @xmath103 . note that it vanishes in the limit of infinitely many agents , i.e. when @xmath104 . if we consider a spherically symmetric probability distribution @xmath105 at time @xmath106 , i.e. independent of the angle @xmath107 at @xmath106 , then this symmetry is conserved by the dynamics according to eq . ( [ pol - f - p ] ) and one is left with a radial distribution function @xmath108 . the fokker - planck equation ( [ pol - f - p ] ) thus further simplifies and reads @xmath109p(r , t)\quad . \label{f_p_r}\ ] ] this is the diffusion equation in two dimensions with diffusion constant @xmath103 , expressed in polar coordinates , for a spherically symmetric probability distribution . this is the case on which we specifically focus in the following . within our approximations around the reactive fixed point , the probability distribution @xmath110 is thus the same as for a system performing a two - dimensional random walk in the @xmath56 variables . intuitively , such behavior is expected and its origin lies in the neutrally stable character of the cycles of the deterministic solution of the rate equations ( [ re ] ) . the cycling around the reactive fixed point does not yield additional effects when considering spherically symmetric probability distributions . furthermore , due to the existence of the constant of motion ( [ const2 ] ) , the neutral stability does not only hold in the vicinity of the reactive fixed point , but in the whole ternary phase space . we thus expect the distance @xmath49 from the reactive fixed point , a random variable , to obey a diffusion - like equation in the whole phase space ( and not only around the reactive fixed point ) . actually , qualitatively identical behavior is also found in the general case of non - equal reaction rates @xmath22 , @xmath23 , and @xmath6 ( see appendix [ app_gen_rates ] ) , as the constant of motion ( [ const2 ] ) again guarantees the neutral stability of the deterministic cycles . in this subsection , we will use the fokker - planck equation ( [ f_p_r ] ) to investigate the time - dependence of fluctuations around the reactive fixed point . in particular , we are interested in the time evolution of mean deviation from the latter . the average square distance @xmath111 will be found to grow linearly in time , rescaled by the number @xmath3 of individuals , before saturating . the frequency spectrum arising from erratic oscillations around the reactive fixed point is of further interest to characterize the stochastic dynamics : we will show that the frequency ( [ freq_gen_rates ] ) predicted by the deterministic approach emerges as a pole in the power spectrum . ( color online ) the averaged square radius @xmath112 as a function of the rescaled time @xmath113 . the blue ( or dark gray ) curve represents a single trajectory , which is seen to fluctuate widely . the linear black line indicates the analytical prediction ( [ r_time_evol ] ) , the red ( or light gray ) one corresponds to sample averages over @xmath114 realizations in stochastic simulations . hereby , we used a system size of @xmath61 . ] being interested in the vicinity of the reactive fixed point , in this subsection we ignore the fact that the boundary of the ternary phase space is absorbing . then , the solution to the fokker - planck equation ( [ f_p_r ] ) with the initial condition @xmath115 , where the prefactor @xmath116 ensures normalization , is simply a gaussian : @xmath117 this result predicts a broadening of the probability distribution in time , as the average square radius increases linearly with increasing time @xmath15 : @xmath118 as the time scales linearly with @xmath3 , increasing the system size results in rescaling the time @xmath15 and the broadening of the probability distribution takes longer . to capture these findings , we introduce the rescaled variable @xmath119 . in fig . [ radius_time_evol ] , we compare the time evolution of the squared radius @xmath120 obtained from stochastic simulations with the prediction of eq . ( [ r_time_evol ] ) and find a good agreement in the linear regime around the reactive fixed point , where @xmath101 . in fact , for short times , @xmath120 displays a linear time dependence , with systematic deviation from ( [ r_time_evol ] ) at longer times . we understand this as being in part due to the linear approximations used to derive ( [ r_time_evol ] ) , but also , and more importantly , to the fact that so far we ignored the absorbing character of the boundary . we conclude that the latter invalidates the gaussian probability distribution for longer times . this issue , which requires a specific analysis , is the scope of section [ sect - ext - prob ] , where a proper treatment is devised . ( color online ) time evolution of the variances of the densities @xmath11 and @xmath12 when starting at the reactive fixed point . the blue ( or dark gray ) curves correspond to a single realization , while the red ( or light gray ) ones denote averages over 1000 samples . our results are obtained from stochastic simulations with a system size of @xmath61 . the black line indicates the analytical predictions . ] from the finding @xmath121 together with the spherical symmetry of ( [ gaussian ] ) , resulting in @xmath122 , we readily obtain the variances of the densities @xmath11 and @xmath12 : @xmath123 according to these results , the average square deviations of the densities from the reactive fixed point grow linearly in the rescaled time @xmath124 , thus exhibiting the same behavior as we already found for the average squared radius ( [ r_time_evol ] ) . in fig . [ fluct ] , these findings are compared to stochastic simulations for small times @xmath124 , where the linear growth is indeed recovered . ( color online ) variances of the densities @xmath11 and @xmath12 when starting at a cycle away from the fixed point . the black lines denote our analytical results , while the red ( or gray ) ones are obtained by stochastic simulations as averages over 1000 realizations . they are seen to agree for small times , while for larger ones the stochasticity of the system induces erratic oscillations . these results were obtained when starting from @xmath125 . ] we may consider fluctuations of the densities @xmath11 , @xmath12 , and @xmath13 not only around the reactive stationary state , as obtained in ( [ fluct_dens ] ) , but also as the deviations from the deterministic cycles . we consider the latter in the linear approximation around the fixed point , given by eq . ( [ re - lin - sol ] ) . the fluctuations around them are again described by the fokker - planck equation ( [ fokker_planck ] ) with @xmath126 and @xmath75 given by eqs . ( [ alpha_b ] ) , but now @xmath127 , @xmath128 , and @xmath28 are the deviations from the deterministic cycle : @xmath129 , where @xmath130 , @xmath131 , and @xmath132 obey eqs . ( [ re ] ) and characterize the deterministic cycles given by eq . ( [ re - lin - sol ] ) . again performing van kampen s linear noise approximation , we obtain a fokker - planck equation of the type ( [ gen - f - p ] ) , now with the matrices @xmath133 note that the entries of the above matrices now depend on time @xmath15 via @xmath134 and @xmath135 . the fokker - planck equation ( [ gen - f - p ] ) yields equations for the time evolution of the fluctuations : @xmath136 they may be solved numerically for @xmath137 , @xmath138 , and @xmath139 , yielding growing oscillations . in fig . [ fluct_as ] we compare these findings to stochastic simulations . we observe a good agreement for small rescaled times @xmath113 , while the oscillations of the stochastic results become more irregular at longer times . + recently , it has been shown that the frequency predicted by the deterministic rate equations appears in the stochastic system due to a `` resonance mechanism '' @xcite . internal noise present in the system covers all frequencies and induces excitations ; the largest occurring for the `` resonant '' frequency predicted by the rate equations . here , following the same lines , we address the issue of the characteristic frequency in the stochastic cyclic lotka - volterra model . . it agrees with stochastic simulations ( solid ) . the inset shows the erratic oscillations of the density of one of the species for one realization . the system size considered is @xmath140 . [ urn_power_spectrum ] ] in the vicinity of the reactive fixed point , the deterministic rate equations ( [ re ] ) predict density oscillations with frequency @xmath34 given in eq . ( [ freq_gen_rates ] ) . for the stochastic model , we now show that a spectrum of frequencies centered around this value @xmath34 arises . the most convenient way to compute this power spectrum @xmath141 from the fokker - planck equation ( [ gen - f - p ] ) is through the set of equivalent langevin equations @xcite : @xmath142 with the white noise covariance matrix @xmath75 : @xmath143 . from the fourier transform of eq . ( [ langevin ] ) , it follows that @xmath144\cr & = \frac{4}{3n}\frac{1 + 3\omega^2}{(1 - 3\omega^2)^2 } \quad .\end{aligned}\ ] ] the power spectrum has a pole at the characteristic frequency already predicted from the rate equations ( [ re ] ) , @xmath145 . for increasing system size @xmath3 , the power spectrum displays a sharper alignment with this value . in fig . [ urn_power_spectrum ] we compare our results to stochastic simulations and find an excellent agreement , except for the pole , where the power spectrum in finite systems obviously has a finite value . these results were obtained in the vicinity of the reactive fixed point , where the linear analysis ( [ re - lin - sol ] ) applies . as already found in the deterministic description ( see fig . [ dens_comp ] ) , when departing from the center fixed point , nonlinearities will alter the characteristic frequency . so far , within the stochastic formulation , the fluctuations around the reactive steady state were found to follow a gaussian distribution , linearly broadening in time . however , when approaching the absorbing boundary , the latter alters this behavior , see fig . [ radius_time_evol ] . in the following , we will incorporate this effect in our quantitative description . it plays an essential role when discussing the extinction probability , which is the scope of this subsection . the probability that one or more species die out in the course of time is of special interest within population dynamics from a biological viewpoint . when considering meta - populations formed of local patches , such questions were e.g. raised in ref . @xcite . here , we consider the extinction probability @xmath146 that , starting at the reactive fixed point , after time @xmath15 two of the three species have died out . in our formulation , this corresponds to the probability that after time @xmath15 the system has reached the absorbing boundary of the ternary phase space , depicted in fig . [ simplex_as ] . considering states far from the reactive fixed point , we now have to take into account the absorbing nature of the boundary . this feature is incorporated in our approach by discarding the states having reached the boundary , so that a vanishing density of states occurs there . as the normalization is lost , we do no longer deal with probability distributions : in fact , discarding states at the boundary implies a time decay of the integrated density of states [ which , for commodity , we still refer to as @xmath110 ] . ignoring the nonlinearities , the problem now takes the form of solving the fokker - planck equation ( [ f_p_r ] ) for an initial condition @xmath148 , and with the requirement that @xmath110 has to vanish at the boundary . hereby , the triangular shaped absorbing boundary is regarded as the outermost ( degenerate ) cycle of the deterministic solutions . as the linearization in the @xmath149 variables around the reactive fixed point maps the cycles onto circles ( see above ) , the triangular boundary is mapped to a sphere as well . although the linearization scheme on which these mappings rely is inaccurate in the vicinity of the boundary , it is possible to incorporate nonlinear effects in a simple and pragmatic manner , as shown below . however , first let us consider the linearized problem , being a first - passage to a sphere of radius @xmath150 , which is , e.g. , treated in chapter 6 of ref . @xcite . the solution is known to be a combination of modified bessel functions of the first and second kind . actually , the laplace transform of the density of states reads @xmath151 where @xmath103 is the diffusion constant ; @xmath152 and @xmath153 denote the bessel function of the first , resp . second , kind and of order zero ; and @xmath150 is the radius of the absorbing sphere . it is normalized at the initial time @xmath154 ; however , for later times @xmath15 , the total number of states @xmath155 will decay in time , as states are absorbed at the boundary . equipped with this result , we are now in a position to calculate the extinction probability @xmath156 . it can be found by considering the probability current @xmath157 at the absorbing boundary @xcite , namely : @xmath158 whose laplace transform is given by @xmath159 the extinction probability at time @xmath15 is obtained from @xmath157 by integrating over time @xmath160 until time @xmath15 @xcite : @xmath161 . therefore , the laplace transform of @xmath156 reads @xmath162 . again , we notice that we can write this equation in a form that depends on the time @xmath15 only via the rescaled time @xmath113 : @xmath163 hence , for different system sizes @xmath3 , one obtains the same extinction probability @xmath164 provided one considers the same value for the scaling variable @xmath165 ( see fig . [ urn_transition ] ) . we can not solve the inverse laplace transform appearing in eq . ( [ ext_prob_u ] ) , but @xmath166 might be expanded according to @xcite as @xmath167 considering the first three terms in this expansion , i.e contributions up to @xmath168 , yields the approximate result @xmath169 \quad . \label{approx_ext_prob}\ ] ] numerically , we have included higher terms . the results for contributions up to @xmath170 in ( [ exp_i0 ] ) are shown in fig . [ urn_transition ] . an estimate of the effective distance @xmath150 to the absorbing boundary is determined by plugging either @xmath171 , or @xmath172 , or @xmath173 into eq . ( [ r - gen ] ) , which yields @xmath174 . from fig . [ urn_transition ] , we observe the extinction probability to be overestimated by ( [ approx_ext_prob ] ) with @xmath175 . this stems from nonlinearities altering the analysis having led to ( [ approx_ext_prob ] ) . however , adjusting @xmath150 on physical grounds , we are able to capture these effects . another estimate of @xmath176 is obtained by considering the expression eq . ( [ r - x ] ) , which arises from a linear analysis . as the extinction of two species requires @xmath177 , in the expression eq . ( [ r - x ] ) this leads to the estimate @xmath178 for @xmath150 . the comparison with fig . [ urn_transition ] , shows that ( [ approx_ext_prob ] ) together with @xmath179 is a lower bound of @xmath176 . a simple attempt to interpolate between the above estimates is to consider the mean value of these radii , @xmath180 , which happens to yield an excellent agreement with results from stochastic simulations , see fig . [ urn_transition ] . for the latter , we have considered systems of @xmath181 , and @xmath182 individuals . rescaling the time according to @xmath183 , they are seen to collapse on a universal curve . this is well described by eq . ( [ approx_ext_prob ] ) with @xmath184 as the radius of the absorbing boundary . ( color online ) the extinction probability when starting at the reactive fixed point , depending on the rescaled time @xmath113 . stochastic simulations for different system sizes ( @xmath60 : triangles ; @xmath25 : boxes ; @xmath182 : circles ) are compared to the analytical prediction ( [ ext_prob_u ] ) . the left ( blue or dark gray ) is obtained from @xmath174 , the right one ( red or light gray ) from @xmath178 , and the middle ( black ) corresponds to the average of both : @xmath185 . ] to conclude , we consider the mean time @xmath186 that it takes until one species becomes extinct . from @xcite we find for the rescaled mean absorption time @xmath187 which for our best estimate @xmath188 yields a value of @xmath189 . motivated by recent _ in vitro _ and _ in vivo _ experiments aimed at identifying mechanisms responsible for biodiversity in populations of _ escherichia coli _ @xcite , we have considered the stochastic version of the ` rock - paper - scissors ' , or three - species cyclic lotka - volterra , system within an urn model formulation . this approach allowed us to quantitatively study the effect of finite - size fluctuations in a system with a large , yet _ finite _ , number @xmath3 of agents . while the classical rate equations of the cyclic lotka - volterra model predict the existence of one ( neutrally stable ) center fixed point , associated with the _ coexistence _ of all the species , this picture is _ dramatically _ invalidated by the fluctuations which unavoidably appear in a finite system . the latter were taken into account by a fokker - planck equation derived from the underlying master equation through a van kampen expansion . within this scheme , we were able to show that the variances of the densities of individuals grow in time ( first linearly ) until extinction of two of the species occurs . in this context , we have investigated the probability for such extinction to occur at a given time @xmath15 . as a main result of this work , we have shown that this extinction probability is a function of the scaling variable @xmath183 . exploiting polar symmetries displayed by the deterministic trajectories in the phase portrait and using a mapping onto a classical first - passage problem , we were able to provide analytic estimates ( upper and lower bounds ) of the extinction probability , which have been successfully compared to numerical computation . from our results , it turns out that the classical rate equation predictions apply to the urn model with a finite number of agents only for short enough time , i.e. in the regime @xmath190 . as time increases , the probability of extinction grows , asymptotically reaching @xmath191 for @xmath192 , so that , for finite @xmath3 , fluctuations are _ always _ responsible for extinction and thus dramatically jeopardize the possibility of coexistence and biodiversity . interestingly , these findings are in qualitative agreement with those ( both experimental and numerical ) reported in ref . @xcite , where it was found that in a well mixed environment ( as in the urn model considered here ) two species get extinct . while this work has specifically focused on the stochastic cyclic lotka - volterra model , the addressed issues are generic . indeed , we think that our results and technical approach , here illustrated by considering the case of a paradigmatic model , might actually shed further light on the role of fluctuations and the validity of the rate equations in a whole class of stochastic systems . in fact , while one might believe that fluctuations in an urn model should always vanish in the thermodynamic limit , we have shown that this issue should be dealt with due care : this is true for systems where the rate equations predict the existence of an asymptotically stable fixed point , which is always reached by the stochastic dynamics @xcite . in contrast , in systems where the deterministic ( rate equation ) description predicts the existence of ( neutrally stable ) center fixed points , such as the cyclic lotka - volterra model , fluctuations have dramatic consequences and hinder biodiversity by being responsible ( at long , yet finite , time ) for extinction of species @xcite . in this case , instead of a deterministic oscillatory behavior around the linearly ( neutrally ) stable fixed point , the stochastic dynamics always drives the system toward one of its absorbing states . thus , the absorbing fixed points , predicted to be linearly unstable within the rate equation theory , actually turn out to be the _ only stable fixed points _ available at long time . we would like to thank u. c. tuber and p. l. krapivsky for helpful discussions , as well as a. traulsen and j. c. claussen for having made manuscript @xcite available to us prior to publication . m.m . gratefully acknowledges the support of the german alexander von humboldt foundation through the fellowship no . iv - scz/1119205 stp . in section [ stoch_appr ] , when considering the stochastic approach to the cyclic lotka - volterra model , for the sake of clarity we have specifically turned to the situation of equal reaction rates , @xmath59 . in this appendix , we want to provide some details on the general case with unequal rates . while the mathematical treatment becomes more involved , we will argue that the qualitative general situation still follows along the same lines as the ( simpler ) case that we have discussed in detail in sec . [ stoch_appr ] . the derivation of the fokker - planck equation ( [ gen - f - p ] ) is straightforward , following the lines of subsection [ master - f - p ] . the matrix @xmath33 remains unchanged and is given in eq . ( [ a - lin ] ) , for @xmath75 we now obtain : @xmath193 the corresponding fokker - planck equation reads @xmath92+\frac{1}{2}\mathcal{b}_{ij}\partial_i\partial_jp({\bf x},t ) \quad .\ ] ] again , we aim at benefitting from the cyclic structure of the deterministic solutions , and perform a variable transformation to @xmath36 , with @xmath194 given in eq . ( [ s - matrix ] ) . as already found in subsection [ det - eq ] , @xmath33 turns into @xmath195 , such that in the @xmath56 variables the deterministic solutions correspond to circles around the origin . for the stochastic part , entering via @xmath75 , a technical difficulty arises . it is transformed into @xmath196 being no longer proportional to the unit matrix as in the case of equal reaction rates . we can do slightly better by using an additional rotation , @xmath197 , with rotation angle @xmath198 this variable transformation leaves @xmath199 invariant , but brings @xmath200 to diagonal form , with unequal diagonal elements . the stochastic effects thus correspond to _ anisotropic diffusion_. however , for large system size @xmath3 , the effects of the anisotropy are washed out : the system s motion on the deterministic cycles , described by @xmath33 , occurs on a much faster timescale then the anisotropic diffusion , resulting in an averaging over the different directions . to calculate the time evolution of the average deviation from the reactive fixed point , @xmath201 , we start from the the fluctuations in @xmath56 , which fulfill the equations @xmath202 using eq . ( [ r - y ] ) , we obtain @xmath203 the dependence on the deterministic part has dropped out , and the solution to the above equation with initial condition @xmath204 is a linear increase in the rescaled time @xmath119 : @xmath205\frac{t}{n}\quad,\ ] ] valid around the reactive fixed point . as in the case of equal reaction rates , we have a linear dependence @xmath206 , corresponding to a two - dimensional random walk . as a conclusion of the above discussion , the general case of unequal reaction rates qualitatively reproduces the behavior of the before discussed simplest situation of equal rates ( confirmed by stochastic simulations ) . the latter turns out to already provide a comprehensive understanding of the system .
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cyclic dominance of species has been identified as a potential mechanism to maintain biodiversity , see e.g. b. kerr , m. a. riley , m. w. feldman and b. j. m. bohannan [ nature * 418 * , 171 ( 2002 ) ] and b. kirkup and m. a. riley [ nature * 428 * , 412 ( 2004 ) ] . through analytical methods supported by numerical simulations
, we address this issue by studying the properties of a paradigmatic non - spatial three - species stochastic system , namely the ` rock - paper - scissors ' or cyclic lotka - volterra model .
while the deterministic approach ( rate equations ) predicts the coexistence of the species resulting in regular ( yet neutrally stable ) oscillations of the population densities , we demonstrate that fluctuations arising in the system with a _ finite number of agents _ drastically alter this picture and are responsible for extinction : after long enough time , two of the three species die out . as main findings we provide analytic estimates and numerical computation of the extinction probability at a given time .
we also discuss the implications of our results for a broad class of competing population systems .
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in 2003 , belle reported the discovery of a charmoiniumlike neutral states x(3872 ) with mass=@xmath21 mev with width @xmath22 mev @xcite and latter confirmed by do @xcite , cdf @xcite and barbar @xcite . this discovery fed excitement in the charmonium spectroscopy because of unconventional properties of the state . x(3872 ) could not be explained through ordinary meson(@xmath23 ) and baryon ( qqq ) scheme . the conventional theories predicts complicated color neutral structures and search of such exotic structures are as old as quark model @xcite . after the discovery of x(3872 ) , the large number of charge , neutral and vector states have been detected in various experiments , famous as the xyz states . recently , the charge bottomoniumlike resonances @xmath24 and @xmath25 have been reported by belle collaboration in the process @xmath26 and @xmath27 @xcite . moreover , a state reported by besiii collaboration in @xcite as @xmath28 in the @xmath29 reaction , again the besiii collaboration reported a state @xmath30 from invariant mass @xmath31 in the @xmath32 reaction @xcite , whereas the belle @xcite and cleo @xcite reconfirmed the status of the state . the sub structure of the all these states are still a open question , they might driven exotic structure like tetraquark , molecular or hybrid , expected as per theory of qcd , needs theoretical attention . + in the present study , we focus on the molecular structure , as meson - antimeson bound state , just like deuteron . the multiquark structures have been studied since long time @xcite . t@xmath33rnqvist , in @xcite predicted mesonic molecular structures , introduced as @xmath34 by using one pion exchange potential . with heavy flavour mesons , various authors predicted the bound state of @xmath0 and @xmath12 as a possible mesonic molecular structures as well as studied the possibilities of the @xmath4 and @xmath15 as vector - vector molecule @xcite , also it have been studied in various theoretical approaches like potential model @xcite , effective field theory @xcite , local hidden gauge approach @xcite etc .. + in the variational scheme , we have used the potential model approach to study the meson - antimeson bound system . for that , we have used the hellmann potential @xcite ( superposition of the coulomb + yukawa potential ) with one pion exchange potential ( opep ) . we assume that the colour neutral states experience residual force due to confined gluon exchange between quarks of the hadrons ( generally known as residual strong force ) , skyrme - like interaction . as mentioned by greenberg in ref . @xcite and also noted by shimizu in ref @xcite that this dispersive force ( also called london force ) or the attraction between colour singlet hadron comes from the virtual excitation of the colour octet dipole state of each hadron @xcite . indeed , long ago skyrme @xcite in 1959 and then guichon @xcite , in 2004 had remarked that the nucleon internal structure to the nuclear medium does play a crucial role in such strong effective force of the n - n interaction . in the study of the s - wave n - n scattering phase shift , in ref.@xcite , khadkikar and vijayakumar used the colour magnetic part of the fermi - breit unconfined one - gluon - exchange potential responsible for short range repulsion and sigma and pion are used for bulk n - n attraction . in this way , with such assumption of the interaction , the mass spectra of the dimesonic bound states are calculated . + for molecular binding , the ref.@xcite found that the quark exchange alone could not bind the system , led to include one pion exchange . the ref.@xcite mention some additional potential strength required with one pion exchange . whereas , the dynamics at very short distance led to complicated heavy boson exchange models as studied in @xcite . in all these studies @xcite , one common conclusion was extracted that the highly sensitive dependence of the results on the regularisation parameter . to avoid these dependency and complicated heavy boson exchange in this phenomenological study , we used the hellmann potential in accordance to delicate calculation of attraction and repulsion at short distance . the overall hellmann potential represents the residual strong interaction at short distance in flavour of the virtual excitation of the colour octet dipole state of each colour neutral states . the opep is included for long range behaviour of the strong force . the ope potential could be split into two parts ( i ) central term with sipn - isospin factor ( ii ) tensor part . we have analyse the effect of these two parts . by calculating the spin - isospin factor as in @xcite , we have found the symmetry braking in our results which was also discussed by t@xmath33rnqvist in @xcite , whereas , the tensor term to be found play a very crucial role implicit the necessity of it . in that way , bound state of @xmath0 is compared with the state x(3872 ) which also have been predicted as mesonic molecule by the authors of ref.@xcite whereas states @xmath24 and @xmath24 which are close to the @xmath12 and @xmath15 threshold . + to test the internal structure of the state , in general , one have to look for the decay pattern of the state . in ref . @xcite , the hadronic decays of the x(3872 ) have been studied in accordance to its decay mode sensitive to the short or long distance structure of the state . to test the compared states as dimesonic system , we have used the binding energy as input for decay calculation . we have adopt the formula developed by authors of ref . @xcite for the partial width sensitive to the long distance structure of the state , whereas , the formula for the decay mode sensitive to short distance structure of the state is taken from @xcite . in ref . @xcite , authors predicted existence of the neutral spin-2 ( @xmath35=@xmath36 ) partner of x(3872 ) , would be @xmath4 bound state , and in same way expected spin-2 partner of @xmath12 , would be @xmath15 bound state , on the basis of heavy quark spin symmetry and calculated the hadronic and radiative decays . we have used formula of @xcite for radiative decay for predicated states . with calculated binding energy , the decay properties are in good agreement with @xcite . + the article is organize as follows , after the brief introduction we have discussed the hellmann potential and ope potential in sec - ii then in sec - iii we presents our theoretical approach for the calculation of the mass spectra as well as for decay width , in sec - iv we present our results for deuteron as per our model and generalize our approach . in sec - v and vi we presents our results of dimesonic states and finally we summarise our present work . two color neutral states formed the bound states , just like deuteron is the bound state of proton and neutron . in the case of the dimesonic states , the meson and antimeson are takeover as a constituents , forming a bound state . the interaction potential between two color neutral constituents taken as a phenomenological hellmann potential @xcite with one pion exchange potential(opep ) @xcite . the colour neutral states experience residual force due to confined gluon exchange between quarks of the hadrons . the attraction between colour singlet hadron comes from the virtual excitation of the colour octet dipole state of each hadron . when two colour neutral states are come enough close to each other so the quark of the one colour neutral states can feel to quark of the other state creates manifestation of the energy . once the interaction become strong enough then this manifestation of the energy being the source for the creation for the quark anti quark pair . this residual force is analogous to the residual interaction between two electrical neutral atoms , experienced the van der waals force . the overall hellmann potential represents the residual strong interaction at short distance , where , these pseudopotential has the form , namely @xmath37 here , the constant @xmath38 is the residual strength of the strong running coupling constant between the two colour neutral states and b are the strength of the yukawa potential whereas @xmath39 is the relative separation between constituents . the value of the @xmath38 could be determined through the model , such as @xmath40 where @xmath41 and @xmath42 are constituent masses , m=2@xmath41 @xmath42/(@xmath41+@xmath42 ) , @xmath43=1 gev , @xmath44=0.250 gev and @xmath45 is number of flavour @xcite . so , the coulomb interaction is constrained by the value of @xmath38 . the constant b and c appeared in the yukawa potential of eq.(1 ) play very trivial role on overall characteristic of the hellmann potential . in the fig(3 ) ( see appendix - a ) , one can observe that once the coulombic interaction is fixed , the variation of the constant b and c are inversely proportional to each other . as the value of the b increases , it increases the repulsive nature of the potential while the constant c smoothing the curve as well as it increases the strength of the yukawa part - inversely to b. to show such nature of the hellmann potential , in the fig(3 ) , the graph have been plotted for various set of values of the b and c. for the bound state , the value of b can take both positive and negative values . with the negative values of the b the overall potential become attractive . for more detailed discussion on the hellmann potential , we suggest to the readers for ref.@xcite . in our case of the dimesonic bound states calculations , we assume that the hellmann potential at very short distance cares for the delicate cancellation of attraction and repulsion respectively which is mainly taken care by heavy boson exchange in the one boson exchange model . + whereas , the long range behaviour of the interaction part is accomplished by one pion exchange . the model carried the net potential as the hellmann potential plus ope potential plus relativistic correction . we introduced and discussed the relativistic correction after this section . for instance , to fix the model ( or the potential ) for deuteron which is widely believed to be have molecular structure and well studied in obe potential model , we fit the values of the constant @xmath46 and @xmath47 to get approximate binding energy of deuteron . then , to generalize the model to dimesonic system , we arrive on the relation between b and c in accordance of the mass of the system , + if @xmath48 , where @xmath49 is the threshold mass of dimesonic state and @xmath41 mass of deuteron and n is integer number ( n=1,2,3 .. ) , then @xmath50 here , @xmath46 and @xmath47 are the constant , fitted for deuteron binding energy . + the one pion exchange potential ( opep ) taken for long range interaction which is well studied for nn - interaction . the ope potential for nn - interaction takes the form @xcite @xmath51\end{aligned}\ ] ] where @xmath52 is the nucleon - pion coupling constant , @xmath53 and @xmath54 are spin and isospin factors respectively while the @xmath55 and @xmath56 are defined as @xmath57 @xmath58 whereas the @xmath59 is the usual tensor operator expressed as @xmath60 , is mainly responsible for long range tail of the potential and play very crucial role in the nn - interaction . the expression of opep in the eq.(4 ) is for the point like pion . while , in a more realistic picture where the pion itself has its own internal structure , it is natural to introduced the usual form factor due to the dressing of the quarks . by introducing this finite size effect @xcite , we have \(a ) ( b ) ( c ) ( d ) @xmath61 @xmath58 thus , the function @xmath62 and @xmath63 with the finite size effect take the form @xmath64\ ] ] @xmath65\ ] ] now , one pion exchange potential for dimesonic system could be written as @xmath66 @xmath67\end{aligned}\ ] ] such that , the opep becomes for dimesonic systems , @xmath68 where the @xmath69=0.69 is the meson - pion coupling constant , @xmath49 and @xmath70 are the average mass of two constituent of the dimesonic state and pion mass respectively , while @xmath71 is the pion form factor . the mesons and quark masses with their quantum numbers are taken from the listing of particle data group@xcite . the constituent meson - quark coupling constant may derive by using the goldberger - treiman relation on suitable estimates of known @xmath72 coupling constant . the relation between quark - boson and nucleon - boson coupling could expressed as @xcite @xmath73 the effective ope potential can be split into a central and tensor term proportional to @xmath74 @xmath75 and @xmath59 @xmath75 respectively . in ref.@xcite , thomas et.al . mentioned and discussed sign convention and detail calculation of spin - isospin factor . in ref.@xcite , authors showed the inconsistent sign convention adopted by ref.@xcite in the calculation of spin - isospin factor of @xmath76(@xmath36 ) and then derive and explain overall sign for determination of @xmath74 @xmath75 . we are agreed with ref.@xcite and adopt the same . for total spin s , total isospin state i and charge conjugation parity c , the spin - isospin factor for central term are given by @xcite @xmath77 the matrix element of the tensor oerator for different spin state are real numbers and it is well discussed by t@xmath33rnqvist in ref.@xcite . for example , the matrix element of @xmath59 in the case of deuteron @xmath78 with such interaction the opep potential shown in eq(11 ) , one need to solve the coupled channel schr@xmath33dinger equation . in the present study , we have focus only on the s - wave spectra of dimesonic state . as such , we have taken only the s - wave tensor contribution of particular spin state . the matrix element of such tensor operator of dimesonic states are given by @xmath79 the value of the form factor @xmath80 in opep affects the strength of the potential very drastically . the sensitivity of the @xmath80 have discussed by authors of ref.@xcite . in the fig(1 ) , the nature of the opep in different spin - isospin state are shown for different values of @xmath80 ( noted that only the s - wave contribution of the tensor interaction is considered ) . one can analysed from the plot , as the value of the @xmath80 increases the strength of the potential is also increases drastically . in ref.@xcite , t@xmath33rnqvist noted the value of @xmath80 fall in range 0.8 - 1.5 gev for fit the nn scattering data where as in case of mesonic molecule , specially , the heavy meson which are very small compared to the size of the nucleon , the larger value of @xmath80 is expected and the large value of @xmath80 increases the binding energy . ref.@xcite shows the results are very sensitive to @xmath80 and the binding energy no longer monotonically increases with @xmath80 with opep model . for instant , we fixed @xmath80=1.5 gev consistent with nn - scattering data and increase the dominance of the hellmann potential , such that we arrive at relation of the eq(2 ) . + the factors @xmath74 @xmath75 and @xmath59 @xmath75 makes the opep spin and isospin state dependent . as per the literature @xcite , the most attractive channel is ( i , s)=(0,0 ) while the channel ( i , s)=(2,0 ) is the most repulsive . but , as in fig(1 ) , we observed the channel ( i , s)=(1,1 ) is most attractive while ( i , s)=(0,1 ) noted as most repulsive one . the next order of the attractive channels are ( i , s)=(0,0 ) and ( i , s)=(1,2 ) . whereas , the channels ( i , s)=(0,1),(i , s)=(2,0 ) and ( i , s)=(1,0 ) expected to be unbound . the observed change in the nature of the these channels may be due to only the s - wave contribution of the tensor interaction . it clearly indicates the dominant effect of tensor operator at long range part of the potential . the discussed effect of opep is also reflect in our results , tabulated in table - ii and table - iii . .masses of the mesons ( in mev)@xcite [ cols="^,^,^,^,^,^,^,^ " , ] there are large possibilities of dimesonic states with heavy light flavours , also shown in our calculated mass spectrum(see table - iii ) . we have predicted the mass and root mean square radius . for instant with @xmath76 dimesonic states , for the decay properties , we focus on these two states @xmath81 and @xmath82 as spin-2 partner of @xmath0 and @xmath12 molecules respectively and also expected as per heavy quark spin symmetry(hqss ) . the study of decay properties with mass spectra is very important to investigate their sub structure . we have calculated the hadronic and radiative decay of these states . the hadronic decay modes of @xmath81 @xmath83 by using the eq(20)(with @xmath84=0.993 ) , the calculated partial decay width , we have @xmath85 similarly the decay width for @xmath82 with the same formula ( with @xmath84=0.998 ) to be found @xmath86 to understand the electromagnetic branching fraction we need to understand the interaction with photon and s - wave mesons and their contribution due to light and heavy quarks @xcite . in addition to , radiative decay is more sensitive to long distance molecular structure . the radiative decay mode of @xmath81 expressed as @xmath87 thus the decay width for the state calculated as per eq(23 ) ( with @xmath88 = 1 gev ) , we have @xmath89 the calculated partial decay widths are in good agreement with the results calculated in ref@xcite . the radiative decay for the state @xmath82 could also calculated similar to @xmath81 . @xmath90 for this decay mode of @xmath82 , in our model the radiative decay of @xmath82 to be found forbidden with suggestive value of @xmath88 = 1 gev . with very large value of @xmath88 ( @xmath918 ) , we have get the comparable results with ref . moreover , the radiative decay width for @xmath82 calculated in @xcite is @xmath91 10 ev , which is very small . with very large value of @xmath88 , we have found @xmath91 14 ev . ( see the table - iv , for the results of calculated partial widths with comparisons with others ) . in summary , we have calculated the mass spectra of dimesonic states with heavy - light flavour mesons by using the hellmann and one pion exchange potential . the calculated binding energy of dimesonic states are found overestimated which is also expected in variational approach . we have analysed the change in the binding energy due to charge conjugation parity and isospin , agreed with ref.@xcite suggested isospin symmetry braking . we have also discussed the effect of tensor term in one pion exchange . we includes only the s - wave contribution of tensor term and observed shuffling of channels which is different from expected channels as in @xcite . it is remarkably pointing out the effect of the tensor term and its contribution in different processes , led us to conclude that it can not be ignore . whereas , the contribution from the relativistic correction in the potential to the binding energy is @xmath92 which is almost one third to our previous work @xcite . + in addition , we have calculated the decay properties of @xmath0 , @xmath12 , @xmath4 and @xmath15 ( the formalism adopted from ref.@xcite ) , using our calculated binding energy as input where formalism mainly dependent on the masses . to enforced the our predictions , we have attempted the decay calculations as well to test the molecular structure of compared states with dimesonic states on the basis of decay modes which may have responsible for their long and short distance structure , as studied in the literature @xcite . we have found comparable results of decay properties ( with our mass spectra ) @xcite , support the prediction of @xmath0 , @xmath12 , @xmath4 and @xmath15 bound states as mesonic molecules . we support the molecular structure of the x(3872 ) as @xmath0 bound state with @xmath93 dominant decay mode . the long distance radiative decay may gives information of the mixing of charge to neutral channel of the bound state which lack in this study . we have predicted the mass spectra and decay properties of the possible spin-2 partner ( @xmath94 ) of x(3872 ) in charm sector as well as spin-2 partner of @xmath12 in the bottom sector , as miguel et.al studied in @xcite , also expected in heavy quark spin - flavour symmetry . + apart form these states , we have suggest mass spectra of possible dimesonic states as mesonic molecules with @xmath95 , @xmath96 , @xmath97 , @xmath98 and @xmath99 mesons . wei chen et.al in @xcite suggested open flavour tetraquark structure with having strange quark , with mass range 6.9 - 7.3 gev , with @xmath100 or @xmath101 . in ref.@xcite , gui - jun ding suggest @xmath5 molecule . the masses of @xmath102 and @xmath76 dimesonic states with @xmath95 , @xmath96 , @xmath97 and @xmath98 mesons are fall in the same range of 7.1 - 7.3 gev . moreover , we have also predicted the masses of the dimesonic states having @xmath99 mesons as a constituent . we look forward to the experimental facilities and working groups to take more attention for searches of the possible dimesonic states as well as for the confirmations of our theoretical predictions . + * acknowledgements * d. p. rathaud would like to thanks prof . r. c. johnson for the useful discussion . a. k. rai acknowledge the financial support extended by d.s.t . , government of india under serb fast track scheme sr / ftp /ps-152/2012 . \(a ) ( b ) \(c ) ( d ) \(e ) ( f ) \(g ) ( h ) \(i ) ( j ) \(i ) for hadronic decay @xcite : one of the constituent meson decay into two mesons : @xmath103 + here , m is the mesonic molecule ( m = d+b , where d and b are constituents ) decaying into product mesons a , d , e in which one of the meson is the constituent of the dimesonic state , for example , @xmath104 , here , @xmath105 or @xmath106 + from ref @xcite , on the bassis of the tree level approximation the amplitude is given by @xcite @xmath107 where g=0.69 the quarks meson coupling constant , @xmath108= 92.2 mev pion decay constant where as @xmath109=0.35 @xmath110 is the coupling constant to the dimesonic state to the charge or neutral channels @xcite . thus , decay formula reads @xmath111 where @xmath112 is the three momentum of the pion . while @xmath113 and @xmath114 are the the four momenta of ad and ac systems . since the amplitude is dependent on the invariant masses we have @xmath115=@xmath116 and @xmath117=@xmath118 of the final state ea and ad pairs respectively @xcite . here , @xmath119 is the k@xmath120llen function . for the known value of @xmath115 the range of the @xmath117 could be determined by the values of momentum @xmath121 is parallel or antiparallel to @xmath122 as here , @xmath129 is the dimesonic coupling constant , @xmath130 is the fine structure constant , @xmath131gev is the charm quark mass , @xmath132 @xmath133 ( @xmath134 is the light quark mass ) , @xmath70 and @xmath135 is the pion and kaon masses , respectively . whereas , @xmath108 and @xmath136 are the pion and kaon decay constant , respectively . thus the partial radiative decay width given as for the known value of the @xmath138 the range of @xmath139 could be determined by the values of momentum @xmath122 is parallel or antiparallel to @xmath140 as @xmath141 here , @xmath142 @xmath143 and @xmath140 are the photon energy and momentum in the @xmath138 cm frame , respectively . whereas , the amplitude for the @xmath15 read as with @xmath145=@xmath146 for @xmath15 system and could be calculated according to eq.(b8 ) . @xmath42=4.6 gev is mass of the bottom meson . for more detailed of the calculations and formalism of decay calculation we refer to ref.@xcite . al.(particle data group ) , chin . c * 38 * , 090001 ( 2014 ) . choi et al . 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in this work , we have calculated the mass spectra and decay properties of dimesonic ( meson - antimeson ) bound state in the variational scheme . the intermesonic interaction considered as the hellmann potential and one pion exchange potential .
the mass spectra of the @xmath0 , @xmath1 , @xmath2 , @xmath3 , @xmath4 , @xmath5 , @xmath6 , @xmath7 , @xmath8 , @xmath9 , @xmath10,@xmath11 , @xmath12 , @xmath13 , @xmath14 , @xmath15 , @xmath16 , @xmath17 etc .. are calculated .
the states x(3872 ) , @xmath18 , @xmath19 and @xmath20 are compared with @xmath0 , @xmath4 , @xmath12 and @xmath15 dimesonic bound states . to probe the molecular structure of the compared states
, we have calculated the decay properties sensitive to their long and short distance structure of hadorinc molecule , our results suggested the compared states driven the molecular structure . apart from these the other calculated mass spectra of dimesonoic states are predicted and for such bound states , the experimental search are suggested .
pacs numbers : : 12.39.jh , 12.39.pn , 13.25.jx,14.40.rt
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You are an expert at summarizing long articles. Proceed to summarize the following text:
since dorothy denning s seminal 1987 paper on intrusion detection @xcite , ml and data mining(dm ) have steadily gained attention in security applications . darpa s 1998 network intrusion detection evaluation @xcite , and kdd(conference on knowledge discovery and data mining ) cup s 1999 challenge @xcite have raised profile of ml in security contexts . yet , constrained by hardware and system resources@xcite , large - scale ml applications did not receive much attention for many years . in 2008 , acm conference on computer and communications security(ccs ) hosted the 1st artificial intelligence in security(aisec ) workshop , which has since been a dedicated venue at a top - level security conference for the intersection of ml and security . from 2008 , the pace of research and publicity of ml in security started to accelerate in academic communities ( section 2.3 ) , and industry venues ( e.g. black hat , rsa ) also shifted interests . for instance , ml in security was still a topic of minority interest at black hat usa 2014 in august @xcite , but at rsa 2016 in february , the majority of vendors claimed to deploy ml in their products @xcite . a part of this shift may be motivated by the sudden increase in blackswan events like the discovery of crime , beast and heartbleed vulnerabilities . the discovery of these vulnerabilities suggest that organizations may be attacked via previously unknown classes of attacks . to defend against these types of attacks requires monitoring not just for known vectors attacks , but also for behavior suggestive of a compromised machine . the latter requires the gathering and analysis of much larger sets of data . advances in hardware and data processing capacities enabled large - scale systems . with increasing amount of data from growing numbers of information channels and devices , the analytic tools and intelligent behaviors provided by ml becomes increasingly important in security . with darpa s cyber grand challenge final contest looming @xcite , research interest in ml and security is becoming even more conspicuous . now is the crucial time to examine research works done in ml applications and security . to do so , we studied the state - of - art of ml research in security between 2008 and early 2016 , and systematize this research area in 3 ways : 1 . we survey cutting - edge research on applied ml in security , and provide a high - level overview taxonomy of ml paradigms and security domains . 2 . we point to research challenges that will improve , enhance , and expand our understanding , designs , and efficacy of applying ml in security . we emphasize a position which treats security as a game theory problem . while we realize there are different ways to classify existing security problems based on purpose , mechanism , targeted assets , and point of flow of the attack , our sok s section structure is based on the `` security and privacy '' category of 2012 acm computing classification system@xcite , which is a combination of specific use cases(e.g . malware , phishing ) , technique ( e.g. information flow ) , and targeted assets(e.g . web application , proxies ) . we present the state - of - art ml applications in security as the following : section 3 and table 2 & 3 discusses network security , section 4 and table 4 surveys security services , section 5 and table 5 specifies advances in software & applications security , section 6 and table 6 & 7 lays out taxonomy for system security , and section 7 and table 8 , 9 & 10 summarizes progress since 2008 in malware detection , ids , and social engineering . throughout the survey , we share our frameworks for ml system designs , assumptions , and algorithm deployments in security . we focus our survey on security _ applications _ and security - related ml and ai problems on the _ defense _ side , hence our scope excludes theories related to security such as differential privacy and privacy - preservation in ml algorithms@xcite , and excludes ml applications in side channel attacks such as @xcite . partly because there is already a 2013 sok on evolution of sybil defense@xcite in online social networks(osn ) , and partly because we would like to leave it as a small exercise to our readers , we excluded sybil defense schemes in osn as well@xcite . still with a broad base , we propose an alternative position to frame security issues , and we also recommend a taxonomy for ml applications in security use cases . yet , we do not conclude with a terminal list of `` right '' or `` correct '' approaches or methods . we believe that the range of the applications is too wide to fit into one singular use case or analysis framework . instead , we intend this paper as a systematic design and method overview of thinking about researching and developing ml algorithms and applications , that will guide researchers in their problem domains on an individual basis . we target our work to security researchers and practitioners , so we assume that our readers have general knowledge for key security domains and awareness of common ml algorithms , and we also define terms when needed . theory + + & 3.1/4/5/6.1/6.2/7.1/7.2/7.3&5/6.1/7.2&3.1/3.2/5/6.1/6.2/7.1/7.2&5/6.1 & + & 3.1/3.2/4/5/6.1/6.2/7.1/7.2/7.3&7.1/7.3&3.1/5&&6.2 + & & & 7.1&&6.2/7.2 + + & 6.1/7.2&7.2&6.2 & & + & 3.1/3.2/5/7.2/7.3&5/7.3&3.1/3.2/6.1/7.1&&6.2 + & 5&&5&5/6.1 & + & 3.1/4/7.2/7.3&7.2&4/7.2&&7.2 + & 6.1/6.2/7.1&6.1/7.1&6.2/7.1 & & + + & 4/5/6.1/7.2&&4/6.2/7.2 & & + & 3.1/6.2/7.3&7.2&3.1&&6.2/7.2 + & 3.2/5/6.1/6.2/7.1/7.2/7.3&5/6.1/7.1/7.2/7.3&3.2/5/6.1/7.1/7.2&5/6.1 & + we agree with assessment of top conferences in @xcite . we systematically went through all proceedings between 2008 and early 2016 of the top 6 network- and computer - security conferences to collect relevant papers . because of kdd s early and consistent publication record on ml applications in security , and its status as a top - level venue for ml and dm applications , we also include kdd s 2008 - 2015 proceedings . to demonstrate the wide - ranging research attention drawn to ml applications in security , we also added chosen selections from the workshop aisec , international conference on machine learning(icml ) , neural information processing systems(nips ) , and internet measurement conference(imc ) papers between 2008 - 2015 , mostly in the `` future development '' section . figure 1 shows the generalization of ml system designs when applied in security , that emerged from our survey of the papers(the legend is on the figure s bottom left ) . in different use cases , the system components may embody different names , but their functionalities and positions are captured in the figure . for example : 1 . * _ knowledge base _ * is baseline of known normality and/or abnormality , depending on use cases , they include but are not limited to blacklist(bl ) , whitelist(wl ) , watchlist ; known malware signatures , system traces , and their families ; initial set of malicious web pages ; existing security policies or rules , etc .. 2 . * _ data sources _ * are where relevant data is collected . they can be either off - line or live online data feed , e.g. malware traces collected after execution(off - line ) , url stream(online ) . * _ training data _ * are labeled data which are fed to classifiers in training . they can be standard research datasets , new data(mostly from industry ) labeled by human , synthetic datasets , or a mix . _ pre - processor and feature extractor _ * construct features from data sources , for example : url aggregators , graph representations , smtp header extractions , n - gram model builders . dynamic analyzer and static analyzer are used most often in malware - related ml tasks , and human feedback loop is added when the system s design intends to be semi - supervised or human - in - the - loop(hitl ) . theory + + & 58(49%)&7(5.9%)&24(20%)&2(1.7%)&0(0% ) + & 18(15%)&4(3.4%)&3(2.5%)&0(0%)&1(0.85% ) + & 0(0%)&0(0%)&0(0%)&0(0%)&2(1.7% ) + + & 4(3.4%)&1(0.85%)&1(0.85%)&0(0%)&0(0% ) + & 17(14.4%)&4(3.4%)&11(9.3%)&0(0%)&1(0.85% ) + & 4(3.4%)&0(0%)&1(0.85%)&2(1.7%)&0(0% ) + & 31(26%)&2(1.7%)&9(7.6%)&0(0%)&2(1.7% ) + & 20(17%)&4(3.4%)&5(4.2%)&0(0%)&0(0% ) + + & 16(13.6%)&0(0%)&7(6%)&0(0%)&0(0% ) + & 9(7.6%)&1(0.85%)&5(4.2%)&0(0%)&3(2.5% ) + & 51(43.2%)&10(8.5%)&15(12.7%)&2(1.7%)&0(0% ) + table 1 shows a matrix with rows indicating different ways of classifying the security problems , and the columns showing well - understood ml paradigms . based on the threat models and modeling purposes presented in the papers , we qualitatively group the attacker into three groups . if there are multiple attacker types in one section , the section s numbering appears multiple times accordingly . 1 . * _ passive _ * attackers make no attempt to evade detections ; their behaviors fit into descriptions of the threat models . _ semi - aggressive _ * attackers have knowledge of the detectors , and only attempt to evade detections . * _ active _ * attackers do not only have knowledge of the detectors and attempt to evade detections , but also actively try to poison , mislead , or thwart detection . _ knowledge _ * of attackers , is the information in at least one of the five aspects : the learning algorithms themselves , the algorithms feature spaces , the algorithm s parameters , training and evaluation data - regardless of being labeled or not - used by the algorithms , and decision feedback given by the algorithms @xcite . influenced by @xcite , we extend their definitions , and qualitatively categorize attackers primary purpose as to compromise _ confidentiality , availability _ or _ integrity _ of legitimate systems , services , and users . 1 . attacks on * _ confidentiality _ * compromise the confidential or secret information of systems , services , or users ( e.g. password crackers ) . 2 . attacks on * _ availability _ * make systems and services unusable with unwanted information , requests , or many errors in defense schemes ( e.g. ddos , spam ) . 3 . attacks on * _ integrity _ * masquerade maliciously intentions as benign intentions in systems , services , and users ( e.g. malware ) . we also define ml paradigms shown in the matrix : 1 . * _ supervised _ * learning uses labeled data for training . _ semi - supervised _ * learning uses both labeled and unlabeled data for training . _ unsupervised _ * learning has no labeled data available for training . _ human - in - the - loop(hitl ) _ * learning incorporates active human feedback to algorithm s decisions into the knowledge base and/or algorithms . _ game theory(gt)_*-based learning considers learning as a series of strategic interactions between the model learner and actors with conflicting goals . the actors can be data generators , feature generators , chaotic human actors , or a combination@xcite . for `` means of attacks '' in table 1 , server , network , and user are straightforward and intuitive , so here we only describe `` client app '' and `` client machine '' . * _ client app _ * is any browser - based means of attack on any client device , and * _ client machine _ * is any non - browser - based means of attack on any client device . as shown in table 1 , the majority of surveyed papers in different security domains use supervised learning to deal with passive or semi - aggressive attackers . however , the core requirement of supervised learning - labeled data - is not always viable or easy to obtain , and authors have repeatedly written about the difficulty of obtaining labeled data for training . based on this observation , we conclude that _ exploring semi - supervised and unsupervised learning approaches would expand the research foundation of ml applications in security domains , because semi - supervised and unsupervised learning can utilize unlabeled datasets which had not been used by supervised learning approaches before . _ moreover , during our survey , we realized that many ml applications in security assume that training and testing data come from the same distribution ( in statistical terms , this is the assumption of stationarity ) . however , in the real world , it is highly unlikely that data are stationary , let alone that the data could very well be generated by an adversarial data generator producing training and/or testing data sets , as the case in @xcite , or simply be generated responding to specific models as in @xcite . our observation from the comprehensive survey confirmed @xcite s statement , and we propose that _ gt - based learning approaches and hitl learning system designs should be explored more , in order to design more efficient security defense mechanisms to deal with active and unpredictable adversaries . at the same time , human knowledge and judgment in htil should go beyond feature engineering , to providing feedback to decisions made by ml models_. some theory - leaning papers have modeled spam filtering as bayesian games or stackelberg games@xcite . use cases in data sampling , model training with poisoned or low - confidence data have also been briefly explored in literature@xcite . based on seminal works and establishments in notable venues , the gradually increasing levels of interest in ml research applied to security is fairly visible . here we gathered some milestone events : 1 . 1987 : denning published `` an intrusion detection system '' @xcite , first framing security as a learning problem 2 . 1998 : darpa ids design challenge@xcite 3 . 1999 : kdd cup ids design challenge@xcite 4 . 2008 : ccs hosted the 1st aisec workshop . continues to operate each year@xcite 5 . 2007 , 2008 : twice , kdd hosted the international workshop on privacy , security , and trust(pinkdd)@xcite 6 . 2010 , 2012 : twice , kdd hosted intelligence and security informatics workshop(isi)@xcite 7 . 2011 : `` adversarial machine learning '' published in 4th aisec@xcite 8 . 2012 : `` privacy and cybersecurity : the next 100 years '' by landwehr et al published@xcite 9 . 2013 : manifesto from dagstuhl perspectives workshop published as `` machine learning methods for computer security '' by joseph et al @xcite . 10 . 2014 : kdd hosted its 1st `` security & privacy '' session in the main conference program@xcite 11 . 2014 : icml hosted its 1st , and so far the only workshop on learning , security , and privacy(lsp)@xcite 12 . 2016 : aaai hosted its 1st artificial intelligence for cyber security workshop(aisc)@xcite despite the surge of research interests and industry applications in the intersection of ml and security , few surveys or overviews were published after 2008 , the watershed year of increasing interest in this particular domain . in 2013 @xcitesurveyed server - side web application security , @xcite surveyed data mining applied to security in the cloud focusing on intrusion detection , @xcite discussed an ml perspective in network anomaly detection . while they are helpful and informative , the former two are limited by their scope and perspective , and the latter serves as a textbook , hence absent the quintessential of survey - mapping the progresses and charting the state - of - art . a collection of papers in 2002 and 2012 @xcite discussed applications of dm in computer security , but lacks a systematic survey on ml applications in resolving security issues . @xcite briefly compared two network anomaly detection techniques , but limited in scope . @xcite of 2009 conducted a comprehensive survey in anomaly detection techniques , some involving discussions of security domains . the dagstuhl manifesto in 2013 @xcite articulated the status quo and looked to the future of ml in security , but the majority of the literature listed were published before 2008 . @xcite of 2010 highlighted use cases and challenges for ml in network intrusion detection , but did not incorporate a high - level review of ml in security in recent years . research works on botnets among our surveyed literature focuses mainly on designing systems to detect command - and - control(c&c ) botnets , where many bot - infected machines are controlled and coordinated by few entities to carry out malicious activities@xcite . those systems need to learn decision boundaries between human and bot activities , therefore ml - based classifiers are at the core of those systems , and are often trained by labeled data in supervised learning environments . the most popular classifier is support vector machines(svms ) with different kernels , while spatial - temporal time series analysis and probabilistic inferences are also notable techniques employed in ml - based classifiers . topic clustering , mostly seen in natural language processing(nlp ) , is used to build a large - scale system to identify bot queries @xcite . in botnet detection literature , 3 core assumptions are widely shared : 1 . botnet protocols are mostly c&c @xcite 2 . individual bots within same botnets behave similarly and can be correlated to each other @xcite 3 . botnet behaviors are different and distinguishable from legitimate human user , e.g. human behaviors are more complex@xcite other stronger assumptions include that bots and humans interact with different server groups @xcite , and content features from messages generated by bots and human are independent @xcite . while classification techniques differ , wls , bls , hypothesis testing , and a classifier @xcite are usual system components . attempts have been made to abstract state machine models of network to simulate real - world network traffic and create honeypots @xcite . ground truths are often heuristic @xcite , labeled by human experts , or a combination - even at large scale , human labeled ground truths are used , for example in @xcite , game masters visual inspections serve as ground truth to detect bots in online games . in retrospect , the evolution of botnet detection is clear : from earlier and more straightforward uses of classification techniques such as clustering and nb , the research focus has expanded from the last step of classification , to the important preceding step of constructing suitable metrics , that measures and distinguishes bot - based and human - based activities@xcite . classifying dns domains that distribute or host malware , scams , and malicious content has drawn research interest especially in passive dns analysis . there are two main approaches : reputation system@xcite and classifier@xcite . reputation system scores benign and malicious domains and dns hosts , and a ml - based classifier learns boundaries between the two . nonetheless , both reputation system and classifier use various decision trees , random forest(rf ) , nave bayes(nb ) , svm , and clustering techniques for mostly supervised learning - based scoring and classification . many features used are from protocols and network infrastructures , e.g. border gateway protocol(bgp ) and updates , automated systems(as ) , registration , zone , hosts , and public bls . similar to botnet detectors , variations of bl , wl , and honeypots@xcite are used in similar functions as knowledge bases , while ground truths are often taken from public bls , limited wls , and anti - virus(av ) vendors such as mcafee and norton @xcite . but before any ml attempts take place , most studies would assume the following : 1 . malicious uses of dns are distinct and distinguishable from legitimate dns services . 2 . the data collection process - regardless of different names such as data flow , traffic recorder , or packet assembler - follows a centralized model . in other words , all the traffic / data / packets flow through certain central node or nodes to be collected . stronger assumptions include that as hijackers can not manipulate as path before it reaches them@xcite , and maliciousness will trigger an accurate ip address classifier to fail@xcite . besides analyzing the status quo , @xcite showed efforts to preemptively protect network measurement integrity and predict potentially malicious activities from web domains and ip address spaces . both offense and defense for access control , authentication , and authorization reside within the domain of security services . defeating audio and visual captchas(completely automated public turing test to tell computers and humans apart)@xcite , cracking passwords@xcite , measuring password strengths@xcite , and uncovering anonymity@xcite are 4 major use cases . on the offense , specialized ml domains such as computer vision , signal processing , and nlp automate attacks on user authentication services i.e. textual or visual passwords and captchas , and uncover hidden identities and services . on the defense side , entropy - based and ml - based systems calculate password strengths . other than traditional user authentication schemes , behavioral metrics of users are also introduced . following the generalized ml pipeline shown in figure 1 , the `` classifier '' is replaced by `` recognition engine '' in the password cracking process , and `` user differentiation engine '' in authentic metric engineering @xcite . hence the process becomes : `` data source @xmath0 pre - process & feature extraction @xmath0 recognition or user differentiation engine @xmath0 decision '' for ml - based security services . a noteworthy trend to observe , is that attacks on captchas are getting more generalized - from utilizing svm in 2008 to attack a specific type of text captcha@xcite , in 2015 a generic attach approach to attack text - based captcha @xcite was proposed . ml - based attacks on textual and visual captcha typically follow the 4-step process : 1 . _ segmentation _ : e.g. signal to noise ratio(snr ) for audio ; hue , color , value(hsv ) for visual @xcite 2 . _ signal or image representation _ : e.g. discrete fourier transformation(audio)@xcite , letter binarization(visual ) @xcite 3 . _ feature extraction _ : e.g. spectro - temporal features , character strokes @xcite 4 . _ recognition _ : k - nearest neighbor(knn ) , svm(rbf kernel ) , convolutional neural networks(cnn ) @xcite on the side of password - related topics in security services , there are 2 password models : whole - string markov models , and template - based models @xcite . concepts in statistical language modeling , such as natural language encoder and n - grams associated with markov models(presented as directed graphs with nodes labeled by n - grams ) , and context - free grammars are common probabilistic foundations to build password strength meters and password crackers @xcite . ml research in software and applications security mostly concentrate on web application security in our survey , and have used supervised learning to train popular classifiers such as nb and svm to detect web - based malware and javascript(js ) code@xcite , filter unwanted resources and requests such as malicious advertisements@xcite , predict unwanted resources and requests(e.g . future blacklisted websites)@xcite , and quantify web application vulnerabilities@xcite . while @xcite explored building web application anomaly detector with scarce training data , most use cases follow the supervised paradigm assuming plentiful labeled data : data source(web applications , static / dynamic analyzers ) @xmath0 feature extraction(often with specific pre - filter , metrics , and de - obfuscator if needed ) @xmath0 classifiers trained with labeled data . apart from this supervised setting , if a human expert s feedback is added after classifiers decisions@xcite , it forms a semi - supervised system . regardless of system designs , the usual assumption holds : _ malicious activities or actors are different from normal and benign ones likely do not change much_. the knowledge bases of normality and abnormality can vary , from historical regular expression lists@xcite to other publicly available detectors@xcite . graph - based algorithms@xcite and image recognition@xcite are both used in resource filtering , but in detecting js malware and evasions and quantifying leaks , having suitable measurements of similarities is a significant focal point . indeed , from @xcite , ml - based classifiers do well in finding similarities between mutated malicious code snippets , while the same code pieces could evade static or dynamic analyzer detections . as landwehr noticed@xcite , ml can be applied in spm . however , in automatic fingerprinting of operating systems(os ) , c4.5 decision tree , svm , rf , knn - some most commonly used ml - based classifiers in security - failed to distinguish remote machine instances with coarse- and fine - grained differences , as the algorithms can not exploit semantic knowledge of protocols or send multi - packet probes @xcite . yet by taking advantage of semantic and syntactic features , plus semi - supervised system design , @xcite showed that svm(optimized by sequential minimal optimization[smo ] algorithm ) , knn , and nlp techniques do well in android spm . on the other hand , in vulnerability management , @xcite , clustering techniques have done well in predicting future incidents and infer vulnerability patterns in code , as well as nb , svm , and rf in ranking risks and identifying proper permission levels . both vulnerability management and spm also focus on devising proper metrics for ml applications : from heuristics based on training set @xcite , jaro distance @xcite , to outside reputation system oracles @xcite , metrics are needed to compare dependency graphs , string similarities , and inferred vulnerability patterns . in most use cases , because of the need for labeled data to train supervised learning systems , many systems follow the generalized training process in figure 1 : `` knowledge base @xmath0 offline trainer @xmath0 online or offline classifier '' . when policy management decisions need feedback , a hitl design is in place where end human users feedback is directed to knowledge base . one distinguishing tradition in ml applications research in this domain , is a strong emphasis on measurement - selecting or engineering proper similarity or scoring metrics are often important points of discussion in research literature . from earlier uses of heuristics in clustering algorithms , to more recent semantic connectivity measurement applied in semi - supervised systems , both the metrics and the system designs for vulnerability and security policy management have evolved to not only identify , but also to infer and predict future vulnerable instances . compared to other security domains , ml research in information flow and ddos focus more on evasion tactics and limits of ml systems in adversarial environments . hence we grouped together the two sub - domains , and marked studies in table 7 with `` ( if ) '' and `` ( ddos ) '' accordingly . for ddos@xcite , the usual assumption is that _ patterns of attack and abuse traffic are different from normal traffic _ @xcite , but @xcite challenged it by proposing an adversary who can generate attributes that look as plausible as actual attributes in benign patterns , and caused failure in ml - based automated signature generation to distinguish benign and malicious byte sequences . then , @xcite introduced gt to evaluate ddos attack and defense in real - world . for information flow@xcite , assumptions can take various forms . in pdf classifiers based on document structural features , it is _ malicious pdf has different document structures than good pdfs _ @xcite ; in android privacy leak detector , it is _ _ the majority of an android application s semantically similar peers has similar privacy disclosure scenarios__@xcite . but @xcite poses semi - aggressive and active attackers with some information about the data , feature sets , and/or algorithms , and then attackers successfully evade ml - based pdf classifiers . another example is , pdf malware could be classified @xcite , and then a generic and automated evasion technique based on genetic programming is successfully experimented@xcite . overall , while using svm , rf , and decision trees trained with labeled data to detect and predict ddos and malicious information and data flows , ml applications in information flow and ddos challenge the usual assumption of stationary adversary behaviors . from collecting local information only , to proposing a general game theory - based framework to evaluate ddos attacks and defense , and from using static method to detect malicious pdf file to generic automated evasion , the scope of ml applications in both ddos and if have expanded and generalized over the years . program - centric or system - centric , there are 3 areas that draw most ml application research attention in malware : malware detection@xcite , classifying unknown malware into families@xcite , and auto - extract program or protocol specifications @xcite . realizing the signature and heuristic - based malware detectors can be evaded by obfuscation and polymorphism @xcite , more behavior - based matching and clustering systems and algorithms have been researched . figure 1 already shows a generalized ml system design for malware detection and classification , and a more detailed description is below : 1 . collect malware artifacts and samples , analyze them , execute them in a controlled virtual environment to collect traces , system calls , api calls , etc . @xcite . or , directly use information from already completed static and/or dynamic analyses . 2 . decide or devise similarity measurements between generalized binaries , system call graphs(scg ) , function call graphs(fcg ) , etc . , then extract features @xcite 3 . classify malware artifacts into families in - sample , or cluster them with known malware families . the classifiers and clustering engines are usually trained with labeled data@xcite . popular ones are svm and rf for classification , and hidden markov model(hmm ) and knn alongside different clustering techniques . even in the use case of auto - extract specifications , supervised learning with labeled data is needed when behavior profiles , state machine inferences , fuzzing , and message clustering are present . evasion techniques of detectors and poisoning of ml algorithms are also discussed , and typical evasion techniques include obfuscation , polymorphism , mimicry , and reflecting set generation@xcite . malware detection and matching based on structural information and behavior profiles@xcite show a tendency to use graph - based clustering and detection algorithms , and similarity measurement used in these algorithms have ranged from jaccard distance to new graph - based matching metrics . while clustering techniques have been mostly used in malware detection , a nearest neighbor technique is explored to evade malware detection . spams , malicious webpages and urls that redirect or mislead un - suspecting users to malware , scams , or adult content @xcite is perhaps as old as civilian use of the internet . research literature mostly focus on 3 major areas : detecting phishing malicious urls@xcite , filtering spam or fraudulent content @xcite , and detecting malicious user account behaviors@xcite . moreover , because phishing is a classic social engineering tactic , it is often the gateway of many studies to detect malicious urls , spam , and fraudulent content . to identify malicious urls , ml - based classifiers draw features from webpage content(lexical , visual , etc . ) , url lexical features , redirect paths , host - based features , or some combinations of them . such classifiers usually act in conjunction with knowledge bases which are usually in - browser url bls or from web service providers . if the classifier is fed with url - based features , it is common to set an url aggregator as a pre - processor before extracting features . mostly using supervised learning paradigm , nb , svm with different kernels , and lr are popular ml classifiers for filtering spam and phishing . meanwhile , gt - based learning to deal with active attackers is also evaluated in spam filtering . @xcite evaluates a bayesian game model where the defense is not fully informed of the attacker s objectives and the active adversary can exercise control over data generation , @xcite proposes a stackelberg game where spammer reacts to the learner s moves . stronger assumptions also exist : for example , @xcite assumes spammers phone blocks follow a beta distribution as conjugate prior for bernoulli and binomial distribution . another social engineering tactic is spoofing identities with fake or compromised user accounts , and detection of such malicious behaviors utilize features from user profiles , spatial- , temporal- , and spatial - temporal patterns , and user profiles are used in particular to construct normality . graph representation and trust propagation models are also deployed to distinguish genuine and malicious accounts with different behavior and representations@xcite . tracing the chronology of applying ml to defend against social engineering , one trend is clear : while content- , lexical- , and syntactic - based features are still being widely used , constructing graph representations and exploring temporal patterns of redirect paths , events , accounts , and behaviors have been on the rise as feature spaces for ml applications in defend against social engineering efforts . accordingly , the ml techniques have also changed from different classification schemes to graphic models . it is also noteworthy that in @xcite , addressing adversarial environments challenges to ml systems is elaborated as primary research areas , instead of a short discussion point . from feature sets to algorithms and systems , ids has been extensively studied . however , as @xcite cautioned , ml can be easily conflated with anomaly detection . while both are applied to build ids , important difference is that ml aims to generalize expert - defined distinctions , but anomaly detection focuses on finding unusual patterns , while attacks are not necessarily anomalous . for example , @xcite distinguished n - gram model s different use cases : anomaly detection uses it to construct normality(hence more appropriate when no attack is available for learning ) , and ml classifiers learn to discriminate between benign and malicious n - grams(hence more appropriate when more labeled data is present ) . since 2008 , works at top venues have added to the rigor for ml applications in ids . for example , a common assumption of ids is : _ anomalous or malicious behaviors or traffic flows are fundamentally different from normal ones _ , but @xcite challenges the assumption by studying low - cardinality intrusions where attackers do nt send a large number of probes . to address adversarial learning environment and minimal labels in training data , semi - supervised paradigms , especially active learning , are also used@xcite . heterogeneous designs of ids in different use cases give rise to many ad - hoc evaluations in research works , and a reproducibility and comparison framework was proposed to address the issue@xcite . meanwhile , techniques such as graph - based community detection@xcite , time series - based methods@xcite , and generalized support vector data description in cyber - physical system and adversarial environment for auto - feature selection@xcite , have also emerged . although they carry different assumptions of normality and feature representations , the supervised ml system design remains largely the same . besides the fact the more techniques and use cases have been proposed , the focus of research in ids had evolved from discovering new techniques and use cases , to rigorously evaluating fundamental assumptions and workflows of ids . for example , while feature selection has stayed as a major component , there are re - examination of assumptions and measurements on what constitutes normality and abnormality@xcite , alternative to more easily acquire data and use low - confidence data for ml systems@xcite , and proposal on validating reproducibility of results from different settings@xcite . one key goal of our sok survey is to help researchers look into the future . ml applications in security domains are attracting academic research attention as well as industrial interest , and this presents a valuable opportunity for researchers to navigate the landscapes between ml theories and security applications . there are also opportunities to explore if there are some types of ml paradigms that are especially well suited to particular security problems . apart from highlighting that 1 ) semi - supervised and unsupervised ml paradigms are more effective in utilizing unlabeled data , hence ease the difficulty of obtaining labeled data , and 2 ) gt - based ml paradigms and hitl ml system designs will become more influential in dealing with semi - aggressive and aggressive attackers , we also share the following seven speculations of future trends , based on our current sok . 1 . * metric learning * : measurement has become more and more conspicuous for ml research in security , mostly in similarity measurement for clustering algorithms@xcite . proper measurements and metrics are also used to construct ground truths to evaluate ml - based classifiers , and also have important roles in feature engineering@xcite . given the ubiquitous presence of metrics and the complex nature of constructing them , ml applications in security will benefit much from metric learning . * nlp * : malicious content , spam , and malware analysis and detections have used tools from statistical language modeling(e.g . n - gram - based representation for strings in code and http request)@xcite , as textual information explodes , nlp will become more widely used beyond source filtering and clustering e.g. @xcite use n - gram models to infer state machines of protocols . * upstream movement of ml in security defense designs*. in malware detection and classifications , behavior- and signature - based malware classifiers have used inputs from static and dynamic binary analysis as features@xcite , and @xcite already shows rnn can be applied to automatically recognize functions in binary analysis . we also see ml algorithms applied in vulnerability , device , and security policy management , ddos mitigation , information flow quantifications , and network infrastructure@xcite . hence , it is reasonable to expect that more ml systems and algorithms will move upstream in more security domains . * scalability * : with increasing amount of data from growing numbers of information channels and devices , scale of ml - based security defenses will become a more important aspect in researching ml applications in security @xcite . as a result , large - scale systems will enable * distributed graph algorithms * in malware analysis , as path hijacker tracing , cyber - physical system fault correlation , etc .. @xcite 5 . * specialized probabilistic models * will be applied beyond the context of classifiers , e.g. access control@xcite . high fp rates have always been a concern for system architects and algorithm researchers @xcite . * reducing fp rates * will grow from an ad - hoc component in various system designs , to independent formal frameworks , algorithms , and system designs . * privacy enforcement * was framed as a learning problem recently in @xcite , in the light of many publications on privacy - preservation in ml algorithms , and privacy enhancement by probabilistic models@xcite . this new trend will become more prominent . in this paper , we analyzed ml applications in security domains by surveying literature from top venues of our field between 2008 and early 2016 . we attempted to bring clarity to a complex field with intersecting expertises by identifying common use cases , generalized system designs , common assumptions , metrics or features , and ml algorithms applied in different security domains . we constructed a matrix showing the intersections of ml paradigms and three different taxonomy structures to classify security domains , and show that while much research has been done , explorations in gt - based ml paradigms and hitl ml system designs are still much desired ( and under - utilized ) in the context of active attackers . we point out 7 promising areas of research based on our observations , and argue that while ml applications can be powerful in security domains , it is critical to match the ml system designs with the underlying constraints of the security applications appropriately . we would like to thank megan yahya , krishnaprasad vikram , and scott algatt for their time and valuable feedback . c. landwehr , d. boneh , j. c. mitchell , s. m. bellovin , s. landau , and m. e. lesk , `` privacy and cybersecurity : the next 100 years , '' _ proceedings of the ieee _ special centennial issue , 2012 . t. ahmed , b. oreshkin , and m. coates , `` machine learning approaches to network anomaly detection , '' in _ proceedings of the 2nd usenix workshop on tackling computer systems problems with machine learning techniques _ , 2007 . p. g. kelley , s. komanduri , m. l. mazurek , r. shay , t. vidas , l. bauer , n. christin , l. f. cranor , and j. lopez , `` guess again ( and again and again ) : measuring password strength by simulating password - cracking algorithms , '' in _ sp 2012_. r. wang , w. enck , d. reeves , x. zhang , p. ning , d. xu , w. zhou , and a. m. azab , `` easeandroid : automatic policy analysis and refinement for security enhanced android via large - scale semi - supervised learning , '' in _ usenix security 2015_. k. lu , z. li , v. p. kemerlis , z. wu , l. lu , c. zheng , z. qian , w. lee , and g. jiang , `` checking more and alerting less : detecting privacy leakages via enhanced data - flow analysis and peer voting . '' in _ ndss 2015_.
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the idea of applying machine learning(ml ) to solve problems in security domains is almost 3 decades old . as information and communications
grow more ubiquitous and more data become available , many security risks arise as well as appetite to manage and mitigate such risks . consequently , research on applying and designing ml algorithms and systems for
security has grown fast , ranging from intrusion detection systems(ids ) and malware classification to security policy management(spm ) and information leak checking . in this paper
, we systematically study the methods , algorithms , and system designs in academic publications from 2008 - 2015 that applied ml in security domains .
98% of the surveyed papers appeared in the 6 highest - ranked academic security conferences and 1 conference known for pioneering ml applications in security .
we examine the generalized system designs , underlying assumptions , measurements , and use cases in active research .
our examinations lead to 1 ) a taxonomy on ml paradigms and security domains for future exploration and exploitation , and 2 ) an agenda detailing open and upcoming challenges . based on our survey
, we also suggest a point of view that treats security as a game theory problem instead of a batch - trained ml problem .
* keywords * : security , machine learning , large - scale applications , game theory , security policy management
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dans le modle cosmologique , dit de concordance car il est en conformit avec tout un ensemble de donnes observationnelles , la matire ordinaire do nt sont constitus les toiles , le gaz , les galaxies , etc . ( essentiellement sous forme baryonique ) ne forme que 4% de la masse - nergie totale , ce qui est dduit de la nuclosynthse primordiale des lments lgers , ainsi que des mesures de fluctuations du rayonnement du fond diffus cosmologique ( cmb ) le rayonnement fossile qui date de la formation des premiers atomes neutres dans lunivers . nous savons aussi quil y a 23% de matire noire sous forme _ non baryonique _ et do nt nous ne connaissons pas la nature . et les 73% qui restent ? et bien , ils sont sous la forme dune mystrieuse nergie noire , mise en vidence par le diagramme de hubble des supernovae de type ia , et do nt on ignore lorigine part quelle pourrait tre sous la forme dune constante cosmologique . le contenu de lunivers grandes chelles est donc donn par le `` camembert '' de la figure [ fig1 ] do nt 96% nous est inconnu ! la matire noire permet dexpliquer la diffrence entre la masse dynamique des amas de galaxies ( cest la masse dduite du mouvement des galaxies ) et la masse de la matire lumineuse qui comprend les galaxies et le gaz chaud intergalactique . mais cette matire noire ne fait pas que cela ! nous pensons quelle joue un rle crucial dans la formation des grandes structures , en entranant la matire ordinaire dans un effondrement gravitationnel , ce qui permet dexpliquer la distribution de matire visible depuis lchelle des amas de galaxies jusqu lchelle cosmologique . des simulations numriques trs prcises permettent de confirmer cette hypothse . pour que cela soit possible il faut que la matire noire soit non relativiste au moment de la formation des galaxies . on lappelera matire noire _ froide _ ou cdm selon lacronyme anglais , et il y a aussi un nom pour la particule associe : un wimp pour `` weakly interacting massive particle '' . il ny a pas dexplication pour la matire noire ( ni pour lnergie noire ) dans le cadre du modle standard de la physique des particules . mais des extensions au - del du modle standard permettent de trouver des bons candidats pour la particule ventuelle de matire noire . par exemple dans un modle de super - symtrie ( qui associe tout fermion un partenaire super - symtrique qui est un boson et rciproquement ) lun des meilleurs candidats est le _ neutralino _ , qui est un partenaire fermionique super - symtrique dune certaine combinaison de bosons du modle standard . , qui fut introduit dans une tentative pour rsoudre le problme de la violation cp en physique des particules , est une autre possibilit . il y a aussi les tats de kaluza - klein prdits dans certains modles avec dimensions suplmentaires . quant lnergie noire , elle apparat comme un milieu de densit dnergie _ constante _ au cours de lexpansion , ce qui implique une violation des `` conditions dnergie '' habituelles avec une pression ngative . lnergie noire pourrait tre la fameuse constante cosmologique @xmath0 queinstein avait introduite dans les quations de la relativit gnrale afin dobtenir un modle dunivers statique , puis quil avait abandonne lorsque lexpansion fut dcouverte . depuis zeldovich on interprte @xmath0 comme lnergie du vide associe lespace - temps lui - mme . le problme est que lestimation de cette nergie en thorie des champs donne une valeur @xmath1 fois plus grande que la valeur observe ! on ne comprend donc pas pourquoi la constante cosmologique est si petite . malgr lnigme de lorigine de ses constituents , le modle @xmath0-cdm est plein de succs , tant dans lajustement prcis des fluctuations du cmb que dans la reproduction fidle des grandes structures observes . une leon est que la matire noire apparat forme de particules ( les wimps ) grande chelle . la matire noire se manifeste de manire clatante dans les galaxies , par lexcs de vitesse de rotation des toiles autour de ces galaxies en fonction de la distance au centre cest la clbre courbe de rotation ( voir la figure [ fig2 ] ) . les mesures montrent qu partir dune certaine distance au centre la courbe de rotation devient pratiquement plate , cest - - dire que la vitesse devient constante . daprs la loi de newton la vitesse dune toile sur une orbite circulaire ( keplerienne ) de rayon @xmath2 est donne par @xmath3 o @xmath4 est la masse contenue dans la sphre de rayon @xmath2 . pour obtenir une courbe de rotation plate il faut donc supposer que la masse crot proportionnellement @xmath2 ( et donc que la densit dcrot comme @xmath5 ) , ce qui nest certainement pas le cas de la matire visible . on est oblig dinvoquer lexistence dun gigantesque halo de matire noire invisible ( qui ne rayonne pas ) autour de la galaxie et do nt la masse dominerait celle des toiles et du gaz . cette matire noire peut - elle tre faite de la mme particule que celle suggre par la cosmologie ( un wimp ) ? des lments de rponse sont fournis par les simulations numriques de cdm en cosmologie qui sont aussi valables lchelle des galaxies , et qui donnent un profil de densit universel pour le halo de matire noire . a grande distance ce profil dcroit en @xmath6 soit plus rapidement que ce quil faudrait pour avoir une courbe plate , mais ce nest pas trs grave car on peut supposer que la courbe de rotation est observe dans un rgime intermdiaire avant de dcrotre . plus grave est la prdiction dun pic central de densit au centre des galaxies , o les particules de matire noire tendent sagglomrer cause de la gravitation , avec une loi en @xmath7 pour @xmath2 petit . or les courbes de rotation favorisent plutt un profil de densit sans divergence , avec un coeur de densit constante . dautres problmes rencontrs par les halos simuls de cdm sont la formation dune multitude de satellites autour des grosses galaxies , et la loi empirique de tully et fisher qui nest pas explique de faon naturelle . cette loi montre dans la figure [ fig3 ] relie la luminosit des galaxies leur vitesse asymptotique de rotation ( qui est la valeur du plateau dans la figure [ fig2 ] ) par @xmath8 . noter que cette loi ne fait pas rfrence la matire noire ! la vitesse et la luminosit sont bien sr celles de la matire ordinaire , et la matire noire semble faire ce que lui dicte la matire visible . mais le dfi le plus important de cdm est de pouvoir rendre compte dune observation tonnante appele _ loi de milgrom _ @xcite , selon laquelle la matire noire intervient uniquement dans les rgions o le champ de gravitation ( ou , ce qui revient au mme , le champ dacclration ) est plus _ faible _ quune certaine acclration critique mesure la valeur `` universelle '' @xmath9 . tout se passe comme si dans le rgime des champs faibles @xmath10 , la matire ordinaire tait acclre non par le champ newtonien @xmath11 mais par un champ @xmath12 donn simplement par @xmath13 . la loi du mouvement sur une orbite circulaire donne alors une vitesse _ constante _ et gale @xmath14 . ce rsultat nous rserve un bonus important : puisque le rapport masse - sur - luminosit @xmath15 est approximativement le mme dune galaxie lautre , la vitesse de rotation doit varier comme la puissance @xmath16 de la luminosit @xmath17 , en accord avec la loi de tully - fisher ! pour avoir une rgle qui nous permette dajuster les courbes de rotation des galaxies il nous faut aussi prendre en compte le rgime de champ fort dans lequel on doit retrouver la loi newtonienne . on introduit une fonction dinterpolation @xmath18 dpendant du rapport @xmath19 et qui se ramne @xmath20 lorsque @xmath10 , et qui tend vers 1 quand @xmath21 . notre rgle sera donc @xmath22 ici @xmath12 dsigne la norme du champ de gravitation @xmath23 ressenti par les particules dpreuves . une formule encore plus oprationnelle est obtenue en prenant la divergence des deux membres de ce qui mne lquation de poisson modifie , o @xmath24 est le laplacien et @xmath25 le potentiel newtonien local . loprateur @xmath26 appliqu une fonction scalaire est le gradient , appliqu un vecteur cest la divergence : @xmath27 . par convention , on note les vecteurs en caractres gras . ] : @xmath28 = -4 \pi \ , g\,\rho_\text{b } \,,\ ] ] do nt la source est la densit de matire baryonique @xmath29 ( le champ gravitationnel est irrotationnel : @xmath30 ) . on appellera lquation la formule mond pour `` modified newtonian dynamics '' . le succs de cette formule ( on devrait plus exactement dire cette _ recette _ ) dans lobtention des courbes de rotation de nombreuses galaxies est impressionnant ; voir la courbe en trait plein dans la figure [ fig2 ] . cest en fait un ajustement un paramtre libre , le rapport @xmath15 de la galaxie qui est donc _ mesur _ par notre recette . on trouve que non seulement la valeur de @xmath15 est de lordre de 1 - 5 comme il se doit , mais quelle est remarquablement en accord avec la couleur observe de la galaxie . beaucoup considrent la formule mond comme `` exotique '' et reprsentant un aspect mineur du problme de la matire noire . on entend mme parfois dire que ce nest pas de la physique . bien sr ce nest pas de la physique _ fondamentale _ cette formule ne peut pas tre considre comme une thorie fondamentale , mais elle constitue de lexcellente physique ! elle capture de faon simple et puissante tout un ensemble de faits observationnels . au physicien thoricien dexpliquer pourquoi . la valeur numrique de @xmath31 se trouve tre trs proche de la constante cosmologique : @xmath32 . cette concidence cosmique pourrait nous fournir un indice ! elle a aliment de nombreuses spculations sur une possible influence de la cosmologie dans la dynamique locale des galaxies . face la `` draisonnable efficacit '' de la formule mond , trois solutions sont possibles . 1 . la formule pourrait sexpliquer dans le cadre cdm . mais pour rsoudre les problmes de cdm il faut invoquer des mcanismes astrophysiques compliqus et effectuer un ajustement fin des donnes galaxie par galaxie . 2 . on est en prsence dune modification de la loi de la gravitation dans un rgime de champ faible @xmath10 . cest lapproche traditionnelle de mond et de ses extensions relativistes . la gravitation nest pas modifie mais la matire noire possde des caractristiques particulires la rendant apte expliquer la phnomnologie de mond . cest une approche nouvelle qui se prte aussi trs bien la cosmologie . la plupart des astrophysiciens des particules et des cosmologues des grandes structures sont partisans de la premire solution . malheureusement aucun mcanisme convainquant na t trouv pour incorporer de faon naturelle la constante dacclration @xmath31 dans les halos de cdm . dans la suite nous considrerons que la solution 1 . est dores et dj exclue par les observations . les approches 2 . de gravitation modifie et 3 . que lon peut qualifier de _ matire noire modifie _ croient toutes deux dans la pertinence de mond , mais comme on va le voir sont en fait trs diffrentes . notez que dans ces deux approches il faudra expliquer pourquoi la matire noire semble tre constitue de wimps lchelle cosmologique . cette route , trs dveloppe dans la littrature , consiste supposer quil ny a pas de matire noire , et que reflte une violation fondamentale de la loi de la gravitation . cest la proposition initiale de milgrom @xcite un changement radical de paradigme par rapport lapproche cdm . pour esprer dfinir une thorie il nous faut partir dun lagrangien . or il est facile de voir que dcoule dun lagrangien , celui - ci ayant la particularit de comporter un terme cintique non standard pour le potentiel gravitationnel , du type @xmath33 $ ] au lieu du terme habituel @xmath34 , o @xmath35 est une certaine fonction que lon relie la fonction @xmath18 . ce lagrangien a servi de point de dpart pour la construction des thories de la gravitation modifie . on veut modifier la relativit gnrale de faon retrouver mond dans la limite non - relativiste , cest - - dire quand la vitesse des corps est trs faible par rapport la vitesse de la lumire @xmath36 . en relativit gnrale la gravitation est dcrite par un champ tensoriel deux indices appel la mtrique de lespace - temps @xmath37 . cette thorie est extrmement bien vrifie dans le systme solaire et dans les pulsars binaires , mais peu teste dans le rgime de champs faibles qui nous intresse ( en fait la relativit gnrale est le royaume des champs gravitationnels forts ) . la premire ide qui vient lesprit est de promouvoir le potentiel newtonien @xmath25 en un champ scalaire @xmath38 ( sans indices ) et donc de considrer une thorie _ tenseur - scalaire _ dans laquelle la gravitation est dcrite par le couple de champs @xmath39 . on postule , de manire _ ad - hoc _ , un terme cintique non standard pour le champ scalaire : @xmath40 o @xmath41 est reli @xmath18 , et on choisit le lagrangien deinstein - hilbert de la relativit gnrale pour la partie concernant la mtrique @xmath37 . tout va bien pour ce qui concerne le mouvement des toiles dans une galaxie , qui reproduit mond . mais notre thorie tenseur - scalaire est une catastrophe pour le mouvement des photons ! en effet ceux - ci ne ressentent pas la prsence du champ scalaire @xmath38 cens reprsenter la matire noire . dans une thorie tenseur - scalaire toutes les formes de matire se propagent dans un espace - temps de mtrique _ physique _ @xmath42 qui diffre de la mtrique deinstein @xmath37 par un facteur de proportionalit dpendant du champ scalaire , soit @xmath43 . une telle relation entre les mtriques est dite conforme et laisse invariants les cnes de lumire de lespace - temps . les trajectoires de photons seront donc les mmes dans lespace - temps physique que dans lespace - temps deinstein ( cela se dduit aussi de linvariance conforme des quations de maxwell ) . comme on observe dnormes quantits de matire noire grce au mouvement des photons , par effet de lentille gravitationnelle , la thorie tenseur - scalaire est limine . pour corriger cet effet dsastreux du mouvement de la lumire on rajoute un nouvel lment notre thorie . puisque cest cela qui cause problme on va transformer la relation entre les mtriques @xmath42 et @xmath37 . une faon de le faire est dy insrer ( encore de faon _ ad - hoc _ ) un nouveau champ qui sera cette fois un vecteur @xmath44 avec un indice . on aboutit donc une thorie dans laquelle la gravitation est dcrite par le triplet de champs @xmath45 . cest ce quon appelle une thorie _ tenseur - vecteur - scalaire _ ( teves ) . la thorie teves a t mise au point par bekenstein et sanders @xcite . comme dans la thorie tenseur - scalaire on aura la partie deinstein - hilbert pour la mtrique , plus un terme cintique non standard @xmath40 pour le champ scalaire . quant au champ vectoriel on le munit dun terme cintique analogue celui de llectromagntisme , mais dans lequel le rle du potentiel lectromagntique @xmath46 est tenu par notre champ @xmath44 . la thorie teves rsultante est trs complique et pour linstant non relie de la physique microscopique . il a t montr que cest un cas particulier dune classe de thories appeles thories einstein-_ther _ dans lesquelles le vecteur @xmath44 joue le rle principal , en dfinissant un rfrentiel priviligi un peu analogue lther postul au xix@xmath47 sicle pour interprter la non - invariance des quations de maxwell par transformation de galile . si elle est capable de retrouver mond dans les galaxies , la thorie teves a malheureusement un problme dans les amas de galaxies car elle ne rend pas compte de toute la matire noire observe . cest en fait un problme gnrique de toute extension relativiste de mond . cependant ce problme peut tre rsolu en supposant lexistence dune composante de matire noire _ chaude _ sous la forme de neutrinos massifs , ayant la masse maximale permise par les expriences actuelles soit environ @xmath48 . rappelons que toute la matire noire ne peut pas tre sous forme de neutrinos : dune part il ny aurait pas assez de masse , et dautre part les neutrinos tant relativistes auraient tendance lisser lapparence des grandes structures , ce qui nest pas observ . nanmoins une pince de neutrinos massifs pourrait permettre de rendre viables les thories de gravitation modifie . de ce point de vue les expriences prvues qui vont dterminer trs prcisment la masse du neutrino ( en vrifiant la conservation de lnergie au cours de la dsintgration dune particule produisant un neutrino dans ltat final ) vont jouer un rle important en cosmologie . teves a aussi des difficults lchelle cosmologique pour reproduire les fluctuations observes du cmb . l aussi une composante de neutrinos massifs peut aider , mais la hauteur du troisime pic de fluctuation , qui est caractristique de la prsence de matire noire sans pression , reste difficile ajuster . une alternative logique la gravit modifie est de supposer quon est en prsence dune forme particulire de matire noire ayant des caractristiques diffrentes de cdm . dans cette approche on a lambition dexpliquer la phnomnologie de mond , mais avec une philosophie nouvelle puisquon ne modifie pas la loi de la gravitation : on garde la relativit gnrale classique , avec sa limite newtonienne habituelle . cette possibilit merge grce lanalogue gravitationnel du mcanisme physique de polarisation par un champ extrieur et quon va appeler `` polarisation gravitationnelle '' @xcite . la motivation physique est une analogie frappante ( et peut - tre trs profonde ) entre mond , sous la forme de lquation de poisson modifie , et la physique des milieux dilectriques en lectrostatique . en effet nous apprenons dans nos cours de physique lmentaire que lquation de gauss pour le champ lectrique ( cest lune des quations fondamentales de maxwell ) , est modifie en prsence dun milieu dilectrique par la contribution de la polarisation lectrique ( voir lappendice ) . de mme , mond peut - tre vu comme la modification de lquation de poisson par un milieu `` digravitationnel '' . explicitons cette analogie . on introduit lanalogue gravitationnel de la susceptibilit , soit @xmath49 qui est reli la fonction mond par @xmath50 . la `` polarisation gravitationnelle '' est dfinie par @xmath51 la densit des `` masses de polarisation '' est donne par la divergence de la polarisation soit @xmath52 . avec ces notations lquation devient @xmath53 qui apparat maintenant comme une quation de poisson ordinaire , mais do nt la source est constitue non seulement par la densit de matire baryonique , mais aussi par la contribution des masses de polarisation @xmath54 . il est clair que cette criture de mond suggre que lon est en prsence non pas dune modification de la loi gravitationnelle , mais dune forme nouvelle de matire noire de densit @xmath54 , cest - - dire faite de moments dipolaires aligns dans le champ de gravitation . ltape suivante serait de construire un modle microscopique pour des diples gravitationnels @xmath55 ( tels que @xmath56 ) . lanalogue gravitationnel du diple lectrique serait un vecteur @xmath57 sparant deux masses @xmath58 . on se heurte donc un problme svre : le milieu dipolaire gravitationnel devrait contenir des masses ngatives ! ici on entend par masse lanalogue gravitationnel de la charge , qui est ce quon appelle parfois la masse grave . ce problme des masses ngatives rend _ a priori _ le modle hautement non viable . nanmoins , ce modle est intressant car il est facile de montrer que le coefficient de susceptibilit gravitationnelle doit tre ngatif , @xmath59 , soit loppos du cas lectrostatique . or cest prcisment ce que nous dit mond : comme la fonction @xmath18 interpole entre le rgime mond o @xmath60 et le rgime newtonien o @xmath61 , on a @xmath62 et donc bien @xmath59 . il est donc tentant dinterprter le champ gravitationnel plus intense dans mond que chez newton par la prsence de `` masses de polarisation '' qui _ anti - crantent _ le champ des masses gravitationnel ordinaires , et ainsi augmentent lintensit effective du champ gravitationnel ! dans le cadre de ce modle on peut aussi se convaincre quun milieu form de diples gravitationnels est intrinsquement instable , car les constituants microscopiques du diple devraient se repousser gravitationnellement . il faut donc introduire une force interne dorigine _ non - gravitationnelle _ , qui va supplanter la force gravitationnelle pour lier les constituants dipolaires entre eux . on pourrait qualifier cette nouvelle interaction de `` cinquime force '' . pour retrouver mond , on trouve de faon satisfaisante que ladite force doit dpendre du champ de polarisation , et avoir en premire approximation la forme dun oscillateur harmonique . par leffet de cette force , lquilibre , le milieu dipolaire ressemble une sorte d``ther statique '' , un peu limage du dilectrique do nt les sites atomiques sont fixes . les arguments prcdents nous laissent penser que mond a quelque chose voir avec un effet de polarisation gravitationnelle . mais il nous faut maintenant construire un modle cohrent , reproduisant lessentiel de cette physique , et _ sans _ masses graves ngatives , donc respectant le principe dquivalence . il faut aussi bien sr que le modle soit _ relativiste _ ( en relativit gnrale ) pour pouvoir rpondre des questions concernant la cosmologie ou le mouvement de photons . on va dcrire le milieu comme un fluide relativiste de quadri - courant @xmath63 ( o @xmath64 est la densit de masse ) , et muni dun quadri - vecteur @xmath65 jouant le rle du moment dipolaire . le vecteur de polarisation est alors @xmath66 . on dfinit un principe daction pour cette matire dipolaire , que lon rajoute laction deinstein - hilbert , et la somme des actions de tous les champs de matire habituels ( baryons , photons , etc ) . on inclue dans laction une fonction potentielle dpendant de la polarisation et cense dcrire une force interne au milieu dipolaire . par variation de laction on obtient lquation du mouvement du fluide dipolaire , ainsi que lquation dvolution de son moment dipolaire . on trouve que le mouvement du fluide est affect par la force interne , et diffre du mouvement godsique dun fluide ordinaire . ce modle ( propos dans @xcite ) reproduit bien la phnomnologie de mond au niveau des galaxies . il a t construit pour ! mais il a t aussi dmontr quil donne satisfaction en cosmologie o lon considre une perturbation dun univers homogne et isotrope . en effet cette matire noire dipolaire se conduit comme un fluide parfait sans pression au premier ordre de perturbation cosmologique et est donc indistinguable du modle cdm . en particulier le modle est en accord avec les fluctuations du fond diffus cosmologique ( cmb ) . en ce sens il permet de rconcilier laspect particulaire de la matire noire telle quelle est dtecte en cosmologie avec son aspect `` modification des lois '' lchelle des galaxies . de plus le modle contient lnergie noire sous forme dune constante cosmologique @xmath0 . il offre une sorte dunification entre lnergie noire et la matire noire _ la _ mond . en consquence de cette unification on trouve que lordre de grandeur naturel de @xmath0 doit tre compatible avec celui de lacclration @xmath31 , cest - - dire que @xmath67 , ce qui est en trs bon accord avec les observations . le modle de matire noire dipolaire contient donc la physique souhaite . son dfaut actuel est de ne pas tre connect de la physique microscopique fondamentale ( _ via _ une thorie quantique des champs ) . il est donc moins fondamental que cdm qui serait motiv par exemple par la super - symtrie . ce modle est une description effective , valable dans un rgime de champs gravitationnels faibles , comme la lisire dune galaxie ou dans un univers presque homogne et isotrope . lextrapolation du modle au champ gravitationnel rgnant dans le systme solaire nest pas entirement rsolue . dun autre ct le problme de comment tester ( et ventuellement falsifier ) ce modle en cosmologie reste ouvert . m. milgrom , astrophys . j. * 270 * , 365 ( 1983 ) . bekenstein , phys . rev . d * 70 * , 083509 ( 2004 ) . sanders , mon . not . 363 * , 459 ( 2005 ) . l. blanchet , class . * 24 * , 3529 ( 2007 ) . l. blanchet and a. le tiec , phys . d * 78 * , 024031 ( 2008 ) ; and submitted , arxiv:0901.3114 ( 2009 ) . un dilectrique est un matriau isolant , qui ne laisse pas passer les courants , car tous les lectrons sont rattachs des sites atomiques . nanmoins , les atomes du dilectrique ragissent la prsence dun champ lectrique extrieur : le noyau de latome charg positivement se dplace en direction du champ lectrique , tandis que le barycentre des charges ngatives cest - - dire le nuage lectronique se dplace dans la direction oppose . on peut modliser la rponse de latome au champ lectrique par un diple lectrique @xmath68 qui est une charge @xmath69 spare dune charge @xmath70 par le vecteur @xmath71 , et align avec le champ lectrique . la densit des diples nous donne la polarisation @xmath72 . le champ cre par les diples se rajoute au champ extrieur ( engendr par des charges extrieures @xmath73 ) et a pour source la densit de charge de polarisation qui est donne par la divergence de la polarisation : @xmath74 . ainsi lquation de gauss ( qui scrit normalement @xmath75 ) devient en prsence du dilectrique @xmath76 en utilisant les conventions habituelles , avec @xmath77 . on introduit un coefficient de susceptibilit lectrique @xmath78 qui intervient dans la relation de proportionalit entre la polarisation et le champ lectrique : @xmath79 , ainsi : @xmath80 . la susceptibilit est positive , @xmath81 , ce qui implique que le champ dans un dilectrique est plus faible que dans le vide . cest leffet d_crantage _ de la charge par les charges de polarisation . ainsi garnir lespace intrieur aux plaques dun condensateur avec un matriau dilectrique diminue lintensit du champ lectrique , et donc augmente la capacit du condensateur pour une tension donne .
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pour lastrophysicien qui aborde le puzzle de la matire noire , celle - ci apparat sous deux aspects diffrents : dune part en cosmologie , cest - - dire trs grandes chelles , o elle semble tre forme dun bain de particules , et dautre part lchelle des galaxies , o elle est dcrite par un ensemble de phnomnes trs particuliers , qui paraissent incompatibles avec sa description en termes de particules , et qui font dire certains que lon est en prsence dune modification de la loi de la gravitation .
rconcilier ces deux aspects distincts de la matire noire dans un mme formalisme thorique reprsente un dfi important qui pourrait peut - tre conduire une physique nouvelle en action aux chelles astronomiques .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
quantum correlations , which lie in the foundations of quantum theory , have been of renewed interest during the last two decades as the field of quantum information science emerged and matured . quantum entanglement , as a kind of quantum correlations , has been playing central roles in quantum information and computation @xcite . the negativity is one of the best known and most popular tools of quantifying bipartite quantum entanglement . it can be computed easily for arbitrary states of a composite system . therefore , it has received various studies @xcite . however , quantum entanglement does not account for all of the non - classical properties of quantum phenomenons . therefore , numerous quantifiers of quantum correlations have been further proposed to reveal the non - classical correlations that can not be fully captured by quantum entanglement @xcite . among the various measures , it has been shown that quantum discord provides a larger region of quantum states with non - classical correlations , and a non - zero discord quantum state without entanglement may be responsible for the efficiency of a quantum computer . therefore , much attention has been paid to study the quantum discord in various quantum systems @xcite . however , due to the complicated optimization , it is usually complicated to calculate quantum discord analytically . even for qubit - qubit quantum systems , the analytical formula of quantum discord can only be obtained for a few cases , and a general method is still lacking . as a variant of quantum discord , geometric discord ( gd ) was introduced by dakic and coworkers @xcite . due to its simplicity in calculation , it has attracted a lot of research interests @xcite . it is well known that quantum systems inevitably interact with the environment when quantum information is processed in the real - world , which brings about the loss of initially presented quantum properties @xcite . this phenomenon can be characterized by decoherence , which can be illustrated by the example of faithfully transmitting an unknown quantum state through a noisy quantum channel . during the transmission , the carrier of the information interacts with the channel and gets correlated with other degrees of freedom . this gives rise to the phenomenon of so - called decoherence on the subspace of the information carrier . recent studies on open quantum systems highlight the existence of two different classes of dynamical behaviours known as markovian and non - markovian regimes @xcite . in the quantum domain , under certain assumptions , markovian dynamics lead to a master equation in the lindblad form . however , in certain scenarios such approximations are not justified and one needs to go beyond perturbation theory . it is clear that , due to the general complexity of the problem to be studied , exact solutions exist only for simple open quantum systems models such as the well - known jaynes - cummings model , quantum brownian motion model , and certain pure dephasing models @xcite . for the deeper understanding of quantum phenomenons , it is desirable to investigate the behaviours of the quantum properties under the action of decoherence . the previous work showed that quantum discord and entanglement behave differently under the effect of the environment @xcite . in particular , for certain quantum states , the phenomenon of entanglement sudden death does not occur for quantum discord . there exists some studies on the quantum correlations of qubit - qubit and qubit - qutrit systems under the effect of local quantum noises in the markovian environment @xcite . the importance of quantum states with higher dimensions is gradually recognized in recent years @xcite . therefore , a deep understanding of the quantum correlations dynamics of the high - dimensional quantum systems beyond qubits is very practical . song et al . @xcite have studied the distillability in qutrit - qutrit systems under local dephasing channels , and found that the distillability will be decrease after a finite time . m. ali studied the distillability sudden death in qutrit - qutrit systems under global and multilocal dephasing @xcite , and amplitude damping channels @xcite . he has found that the quantum entanglement will decrease after a finite time under dicoherence . guo et al . @xcite studied the dynamics of quantum correlations of qubit - qutrit systems under various decoherent channels . they have shown that the decoherent channels bring with different influences for the dynamics of quantum correlations measured by negativity , quantum discord and gd . the influences depend on the initial state parameters and the properties of the decoherent channels . in this work , we will study the quantum correlations in qutrit - qutrit systems under identical and non - identical local noise channels . we consider four kinds of local noise channels on each qutrit , including dephasing , trit - flip , trit - phase - flip , and depolarising channels . the remainder of this paper is organized as follows : in sec . [ sec:2 ] , we will introduce the negativity and the gd as our measures of quantum correlations . in sec . [ sec:3 ] , the evolution of the quantum correlations for a qutrit - qutrit system under the identical and non - identical local noise channels are discussed . finally , we conclude in sec . [ sec:4 ] . in this section , we will describe the gd and the negativity in a brief manner . the negativity is a useful entanglement measure for arbitrary bipartite state quantum systems @xcite , especially for free entangled states . for a given density matrix @xmath0 , its negativity will be @xmath1 where @xmath2 is the partial transpose of @xmath0 with respect to subsystem @xmath3 and @xmath4 represents the trace norm . if @xmath5 , then the bipartite state is free entangled . [ [ gd ] ] gd ~~ geometric measure of quantum discord in a bipartite state can be described as the distance of the state from the closest zero - discord state @xcite @xmath6 where @xmath7 is the hilbert - schmidt norm and @xmath8 is the set of all zero discord states . for a general bipartite state of @xmath9 dimensions , the density matrix can be written as @xmath10,\ ] ] where @xmath11 ( @xmath12 ) and @xmath13 ( @xmath14 ) are the generators of @xmath15 and @xmath16 groups ( special unitary groups of dimensions @xmath17 and @xmath18 , respectively ) , satisfying the relation @xmath19 . the lower bound @xcite of the gd has been derived as @xmath20 where @xmath21 with bloch vector @xmath22 with elements @xmath23 , matrix @xmath24 is the correlation matrix with elements @xmath25 and @xmath26 are the eigenvalues of the matrix @xmath27 arranged in non - increasing order . we will use the lower bound as the estimation of the gd in the following . let us consider a qutrit - qutrit system with each particle interacts independently with the local noise channels . in this case , the evolution of a quantum state can be described by the lindblad equation , which can be written in terms of the kraus operators as @xmath28 where @xmath29 and @xmath30 are the kraus operators characterizing the local noise channels on @xmath3 and @xmath31 , respectively . they satisfy @xmath32 and @xmath33 @xcite . to study the dynamics of quantum correlations for a qutrit - qutrit system , we choose the bell state @xmath34 as the initial quantum state . in the following , we will study the dynamics of the quantum correlations of @xmath35 under the effect of four different noise channels . physically , dephasing corresponds to any process of losing coherence without the exchange of energy . under the action of dephasing noises , the off - diagonal elements of the density matrix decay exponentially against time . for two local dephasing noises on qutrits @xmath3 and @xmath31 , the kraus operators are given by @xmath36 where @xmath37 @xmath38 or @xmath31 , @xmath39 and @xmath40 with @xmath41 and @xmath42 denoting the decay rates of qutrits @xmath3 and @xmath31 , respectively . for trit - flip channels of qutrits @xmath3 and @xmath31 , the kraus operators are given by @xmath43 where @xmath44 the kraus operators are @xmath45 where @xmath46 the kraus operators are @xmath47 where @xmath48 based on the numerical simulations , we give our results in figs . [ fig1 ] , [ fig2 ] , [ fig3 ] , and [ fig4 ] for each two identical dephasing , trit - flip , trit - phase - filp , and depolarizing channels , respectively . one can observe that under the effect of two identical noise channels , as @xmath49 or @xmath50 increases , the negativity and the gd decrease rapidly ( see @xmath51 and @xmath52 in the figures ) , which indicates that quantum correlation is strongly affected by large decay rates @xcite . by observing figs . [ fig1 ] and [ fig2 ] , one can see that the gd is more robust than the negativity ageist two identical dephasing and trit - flip noise channels . however , as shown in figs . [ fig3 ] and [ fig4 ] , for two identical tirt - phase - flip and depolarizing channels , the negativity becomes more robust than the gd . to study the effect of the decay rates of the channels , we draw the negativity and the gd versus the decay rates @xmath49 or @xmath50 in three - dimension figures with a fixed time @xmath53 ( see @xmath51 and @xmath52 in the figures ) . it can been seen that in the case where @xmath49 and @xmath50 are large , the negativity and the gd are influenced strongly during the decoherence . in this section , we discuss the negativity and the gd under non - identical local noise channels , namely , the noise channels are of different types and decay rates . the kraus operators have been given in sec . [ idt ] . the results are drawn in figs . [ fig5 ] , [ fig6 ] , [ fig7 ] , [ fig8 ] , [ fig9 ] , and [ fig10 ] for dephasing - trit flip channels , dephasing - trit phase flip channels , dephasing - depolarising channels , trit flip - trit phase flip channels , trit flip - depolarising channels , and trit phase flip - depolarising channels respectively . it can be seen that under the effect of two different local noise channels , the negativity and the gd behave similar to the case where the channels are identical . one can observe that in all cases of non - identical local noise channels the gd is more robust than the negativity . same in the identical channels , as the time elapsed the state losses its quantum entanglement . to study just the effect of rate of the channels we draw the negativity and the gd versus the rates of two channels in three dimensions at a fixed time @xmath53 ( see @xmath54 and @xmath55 in the figures ) . as the rates of the channels are large , the negativity and the gd decease rapidly . to summarize , in this work , we have studied the quantum correlations of a qutrit - qutrit quantum system under identical and non - identical local noise channels by using the negativity and the gd . we have considered four kinds of local noise channels on each qutrit , including dephasing , trit - flip , trit - phase - flip , and depolarising channels . meanwhile , we have further investigated the effect of various combinations local noise channels on the quantum correlation . the results show that the quantum correlations decrease monotonically with time under decoherence . meanwhile , when the time is fixed , negativity and geometric discord decrease rapidly when the decay rates are large . we expect our work may find further theoretical and experimental applications . bostrom , k. , felbinger , t. : phys . lett . * 89 * , 187902 ( 2002 ) . yu , g. : int . . phys . * 51 * , 2954 ( 2012 ) . deng , f. g. , li , x. h. , at all . a * 72 * , 359 ( 2006 ) . yang , y. g. , wen , q. y. : sci . china ser . g 5 * 50 * , 558 ( 2007 ) . guhne , a. , hyllus , p. : int . j. theor 42 * , 1001 ( 2003 ) . nakano , t. , piani , m. , adesso , g. : phys . a * 72 * , 012117 ( 2013 ) eltschka , c. , siewert , j. : phys . lett . * 111 * , 100503 ( 2013 ) . datta , a. : phys . a * 81 * , 052312 ( 2010 ) . ferrie , c. , morris , r. , emerson , j. : phys . a * 82 * , 044103 ( 2010 ) . dajka , j. , mierzejewski , m. , uczka , j. , blattmann , r. , hanggi , p. : j. phys . a : math . theor . * 45 * , 485306 ( 2012 ) . nielson , m.a . , chuang , i.l . : cambridge university press , cambridge ( 2000 ) . rudolph , o. : phys . a * 67 * , 032312 ( 2003 ) . moradi , s. : quant . inf . comp . * 11 * , 957 ( 2011 ) . , song , h.s . a * 72 * , 022333 ( 2005 ) . vidal , g. , werner , r.f . a * 65 * , 032314 - 032317 ( 2002 ) . horodecki , r. , horodecki , p. , horodecki , m. , horodecki , k. : rev . phys . * 81 * , 865 ( 2009 ) ollivier , h , . zurek , w.h . lett . * 88 * , 017901 ( 2001 ) . brodutch , a , . terno , d.r . a * 83 * , 010301(r ) ( 2011 ) . datta , a. , shaji , a. , caves , c.m . . lett . * 100 * , 050502 ( 2008 ) . lanyon , b.p . , barbieri , m. , almeida , m.p . , white , a.g . : * 101 * , 200501 ( 2008 ) . okrasaa , m.,walczak , z. : euro . . lett . * 96 * , 60003 ( 2011 ) . zhang . y. , zhao . h. : quantum inf . process . ( published online ) . he , j. , tao liu . , w. : int . j. theor . * 52 * , 3381 ( 2013 ) . liu , x. , ma , j. , xi , z.,wang , x. : phys . a * 83 * , 012327 ( 2011 ) . chen , q. , zhang , c. , yu , s. , yi , x.x . , oh , c.h . : phys . rev . a * 84 * , 042313 ( 2011 ) . ali , m. : j. phys . a : math . theor . * 43 * , 495303 ( 2010 ) . dakic , b. , vedral , v. , brukner , c. : phys . rev . lett . * 105 * , 190502 ( 2010 ) . rana , s. , parashar , p. : phys . a * 85 * , 024102 ( 2012 ) . vidal , g. , werner , r. f. : phys . a * 65 * , 032314 ( 2002 ) . doustimotlagh , n. , wang , s. , you , c. , long , g. l. : epl , * 106 * , 60003 ( 2014 ) . wang , j. , jing , j. : phys . a * 82 * , 032324 ( 2010 ) . wang , j. , jing , j. : ann . phys . * 327 * , 283 ( 2012 ) . hu , m.l . , fan , h. : ann . phys . * 327 * , 851 ( 2012 ) . miao , c. , yang , m. , cao , z , l. : int . j. theor . phys . * 52 * , 1780 ( 2013 ) . ramzan , m. , khan , m.k . inf . process . * 11 * , 443 ( 2012 ) . ren , b.c . , wei , h.r . , deng , f.g . : quantum inf . process . * 13 * , 1175 ( 2014 ) . guo , j.l . , li . h. , long . g. l. : quantum inf . process . * 12 * , 3421 ( 2013 ) . chou , c.h.,yu , t. , hu , b.l . e * 77 * , 011112 ( 2008 ) . an , j.h . , zhang , w.m . a * 76 * , 042127 ( 2007 ) . an , j.h . , feng , m. , zhang , w.m . : quantum inf . * 9 * , 0317 ( 2009 ) . c. , bylicka.b . , chruscinski . d. , maniscalco . s. : arxiv:1402.4975v3 . li , j.g . , zou , j. , shao , b. : phys . a * 82 * , 042318 ( 2010 ) . maziero , j. , cleri , l.c . , serra , r.m . , vedral , v. : phys . rev . a * 80 * , 044102 ( 2009 ) . fanchini , f.f.,werlang , t. , brasil , c.a . , arruda , l.g.e . , caldeira , a.o . a * 81 * , 052107 ( 2010 ) . d. : arxiv:1403.2446v1 . wang , s.,yao , l. , long , g.l.:phys . a * 87 * , 062305 ( 2013 ) . song , w. , chen , l. , zhu , s , l. : phys . a * 80 * , 012331 ( 2009 ) . mazhar , a. : phys . rev . a * 81 * , 042303 ( 2010 ) . mazhar , a. : j. phys . b : at . mol . phys * 43 * , 045504 ( 2010 ) . hassan , a.s.m . , lari , b. , joag , p.s . a * 85 * , 024302 ( 2012 ) . xu , j. : j. phys . * 45 * , 405304 ( 2012 ) . hassan , a.s.m . , joag , p.s . : j. phys . a : math . theor . * 45 * , 345301 ( 2012 ) .
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due to decoherence , realistic quantum systems inevitably interact with the environment when quantum information is processed , which causes the loss of quantum properties . as a fundamental issue of quantum properties ,
quantum correlations have attracted a lot of interests in recent years .
because of the importance of high dimensional systems in quantum information , in this work , we study the quantum correlations affected by the markovian environment by considering the quantum correlations of qutrit - qutrit quantum systems measured by the negativity and the geometric discord .
the local noise channels covered in this work includes dephasing , trit - flip , trit - phase - flip , and depolarising channels .
we have also investigated the cases where the local decoherence channels of two sides are identical and non - identical .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
scene flow is a three - dimensional motion field of the surface in world space , or in other words , it shows the three - dimensional displacement vector of each surface point between two frames . as most computer vision issues are , scene flow estimation is essentially an ill - posed energy minimization problem with three unknowns . prior knowledge in multiple aspects is required to make the energy function solvable with just a few pairs of images . hence , it s essential to fully make use of information from the data source and to weigh different prior knowledge for a better performance . the paper attempts to reveal clues by providing a comprehensive literature survey in this field . scene flow is first introduced by vedula in 1999 @xcite and has made constant progress over the years . diverse data sources has emerged thus scene flow estimation do nt need to set up the complicated array of cameras . the conventional framework derived from optical flow field @xcite has extended to this three - dimensional motion field estimation task , while diverse ideas and optimization manners has improved the performance noticeably . the widely concerned learning based method has been utilized for scene flow estimation @xcite , which brings fresh blood to this integrated field . moreover , a few methods have achieved real - time estimation with gpu implementation at the qvga(@xmath0 ) resolution @xcite , which insure a promising efficiency . the emergence of these methods stands for the fact that scene flow estimation will be widely utilized and applied to practice soon in the near future . the paper is organized as follows . section [ sec : background ] illustrates the relevant issues , challenges and applications of scene flow as a background . section [ sec : taxonomy ] provides classification of scene flow in terms of three major components . emerging datasets that are publicly available and the diverse evaluation protocols are presented and analyzed in section [ sec : evaluation ] . section [ sec : discussion ] arises few questions to briefly discuss the content mentioned above , and the future vision is provided . finally , a conclusion is presented in section [ sec : conclusion ] . we provide relevant issues , major challenges and applications as the background information for better understanding this field . scene flow estimation is an integrated task , which is relevant to multiple issues . firstly , optical flow is the projection of scene flow onto an image plane , which is the basis of scene flow and has made steady progress over the years . the basic framework and innovations of scene flow estimation mainly derives from optical flow estimation field . secondly , in a binocular setting , scene flow can be simply acquired by coupling stereo and optical flow , which makes the stereo matching an essential part for scene flow estimation . most scene flow estimation methods with promising performance are initialized with a robust optical flow method or a stereo matching method . and the innovation in scene flow mostly derived from these two fields . hence , we provides the changes and trend in the relevant issues as heuristic information . optical flow is a two - dimensional motion field . the global variational horn - schunck(h - s ) method and the local total - least - square(tls ) lucas - kanade(l - k ) method have led the optical flow field and scene flow field over the years @xcite . early works was studied and categorized by barron and otte with quantitative evaluation models @xcite . afterwards , brox implemented the coarse - to - fine strategy to deal with large displacement @xcite , while sun studied the statistics of optical flow methods to find the best way for modeling @xcite . baker proposed a thorough taxonomy of current optical flow methods and introduced the middlebury dataset for evaluation @xcite , and comparisons between error evaluation methodologies , statistics and datasets are presented as well . currently , optical flow estimation has reached to a promising status . a segmentation - based method with the approximate nearest neighbor field to handle large displacement ranks the top of middlebury dataset in terms of both endpoint error(epe ) and average angular error(aae ) currently @xcite , where epe varies from 0.07@xmath1 to 0.41@xmath1 in different data and aae varies from 0.99@xmath2 to 2.39@xmath2 . a similar method reached promising results as well @xcite . moreover , there are a variety of methods which achieve top - tier performance and solve different problems respectively . rushwan utilized a tensor voting method to preserve discontinuity @xcite . xu introduced a novel extended coarse - to - fine optimization framework for large displacement @xcite , while stoll combines the feature matching method with variational estimation to keep small displacement area from being compromised @xcite . he also introduced a multi - frame method utilizing trilateral filter @xcite . to handle non - rigid optical flow , li proposed a laplacian mesh energy formula which combines both laplacian deformation and mesh deformation @xcite . stereo matching is essential to scene flow estimation under binocular setting . a stereo algorithm generally consists of four parts:(1 ) matching cost computation , ( 2 ) cost aggregation , ( 3 ) estimation and optimization and ( 4 ) refinement . it is categorized into local methods and global methods depending on how the cost aggregation and computation are performed . local methods suffer from the textureless region , while global methods is computationally expensive . a semi - global - matching(sgm ) method combines local smoothness and global pixel - wise estimation and leads to a dense matching result at low runtime @xcite , which is commonly utilized as the modification . a comprehensive review is presented by scharstein in 2001 @xcite . the upper rank algorithms of middlebury stereo dataset and kitti stereo dataset @xcite are mainly occupied by unpublished papers , indicating the rapid development in this field . learning methods are utilized with promising efficiency and accuracy @xcite . besides , zhang proposed a mesh - based approach considering the high speed of rendering and ranks the top among the published papers @xcite , while segmentation - based methods are proven to tackle the textureless problem @xcite . the complex scene and the limited image capture approach post challenges in diverse ways , which are discussed as follows . what happens when an unstoppable force meets an immovable object ? the relationship between accuracy and efficiency is just like the omnipotence paradox . to achieve better accuracy , sufficient and complicated prior knowledge is obliged , while in terms of efficiency , the data need to be listed down to a reasonable scale and the calculation scheme should be as simple as possible . we can not only consider the enhancement of efficiency and accuracy , but also value the trade - off between these two . the trade - off is discussed in section [ sec : discussion ] , and the performance is illustrated in figure [ fig : accuracy - efficiency ] . occlusion is common in a complex scene with multiple moving objects . it occurs between views and frames as figure [ fig : occlusion ] illustrates . it violates the data consistency assumption and may lead to mismatching on account of missing information of the occluded object . besides , occlusion may perturb the consistency between frames and affect the multi - frames tracking method . [ htbp ] the occlusion between views can be handled well under the multi - view stereopsis with abundant prior knowledge , while temporal constraint may provide robust temporal coherence and prediction to alleviate occlusion between frames . large displacement occurs frequently when an object is moving at a high speed or under a limited frame - rate . moreover , articulated motion may lead to large displacement as well . this kind of problem is hard to tackle on account that the scene flow algorithms normally assume the constancy and smoothness within a small region , large displacement may make the solution to energy function trapped into a local minimum which leads to enormous errors propagated by iteration procedure . brox implemented the coarse - to - fine method along with a gradient constancy assumption to alleviate the impact caused by large displacement in the optical flow field @xcite . currently , several matching algorithms have been introduced to handle this issue specifically and achieved promising results @xcite . brightness constancy does nt obey the illumination - varying environment . however , this issue is common in an outdoor scene , e.g. , drifting clouds that block the sunlight , sudden reflection from a window , and lens flares . furthermore , it will be a disaster at night when lights start to flash . in the optical flow field , additional assumptions such as gradient constancy and some more complicated constraints have been added to make it more robust to the illumination changes @xcite . schuchert specifically studied range flow estimation under varying illumination @xcite . in his paper , pre - filtering and changes of brightness model improve the accuracy . gotardo introduced an albedo consistency assumption as a revision @xcite . a relighting procedure was proposed as a key element to handle the multiplexed situation in his paper as well . the lack of texture may make the scene flow estimation still an ill - posed problem , which is a challenge for discovering consistency . it is also a challenge to stereo matching , which may lead to enormous errors in the binocular - based scene flow estimation . the textureless region is still a major distribution of the estimation error . to overcome this problem , different scene representations have been utilized . for example , popham introduced a patch - based method @xcite . the motion of each patch does nt only rely on the texture information , but utilizes the motion from neighbor patches . this makes it more robust for a textureless region . as a solution to the occlusion issue , segmentation - based method is valid because it assumes uniform motion among the small regions to deal with the ambiguousness @xcite . scene flow estimation is a comprehensive problem . motion information reveals the temporal coherence between two moments . in a long sequence , scene flow can be utilized to get the initial value for the next frame and serve as a constraint in its relevant issue fields . scene flow can not only profit from its relevant issues , but also facilitate them mutually . gotardo captured three - dimensional scene flow to provide delicate geometric details @xcite , while liu utilized scene flow as a soft constraint for stereo matching and a prediction for next frame disparity estimation @xcite . ghuffar combined local estimation and global regularization in a tls framework and utilized scene flow for segmentation and trajectory generation @xcite . beyond that , scene flow can be a valuable input or mobile robotics and autonomous driving field , which consist of multiple task such as obstacle avoidance and scene understanding . frank first fused optical flow and stereo by means of kalman filter for obstacle avoidance @xcite . alcantarilla combined scene flow estimation with the visual slam to enhance the robustness and accuracy @xcite . herbst got object segmentation with the rgb - d scene flow estimation result @xcite , aiming to achieve autonomous exploration of indoor scenes . menze utilized scene flow to reason objects by regarding the scene as a set of rigid objects @xcite . autonomous driving could make use of both the geometric information that represents distance and the scene flow information that represents motion for multiple tasks . in addition , scene flow can be utilized to serve as a feature as the histogram of optical flow(hof ) @xcite or the motion boundary histogram(mbh ) @xcite feature descriptors for object detection and recognition , e.g. , facial expression , gesture , and body motion recognition . it may enrich the information in the descriptor with additional depth dimension and can be applied for motion like rotation or dolly moves that optical flow ca nt handle . for instance , in 2009 , furukawa recorded the motion model of the facial expression using scene flow estimation@xcite . scene flow estimation is viewed as an ill - posed problem , which includes three main steps : data acquisition , energy function modeling , energy minimization and optimization . the general taxonomy is depicted in figure [ fig : overall ] . the energy function consists of data term and regularization as equation [ eq : energy ] illustrates . @xmath3 data terms derive from different data sources assuming brightness constancy(bc ) or gradient constancy(gc ) as local constraint . while there are three unknown parameters , regularization terms need to be added to regularize the ill - posed problem and provide spatial coherence . theoretically , the more regularization terms are , the more robust and accurate the estimation is . however , miscellaneous regularization terms may lead to redundancy , intractability and over - fitting , and that s why the design and solution of regularization term are key to a method . hence , in this section , existing methods are categorized in terms of three fundamental properties that distinguish the major algorithms : _ scene representation _ presented the diverse representations for both scene and scene flow . _ data source _ describes the major data acquisition manner and the corresponding data term choices . _ calculation scheme _ mainly discuss the idea for estimation and optimization manner , including diverse choices of regularization terms and implement . over the years , diverse ways to represent the scene have emerged with different emphasis , which can be broadly categorized into _ depth / disparity _ , _ point cloud _ , _ mesh _ and _ patch_. the convenient way to represent the scene is to couple color image and depth map as color - d information , where d stands for depth information with rgb - d data or disparity information under a binocular setting . scene flow under this sort of representation is known as 2.5d scene flow or 2d parameterization of scene flow . it consists of optical flow component which is measured in pixels , and disparity or depth change component which is measured in pixels or @xmath4 . the binocular - based scene flow @xcite can be presented as @xmath5 , where @xmath6 is the 2d optical flow , and the @xmath7 stands for disparity change . likewise , in terms of rgb - d scene flow @xcite , the motion field consists can be presented @xmath8 , where @xmath9 stands for the depth change . particularly , disparity value can be converted into depth value as equation [ eq : disparitytodepth ] illustrates . @xmath10 where @xmath11 is the focal length of the camera , and @xmath12 is the camera baseline value . to truly present the three - dimensional scene , the image pixel need to be projected into the scene space . the projection is illustrated in figure [ fig:3d - mapping ] and presented in equation [ eq:3d - mapping ] . @xmath13 where @xmath14 is the image pixel @xmath15 , @xmath16 is the projection @xmath17 from a pixel value @xmath14 and a depth value @xmath18 to a 3d point @xmath19 . @xmath20 and @xmath21 stands for the camera focal length , and @xmath22 and @xmath23 are the principle points . formulation [ eq:3d - mapping ] can also be presented as : @xmath24 where matrix @xmath25 is known as the camera projection matrix . hence , scene flow under point cloud representation @xcite can be presented as @xmath26 , which truly reveal the three - dimensional displacement . meshes represent a surface as a set of planar polygons , e.g. , triangle , which connected to each other as is shown in figure [ fig : mesh ] . this representation is a efficient way for rendering , and it occupies less memory . mesh is essentially sort of point cloud representation as the vertex can be viewed as a point in the three - dimensional point cloud . scene flow estimation methods with a mesh representation @xcite are only under a multi - view setting and the geometry estimation is given simultaneously . the motion of vertice is solved with a point cloud methods , while the rest part are solved by interpolating along the meshes . in terms of patch representation , the surface is represented by collections of small planar or sphere patches . each patch is six - dimensional in terms of three - dimensional patch center position and three - dimensional patch direction . patch representation is similar to mesh representation with different emphasis , where mesh representation focuses on the precision and deformable property of each vertex , and patch representation values the local consistency in terms of rigidity and motion within a small neighborhood region . early paper viewed patch as a surface element(surfel ) under a multi - view setting@xcite . a few binocular - based scene flow methods utilized patches to fit the surface of the scene @xcite on account that this kind of patch - based methods are common in stereo matching field . in addition , hornacek uniquely exploited a pair of rgb - d data to seek patch correspondences in the 3d world space and leads to dense body motion field including both translation and rotation @xcite . over the years , scene flow has been estimated under three main kinds of data source : a calibrated multi - camera system , the binocular stereo camera or the rgb - d camera which consists of both rgb color information and depth information . moreover , the emerging light field has been applied into the scene flow estimation with promising performance @xcite . data terms under different data sources differs from each other . hence , in this section , scene flow estimation methods are categorized into these four kinds : _ multi - view stereopsis _ , _ binocular setting _ , _ rgb - d data _ and _ light field data_. most of the algorithms in the early 2000s assume a multi - view system , with multiple cameras set in a complex calibrated scene . multi - view scene flow estimation is usually along with 3d geometry reconstruction simultaneously . ample data sources and diverse prior knowledge ensure the robustness of the estimation , and occlusion issue can be handled well . however , it is commonly at a high computational cost with an intricate full - view scene to deal with . vedula proposed two choices for regularization and distinguished three scenarios in 1999 @xcite , which guides the multi - view scene flow estimation till now . in his paper , a multi - view scene flow can be estimated from one optical flow and the known surface geometry . the equation is formulated in equation [ eq : vedula ] . @xmath27 where @xmath19 is the three - dimensional scene point , @xmath15 is the two - dimensional image pixel , @xmath28 is the scene flow , @xmath29 is the optical flow , and @xmath30 is the inverse jacobian which can be estimated from the surface gradient @xmath31 . afterwards , zhang proposed two systems for estimation @xcite , where ims assumed each small patch undergoes 3d affine motion , and egs used segmentation to keep the boundary . these papers modeled energy function with multiple constraint , which provided a basic estimation process . similarly , pons presented a common variational framework with local similarity criteria constraint @xcite . henceforth , different scene representations were introduced to describe the surface @xcite . diverse multi - frame tracking methods mentioned for sparse estimation are utilized as well to build the temporal coherence @xcite . moreover , letouzey added an rgb - d camera into the multi - view system with a mesh representation @xcite , aiming to enrich the geometry information with the depth data constraint . binocular setting is regarded as a basic and simplified version of multi - view system , while the difference between the two is that binocular scene flow estimation is usually along with disparity estimation between two views but not the full - view 3d geometry knowledge . the relevance between views and frames is illustrated in figure [ fig : binocular - a ] , and figure [ fig : binocular - b ] depicts the basic data terms in a binocular setting which consists of stereo consistency terms in time @xmath32 and @xmath33 along with optical flow consistency terms in both views . the specific formulation is presented in equation [ eq : binoculardataterms ] . @xmath34 where [ eq : binoculardataterms ] @xmath35 @xmath36 and @xmath37 are the optical flow consistency terms that assume the brightness of the same pixel stay constant between frames . similarly , @xmath38 and @xmath39 are the stereo consistency terms that assume brightness constancy between views . moreover , @xmath40 is the cross term to constrain the constancy between both frames and views . most binocular - based methods fused stereo and optical flow estimation into a joint framework @xcite . on the contrary , others decoupled motion from disparity estimation to estimate scene flow with stereo matching method replaceable at will @xcite , and basha utilized a point cloud scene representation as a three - dimensional parametrization version of scene flow @xcite . moreover , local rigidity prior was presented along with segmentation prior and achieved promising results @xcite . specifically , valgaerts introduced a variational framework for scene flow estimation under an uncalibrated stereo setup by embedding an epipolar constraint @xcite , which makes it possible for scene flow estimation under two arbitrary cameras . in 2016 , richardt has made it a reality to compute dense scene flow from two handheld cameras with varying camera settings @xcite . scene flow was estimated under a variational framework with a daisy descriptor @xcite for wide - baseline matching . table [ tab : binoculardataterm ] enumerates some typical methods under a binocular setting with diverse choices of data terms . most methods chose optical flow consistency terms in both views and stereo consistency terms in both time @xmath32 and time @xmath33 @xcite , and few methods only take parts of terms mentioned above @xcite . cross term was utilized in @xcite . moreover , huguet @xcite and hung @xcite utilized additional gradient constancy assumption besides intensity to enhance robustness against illumination changes , which turns the image intensity value @xmath41 in energy function into image gradient @xmath42 . additionally , extra rgb constancy terms @xmath43 and @xmath44 are taken in hung s paper as well , which extends gray value intensity into three - channel information . however , it is proposed that image gradient is sensitive to noise and is view dependent @xcite . hence , the necessity of additional assumptions like gradient constancy remains further research to balance the pros and cons . .typical methods under the binocular setting [ cols="<,<,<,<",options="header " , ] we calculated the statistics of the ground truth optical flow in terms of average magnitude and the max magnitude to show the challenging level and large displacement situation of each dataset , as figure [ fig : datasetstatistic ] presents . where the bigger the average magnitude and standard deviation are , the more challenging the dataset is , and the bigger the max magnitude is , the more a robust large displacement handling is required . [ htbp ] we can see clearly that the up - to - date dataset is more challenging than datasets published before . along with information provided in table [ tab : dataset ] , we provide suggestion for datasets with different purpose as follows : + * comprehensiveness * taken multiple issues , e.g. , categories of ground truth data , image resolution , challenging level , naturalism , scale , popularity , into consideration , we recommended mpi sintel dataset and freiburg dataset for their comprehensive property . + * challenging * flyingthings3d subset of freiburg dataset shows the characteristic of large displacement , complex occlusion and diverse changes between frames , which is really challenging for scene flow estimation . monkaa subset shows the similar characteristic which is recommended as well . + * public popularity * middlebury , kitti(2012&2015 ) and mpi sintel dataset provide evaluation protocols and online ranking that can evaluate the performance of a method conveniently . moreover , the evaluation can be compared with top tier methods in optical flow estimation field and stereo matching field to indicate the superiority of scene flow estimation . + * multi - view and rgb - d data source * for multi - view stereopsis , basha rotating sphere and kitti2012/2015 provide multi - view extension for evaluation . to be noted , basha provides only point cloud scene flow ground truth for quantitative analysis , while the ground truth of kitti2012/2015 is sparse . in terms of rgb - d scene flow estimation , mpi sintel dataset and basha rotating sphere dataset provides depth ground truth so that disparity - to - depth conversion is no need . in addition , long range of distance may make the depth map transferred from disparity map unclear for visualization , which make image - based algorithm hard to work . the provided depth visualization map is a good option . + * large scale for learning * freiburg dataset is currently the only dataset with @xmath45 order of scale that is designed for training optical flow , disparity , and scene flow . protocols usually varies from each other based on different datasets . the previous datasets like middlebury and rotating sphere usually use rmse / nrmse and aae for evaluation , while newly - introduced datasets like kitti , mpi sintel and freiburg utilize epe for evaluation . we recommend researchers to use epe as a overall protocol , while rmse and aae can be supplementary means that reveal error distribution and angular error . + * should we evaluate the 3d error ? * as equation [ eq : errorprojection ] presents , the disparity in time @xmath33 or the disparity change do have influence on scene flow estimation . hence , we highly recommended that for 2d parametrization scene flow estimation methods , the accuracy of disparity @xmath33 or the disparity change should be evaluated and provided . considering the fact that equation [ eq : nrmse - wedel ] and [ eq : aae - wedel ] is rarely utilized , and rotating sphere of basha is the only dataset that provide three - dimension point cloud ground truth , we recommend researchers to provide epe for optical flow , disparity in time @xmath32 and time @xmath33 as a common protocol for scene flow evaluation . with 17 years of development , scene flow estimation has reached a promising status , while there are still many issues remains to be solved . in this section , we discuss the limitation of existing datasets , and present a brief vision on modification of algorithms . the survey of existing evaluation methodologies reveals problems and limitations as elaborated below : + * size * on account that real - time scene flow has been achieved with gpu implementation , high resolution data can be introduced for more challenging work . + * ground truth * point cloud is the real three - dimensional parametrization representation for scene flow , which reveals the difference between scene flow and optical flow significantly . hence , the ground truth for scene flow under the point cloud representation is necessary . meanwhile , occlusion , textureless region and discontinuity region ground truth are essential for evaluation due to the fact that errors mainly exist near these regions . + * data source * current datasets mainly focus on scene flow under the binocular setting , while multi - view extension and specific datasets for rgb - d and light field based scene flow is required . otherwise , the properties of missing data in rgb - d cameras and the abilities like refocusing of light field cameras will be neglected with current datasets . + * protocol * a protocol for evaluating performances under different datasets remains vacant . moreover , the three - dimensional protocol is nt been applied due to the limitation of point cloud ground truth . by checking the error map provided by kitti benchmark , it s clear that inaccuracy mainly exists in the boundaries of objects . since this is a common issue for all computer vision tasks , edge - preserving and reasonable filtering is the first priority . gpu implementation has shown a great efficiency improvement , and the duality - based optimization has proved to enhance the efficiency of global variational methods without accuracy sacrifice . these kind of methods may be a routine in the future for better efficiency . with the development of a robust and efficient estimation between two frames , some papers have studied motion estimation under a long sequence @xcite . the multi - fames estimation with temporal prior knowledges deserves more attention . a robust temporal constraint can benefit the methods with a better initial value or a better feature @xcite to match . the challenges like varying illumination and occlusion can be handled with the help of it . the emerging learning based methods and light field technique has brought fresh blood to scene flow estimation . learning method with cnn shows an upward tendency in the relevant issues of scene flow estimation like stereo matching and optical flow with promising accuracy and computational cost @xcite . with the help of the up - to - date large scale training dataset @xcite , learning - based method has a profound potential to achieve an accurate and fast estimation . light field camera provides more data than existing data source , which brings diverse possibilities for this field . similar to the emergence of rgb - d cameras , this new source of data may lead to a new attractive branch . on account of the fact that scene flow estimation relies highly on texture and intensity information , application will suffer in the night or an insufficient illumination circumstance . moreover , the car headlights and lighting on the building that are frequent in the autonomous driving scene may interfere motion estimation significantly . hence , the scene flow estimation with insufficient illumination is worth studying . this paper presents a comprehensive and up - to - date survey on both scene flow estimation methods and the evaluation methodologies for the first time after 17 years since scene flow was introduced . we have discussed most of the estimation methods so researchers could have a clear view of this field and get inspired for their studies of interest . the representative methods are highlighted so the differences between these methods are clear , and the similarities between top - tier methods can be seen as a tendency for modification . the widely used benchmarks have been analyzed and compared , so are multiple evaluation protocols . this paper provides sufficient information for researchers to choose the appropriate datasets and protocols for evaluating performance of their algorithms . there are still ample rooms for future research on accuracy , efficiency and multiple challenges . we wish our work could arise public interest in this field and bring it to a new stage . this work was supported by projects of national natural science foundation of china [ 61401113 ] ; and natural science foundation of heilongjiang province of china [ lc201426 ] . n. mayer , e. ilg , p. husser , p. fischer , d. cremers , a. dosovitskiy , t. brox , a large dataset to train convolutional networks for disparity , optical flow , and scene flow estimation , arxiv preprint arxiv:1512.02134 . c. rabe , t. muller , a. wedel , u. franke , dense , robust , and accurate motion field estimation from stereo image sequences in real - time , in : european conference on computer vision ( eccv ) , springer - verlag , 2010 , pp . 582595 . m. jaimez , m. souiai , j. gonzalez - jimenez , d. cremers , a primal - dual framework for real - time dense rgb - d scene flow , in : ieee international conference on robotics and automation ( icra ) , 2015 , pp . 98104 . o. u. n. jith , s. a. ramakanth , r. v. babu , optical flow estimation using approximate nearest neighbor field fusion , in : ieee international conference on acoustics , speech and signal processing ( icassp ) , 2014 , pp . 6736577 . h. a. rashwan , m. a. garca , d. puig , variational optical flow estimation based on stick tensor voting , ieee transactions on image processing a publication of the ieee signal processing society 22 ( 7 ) ( 2013 ) 25892599 . c. zhang , z. li , y. cheng , r. cai , h. chao , y. rui , meshstereo : a global stereo model with mesh alignment regularization for view interpolation , in : ieee international conference on computer vision ( iccv ) , 2015 , pp . 20572065 . f. alcantarilla , j. j. yebes , j. almaz , x00e , l. m. bergasa , on combining visual slam and dense scene flow to increase the robustness of localization and mapping in dynamic environments , in : ieee international conference on robotics and automation ( icra ) , 2012 , pp . 12901297 . a. wedel , c. rabe , t. vaudrey , t. brox , u. franke , d. cremers , efficient dense scene flow from sparse or dense stereo data , in : european conference on computer vision ( eccv ) , vol . 5302 lncs , 2008 , pp . 739751 . y. zhang , c. kambhamettu , integrated 3d scene flow and structure recovery from multiview image sequences , in : ieee conference on computer vision and pattern recognition ( cvpr ) , vol . 2 , 2000 , pp . 674681 . d. ferstl , c. reinbacher , g. riegler , r. m , x00fc , ther , h. bischof , atgv - sf : dense variational scene flow through projective warping and higher order regularization , in : international conference on 3d vision , vol . 1 , 2014 , pp . 285292 . j. park , t. h. oh , j. jung , y .- w . tai , i. s. kweon , a tensor voting approach for multi - view 3d scene flow estimation and refinement , in : european conference on computer vision ( eccv ) , vol . 7575 lncs , 2012 , pp . 288302 . r. l. carceroni , k. n. kutalakos , multi - view scene capture by surfel sampling : from video streams to non - rigid 3d motion , shape and reflectance , in : ieee international conference on computer vision ( iccv ) , vol . 2 , 2001 , pp . 6067 vol.2 . j. p. pons , r. keriven , o. faugeras , g. hermosillo , variational stereovision and 3d scene flow estimation with statistical similarity measures , in : ieee international conference on computer vision ( iccv ) , 2003 , pp . 597602 vol.1 . j. p. pons , r. keriven , o. faugeras , multi - view stereo reconstruction and scene flow estimation with a global image - based matching score , international journal of computer vision 72 ( 2 ) ( 2007 ) 179193 . l. valgaerts , a. bruhn , h. zimmer , j. weickert , c. stoll , c. theobalt , joint estimation of motion , structure and geometry from stereo sequences , in : european conference on computer vision ( eccv ) , vol . 6314 lncs , 2010 , pp . 568581 . x. zhang , d. chen , z. yuan , n. zheng , dense scene flow based on depth and multi - channel bilateral filter , in : x. zhang , d. chen , z. yuan , z. hu ( eds . ) , asian conference on computer vision ( accv ) , 2012 , pp . 140151 . m. jaimez , m. souiai , j. st , x00fc , ckler , j. gonzalez - jimenez , d. cremers , motion cooperation : smooth piece - wise rigid scene flow from rgb - d images , in : international conference on 3d vision ( 3dv ) , 2015 , pp . 6472 . m. w. tao , s. hadap , j. malik , r. ramamoorthi , depth from combining defocus and correspondence using light - field cameras , in : ieee international conference on computer vision ( iccv ) , 2013 , pp . 673680 . v. lempitsky , s. roth , c. rother , fusionflow : discrete - continuous optimization for optical flow estimation , in : computer vision and pattern recognition , 2008 . cvpr 2008 . ieee conference on , ieee , 2008 , pp . z. lv , c. beall , p. f. alcantarilla , f. li , z. kira , f. dellaert , a continuous optimization approach for efficient and accurate scene flow , in : european conference on computer vision , springer , 2016 , pp . 757773 . a. dosovitskiy , p. fischer , e. ilg , h. p , xe , usser , c. hazirbas , v. golkov , p. v. d. smagt , d. cremers , t. brox , flownet : learning optical flow with convolutional networks , in : ieee international conference on computer vision ( iccv ) , 2015 , pp . 27582766 . t. vaudrey , c. rabe , r. klette , j. milburn , differences between stereo and motion behavior on synthetic and real - world stereo sequences , in : international conference of image and vision computing new zealand ( ivcnz ) , 2008 , pp . 16 . j. gall , c. stoll , e. d. aguiar , c. theobalt , b. rosenhahn , h. p. seidel , motion capture using joint skeleton tracking and surface estimation , in : ieee conference on computer vision and pattern recognition ( cvpr ) , 2009 , pp .
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this paper is the first to review the scene flow estimation field to the best of our knowledge , which analyzes and compares methods , technical challenges , evaluation methodologies and performance of scene flow estimation .
existing algorithms are categorized in terms of scene representation , data source , and calculation scheme , and the pros and cons in each category are compared briefly . the datasets and evaluation protocols are enumerated , and the performance of the most representative methods is presented .
a future vision is illustrated with few questions arisen for discussion .
this survey presents a general introduction and analysis of scene flow estimation .
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frobenius splittings of algebraic varieties appear prominently in tight closure theory , have emerged as a fundamental tool in the study of the representation theory of algebraic groups , and have tantalizing links to concepts in the minimal model program . suppose that @xmath6 is a reduced ring of characteristic @xmath7 with normalization in its total ring of fractions @xmath8 . it follows from ( * ? ? ? * exercise 1.2.4(e ) ) that if @xmath9 is frobenius split then so is @xmath10 . the goal of this paper is to study to what extent the converse holds . of course , there are many non - frobenius split affine varieties whose normalizations are regular and thus frobenius split ( for example , the cusp @xmath11 in any characteristic ) and so we focus our attention not just on the ring , but on the ring and a choice of a potential frobenius splitting . by definition , frobenius splitting _ on a variety @xmath12 is an @xmath13-linear map @xmath14 which sends the section @xmath15 to @xmath15 . when @xmath0 is affine , the existence of a frobenius splitting is equivalent to existence of a map @xmath14 such that @xmath16 for some section @xmath17 ( in other words , to require that @xmath4 is surjective ) . on a normal @xmath12 , surjective maps @xmath18 are very closely related to boundary divisors @xmath19 such that @xmath20 is log canonical , see @xcite and @xcite . based upon our intuition with non - normal log canonical singularities ( which are usually called _ semi - log canonical _ ) one should be able to detect surjective @xmath18 by studying the normalization of @xmath0 . in fact , this correspondence between log canonical pairs @xmath20 and surjective @xmath4 suggests that the following question should have an affirmative answer . [ quest.motivation ] suppose that @xmath0 is an affine variety in characteristic @xmath7 with normalization @xmath21 , and set @xmath22 to be the @xmath23-iterated frobenius . given a map @xmath24 , it always extends to a unique map @xmath25 . if @xmath26 is surjective , does it follow that @xmath4 is also surjective ? perhaps unfortunately , this question has a negative answer . a counterexample is given by the scheme @xmath27/(x^2y - z^2)$ ] , ( * ? ? ? * example 8.4 ) . however , that counterexample possesses substantial inseparability : if @xmath28 is the non - normal locus of @xmath0 and @xmath29 is its pre - image inside @xmath30 , then the induced map @xmath31 is generically inseparable . we show that if we can avoid this inseparability and also a certain variant of wild ramification , then the question has a positive answer . in particular , we say that a ring possesses _ _ if it avoids these positive characteristic pathologies ( see definition [ def.hst ] for a precise definition ) . our main theorem follows : [ thm : maintheorem ] [ theorem [ thm : main ] ] suppose that @xmath0 is a reduced @xmath5-finite affine scheme having . further suppose that @xmath24 is an @xmath13-linear map . if the unique extension @xmath25 is surjective , then @xmath4 is also surjective . the main theorem should also be viewed as complementary to ( * ? ? ? * theorem 6.26 ) . in the context of a finite surjective map @xmath32 between _ normal _ varieties , that result answers the same question for @xmath14 and its extension @xmath33 ( if it exists ) . also see @xcite where 1-dimensional non - normal @xmath5-pure rings are studied . we now discuss in more detail the motivation for this theorem . in this context , maps @xmath34 correspond to @xmath35-divisors @xmath19 on @xmath30 such that @xmath36 is @xmath35-cartier , see @xcite and section [ sec : prelim ] . the condition @xmath4 being surjective ( _ i.e. _ the pair @xmath37 being @xmath5-pure ) corresponds to the pair @xmath37 having log canonical singularities @xcite , @xcite . on a characteristic zero variety @xmath0 that is s2 ( _ i.e. _ serre s second condition ) and g1 ( _ i.e. _ gorenstein in codimension @xmath15 ) with normalization @xmath38 and conductor divisor @xmath39 , if there exists a @xmath40-divisor @xmath41 on @xmath0 such that @xmath42 is @xmath35-cartier and furthermore that @xmath43 is log canonical , then @xmath44 is called _ semi - log canonical_. via the map-(@xmath4 ) divisor-(@xmath19 ) correspondence , our main theorem should be interpreted as saying : ll as an explicit example of this statement , and of the translation between maps @xmath45 and divisors , we have the following corollary of our main theorem , a criterion for certain non - normal algebraic varieties to be @xmath5-split . * corollary [ cor.hstnormalizationqgorenstein ] . * _ suppose that @xmath12 is an affine @xmath5-finite scheme satisfying and which is also s2 , g1 and @xmath40-gorenstein with index not divisible by the characteristic @xmath46 . set @xmath30 to be the normalization of @xmath0 and set @xmath29 to be the divisor on @xmath30 corresponding to the conductor ideal , _ i.e. _ @xmath47 . then @xmath0 is @xmath5-pure ( equivalently @xmath5-split ) if and only if @xmath48 is @xmath5-pure . _ in the above corollary , if one replaces the two occurrences of the word @xmath5-pure by semi - log canonical and log canonical ( respectively ) , one obtains a well known criterion for a scheme having semi - log canonical singularities , see @xcite . more generally , the correspondence between log canonical and @xmath5-pure singularities is still conjectural , although an area of active research . [ conj.fpurevslogcanonical ] suppose that @xmath49 is a normal variety of characteristic zero and that @xmath19 is a divisor on @xmath49 such that @xmath50 is @xmath35-cartier . the pair @xmath51 is log canonical if and only if @xmath51 has dense @xmath5-pure type pair @xmath52 has dense @xmath5-pure type if the map corresponding to @xmath19 is surjective after reduction to characteristic @xmath7 for infinitely many @xmath7 . see @xcite for more details . ] . see @xcite and @xcite for connections between this conjecture and open conjectures in arithmetic geometry ( these connections will not be utilized in this paper ) . the setting for our applications will necessarily deal with geometry on non - normal schemes , and as such we will be dealing with an extended notion of divisors , see @xcite . for us , the important divisors are the @xmath35-almost cartier divisor ( or simply _ @xmath35-ac divisors _ ) , and we will review the definitions in detail in section [ sec : divisors ] . our main result implies the following : * corollary [ cor.fpurevsslc ] . * _ assume now that @xmath0 is an s2 , g1 and seminormal variety in characteristic zero . further suppose that @xmath19 is an @xmath35-ac divisor ( _ i.e. _ @xmath35-weil sheaf ) on @xmath53 assuming that conjecture [ conj.fpurevslogcanonical ] holds , @xmath20 has semi - log canonical singularities if and only if it has dense @xmath5-pure type . _ another application of theorem [ thm : maintheorem ] is to give a statement of inversion of adjunction for divisors on characteristic @xmath7 schemes with that closely aligns with the characteristic @xmath54 picture , compare with @xcite . inversion of adjunction for characteristic @xmath7 schemes was studied first in @xcite , see also takagi @xcite and @xcite . however the direct analog of kawakita s result is not possible in characteristic @xmath7 ( * ? ? ? * example 8.4 ) . the culprit is the same conditions of inseparability and wild ramification which obstruct a positive answer to the motivating question . however : * corollary [ cor.invofadj ] . * _ suppose that @xmath0 is a normal scheme of characteristic @xmath7 , @xmath19 is a @xmath35-divisor on @xmath0 and @xmath55 is a reduced integral weil divisor on @xmath0 , with no common components with @xmath19 , such that @xmath56 is @xmath35-cartier with index not divisible by @xmath46 . denote by @xmath57 the normalization of @xmath55 and @xmath58 the natural map . there exists a canonically determined @xmath35-divisor @xmath59 on @xmath57 such that @xmath60 . furthermore , if @xmath61 has , then @xmath62 is @xmath5-pure near @xmath55 if and only if @xmath63 is @xmath5-pure . _ -12pt_acknowledgements : _ the authors would like to thank bryden cais , neil epstein and kevin tucker for inspiring conversations related to this project . the authors would also like to thank the referee , shunsuke takagi and kevin tucker for numerous useful comments on an earlier draft of this manuscript . this work was initiated when the second author was visiting the university of utah during fall 2010 . throughout this article , all rings will be commutative with unity , noetherian , excellent and , unless otherwise specified , of prime characteristic and all schemes will be separated and noetherian . we frequently use the following notations and conditions on rings and schemes . finally , in order to avoid confusion , we remark that we use the notation @xmath64 to denote the ring of integers @xmath65 localized at the prime ideal @xmath66 . for a reduced ring @xmath6 ( resp . @xmath0 is a reduced scheme ) , we use @xmath67 ( resp . @xmath68 ) to denote its total ring of fractions . furthermore , we use @xmath8 ( resp . @xmath30 ) to denote the normalization of @xmath6 in @xmath67 ( resp . the normalization of @xmath0 in @xmath68 ) . a ring is called s2 if it satisfies serre s second condition , _ i.e. _ the localization at any prime in @xmath6 of height at least @xmath69 or @xmath15 has depth at least @xmath69 or @xmath15 respectively . it is called g1 if it is gorenstein in codimension 1 . a finite extension of reduced rings @xmath70 is called _ subintegral _ ( resp . _ weakly subintegral _ ) if it induces a bijection on prime spectra such that the residue field maps are all isomorphisms ( resp . purely inseparable ) . a reduced ring @xmath6 is called _ seminormal _ if it possesses no proper subintegral extensions inside its own field of fractions . any reduced ring @xmath6 has a _ seminormalization _ @xmath71 which is the unique largest subintegral extension of @xmath6 which is contained inside @xmath67 . for an introduction to seminormalization , see @xcite or @xcite . it is important to note that if @xmath6 is seminormal then the conductor ideal @xmath72 is a radical ideal in both @xmath6 and @xmath8 , see @xcite or @xcite . for rings of characteristic @xmath7 , we denote by @xmath73 the frobenius homomorphism sending @xmath74 to its @xmath46-th power . given any @xmath6-module , @xmath75 , one can view @xmath75 as an @xmath6-module via the frobenius map and obtain a new @xmath6-module @xmath76 , called the _ frobenius pushforward of @xmath75_. this is the @xmath6-module with the same underlying additive group as @xmath75 but scalar multiplication is defined by the rule @xmath77 for @xmath78 , @xmath79 . one can iterate the frobenius and for each @xmath23 we get a frobenius push forward @xmath80 whose @xmath6-module structure is defined similarly . for a scheme @xmath0 over a field of characteristic @xmath46 , one can ask about the @xmath13-module structure of @xmath81 . connections between this question and the geometry of @xmath0 has been a strong guiding force in the study of such characteristic @xmath7 schemes . for example , @xmath0 is regular if and only if @xmath82 is a flat @xmath13-module @xcite . we now review some of these relationships . a scheme @xmath0 is _ f - finite _ provided @xmath81 is coherent , _ i.e. _ a finitely generated @xmath13-module for some ( equivalently all ) @xmath83 . [ cols= " < , < " , ] the class of @xmath5-finite schemes is particularly nice because @xmath5-finite schemes are abundant , namely varieties over a perfect field are all @xmath5-finite . @xmath5-finite schemes are always locally excellent @xcite and they always locally have dualizing complexes @xcite . as in the introduction , we say @xmath0 is _ @xmath5-split _ provided there is a map @xmath84 which sends @xmath15 to @xmath15 and we say @xmath0 is _ @xmath5-pure _ provided the frobenius map on @xmath13 is a pure morphism . for @xmath5-finite affine schemes , @xmath5-splitting and @xmath5-purity coincide , see for example @xcite and @xcite . to describe how these notions relate to characteristic @xmath7 geometry , we use the setting of pairs . for a more detailed treatment see @xcite . a _ prime divisor _ on a normal irreducible scheme @xmath49 is a reduced irreducible subscheme of codimension 1 and a _ weil divisor _ is any element of the free abelian group generated by the prime divisors . this abelian group is denoted by @xmath85 . a @xmath35-divisor is an element of @xmath86 , _ i.e. _ a divisor with rational coefficients . an element @xmath87 is called an _ integral divisor _ if it is contained within @xmath88 ( in other words , if it is a weil divisor , and we wish to emphasize that its coefficients are integral ) . cartier divisor _ is a weil divisor that is locally principal and a @xmath35-cartier divisor is a @xmath40-divisor @xmath89 such that @xmath90 is integral and cartier for some @xmath91 . for a @xmath40-cartier divisor @xmath89 , the smallest positive integer @xmath92 such that @xmath90 integral and cartier is called the _ index of @xmath89_. see @xcite for additional discussion of @xmath40-divisors in this context . we now discuss divisors on non - normal s2 schemes . one should note that our main theorem can be proven without appeal to these objects , and our main corollary [ cor.fpurevsslc ] is interesting even when the divisor @xmath93 . thus the reader not already familiar with divisors on non - normal schemes may wish to skip to section [ sec.extendingnormal ] at this point . @xcite , @xcite [ def.almostcartierdivisor ] for an s2 equidimensional reduced scheme @xmath0 , an _ ac divisor _ ( or _ almost cartier divisor _ ) is a coherent @xmath13-module @xmath94 satisfying the following properties * @xmath95 is s2 . * @xmath96 , abstractly , for all points @xmath97 of codimension 0 or 1 . as was pointed out by the referee , the terminology _ almost cartier _ is misleading since any weil divisor on a normal scheme is almost cartier ( the terminology is taken from @xcite ) . therefore , in order to avoid this confusion , we instead call almost cartier divisors by the name ac divisors . these divisors form an _ additive _ group via tensor product up to s2-ification , ( * ? ? ? * part 2 , section 5.10 ) , which we denote by @xmath98 . given @xmath99 , we sometimes use @xmath100 to denote the sheaf @xmath94 defining @xmath89 . note that given any @xmath101 non - zero at any minimal prime , we use @xmath102 to denote the element of @xmath98 corresponding to @xmath103 . by a _ @xmath35-ac divisor _ _ @xmath64-ac divisor _ ) we mean an element of @xmath104 ( resp . @xmath105 ) . we say that a divisor @xmath94 of @xmath98 is _ effective _ if @xmath106 and we say that @xmath107 ( resp . @xmath108 ) is _ effective _ if @xmath109 for some effective @xmath99 and some @xmath110 ( resp . @xmath64 ) . we say that two ac divisors @xmath111 and @xmath112 are _ linearly equivalent _ if there is @xmath113 in @xmath68 such that @xmath114 . we call an ac divisor _ cartier _ if the sheaf @xmath95 is a line bundle . we say that an element @xmath107 ( resp . @xmath115 ) is _ @xmath40-cartier _ ( resp . _ @xmath64-cartier _ ) if there exists @xmath116 ( resp . @xmath117 ) such that @xmath118 for some cartier divisor @xmath28 in @xmath98 . we say that two elements @xmath119 ( resp . @xmath108 ) are _ @xmath40-linearly equivalent _ ( resp . @xmath64-linearly equivalent ) , denoted @xmath120 ( resp . @xmath121 ) , if there exists a non - zero integer @xmath122 ( resp . @xmath123 ) such that @xmath124 and there exists an element @xmath101 such that @xmath125 . [ rem.pathologiesfordivisorsonnonnormal ] one should note that two distinct ac divisors of @xmath98 can be identified in @xmath126 , see ( * ? ? ? * page 172 ) , and in particular , the natural map @xmath127 is generally not injective unlike the case when @xmath0 is normal . working with @xmath108 is not common . however , this group behaves much better than @xmath126 in characteristic @xmath7 for our purposes , see theorem [ thmdivisormapcornonnormal ] . when working on normal varieties , this distinction is less important because @xmath108 as a subset of @xmath126 is simply the set of divisors whose coefficients do not have @xmath46 in their denominators . for non - normal varieties however , the natural map from @xmath108 to @xmath126 is not generally an injection . therefore , when working in characteristic @xmath7 on _ non - normal _ varieties , we will generally work with @xmath108 . suppose that @xmath0 is a reduced equidimensional scheme which is s2 and g1 . further suppose that @xmath0 possesses a _ canonical module @xmath128 _ , _ i.e. _ a module isomorphic to the first non - zero cohomology of a dualizing complex at each point . by a canonical divisor @xmath129 we mean any embedding of @xmath130 up to multiplication by a unit of @xmath131 , see @xcite and @xcite . notice that the condition that @xmath0 is g1 is exactly the condition needed to guarantee definition [ def.almostcartierdivisor](ii ) . by a pair , we mean a tuple @xmath52 where @xmath49 is a s2 and g1 scheme and @xmath19 is an element of @xmath126 or @xmath108 . a pair is called _ log @xmath35-gorenstein _ _ log @xmath64-gorenstein _ ) provided @xmath50 is @xmath35-cartier ( resp . @xmath64-cartier ) . when @xmath93 the ` log ' is omitted and the scheme @xmath49 is just called @xmath35-gorenstein ( resp . @xmath64-gorenstein ) . now suppose that @xmath49 is normal and @xmath132 is a log resolution , which always exists in characteristic zero by @xcite . decompose @xmath133 where @xmath134 is prime and @xmath135 is a canonical divisor that agrees with @xmath136 wherever @xmath137 is an isomorphism . the pair @xmath52 is _ log canonical _ provided @xmath138 for all @xmath139 . if @xmath0 is an s2 but not necessarily normal scheme , denote by @xmath21 its normalization and @xmath29 the divisor associated to the conductor @xmath140 in @xmath30 ; _ i.e. _ @xmath47 . if additionally @xmath0 is g1 , we say @xmath141 is _ semi - log canonical _ provided @xmath42 is @xmath35-cartier and @xmath142 is log canonical . a more detailed treatment of these concepts in characteristic @xmath54 can be found in ( * ? ? ? * chapter 16 ) or ( * ? ? ? * chapter 3 ) . we call an additive map , @xmath144 , _ @xmath143-linear _ provided it locally satisfies @xmath145 . these are identified with the set of @xmath13-module homomorphisms in @xmath146 . such maps @xmath4 naturally correspond to effective @xmath35-divisors @xmath147 such that @xmath148 is @xmath35-cartier . in the normal setting , variants of this correspondence have appeared in many places such as @xcite and @xcite , this correspondence was recently formalized in @xcite . however , we need a version of this correspondence in the non - normal setting as well . first suppose that @xmath12 is a s2 and g1 reduced and equidimensional affine scheme which is also the spectrum of a semi - local ring . the usual arguments still imply that @xmath149 . therefore any section @xmath150 which is non - zero at each generic point of @xmath0 , induces an effective ac divisor @xmath151 via ( * ? ? ? * proposition 2.9 ) and ( * ? ? ? * remark 2.9 ) ( for example , choose the embedding of @xmath152 into @xmath68 which sends @xmath4 to @xmath15 ) . we define the divisor @xmath147 associated to @xmath4 to be the element @xmath153 . conversely , suppose we are given an effective @xmath64-ac divisor @xmath19 such that @xmath154 for some integer @xmath155 which is not divisible by @xmath46 and for some @xmath99 . additionally suppose that @xmath156 . thus @xmath89 corresponds to a section @xmath157 , up to multiplication by a unit of @xmath131 . without loss of generality , we may assume that @xmath158 for some @xmath159 and so we may view @xmath160 as an element @xmath161 . these two observations lead us to the following correspondence . [ thmdivisormapcornonnormal ] for @xmath6 a semilocal @xmath5-finite reduced , s2 , and g1 ring and set @xmath12 . we have the following bijection of sets : @xmath162 here the equivalence relation @xmath163 on the right is generated by two equivalences . * we say that @xmath164 are equivalent if there exists a unit @xmath165 in @xmath166 such that @xmath167 . * we say that @xmath150 and @xmath168 in @xmath169 are equivalent . the proof is more subtle than in the normal case : if @xmath170 and @xmath171 are elements of @xmath172 that determine the same @xmath64-divisor , it does not imply that one is a unit multiple of the other as is the case for normal rings . the reason for this is that there can be torsion in @xmath98 . we certainly know that every @xmath143-linear map @xmath4 as described above induces an effective @xmath19 in @xmath108 by the procedure described above . furthermore , this procedure is surjective by ( * ? ? ? * proposition 2.9 ) . thus we simply have to show that the two equivalence relations described in ( i ) and ( ii ) above are sufficient to induce a bijection . so suppose that @xmath173 and @xmath174 induce the same divisor @xmath19 . by using ( ii ) , we may assume that @xmath175 . thus we have two sections @xmath176 inducing the same @xmath64-ac divisors in @xmath177 , say @xmath178 and @xmath179 respectively . it follows that there is an integer @xmath155 not divisible by @xmath7 such that @xmath180 . by making @xmath92 bigger if necessary , we may take @xmath181 for some @xmath182 . as a section , we claim that @xmath183 induces the divisor @xmath184 in @xmath98 via ( * ? ? ? * proposition 2.9 ) . we now prove this claim . since divisors are determined in codimension @xmath15 , we may assume that @xmath12 where @xmath6 is a @xmath15-dimensional gorenstein ring . because @xmath6 is gorenstein , by duality for a finite map , @xmath185 is isomorphic to @xmath186 ( abstractly ) . choose @xmath187 to be an @xmath186-module generator of @xmath185 . we may thus write @xmath188 for some @xmath189 . notice @xmath190 which corresponds to the sheaf @xmath191 . it follows that @xmath192 therefore , because @xmath193 generates @xmath194 as an @xmath195-module by ( * ? ? ? f ) or ( * ? ? ? * lemma 3.9 ) , @xmath196 which proves the claim . thus @xmath197 and @xmath198 agree up to multiplication by a unit and so @xmath170 and @xmath199 are indeed related by relations ( i ) and ( ii ) . we also have to verify that if @xmath170 and @xmath199 are related by conditions ( i ) or ( ii ) , then they induce the same element of @xmath108 . certainly condition ( i ) is harmless . to check condition ( ii ) , one simply has to tensor equation ( [ eqndivisortopower ] ) by @xmath200 . if @xmath6 is not semilocal but additionally there is a quasi - isomorphism @xmath201 ( which occurs for example , if @xmath6 is essentially of finite type over a field , or more generally a gorenstein local ring ) , then the theorem above still holds as long as one restricts the left - hand - side to @xmath202 of course , the @xmath203 direction of the correspondence in theorem [ thmdivisormapcornonnormal ] always exists . alternately , see ( * ? ? ? * remark 9.5 ) . suppose that @xmath6 is a reduced @xmath5-finite local ring of characteristic @xmath7 . fix a map @xmath204 . the pair @xmath205 is called _ @xmath5-pure _ if @xmath4 is surjective . suppose that @xmath0 is an s2 , g1 and reduced @xmath5-finite scheme and @xmath206 corresponds at each point @xmath207 to a map @xmath208 via theorem [ thmdivisormapcornonnormal ] . the pair @xmath44 is called _ @xmath5-pure _ if @xmath209 is @xmath5-pure for all @xmath207 . while there is substantial freedom in the choice of @xmath210 associated to @xmath41 at each point , it is straightforward to verify that if two maps @xmath210 and @xmath211 are related via the two conditions ( i ) and ( ii ) in theorem [ thmdivisormapcornonnormal ] , then @xmath210 is surjective if and only @xmath211 is surjective . for a finite inclusion of rings @xmath212 and an @xmath6-linear map @xmath213 one can ask when there is a @xmath55-linear map @xmath214 so that @xmath215 . the study of such extensions of maps in the case where @xmath6 and @xmath55 are both normal domains is carefully laid out in @xcite . however , even in that case , an arbitrary @xmath143-linear map on @xmath6 need not extend to a @xmath143-linear map on @xmath55 . * example 3.4 ) consider for example @xmath216 , @xmath217 and the inclusion @xmath218 \subset \mbf{f}_3[x ] = s$ ] . an @xmath6-basis for @xmath219 is @xmath220 and an @xmath55-basis for @xmath221 is @xmath222 . any map @xmath223 is defined by the images @xmath224 . a map @xmath225 which extends @xmath4 must agree with @xmath4 on @xmath220 . however , @xmath226 since @xmath26 is @xmath143-linear and so @xmath227 . thus when @xmath228 is not divisible by @xmath229 , @xmath4 can not extend . generic separability is necessary to guarantee extensions of maps ( * ? ? ? * proposition 5.1 ) . when the inclusion of fraction fields is not generically separable then only the zero map extends . this separability allows one to use the trace map as a vehicle for understanding such extensions in the normal case . in the next section , where we address out motivating question , we will see that the non - normal setting is even more complicated . for an affine variety @xmath12 denote by @xmath230 the normalization . if we set @xmath140 to be the conductor ( the largest ideal of @xmath6 that is also an ideal of @xmath8 ) the inclusion @xmath231 extends to the following commutative diagram where the inclusions are the obvious ones and the other maps are the natural surjections . \(a ) at ( -1,1 ) @xmath8 ; ( b ) at ( -1,-1)@xmath6 ; ( c ) at ( 1,0)@xmath232 ; ( d ) at ( 1,-2)@xmath233 ; ( b ) edge ( a ) ; ( a ) edge ( c ) ; ( b ) edge ( d ) ; ( d ) edge ( c ) ; recall also the following lemma . * exercise 1.2.4(e ) ) [ lem : extendingmapsundernormalization ] suppose that @xmath6 is a non - normal reduced ring , @xmath8 is its normalization and @xmath140 is the conductor ideal . any @xmath6-linear map @xmath204 is compatible with @xmath140 ( _ i.e. _ @xmath234 ) and extends uniquely to an @xmath8-linear map @xmath235 . a proof can be found in ( * ? ? ? * propositions 7.10 , 7.11 ) . suppose we are given a map @xmath204 as above . lemma [ lem : extendingmapsundernormalization ] implies that there exists a map @xmath235 and furthermore that the induced map @xmath236 extends to an @xmath237-linear map @xmath238 . in order to prove our main theorem , we will relate the surjectivity of @xmath4 to that of @xmath26 by studying the surjectivity of @xmath239 verses @xmath240 . following the ideas of @xcite , it is natural to attempt to apply the trace map ( for the inclusion @xmath241 ) to solve this problem . more generally , for a finite inclusion @xmath242 of reduced rings @xmath243 and @xmath29 where each minimal prime of @xmath29 lies over a minimal prime of @xmath243 , one can ask whether the trace map @xmath244 restricts to a _ surjective _ map from @xmath29 to @xmath243 ( here @xmath244 is defined to be the sum of the individual field trace maps ) . if this were always the case in our setting , one would have a _ surjective _ trace map @xmath245 and one could use this show that surjectivity of @xmath4 directly . however , the next examples show the trace map can fail to be surjective ( or even fail to induce a map from @xmath29 to @xmath243 ) for inclusions @xmath242 . suppose that @xmath246 is a perfect field of characteristic @xmath69 and consider the ring @xmath247/(xz^2 - y^2 ) \cong k[a^2 , ab , b ] \subseteq k[a , b]$ ] . the conductor ideal is @xmath248 and so @xmath249 \cong r/\goth{c } \subseteq r^{\textnormal{n}}/\goth{c } \cong k[a]$ ] is generically purely inseparable . thus the trace map @xmath245 is the zero map ( and in particular , not surjective ) . likewise , one can construct similar examples of galois extensions @xmath241 of dedekind domains which are generically separable ( and so the trace map is non - zero ) but which have wild ramification and so the trace map is not surjective . consider the extension @xmath250 \subseteq k[x ] = b$ ] where the characteristic of @xmath246 is _ not _ equal to @xmath69 . the trace map @xmath251 yields @xmath252 . however , we also have the inclusion @xmath253 \subseteq k[x ] = b$ ] noting that @xmath254 $ ] and @xmath255 $ ] have the same fraction field , @xmath256 . therefore @xmath257 . by using pushout diagrams of schemes ( as in @xcite ) one can construct a ring @xmath6 where @xmath258 . [ xmp1 ] let @xmath246 be any field and consider rings @xmath259 \oplus k[y ] \mid \text{$s$ and $ t$ have the same constant term } \}\ ] ] and @xmath260 \oplus k[y ] \mid \text{$u$ and $ v$ have the same constant term } \}\ ] ] over @xmath246 both having a node at the origin . consider the normalizations @xmath261 \oplus k[y]$ ] and @xmath262\oplus k[y]$ ] respectively . one can see the conductor of @xmath243 in @xmath263 is the ideal made up of all pairs @xmath264 with zero constant term . likewise the conductor of @xmath29 in @xmath265 is the ideal made up of all pairs @xmath266 with no constant term . in this example , we consider the trace on each irreducible component , and then add the resulting trace maps . somewhat abusively , we call this sum the `` trace '' also , and denote it by @xmath267 . clearly this trace map sends the conductor to the conductor but @xmath268 is not contained in @xmath243 because @xmath269 . compare with the question `` trace map attached to a finite homomorphism of noetherian rings '' of bryden cais on asked on december 3rd , 2009 . in light of these examples , the condition that @xmath270 is too restrictive . however , the following ( recursive ) definition , which is substantially weaker , will be exactly what we want in order to answer our motivating question . [ def.hst ] suppose that @xmath6 is a reduced local ring and @xmath12 . define @xmath230 to be the normalization with conductor @xmath140 . we set @xmath39 and @xmath271 to be the subschemes defined by @xmath140 and set @xmath272 and @xmath273 to be the associated reduced subschemes . we say @xmath0 has _ hereditary surjective trace _ provided that there is some irreducible component @xmath274 of @xmath273 dominated by an irreducible component @xmath275 of @xmath272 such that : * the induced trace map @xmath276 is surjective and * @xmath274 also has ( this condition is vacuous if @xmath274 is normal ) . in the language of commutative algebra , a ring having _ _ means that there exist minimal associated prime ideals of @xmath140 ( which is an ideal of both @xmath6 and @xmath8 ) , @xmath277 and @xmath278 such that @xmath279 satisfying ( i ) is surjective and ( ii ) @xmath280 also has . we say that a ( non - local ) scheme @xmath12 has _ _ if it has at every point . observe that the dimension of @xmath28 is strictly smaller than that of @xmath0 , and so the recursive process of definition [ def.hst ] will stop after finitely many steps . [ rem.strongerhst ] it would also be natural to require conditions ( i ) and ( ii ) for _ every _ irreducible component @xmath274 of @xmath273 dominated by an irreducible component @xmath275 of @xmath272 . however , we will not need this stronger condition . we consider the following special case of our main result whose proof we feel is illuminating . [ xmp : basecase ] recall that any complete one dimensional seminormal ring of characteristic @xmath7 with algebraically closed residue field @xmath246 is isomorphic to the completion of the coordinate ring of some set of coordinate axes in @xmath281 by @xcite . suppose that @xmath6 is an @xmath5-finite complete two - dimensional s2 seminormal ring with algebraically closed residue field . use @xmath8 to denote the normalization of @xmath6 and suppose that @xmath282 is a surjective map which is compatible with the conductor ideal @xmath140 . because @xmath6 is s2 , @xmath283 is equidimensional and reduced . therefore , @xmath283 is also @xmath5-pure and thus seminormal and so it is isomorphic to a direct sum of completions of coordinate rings of coordinate axes in various @xmath284 . set @xmath28 to be the pull - back of the diagram @xmath285 in other words , @xmath28 is the ring @xmath286 notice that @xmath287 is subintegral by construction , see @xcite . therefore , @xmath288 since @xmath6 is seminormal . this implies @xmath289 and so @xmath290 is also isomorphic to coordinate axes . given any component @xmath291 of @xmath292 and a component @xmath293 of @xmath294 dominating it , suppose that @xmath295 $ ] . provided @xmath46 does not divide @xmath92 we see that definition [ def.hst](i ) holds . part ( ii ) holds vacuously as the components here are normal . recall a small lemma about extending maps in local rings . * observation 5.1 ) [ lem : local ] let @xmath6 be a local ring and @xmath296 a proper ideal . suppose there is a surjective @xmath6-linear map @xmath297 which is the restriction of an @xmath6-linear map @xmath298 . then @xmath299 is surjective . finally , we prove our main theorem . [ thm : main ] suppose that @xmath0 is a reduced affine @xmath5-finite scheme having . further suppose that @xmath24 is an @xmath13-linear map . if the unique extension @xmath25 is surjective , then @xmath4 is also surjective . we proceed by induction , the case where @xmath0 is zero - dimensional is obvious since then @xmath300 . furthermore , the statement is local so we may assume that @xmath12 where @xmath6 is a local @xmath5-finite ring of characteristic @xmath7 and @xmath230 . we aim to show that an @xmath6-linear map @xmath213 which extends to a surjective @xmath8-linear map @xmath301 is also surjective . notice that @xmath140 is radical in @xmath8 ( and thus also in @xmath6 ) because @xmath26 is surjective and @xmath140 is @xmath26-compatible . modulo @xmath140 , one has maps @xmath302 and @xmath303 by lemma [ lem : extendingmapsundernormalization ] . \(a ) at ( -1,1 ) @xmath8 ; ( b ) at ( -1,-1)@xmath6 ; ( c ) at ( 1,0)@xmath232 ; ( d ) at ( 1,-2)@xmath233 ; ( fa ) at ( -3,1)@xmath304 ; ( fb ) at ( -3,-1)@xmath305 ; ( fc ) at ( 3,0)@xmath306 ; ( fd ) at ( 3,-2)@xmath307 ; ( b ) edge ( a ) ; ( a ) edge ( c ) ; ( b ) edge ( d ) ; ( d ) edge ( c ) ; ( fa ) edge node[above]@xmath308 ( a ) ; ( fb ) edge node[above]@xmath309 ( b ) ; ( fb ) edge ( fa ) ; ( fc ) edge node[above]@xmath310 ( c ) ; ( fd ) edge node[above]@xmath311 ( d ) ; ( fd ) edge ( fc ) ; we will examine the behavior of the maps @xmath312 . when @xmath26 is surjective then so too is @xmath313 and one can ask whether the surjectivity of @xmath313 implies surjectivity of @xmath314 . this is equivalent to the surjectivity of @xmath4 by lemma [ lem : local ] . since @xmath6 has , there are components of @xmath292 and @xmath294 which have a surjective trace map between their normalizations . specifically , let @xmath315 and @xmath316 be such that @xmath317 and @xmath318 are the components in question . furthermore , note that @xmath280 is local and @xmath319 is semilocal . we consider the normalization of these rings and , by our hypothesis , we know that the trace map @xmath320 is surjective . there are induced @xmath6-linear maps @xmath321 since minimal primes of a ring are always compatible . furthermore because @xmath313 is surjective so too is @xmath322 . so we have the following diagram . \(a ) at ( -1,1 ) @xmath232 ; ( b ) at ( -1,-1)@xmath233 ; ( c ) at ( 1,0)@xmath319 ; ( d ) at ( 1,-2)@xmath280 ; ( fa ) at ( -3,1)@xmath306 ; ( fb ) at ( -3,-1)@xmath307 ; ( fc ) at ( 3,0)@xmath323 ; ( fd ) at ( 3,-2)@xmath324 ; ( b ) edge ( a ) ; ( a ) edge ( c ) ; ( b ) edge ( d ) ; ( d ) edge ( c ) ; ( fa ) edge node[above]@xmath310 ( a ) ; ( fb ) edge node[above]@xmath311 ( b ) ; ( fb ) edge ( fa ) ; ( fc ) edge node[above]@xmath325 ( c ) ; ( fd ) edge node[above]@xmath326 ( d ) ; ( fd ) edge ( fc ) ; since @xmath233 is local , by lemma [ lem : local ] , @xmath314 is surjective if and only if @xmath327 is . by lemma [ lem : extendingmapsundernormalization ] , @xmath328 extends to a map @xmath329 and @xmath330 extends to a map @xmath331 for clarity , the diagram for @xmath332 is below . \(a ) at ( -2,2)@xmath333 ; ( b ) at ( -2,0)@xmath334 ; ( c ) at ( 2,2 ) @xmath335 ; ( d ) at ( 2,0)@xmath280 ; ( b ) edge ( a ) ; ( a ) edge node[above]@xmath336 ( c ) ; ( b ) edge node[above]@xmath326 ( d ) ; ( d ) edge ( c ) ; note that @xmath337 is surjective as @xmath322 is . by construction , @xmath280 has , and since it is lower dimensional , by our inductive hypothesis it is sufficient to show that @xmath332 is surjective . now we are in a situation to use the surjective trace map @xmath320 . the following commutative diagram shows that @xmath338 is surjective as the top square is commutative by ( * ? ? ? * corollary 4.2 ) . see also ( * ? ? ? * proposition 4.1 , theorem 5.6 ) . this completes the proof . ( fb ) at ( -1.5,0)@xmath339 ; ( fc ) at ( -1.5,1.5)@xmath333 ; \(b ) at ( 1.5,0)@xmath340 ; ( c ) at ( 1.5,1.5)@xmath335 ; \(b ) edge node[right]@xmath341(c ) ; ( fb ) edge node[left]@xmath342(fc ) ; ( fb ) edge node[below]@xmath343 ( b ) ; ( fc ) edge node[above]@xmath336 ( c ) ; we use the map - divisor correspondence described in theorem [ thmdivisormapcornonnormal ] to study schemes with and semi - log canonical singularities . while @xmath5-pure schemes are known to have log canonical singularities , there is only a conjectural converse , see conjecture [ conj.fpurevslogcanonical ] in the introduction . we pause to prove a few needed lemmas . we refer the reader to @xcite for a detailed description of the reduction to characteristic @xmath344 process . we also acknowledge the following abuse of notation , by @xmath344 we technically are referring to an open and zariski dense set of maximal ideals in @xmath345 , a finitely generated @xmath65-algebra used in the reduction to characteristic @xmath344 process . again , see the aforementioned reference for more details . see ( * ? ? ? * theorem 2.3.6 ) for the relevance of the algebraically closed base - field assumption . suppose we are given components of the conductor subschemes @xmath346 and @xmath347 with an degree @xmath348 finite morphism @xmath349 . performing reduction to characteristic @xmath344 ( in particular @xmath350 ) , the trace map @xmath351 is clearly surjective . continuing recursively , we can require that @xmath352 for @xmath353 s determined by a finite set of varieties . this is certainly enough to guarantee . in fact , we can guarantee the stronger variant of found in remark [ rem.strongerhst ] . [ lem.divofextensionsplusconductor ] suppose that @xmath12 is an s2 , g1 seminormal scheme with normalization @xmath21 . further suppose that @xmath24 corresponds to a divisor @xmath147 . denote by @xmath25 the extension of @xmath26 to @xmath30 . the @xmath35-divisor on @xmath30 corresponding to @xmath26 , denoted @xmath354 , satisfies @xmath355 where @xmath29 is the divisor corresponding to the conductor on @xmath30 , _ i.e. _ @xmath47 ( the conductor is pure codimension 1 because @xmath0 is s2 ) . first we explain how to pull back divisors via @xmath160 . this pull - back process is completely determined in codimension 1 ( which is reasonable , since the divisors are determined in codimension 1 ) . thus , suppose that we are given a @xmath356 . by construction , since @xmath95 is ac , it is easy to pull - back ( work outside a set of codimension 2 , or see @xcite ) . we define @xmath357 . it is straightforward to verify that this is well defined . the statement of the lemma can also be checked in codimension 1 and so we assume that @xmath6 is 1-dimensional and local . write @xmath358 for some @xmath359 ( we can do this because @xmath19 is @xmath64-ac ) . the pullback of @xmath147 is then just @xmath360 . we claim it is sufficient to check the statement when @xmath361 ( so that @xmath362 ) . to see this claim , choose @xmath363 such that @xmath364 is zero ( which we can do since we have reduced to the case where @xmath6 is gorenstein ) . this @xmath365 generates @xmath366 as an @xmath82-module and so @xmath367 for some @xmath368 . it follows that @xmath369 . thus @xmath370 . therefore , if @xmath371 , we can add @xmath372 to both sides of the equation , which proves the claim . in this context , @xmath6 is gorenstein and local and @xmath8 is regular and semi - local . it follows that @xmath373 is a free @xmath8-module . fix @xmath374 to be a generator ( and assume it sends @xmath15 to some @xmath375 which generates @xmath140 as an @xmath8-module ) . also notice that the assumptions imply that @xmath4 generates @xmath185 as an @xmath186-module . furthermore , for any @xmath376-module generator @xmath377 , we know that we have @xmath378 up to multiplication by a unit ( which we can then absorb into @xmath365 obtaining a true equality ) . at the level of the field of fractions , @xmath4 and @xmath26 are the same map and @xmath187 is multiplication by @xmath379 . thus , since @xmath378 , we have @xmath380 again at the level of the field of fractions . therefore @xmath381 . this implies that @xmath354 is the divisor of @xmath140 as desired . [ cor.hstnormalizationqgorenstein ] suppose that @xmath12 is an affine @xmath5-finite scheme satisfying and which is also s2 , g1 and @xmath64-gorenstein . set @xmath30 to be the normalization of @xmath0 and set @xmath29 to be the divisor on @xmath30 corresponding to the conductor ideal , _ i.e. _ @xmath47 . then @xmath0 is @xmath5-pure if and only if @xmath48 is @xmath5-pure . since @xmath0 is @xmath64-gorenstein , by working sufficiently locally , we may assume that the zero divisor on @xmath0 corresponds to a map @xmath382 . therefore lemma [ lem.divofextensionsplusconductor ] implies that @xmath383 corresponds to the divisor @xmath29 . an application of theorem [ thm : main ] completes the proof . if @xmath12 is a curve singularity with a node at @xmath207 , then @xmath30 is smooth and the conductor ideal is simply the ideal of @xmath229 . in particular , if one takes a @xmath384-module generator @xmath385 and extends it to a map @xmath386 , the divisor @xmath147 is zero while the divisor @xmath354 is the divisor of the origin with coefficient @xmath15 . however , now suppose that a curve @xmath12 has a cusp singularity at @xmath207 . note @xmath30 is still smooth and fix @xmath387 to be the preimage of @xmath229 . the conductor ideal is the _ square _ of the ideal of the point of @xmath388 . in particular , if one takes a @xmath384-module generator @xmath385 and extends it to a map @xmath386 , the divisor @xmath147 is zero while the divisor @xmath354 is the divisor of the origin with coefficient @xmath69 . as before , consider a local non - normal s2 affine reduced scheme @xmath12 and the natural map @xmath21 on the normalization . write @xmath230 and let @xmath28 be the subscheme associated to the conductor in @xmath0 and @xmath29 the divisor associated to the conductor in @xmath30 , _ i.e. _ @xmath47 . [ cor.fpurevsslc ] assume conjecture [ conj.fpurevslogcanonical ] holds . suppose that @xmath0 is a seminormal , s2 and g1 pair of finite type over an algebraically closed field @xmath246 of characteristic zero and @xmath107 is such that @xmath389 is @xmath35-cartier . the pair @xmath20 has semi - log canonical singularities if and only if it has dense @xmath5-pure type . we now refer the reader to both @xcite and @xcite for a detailed description of the reduction to characteristic @xmath344 process in this context . as before , we also acknowledge the following abuse of notation , by @xmath344 we technically are referring to an open and zariski dense set of maximal ideals in @xmath345 , a finitely generated @xmath65-algebra used in the reduction to characteristic @xmath344 process . again , see the aforementioned references for more details . since the s2 property can be detected by examining the support of finitely many @xmath390 modules , the reductions to characteristic @xmath344 are also s2 . likewise because @xmath0 is g1 , there is a subset @xmath391 of codimension greater than @xmath15 , such that @xmath392 is gorenstein . thus the same can be preserved after reduction to characteristic @xmath344 . finally , since @xmath0 is seminormal , so are its reductions to characteristic @xmath344 , to see this use ( * ? ? ? * corollary 2.7(vii ) ) . set @xmath21 to be the normalization and fix @xmath271 and @xmath39 to be the subschemes defined by the conductor , respectively . we can reduce these schemes and subschemes to characteristic @xmath344 as well . assume that @xmath20 is semi - log canonical which implies that @xmath393 is log canonical . for a zariski - dense set of characteristic @xmath344 , by assumption we have that @xmath394 is @xmath5-pure . by working on sufficiently small affine charts , we also assume that @xmath395 and the same holds for @xmath396 in characteristic @xmath344 . in fact , we may even represent @xmath397 and assume that it is @xmath64-cartier ( or @xmath64-linearly equivalent to zero ) . it may also be helpful to the reader to notice that , under either hypothesis , @xmath398 has no common components with @xmath29 and so pathologies discussed in remark [ rem.pathologiesfordivisorsonnonnormal ] can be avoided by viewing @xmath19 as an element of @xmath399 , the @xmath40-weil divisorial sheaves are the @xmath400 which equal @xmath13 along the non - normal locus of @xmath0 . associated @xmath40-divisors may be treated more like @xmath40-divisors on normal varieties ( in particular , the subgroup of weil divisorial sheaves has no torsion ) . ] , see @xcite . for @xmath344 , we may assume that the index of @xmath396 is not divisible by the characteristic @xmath7 . thus @xmath401 induces a map @xmath402 , which extends to @xmath403 . the divisor associated to @xmath26 is thus @xmath404 and so @xmath26 is surjective by lemma [ lem.divofextensionsplusconductor ] . now @xmath405 has for @xmath344 , and so @xmath4 is surjective for a zariski - dense set of @xmath344 . this proves the ( @xmath406 ) direction . conversely , suppose that @xmath20 has dense @xmath5-pure type . but again by lemma [ lem.divofextensionsplusconductor ] this implies that @xmath393 also has dense @xmath5-pure type , which implies that @xmath393 is log canonical by @xcite and so @xmath20 is semi - log canonical by definition . we first review the inversion of adjunction statement we are concerned with . fix a pair @xmath62 where @xmath0 is a normal scheme , @xmath55 a reduced integral weil divisor and @xmath19 an effective @xmath35-divisor , with no common components with @xmath55 , such that @xmath407 is @xmath40-cartier . set @xmath408 to be the normalization of @xmath55 and recall that there is a canonically defined divisor @xmath59 called the _ different of @xmath19 on @xmath57 _ which satisfies @xmath409 . in general , adjunction and inversion of adjunction is the comparison of the singularities of a pair @xmath410 with the singularities of @xmath63 . the implication `` @xmath62 is log canonical @xmath406 @xmath411 is log canonical '' is called the _ adjunction direction_. the converse implication ( at least near @xmath55 ) is known as _ inversion of adjunction _ ; see @xcite , ( * ? ? ? * chapter 17 ) . the direct analog of the adjunction direction is known in characteristic @xmath7 ( * ? ? ? 8.2(iv ) ) . in particular , in characteristic @xmath7 , if additionally the index of @xmath407 is not divisible by @xmath7 then there exists a canonically determined divisor @xmath59 , called the _ @xmath5-different _ , such that @xmath412 and furthermore if @xmath62 is @xmath5-pure , then so is @xmath63 . the @xmath5-different @xmath59 is constructed as follows : we work locally and so may assume that @xmath12 is the spectrum of a local ring . by hypothesis , there exists a map @xmath413 corresponding to @xmath414 . the ideal @xmath415 is @xmath416-compatible ( see for example , ( * ? ? ? * section 7 ) ) , and so there is an induced map @xmath417 . now , @xmath55 is not necessarily normal ( or s2 or g1 ) so it is difficult interpret @xmath4 as a divisor . however , by lemma [ lem : extendingmapsundernormalization ] , @xmath4 extends to a map @xmath418 . finally , we associate to this map the divisor @xmath419 . in ( * ? ? * example 8.4 ) an example is produced where @xmath62 is not @xmath5-pure ( _ i.e. _ @xmath416 is not surjective ) but @xmath63 is @xmath5-pure ( _ i.e. _ @xmath26 is surjective ) . in other words , inversion of adjunction fails . this counterexample is loaded with exactly the pathologies that are avoided by rings having and the following corollary is easy to prove . [ cor.invofadj ] suppose that @xmath0 is a normal scheme of characteristic @xmath7 , @xmath19 is a @xmath35-divisor on @xmath0 and @xmath55 is a reduced integral weil divisor on @xmath0 , with no common components with @xmath19 , such that @xmath56 is @xmath35-cartier with index not divisible by @xmath46 . denote by @xmath57 the normalization of @xmath55 and @xmath58 the natural map . there exists a canonically determined @xmath35-divisor @xmath59 on @xmath57 such that @xmath420 . furthermore , if @xmath55 has , then @xmath62 is @xmath5-pure near @xmath55 if and only if @xmath63 is @xmath5-pure . we need only prove the final statement . using the notation above , it follows from lemma [ lem : local ] that @xmath416 is surjective if and only if @xmath4 is surjective . by our main theorem , @xmath4 is surjective if and only if @xmath26 is surjective because @xmath55 has . : _ generalized divisors on gorenstein schemes _ , proceedings of conference on algebraic geometry and ring theory in honor of michael artin , part iii ( antwerp , 1992 ) , vol . 8 , 1994 , pp . 287339 . mr1291023 ( 95k:14008 ) : _ flips and abundance for algebraic threefolds _ , socit mathmatique de france , paris , 1992 , papers from the second summer seminar on algebraic geometry held at the university of utah , salt lake city , utah , august 1991 , astrisque no . 211 mr1225842 ( 94f:14013 ) : _ positivity in algebraic geometry . ii _ , ergebnisse der mathematik und ihrer grenzgebiete . 3 . folge . a series of modern surveys in mathematics [ results in mathematics and related areas . 3rd series . a series of modern surveys in mathematics ] , vol . 49 , springer - verlag , berlin , 2004 , positivity for vector bundles , and multiplier ideals . mr2095472 ( 2005k:14001b ) : _ gluing schemes and a scheme without closed points _ , proceedings of the 2002 john h. barrett memorial lectures conference on algebraic and arithmetic geometry ( p. t. y. kachi , s. mulay , ed . ) , contemporary mathematics , vol . 386 , american mathematical society , providence , ri , 2005 , pp . 157172
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if @xmath0 is frobenius split , then so is its normalization and we explore conditions which imply the converse . to do this , we recall that given an @xmath1-linear map @xmath2 , it always extends to a map @xmath3 on the normalization of @xmath0 . in this paper
, we study when the surjectivity of @xmath3 implies the surjectivity of @xmath4 . while this does nt occur generally , we show it always happens if certain tameness conditions are satisfied for the normalization map .
our result has geometric consequences including a connection between @xmath5-pure singularities and semi - log canonical singularities , and a more familiar version of the ( @xmath5-)inversion of adjunction formula .
frobenius map , frobenius splitting , f - purity , semi - log canonical , log canonical , inversion of adjunction , normalization , seminormal
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a @xmath0 is a finite set @xmath1 paired with a finite set @xmath2 of unordered pairs @xmath3 with @xmath4 . a simple graph has no multiple connections and no self - loops : every @xmath5 appears only once and no @xmath6 is in @xmath2 . elements in @xmath1 are called , elements in @xmath2 are called . given a simple graph @xmath0 , denote by @xmath7 the of a vertex @xmath8 . it is a subgraph generated by the set of vertices directly connected to @xmath8 . denote by @xmath9 the set of complete @xmath10 subgraphs of @xmath11 . elements in @xmath9 are also called . the set @xmath12 for example is the set of all triangles in @xmath11 . of course , @xmath13 and @xmath14 . if the cardinality of @xmath9 is denoted by @xmath15 , the of @xmath11 is defined as @xmath16 , a finite sum . for example , if no tetrahedral subgraphs @xmath17 exist in @xmath11 , then @xmath18 , where @xmath19 is the , the number of vertices , @xmath20 is the , the number of edges and @xmath21 the number of @xmath22 . is defined inductively as @xmath23 with @xmath24 . cyclic graphs , trees or the dodecahedron are examples of graphs of dimension @xmath25 , a triangle @xmath22 , an octahedron or icosahedron has dimension @xmath26 . a tetrahedron has dimension @xmath27 . a complete graph @xmath10 on @xmath28 vertices has dimension @xmath29 . dimension is defined for any graph but can become a fraction . for a truncated cube @xmath11 for example , each unit sphere @xmath7 is a graph of @xmath27 vertices and one edge , a graph of dimension @xmath30 so that @xmath31 . the euler characteristic of this graph @xmath11 is @xmath32 . a is a function on @xmath9 which is antisymmetric in its @xmath33 arguments . the set @xmath34 of all @xmath29-forms is a vector space of dimension @xmath15 . the remaining sign ambiguity can be fixed by introducing an orientation on the graph : a @xmath10 subgraph is called a if it is not contained in a larger @xmath35 graph . an attaches a @xmath29-form @xmath36 to each maximal simplex with value @xmath25 . it induces forms on smaller dimensional faces . if @xmath36 cancels on intersections of maximal graphs , it is a `` volume form '' and @xmath11 is called . an icosahedron for example has triangles as maximal simplices . it is orientable . a wheel graph @xmath37 in which two opposite edges are identified models a mbius strip and is not orientable . a @xmath38-form is a function on @xmath39 and also called a . call @xmath40 the . it is defined as a @xmath25-form if @xmath11 has an orientation . without an orientation , we can still look at the @xmath41 if @xmath42 is an edge attached to @xmath43 . define the @xmath44 and the @xmath45 . a vertex @xmath8 is a if @xmath46 . if @xmath47 and @xmath11 is @xmath29-dimensional , a vertex @xmath8 is an if @xmath7 is a @xmath48-dimensional graph for which every point is an interior point within @xmath7 ; for @xmath49 we ask @xmath7 to be connected . the base induction assumption is that an interior point of a one - dimensional graph has two neighbors . a vertex @xmath8 of a @xmath29-dimensional graph @xmath11 is a if @xmath7 is a @xmath48-dimensional ( for @xmath49 connected ) graph in which every vertex is either a boundary or interior point and both are not empty . the seed assumption is that for @xmath50 , the graph @xmath7 has one vertex . a @xmath29-dimensional graph @xmath51 is a @xmath52 if every @xmath53 is an interior point or a boundary point . glue two copies of @xmath51 along the boundary gives a graph @xmath11 without boundary . a wheel graph @xmath54 is an example of a @xmath26-dimensional graph with boundary if @xmath55 . the boundary is the cyclic one dimensional graph @xmath56 . cut an octahedron in two gives @xmath57 . for an oriented graph @xmath11 , the @xmath58 is defined as @xmath59 , where @xmath60 denotes a variable taken away . for example @xmath61 is a function on triangles called the of a @xmath25-form @xmath62 . a form is if @xmath63 . it is if @xmath64 . the vector space @xmath65 of closed forms modulo exact forms is a of dimension @xmath66 , the . example : @xmath67 is the number of . the is @xmath68 . for a @xmath29-form define the @xmath69 . let @xmath70 be the number of @xmath10 subgraphs of @xmath71 . especially , @xmath72 is the @xmath73 of @xmath8 , the order of @xmath7 . the local quantity @xmath74 is called the of the graph at @xmath8 . the sum is of course finite . for a @xmath26-dimensional graph without boundary , where @xmath7 has the same order and size , it is @xmath75 . for a 1-dimensional graph with or without boundary and trees in particular , @xmath76 . for an arbitrary finite simple graph we have @xcite for an arbitrary finite simple graph and injective @xmath77 , we have @xcite assume @xmath62 is a @xmath48-form and @xmath11 is an oriented @xmath29-dimensional graph with boundary , then with boundary @xmath52 , the later remains a graph . in general it is only a , an element in the group of integer valued functions on @xmath78 usually written as @xmath79 . ] the * transfer equations * are . by definition of curvature , we have @xmath80 since the sums are finite , we can change the order of summation . using the transfer equations we get @xmath81 the number of @xmath29 simplices @xmath82 in the exit set @xmath83 and the number of @xmath29 simplices @xmath84 in the entrance set @xmath85 are complemented within @xmath7 by the number @xmath86 of @xmath29 simplices which contain both vertices from @xmath83 and @xmath85 . by definition , @xmath87 . the index @xmath88 is the same for all injective functions @xmath77 . the * intermediate equations * are . let @xmath89 . because replacing @xmath62 and @xmath90 switches @xmath91 with @xmath92 and the sum is the same , we can prove @xmath93 instead . the transfer equations and intermediate equations give @xmath94 = 2v_0 + \sum_{k=0}^{\infty } ( -1)^k 2 v_{k+1 } = 2 \chi(g ) \ ; .\end{aligned}\ ] ] denote a @xmath29-simplex graph @xmath95 by @xmath96 . from @xmath97 and algebraic boundary @xmath98 = \sum_k ( -1)^k ( x_0 , ... , ,x_n))$ ] , stokes theorem is obvious for a single simplex : @xmath99 gluing @xmath29-dimensional simplices cancels boundary . a @xmath29-dimensional graph with boundary is a union of @xmath29-dimensional simplices identified along @xmath48- dimensional simplices . a @xmath29-dimensional oriented graph with boundary can be built by gluing cliques as long as the orientation @xmath29-form can be extended . we also used that the * boundary as a graph * agrees with the * algebraic boundary * if differently oriented boundary pieces cancel . here are families of graphs , where the curvature is indicated at every vertex : for the history of the classical stokes theorem , see @xcite . the history of topology @xcite . the collection @xcite contains in particular an article on the history of graph theory . a story about euler characteristic and polyhedra is told in @xcite . for an introduction to gauss - bonnet with historical pointers to early discrete approaches see @xcite . for poincar - hopf , the first volume of @xcite or @xcite . for morse theory and reeb s theorem @xcite . poincar proved the index theorem in chapter viii of @xcite . hopf extended it to arbitrary dimensions in @xcite . gauss - bonnet in higher dimensions was proven first independently by allendoerfer @xcite and fenchel @xcite for surfaces in euclidean space and extended jointly by allendoerfer and weil @xcite to closed riemannian manifolds . chern gave the first intrinsic proof in @xcite . y. tong m. desbrun , e. kanso . discrete differential forms for computational modeling . in j. sullivan g. ziegler a. bobenko , p. schroeder , editor , _ discrete differential geometry _ , oberwohlfach seminars , 2008 .
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by proving graph theoretical versions of green - stokes , gauss - bonnet and poincar - hopf , core ideas of undergraduate mathematics can be illustrated in a simple graph theoretical setting . in this pedagogical exposition
we present the main proofs on a single page and add illustrations . while discrete stokes is at least 100 years old , the other two results for graphs were found only recently .
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humans can effortlessly and rapidly recognize surrounding objects @xcite , despite the tremendous variations in the projection of each object on the retina @xcite caused by various transformations such as changes in object position , size , pose , illumination condition and background context @xcite . this invariant recognition is presumably handled through hierarchical processing in the so - called ventral pathway . such hierarchical processing starts in v1 layers , which extract simple features such as bars and edges in different orientations @xcite , continues in intermediate layers such as v2 and v4 , which are responsive to more complex features @xcite , and culminates in the inferior temporal cortex ( it ) , where the neurons are selective to object parts or whole objects @xcite . by moving from the lower layers to the higher layers , the feature complexity , receptive field size and transformation invariance increase , in such a way that the it neurons can invariantly represent the objects in a linearly separable manner @xcite . another amazing feature of the primates visual system is its high processing speed . the first wave of image - driven neuronal responses in it appears around 100 ms after the stimulus onset @xcite . recordings from monkey it cortex have demonstrated that the first spikes ( over a short time window of 12.5 ms ) , about 100 ms after the image presentation , carry accurate information about the nature of the visual stimulus @xcite . hence , ultra - rapid object recognition is presumably performed in a feedforward manner @xcite . moreover , although there exist various intra- and inter - area feedback connections in the visual cortex , some neurophysiological @xcite and theoretical @xcite studies have also suggested that the feedforward information is usually sufficient for invariant object categorization . appealed by the impressive speed and performance of the primates visual system , computer vision scientists have long tried to `` copy '' it . so far , it is mostly the architecture of the visual system that has been mimicked . for instance , using hierarchical feedforward networks with restricted receptive fields , like in the brain , has been proven useful @xcite . in comparison , the way that biological visual systems learn the appropriate features has attracted much less attention . all the above - mentioned approaches somehow use non biologically plausible learning rules . yet the ability of the visual cortex to wire itself , mostly in an unsupervised manner , is remarkable @xcite . here , we propose that adding bio - inspired learning to bio - inspired architectures could improve the models behavior . to this end , we focused on a particular form of synaptic plasticity known as spike timing - dependent plasticity ( stdp ) , which has been observed in the mamalian visual cortex @xcite . briefly , stdp reinforces the connections with afferents that significantly contributed to make a neuron fire , while it depresses the others @xcite . a recent psychophysical study provided some indirect evidence for this form of plasticity in the human visual cortex @xcite . in an earlier study @xcite , it is shown that a combination of a temporal coding scheme where in the entry layer of a spiking neural network the most strongly activated neurons fire first with stdp leads to a situation where neurons in higher visual areas will gradually become selective to complex visual features in an unsupervised manner . these features are both salient and consistently present in the inputs . furthermore , as learning progresses , the neurons responses rapidly accelerates . these responses can then be fed to a classifier to do a categorization task . in this study , we show that such an approach strongly outperforms state - of - the - art computer vision algorithms on view - invariant object recognition benchmark tasks including 3d - object @xcite and eth-80 @xcite datasets . these datasets contain natural and unsegmented images , where objects have large variations in scale , viewpoint , and tilt , which makes their recognition hard @xcite , and probably out of reach for most of the other bio - inspired models @xcite . yet our algorithm generalizes surprisingly well , even when `` simple classifiers '' are used , because stdp naturally extracts features that are class specific . this point was further confirmed using mutual information @xcite and representational dissimilarity matrix ( rdm ) @xcite . moreover , the distribution of objects in the obtained feature space was analyzed using hierarchical clustering @xcite , and objects of the same category tended to cluster together . the algorithm we used here is a scaled - up version of the one presented in @xcite . essentially , many more c2 features and iterations were used . our code is available upon request . we used a five - layer hierarchical network @xmath0 @xmath1 @xmath2 @xmath1 @xmath3 @xmath1 @xmath4 @xmath1 @xmath5 , largely inspired by the hmax model @xcite ( see fig . [ model_figure ] ) . specifically , we alternated simple cells that gain selectivity through a sum operation , and complex cells that gain shift and scale invariance through a max operation . however , our network uses spiking neurons and operates in the temporal domain : when presented with an image , the first layer s @xmath0 cells , detect oriented edges and the more strongly a cell is stimulated the earlier it fires . these @xmath0 spikes are then propagated asynchronously through the feedforward network . we only compute the first spike fired by each neuron ( if any ) , which leads to efficient implementations . the justification for this is that later spikes are probably not used in ultra - rapid visual categorization tasks in primates @xcite . we used restricted receptive fields and a weight sharing mechanism ( i.e. convolutional network ) . in our model , images are presented sequentially and the resulting spike waves are propagated through to the @xmath3 layer , where stdp is used to extract diagnostic features . more specifically , the first layer s @xmath0 cells detect bars and edges using gabor filters . here we used @xmath6 convolutional kernels corresponding to gabor filters with the wavelength of 5 and four different preferred orientations ( @xmath7 ) . these filters are applied to five scaled versions of the original ) . hence , for each scaled version of the input image we have four @xmath0 maps ( one for each orientation ) , and overall , there are 4@xmath85 @xmath9 20 maps of @xmath10 cells ( see the @xmath10 maps of fig . [ model_figure ] ) . evidently , the @xmath0 cells of larger scales detect edges with higher spatial frequencies while the smaller scales extract edges with lower spatial frequencies . indeed , instead of changing the size and spatial frequency of gabor filters , we are changing the size of input image . this is a way to implement scale invariance at a low computational cost . each @xmath10 cell emits a spike with a latency that is inversely proportional to the absolute value of the convolution . thus , the more strongly a cell is stimulated the earlier it fires ( intensity - to - latency conversion , as observed experimentally @xcite ) . to increase the sparsity at a given scale and location ( corresponding to one cortical column ) , only the spike corresponding to the best matching orientation is propagated ( i.e. a winner - take - all inhibition is employed ) . in other word , for each position in the four @xmath0 orientation maps of a given scale , the @xmath0 cell with highest convolution value emits a spike and prevents the other three @xmath0 cells from firing . for each @xmath10 map , there is a corresponding @xmath11 map . each @xmath11 cell propagates the first spike emitted by the @xmath10 cells in a @xmath12 square neighborhood of the @xmath10 map which corresponds to one specific orientation and one scale ( see the @xmath11 maps of fig . [ model_figure ] ) . @xmath11 cells thus execute a maximum operation over the @xmath10 cells with the same preferred feature across a portion of the visual field , which is a biologically plausible way to gain local shift invariance @xcite . the overlap between the afferents of two adjacent @xmath2 cells is just one @xmath0 row , hence a subsampling over the @xmath0 maps is done by the @xmath2 layers as well . therefore , each @xmath2 map has @xmath13 fewer cells than the corresponding @xmath0 map . @xmath14 features correspond to intermediate - complexity visual features which are optimum for object classification @xcite . each @xmath14 feature has a prototype @xmath14 cell ( specified by a @xmath11-@xmath14 synaptic weight matrix ) , which is a weighted combination of bars ( @xmath11 cells ) with different orientations in a @xmath15 square neighborhood . each prototype @xmath14 cell is retinotopically duplicated in the five scale maps ( i.e. weight - sharing is used ) . within those maps , the @xmath14 cells can integrate spikes only from the four @xmath11 maps of their corresponding processing scales . this way , a given @xmath14 feature is simultaneously explored in all positions and scales ( see @xmath14 maps of fig . [ model_figure ] with same feature prototype but in different processing scales specified by different colors ) . indeed , duplicated cells in all positions of all scale maps integrate the spike train in parallel and compete with each other . the first duplicate reaching its threshold , if any , is the winner . the winner fires and prevents the other duplicated cells in all other positions and scales from firing through a winner - take - all inhibition mechanism . then , for each prototype , the winner @xmath14 cell triggers the unsupervised stdp rule and its weight matrix is updated . the changes in its weights are applied over all other duplicate cells in different positions and scales ( weight sharing mechanism ) . this allows the system to learn frequent patterns , independently of their position and size in the training images . the learning process begins with @xmath14 features initialized by random numbers drawn from a normal distribution with mean @xmath16 and std @xmath17 , and the threshold of all @xmath14 cells is set to 64 ( @xmath18 ) . through the learning process , a local inhibition between different @xmath14 prototype cells is used to prevent the convergence of different @xmath14 prototypes to similar features : when a cell fires at a given position and scale , it prevents all the other cells ( independently of their preferred prototype ) from firing later at the same scale and within a neighborhood around the firing position . thus , the cell population self - organizes , each cell trying to learn a distinct pattern so as to cover the whole variability of the inputs . moreover , we applied a k - winner - take - all strategy in @xmath14 layer to ensure that at most two cells can fire for each processing scale . this mechanism , only used in the learning phase , helps the cells to learn patterns with different real sizes . without it , there is a natural bias toward small " patterns ( i.e. , large scales ) , simply because corresponding maps are larger , and so likeliness of firing with random weights at the beginning of the stdp process is higher . a simplified version of stdp is used to learn the @xmath19 weights as follows : @xmath20 where @xmath21 and @xmath22 respectively refer to the index of post- and presynaptic neurons , @xmath23 and @xmath24 are the corresponding spike times , @xmath25 is the synaptic weight modification , and @xmath26 and @xmath27 are two parameters specifying the learning rate . note that the exact time difference between two spikes ( @xmath28 ) does not affect the weight change , but only its sign is considered . these simplifications are equivalent to assuming that the intensity - to - latency conversion of @xmath10 cells compresses the whole spike wave in a relatively short time interval ( say , @xmath29 ms ) , so that all presynaptic spikes necessarily fall close to the postsynaptic spike time , and the time lags are negligible . the multiplicative term @xmath30 ensures the weights remain in the range [ 0,1 ] and maintains all synapses in an excitatory mode . the learning phase starts by @xmath31 which is multiplied by 2 after each 400 postsynaptic spikes up to a maximum value of @xmath32 . a fixed @xmath33 ratio ( -4/3 ) is used . this allows us to speed up the convergence of @xmath14 features as the learning progresses . initiation of the learning phase with high learning rates would lead to erratic results . for each @xmath14 prototype , a @xmath34 cell propagates the first spike emitted by the corresponding @xmath14 cells over all positions and processing scales , leading to the global shift- and scale - invariant cells ( see the @xmath34 layer of fig . [ model_figure ] ) . to study the robustness of our model with respect to different transformations such as scale and viewpoint , we evaluated it on the _ 3d - object _ and _ eth-80 _ datasets . the 3d - object is provided by savarese et al . at cvglab , stanford university @xcite . this dataset contains 10 different object classes : bicycle , car , cellphone , head , iron , monitor , mouse , shoe , stapler , and toaster . there are about 10 different instances for each object class . the object instances are photographed in about 72 different conditions : eight view angles , three distances ( scales ) , and three different tilts . the images are not segmented and the objects are located in different backgrounds ( the background changes even for different conditions of the same object instance ) . figure [ objects ] presents some examples of objects in this dataset . the eth-80 dataset includes 80 3d objects in eight different object categories including apple , car , toy cow , cup , toy dog , toy horse , pear , and tomato . each object is photographed in 41 viewpoints with different view angles and different tilts . figure s1 in supplementary information provides some examples of objects in this dataset from different viewpoints . for both datasets , five instances of each object category are selected for the training set to be used in the learning phase . the remaining instances constitute the testing set which is not seen during the learning phase , but is used afterward to evaluate the recognition performance . this standard cross - validation procedure allows to measure the generalization ability of the model beyond the specific training examples . note that for 3d - object dataset , the original size of all images were preserved , while the images of eth-80 dataset are resized to @xmath35 pixels in height while preserving the aspect ratio . the images of both datasets were converted to grayscale values . [ cols="^,^,^,^,^,^,^,^,^,^,^,^",options="header " , ] in an other experiment , we analyzed the class dependency of the @xmath4 features for our model . to this end , the 50 most informative features , when classifying a specific class against all the other classes , are selected by employing the mutual information technique . in other words , for each class , we selected those 50 features which have the highest activity for samples of that class and have less activity for other classes . afterwards , the number of common features among the informative features of each pair of classes are computed as provided in table [ table_example2 ] . on average , there are only about 5.4 common features between pairs of classes . although there are some common features between any two classes , their co - occurrence with the other features help the classifier to separate them from each other . in this way , our model can represent various object classes with a relatively small number of features . indeed , exploiting the intermediate complexity features , which are not common in all classes and are not very rare , can help the classifier to discriminate instances of different classes @xcite . in a previous study @xcite , it has been shown that using the hmax model with random dot patterns in the @xmath14 layer can reach a reasonable performance , comparable to the one obtained with random patches cropped from the training images . it seems that this is due to the dependency of hmax to the application of a powerful classifier . indeed , the use of both random dot or randomly selected patches transform the images into a complex and nested feature space and it is the classifier which looks for a complex signature to separate object classes . the deficiencies emerge when the classification problem gets harder ( such as invariant or multiclass object recognition problems ) and then even a powerful classifier is not able to discriminate the classes @xcite . here , we show that the superiority of our model is due to the informative feature extraction through a bio - inspired learning rule . to this end , we have compared the performances on 3d - object dataset obtained with random features versus stdp features , as well as a very simple classifier versus svm . to generate random features , we have set the weight matrix of each @xmath14 feature of our model with random values . first , we have computed the mean and standard deviation ( std ) ( @xmath36 ) of the number of active ( nonzero ) weights in the features learned by stdp . second , for each random feature , the number of active weights , @xmath37 , is computed by generating a random number based on the obtained mean and std . finally , a random feature is constructed by uniformly distributing the @xmath37 randomly generated values in the weight matrix . in addition , we designed a simple classifier comprised of several one - versus - one classifiers . for each binary classifier , two subset of @xmath34 features with high occurrence probabilities in one of the two classes are selected . in more details , to select suitable features for the first class , the occurrence probabilities of @xmath34 features in this class are divided by the corresponding occurrence probabilities in the second class . then , a feature is selected if this ratio is higher than a threshold . the optimum threshold value is computed by a trial and error search in which the performance over the training samples is maximized . to assign a class label to the input test sample , we performed an inner product on the feature value and feature probability vectors . finally , the class with the highest probability is reported to the combined classifier . the combined classifier selects the winner class based on a simple majority voting . for 500 random features , using the svm and the simple classifier , our model reached classification performances of 71% and 21% on average , respectively . whereas , for the learned @xmath14 features , both the svm and simple classifiers attained reasonable performances of 96% and 79% , respectively . based on these results , it can be concluded that the features obtained through the bio - inspired unsupervised learning projects the objects into an easily separable space , while the feature extraction by selection of random patches ( drawn from the training images ) or by generation of random patterns leads to a complex object representation . position and scale invariance in our model are built - in , thanks to weight sharing and scaling process . conversely , view - invariance must be obtained through the learning process . here , we used all images of five object instances from each category ( varied in all dimensions ) to learn the @xmath14 visual features , while images of all other object instances of each category were used to test the network . hence , the model was exposed to all possible variations during the learning to gain view - invariance . moreover , near or opposite views of the same object shares some features which are suitable for invariant object recognition . for instance , consider the overall shape of a head , or close views of a bike wheel which could be a complete circle or an ellipse . regarding the fact that stdp tends to learn more frequent features in different images , different views of an object could be invariantly represented based on more common features . our model appears to be the best choice when dealing with few object classes , but huge variations in view points . as pointed out in previous studies , both hmax and deepconvnet models could not handle these variations perfectly @xcite . conversely , our model is not appropriate to handle many classes , which requires thousands of features , like in the imagenet contest , because its time complexity is roughly in @xmath38 , where @xmath37 is the number of features ( briefly : since the number of firing neurons per image is limited , if the number of features is doubled , reaching convergence will take roughly twice as many images , and the processing time for each of them will be doubled as well ) . for example , extracting 4096 features in our model , the same number of features in deepconvnet , would take about 67 times it took us to extract 500 . however , parallel implementation of our algorithm could speed - up the computation time by several orders of magnitude @xcite . even in this case , we do not expect to outperform the deepconvnet model on the imagenet database , since only the shape similarities are taken into account in our model and the other cues such as color or texture are ignored . importantly , our algorithm has a natural tendency to learn salient contrasted regions @xcite , which is desirable as these are typically the most informative @xcite . most of our @xmath4 features turned out to be class - specific , and we could guess what they represent by doing the reconstructions ( see fig . [ features ] and fig . since each feature results from averaging multiple input images , the specificity of each instance is averaged out , leading to class archetypes . consequently , good classification results can be obtained using only a few features , or even using ` simple ' decision rules like feature counts @xcite and majority voting ( here ) , as opposed to a ` smart classifier ' such as svm . there are some similarities between stdp - based feature learning , and non - negative matrix factorization @xcite , as first intuited in @xcite , and later demonstrated mathematically in @xcite . within both approaches , objects are represented as ( positive ) sums of their parts , and the parts are learned by detecting consistently co - active input units . our model could be efficiently implemented in hardware , for example using address event representation ( aer ) @xcite . with aer , the spikes are carried as addresses of sending or receiving neurons on a digital bus . time ` represents itself ' as the asynchronous occurrence of the event @xcite . thus the use of stdp will lead to a system which effectively becomes more and more reactive , in addition to becoming more and more selective . furthermore , since biological hardware is known to be incredibly slow , simulations could run several order of magnitude faster than real time @xcite . as mentioned earlier , the primate visual system extracts the rough content of an image in about 100ms . we thus speculate that some dedicated hardware will be able to do the same in the order of a millisecond or less . recent computational @xcite , psychophysical @xcite , and fmri @xcite experiments demonstrate that the informative intermediate complexity features are optimal for object categorization tasks . but the possible neural mechanisms to extract such features remain largely unknown . the hmax model ignores these learning mechanisms and imprints its features with random crops from the training images @xcite , or even uses random filters @xcite . most individual features are thus not very informative , yet in some cases , a ` smart ' classifier such as svm can efficiently separate the high - dimensional vectors of population responses . many other models use supervised learning rules @xcite , sometimes reaching impressive performance on natural image classification tasks @xcite . the main drawback of these supervised methods , however , is that learning is slow and requires numerous labeled samples ( e.g. , about 1 million in @xcite ) , because of the credit assignment problem @xcite . this contrasts with humans who can generalize efficiently from just a few training examples @xcite . we avoid the credit assignment problem by keeping the @xmath4 features fixed when training the final classifier ( that being said , fine - tuning them for a given classification problem would probably increase the performance of our model @xcite ; we will test this in future studies ) . even if the efficiency of such hybrid unsupervised - supervised learning schemes has been known for a long time , few alternative unsupervised learning algorithms have been shown to be able to extract complex and high - level visual features ( see @xcite ) . finding better representational learning algorithms is thus an important direction for future research and seeking for inspiration in the biological visual systems is likely to be fruitful @xcite . we suggest here that the physiological mechanism known as stdp is an appealing start point . considering the time relation among the incoming inputs is an important aspect of spiking neural networks . this property is critical to promote the existing models from static vision to continuous vision @xcite . a prominent example is the trace learning rule @xcite , suggesting that the invariant object representation in ventral visual system is instructed by the implicit temporal contiguity of vision . also , in various motion processing and action recognition problems @xcite , the important information lies in the appearance timing of input features . our model has this potential to be extended for continuous and dynamic vision something that we will further explore . to date , various bio - inspired network architectures for object recognition have been introduced , but the learning mechanism of biological visual systems has been neglected . in this paper , we demonstrate that the association of both bio - inspired network architecture and learning rule results in a robust object recognition system . the stdp - based feature learning , used in our model , extracts frequent diagnostic and class specific features that are robust to deformations in stimulus appearance . it has previously been shown that the trivial models can not tolerate the identity preserving transformations such as changes in view , scale , and position . to study the behavior of our model confronted with these difficulties , we have tested our model over two challenging invariant object recognition databases which includes instances of 10 different object classes photographed in different views , scales , and tilts . the categorization performances indicate that our model is able to robustly recognize objects in such a severe situation . in addition , several analytical techniques have been employed to prove that the main contribution to this success is provided by the unsupervised stdp feature learning , not by the classifier . using representational dissimilarity matrix , we have shown that the representation of input images in @xmath34 layer are more similar for within - 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based orientation selectivity in the early visual system in natural continuous and saccadic vision : a computational model . , journal of computational neuroscience 32 ( 3 ) ( 2012 ) 42541 . http://dx.doi.org/10.1007/s10827-011-0361-9 [ ] . p. fldik , learning invariance from transformation sequences , neural computation 3 ( 1991 ) 194200 . escobar , g. s. masson , t. vieville , p. kornprobst , action recognition using a bio - inspired feedforward spiking network , international journal of computer vision 82 ( 3 ) ( 2009 ) 284301 . here we provide the results of feature analysis techniques such as rdm and hierarchical clustering on eth-80 dataset for both hmax and our model . some sample images of eth-80 dataset are shown in fig . [ eth_samples ] . in fig . [ rdms_supliment_tims ] and fig . [ rdms_supliment_hmax ] the rdms of @xmath39 features of our model and hmax in eight view angels are presented , respectively . it can be seen that our model can better represent classes with high shape similarities such as tomato , apple , and pear or cow , horse , and dog with respect to the hmax model . also , the hierarchical clustering of whole training data based on their representations on feature spaces of our model and hmax are demonstrated in fig . [ cluster_tim_eth80 ] and fig.[cluster_hmax_eth80 ] , respectively . as for the 3d - object dataset , hmax feature extraction leads to a nested representation of different object classes which causes a poor classification accuracy . here again a huge number of images which belong to different classes are assigned to a large cluster with lower than 0.14 internal dissimilarities . on the other hand , our model has distributed images of different classes in different regions of @xmath39 feature space . note that the largest cluster of our model includes the instances of tomato , apple , and pear classes which their shapes are so similar .
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retinal image of surrounding objects varies tremendously due to the changes in position , size , pose , illumination condition , background context , occlusion , noise , and nonrigid deformations . but
despite these huge variations , our visual system is able to invariantly recognize any object in just a fraction of a second . to date , various computational models have been proposed to mimic the hierarchical processing of the ventral visual pathway , with limited success . here , we show that the association of both biologically inspired network architecture and learning rule significantly improves the models performance when facing challenging invariant object recognition problems .
our model is an asynchronous feedforward spiking neural network .
when the network is presented with natural images , the neurons in the entry layers detect edges , and the most activated ones fire first , while neurons in higher layers are equipped with spike timing - dependent plasticity .
these neurons progressively become selective to intermediate complexity visual features appropriate for object categorization .
the model is evaluated on _ 3d - object _ and _ eth-80 _ datasets which are two benchmarks for invariant object recognition , and is shown to outperform state - of - the - art models , including deepconvnet and hmax .
this demonstrates its ability to accurately recognize different instances of multiple object classes even under various appearance conditions ( different views , scales , tilts , and backgrounds ) .
several statistical analysis techniques are used to show that our model extracts class specific and highly informative features .
* keywords : * view - invariant object recognition , visual cortex , stdp , spiking neurons , temporal coding
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You are an expert at summarizing long articles. Proceed to summarize the following text:
recent development in the cooling and trapping technique of atoms and the following success in achieving the bose - einstein condensation of alkali atoms@xcite have now opened up a new era in the progress in quantum physics . while the condensed system serves as a testing ground for a research in fundamental problems of quantum mechanics , it also offers a new example of finite quantum many - body systems such as hadrons , nuclei and microclusters . an important characteristic of the trapped alkali atoms is that it is a dilute system of weakly interacting particles and an ideal place to test genuine properties of condensed systems predicted by theories . along with a further progress in the study of bose - einstein condensed systems , a similar technique is being extended to create a degenerate gas of fermionic atoms , where a number of theoretical studies have been made@xcite . to realize such a degenerate fermionic system that requires a still lower temperature than bose systems , the technique of sympathetic cooling has been investigated@xcite : the cooling mechanism through the collisions with coexisting cold bose particles in a polarized boson - fermion mixture , where the fermion - fermion interaction is less effective . the mixture of bose and fermi particles is itself an interesting system for a study : the hydrogen - deuterium system has been studied already at the early stage of these investigations@xcite . recently , theoretical studies have been made for the trapped boson - fermion system of alkali atoms and proposed many interesting properties : the exotic density configurations through the repulsive or attractive boson - fermion interactions @xcite , the decaying processes after the removal of the confining trap @xcite , and the metastability of the @xmath1li-@xmath2li mixture @xcite . in this paper , we study static properties of a mixed bose - fermi system of trapped potassium atoms , where one fermionic ( @xmath3k ) and two bosonic isotopes ( @xmath4k and @xmath5k ) are known as candidates for a realization of such a degenerate quantum system . trapping of the fermionic isotope @xmath3k has already been reported@xcite . these isotopes are also of an interest since the boson - fermion interaction may be repulsive or attractive depending on their choice@xcite . below we first describe a set of equations for the mixed system at @xmath6 , where the bosonic part is given by a gross - pitaevskii ( gp ) equation and including boson - fermion interaction , and the fermionic part is by the thomas - fermi ( tf ) equation . the use of the gp equation allows one to study a system with a net attractive potential for bosons where the thomas - fermi - type ( tf ) approximation is not applicable . the applicability of these equations for the trapped potassium system was discussed in @xcite . in sec . iii , we discuss qualitative properties of solutions from the gp and tf equations analytically ; especially , their dependence on the boson / fermion number is estimated . in ref . @xcite , general features of solution including a deformed system is discussed in the repulsive boson - fermion interaction . on the other hand , to study more realistic system , we concentrate on the specific systems of potassium isotopes , i.e.,@xmath4k-@xmath3k system and @xmath5k-@xmath3k system using empirically derived atomic interactions . we also consider the effect of changing parameters of the trapped potential . finally , we mention the possibility of trapping fermions only through the attractive interactions by the trapped bose particles . we consider a spin - polarized system of bosons and fermions at @xmath6 described by a hamiltonian @xmath7 : @xmath8 where @xmath9 and @xmath10 respectively denote boson and fermion field operators with masses @xmath11 and @xmath12 . because of diluteness , the particle interactions have been approximated by the @xmath13-wave - dominated contact potential @xmath14 . the strengths of the coupling constants @xmath15 ( boson - boson ) and @xmath16 ( boson - fermion ) are given by @xmath17 where @xmath18 and @xmath19 are the @xmath13-wave scattering lengths for the boson - boson and boson - fermion scatterings and @xmath20 is a reduced mass . the fermion - fermion interaction has been neglected because of diluteness and particle polarization . the chemical potentials @xmath21 and @xmath22 in ( [ eqha ] ) and ( [ eqhb ] ) are determined from the condensed boson / fermion numbers @xmath23 and @xmath24 through the ground state expectation values : @xmath25 in the mean - field approximation at @xmath6 , the bosons occupy the lowest single - particle state @xmath26 , and the energy of the system is just given by a replacement of field operator @xmath9 in the hamiltonian with its expectation value : @xmath27 that describes the order parameter of the bose - einstein condensate . ( we here neglect quantities of the order @xmath28 . ) in this approximation , the boson density @xmath29 is given by @xmath30 . in the same approximation , the fermion wave function is given by a slater determinant ; the single - particle states are determined , e.g. , by the hartree - fock self - consistent equation . in the present approximation of neglecting fermion - fermion interactions , we only have to solve for the single particle states under the effective potential for fermions : the trapping potential and the boson - fermion interaction . in the actual calculation , we however adopt a semiclassical ( tf ) description for fermion density , which is known to provide a good approximation as far as the number of atoms is sufficiently large@xcite . we thus obtain a set of equations for @xmath31 and @xmath32 @xcite : @xmath33 \phi(\vec{r } ) = \mu_b \phi(\vec{r } ) , \label{gpeq}\\ % & & \frac{\hbar^2}{2 m_f } [ 6 \pi^2 n_f(\vec{r})]^{2/3 } + \frac{1}{2 } m_f \omega_f^2 r^2 + h\ , n_b(\vec{r } ) = \mu_f , \label{fermieq}\end{aligned}\ ] ] where the boson and fermion densities are defined by @xmath34 and @xmath35 . before the study of numerical results for ( [ gpeq ] ) and ( [ fermieq ] ) , we discuss about their solutions qualitatively . \i ) _ radius of the boson / fermion distribution _ if we neglect the boson - fermion interaction , we would obtain a much broader distribution for fermions as compared with that for bosons because of the exclusion principle . to see this , we consider a free boson / fermion system trapped in a harmonic oscillator potential . the mean square radius of each system is obtained from the virial theorem : @xmath36 where the harmonic - oscillator lengths @xmath37 are @xmath38 and @xmath39 are the total boson / fermion numbers . the @xmath40 in ( [ eqd ] ) is the number of fermions in the last occupied shell that is obtained as a solution of @xmath41 . the ratio of the root - mean - square ( rms ) radii @xmath42 thus increases as the power @xmath43 when @xmath44 and @xmath45 . in fact , for a repulsive boson - boson interaction , the boson distribution become broader than that for a free system just discussed , but this effect does not change the qualitative structure as shown in sec . v. on the other hand , the strong boson - fermion interaction may give qualitative changes of the density profiles such as the phase separation and the fermion collapse as shown in @xcite . we also discussed about them from different points of view in iii ) in this section . ii)_scaled gp and tf equations and changes of the density distribution for the different @xmath39 _ for qualitative estimations of the density distribution , we should use the scaled dimensionless variables @xcite : @xmath46 where @xmath39 , @xmath37 are the total boson / fermion numbers and the harmonic - oscillator lengths defined in eqs . ( [ eqh ] ) and ( [ eqc ] ) . using these scaled variables , the gp and tf equations in eqs . ( [ gpeq ] ) and ( [ fermieq ] ) become @xmath47 \tilde{\varphi_b}(\vec{x } ) = \tilde{\mu}_b \tilde{\varphi_b}(\vec{x } ) \label{scgpeq}\\ % & & \frac{1}{r_m } [ 6 \pi^2 n_{f } \rho_f(\vec{x})]^{\frac{2}{3 } } + r_m r_\omega^2 x^2 + \tilde{h}\ , n_b\ , \rho_b(\vec{x } ) = \tilde{\mu}_f , \label{scfermieq}\end{aligned}\ ] ] where @xmath48 and @xmath49 . the scaled coupling constants @xmath50 and @xmath51 in eqs . ( [ scgpeq ] ) and ( [ scfermieq ] ) are defined by @xmath52 where eq . ( [ eqe ] ) has been used to obtain the @xmath53-representations , and the last factor comes from @xmath54 . from eqs . ( [ scgpeq ] ) and ( [ scfermieq ] ) , we should notice that the scaled equations include six independent parameters , ( @xmath55 , @xmath56 , @xmath24 , @xmath57 , @xmath58 , @xmath59 ) ; for two systems where these parameters take the same values , their physical properties become similar under the scaling relations ( [ eqb ] ) . here , based on eqs . ( [ scgpeq ] ) and ( [ scfermieq ] ) , we discuss about changes of the density distribution by the scaled parameters . in this subsection , we assume @xmath60 and the large particle numbers @xmath61 ; for the effect of @xmath62 , the discussion will be given in sec . v. let us consider the fermion tf equation ( [ scfermieq ] ) , first . for the chemical potential @xmath63 , we use the result in the system without boson - fermion interaction : @xmath64 . it should be a good approximation when @xmath65 . using it in eq . ( [ scgpeq ] ) , we obtain @xmath66 at the central region ( @xmath67 ) . on the other hand , the boson equation ( [ scgpeq ] ) gives @xmath68 under the tf approximation valid in @xmath69 . using these estimates , we can obtain the ratio @xmath0 of the boson - boson / boson - fermion interaction effects in the gp equation ( [ scgpeq ] ) : @xmath70 the small @xmath0 ( @xmath71 ) indicates the small contribution from the boson - fermion interaction for the boson part , and we obtain @xmath72 . however , when @xmath73 , the boson - fermion interaction is not negligible , and the boson density profile can be quite different from that in non - interacting case ( @xmath74 ) . in @xmath4k-@xmath3k system discussed in sec . v , the scaled parameters are @xmath75 in @xmath76 , @xmath55 and @xmath56 are in the same order , but @xmath24 is a factor of 2 larger than @xmath56 , so eq . ( [ eqi ] ) gives @xmath77 . it explains the rather small change of the boson density profile @xmath72 in @xmath4k-@xmath3k system , as shown in the numerical results in sec . v. it is very interesting to apply the above estimation for the results by @xcite and compare them with the results in this paper . one of the parameter set with which the boson density profiles changed drastically is @xmath78 where @xmath79 has been taken and the qualitative changes of the boson density profile have been seen @xmath80 @xcite . it should be noticed that the parameters in ( [ eqf ] ) are in the same order when @xmath81 , and it means @xmath82 in that case . for this reason , the boson density profile changed drastically from that of @xmath83 with these parameters . to check the relation between the boson profile change and the ratio @xmath0 in eq . ( [ eqi ] ) a little more , we solved eqs . ( [ scgpeq ] ) and ( [ scfermieq ] ) numerically for a ) @xmath84 and b ) @xmath85 with other parameters fixed as in ( [ eqf ] ) , and , for @xmath86 , two values were taken @xmath87 . the resultant boson densities are shown in fig . 1 : @xmath88 has very different profile from @xmath89 in case a ) , but the very small change is found between them in case b ) . those results shows the clear relation between @xmath0 and the boson density profile . in summary , the parameter regions exist where the boson density profile changes very large or not , and these regions are discriminated by the value of @xmath0 . we should comment that the results in this paper are complementary with those with ( [ eqf ] ) in @xcite in the scale of the ratio @xmath0 . \iii ) _ boson - fermion interaction effects for the fermion distribution function _ we next consider the effect of the boson - fermion interaction on the fermion distribution , especially the phase separation phenomena in the strongly repulsive boson - fermion interaction case , which has been proposed in @xcite . here we assume the tf approximation for bosons ( neglecting the kinetic term in eq.([fermieq ] ) ) . as discussed in the literature @xcite , the effective potential for fermions receives an additional repulsive or attractive contribution from bosons within the range @xmath90 depending on the sign of the interaction . for a strongly repulsive boson - fermion interaction ( @xmath91 ) , the fermion will be squeezed out from the center , so that the two kind of particles tend to make a separate phase @xcite . here we give analytical estimation for this interesting phenomena . as shown in @xcite , in the phase - separated system , the fermion are almost completely pushed away outside the boson distribution , so that the critical ratio @xmath92 for the phase separation can be estimated from the vanishing fermion density at the center @xmath93 . neglecting the boson - fermion interaction in the boson tf equation , we obtain @xmath94 where @xmath95 . in fig . 2 , the function @xmath96 in ( [ eqa ] ) is plotted when @xmath97 with the parameters used in @xcite : ( @xmath98 , @xmath99 , @xmath100 , @xmath101 , @xmath102 , @xmath103 ) = ( @xmath104 , @xmath105 , @xmath106 , @xmath105 , @xmath107 , @xmath107 ) @xcite . from the crossing point of @xmath108 and @xmath97 ( dashed line ) in fig . 2 , the critical point can be read off : @xmath109 , which is almost consistent with the result given in @xcite ( @xmath110 ) . for an attractive interaction ( @xmath111 ) , fermions tend to concentrate and increase the overlap with bosons . thus we may expect a coherence of the two kinds of particles to occur in this case . for large attraction , another interesting phenomena , fermion collapse in mixture has been proposed @xcite . in the numerical calculation , we solved the set of equations ( [ gpeq ] ) and ( [ fermieq ] ) by reducing them into the following equation for the order parameter : @xmath112 f(t)=0 , \label{numeric}\ ] ] where @xmath113 and the scaled parameters @xmath114 , @xmath50 , @xmath51 and @xmath115 have been defined in ( [ eqb ] ) . the function @xmath116 is related to the order parameter by @xmath117 . the boundary conditions for @xmath118 are given by the value at @xmath119 and by the asymptotic condition @xmath120 . we first give initial values for @xmath121 and @xmath22 and solved the equation ( [ numeric ] ) numerically by means of the relaxation method . finally , the fermion density @xmath122 is obtained from eq.([fermieq ] ) . the details of the calculation will be given elsewhere@xcite . we consider the potassium atoms where there exist two bosonic ( @xmath4k , @xmath5k ) and one fermionic ( @xmath3k ) isotopes . the precise values of the interatomic interaction are not known , but the recent estimate from molecular scattering@xcite suggest a repulsive interaction between @xmath4k and @xmath3k , and an attractive one between @xmath5k and @xmath3k , although the values still have rather large ambiguities . we take up the following values for the @xmath13-wave scattering lengths @xcite : @xmath123 , @xmath124 , @xmath125 , @xmath126 . we take the atomic masses to be the same for all the isotopes : @xmath127 . the angular frequency @xmath128 of the bosonic external potential is fixed at @xmath129 , while @xmath130 is allowed to vary . in this case , it is corresponding to @xmath131 , @xmath132 for @xmath4k-@xmath3k system and @xmath133 , @xmath134 for @xmath5k-@xmath3k system . first , we consider the @xmath4k-@xmath3k system where the boson - fermion interaction is repulsive . in this case , because of @xmath135 , phase separation do not occur . 3 shows the density - distribution functions of bosons ( a ) and fermions ( b ) for @xmath136 , @xmath137 and @xmath138 . the dashed line shows the result for @xmath139 ( @xmath83 ) as compared with the solid line for @xmath140 ( @xmath141 ) . as seen from the figure , the fermions have a much broader distribution than bosons even for a much less number of particles . that has been discussed in sec . iii - i ) . the boson - fermion interaction is seen to squeeze out the fermions leading to a fermion - density depletion at the center , while its effect on the boson distribution is negligible . this also can be seen in table i where some observables are listed for the two cases @xmath142=(1000,1000 ) and ( 10000,1000 ) with / without the boson - fermion interaction . the robust structure of the boson distribution just mentioned is reflected in the almost constant values of the rms radius and other observables for the bosonic part of the mixed condensate against the switching on / off of the boson - fermion interaction . to understand this robustness , we apply the estimation in sec . iii - ii ) : the ( 1000,1000 ) system just corresponds to the parameters of eq . ( [ eqj ] ) , so the @xmath0 in ( [ eqi ] ) becomes @xmath77 . for ( 10000,1000 ) case , the scaled parameters are ( @xmath98 , @xmath99 , @xmath100 , @xmath101 ) = ( @xmath143 , @xmath144 , @xmath145 , @xmath146 ) , and the ratio @xmath147 in eq . ( [ eqi ] ) gives additional factor @xmath148 , so we obtain @xmath149 . it shows that ( 10000,1000 ) system is more robust than ( 1000,1000 ) case . it is supported in ref . @xcite that the tf approximation yields qualitative correct result . using the method in @xcite , we can also verify that the correction for the tf approximation is small for 1000 fermion case . we next turn to the @xmath5k-@xmath3k system where the boson - fermion interaction is attractive . because of the large value of the strength , one may expect a sizable effect of the boson - fermion interaction for this system . moreover , the strong boson - boson interaction makes a stronger overlap of bosons and fermions and thus the attractive interaction are more efficient . so the system might be expected to be unstable against collapse . if the effect of the boson - fermion interaction is too strong . actually , @xmath142=(1000,1000 ) and ( 10000,1000 ) , we obtained a distribution as given in fig . 4 and table ii . the effect of the attraction on the fermions is larger than the @xmath4k-@xmath3k system , although it is not strong enough to cause an appreciable change in the boson distribution . the stability condition for the fermion collapse can be derived from the scaling law in refs . @xcite : @xmath150 in the present case , we obtain @xmath151 and @xmath152 , so that the stability condition ( [ eqm ] ) is well satisfied , and we have found the parameters of @xmath5k-@xmath3k system is in ( quasi ) stable region . is the bosonic distribution always robust against a mixture of fermions ? aside from the possibility of changing the interaction strength via feshbach resonances@xcite , we have a possibility of enhancing the boson - fermion interaction effect using the different confining potential for bosons and fermions separately ( i.e. @xmath153 ) . fig . 5 shows the boson / fermion distribution for ( a ) @xmath4k-@xmath3k ( @xmath154 ) and ( b ) @xmath5k-@xmath3k ( @xmath155 ) : @xmath156 , @xmath157 . for a large value of the ratio @xmath158 , the fermions are trapped in a strong confining potential and thus have a large overlap with bosons ; the boson distribution ( solid lines ) is sensitive to the value of @xmath59 even though the bosonic parameters are fixed . this may be traced back to a rather large fraction of the boson - fermion interaction energy in the total boson energies : for the @xmath5k-@xmath3k system at @xmath159 , for instance , the ratio of the boson - fermion interaction energy to the total boson energy becomes @xmath160 @xcite , while it is less than 1 per cent to the total fermion energy . the change of the boson - density distribution can be estimated from the ratio @xmath0 in ( [ eqi ] ) . when @xmath62 , it becomes @xmath161 so that the boson - density distribution become more sensitive for large values of @xmath59 . the effect of the boson - fermion interaction on fermions may show up in the opposite extreme , i.e. , for vanishing @xmath59 . in this case , the fermions can keep themselves from escape only through the attractive boson - fermion interaction . this possibility can be estimated from the chemical potential : when fermions are trapped by bosons , the @xmath22 should take the negative value . taking the tf approximation for the bosonic part ( @xmath162 , @xmath163 ) and setting @xmath164 for the fermionic part , we obtain the total fermion number represented by @xmath22 : @xmath165 when @xmath166 in ( [ eqk ] ) , we obtain the maximum number for the boson - trapped fermions : @xmath167 where we used the parameters of @xmath5k-@xmath3k system to obtain the last term . from eq . ( [ eql ] ) , estimations can be obtained for the boson - trapped fermion number : for @xmath136 , only 18 fermions are bound , while for @xmath168 more than 4000 fermions will be trapped within the bose condensate . this qualitative estimate is verified also in the numerical calculation for @xmath136 , although one should actually take the shell effect into account for such a small number of fermions@xcite . in the present paper , we studied static properties of a mixed system of bosons ( @xmath4k or @xmath5k ) and fermions ( @xmath3k ) in the trapping potential at temperature zero . we solved a coupled gp and tf equations for a different combination of boson and fermion particle numbers and also for a repulsive or an attractive interaction of bosons and fermions depending on the combination of potassium isotopes . the analytical estimations have been done generally for the rms radii of the boson / fermion distribution , the parameter - dependence of their profile and the phase separation , and we have found the feasible conditions for them . these analytical estimations have been applied to the numerical results for the potassium system and also compared with the discussion done in @xcite : consequently , the consistency was checked for these estimations . for similar particle numbers and external potential parameters , the fermions have a much more extended distribution and larger energies than bosons because of the exclusion principle . we studied the effect of the boson - fermion interaction on the density distribution of both kinds of particles . for a repulsive interaction , the fermions tend to be pushed out , but the effect is not very large for the parameters of @xmath5k-@xmath3k system adopted in the present paper . in the case of the attractive interaction , a coherence of @xmath5k and @xmath3k occurs and is strongly enhanced if the trapping potential is adjusted so as to make them overlap . it was shown also that the fermions could be kept trapped without external potential due only to an attractive interaction with trapped bose particles . one may as well consider the opposite case where the bosons are trapped only via an attractive interaction with surrounding trapped fermi particles . this will be discussed in a future publication@xcite . m. h. anderson et al . , science * 269 * , 198 ( 1995 ) ; k. b. davis et al . , phys . lett . * 75 * , 3969 ( 1995 ) ; c. c. bradley et al . lett . * 75 * , 1687 ( 1995 ) . k. burnett , contemp . phys . * 37 * , 1 ( 1996 ) ; m. lewenstein and l. you , adv . atom . phys . , * 36 * , 221 ( 1996 ) ; a. s. parkins and h. d. f. walls , phys . reports * 303 * , 1 ( 1998 ) . f. dalfovo , s. giorgini , l. p. pitaevskii and s. stringari rev . phys.*71 * , 463 ( 1999 ) . j. schneider and h. wallis , phys . rev.*a57 * , 1253 ( 1997 ) ; d. a. butts and d. s. rokhsar , phys . rev.*a55 * , 4346 ( 1997 ) ; g. m. bruun and k. burnett , phys . rev . * a58 * , 2427 ( 1998 ) . c. j. myatt et al . lett . * 78 * , 586(1997 ) ; d. s. hall et al . * 81 * , 1539 ( 1998 ) ; w. geist , l. you and t. a. b. kennedy , phys . rev . * a59 * , 1500 ( 1999 ) . i. f. silvera ; physica * 109 & 110 * , 1499 ( 1982 ) ; j. oliva ; phys . rev . * b38 * , 8811 ( 1988 ) . k. mlmer , phys . . lett . * 80 * , 3419 ( 1998 ) . n. nygaard and k. mlmer , phys . rev . * a59 * , 2974 ( 1999 ) . m. amoruso et al , eur . j. * d4 * , 261 ( 1998 ) ; l. vichi , et al . , j. phys . * b31 * , l899 ( 1998 ) . m. amoruso , c. minniti and m. p. tosi , to be published in eur . phys . j. * d*. r. s. williamson iii and t. walker , j. opt . . am . * b12 * , 1394 ( 1995 ) ; f. s. cataliotti , et al . , phys . rev . * a57 * , 1136 ( 1998 ) . h. m. j. m. boesten , et al . , phys . rev . * a54 * , r3726 ( 1996 ) . r. cot et al . * a57 * , r4118 ( 1998 ) . in @xcite , @xmath169 was used , where the phase separation was found in @xmath170 . t. miyakawa et al . , in preparation . p. ring and p. shuck , _ the nuclear many - body problem _ ( springer - verlag , 1980 ) . e. tiesinga et al . , phys . rev.*a46 * , r1167 ( 1992 ) ; ibid.*a47 * , 4114(1993 ) . the kinetic ( @xmath171 ) , external potential ( @xmath172 ) , boson - boson ( @xmath173 ) and boson - fermion ( @xmath174 ) interaction energies per particle are @xmath175 , @xmath176 , @xmath177 , and @xmath178 , respectively , in th unit of @xmath179 . . results for the @xmath4k-@xmath3k system with ( @xmath180 ) and without ( @xmath139 ) the boson - fermion interaction for @xmath156 and @xmath181 . contributions to the equilibrium energy ( measured in the unit of @xmath182 ) per particle , the central density @xmath183 ( @xmath184 ) and the rms radii are given for each of the constituent isotopes . other parameters are fixed as @xmath185 , @xmath186 . [ cols="^,^,^,^,^,^,^,^,^,^",options="header " , ]
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static properties of a bose - fermi mixture of trapped potassium atoms are studied in terms of coupled gross - pitaevskii and thomas - fermi equations for both repulsive and attractive bose - fermi interatomic potentials .
qualitative estimates are given for solutions of the coupled equations , and the parameter regions are obtained analytically for the boson - density profile change and for the boson / fermion phase separation .
especially , the parameter ratio @xmath0 is found that discriminates the region of the large boson - profile change .
these estimates are applied for numerical results for the potassium atoms and checked their consistency .
it is suggested that a small fraction of fermions could be trapped without an external potential for the system with an attractive boson - fermion interaction .
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since the seminal work of koenker and bassett ( 1978 ) , quantile regression has received substantial scholarly attention as an important alternative to conventional mean regression . indeed , there now exists a large literature on the theory of quantile regression ( see , for example , koenker ( 2005 ) , yu _ et al_. ( 2003 ) , and buchinsky ( 1998 ) for an overview ) . notably , quantile regression can be used to analyse the relationship between the conditional quantiles of the response distribution and a set of regressors , while conventional mean regression only examines the relationship between the conditional mean of the response distribution and the regressors . quantile regression can thus be used to analyse data that include censored responses . powell ( 1984 ; 1986 ) proposed a tobit quantile regression ( tqr ) model utilising the equivariance of quantiles under monotone transformations . hahn ( 1995 ) , buchinsky and hahn ( 1998 ) , bilias _ et al_. ( 2000 ) , chernozhukov and hong ( 2002 ) , and tang _ et al_. ( 2012 ) considered alternative approaches to estimate tqr . more recent works in the area of censored quantile regression include wang and wang ( 2009 ) for random censoring using locally weighted censored quantile regression , wang and fygenson ( 2009 ) for longitudinal data , chen ( 2010 ) and lin _ et al_. ( 2012 ) for doubly censored data using the maximum score estimator and weighted quantile regression , respectively , and xie _ et al_. ( 2015 ) for varying coefficient models . in the bayesian framework , yu and stander ( 2007 ) considered tqr by extending the bayesian quantile regression model of yu and moyeed ( 2001 ) and proposed an estimation method based on markov chain monte carlo ( mcmc ) . a more efficient gibbs sampler for the tqr model was then proposed by kozumi and kobayashi ( 2011 ) . further extensions of bayesian tqr have also been considered . kottas and krnjaji ( 2009 ) and taddy and kottas ( 2012 ) examined semiparametric and nonparametric models using dirichlet process mixture models . reich and smith ( 2013 ) considered a semiparametric censored quantile regression model where the quantile process is represented by a linear combination of basis functions . to accommodate nonlinearity in data , zhao and lian ( 2015 ) proposed a single - index model for bayesian tqr . furthermore , kobayashi and kozumi ( 2012 ) proposed a model for censored dynamic panel data . for variable selection in bayesian tqr , ji _ et al_. ( 2012 ) applied the stochastic search , alhamzawi and yu ( 2014 ) considered a @xmath2-prior distribution with a ridge parameter that depends on the quantile level , and alhamzawi ( 2014 ) employed the elastic net . as in the case of ordinary least squares , standard quantile regression estimators are biased when one or more regressors are correlated with the error term . many authors have analysed quantile regression for uncensored response variables with endogenous regressors , such as amemiya ( 1982 ) , powell ( 1983 ) , abadie _ et al_. ( 2002 ) , kim and muller ( 2004 ) , ma and koenker ( 2006 ) , chernozhukov and hansen ( 2005 ; 2006 ; 2008 ) , and lee ( 2007 ) . extending the quantile regression model to simultaneously account for censored response variables and endogenous variables is a challenging issue . in the case of the conventional tobit model with endogenous regressors , a number of studies were published in the 1970s and 1980s , such as nelson and olsen ( 1978 ) , amemiya ( 1979 ) , heckman ( 1978 ) , and smith and blundell ( 1986 ) , with more efficient estimators proposed by newey ( 1987 ) and blundell and smith ( 1989 ) . on the contrary , few studies have estimated censored quantile regression with endogenous regressors . while blundell and powell ( 2007 ) introduced control variables as in lee ( 2007 ) to deal with the endogeneity in censored quantile regression , their estimation method involved a high dimensional nonparametric estimation and can be computationally cumbersome . chernozhukov _ et al_. ( 2014 ) also introduced control variables to account for endogeneity . they proposed using quantile regression and distribution regression ( chernozhukov _ et al_. , 2013 ) to construct the control variables and extended the estimation method of chernozhukov and hong ( 2002 ) . in the bayesian framework , mean regression models with endogenous variables have garnered a great deal of research attention from both the theoretical and the computational points of view ( _ e.g . _ rossi _ et al_. , 2005 ; hoogerheide _ et al_. , 2007a , 2007b ; conely _ et al_. , 2008 ; lopes and polson , 2014 ) . however , despite the growing interest in and demand for bayesian quantile regression , the literature on bayesian quantile regression with endogenous variables remains sparse . lancaster and jun ( 2010 ) utilised the exponentially tilted empirical likelihood and employed the moment conditions used in chernozhukov and hansen ( 2006 ) . in the spirit of lee ( 2007 ) , ogasawara and kobayashi ( 2015 ) employed a simple parametric model using two asymmetric laplace distributions for panel quantile regression . however , these methods are only applicable to uncensored data . furthermore , the model of ogasawara and kobayashi ( 2015 ) can be restrictive because of the shape limitation of the asymmetric laplace distribution , which can affect the estimates . indeed , the modelling of the first stage error in this approach remains to be discussed . based on the foregoing , this study proposes a flexible parametric bayesian endogenous tqr model . the @xmath0-th quantile regression of interest is modelled parametrically following the usual bayesian quantile regression approach . following lee ( 2007 ) , we introduce a control variable such that the conditional quantile of the error term is corrected to be zero and the parameters are correctly estimated . as in the approach of lee ( 2007 ) , the @xmath1-th quantile of the error term in the regression of the endogenous variable on the exogenous variables , which is often called the first stage regression , is also assumed to be zero . we discuss the modelling approach for the first stage regression and consider a number of parametric and semiparametric models based on the extensions of ogasawara and kobayashi ( 2015 ) . specifically , following wichitaksorn _ et al_. ( 2014 ) and naranjo _ et al_. ( 2015 ) , we employ the first stage regression models based on the asymmetric laplace distribution , skew normal distribution , and asymmetric exponential power distribution , for which the @xmath1-th quantile is always zero and is modelled by the regression function . to introduce more flexibility into the tail behaviour of the models based on the asymmetric laplace and skew normal distributions , we also consider a semiparametric extension using the dirichlet process mixture of scale parameters as in kottas and krnjaji ( 2011 ) . the value of @xmath1 is a priori unknown , while the choice of @xmath1 can affect the estimates . in this study , hence , @xmath1 is treated as a parameter to incorporate uncertainty and is estimated from the data . the performance of the proposed models is demonstrated in a simulation study under various settings , which is a novel contribution of the present study . we also illustrate the influence of the prior distributions on the posterior in the cases where valid and weak instruments are used . the rest of this paper is organised as follows . section [ sec : tobit ] introduces the standard bayesian tqr model with a motivating example . then , section [ sec : approach ] proposes bayesian tqr models to deal with the endogenous variables . the mcmc methods adopted to make inferences about the models are also described . the simulation study under various settings is presented in section [ sec : sim ] . the models are also illustrated by using the real data on the working hours of married women in section [ sec : real ] . finally , we conclude in section [ sec : conc ] . suppose that the response variables are observed according to @xmath3 then , consider the @xmath0-th quantile regression model for @xmath4 given by @xmath5 where @xmath6 is the vector of regressors , @xmath7 is the coefficient parameter , and @xmath8 is the error term whose @xmath0-th quantile is zero . the @xmath0-th conditional quantile of @xmath9 is modelled as @xmath10 . the equivariance under the monotone transformation @xmath11 of quantiles implies that the @xmath0-th conditional quantile of @xmath12 is given by @xmath13 the tqr model can be estimated by minimising the sum of asymmetrically weighted absolute errors @xmath14 where @xmath15 and @xmath16 denotes the indicator function ( powell , 1986 ) . the bayesian approach assumes that @xmath17 follows the asymmetric laplace distribution , since minimising ( [ eqn : check ] ) is equivalent to maximising the likelihood function of the asymmetric laplace distribution ( koenker and machado , 1999 ; chernozhukov and hong , 2003 ) . the probability density function of the asymmetric laplace distribution , denoted by @xmath18 , is given by @xmath19 where @xmath20 is the scale parameter and @xmath21 is the shape parameter ( yu and zhang , 2005 ) . the mean and variance are given by @xmath22=\sigma\frac{1 - 2p}{p(1-p)}$ ] and @xmath23 . the @xmath0-th quantile of this distribution is zero , @xmath24 . assuming the prior distributions for the parameters , the parameters are estimated by using the mcmc method ( _ e.g . _ yu and stander , 2007 ; kozumi and kobayashi , 2011 ) . posterior consistency of bayesian quantile regression based on the asymmetric laplace distribution was shown by sriram _ et al_. ( 2013 ) . estimates under the standard bayesian tqr model are biased when endogenous variables are included as regressors . consider a simple motivating example where the dataset was generated from @xmath25 for @xmath26 , where @xmath27 , @xmath28 , @xmath29 and @xmath30.\ ] ] see also chernozhukov _ et al_. ( 2014 ) . note that @xmath31 expresses the level of endogeneity . while @xmath32 is an exogenous variable when @xmath33 , @xmath32 is endogenous when @xmath34 . since @xmath35 , the model can be rewritten as @xmath36 therefore , the standard model that models the conditional quantile of @xmath9 as @xmath37 produces biased estimates . figure [ fig : fig1 ] shows the posterior distributions of @xmath38 , @xmath39 , and @xmath40 for the standard model for @xmath41 obtained by using the method of kozumi and kobayashi ( 2011 ) . the vertical lines in the figure indicate the true values . in the case of @xmath33 , the posterior distributions are concentrated around the true values . however , in the case of @xmath42 , the posterior distributions are concentrated away from the true values . , @xmath39 , and @xmath40 using the standard bayesian tobit median regression ] we propose the following model to deal with the endogenous variables : @xmath43 for @xmath44 , where @xmath6 is the vector of the exogenous variables whose the first element is @xmath45 , @xmath46 is the endogenous variable , @xmath47 , and @xmath48 is the exogenous variable not included in @xmath6 , which is also called the instrumental variable . the term @xmath49 in ( [ eqn:2nd ] ) is called the control variable and is introduced to account for endogeneity . note that @xmath50 indicates @xmath46 is endogenous . we refer to ( [ eqn:1st ] ) as the first stage regression and to ( [ eqn:2nd ] ) as the second stage regression . a similar form is found in lopes and polson ( 2014 ) in the context of the instrumental variable regression for means by using the cholesky - based prior . following lee ( 2007 ) , the error term @xmath8 of the standard bayesian tqr is decomposed into the terms @xmath51 and @xmath52 . it is assumed that relationship ( [ eqn:1st ] ) is specified correctly and the quantile independence of @xmath52 on @xmath53 conditional on @xmath54 : @xmath55 as in lee ( 2007 ) , we also assume @xmath56 where the @xmath1-th conditional quantile of @xmath54 is zero for some @xmath57 . we are mainly concerned with modelling the first stage error that satisfies ( [ eqn : alpha ] ) . a simple and convenient approach is to assume @xmath58 as in ogasawara and kobayashi ( 2015 ) , since ( [ eqn : alpha ] ) is always satisfied for the asymmetric laplace distribution . however , the asymmetric laplace distribution has limitations , such as peaky density , restrictive tail behaviour , and skewness . when a model lacks fit to the data , the estimate of the conditional quantile would be away from the value such that ( [ eqn : alpha ] ) truly holds . then , assuming @xmath54 is homoskedastic , the estimate of the intercept , @xmath59 , may be biased as well . consequently , the estimate of @xmath60 would be affected through the introduced term @xmath51 . when @xmath54 is heteroskedastic , the entire coefficient vector would be affected . therefore , we consider some alternative models for the first stage error distribution . recently , wichitaksorn _ et al_. ( 2014 ) considered a class of parametric distributions with a quantile constraint of the form ( [ eqn : alpha ] ) , including the asymmetric laplace distribution , and applied them in the context of quantile modelling . furthermore , zhu and zinde - walsh ( 2009 ) , zhu and galbraith ( 2011 ) , and naranjo _ et al_. ( 2015 ) considered a flexible parametric distribution with the quantile constraint . based on these studies , we also consider the following two distributions to model the first stage error . first , we consider the skew normal distribution denoted by @xmath61 , where @xmath62 is the scale parameter and @xmath57 is the shape parameter . the probability density function is given by @xmath63 when @xmath64 , the distribution reduces to @xmath65 . the mean and variance are given by @xmath66=\sqrt{\frac{\phi}{2\pi}}\frac{1 - 2\alpha}{\alpha(1-\alpha)}$ ] and @xmath67 ( see wichitaksorn _ et al_. , 2014 ) . when the actual error distribution is close to the normal distribution , this distribution would lead to better performance than the asymmetric laplace distribution . however , just as the asymmetric laplace distribution , the skewness and the quantile level of the mode are controlled by the single parameter @xmath1 . second , we consider the asymmetric exponential power distribution treated by zhu and zinde - walsh ( 2009 ) , zhu and galbraith ( 2011 ) , and naranjo _ et al_. ( 2015 ) . the probability density function of the asymmetric exponential power distribution , denoted by @xmath68 , is given by @xmath69 where @xmath62 is the scale parameter , @xmath57 is the skewness parameter , @xmath70 is the shape parameter for the left tail , and @xmath71 is the shape parameter for the right tail . after some reparameterisation , the distribution reduces to the asymmetric laplace distribution when @xmath72 and to the skew normal distribution when @xmath73 . the tails of the asymmetric exponential power distribution are controlled separately by @xmath74 and @xmath75 , respectively , and the overall skewness is controlled by @xmath1 . although the distribution is more flexible than the above two distributions , the posterior computation using mcmc would be inefficient , because it includes two additional shape parameters and it has no convenient mixture representation , apart from the mixture of uniforms that is inefficient , to facilitate an efficient mcmc algorithm . the computational efficiency is also compared in section [ sec : sim ] . in addition to the three parametric models , we also consider the semiparametric extension of the models based on the asymmetric laplace and skew normal distributions to achieve both flexibility and computational efficiency . more specifically , the following two models using the dirichlet process mixtures of scales are considered : @xmath76 where @xmath77 denotes the dirichlet process with the precision parameter @xmath78 and the base measure @xmath79 . for both models , we set @xmath80 as it is computationally convenient . while those mixture models have the same limitation as the parametric versions in terms of skewness , they extend the tail behaviour of the error distribution preserving ( [ eqn : alpha ] ) ( kottas and krnjaji , 2009 ) . hereafter , the models with the asymmetric laplace , skew normal , and asymmetric exponential power first stage errors are respectively denoted by al , sn , and aep , and those with the dirichlet process mixtures are denoted by aldp and sndp . we must take care when selecting the @xmath1 value in ( [ eqn : alpha ] ) , as it is a part of the model specification and can thus affect the estimates ( lee , 2007 ) . we treat @xmath1 as a parameter and estimate its value along with the other parameters . since @xmath1 determines the quantile level of the mode for all models considered here , our approach to modelling the first stage regression can also be regarded as a kind of mode regression ( see wichitaksorn _ et al_. , 2014 ) . to gain further flexibility , we might extend the model through a fully nonparametric mixture . several semiparametric models in the context of bayesian quantile regression with exogenous variables have been proposed by kottas and gelfand ( 2001 ) , kottas and krnjaji ( 2009 ) , and reich _ et al_. ( 2010 ) . for example , kottas and krnjaji ( 2009 ) considered the nonparametric mixture of uniform distributions for any unimodal density on the real line with the quantile restriction at the mode using the dirichlet process mixture ( see also kottas and gelfand , 2001 ) . in the more flexible model proposed by reich _ et al_. ( 2010 ) , the mode of the error distribution does not have to coincide with zero . this is achieved by using a nonparametric mixture of the quantile - restricted two - component mixtures of normal distributions . however , their approaches are not directly applicable in the present context where the value of @xmath1 is estimated . if we were to estimate the quantile level for which the quantile restriction holds , the computation under the former model is expected to be extremely inefficient and unstable as the model involves many indicator functions , and @xmath1 and the intercept would be highly correlated . the intercept would not be identifiable in the latter model . we could further extend the model to account for heteroskedasticity such that @xmath81 for @xmath44 , where @xmath82 for all @xmath83 and the first element of @xmath84 is fixed to one ( _ e.g . _ reich , 2010 ) . in this case , the @xmath1-th quantile of @xmath32 is given by @xmath85 as in the usual quantile regression . however , since the first stage regression model is built based on ( [ eqn : alpha ] ) , models ( [ eqn:1st ] ) and ( [ eqn : hetero ] ) would produce identical estimates . we next turn to the model of the new second stage error , @xmath52 , in ( [ eqn:2nd ] ) . since the @xmath0-th conditional quantile of @xmath52 is now zero , we assume that @xmath86 as in the standard bayesian quantile regression approach . we utilise the location scale mixture of normals representation for the asymmetric laplace distribution to facilitate an efficient mcmc method following kozumi and kobayashi ( 2011 ) ( see also kotz _ et al_. , 2001 ) . the model is expressed in the hierarchical form given by @xmath87 for @xmath44 , where @xmath88 , @xmath89 , @xmath90 denotes the exponential distribution with mean @xmath91 , and @xmath92 the coefficient parameter @xmath93 is common to all first stage regression specifications . first , we assume the normal prior for @xmath93 , since it is computationally convenient for the al , sn , aldp , and sndp models . since we do not have information on the coefficient values , the variances are set such that the prior distributions are relatively diffuse . our default choice is @xmath94 . for the scale parameters , @xmath95 for the al , sn , and aep distributions , a relatively diffuse inverse gamma distribution is assumed and the default choice is set to @xmath96 . for aep , we assume @xmath97 , where @xmath98 denotes the normal distribution with the mean @xmath99 and variance @xmath100 truncated on the interval @xmath101 . a similar prior specification is found in naranjo _ et al_. ( 2015 ) . for all models , @xmath102 is assumed . for the semiparametric models , we need to specify the parameters of the inverse gamma base measure . assuming that the data have been rescaled , @xmath103 and @xmath104 are chosen such that the variance of @xmath54 takes values between @xmath105 and @xmath106 with high probability ( _ e.g . _ ishwaran and james , 2002 ) . our default choice is @xmath107 and @xmath108 for aldp and @xmath109 for sndp . under this choice , when @xmath64 for aldp , @xmath110 as @xmath111 . similarly , when @xmath112 , @xmath113 . for sndp , @xmath114 when @xmath64 and @xmath115 when @xmath112 . for the precision parameter of the dirichlet process , @xmath116 , we assume @xmath117 such that both small and large values for @xmath116 , hence the number of clusters , are allowed . for the coefficient parameters in the second stage , @xmath7 and @xmath118 , we also assume relatively diffuse normal distributions . our default choice of prior is @xmath119 . similar to @xmath95 in the parametric first stage , we assume an inverse gamma prior for the scale of the al pseudo likelihood . our default choice is @xmath96 . the parameter @xmath120 accounts for the endogeneity and we need to take care in prior elicitation . when the data follow the bivariate normal distribution , as in the motivating example ( [ eqn : example ] ) , @xmath120 is equal to @xmath121 , where @xmath31 is the correlation coefficient and @xmath122 and @xmath123 are the standard deviations of the first and second stage errors , respectively . in this case , we may follow lopes and polson ( 2014 ) to determine the variance of the normal prior implied from an inverse wishart prior for the covariance matrix . however , we do not limit ourselves to normal data as the quantile regression approach is suitable for heteroskedastic and non - normal data , and the non - normal models are used in the first stage . in the literature on bayesian non - normal selection models , the prior distribution of @xmath120 is normal typically with a very small variance , such as @xmath124 ( _ e.g . _ munkin and trivedi , 2003 , 2008 ; deb _ et al_. , 2006 ) . on the other hand , we use a more diffused prior to reflect our ignorance about @xmath120 and set our default choice of prior to be @xmath125 . when the instrument is weak , it is expected that our quantile regression models face the problem of prior sensitivity and that the posterior distributions exhibit sharp behaviour , as in the case of the bayesian instrumental variable regression model . section [ sec : sim ] considers the alternative choices of the hyperparameters to study the prior sensitivity . the proposed models are estimated by using the mcmc method based on the gibbs sampler . we describe the gibbs sampler for the semiparametric models with aldp and sndp , which is an extension of the gibbs sampler described in kozumi and kobayashi ( 2011 ) and ogasawara and kobayashi ( 2015 ) . the algorithms for the al and sn models can be obtained straightforwardly . we also mention the algorithm for the aep model . the variables involved in the dirichlet process are sampled by using the retrospective sampler ( papaspiliopoulos and roberts , 2008 ) and the slice sampler ( walker , 2007 ) . first , we introduce @xmath126 and @xmath127 , such that @xmath128 . then , as in walker ( 2007 ) , the gibbs sampler is constructed by working on the following joint densities @xmath129 where @xmath130 , @xmath131 , @xmath132 , and @xmath133 denotes the beta distribution with the parameters @xmath116 and @xmath134 ( sethuraman , 1994 ) . we also let @xmath135 denote the minimum integer such that @xmath136 . for the aldp model , we utilise the mixture representation for the asymmetric laplace distribution to sample @xmath93 efficiently such that @xmath137 , @xmath138 , @xmath44 , where @xmath139 and @xmath140 are defined as in ( [ eqn : theta ] ) . let us denote @xmath89 and @xmath141 . our gibbs sampler proceeds by alternately sampling @xmath142 , @xmath143 , @xmath144 , @xmath145 , @xmath116 , @xmath93 , @xmath146 , @xmath1 , @xmath147 , @xmath148 , @xmath91 , and @xmath149 . * * sampling @xmath142 : * generate @xmath150 from @xmath151 for @xmath44 . * * sampling @xmath143 : * generate @xmath152 from @xmath153 where @xmath154 for @xmath155 . * * sampling @xmath144 : * generate @xmath156 from the multinomial distribution with probabilities @xmath157 for @xmath44 . * * sampling @xmath145 : * generate @xmath158 from @xmath159 where @xmath160 + d_0.\ ] ] * * sampling @xmath116 : * assuming the gamma prior , @xmath161 , we use the method described by escobar and west ( 1995 ) to sample @xmath116 . by introducing @xmath162 , the full conditional distribution of @xmath116 is the mixture of two gamma distributions given by @xmath163 where @xmath164 is the number of distinct clusters and @xmath165 . * * sampling @xmath93 : * assuming @xmath166 , @xmath93 is sampled from @xmath167 where @xmath168^{-1},\\ { \mathbf{g}}_1&=&{\mathbf{g}}_1\left[\sum_{i=1}^n{\mathbf{z}}_i\left(-\frac{\eta_p(y_i^*-{\mathbf{x}}_i'{\text{\boldmath{$\beta$}}}_p - \eta_pd_i-\theta_pg_i)}{\tau_p^2\sigma g_i}+\frac{d_i-\theta_\alpha h_i}{\tau_\alpha^2\phi_{k_i}h_i } \right)+{\mathbf{g}}_0^{-1}{\mathbf{g}}_0\right ] , \end{aligned}\ ] ] as the density of the full conditional distribution denoted by @xmath169 is given by @xmath170 * * sampling @xmath146 : * the full conditional distribution of @xmath171 is the generalised inverse gaussian distribution , denoted by @xmath172 . the probability density function of @xmath173 is given by @xmath174 where @xmath175 is the modified bessel function of the third kind ( barndorff - nielsen and shephard , 2001 ) . for @xmath44 , we sample @xmath171 from @xmath176 where @xmath177 * * sampling @xmath1 : * the density of the full conditional distribution of @xmath1 is given by @xmath178 where @xmath179 and@xmath180 denote the full conditional and prior density of @xmath1 , respectively . we use the random walk metropolis hastings ( mh ) algorithm to sample from this distribution . * * sampling @xmath147 : * the full conditional distribution of @xmath4 is given by @xmath181 * * sampling @xmath148 : * we sample @xmath89 in one block . assuming @xmath182 , the full conditional distribution is given by @xmath183 where @xmath184^{-1 } , \quad \tilde{{\mathbf{b}}}_1=\tilde{{\mathbf{b}}}_1\left[\sum_{i=1}^n\frac{\tilde{{\mathbf{x}}}_i(y_i^*-\theta_pg_i)}{\tau_p^2\sigma g_i}+ \tilde{{\mathbf{b}}}_0^{-1}\tilde{{\mathbf{b}}}_0\right].\ ] ] * * sampling @xmath91 : * assuming @xmath185 , we sample @xmath91 from @xmath186 where @xmath187 and @xmath188 . * * sampling @xmath149 : * similar to @xmath171 , @xmath189 is sampled from @xmath190 where @xmath191 the gibbs sampler for sndp consists of sampling @xmath142 , @xmath143 , @xmath144 , @xmath145 , @xmath116 , @xmath93 , @xmath1 , @xmath147 , @xmath148 , @xmath91 , and @xmath149 . the sampling algorithms for @xmath142 , @xmath143 , @xmath116 , @xmath147 , @xmath148 , @xmath91 , and @xmath149 remain the same as in the case of aldp . the sampling scheme of @xmath144 and @xmath1 can be obtained by replacing @xmath192 with @xmath193 . similar to the case of aldp , the density of the full conditional distribution is given by @xmath194 where @xmath195^{-1 } , \\ { \mathbf{g}}_1({\text{\boldmath{$\gamma$}}})&=&{\mathbf{g}}_1({\text{\boldmath{$\gamma$}}})\left[\sum_{i=1}^n{\mathbf{z}}_i \left ( -\frac{\eta_p(y_i^*-{\mathbf{x}}_i'{\text{\boldmath{$\beta$}}}_p-\eta_pd_i-\theta_p g_i)}{\tau_p^2\sigma g_i}+\frac{4d_i(\alpha - i(d_i\leq{\mathbf{z}}_i'{\text{\boldmath{$\gamma$}}}))^2}{\phi_{k_i}}\right ) + { \mathbf{g}}_0^{-1}{\mathbf{g}}_0\right ] , \end{aligned}\ ] ] which is similar to the density of the normal distribution . therefore , we sample @xmath93 by using the mh algorithm with the proposal distribution given by @xmath196 . since no convenient representation for the aep distribution is available , the full conditional distributions of the parameters in the first stage regression , @xmath93 , @xmath95 , @xmath1 , @xmath74 , and @xmath75 , are not in the standard forms . therefore , we employ the adaptive random walk mh algorithm . although naranjo _ et al_. ( 2015 ) proposed the scale mixture of uniform representation for the aep distribution , the algorithm based on this representation would be inefficient , because it consists of sampling from a series of distributions that are truncated on some intervals such that the mixture representation holds and such intervals move quite slowly as sampling proceeds ( see also kobayashi , 2015 ) . since the additional shape parameters in aep free up the role of @xmath1 , @xmath1 controls the overall skewness by allocating the weights on the left and right sides of the mode . hence , the mcmc sample would exhibit relatively high correlation between @xmath1 and @xmath59 . the models considered in the previous section are demonstrated using simulated data . the aims of this section are ( 1 ) to compare the performance of the proposed models ( section [ sec : default ] ) , ( 2 ) to study the sensitivity to the prior settings , and ( 3 ) to illustrate the behaviour of the posterior distribution when the instrument is weak ( section [ sec : alt ] ) . the data are generated from the model given by @xmath197 for @xmath26 , where @xmath198 assuming that a valid instrument is available , @xmath199 , @xmath200 , and @xmath201 . the performance of the models is compared by considering the various settings for @xmath54 , while the distributions of @xmath52 are kept relatively simple in order that the true values of the quantile regression coefficients are tractable . the following five settings are considered : + * setting 1 * @xmath202 , @xmath203 , + * setting 2 * @xmath204 , @xmath205 , + * setting 3 * @xmath206 , @xmath205 , + * setting 4 * @xmath207 , @xmath203 , + * setting 5 * @xmath208 , @xmath205 , + where @xmath209 denotes the skew @xmath210 distribution with the location parameter @xmath99 , scale parameter @xmath100 , skewness parameter @xmath211 , @xmath212 , and degree of freedom @xmath213 ( see azzalini and capitanio , 2003 ; fr@xmath214hwirth - schnatter and pyne , 2010 ) , and we set @xmath215 . in setting 1 , the error terms follow the bivariate normal distribution as in the motivating example in section [ sec : tobit ] . setting 2 considers the fat tailed first stage regression . setting 3 considers a more difficult situation where the first stage error is fat tailed and skewed . setting 4 replaces the first stage error of setting 1 with the heteroskedastic error with respect to the instrument . setting 5 is also a challenging situation where the first stage error is fat tailed , skewed , and heteroskedastic . in settings 3 and 5 , the location parameters of the first stage error distributions are set such that the mode of @xmath54 is zero and the quantile level of the mode is @xmath216 . the average censoring rates for the settings are around @xmath217 . for each setting , the data are replicated @xmath218 times . we first estimated the proposed models under the default prior specifications ( see section [ sec : prior ] ) for @xmath219 and @xmath220 by running the mcmc for @xmath221 iterations and discarding the first @xmath222 draws as the burn - in period . the standard bayesian tqr model was also estimated . the bias and root mean squared error ( rmse ) of the parameters were computed over the @xmath218 replications . to assess the efficiency of the mcmc algorithm , we also recorded the inefficiency factor , which was defined as a ratio of the numerical variance of the sample mean of the markov chain to the variance of the independence draws ( chib , 2001 ) . table [ tab : default ] presents the biases , rmses , and median inefficiency factors for the parameters over the @xmath218 replications . first , we examined the inefficiency factors . overall , our sampling algorithms appear to be efficient , especially for al , sn , aldp , and sndp . the table shows that the inefficiency factors for al , sn , aldp , and sndp are reasonably small for @xmath223 , @xmath118 , @xmath120 , @xmath224 , and @xmath225 . since @xmath1 and @xmath59 determine the quantile level of the mode and location of the mode , respectively , the mcmc sample exhibits correlation between @xmath1 and @xmath59 and this results in higher inefficiency factors for them . hence , the inefficiency factors for @xmath226 tend to be higher than those for the other parameters . this pattern is more profound in the case of aep where the inefficiency factors for @xmath1 , @xmath59 , and @xmath226 are quite high . since the additional shape parameters in aep free up the role of @xmath1 , the mcmc sample exhibits higher correlation between @xmath1 and @xmath59 . furthermore , the inefficiency factors for the other parameters for aep are also higher than those for the other endogenous models . next , we turn to the performance of the models . as expected , tqr produces biased estimates in all cases . the rmses for the proposed endogenous models are generally larger for @xmath219 , which is below the censoring point , than for @xmath41 . the al and aldp models result in similar performance . the aep model shows the largest rmses for @xmath59 and @xmath226 among the proposed models for all cases . combined with the high inefficiency factors for those parameters , the convergence of the mcmc algorithm for aep may be difficult to ensure in the given simulation setting . this finding suggests a considerable practical limitation and , thus , aep will not be considered henceforth . the same limitation applies to the potentially more flexible nonparametric models discussed in section [ sec:1st ] . table [ tab : default ] also shows that the estimation of the first stage regression can influence the second stage parameters . for example , in setting 1 , the rmses for @xmath59 for sn and sndp are smaller than those for al and aldp , as the true model is the normal and thus sn and sndp produce smaller rmses for @xmath226 . similarly , in setting 4 , the rmses for @xmath226 for sn and sndp are smaller than those for al and aldp . in addition , the heteroskedasticity in the first stage influences the performance of the slope parameters , resulting in slightly smaller rmses for @xmath223 for sn and sndp than for al and aldp . however , the performance of the sn model becomes worse when the first stage error is fat tailed , since the skew normal distribution can not accommodate a fat tailed distribution . while the results in setting 2 are somewhat comparable across the models , the table shows that sn results in larger biases and rmses in setting 3 and , especially , setting 5 . in setting 3 , sn results in larger rmses for @xmath226 than for al , aldp , and sndp . in setting 5 , given the heteroskedasticity of the first stage , the biases and rmses for the intercept and slope parameters for sn are larger than those for al , aldp , and sndp . on the other hand , compared with sn , the semiparametric sndp model is able to cope with fat tailed errors and this produces results comparable with those for al and aldp . while the models result in reasonable overall performance , the results for settings 3 and 5 also illustrate the limitation of our modelling approach to some extent . in setting 3 , the models exhibit some bias in @xmath226 because of the lack of fit in the first stage . this lack of fit , which is represented by the bias for @xmath59 , is reflected in the bias for @xmath226 . the entire coefficient vector may be influenced by this lack of fit in the first stage in the presence of heteroskedasticity as in setting 5 . the lack of fit in the first stage is also indicated by the biases in @xmath1 . this finding implies that an inflexible first stage model can fail to estimate the true quantile such that ( [ eqn : alpha ] ) holds and that choosing the value of @xmath1 a priori could lead to biased estimates ( see the discussion in section [ sec:1st ] ) . for comparison purposes , we consider two alternative specifications for the inverse gamma base measure for the semiparametric models . the following slightly less diffuse settings than the default are considered . for aldp , we consider @xmath227 such that @xmath228 and @xmath229 such that @xmath230 when @xmath64 . for sndp , we consider @xmath231 such that @xmath232 and @xmath233 such that @xmath234 . for the other parameters , we use the default prior specifications . table [ tab : base ] presents the biases and rmses for aldp and sndp under the alternative base measures for @xmath219 and @xmath41 . the results in table [ tab : base ] are essentially identical to those in table [ tab : default ] , suggesting that the default choice of the base measures provides reasonable performance . next , the two alternative prior specifications for @xmath120 , @xmath91 , and @xmath95 are considered to study the prior sensitivity . the first alternative specification considers the more diffuse priors given by @xmath235 , @xmath236 , and @xmath237 . the second alternative specification is the even more diffuse setting given by @xmath238 , @xmath239 , and @xmath240 . for aldp and sndp , the default base measures are used . for @xmath7 , @xmath118 , and @xmath93 , we use the default specification . table [ tab : prior ] presents the biases and rmses for al , sn , aldp , and sndp under the five simulation settings for @xmath219 and @xmath220 , showing that the result is robust with respect to the choice of hyperparameters . we also considered some different prior choices for @xmath241 and @xmath93 , and obtained robust results . these findings thus confirm the robustness of the results with respect to the choice of base measures and prior distributions provided that a valid instrument is available . in the context of mean regression models , however , when the instrument is weak , the posterior distribution is known to exhibit sharp behaviour in the vicinity of non - identifiability ( hoogerheide _ et al_. , 2007b ) and the posterior distribution is greatly affected by the prior specification ( _ e.g . _ lopes and polson , 2014 ) . here , we illustrate the behaviour of the posterior distribution by using a weak instrument . the data are generated from ( [ eqn : sim1 ] ) without the regressor : @xmath242 for @xmath26 , where @xmath243 , @xmath244 , @xmath245 , @xmath202 , and @xmath203 . the al and sn models are estimated for @xmath219 by running the mcmc for 20000 iterations and discarding the first 5000 draws as the burn - in period under the three prior specifications previously considered . figure [ fig : fig2 ] presents the joint posterior distribution of @xmath246 and @xmath247 for al and sn under the three prior specifications and shows that the posterior distribution is greatly affected by the prior specification . the posterior distribution of @xmath40 becomes more diffuse as @xmath248 approaches zero . this trend becomes more profound as we use more diffuse prior distributions , producing star shapes . the figure also suggests that the prior distribution can act as an informative prior about the linear relationship between @xmath40 and @xmath249 . similar results were also obtained under different prior specifications for @xmath7 , @xmath118 , and @xmath93 as well as for aldp and sndp . and @xmath247 for al ( top row ) and sn ( bottom row ) ] the proposed endogenous models are applied to the dataset on the labour supply of married women of mroz ( 1987 ) . the dataset includes observations on @xmath250 individuals . the response variable is the total number of hours in every @xmath218 hours the wife worked for a wage outside the home during 1975 . in the data , 325 of the 753 women worked zero hours and the corresponding responses are treated as left censored at zero . hence , the censoring rate is approximately @xmath251 . the regressors of our model include years of education ( _ educ _ ) , years of experience ( _ exper _ ) and its square ( _ expersq _ ) , age of the wife ( _ age _ ) , number of children under 6 years old ( _ kidslt6 _ ) , number of children equal to or greater than 6 years old ( _ kidsge6 _ ) , and non - wife household income ( _ nwifeinc _ ) . we treat _ nwifeinc _ as an endogenous variable because it may be correlated with the unobserved household preference for the labour force participation of the wife . as an instrument , we include the years of education of the husband ( _ huseduc _ ) , since this can influence both his income and the non - wife household income , but it should not influence the decision of the wife to participate in the labour force . smith and blundell ( 1986 ) considered a similar setting where non - wife income was considered to be endogenous and the education of the husband was employed as the instrumental variable . they applied the endogenous tobit model to data derived from the 1981 family expenditure survey in the united kingdom . using the default prior specifications , the aldp and sndp models are estimated for @xmath252 by running the mcmc for @xmath253 iterations and discarding the first @xmath254 draws as the burn - in period . convergence is monitored by using the trace plots and gelman - rubin statistic for two chains with widespread starting values ( gelman _ et al_. , 2014 ) . the upper bounds of the gelman - rubin confidence intervals for the selected parameters , @xmath255 , @xmath118 , @xmath120 , @xmath256 , @xmath257 , and @xmath1 , for sndp in the case of @xmath219 are @xmath258 , @xmath258 , @xmath258 , @xmath259 , @xmath259 , and @xmath260 , respectively . figure [ fig : fig3 ] presents the post burn - in trace plots for these parameters and shows the evidence of convergence of the chains . ] first , we present the results for the representative quantiles , @xmath219 , @xmath220 , and @xmath261 . table [ tab : real ] shows the posterior means , 95% credible intervals , and inefficiency factors for aldp and sndp for these quantiles . the table shows that the sampling algorithm worked efficiently as the inefficiency factors are reasonably small . the posterior means for the instrument , _ huseduc _ , are positive and the 95% credible intervals do not include zero for all cases for both models , implying that _ huseduc _ is a valid instrument . for @xmath41 , the posterior means for @xmath120 are @xmath262 and @xmath263 for aldp and sndp , respectively , and the 95% credible intervals do not include zero . therefore , it is suggested that non - wife income be treated as an endogenous variable for the median regression . to study the endogeneity in non - wife household income across quantiles , the posterior distributions of @xmath120 are presented . the results across the quantiles can be best understood by plotting the posterior distributions as a function of @xmath0 . figure [ fig : fig4 ] shows the posterior means and 95% credible intervals of @xmath120 for aldp and sndp for @xmath252 . the figure shows that the two models produced similar results and that the posterior distributions of @xmath120 are concentrated away from zero for the mid quantiles . specifically , for @xmath264 , the 95% credible intervals do not include zero for either model . there are notable peaks around @xmath265 , where the posterior means of @xmath120 under the default prior specifications are @xmath266 and @xmath267 with the 95% credible intervals @xmath268 and @xmath269 for aldp and sndp , respectively . this is an interesting result considering that the censoring rate is @xmath251 . the result implies that the effect of the endogeneity of non - wife income is the most profound when the wife is about to decide whether to enter the labour force . when the opportunity cost of labour supply is very high ( lower quantile ) or the wife works on a more regular basis ( higher quantile ) , such endogeneity diminishes . smith and blundell ( 1986 ) also reported that non - wife income is endogenous by using the endogenous tobit regression model . the mean of our dataset is @xmath270 , which approximately corresponds to the @xmath271-th quantile . for @xmath272 , the posterior mean of @xmath120 for aldp is @xmath273 with the 95% credible interval @xmath274 and that for sndp is @xmath275 with the 95% credible interval @xmath276 . the figure also shows the posterior means and 95% credible intervals under the two alternative prior specifications considered in section [ sec : alt ] , confirming that our results are robust with respect to the prior specifications . under the default and alternative priors for @xmath252 ] figure [ fig : fig5 ] compares the posterior means and 95% credible intervals of @xmath241 for sndp , aldp , and tqr for @xmath252 . the results for sndp and aldp are quite similar . the figure clearly shows that the posterior distributions for the key variable , _ nwifeinc _ , for the proposed models and tqr exhibit some differences for @xmath264 , where _ nwifeinc _ is indicated to be endogenous . the difference becomes the most profound around @xmath265 for which the posterior mean for _ nwifeinc _ is @xmath278 for aldp , @xmath279 for sndp , and @xmath280 for tqr , implying a stronger effect of non - wife income when endogeneity is taken into account . the posterior distributions for _ nwifeinc _ for aldp and sndp are more dispersed than that for tqr for all @xmath0 . while the 95% credible intervals include zero for all models for the upper quantiles , for the lower quantiles , such as @xmath219 , those for aldp and sndp include zero and those for tqr do not . differences in the results are also observed for other variables . for @xmath265 , the posterior means for _ educ _ and _ age _ are respectively @xmath281 and @xmath282 for aldp , @xmath283 and @xmath284 for sndp , and @xmath285 and @xmath286 for tqr . for the upper quantiles , @xmath287 , the 95% credible intervals for _ educ _ include zero for the proposed models , while those for tqr do not , implying that an additional year of education does not increase the working hours for those quantiles when the endogeneity from non - wife income is taken into account . for _ expersq _ , the endogenous models result in slightly more dispersed posterior distributions for @xmath288 . the posterior means for @xmath265 are @xmath289 , @xmath290 , and @xmath291 for aldp , sndp , and tqr , respectively . for _ kidsge6 _ , the posterior means for @xmath265 are @xmath292 , @xmath293 , and @xmath294 for aldp , sndp , and tqr , respectively . however , the 95% credible intervals include zero for all @xmath0 for all models . on the other hand , the figure also shows that the models produced similar results for _ exper _ and _ kidslt6 _ for all @xmath0 . for aldp , sndp , and tqr for @xmath252 ] we proposed bayesian endogenous tqr models using parametric and semiparametric first stage regression models built around the zero @xmath1-th quantile assumption . the value of @xmath1 determines the quantile level of the mode of the error distribution and is estimated from the data . from the simulation study , the al , aldp , and sndp models worked relatively well for the various situations , while they faced the same limitation pointed out by kottas and krnjaji ( 2011 ) . on the other hand , the sn model could not accommodate the fat tailed first stage errors . although aep could be a promising model in terms of flexibility , the inefficiency of the mcmc algorithm largely limits its applicability in practice . the development of a more convenient mixture representation for the aep distribution is thus required . from application to data on the labour supply of married women , the effect of the endogeneity in non - wife income was found to be the most profound for the quantile level close to the censoring rate . for this quantile , some differences in the parameter estimates between the endogenous and standard models were found , such as the stronger effect of non - wife income on working hours . this study only considered the case of continuous endogenous variables . we are also interested in incorporating endogenous binary variables into a bayesian quantile regression model . an important extension might therefore be addressing multiple endogenous dummy variables to represent selection among multiple alternatives , such as the choice of a hospital and insurance plan , as considered in geweke _ et al_. ( 2003 ) and deb _ et al_. ( 2006 ) . however , such an extension would be challenging with respect to the assumptions that must be imposed on the multivariate error terms . we leave these issues to future research . the author would like to thank the seminar participants at the second baysm and esobe 2014 and the anonymous referees for the valuable comments to improve the manuscript . the computational results were obtained by using ox version 6.21 ( doornik , 2007 ) . this study was supported by jsps kakenhi grant numbers 25245035 , 26245028 , 26380266 , and 15k17036 . azzalini , a. and capitanio , a. ( 2003 ) . `` distributions generated by perturbation of symmetry with emphasis on a multivariate skew @xmath210-distribution , '' _ journal of royal statistical society series b _ , * 65 * , 367389 . barndorff - nielsen , o. e. and shephard , n. ( 2001 ) . `` non - gaussian ornstein - uhlenbeck - based models and some of their uses in financial econometrics , '' _ journal of royal statistical society series b _ , * 63 * , 167241 . deb , p. , munkin , k. m. , and trivedi , k. p. ( 2006 ) . `` bayesian analysis of the two - part model with endogeneity : application to health care expenditure , '' _ journal of applied econometrics _ , * 21 * , 10811099 . fr@xmath214hwirth - schnatter , s. and pyne , d. ( 2010 ) . `` bayesian inference for finite mixtures of univariate and multivariate skew - normal and skew-@xmath210 distributions , '' _ biostatistics _ , * 11 * , 317336 . hoogerheide , l. f. , kleibergen , f. , van dijk , h. k. ( 2007a ) . `` natural conjugate priors for the instrumental variables regression model applied to the angrist - krueger data , '' _ journal of econometrics_**138 * * , 63103 . hoogerheide , l. , kaashoek , j. f. , and van dijk , h. k. ( 2007b ) . `` on the shape of posterior densities and credible sets in instrumental variable regression models with reduced rank : an application of flexible sampling methods using neural networks , '' _ journal of econometrics _ , * 139 * , 154180 . ishwaran , h. and james , l. f. ( 2002 ) . `` approximate dirichlet process computing in finite normal mixtures : smoothing and prior information , '' _ journal of computational and graphical statistics _ , * 11 * , 508532 . kobayashi , g. ( 2015 ) . `` skew exponential power stochastic volatility model for analysis of skewness , non - normal tails , quantiles and expectiles , '' _ computational statistics _ , doi:10.1007/s00180 - 015 - 0596 - 4 . , kozubowski , t. j. , and podgrski , k. ( 2001 ) . _ the laplace distribution and generalizations : a revisit with applications to communications , economics , engineering , and finance _ , birkh@xmath295user , boston . munkin , m. k. and trivedi , p. k. ( 2003 ) . `` bayesian analysis of a self - selection model with multiple outcomes using simulation - based estimation : an application to the demand for healthcare , '' _ journal of econometrics _ , * 114 * , 197220 . ogasawara , k. and kobayashi , g. ( 2015 ) . `` the impact of social workers on infant mortality in inter - war tokyo : bayesian dynamic panel quantile regression with endogenous variables , '' _ cliometrica _ , * 9 * , 97130 . wichitaksorn , n. , choy , s. t. b. , and gerlach , r. ( 2014 ) . `` a generalized class of skew distributions and associated robust quantile regression models , '' _ the canadian journal of statistics _ , * 42 * , 5779596 . zhu , d. and galbraith , j. w. ( 2011 ) . `` modeling and forecasting expected shortfall with the generalized asymmetric student-@xmath210 and asymmetric exponential power distributions , '' _ journal of empirical finance _ , * 18 * , 765778 .
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this study proposes @xmath0-th tobit quantile regression models with endogenous variables . in the first stage regression of the endogenous variable on the exogenous variables , the assumption that the @xmath1-th quantile of the error term is zero
is introduced .
then , the residual of this regression model is included in the @xmath0-th quantile regression model in such a way that the @xmath0-th conditional quantile of the new error term is zero .
the error distribution of the first stage regression is modelled around the zero @xmath1-th quantile assumption by using parametric and semiparametric approaches .
since the value of @xmath1 is a priori unknown , it is treated as an additional parameter and is estimated from the data .
the proposed models are then demonstrated by using simulated data and real data on the labour supply of married women . * keywords * : asymmetric laplace distribution ; bayesian tobit quantile regression ; dirichlet process mixture ; endogenous variable ; markov chain monte carlo ; skew normal distribution ;
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the allen brain atlas ( aba , @xcite ) put neuroanatomy on a genetic basis by releasing voxelized , _ in situ _ hybridization data for the expression of the entire genome in the mouse brain ( ) . these data were co - registered to the allen reference atlas of the mouse brain ( ara , @xcite ) . about 4,000 genes of special neurobiological interest were proritized . for these genes an entire brain was sliced coronally and processed ( giving rise to the coronal aba ) . for the rest of the genome the brain was sliced sagitally , and only the left hemisphere was processed ( giving rise to the sagittal aba ) . + from a computational viewpoint , gene - expression data from the the aba can be studied collectively , thousands of genes at a time . indeed the collective behaviour of gene - expression data is crucial for the analysis of @xcite , in which the brain - wide correlation between the aba and cell - type - specific microarray data was studied . these microarray data characterize the transcriptome of @xmath0 different cell types , microdissected from the mouse brain , and collated in @xcite . however , for a given cell characterized in this way , it is not known where other cells of the same type are located in the brain . a linear model was proposed in @xcite ( see also @xcite ) , and used to estimate the region - specificity of cell types by linear regression with positivity constraint . the model was fitted using the coronal aba only , which allowed to obtain brain - wide results . however , this restriction implies that only one ish expression profile per gene was used to fit the model . this poses the problem of the error bars on the results of the model . since all the ish data in the aba were co - registered to the voxelized ara , so that data for the sagittal and coronal atlas can be treated computationally in the same way . however , the aba does not specify from which cell type(s ) the expression of each gene comes . + * gene expression energies from the allen brain atlas . * in the aba , the adult mouse brain is partitioned into @xmath1 cubic voxels of side 200 microns , to which ish data are registered @xcite for thousands of genes . for computational purposes , these gene - expression data can be arranged into a voxel - by - gene matrix . for a cubic labeled @xmath2 , the _ expression energy _ @xcite of the gene @xmath3 is a weighted sum of the greyscale - value intensities evaluated at the pixels intersecting the voxel : @xmath4 the analysis of @xcite is restricted to digitized image series from the coronal aba , for which the entire mouse brain was processed in the aba pipeline ( whereas only the left hemisphere was processed for the sagittal atlas ) . + * cell - type - specific transcriptomes and density profiles . * on the other hand , the cell - type - specific microarray reads collated in @xcite ( for @xmath5 different cell - type - specific samples studied in @xcite ) can be arranged in a type - by - gene matrix denoted by @xmath6 , such that @xmath7 and the columns are arranged in the same order as in the matrix @xmath8 of expression energies defined in eq . [ expressionenergy ] . + we proposed the following linear model in @xcite for a voxel - based gene - expression atlas in terms of the transcriptome profiles of individual cell types and their spatial densities : @xmath9 where the index @xmath10 denotes labels cell type , and @xmath11 denotes its ( unknown ) density at voxel labeled @xmath2 . the values of the cell - type - specific density profiles were computed in @xcite by minimizing the value of the residual term over all the ( positive ) density profiles , which amounts to solving a quadratic optimization problem ( with positivity constraint ) at each voxel . these computations can be reproduced on a desktop computer using the matlab toolbox ( bgea ) @xcite . for other applications of the toolbox see @xcite ( marker genes of brain regions ) , @xcite for co - expression properties of some autism - related genes , and @xcite for computations of stereotactic coordinates ) . the optimization procedure in our model is deterministic . on the other hand , decomposing the density of a cell type into the sum of its mean and gaussian noise is a difficult statistics problem ( see @xcite ) . some error estimates on the value of @xmath11 were obained in @xcite using sub - sampling techniques ( i.e. sub - sampling the data repeatedly by keeping only a random 10% of the coronal aba ) . this induced a ranking of the cell types based on the stability of the results against sub - sampling . however , the 10 % fraction is arbitrary ( even though it is close to the fraction of the genome covered by our coronal data set ) . + , defined in eq . [ meanfittingdef ] , for medium spiny neurons , labeled @xmath12 in our data set . the restriction to the left hemisphere comes from the use we made of sagittal image series , which cover the left hemisphere only.,scaledwidth=99.0% ] in the present work we simulated the variability of the spatial density of cell types by integrating the digitized sagittal image series into the data set . for gene labeled @xmath3 , the aba provides @xmath13 expression profiles , where @xmath13 is the number of image series in the aba for this gene . hence , instead of just one voxel - by - gene matrix , the aba gives rise to a family of @xmath14 voxel - by - gene matrices , with voxels belonging to the left hemisphere . a quantity computed from the coronal aba can be recomputed from any of these matrices , thereby inducing a distribution for this quantity . this is a finite but prohibitively large number of computations , so we took a monte carlo approach based on @xmath15 random choices of images series , described by the following pseudo - code : + the larger @xmath15 is , the more precise the estimates for the distribution of the spatial density of cell types will be . the only price we have to pay for th e integration of the sagittal aba is the restriction of the results to the left hemisphere in step 2 of the pseudo - code . the average density across random draws of image series for cell type labeled @xmath10 reads : @xmath16}(v ) . \label{meanfittingdef}\ ] ] ) , for medium spiny neurons , labeled @xmath12 , based on @xmath17 random draws . the right - most peak , corresponding to the striatum , is well - decoupled from the others , furthermore the other peaks are all centered close to zero ( making most of them almost invisible ) . medium spiny neurons have @xmath18 percent of their densities supported in the striatum , without any region gathering more than 5 percent of the signal in any of the random draws.,scaledwidth=110.0% ] a heat map of this average for medium spiny neurons ( extracted from the striatum ) is presented on fig . [ meanfittings ] . it is optically very similar to the ( left ) striatum , which allows the model to predict that medium spiny neurons are specific to the striatum ( which confirms prior neurobiological knowledge and therefore serves as a proof of concept for the model ) . + to compare the results to classical neuroanatomy , we can group the voxels by region according to the ara . since the number of cells of a given type in an extensive quantity , we compute the fraction of the total density contributed by a given brain region denoted by @xmath19 ( see the legend of fig . [ fittingdistrs ] for a list of possible values of @xmath19 ) : @xmath20}(t ) = \frac{1}{\sum_{v\in\mathrm{left\;hemisphere}}\rho_{[i],t}(v ) } \sum_{v\in v_r } \rho_{t,[i]}(v ) . \label{fittingdistrdef}\ ] ] we can plot the distribution of these @xmath15 values for a given cell type and all brain regions ( see fig . [ meanfittings ] for medium spiny neurons , which gives rise to the best - decoupled right - most peak in the distribution of simulated densities ) . moreover , we estimated the densities of the contribution of each region in the coarsest version of the ara to the total density of each cell type in the data set . for most cell types , this confirms the ranking of cell types by stability obtained in @xcite , but based on error bars obtained from the same set of genes in every fitting of the model ( see the accompanying preprint @xcite for exhaustive results for all cell types in the panel ) . the most stable results against sub - sampling tend to correspond to cell types for which the anatomical distribution of results is more peaked . the present analysis can be repeated when the panel of cell - type - specific microarray expands . 9 p. grange , j.w . bohland , b.w . okaty , k. sugino . h. bokil , s.b . nelson , l. ng , m. hawrylycz and p.p . mitra , _ cell - type based model explaining coexpression patterns of genes in the brain _ , proceedings of the national academy of sciences 2014 111 ( 14 ) 53975402 . p. grange , j.w . bohland , b.w . okaty , k. sugino . h. bokil , s.b . nelson , l. ng , m. hawrylycz and p.p . mitra , _ cell - type - specific transcriptomes and the allen atlas ( ii ) : discussion of the linear model of brain - wide densities of cell types _ , . y. ko , s.a . ament , j.a . eddy , j. caballero , j.c . earls , l. hood and n.d . price ( 2013 ) , _ cell - type - specific genes show striking and distinct patterns of spatial expression in the mouse brain _ , proceedings of the national academy of sciences , 110 ( 8) , 30953100 . tan , l. french and p. pavlidis ( 2013 ) , _ neuron - enriched gene expression patterns are regionally anti - correlated with oligodendrocyte - enriched patterns in the adult mouse and human brain , _ frontiers in neuroscience , 7 . p. grange , m. hawrylycz and p.p . mitra ( 2013 ) , _ computational neuroanatomy and co - expression of genes in the adult mouse brain , analysis tools for the allen brain atlas _ , quantitative biology , 1(1 ) : 91100 . ( doi ) 10.1007/s40484 - 013 - 0011 - 5 . p. grange and p.p . mitra ( 2012 ) , _ computational neuroanatomy and gene expression : optimal sets of marker genes for brain regions _ , ieee , in ciss 2012 , 46th annual conference on information science and systems ( princeton ) .
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the allen brain atlas ( aba ) of the adult mouse consists of digitized expression profiles of thousands of genes in the mouse brain , co - registered to a common three - dimensional template ( the allen reference atlas ) .
this brain - wide , genome - wide data set has triggered a renaissance in neuroanatomy .
its voxelized version ( with cubic voxels of side 200 microns ) can be analyzed on a desktop computer using matlab .
on the other hand , brain cells exhibit a great phenotypic diversity ( in terms of size , shape and electrophysiological activity ) , which has inspired the names of some well - studied cell types , such as granule cells and medium spiny neurons .
however , no exhaustive taxonomy of brain cells is available .
a genetic classification of brain cells is under way , and some cell types have been characterized by their transcriptome profiles .
however , given a cell type characterized by its transcriptome , it is not clear where else in the brain similar cells can be found .
the aba can been used to solve this region - specificity problem in a data - driven way : rewriting the brain - wide expression profiles of all genes in the atlas as a sum of cell - type - specific transcriptome profiles is equivalent to solving a quadratic optimization problem at each voxel in the brain .
however , the estimated brain - wide densities of 64 cell types published recently were based on one series of co - registered coronal _ in situ _ hybridization ( ish ) images per gene , whereas the online aba contains several image series per gene , including sagittal ones . in the presented work ,
we simulate the variability of cell - type densities in a monte carlo way by repeatedly drawing a random image series for each gene and solving optimization problems .
this yields error bars on the region - specificity of cell types . + _ prepared for the international conference on mathematical modeling in physical sciences , 5th-8th june 2015 , mykonos island , greece . _
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the @xmath0-function introduced by fox @xcite , will be represented and defined in the following manner : @xmath1 & = h_{p , q}^{m , n } \left\lbrack x \bigg| \begin{array}{@{}l@ { } } ( a_{1 } , \alpha_{1 } ) , \ldots , ( a_{p } , \alpha_{p})\\[.2pc ] ( b_{1 } , \beta_{1 } ) , \ldots , ( b_{q } , \beta_{q } ) \end{array}\right\rbrack\nonumber\\[.2pc ] & = \frac{1}{2 \pi i } \int_{l } \frac{\prod_{j = 1}^{m } \gamma ( b_{j } - \beta_{j } \xi ) \prod_{j = 1}^{n } \gamma ( 1 - a_{j } + \alpha_{j } \xi)}{\prod_{j = m + 1}^{q } \gamma ( 1 - b_{j } + \beta_{j } \xi ) \prod_{j = n + 1}^{p } \gamma ( a_{j } - \alpha_{j } \xi ) } x^{\xi } \ { \rm d}\xi.\end{aligned}\ ] ] for the nature of contour @xmath2 in ( 1.1 ) , the convergence , existence conditions and other details of the @xmath0-function , one can refer to @xcite . the general class of polynomials introduced by srivastava @xcite is defined in the following manner : @xmath3 = \sum\limits_{k = 0}^{[v / u ] } \frac{(-v)_{uk } a ( v , k)}{k ! } x^{k } , \quad v = 0 , 1 , 2 , \ldots,\ ] ] where @xmath4 is an arbitrary positive integer and coefficients @xmath5 are arbitrary constants , real or complex . @xmath6^{-\nu } h_{p , q}^{m , n } [ y \ { x + a + ( x^{2 } + 2ax)^{1/2 } \}^{-\mu}]\nonumber\\ & \qquad\qquad \times s_{v}^{u } [ z \ { x + a + ( x^{2 } + 2ax)^{1/2 } \}^{-\alpha } ] \hbox{d}x\nonumber\end{aligned}\ ] ] @xmath7 } ( -v)_{uk } a ( v , k ) \frac{(z / a^{\alpha})^{k}}{k ! } h_{p + 2 , q + 2}^{m , n + 2}\nonumber\\[.2pc ] & \quad\ , \times \left\lbrack ya^{-\mu } \bigg| \begin{array}{@{}l@ { } } ( -\nu - \alpha k , \mu ) , ( 1 + \lambda - \nu - \alpha k , \mu ) , ( a_{1 } , \alpha_{1 } ) , \ldots , ( a_{p } , \alpha_{p})\\[.2pc ] ( b_{1 } , \beta_{1}),\ldots,(b_{q } , \beta_{q } ) , ( 1 - \nu - \alpha k , \mu ) , ( -\nu - \alpha k - \lambda , \mu ) \end{array } \!\right\rbrack,\end{aligned}\ ] ] where 1 . @xmath8 , 2 . @xmath9 . to obtain the result ( 2.1 ) , we first express fox @xmath0-function involved in its left - hand side in terms of contour integral using eq . ( 1.1 ) and the general class of polynomials @xmath10 $ ] in series form given by eq . ( 1.2 ) . interchanging the orders of integration and summation ( which is permissible under the conditions stated with ( 2.1 ) ) and evaluating the @xmath11-integral with the help of the result given below @xcite : @xmath12^{-\nu } { \rm d}x\\ & \quad\ = 2 \nu a^{-\nu } \left(\frac{1}{2}a\right)^{z } [ \gamma ( 1 + \nu + z)]^{-1 } \gamma ( 2z ) \gamma ( \nu - z),\quad 0 < \hbox{re } ( z ) < \nu,\end{aligned}\ ] ] we easily arrive at the desired result ( 2.1 ) if in the integral ( 2.1 ) we reduce @xmath10 $ ] to unity and fox @xmath0-function to gauss hypergeometric function @xcite , we arrive at the following result after a little simplification : @xmath13^{-\nu}\nonumber\\ & \qquad\ \times { _ { 2}f_{1 } } ( a , b ; c ; y ( x + a + ( x^{2 } + 2ax)^{1/2})^{-1}){\rm d}x\nonumber\\ & \quad\ = 2^{1 - \lambda } \nu \gamma ( 2 \lambda ) a^{\lambda - \nu } \frac{\gamma ( \nu - \lambda)}{\gamma ( \nu + \lambda + 1)}\nonumber\\ & \qquad\ \times { _ { 4}f_{3 } } ( a , b , \nu - \lambda , \nu + 1 ; c , \nu , \nu + \lambda + 1 ; y / a),\end{aligned}\ ] ] where @xmath14 the importance of the result given by ( 3.1 ) lies in the fact that it not only gives the value of the integral but also ` augments ' the coefficients in the series in the integrand to give a @xmath15 series as the integrated series . a number of other integrals involving functions that are special cases of fox @xmath0-function @xcite and/or the general class of polynomials @xcite can also be obtained from ( 2.1 ) but we do not record them here . the authors are thankful to the worthy referee for his very valuable suggestions . the first author is thankful to the university grants commission , new delhi for providing necessary financial assistance to carry out the present work . the authors are thankful to k c gupta , jaipur for his useful suggestions .
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in the present paper we derive a unified new integral whose integrand contains products of fox @xmath0-function and a general class of polynomials having general arguments . a large number of integrals involving various simpler functions follow as special cases of this integral .
= msam10 at 10pt = mtgub at 10.4pt = tibi at 10.4pt [ theore]*theorem * [ theore]proposition [ theore]lemma [ theore]definition [ theore]corollary [ theore]remark [ theore]example
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You are an expert at summarizing long articles. Proceed to summarize the following text:
when an assembly of particles are placed in an asymmetric potential , a net dc particle flow can arise due to a ratchet effect that occurs when an ac drive is applied or when the substrate is periodically switched on and off in the presence of a thermal bath @xcite . ratchet effects on asymmetric substrates have been extensively studied in colloidal systems @xcite , granular matter @xcite , and polymers @xcite . ratchet effects also appear in ac - driven vortices in type - ii superconductors in the presence of an asymmetric substrate @xcite , such as a quasi - one - dimensional periodic array produced by asymmetrically modulating the sample thickness @xcite , etching funnel - shaped channels for vortex flow @xcite , introducing asymmetry to the sample edges @xcite , or adding periodic pinning arrays in which the individual pinning sites have some form of intrinsic asymmetry @xcite . at lower vortex densities when collective interactions between vortices are weak , the ratchet effect produces a dc flow of vortices in the easy flow direction of the asymmetric substrate ; however , when collective effects are present it is possible to have reversals of the ratchet effect where for one set of parameters the vortices move in the easy direction while for another set of parameters they move in the hard direction @xcite . a ratchet effect can also be produced by a pinning array containing symmetric pinning sites arranged with a density gradient . @xcite first studied vortex ratchet effects for random gradient array pinning geometries and found that the vortices undergo a net dc flow in the easy direction . experiments and simulations later showed that for a square array of pinning sites with constant pinning density but with a gradient in pinning site size , a variety of forward and reverse vortex ratchet behaviors occur @xcite . experiments on triangular pinning arrays with a density gradient also revealed a forward ratchet effect at low fields with a reversal at higher fields @xcite . . pinning gradients run along the @xmath0 direction . ( a ) conformal pinning ( conf ) . ( b ) random pinning with a periodic gradient ( randg ) . ( c ) square pinning with a gradient ( squareg ) . for each sample we apply an ac drive along the @xmath0 direction and measure the average net vortex displacements @xmath1 in the @xmath0 direction . , width=336 ] recently a new type of pinning geometry was proposed that is constructed by conformally transforming a triangular pinning lattice to create what is called a conformal pinning array , abbreviated conf in this work @xcite . as in the original triangular lattice , each pinning site in the transformed array has six neighbors separated by 60@xmath2 ; however , the distance to each neighbor is no longer constant , producing a density gradient in the pinning sites @xcite . in fig . [ fig:1](a ) we show a periodic lattice composed of three conformal pinning arrays with the same orientation . experimental structures with nearly conformal geometries have been observed for magnetically interacting particles subjected to a gravitational force , and due to the arching nature of the conformal array , the magnetic conformal crystals were dubbed gravity s rainbow structures @xcite . similar conformal structures have also been studied in foams @xcite and in charged particle ordering in confined geometries @xcite . in the superconducting system , an arrangement of two conf arrays placed with their minimum pinning density regions in the center of the sample produces an enhanced critical current or depinning force compared to an equivalent number of pinning sites placed in a uniform periodic , uniform random , or random density gradient array @xcite . the enhancement results both from the natural density gradient formed by the vortices as they enter the sample from the edges and form a bean state , and from the preservation of local six - fold ordering in the conf array @xcite . the suppression of easy vortex flow channels which arise in periodic and random arrays by the arching conformal structure also plays a role in the enhancement . only at the integer matching fields do the periodic pinning arrays produce higher critical currents than the conf array @xcite . subsequent experimental studies confirmed that the conf array produces enhanced pinning over a wide range of fields compared to uniform periodic pinning or uniform random pinning arrangements @xcite . other experiments have shown enhanced critical currents in systems with periodic arrays when a gradient in the pinning density is introduced @xcite . there have also been studies of hyperbolic - tesselation arrays which have a gradient in the pinning density @xcite . since conformal pinning arrays have an intrinsic asymmetry , it is natural to ask whether a ratchet effect can occur under application of an ac drive , and if so , whether this ratchet effect would be enhanced compared to other pinning array geometries with density gradients , or whether ratchet reversals could be possible . in fig . [ fig:1 ] we illustrate three examples of the gradient pinning array geometries we consider in this work : conformal pinning ( conf ) [ fig . [ fig:1](a ) ] , random pinning with a gradient ( randg ) [ fig . [ fig:1](b ) ] , and a square pinning array containing a gradient along the @xmath0 direction ( squareg ) [ fig . [ fig:1](c ) ] . in each case , both the gradient and the applied ac drive are along the @xmath0 direction , while the easy - flow direction for vortex motion is in the negative @xmath0 direction . in addition to superconducting vortex realizations of these geometries , similar pinning arrangements could also be created in colloidal systems using optical trap arrays . vs time measured in ac drive cycles for the systems in fig . 1 at @xmath3 , @xmath4 , and @xmath5 bottom black curve : conf array ; middle green curve : squareg array ; upper red curve : randg array . here the conf array produces a ratchet that is four times more effective than the squareg array and 20 times more effective than the randg array . , width=336 ] we consider a two - dimensional system with periodic boundary conditions in the @xmath0 and @xmath6 directions . the sample size is @xmath7 with @xmath8 , where distance is measured in units of the london penetration depth @xmath9 . the applied magnetic field is perpendicular to the system in the @xmath10 direction . our results apply to the london limit regime in which the vortices can be treated as rigid objects , when the coherence length @xmath11 is much smaller than @xmath9 . the pinning sites are modeled as in previous studies of conformal pinning arrays @xcite by non - overlapping parabolic circular traps with radius @xmath12 and a maximum pinning force of @xmath13 . we place the @xmath14 pinning sites in a conformal array ( conf ) , as described in previous work @xcite , in a random arrangement with a gradient ( randg ) , or in a square array with a density gradient along the @xmath0 direction ( squareg ) . the width of each pinning array segment is @xmath15 , and the segments are repeated three times across the sample as shown in fig . [ fig:1 ] . the total density of the pinning sites is @xmath16 . the sample contains @xmath17 vortices and we measure the magnetic field in units of @xmath18 , where @xmath19 is the matching field at which there is one vortex per pinning site . we obtain the initial vortex configuration by annealing from a high temperature molten state and cooling to @xmath20 or to a low but finite fixed temperature . after annealing , we apply an ac driving force to all the vortices . the dynamics of an individual vortex @xmath21 is obtained by integrating the following overdamped equation of motion : @xmath22 here @xmath23 is the damping constant which is set equal to 1 . the repulsive vortex - vortex interaction force is given by @xmath24 , where @xmath25 is the location of vortex @xmath21 , @xmath26 is the modified bessel function , @xmath27 , @xmath28 , @xmath29 , @xmath30 is the flux quantum , and @xmath31 is the permittivity . the vortex - pinning interaction force is @xmath32 , where @xmath33 is the heaviside step function , @xmath34 is the pinning radius , @xmath13 is the pinning strength , @xmath35 is the location of pinning site @xmath36 , @xmath37 , and @xmath38 . all forces are measured in units of @xmath39 and lengths in units of @xmath9 . thermal forces are represented by langevin kicks @xmath40 with the properties @xmath41 and @xmath42 , where @xmath43 is the boltzmann constant . the ac driving force is @xmath44 where @xmath45 is the ac amplitude . to characterize the ratchet effect we measure the average net displacement of all vortices from their starting positions as a function of time , @xmath46 , where @xmath47 is the @xmath0 position of vortex @xmath21 at time @xmath48 and @xmath49 is an initial reference time . this measure produces a sinusoidal signal , as shown in fig . [ fig:2 ] ; the presence of a net drift indicates that a ratchet effect is occurring . we condense this information into a single number @xmath50 , the value of @xmath1 at @xmath51 ac drive cycles . except where otherwise noted , we consider a fixed ac frequency of @xmath52 and a time step of @xmath53 , so that a single drive cycle has a period of 8000 simulation time steps . in fig . [ fig:2 ] we plot the average net displacement @xmath1 versus time for conf , randg , and squareg arrays with @xmath54 , @xmath55 , and @xmath56 during 100 ac drive cycles . here the overall drift of each curve indicates that all the arrays produce a ratchet effect with the vortices translating in the negative @xmath0 direction . the conf array generates the largest ratchet effect , with the vortices translating distances up to @xmath57 per drive cycle . the ratchet effect for the conf array is about four times larger than that of the squareg array and 20 times larger than that of the randg array . , @xmath58 , and @xmath59 , highlighting the enhanced effectiveness of the ratchet mechanism in the conf array . ( a ) the trajectories for the positive half of the ac drive cycle in the randg array showing the formation of disordered flow channels . ( b ) the negative half of the ac drive cycle in the randg array has a similar pattern and density of flow channels . ( c ) in the positive half of the ac drive cycle for the conf array , the vortices can not move past the densely pinned region . ( d ) in the negative half of the ac drive cycle in the conf array , the vortices can easily funnel between the arches in the conformal array . ( e ) in the positive half of the ac drive cycle for the squareg array , vortices can slip through the interstitial regions between pinned vortices . ( f ) similar interstitial motion occurs in the negative half of the ac drive cycle for the squareg array . , width=336 ] the relative effectiveness of the different arrays can be more clearly understood by plotting the trajectories of the vortices during the positive and negative portions of a single ac cycle . figure [ fig:3](a , b ) shows the trajectories for both halves of the ac cycle in a a randg array with @xmath3 , @xmath58 and @xmath59 . under both positive and negative drive , the vortices form disordered flow paths with a similar density that is independent of the driving direction . in contrast , in fig . [ fig:3](c ) during the positive portion of the ac driving cycle for the conf array , almost no vortices can cross the densely pinned regions of the sample ; instead , the vortices either become trapped at pining sites or remain localized in interstitial cages formed by the pinned vortices . figure [ fig:3](d ) shows that in the negative portion of the cycle for the conf array , numerous vortices move into the interstitial regions and funnel through the conformal arch structures , producing significant vortex motion in the negative @xmath0 direction . in the squareg system , [ fig:3](e , f ) show that vortex motion is strongly suppressed for both directions of drive and occurs only when interstitial vortices manage to squeeze between equally spaced occupied pinning sites . the barrier to this type of vortex motion is the same in each half of the cycle . in contrast , for the conf array the perpendicular spacing between pinned vortices in the sparse portion of the array is larger than the equivalent spacing between pinned vortices in the dense portion of the array , so the interstitial vortices experience much different effective caging barriers when entering the sparse side of the array than when entering the dense side of the array . in the randg arrays , channels of easy vortex flow occur somewhere in the sample with equal probability for both the positive and negative portions of the ac drive cycle . , the average net displacement per vortex after 50 ac drive cycles , vs @xmath13 for the conf ( red circles ) , randg ( yellow squares ) , and squareg ( green diamonds ) arrays . here @xmath60 and @xmath61 . in general , the ratchet effect is suppressed for weak pinning and for strong pinning . ( a ) at @xmath3 , the conf array exhibits the strongest ratchet effects , followed by the squareg array . the randg array has the weakest ratchet effect . ( b ) at @xmath62 the ratchet effect extends to higher values of @xmath13 in all the systems . the conf ratchet is still the most effective . ( c ) at @xmath63 , the squareg ratchet is more effective than the conf or randg ratchets . , width=336 ] in fig . [ fig:4](a ) we plot @xmath50 , the average net displacement per vortex after 50 ac drive cycles , versus @xmath13 for conf , randg , and squareg samples with @xmath3 and @xmath60 . for weak pinning @xmath64 , the vortices move elastically and easily slide over the pinning sites so that there is no ratchet effect in any of the arrays . for @xmath65 most of the vortices become increasingly pinned and the ratchet effect is reduced . the optimal ratchet effect occurs for the conf array at @xmath66 , where there is a mixture of pinned vortices coexisting with vortices that move temporarily through the interstitial regions as illustrated in fig . [ fig:3](c , d ) . the squareg array has a weaker ratchet effect in the range @xmath67 , with a relatively sharp cutoff at the upper end of this range that occurs when the ability of the pinned vortices to shift inside the pinning sites is reduced , preventing the interstitial vortices from slipping between occupied pinning and causing the motion to become localized , as shown in fig . [ fig:3](e , f ) . there is a weak ratchet effect for the randg array with an extremum at @xmath68 where the combination of the ac drive and the vortex - vortex interactions causes a portion of the vortices to depin . the maximum magnitude of the ratchet effect for the randg array is smaller than that for the squareg array ; however , the effect occurs over a wider range of @xmath13 . figure [ fig:4](b ) shows @xmath50 versus @xmath69 for @xmath62 , where there are more interstitial vortices . here the range of @xmath13 over which the ratchet effect occurs for the conf and randg arrays extends up to @xmath70 , with the ratcheting for @xmath71 completely dominated by the flow of interstitial vortices . for the squareg array the ratchet effect is lost for @xmath72 , the point at which the interstitial vortices can no longer slip through the one - dimensional interstitial channels of the array . in fig . [ fig:4](c ) we plot @xmath50 versus @xmath69 at @xmath63 , where there are few interstitial vortices . here most of the motion occurs when vortices jump from one pinning site to another . the ratchet effect for all three arrays vanishes for @xmath73 when vortex hopping is suppressed . at this vortex density , the ratchet effect is most pronounced for the squareg array , where the vortices are able to hop along one - dimensional channels of pinning sites . vs ac amplitude @xmath45 for conf ( red circles ) , randg ( yellow squares ) , and squareg ( green diamonds ) arrays with @xmath4 and @xmath61 . ( a ) at @xmath3 , the ratchet effect is reduced at low @xmath45 when the vortices are pinned as well as at higher @xmath45 when the vortices move rapidly over the pinning array . ( b ) @xmath62 . ( c ) at @xmath63 the squareg array produces the most effective ratchet . , width=336 ] in fig . [ fig:5](a ) we plot @xmath50 versus the ac drive amplitude @xmath45 for conf , randg , and squareg samples with @xmath4 and @xmath3 . for @xmath74 , the vortices are mostly pinned and the ratchet effect is absent for all three of the pinning geometries . at intermediate @xmath45 the conf array has the strongest ratchet effect , with an extremum in @xmath50 at @xmath75 . the squareg array has the next most effective ratchet effect , with an optimal magnitude at @xmath76 . for higher values of @xmath45 , the vortices are all in motion during some portion of the driving cycle and the ratchet effect gradually decreases to zero with increasing @xmath45 . we observe a similar trend at @xmath62 as shown in fig . [ fig:5](b ) . here the ratchet effect for the conf array extends up to much larger values of @xmath45 ; however , the maximum value of @xmath77 is slightly smaller than for the @xmath3 case . for @xmath63 in fig . [ fig:5](c ) , the dominant motion is hopping of vortices from pinning site to pinning site . here the ratchet effect is strongest for the squareg array , similar to what is shown in fig . [ fig:4](c ) . vs @xmath18 for conf ( red circles ) , randg ( yellow squares ) , and squareg ( green diamonds ) arrays with @xmath60 . ( a ) at @xmath4 the ratchet effect is negative for the entire range of @xmath18 . ( b ) at @xmath58 there is a reversal in the ratchet effect for the conf array for @xmath78 . , width=336 ] in fig . [ fig:6](a ) we plot @xmath50 versus @xmath18 for conf , randg , and squareg arrays with @xmath4 and @xmath60 . here the conf array outperforms the randg array for all fields and the squareg array for @xmath79 . for @xmath80 the vortex - vortex interactions become dominant and the ratchet effect is suppressed in all the arrays . in the squareg array , due to the periodic ordering along the @xmath6 direction , some commensuration effects occur , such as enhanced pinning near @xmath3 which locally suppresses the ratchet effect . vs @xmath18 for the conf array in fig . [ fig:6](b ) highlighting the vortex ratchet reversal effect from negative for @xmath81 to positive for @xmath82 to negative again at higher fields . ( b ) @xmath1 vs time in ac drive cycle numbers for the system in ( a ) at @xmath83 ( lower blue curve ) where the vortex motion is in the negative @xmath0 direction and at @xmath84 ( upper red curve ) where the motion is in the positive @xmath0 direction . , width=336 ] figure [ fig:6](b ) shows @xmath50 versus @xmath18 for @xmath58 . in this case , the ratchet effect for the squareg array is lost for @xmath85 when the vortices become strongly pinned at the pinning sites . in general , the ratchet effect for the conf array is stronger than that for the squareg and randg arrays , with a local extremum for the ratchet effect in the negative or normal direction occurring at @xmath86 . the squareg array has a local extremum in @xmath50 in the negative direction at @xmath62 , followed by a sharp drop in @xmath50 for @xmath87 . we find a ratchet reversal in the conf array , where @xmath50 switches from negative to positive over the range @xmath88 . there is a local maximum in the positive ratchet effect at @xmath84 . in fig . [ fig:7](a ) we show a highlight of @xmath50 versus @xmath18 from fig . [ fig:6](b ) for the conf array indicating that two reversals in the ratchet effect occur . figure [ fig:7](b ) illustrates @xmath1 vs time in ac drive cycles for the system in fig . [ fig:7](a ) at @xmath83 , where the motion is in the negative @xmath0 direction , and at @xmath84 , where the motion is in the positive @xmath0 direction , showing more clearly the change in the net direction of vortex motion . vs @xmath45 for conf arrays with @xmath58 . ( a ) at @xmath84 , there is a transition from a negative ratchet effect at low @xmath45 to a positive ratchet effect , followed by a second transition back to a negative ratchet effect . ( b ) at @xmath89 the ratchet effect is always negative ; however , there is a local minimum and a local maximum of the ratchet effect . ( c ) at @xmath90 , the ratchet effect is always negative and has few features . , width=336 ] in fig . [ fig:8](a ) we plot @xmath50 versus @xmath45 for a conf array with @xmath91 at @xmath84 , where there are multiple reversals in the ratchet effect . for @xmath92 there is no ratchet effect since the vortices move only small distances . a negative ratchet effect occurs for @xmath93 , while for @xmath94 there is a positive ratchet effect with a maximum amplitude at @xmath60 . there is another transition to a weaker negative ratchet effect for @xmath95 , and @xmath50 gradually approaches zero for high values of @xmath45 . figure [ fig:8](b ) shows that at @xmath89 , the ratchet effect is always negative ; however , there are still local features in the response such as at @xmath96 where the negative ratchet effect is strongly reduced . in fig . [ fig:8](c ) at @xmath90 , the ratchet effect is strongly negative with an extremum in @xmath50 near @xmath97 . the ratchet effect goes to zero for increasing @xmath45 . vs @xmath13 for conf arrays with @xmath84 . ( a ) at @xmath60 there are multiple reversals as @xmath13 increases . ( b ) at @xmath98 there are again multiple reversals and the positive ratchet effect extends over a wider range of @xmath13 . ( c ) at @xmath99 there is a weak negative ratchet effect . , width=336 ] in fig . [ fig:9](a ) we show @xmath50 versus @xmath13 for a conf array at @xmath84 and @xmath60 . there is a negative ratchet effect for @xmath100 , a positive ratchet effect for @xmath101 , and a much larger negative ratchet effect for @xmath102 . at intermediate @xmath69 when there is a positive ratchet effect , vortices can be temporarily trapped by pinning sites . the negative ratchet effect for large @xmath69 arises from the interstitial flow of vortices , and @xmath50 saturates at large @xmath69 since the caging barrier experienced by interstitial vortices from the neighboring pinned vortices does not increase with increasing @xmath69 . in fig . [ fig:9](b ) , at @xmath98 there is a negative ratchet effect for @xmath103 , a positive ratchet effect for @xmath104 , and another negative ratchet regime for @xmath105 . the vortices at the pinning sites remain permanently pinned for @xmath105 . the positive ratchet effect is larger and extends out to higher values of @xmath69 for the @xmath106 system compared to the @xmath107 system . figure [ fig:9](c ) shows that at @xmath99 , there is a weak negative ratchet effect for all values of @xmath13 . although we focus here on the conf array , we also found that some weak ratchet reversals are possible in the squareg array ; however , we did not observe a vortex ratchet reversal for the randg array . , the fraction of sixfold - coordinated particles , vs time in ac drive cycles for conf arrays from fig . [ fig:9](a ) with @xmath108 and @xmath107 . light orange lines indicate the phase of the drive cycle . mp is the maximum positive drive and mn is the maximum negative drive . ( a ) at @xmath109 the ratchet effect is negative . the system is most ordered whenever the magnitude of the ac drive is maximum ; however , the ordering peaks for the negative portions of the drive cycle are slightly higher than those for the positive portions of the drive cycle . ( b ) at @xmath58 the ratchet effect is positive . the system is most ordered whenever the magnitude of the ac drive is close to zero , but the net motion is determined by the relatively larger ordering at mp points compared to mn points . , width=336 ] in order to better understand the vortex dynamics and ordering during an individual ac cycle , in fig . [ fig:10 ] we plot the time series of the fraction of sixfold - coordinated vortices , @xmath110 , versus time . here @xmath111 , the coordination number of vortex @xmath21 , is obtained from a voronoi construction . superimposed over the plot is a curve showing the phase of the ac drive , and the points at which the drive reaches its maximum positive value are marked mp while those at which the drive reaches its maximum negative value are marked mn . figure [ fig:10](a ) shows @xmath112 versus time for the system from fig . [ fig:9](a ) with @xmath84 and @xmath113 at @xmath109 where there is a negative ratchet effect . here @xmath114 at the start of each drive cycle when the drive magnitude is zero , deceases slightly when the drive becomes positive and the system disorders , then reaches its highest values of @xmath115 in the mp portions of the drive cycle and @xmath116 in the mn portions of the drive cycle . when the magnitude of the ac drive is maximum , all the vortices move elastically , and since they are slightly more ordered during the negative cycle of the drive , they can slide slightly further in the negative @xmath0 direction than in the positive @xmath0 direction , giving a negative ratchet effect . at @xmath58 in fig . [ fig:10](b ) , the ratchet effect is positive and the vortex ordering is reversed . the vortices are now the most ordered when the magnitude of the ac drive is close to zero , and they are disordered when the ac drive magnitude reaches a maximum . during the mp portion of the drive cycle , @xmath117 , while in the mn portion of the drive cycle the system is more disordered with @xmath118 . the more ordered vortices are able to slide slightly further in the positive @xmath0 direction , resulting in a net positive ratchet effect . there is also an asymmetry in the ordering at the zero force portions of the drive cycle . the value of @xmath119 at cycle times of @xmath120 and @xmath121 is smaller than that at times of @xmath122 and @xmath123 ; however , since the vortices are not moving during this portion of the cycle , this asymmetry does not produce a preferred direction of motion . in general we find that the ordering of the vortices at the mp and mn points of the drive determines the direction of the ratchet motion , with the net ratchet effect occurring in whichever drive direction generates the most ordered vortex arrangement . vs @xmath124 for conf ( red circles ) , randg ( yellow squares ) , and squareg ( green diamonds ) arrays with @xmath3 and @xmath60 . ( a ) at @xmath4 , thermal effects reduce the ratchet effect . ( b ) at @xmath125 thermal effects can increase the ratchet effect over a range of @xmath124 . , width=336 ] we next consider thermal effects on the ratchet response . for weak pinning , the addition of thermal fluctuations monotonically decreases the ratchet effect for all three geometries , as shown in fig . [ fig:11](a ) for @xmath4 , @xmath60 , and @xmath3 . as before , the ratchet effect is most pronounced for the conformal array . as @xmath13 increases , the vortices become more strongly pinned , and the addition of thermal fluctuations can increase the ratchet effect by permitting vortices to escape from pinning sites or interstitial caging sites via thermal activation . in fig . [ fig:11](b ) we plot @xmath50 versus @xmath124 in samples with @xmath69 increased to @xmath126 , showing a strong ratchet effect in the conf array . here , the ratchet effect is lost at small @xmath124 since the vortices are strongly pinned , and the ratchet effect also disappears for high values of @xmath124 when the thermal fluctuations become so strong that the vortices enter a molten state that interacts too weakly with the substrate for an asymmetry in the response to positive and negative drives to be noticeable . the largest ratchet signatures appear for intermediate @xmath124 . in general , when @xmath13 increases , the point at which the magnitude of the ratchet effect is largest shifts to higher values of @xmath127 . versus ac cycle period in simulation time steps for conf ( red circles ) , randg ( yellow squares ) , and squareg ( green diamonds ) arrays with @xmath3 , @xmath60 , and @xmath4 . @xmath50 increases linearly with the drive cycle period . , width=336 ] we find that @xmath50 increases linearly as the period of the ac driving cycle increases , as illustrated in fig . [ fig:12 ] for conf , randg , and squareg arrays with @xmath3 , @xmath4 , and @xmath60 . as the other parameters are varied , we find a robust increase in the magnitude of the ratchet effect with decreasing ac frequency . vs time measured in ac drive cycles for colloidal particles interacting with a conf array ( bottom black curve ) , a squareg array ( middle green curve ) , and a randg array ( upper red curve ) with @xmath128 , @xmath4 , @xmath129 , and @xmath130 . as in the vortex case shown in fig . [ fig:2 ] , the conf array produces the strongest ratchet effect . , width=336 ] vs colloid - colloid interaction strength @xmath131 for the conf colloid system in fig . [ fig:13 ] with @xmath128 , @xmath4 , and @xmath129 . ( b ) the corresponding fraction of sixfold coordinated colloids @xmath112 vs @xmath131 . here the maximum ratchet effect occurs when @xmath132 , indicating that although colloid - colloid interactions remain important , the system is in a disordered state . when the system forms a crystalline state with @xmath133 , the ratchet effect disappears . , width=336 ] ratchet effects can be generated in systems of colloidal particles interacting with various types of periodic arrays of traps that are created using optical means @xcite . the ability to make structures similar to conformal lattices has been demonstrated by xiao _ @xcite , who examined a colloidal ratchet effect on optical traps forming fibonacci spirals . in that case the ratchet effect is induced by rotating the potential through a three - step cycle . here we consider the ac - driven motion of colloidal particles over a conf array . the equation of motion for colloids is similar to that given in eqn . 1 for vortices , except that the pairwise repulsive colloid - colloid interaction potential has the form @xmath134 , where @xmath135 , @xmath136 is the dimensionless interaction strength , @xmath137 is the effective charge of the colloidal particles , @xmath138 is the solvent dielectric constant , and @xmath139 is the screening length which we set equal to @xmath121 . the number of colloids in the sample is @xmath140 . for our parameters , the interactions between colloids for @xmath141 is much larger than the interactions between vortices separated by the same distance , while for @xmath142 the colloidal interaction strength falls off much more rapidly than the vortex - vortex interaction strength , so that nearest neighbor interactions are dominant in the colloidal system . we have conduced a series of simulations for colloidal particles moving through conf , randg , and squareg pinning landscapes under an ac driving force , and find results very similar to those obtained in the vortex system . for example , in fig . [ fig:13 ] we plot @xmath143 versus time for colloids interacting with conf , squareg , and randg arrays for @xmath128 , @xmath144 , @xmath129 , and @xmath130 , where we observe that just as in the vortex case , the conf array produces the most pronounced ratchet effect , the squareg array shows a weak ratchet effect , and the randg array does not exhibit a ratchet effect . since the effective charge on the colloids can be changed experimentally , it is possible to hold the substrate strength fixed and modify how strongly the colloids interact with one another . in fig . [ fig:14](a ) we plot @xmath50 versus @xmath131 for the conf array from fig . [ fig:13 ] , and in fig . [ fig:14](b ) we show the corresponding fraction of sixfold coordinated colloids @xmath112 versus @xmath131 . for @xmath145 the colloids all become pinned in the pinning sites since @xmath146 . as @xmath131 increases , the colloid - colloid interactions become important and a ratchet effect arises with a maximum amplitude near @xmath147 . the largest ratchet effect is associated with a sixfold ordering fraction of @xmath132 , indicating that the colloids are still disordered , with some colloids trapped in pinning sites and others occupying interstitial regions between pins . for @xmath148 the ratchet effect begins to diminish with increasing @xmath149 while simultaneously @xmath112 increases , indicating an increase in the ordering of the colloids . for @xmath150 , the colloids form a rigid triangular lattice as indicated by the fact that @xmath133 , and the ratchet effect disappears . these results show that in order for a ratchet effect to appear in the gradient pinning arrangements , it is generally necessary for plasticity or defects in the colloid or vortex lattice to appear . our results with the colloidal system indicate that pronounced ratchet effects should be realizable in a variety of systems where assemblies of interacting particles are driven with an ac drive over conformal array substrates . we examine ratchet effects for ac driven vortices interacting with a conformal pinning array and with square and random pinning arrays containing a gradient along one direction . in general , the conformal pinning array produces the most pronounced ratchet effect , particularly for fields greater than the first matching field . the enhanced effectiveness of the conformal ratchet results in part from the fact that the pinning sites in the low density portion of the array are widely spaced not only parallel to but also perpendicular to the net pinning gradient direction , permitting the easy flow of interstitial vortices through the sparse portion of the array . in contrast , for the square pinning array with a gradient , the perpendicular distance between pinning sites is constant throughout the array , producing the same barrier for interstitial motion in both the sparse and dense portions of the array and reducing the relative magnitude of the ratchet effect for fields at which interstitial vortices are present . for the random pinning array with a gradient , channels of easy vortex flow form for driving in either direction , significantly reducing the effective asymmetry of the array . we find that the conformal array exhibits a series of vortex ratchet reversals as a function of vortex density , ac drive amplitude , and pinning strength , and we show that the direction of the ratchet is determined by the amount of order present in the vortex lattice at different phases of the ac driving cycle . finally , we demonstrate that the conformal array also produces a larger ratchet effect compared to square and random pinning arrays with a gradient in systems of colloidal particles , suggesting that pronounced ratchet effects should be a general feature of particles moving over conformal arrays . p. pieranski , in _ phase transitions in soft condensed matter _ , edited by t. riste and d. sherrington ( plenum , new york , 1989 ) , p. 45 ; f. rothen , p. pieranski , n. rivier , and a. joyet , eur . j. phys . * 14 * , 227 ( 1993 ) .
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a conformal transformation of a uniform triangular pinning array produces a structure called a conformal crystal which preserves the six - fold ordering of the original lattice but contains a gradient in the pinning density .
here we use numerical simulations to show that vortices in type - ii superconductors driven with an ac drive over gradient pinning arrays produce the most pronounced ratchet effect over a wide range of parameters for a conformal array , while square gradient or random gradient arrays with equivalent pinning densities give reduced ratchet effects . in the conformal array , the larger spacing of the pinning sites in the direction transverse to the ac drive permits easy funneling of interstitial vortices for one driving direction , producing the enhanced ratchet effect . in the square array ,
the transverse spacing between pinning sites is uniform , giving no asymmetry in the funneling of the vortices as the driving direction switches , while in the random array , there are numerous easy - flow channels present for either direction of drive .
we find multiple ratchet reversals in the conformal arrays as a function of vortex density and ac amplitude , and correlate the features with a reversal in the vortex ordering , which is greater for motion in the ratchet direction .
the enhanced conformal pinning ratchet effect can also be realized for colloidal particles moving over a conformal array , indicating the general usefulness of conformal structures for controlling the motion of particles .
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apart from providing independent supporting evidence of the big bang , the detection and subsequent study of cosmological chemical evolution provides the empirical details of galaxy formation and evolution . how primordial and processed gas is consumed by star formation , the dominant feedback processes and merging scenarios , may all contribute to the overall evolution of chemical abundances . one high - precision probe of this evolution is the spectroscopic study of damped lyman-@xmath0 systems ( dlas ) : absorbers with neutral hydrogen column densities @xmath2 . although these observations demonstrate that dlas arise along lines of sight through distant galaxies , they do not directly disclose details such as the galaxy s morphology , luminosity , mass or age . there exists substantial evidence that dlas arise in a variety of galaxy types . at low-@xmath3 ( @xmath4 ) , kinematic and hi 21-cm absorption studies ( e.g. @xcite ; @xcite ) suggest a significant contribution from spiral galaxies , a view supported at high-@xmath3 by kinematic modelling and abundance studies @xcite . however , direct imaging at low-@xmath3 ( e.g. * ? ? ? * ) reveals that dla hosts are a mix of irregulars , spirals and low surface - brightness galaxies ( lsbs ) . a recent @xmath5 hi 21-cm emission study @xcite supports this . @xcite argue that the number of dlas per redshift interval and the @xmath6 distribution imply that dlas at @xmath7 are a mix of spirals and lsbs whereas , at higher @xmath3 , they are more likely to be dwarfs . this is supported by fitting of chemical evolution models to dla metal abundances ( e.g. * ? ? ? * ) and by recent 21-cm absorption searches at high redshift @xcite . though further work is clearly needed , the direct and indirect evidence for a ` mixed bag ' of dla hosts is already compelling . evidence for an increase in dla metallicities , [ m / h ] , with cosmic time has emerged only gradually @xcite , the latter reference using over 100 dlas to provide the strongest statistical evidence so far . such large samples are required due to the huge scatter ( @xmath8 ) in [ m / h ] at a given epoch ( see fig . [ fig:1 ] ) , a diversity expected given the variety of dla hosts discussed above . however , the diversity in [ m / h ] could also significantly _ bias _ any estimate of chemical evolution , as could several observational selection effects ( e.g. * ? ? ? * ) . in @xcite we suggest that by selecting those dlas in which h@xmath1 absorption is detected , one may reduce or possibly avoid some of the biases besetting the general dla population . there are currently 9 confirmed h@xmath1-bearing dlas ( see below ) and , typically , h@xmath1 is detected in only a few velocity components . these h@xmath1-bearing components seem distinct from the others , showing lower temperatures and higher dust depletion factors [ m / fe ] @xcite . therefore , h@xmath1-bearing dlas might be a less biased tracer of chemical evolution than the general dla population since they may allow one to focus on a narrower range of physical conditions throughout cosmic time . , dust - depletion [ m / fe ] , and molecular fraction ( @xmath9 ) evolution for confirmed ( filled circles ) and tentative ( hollow circle ) h@xmath1 detections compared with values from the @xcite dlas ( small diamonds ) . the values adopted are discussed in the text . linear fits to the h@xmath1-bearing dlas ( long dashes ) and general dlas ( short dashes ) were obtained by quadrature addition of a constant to the individual error bars such that @xmath10 . due to its much lower @xmath11 and potentially high photo - dissociation rate @xcite , we do not include 0515@xmath124414 in these fits ( but see table [ tab:1]).,scaledwidth=70.0% ] h@xmath1 is detected in dlas via the lyman and werner - band uv absorption lines which generally lie in the hi lyman-@xmath0 forest . a compilation of results from h@xmath1 searches in dlas is given in table 8 of ( * ? ? ? * , hereafter l03 ) , for which 7 dlas have confirmed h@xmath1 detections and metallicity measurements : 0013@xmath12004 ( @xmath13 ) , 0347@xmath123819 , 0405@xmath12443 ( @xmath14 ) , 0528@xmath122505 , 0551@xmath12366 , 1232@xmath15082 and 1444@xmath15014 . the dla towards 0013@xmath12004 @xcite comprises several absorbing components , of which 2 are dominant ( their components @xmath16 and @xmath17 ) : @xmath6 is measured at the mean redshift and we use the mean @xmath18h@xmath19 with error given by the range of @xmath18h@xmath19 in l03 . for the dla towards 1232@xmath15082 we use the @xmath18h@xmath19 value and error from @xcite . we include two further dlas : ( i ) the recent detection towards 0515@xmath124414 @xcite and ( ii ) 0000@xmath122620 , regarded as only a tentative detection by l03 . however , the h@xmath1 identification has been carefully scrutinised by @xcite and relies on two h@xmath1 absorption features , the l(4 - 0)r1 and w(2 - 0)q(1 ) lines , the former appearing relatively clean from lyman-@xmath0 forest blending . a tentative detection of h@xmath1 in 0841@xmath151256 has also been reported @xcite , though confirmation requires future data and analyses . lcccccc & & & + sample & @xmath20 & @xmath21 & @xmath20 & @xmath21 & @xmath22 & @xmath23 + + exc . 0515 & 0.08 & @xmath24 & 0.14 & @xmath25 & 2.2 & 0.28 + inc . 0515 & 0.04 & @xmath26 & 0.03 & @xmath27 & 1.9 & 0.33 + + exc . 0515 & 0.05 & @xmath28 & 0.24 & @xmath29 & 1.7 & 0.47 + inc . 0515 & 0.06 & @xmath30 & 0.25 & @xmath31 & 1.0 & 0.95 + + exc . 0515 & 0.04 & @xmath32 & & & & + inc . 0515 & 0.06 & @xmath33 & & & & + [ tab:1 ] in fig . [ fig:1 ] we plot against @xmath11 the metallicity [ m / h ] , dust - depletion factor [ m / fe ] and molecular fraction @xmath34 $ ] for the 9 h@xmath1-bearing dlas , comparing the former two quantities with dlas in @xcite with zn , si , s or o metallicity over the relevant redshift range . [ m / h ] for the h@xmath1-bearing dlas is [ zn / h ] , except for 0347@xmath123819 and 1232@xmath15082 where it is [ s / h ] and [ si / h ] . for 0515@xmath124414 , [ zn / fe ] is from @xcite . the main results from fig . [ fig:1 ] are summarized by the statistics in table [ tab:1 ] : ( i ) [ m / h ] , [ m / fe ] and @xmath9 for the h@xmath1-bearing dlas are anti - correlated with @xmath11 at the 95% confidence level ( i.e. more significant than for the general dla population ) , ( ii ) [ m / h ] shows a steeper evolution with @xmath11 and a smaller scatter about the slope than the general dla population , ( iii ) [ m / fe ] in h@xmath1-bearing dlas shows strong evolution with @xmath11 while the general dlas show no evidence for evolution , and ( iv ) @xmath9 ranges over @xmath35 and shows a very steep evolution with @xmath11 . the new results ( i)(iii ) support our hypothesis that h@xmath1-bearing dlas form a chemically distinct sub - class and may trace chemical evolution more reliably . the @xmath9@xmath11 correlation was studied by l03 and is discussed further below . the [ m / h]s and @xmath9s in fig . [ fig:1 ] are measured using the total @xmath6 across the dla profile and are not specific to the h@xmath1-bearing components . [ m / fe ] is generally found to be uniform across most dla profiles @xcite , indicating that [ m / h ] should be uniform . however , the h@xmath1-bearing components typically have much higher [ m / fe ] values than other components in the same dla ( e.g. l03 and 0347@xmath12383 s [ si / fe ] profile in @xcite ) . these components usually dominate the non - refractory metal - line profiles and so , although [ m / h ] and @xmath9 will be systematically underestimated , the effect will not be large . the fitted slopes in fig . [ fig:1 ] are likely to be reasonably robust against this effect , but a larger sample and more detailed study is clearly required . what observational selection effects and biases could contribute to the steep @xmath9@xmath11 evolution ? firstly , the sample is inhomogeneous since the spectra do not all have similar s / n and since the h@xmath1 detection methods and criteria were not uniform . indeed , the weak h@xmath1 lines detected towards 0000@xmath122620 are at the typical non - detection level ( see fig . 16 in l03 ) . the @xmath9@xmath11 correlation is therefore only tentative . secondly , the h@xmath1 detection limit will alter with @xmath3 : equivalent widths increase but lyman-@xmath0 forest blending worsens with increasing @xmath3 . though this is an unlikely culprit for the @xmath35 evolution observed , precise quantification of these competing effects requires numerical simulations . dlas containing large amounts of dust could suppress detection of their background quasars and may therefore be ` missing ' from our sample @xcite . however , since [ m / h ] , [ m / fe ] and @xmath9 are positively correlated with each other ( e.g. l03 ) , this effect is likely to suppress , rather than create , the correlations in fig . [ fig:1 ] . a recent survey for dlas towards radio - selected quasars @xcite also indicates that the number of such ` missing ' quasars is likely to be small . selecting those dlas which exhibit h@xmath1 absorption may focus on systems with a narrower range of physical conditions than the dla population as a whole . tentative support for this conjecture lies in the steeper , tighter [ m / h]@xmath11 anti - correlation observed for the h@xmath1 systems studied here . @xcite recently presented detailed chemical evolution models which give a slope for the [ m / h]@xmath11 relation of @xmath36 . they correct this result for various observational biases to match the shallower slope observed for the general dla population . however , the steep [ m / h]@xmath11 slope observed for h@xmath1-bearing systems could be less affected by these biases and may avoid those introduced by sampling many different ism gas phases . it might therefore be more comparable to the uncorrected slopes in the models . h@xmath1-selected dlas are therefore a candidate for a less biased probe of chemical evolution . that there exists such a large range ( @xmath37 ) in the values of @xmath9 in fig . [ fig:1 ] may not be surprising : @xcite describes a @xmath38}=-1 $ ] photo - ionization model for clouds in local hydrostatic equilibrium . for a representative incident uv background flux and dust - to - metals ratio , the molecular fraction in this model shows a sudden increase of @xmath39 for only a small increase in the total hydrogen density . therefore , the very steep @xmath9@xmath11 correlation could be achieved with a modest increase in dust content at lower redshifts , consistent with the observed [ m / fe]@xmath11 anti - correlation . l03 also discuss how @xmath9 might be very sensitive to local physical conditions . for example , within the schaye model , one expects an anti - correlation between @xmath9 and the intensity of the uv background . however , the behaviour of the uv background flux with redshift over the range @xmath403 is still a matter of considerable uncertainty . the strong decrease in @xmath9 at high redshift may also be consistent with recent hi 21-cm absorption measurements in dlas @xcite , where a generally higher spin / excitation temperature is found at @xmath41 . with an increased sample size and more detailed analyses , the @xmath9@xmath11 anti - correlation , if real , may provide complementary constraints on these problems . , a. c. , mathlin , g. p. , churches , d. k. , & edmunds , m. g. 2000 , in esa sp , vol . 445 , star formation from the small to the large scale , ed . f. favata , a. kaas , & a. wilson ( noordwijk , the netherlands : european space agency ) , 21 , s. a. , molaro , p. , centuri ' on , m. , dodorico , s. , bonifacio , p. , & vladilo , g. 2001 , in deep fields , ed . s. cristiani , a. renzini , & r. williams , eso astrophysics symposia series ( berlin , germany : springer ) , 334
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the chemical abundances in damped lyman-@xmath0 systems ( dlas ) show more than 2 orders of magnitude variation at a given epoch , possibly because dlas arise in a wide variety of host galaxies .
this could significantly bias estimates of chemical evolution .
we explore the possibility that dlas in which h@xmath1 absorption is detected may trace cosmological chemical evolution more reliably since they may comprise a narrower set of physical conditions .
the 9 known h@xmath1 absorption systems support this hypothesis : metallicity exhibits a faster , more well - defined evolution with redshift than in the general dla population . the dust - depletion factor and ,
particularly , h@xmath1 molecular fraction also show rapid increases with decreasing redshift .
we comment on possible observational selection effects which may bias this evolution .
larger samples of h@xmath1-bearing dlas are clearly required and may constrain evolution of the uv background and dla galaxy host type with redshift .
# 1_#1 _ # 1_#1 _ = # 1 1.25 in .125 in .25 in
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the method of cosmic crystallography was developed by lehoucq , lachize - rey , and luminet @xcite , and consists of plotting the distances between cosmic images of clusters of galaxies . in euclidean spaces , we take the square of the distance between all pairs of images on a catalogue versus the frequency of occurence of each of these distances . in universes with euclidean multiply connected spatial sections , we have sharp peaks in a plot of distance distributions . it is usual to consider the friedmann - lematre - robertson - walker ( flrw ) cosmological models of constant curvature @xmath0 with simply connected spatial sections . however , models with these spacetime metrics also admit , compact , orientable , multiply connected spatial sections , which are represented by quotient manifolds @xmath1 , where @xmath2 is @xmath3 , @xmath4 or @xmath5 and @xmath6 is a discrete group of isometries ( or rigid motions ) acting freely and properly discontinously on @xmath7 . the manifold @xmath8 is described by a fundamental polyhedron ( fp ) in @xmath7 , with faces pairwise identified through the action of the elements of @xmath6 . so @xmath7 is the universal covering space of @xmath8 and is the union of all cells @xmath9fp@xmath10 , @xmath11 @xmath6 . the repeated images of a cosmic source is the basis of the cosmic cristallography method . the images in a multiply connected universe are connected by the elements @xmath12 of @xmath6 . the distances between images carry information about these isometries . these distances are of two types @xcite : type i pairs are of the form @xmath13 where @xmath14=distance[x , y],\ ] ] for all points @xmath15 and all elements @xmath11 @xmath6 ; type ii pairs of the form @xmath16 if @xmath17=distance[y , g(y ) ] , \label{clifford}\ ] ] for at least some points @xmath15 and some elements @xmath12 of @xmath6 . the cosmic cristallography method puts in evidence type ii pairs . these distances are due to clifford translations , which are elements @xmath18 such that eq . ( [ clifford ] ) holds for _ any _ two points @xmath19 type ii pairs give sharp peaks in distance distributions in euclidean @xcite and spherical spaces @xcite , but they do not appear in hyperbolic space . this is illustrated in fig . [ wt ] for an flrw model with total energy density @xmath20 and having as spatial sections the weeks manifold - coded @xmath21 in @xcite and in table i below - which is the closed , orientable hyperbolic manifold with the smallest volume ( normalized to minus one curvature ) known . the bernui - teixeira ( b - t ) function @xcite is an analytical expression for a uniform density distribution in an open hyperbolic model . , width=302 ] in hyperbolic spaces , the identity ( or trivial motion ) is the only clifford translation . in this case , the cosmic cristallography method by itself can not help us to detect the global topology of the universe . several works have tried to identify multiply connected , or closed , hyperbolic universes by applying variants of the cosmic cristallographical method @xcite , most of which now rely on type i , in the absence of type ii , isometries . it is these variants that we call _ cosmic crystallography of the second degree_. one of these @xcite , proposed by us , consisted of subtracting , from the distribution of distances between images in closed hyperbolic universes , the similar distribution for the open model . it did not pretend to be useful for the determination of a specific topology , but it might reinforce other studies that look for nontrivial topologies . uzan , lehoucq , and luminet @xcite invented the _ collect correlated pairs _ method , that collect type i pairs and plot them so as to produce one peak in function of the density parameters , @xmath22 for matter and @xmath23 for dark energy . gomero et al . @xcite obtained _ topological signatures , _ by taking averages of distance distributions for a large number of simulated catalogues and subtracting from them averages of simulations for trivial topology . here we introduce still another second order crystallographic method , in the absence of clifford translations and sharp peaks . we look for signals of nontrivial topology in statistical parameters of their distance distributions . as commented above on ref . @xcite , these methods are not as powerful as the original clifford crystallography , but will certainly be useful as added tools to help looking for the global shape of the universe . let the metric of the friedmann s open model be written as @xmath24 where @xmath25 is the expansion factor or curvature radius , and @xmath26 is the standard metric of hyperbolic space @xmath27 . we assume a null cosmological quantity , hence the expressions for @xmath25 and other quantities are as in friedmann s open model - see , for example , landau and lifshitz @xcite . to simulate our catalogues we assume for the cosmological density parameter the values @xmath20 and @xmath28 with hubble s constant @xmath29kms@xmath30mpc@xmath30 . the present value of the curvature radius is @xmath31 mpc for @xmath20 and @xmath32mpc for @xmath33 . to generate pseudorandom source distributions in the fp , we first change the coordinates to get a uniform density in coordinate space : @xmath34 with @xmath35 and @xmath36 . our sources are then generated with equal probabilities in @xmath37 space , and their large scale distributions are spatially homogeneous . + + & + name & volume & @xmath38 & @xmath39 & @xmath20 & @xmath40 + & 0.94 & 0.52 & 0.75 & 747 & 379 + & 0.98 & 0.54 & 0.75 & 729 & 357 + & 1.89 & 0.64 & 0.85 & 463 & 247 + & 2.83 & 0.74 & 0.94 & 403 & 199 + & 3.75 & 0.77 & 1.16 & 451 & 237 + & 4.69 & 0.87 & 1.38 & 653 & 273 + & 4.69 & 0.87 & 1.38 & 653 & 273 + & 4.69 & 0.87 & 1.38 & 653 & 273 + we did the simulations for eight spatially compact , hyperbolic models . their space sections are the manifolds listed in table i , which gives their names , volumes , and the circumscribing and inscribing radii ( @xmath39 and @xmath38 ) of their fp s . these and other data on the manifolds were obtained from the snappea program @xcite . in the last two columns , we list the number of cells ( replicas of fp ) needed to assure a complete cover of @xmath27 up to a radius @xmath41 for @xmath20 and @xmath42 for @xmath33 . this corresponds to the redshift ( @xmath43 ) of the last scattering surface . aproximately @xmath44 images were created in each simulated catalogue . manifolds with different volumes will have different numbers of sources in their fp s . in this and remaining figures , a dashed line marks the value of the plotted parameter for the open model , and a solid line marks the bernui - teixeira value . , width=302 ] .,width=302 ] _ for the simulations with @xmath45,title="fig:",width=302 ] _ _ , title="fig:",width=302 ] _ _ .,title="fig:",width=302 ] _ _ .,title="fig:",width=302 ] _ _ , title="fig:",width=302 ] _ _ , title="fig:",width=302 ] _ ten different pseudorandom distributions of sources for each manifold were simulated . from the result of each simulation we calculated the distance distribution and then the latter s mean value @xmath46,$ ] standard deviation @xmath47 skewness @xmath48 and peakedness @xmath49 where @xmath50 is the distance variable , @xmath51 is the expected value operator , and @xmath52 $ ] is the @xmath53-moment about the mean value - cf . @xcite , for example . for each of our chosen eight compact manifolds , we took the averages of these statistical parameters for ten simulations ( obtained by varying the computer pseudorandom seed ) , to get their tendency in comparison with those for the simply connected case . we proceeded in two ways to obtain a distribution in the open model , for comparison with the compact cases . in one of them , a simulated catalogue ( real ones are not yet deep enough for use in cosmic crystallography ) was obtained , with the same cosmological parameters and a pseudorandom distribution of sources inside the observable universe ( redshift @xmath54 ) . in the other , the analytical bernui - teixeira function for a uniform distribution in simply connected universes @xcite was used . we compare the results for eight compact manifolds with those for the simulated simply connected case , and with the b - t function . figures 2 - 9 show these results for @xmath55 , @xmath56 , @xmath57 , and @xmath58 , plotted vs. the manifold volumes , with the density parameters @xmath20 and @xmath33 . it is of course premature to think of a functional dependence on volume ( except for the obvious fact that , if the volume is so large as to enclose the observable universe , then the distribution of sources is indistinguishable from that for an open universe - see fagundes @xcite , for example ) . but we note a tendency of @xmath55 , @xmath56 , and @xmath58 in compact universes to have values below those of the corresponding b - t function , and for @xmath57 to lie above that function . this could give us a signal of nontrivial topology in a future , realistic situation . we expect that the presence of type i pairs of distances is more evident for manifolds with smaller normalized volumes , or for models with smaller @xmath59 . in fact our results are better for @xmath55 and @xmath56 with @xmath20 , where the observable universe is bigger than for @xmath40 . this implies more copies of the fp , and hence more topological effects . the set of these statistical parameters may eventually provide a complementary indication as for the multiply connectedness of a possibly negatively curved cosmos . e. g. is grateful to fundao de amparo pesquisa do estado de so paulo ( fapesp process 01/10328 - 6 ) for a post - doctoral scholarship . h. v. f. thanks conselho nacional de desenvolvimento cientfico e tecnolgico ( cnpq process 300415/84 - 2 ) for partial financial support .
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the cosmic crystallography method of lehoucq et al . @xcite produces sharp peaks in the distribution of distances between the images of cosmic sources .
but the method can not be applied to universes with compact spatial sections of negative curvature .
we apply to the these a second order crystallographic effect , as evidenced by statistical parameters .
pacs number : 98.80.jk keywords : cosmology , large scale structure
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the theory of polymers near surfaces is a very important subject for theoretical investigation . the main reason is the very broad and deep theoretical research possible for these types of systems . modern theories have been developed and brought to a very high standard @xcite . on the other hand there is a large demand on practical interest on studying polymers near surfaces . one of them is surface protection . the theory on polymer brushes @xcite is a typical example of these type of applications . polymers are attached to surfaces to protect the surface from further adsorption of , e.g. , biological active molecules such as proteins . these molecules need to be pretended to from all kind of surface interactions in order to save their biological function @xcite . the ( technical ) problem with polymer brushes is , that a large surface coverage is needed to protect the surface in a most effective way . polymers attached chemically or physically at one end at the surface form brushes , i.e. , the chains are extended and the brush height @xmath0 follows a scaling law @xmath1 ( in good solvent ) , where @xmath2 is the area per chain an is thus related to the grafting density . @xmath3 the degree of polymerization of the chains . the problem is to set up a small value of @xmath2 , or , correspondingly a large grafting density . for low values of the grafting density the chains behave as `` mushrooms '' and the surface protection is incomplete . in earlier papers it has been shown by one of us , that the use of branched chains in much more effective @xcite . chains , polymers and polymeric fractals with a larger connectivity seem to be more appropriate to protect surfaces more effectively . indeed due to their connectivity their occupied area is larger and it turns out that these systems behave more like single chain fractals . typical many body effects , such as occur in conventional polymer brushes do not play a significant role . in the present paper we suggest a different route of surface protection by using microgels and branched structure . microgels had become an important tool in designing polymeric nano structures . these systems can be synthesized @xcite with different structures . indeed the structure of these microgels can range from a fractal state , i.e. , a branched self - similar polymer with a large connectivity , up to almost hard and highly crosslinked spheres . such systems are well designed to study the transition form polymer to colloid behavior by variation of the structure and the crosslinking state . we have shown earlier @xcite , that fractal polymers and gels can interpenetrates each other and screen excluded volume forces , wherever their connectivity is low . in terms of the spectral dimension this is the case , when it is lower than a critical value , i.e. @xmath4 . thus fractals with lower spectral dimensions screen their excluded volume , whereas polymeric fractals with larger spectral dimension saturate . then they form soft balls , which can not interpenetrate each other and are well separated from each other @xcite . this state has an analogon in the case of linear polymers . polymer melts in two dimensions correspond to a saturated state . the individual chains are separated from each other and form on average disks on a hexagonal lattice @xcite . for surface protection in three dimensions it would thus be more effective to use fractal microgels with a large connectivity . then the surface coverage is ruled by the single gel behavior . alternatively crosslinked gels can be used to have the same effect . sufficiently crosslinked gels can not interpenetrate each other . the adsorption behavior can be studied then by the single gel adsorption . the significant parameters are either the connectivity ( spectral dimension @xmath5 ) or the crosslink number @xmath6 . in the following we emphasize mainly on the crosslinked gels rather on the self similar connected polymeric fractals . nevertheless we will consider both cases below . the paper is organized as follows . in section [ 2 ] we introduce the model of self crosslinked polymer chains , which form the microgels . in section [ 3 ] we repeat briefly the scaling behavior of ideal microgels before we consider in section [ 4 ] the effects of excluded volume in the bulk . sections [ 5 ] and [ 6 ] treat the adsorption behavior of the gels close to attracting walls using simple scaling arguments . in section [ 5 ] we will also make some remarks on fractal type gels . flexible interacting macromolecule modeled usually by the edwards hamiltonian in three dimensional space . the edwards hamiltonian consists of two parts , i.e.,the gaussian connectivity of monomers @xmath7 and the self avoidance between monomers @xmath8 thereby @xmath9 is the excluded volume of the monomers , @xmath3 the degree of polymerization , and @xmath10 the kuhn length . the chain configurations are determined by monomer coordinates @xmath11 , where @xmath12 labels all monomers @xmath13 . the edwards hamiltonian is sufficient to describe a free self avoiding walk chain . to study the properties of a microgel crosslinks have to be introduced . the most obvious statistical representation of a microgel is a self crosslinked single chain . such a situation has been studied many years ago by the manchester group in three papers @xcite . the static properties have been computed by variational techniques . in this paper we choose a different route . let us therefore consider a microgel as a self crosslinked polymer of roughly spherical structure , which can be visualized as given in fig.([one ] ) . in this paper we want to describe the crosslinks such that a continuous transition between the free chain and the fully crosslinked state can be represented in the same model . this has been motivated by our earlier work , where soft crosslinks have been introduced . the method of using soft crosslinks is able to interpolate from the free chain to the hard microgel . the state in between is a new kind of branched chain , whose scaling properties have been already described @xcite . to be more precise let us introduce @xmath6 permanently crosslinked monomers , where each of them is characterized by pair of randomly chosen monomer coordinates @xmath14 that form a permanent crosslink . in fact , the whole set of crosslinks @xmath15 determine the random connectivity of the micro network . these definitions and proposals allow us to formulate the partition function of the microgel , i.e. , @xmath16 the partition function describes the gaussian network with self avoiding interactions , and takes into account the total connectivity of the certain crosslink configuration c. if the delta constraint for the permanent crosslinks is represented by a soft gaussian function , i.e. , @xmath17 , the problem can be solved exactly . thus we model the @xmath18-function by gaussian distribution with width @xmath19 in limit @xmath20 and formulate the hamiltonian of the crosslinked chain by @xmath21 although we have shown in previous publications that the partition function can be solved for any value of @xmath22 exactly it is useful to rederive the results previously by the use of scaling arguments . these consideration will reproduce the exact results apart from prefactors . to do so it is useful to recall that the classical random walk contains two elastic contributions , one for stretching and one for compression . the addition of both yield a free energy @xmath23 and by minimization the size of gaussian chain @xmath24 is recovered . let us shortly repeat the results for later use . for the case of soft crosslinks , i.e. , whenever @xmath22 is within the range @xmath25 we have for the relevant part of flory free energy @xmath26 and minimization yields the branched polymer regime @xmath27 the appearance of the typical branched polymer exponent @xmath28 is not surprising in the ideal case , since the constraint can be visualized as springs . then the connectivity is changed . the branched polymer regime in the range of @xmath29 was confirmed also by the exact solution of the crosslinked chain problem , see @xcite . moreover it agrees also with the corresponding branched polymer @xcite in the case of hard crosslinks , whenever @xmath30 the crosslink term must be differently estimated . to do so the crosslink term can be estimated by the size of the random walk through crosslinks . then the relevant part of flory free energy is then given by @xmath31 minimization of the free energy provides the size of the microgel by @xmath32 as also given by the exact results @xcite . formally the latter result can be found from a special choice for @xmath22 from the corresponding result for soft gels ( [ soft ] ) , but note that the way of estimating the free energy contribution of the crosslinks is estimated very differently . thus there can be a different prefactor , which is not accessible by scaling . the exact values for the radius of gyration of the non - interacting but crosslinked chains have been computed exactly in @xcite , where the numerical prefactors can be found . although we have been able to compute the size of the microgel exactly whenever the interactions are not present the self avoiding case appears very difficult . the only possibility for the problem at this stage is to use flory estimates for size @xmath33 . let us first consider the case of soft crosslinks . the distance constraint that forces two arbitrarily chosen polymer segments together shrinks the chain . the shrinkage costs entropy penalty which balances with the distance constraint . the use of the pseudo potentials allows us , however , to cast this in a simple flory free energy to @xmath34 and minimization yields the size of the swollen soft microgel @xmath35 the result is very intriguing . although the gaussian chain size scales the same way as the branched polymer , the swelling behavior produces another excluded volume exponent , resulting in a different swelling behavior as branched chains . in the latter case the swollen branched chain is characterized by @xmath36 . on the other hand it can also be seen that the pure scaling in terms of the variable @xmath37 always present in the gaussian case is destroyed . this becomes clear , since the excluded volume introduces interactions . a similar estimate of the size can be carried out in the case of hard crosslinks . here the elastic term stemming from the pseudo crosslink potential can be estimated the same way as in the case of non interacting networks , which was given by a random walk through the crosslinks . the total excluded volume energy remain the same , because it depends only on the total amount of monomers and not on the special connectivity . then minimization of the free energy @xmath38 yields swollen c*-micro gels of size @xmath39 the size might appear small compared to what is expected intuitively . this comes from the fact that the crosslinks has been chosen totally randomly . in many theories of macroscopic networks the choice of the crosslink pairs is guided by the conformation of the excluded - or random walk chain , i.e. , the often terminated `` zeroth replica '' @xcite . for completeness we mention that the number of crosslinks in excluded volume gels can not be arbitrarily large . a natural limit of the crosslink number is given by the condition that the size must be larger than a fully collapsed ball of space filling density , i.e. @xmath40 . the latter condition yields the upper limit for the number of crosslinks to be @xmath41 . let us first study the adsorption of ideal microgels near a flat surface by naive scaling arguments . to do so , we repeat the scaling idea of de gennes for arbitrarily flexible objects of arbitrary connectivity , but selfsimilarly linked . the connectivity of of such gaussian fractal networks can be described by the spectral dimension @xmath5 . the total number of monomers is therefore @xmath42 , and their ideal size is given by @xmath43 , yielding a fractal dimension of @xmath44 . thus @xmath45 counts the number of monomers through a linear dimension through the fractal object . this way of description includes the wellknown cases of linear chains for @xmath46 and for randomly branched polymers , i.e. , @xmath47 . a simple way of looking at the adsorption conditions is to compare the free energy penalty of confinement of the gaussian structure near the wall with the gain of energy by adsorbing a certain number of monomers . @xmath48 for gaussian chains the confinement is simply given by by @xmath49 . here we have used the symbol @xmath50 for the height of adsorbed layer , @xmath51 is the number of adsorbed monomers , and @xmath52 is the gain of energy per @xmath53 . following de gennes book @xcite we can reproduce the result given there for the linear ideal chain . to do so we must first estimate the number of adsorbed monomers at the surface , we assume that the surface is penetrable for a moment . this yields immediately @xmath54 the main problem is to estimate the confinement free energy for gaussian fractals with a larger connectivity compared to the linear chain . this is definitely not just the inverse of the gaussian free energy of stretching , i.e. , @xmath55 , because upon stretching only the monomers in the shortest path are taking part on the deformation , whereas upon confinement the total number of monomers are concerned . the corresponding confinement free energy must then be of the form @xmath56 this result contains the special case of linear chains , @xmath57 , and the latter agrees with the classical confinement free energy . for the ideal gaussian structures the height of the adsorbed layer scales as @xmath58 the same result can be found by a blob argument . for selfsimilarly branched polymers a blob model can be used @xcite . the manifold is confined to a height @xmath50 . inside blobs of diameter @xmath50 the branched chain is gaussian , i.e. , @xmath59 , where @xmath60 is the number of monomers inside the blob . thus the number of blobs is given by @xmath61 . the confinement free energy is proportional to the number of blobs , i.e. , @xmath62 . employing the same scaling argument as above yields immediately @xmath63 which agrees with the above result , eq.([hhh ] ) . in any case , the above arguments only yield the behavior of ideal chains near the surface . for excluded volume chains and excluded volume manifolds the gaussian elastic entropy penalty must be replaced by the confinement energy . to do so , the manifold can be put between two plates of distance @xmath50 . this procedure yields similar results along to those derived in de gennes book @xcite . to do so , the extension of the manifold between two parallel plates must be computed . it is given by @xmath64 @xcite . this result is consistent with the linear saw chain between two plates . for @xmath46 the two dimensional chain of blobs is recovered . the confinement free energy can be only a function of the ration of the size of the manifold and the distance between the plates , i.e. , @xmath65 , where @xmath33 is the size of the self avoiding manifold @xmath66 . the confinement free energy is then easily found from the condition that the free energy must be an extensive quantity . thus it must scale as @xmath67 , which yields @xmath68 replacing the elastic free energy in the scaling argument above by the confinement energy , yields the physically sensible result @xmath69 which is now independent of the molecular weight for any manifold . moreover for @xmath46 the linear chain result is recovered . moreover the result bears an interesting point in it : the parameter @xmath52 is the gain of energy at adsorption of one segment per thermal energy @xmath70 . thus it is physically reasonable that this parameter is sufficiently small , i.e. , @xmath71 . thus a significant change of the behavior can be expected if the exponent in @xmath50 is larger and smaller than one . it is interesting to note that this happens at @xmath47 which is close to the spectral dimension of randomly branched chains or accidentally for percolation clusters . thus randomly connected manifolds of large connectivity adsorb weaker than objects of low connectivity , as linear chains . this is physically intuitively clear since the number of accessible sites become smaller for increasing connectivity . in the following section we will use the same strategy to discuss the adsorption behavior of microgels as we have seen in the last paragraph on the previous section we have to construct the free energy of confinement . we use the same concept of gels between two plates for energy cost of squeezing . we have seen in the first two sections that we can distinguish between soft and hard microgels by the value of the parameter @xmath22 . for both cases we expect physically different behavior . let us first consider soft gels between plates . to do so we have to determine the size of the gel parallel to the plates . the relevant parts of the free energy is given by @xmath72 where we have argued that the elastic confinement and the `` anisotropic '' excluded volume term balance each other . this yields immediately @xmath73 and the corresponding confinement energy to @xmath74 note that the confinement free energy is determined by the fact that it must be proportional to the total number of monomers , since the free energy is extensive . the same procedure can be employed for hard gels . the relevant free energy takes a similar form as before , apart from the elastic part of the energy . @xmath75 the size of the gel parallel to the plates is given for completeness . it scales as @xmath76 . to find the confinement free energy the same argument yields @xmath77 an important observation is that for @xmath78 ( no crosslink ) linear chain is recovered . finally we are in the position to discuss the adsorption behavior of the gels . to begin with , we employ the same scaling arguments as given in the case of self - similar polymeric fractal . thus we have to consider the competition between confinement , or the entropy penalty of confinement and the energy gain by adsorption . this results in a total free energy of the general form @xmath79 we just summarize the results to be brief . first for soft gels we find @xmath80 similarly for hard gels @xmath81 note that the latter equation contains the free chain result for @xmath78 , i.e. , if no crosslinks are present . we see that in both cases the height of the adsorbed layer depends on the number of crosslinks in a significant and characteristic way . the results are in accordance with the physical intuition . the soft microgels adsorb more easily , because these objects are more flexible . this is also shown by the different exponents of the interaction energy @xmath52 . with use of flory - approximation we had investigated behavior of microgels . the complexity of the distribution function of the non interacting network prevented us to use more refined methods , as they are well known in the case of linear polymer chains . nevertheless we got results which are reasonable and could be checked by experimental methods , at least in their tendency . the cases worked out here have been relevant for penetrable surfaces , i.e. , interfaces . a direct comparison to hard surfaces is not possible , since the number of monomers close to the surface can not be determined by eq . ( [ mono ] ) . the case of linear chains in half space has been studied in detail , see e.g. @xcite for a general reference . crossover exponents and new critical points determine the physics . in the present case of microgels and polymeric manifolds a similar treatment appears very difficult , since the `` bare propagator '' has a very complicated structure , although it is exactly known @xcite . nevertheless we expect that the principal statements can be compared at least qualitatively with experiments . to study the adsorption behavior we first had to calculate the size of the microgel in solution . this has been carried out by employing the flory arguments . the basis for the reliability of the result has been their agreement withe comparison of exact calculations without excluded volume in bulk . then generalization opened the determination of sizes of microgels with excluded volume interaction in bulk systems and near adsorbing flat surfaces . moreover we made straightforward generalizations to fractal type microgels , which could be described by the spectral dimension . the results show a transition from polymeric type of adsorption behavior for soft microgels , i.e. , with small number of crosslinks , or alternatively low spectral dimension , to colloidal adsorption , whenever the crosslink number is large and their coupling is strong . we have seen that the results have been presented from considerations on single gels . this has also experimental interest on surface protection . a layer of adsorbed gels of height @xmath50 at a surface is a single gel problem . unlike linear polymer chains the gels can no longer interpenetrate each other . thus microgels and branched polymers appear more effective in surface protection . another interesting question is also the interplay between vulcanization and adsorption . in the case we studied so far , we had assumed preformed gels and fractals which had then brought to the interacting surfaces . the other case , i.e. , the vulcanization in presence of interacting walls would lead to new types of gels with new structures which depend on the strength of the surface - monomer interaction @xcite . lun des auteurs ( tav ) remercie pierre - gilles de gennes et elie raphal pour leurs remarques interessantes sur ce travail . nous remercions aussi lun des critiques pour avoir attir notre attention sur une methode lgante dobtantion du `` blob '' , qui a permis dameliorer la qualit de ce papier . e. eisenriegler , _ polymers at surfaces _ , world scientific , singapore , 1994 p.g . de gennes , _ simple views on condensed matter _ , world scientific , singapore , 1992 j. rhe , preprint , 1997 t.a . vilgis , p. haronska , macromol . theory simul . , * 4 * , 111 , ( 1995 ) t.a . vilgis , p. haronska , m. benhamou , j. phys.ii , * 4 * , 2139 , 1994 e. zhulina , t.a . vilgis , macromolecules , * 28 * , 1008 , ( 1995 ) m. antonietti , d. ehrlich , k.j . flsch , h. sillescu , m. schmidt , p. lindner , _ macromolecules _ , * 22 * ( 1989 ) 2802 m. antonietti , k.j . flsch , h. sillescu , t. pakula , _ macromolecules _ , * 22 * ( 1989 ) 2812 m. antonietti , w. bremser , k.j . flsch , h. sillescu , _ makromol . , makromol . _ , * 30 * ( 1989 ) 81 m. antonietti , ch . rosenauer , _ macromolecules _ , * 24 * ( 1991 ) 3434 t.a . vilgis , physica a * 153 * , 341 , ( 1988 ) p. haronska , t.a . vilgis , j. chem . phys . , * 102 * , 6586 , ( 1995 ) i. carmesin , k. kremer , j. phys . ( france ) , * 51 * , 915 , ( 1990 ) g.allen , j. burgess , s.f . edwards , d.j . walsh , proc . a , * 334 * , 453 , 465 , 477 , ( 1973 ) m.p . solf , t.a . vilgis , j. phys . a , * 28 * , 6655 , ( 1995 ) m.p . solf , t.a . vilgis , j. phys . i ( france ) , * 6 * , 1451 , ( 1996 ) m.p . solf , t.a . vilgis , phys . e , * 55 * , 3037 , ( 1997 ) t.a . vilgis , macromol . theory simul . , in press , ( 1997 ) p.g . de gennes , _ scaling concepts in polymer physics _ , cornell university press , ithaca , 1979 r. t. deam , s.f . edwards , proc . a * 260 * , 31 , ( 1976 ) m.a . cohen stuart , j.t.f . keurentjes , b.c . bonekamp and j.g.e.m . fraaye , colloids surfaces * 17 * , 91 , ( 1986 ) m. abel , r. lipowsky , unpublished
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the behavior of microgels near surfaces and their adsorption is studied by simple scaling theory .
two different types of microgels can be studied , i.e. , fractal type microgels and randomly crosslinked polymer chains . in the first case
the gel can be described mainly by introducing a spectral dimension .
the second type requires more attention and uses the number of crosslinks as parameter .
the main result is that soft gels with weakly coupled crosslinks and a low number of crosslinks adsorb much better than hard gels , with many crosslinks .
similar results for fractal gels and branched polymer are presented .
fractal gels with low connectivity adsorb easier than gels with a large connectivity dimension .
we discuss also consequences on surface protection by microgels .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
although the sunspot number varies periodically with time with an average period of 11 year , the individual cycle period ( length ) and also the strength ( amplitude ) vary in a random way . it is observed that the stronger cycles have shorter periods and vice versa . this leads to an important feature of solar cycle known as waldmeier effect . it says that there is an anti - correlation between the rise time and the peak sunspot number . we call this as we1 . now instead of rise time if we consider the rise rate then we get very tight positive correlation between the rise rate and the peak sunspot number . we call this as we2 . another important aspect of solar activity are the grand minima . these are the periods of strongly reduced activity . a best example of these is the during during 16451715 . it was not an artifact of few observations , but a real phenomenon ( hoyt & schatten 1996 ) . from the study of the cosmogenic isotope @xmath0c data in tree rings , usoskin et al . ( 2007 ) reported that there are @xmath1 grand minimum during last @xmath2 years . we want to model these irregularities of solar cycle using flux transport dynamo model ( choudhuri et al . 1995 ; dikpati & charbonneau 1999 ; chatterjee et al . 2004 ) . in this model , the turbulent diffusivity is an important ingredient which is not properly constrained . therefore several groups use different value of diffusivity and this leads to two kinds of flux transport dynamo model high diffusivity model and low diffusivity model . in the earlier model , the value of diffusivity usually used is @xmath3 @xmath4 s@xmath5 ( see also jiang et al . 2007 and yeates et al . 2008 for details ) , whereas in the latter model , it is @xmath6 @xmath4 s@xmath5 . we mention that the mixing length theory gives the value of diffusivity as @xmath7 @xmath4 s@xmath5 . another important flux transport agent in this model is the meridional circulation . only since 1990 s we have some observational data of meridional circulation near the surface and therefore we do not know whether the varied largely with solar cycle in past or not . however if the flux transport dynamo is the correct dynamo for the solar cycle , then one can consider the solar cycle period variation as the variation for the because the cycle period is strongly determined by the strength of the meridional circulation in this model . now the periods of the solar cycle indeed had much variation in past , then we can easily say that the had significant variation with the solar cycle . therefore the main sources of randomness in the flux transport dynamo model are the stochastic fluctuations in process of generating poloidal field and the stochastic fluctuations in the meridional circulation . in this paper we explore the effects of fluctuations of the latter . we model last @xmath8 cycles by fitting the periods with variable meridional circulation in a high diffusivity model based on chatterjee et al . ( 2004 ) model . the solid line in fig . [ fit23](a ) shows the variation of the amplitude of @xmath9 used to model the periods of the cycles . note that we did not try to match the periods of each cycles accurately which is bit difficult . we change @xmath9 between two cycles and not during a cycle . in addition , we do not change @xmath9 if the period difference between two successive cycles is less than @xmath10 of the average period . ( in m s@xmath5 ) with time ( in yr ) . the solid line is the variation of @xmath9 used to match the theoretical periods with the observed periods . ( b ) variation of theoretical sunspot number ( dashed line ) and observed sunspot number ( solid line ) with time . ( c ) scatter diagram showing peak theoretical sunspot number and peak observed sunspot number . the linear correlation coefficients and the corresponding significance levels are given on the plot.,scaledwidth=100.0% ] in fig . [ fit23](b ) , we show the theoretical sunspot series ( eruptions ) by dashed line along with the observed sunspot series by solid line . the theoretical sunspot series has been multiplied by a factor to match the observed value . it is very interesting to see that most of the amplitudes of the theoretical sunspot cycle have been matched with the observed sunspot cycle . therefore , we have found a significant correlation between these two ( see fig . [ fit23](c ) ) . this study suggests that a major part of the fluctuations of the amplitude of the solar cycle may come from the fluctuations of the meridional circulation . this is a very important result of this analysis . now we explain the physics of this result based on yeates et al . toroidal field in the flux transport model , is generated by the stretching of the poloidal field in the tachocline . the production of this toroidal field is more if the poloidal field remains in the tachocline for longer time and vice versa . however , the poloidal field diffuses during its transport through the convection zone . as a result , if the diffusivity is very high , then much of the poloidal field diffuses away and very less amount of it reaches the tachocline to induct toroidal field . therefore , when we decrease @xmath9 in high diffusivity model to match the period of a longer cycle , the poloidal field gets more time to diffuse during its transport through the convection zone . this ultimately leads to a lesser generation of toroidal field and hence the cycle becomes weaker . on the other hand , when we increase the value of @xmath9 to match the period of a shorter cycle , the poloidal field does not get much time to diffuse in the convection zone . hence it produces stronger toroidal field and the cycle becomes stronger . consequently , we get weaker amplitudes for longer periods and vice versa . however , this is not the case in low diffusivity model because in this model the diffusive decay of the fields are not much important . as a result , the slower meridional circulation means that the poloidal field remains in the tachocline for longer time and therefore it produces more toroidal field , giving rise to a strong cycle . therefore , we do not get a correct correlation between the amplitudes of theoretical sunspot number and that of observed sunspot number when repeat the same analysis in low diffusivity model based on dikpati & charbonneau ( 1999 ) model . we study the using flux transport dynamo model . we have seen that the stochastic fluctuations in the process and the stochastic fluctuations in the are the two main sources of irregularities in this model . therefore , to study we first introduce suitable stochastic fluctuations in the poloidal field source term of process . we see that this study can not reproduce we1 ( fig . [ pol](a ) ) . however it reproduces we2 ( fig . [ pol](b ) ) . finally we introduce stochastic fluctuations in both the poloidal field source term and the meridional circulation . we see that both we1 and we2 are remarkably reproduced in this case ( see fig . [ both ] ) . we repeat the same study in low diffusivity model based on dikpati & charbonneau ( 1999 ) model . however in this case we are failed to reproduce we1 , only we2 is reproduced . the details of this work can be found in karak & choudhuri ( 2011 ) . we have realized that the is important in modeling many aspects of solar cycle . therefore we check whether a large decrease of the leads to a maunder - like grand minimum . to answer this question , we decrease @xmath9 to a very low value in both the hemispheres . we have done this in the decaying phase of the last sunspot cycle before maunder minimum . we keep @xmath9 at low value for around 1 yr and then we again increase it to the usual value but at different rates in two hemispheres . in northern hemisphere , @xmath9 is increased at slightly lower rate than southern hemisphere . ( in m s@xmath5 ) in northern and southern hemispheres with time . ( b ) the butterfly diagram . ( c ) the dashed and dotted lines show the sunspot numbers in southern and northern hemispheres , whereas the solid line is the total sunspot number . ( d ) variation of energy density of toroidal field at latitude 15@xmath11 at the bottom of the convection zone.,scaledwidth=100.0% ] in fig . [ mm ] , we show the theoretical results covering the maunder minimum episode . fig . [ mm](a ) , shows the maximum amplitude of meridional circulation @xmath9 varied over this period in two hemispheres . in fig . [ mm](b ) , we show the butterfly diagram of sunspot numbers , whereas in fig . [ mm](c ) , we show the variation of total sunspot number along with the individual sunspot numbers in two hemispheres ( see the caption ) . in order to facilitate comparison with observational data , we have taken the beginning of the year to be 1635 . note that our theoretical results reproduce the sudden initiation and the gradual recovery , the north - south asymmetry of sunspot number observed in the last phase of maunder minimum and the cyclic oscillation of solar cycle found in cosmogenic isotope data . we also mention that if we reduce the poloidal field to a very low value at the beginning of the maunder minimum then also we can reproduce maunder - like grand minimum ( choudhuri & karak 2009 ) . however in both the cases , either we need to reduce the or the poloidal field at the beginning of the maunder minimum . however if we reduce the poloidal field little bit , then one can reproduce maunder - like grand minimum at a moderate value of meridional circulation . the details of this study can be found in karak ( 2010 ) . we have shown that with a suitable stochastic fluctuations in the meridional circulation , we are able to reproduce many important irregular features of solar cycle including waldmeier effect and maunder like grand minimum . however we are failed to reproduce these results in low diffusivity model . therefore this study along with some earlier studies ( chatterjee , nandy & choudhuri 2004 ; chatterjee & choudhuri 2006 ; goel & choudhuri 2009 ; jiang , chatterjee & choudhuri 2007 ; karak 2010 ; karak & choudhuri 2011 ; karak & choudhuri 2012 ) supports the high diffusivity model for solar cycle . chatterjee , p. , nandy , d. , & choudhuri , a. r. 2004 , a&a , 427 , 1019 choudhuri , a. r. , chatterjee , p. , & jiang , j. , 2007 , phys . , 98 , 1103 choudhuri , a. r. , & karak , b. b. 2009 , raa 9 , 953 choudhuri , a. r. , schssler , m. , & dikpati , m. 1995 , a&a , 303 , l29 dikpati , m. , & charbonneau , p. 1999 , apj , 518 , 508 jiang , j. , chatterjee , p. , & choudhuri , a. r. 2007 , mnras , 381 , 1527 hoyt , d. v. , & schatten , k. h. , 1996 , sol . phys . , 165 , 181 karak , b. b. 2010 , apj , 724 , 1021 karak , b. b. , & choudhuri , a. r. 2011 , mnras , 410 , 1503 karak , b. b. , & choudhuri , a. r. 2012 , sol . , 278:137 usoskin , i. g. , solanki , s. k. , & kovaltsov , g. a. 2007 , a&a , 471 , 301 yeates , a. r. , nandy , d. , & mackay , d. h. 2008 , apj , 673 , 544
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the sunspot number varies roughly periodically with time .
however the individual cycle durations and the amplitudes are found to vary in an irregular manner .
it is observed that the stronger cycles are having shorter rise times and vice versa .
this leads to an important effect know as the waldmeier effect .
another important feature of the solar cycle irregularity are the grand minima during which the activity level is strongly reduced .
we explore whether these solar cycle irregularities can be studied with the help of the flux transport dynamo model of the solar cycle .
we show that with a suitable stochastic fluctuations in a regular dynamo model , we are able to reproduce many irregular features of the solar cycle including the waldmeier effect and the grand minimum . however , we get all these results only if the value of the turbulent diffusivity in the convection zone is reasonably high .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the study of the surface tension of deconfined quark matter has attracted much attention recently @xcite because a detailed knowledge of it may contribute to a better comprehension of the physics of compact star interiors . in fact , surface tension plays a crucial role in quark matter nucleation during the formation of compact stellar objects , because it determines the nucleation rate and the associated critical size of the nucleated drops @xcite . it is also determinant in the formation of mixed phases at the core of hybrid stars which may arise only if the surface tension is smaller than a critical value of the order of tens of mev / @xmath0 @xcite . also , surface tension affects decisively the properties of the most external layers of a strange star which may fragment into a charge - separated mixture , involving positively - charged strangelets immersed in a negatively charged sea of electrons , presumably forming a crystalline solid crust @xcite . this would happen below a critical surface tension which is typically of the order of a few mev/@xmath0 @xcite . however , in spite of its key role for compact star physics , the surface tension is still poorly known for quark matter . early calculations by berger and jaffe gave rather low values for the surface tension , below @xmath1 @xcite . however , larger values within @xmath2 were used in other works about quark matter droplets in neutron stars @xcite . more recently , values around @xmath3 have been adopted for studying the effect of quark matter nucleation on the evolution of protoneutron stars @xcite . however , much larger values have also been obtained in the literature . estimates given in ref . @xcite give values in the range @xmath4 and values around @xmath5 were suggested on the basis of dimensional analysis of the minimal interface between a color - flavor locked phase and nuclear matter @xcite . in this paper we study the surface tension and the curvature energy of three - flavour quark matter in equilibrium under weak interactions within the nambu - jona - lasinio ( njl ) model . we include the effect of color superconductivity and describe finite size effects within the multiple reflection expansion ( mre ) framework @xcite . we shall consider only the 2sc phase , but we shall see that our conclusions are quite general and the results for other superconducting phases can be easily foreseen within the present model . our calculations result in large values of the surface tension which disfavor the formation of mixed phases at hybrid star cores . the article is organized as follows : in sect . ii we present the quark matter equations of state without finite size effects . then , in sect . iii we introduce the mre formalism for the finite size effects . finally , in sect . iv we present our results and conclusions . in the present work we start from an @xmath6 njl effective model which also includes color superconducting quark - quark interactions . the corresponding lagrangian is given by @xmath7 \label{lagrangian } \\ & + & 2h \!\ ! \sum_{a , a'=2,5,7 } \left [ \left ( \bar \psi \ i \gamma_5 \tau_a \lambda_{a ' } \ \psi_c \right ) \left ( \bar \psi_c \ i \gamma_5 \tau_a \lambda_{a ' } \ \psi \right ) \right ] \nonumber\end{aligned}\ ] ] where @xmath8 is the current mass matrix in flavour space . in what follows we will work in the isospin symmetric limit @xmath9 , and for simplicity we do not include flavour mixing effects . the matrices @xmath10 and @xmath11 with @xmath12 are the gell - mann matrices corresponding to the flavour and color groups respectively , and @xmath13 . in addition , in eq . ( [ lagrangian ] ) we have used the charge conjugate spinors @xmath14 and @xmath15 , where @xmath16 is the dirac conjugate spinor and @xmath17 . the next step is obtaining the grand canonical thermodynamic potential at finite temperature @xmath18 and chemical potentials @xmath19 , where @xmath20 and @xmath21 stand for flavour and color respectively . then , we are able to calculate the relevant thermodynamic quantities . note that first we will show the thermodynamic potential for the bulk system , and later on we will derive the effective potential for the finite size effects . for that purpose , starting from eq . ( [ lagrangian ] ) , it is convenient to perform a standard bosonization of the theory . thus , we introduce the bosonic fields @xmath22 , @xmath23 and @xmath24 corresponding to the sigma and pion mesons , and scalar diquark fields , respectively ; and integrate out the quark fields . in what follows we will work within the mean field approximation ( mfa ) , in which these bosonic fields are expanded around their vacuum expectation values and the corresponding fluctuations are neglected . since the mean field values of the pion fields vanish due to symmetry reasons , in what follows we will only consider those of the sigma and diquark fields . then , @xmath25 and @xmath26 . regarding the diquark mean field , in the present work , we will assume that in the density region of interest only the 2sc phase might be relevant . moreover , due to the color symmetry , one can rotate in color space to fix @xmath27 , @xmath28 . finally , in the framework of the matsubara and nambu - gorkov formalism we obtain the following mfa thermodynamic potential @xmath29 per unit volume ( further calculation details can be found in refs . @xcite ) @xmath30 where @xmath31 is the cut - off of the model and @xmath32 is defined by @xmath33 \nonumber \\ & & - t \ln[1+e^{-(x+y)/t } ] , \label{omeguinha}\end{aligned}\ ] ] with @xmath34 ^ 2 + \delta^2 ) ^{1/2 } , \\ x_{8,9 } = ( [ e \pm ( \mu_{ug } + \mu_{dr})/2 ] ^2 + \delta^2)^{1/2 } \ , \\ y_1 = \mu_{ub } , \quad y_2 = \mu_{db } , \quad y_{3 } = \mu_{sr } , \\ y_{4 } = \mu_{sg } , \quad y_{5 } = \mu_{sb } , \\ y_{6,7 } = ( \mu_{ur}-\mu_{dg})/2 , \quad y_{8,9 } = ( \mu_{ug } - \mu_{dr})/2 . \end{aligned}\ ] ] in the above expressions @xmath35 and @xmath36 , with @xmath37 . as we are working in the isospin limit , then @xmath38 which gives @xmath39 . in principle one has nine different quark chemical potentials , corresponding to the three quark flavours ( u , d and s ) and three quark colors ( r , g , and b ) . nevertheless , as discussed above , with our particular election of the orientation of the gap @xmath40 in the color space , there is a residual color symmetry ( between red and green colors ) . moreover , if we require the system to be in chemical equilibrium , it can be seen that all chemical potentials are not independent from each other , as it will be discussed in next section . the total thermodynamic potential is obtained by adding to @xmath41 the contribution of the electrons and a vacuum constant . namely , @xmath42 where @xmath43 is the thermodynamic potential of the electrons . for them we use the expression corresponding to a free gas of ultra - relativistic fermions @xmath44 it is important to notice that in eq . ( [ qmp ] ) we have subtracted the constant @xmath45 in order to have a vanishing pressure at vanishing temperature and chemical potentials . however , this conventional prescription is merely an arbitrary way to uniquely determine the eos of the njl model without any further assumptions @xcite . in the mit bag model for instance , the pressure in the vacuum is non - vanishing . in view of this , @xmath46 is taken as a free parameter in ref . @xcite , having in mind that tuning this constant is an easy way to control the splitting between the chiral restoration density and the deconfinement density . nevertheless , we shall see below that @xmath46 does nt have a direct influence on the values of the surface tension and the curvature energy . now we are ready to introduce the effects of finite size in the thermodynamic potential . for doing so we consider the multiple reflection expansion formalism ( see refs . @xcite and references therein ) which consists in modifying the density of states for the case of a finite spherical droplet as follows @xmath47 where the surface contribution to the density of states is @xmath48 and the curvature contribution is given in the madsen ansatz @xcite @xmath49\ ] ] to take into account the finite quark mass contribution . the density of states of mre for massive quarks is reduced compared with the bulk one , and for a range of small momentum becomes negative . this non - physical negative values are removed by introducing an infrared ( ir ) cutoff in momentum space @xcite . thus , we have to perform the following replacement in order to obtain the thermodynamic quantities @xmath50 the ir cut - off @xmath51 is the largest solution of the equation @xmath52 with respect to the momentum @xmath53 . after the above replacement , the full thermodynamic potential for finite size spherical droplets reads : @xmath54 multiplying on both sides of the last equation by the volume of the quark matter drop and rearranging terms we arrive to the following form for @xmath55 @xmath56 where the pressure @xmath57 , the surface tension and the curvature energy density , are defined as @xcite @xmath58 @xmath59 and @xmath60 respectively . we are considering a spherical drop , i.e. the area is @xmath61 and the curvature is @xmath62 . as mentioned above , once we have the grand thermodynamic potential @xmath55 then we can obtain the relevant thermodynamic quantities . we can readily obtain the number density of quarks of each flavor and color @xmath63 and the number density of @xmath64 ( assumed to be massless ) @xmath65 then , the corresponding number densities of each flavor , @xmath66 , and of each color , @xmath67 , in the quark phase are given by @xmath68 and @xmath69 respectively . the baryon number density reads @xmath70 . in order to derive the eos from the above formalism it is necessary to impose a suitable number of conditions on the variables @xmath72 and @xmath40 . the first three of these conditions arise from the fact that the thermodynamically consistent solutions correspond to the stationary points of @xmath73 with respect to @xmath74 , @xmath75 , and @xmath40 . then , we have @xmath76 for the remaining conditions one must specify the physical situation in which one is interested in . in this work we are interested in the study of finite size color - superconducting droplets in @xmath71-equilibrium that may form e.g. within the mixed phase of a hybrid star . in such a case , chemical equilibrium is maintained by weak interactions among quarks , e.g. @xmath77 , @xmath78 , @xmath79 . here we consider the situation of no neutrino trapping . then , the lepton number is not conserved and we have four independent conserved charges , namely the electric charge @xmath80 and the three color charges @xmath81 , @xmath82 and @xmath83 . it is more convenient to use the linear combinations @xmath84 , @xmath85 and @xmath86 , where @xmath87 ( the total quark number density ) and @xmath88 and @xmath89 are related with color asymmetries . thus , conserved charges @xmath90 are related to four independent chemical potentials @xmath91 such that @xmath92 . the individual quark chemical potentials @xmath19 are given by @xmath93 \nonumber \\ & & \ ; + \mu_3 ( \lambda_3)_{cc } + \mu_8 ( \lambda_8)_{cc}.\end{aligned}\ ] ] where , as before , @xmath10 and @xmath11 are the gell - mann matrices in flavor and color space respectively . from the @xmath71-equilibrium conditions we have @xmath94 for all colors @xmath21 . then , the electron chemical potential is @xmath95 . finally , the rest of the conditions we need to impose for electrically and color neutral matter are : @xmath96 ( note : remember that we choose a particular orientation of the gap in the color space , which introduces the @xmath97 symmetry . thus , we trivially satisfy @xmath98 , and then @xmath99 is automatically equal to zero . ) in summary , in the case of neutron star quark matter without neutrino trapping , for each value of @xmath100 ( or @xmath101 ) and @xmath18 one can find the values of @xmath102 and @xmath103 by solving eqs . ( [ gapeq ] ) and ( [ colden ] ) supplemented by ( [ beta ] ) . this allows us to obtain the quark matter eos in the thermodynamic region we are interested in and we can evaluate the curvature energy and surface tension of the color superconducting droplets . it is important to remark that , in general , when we numerically solve the set of equations related with all the conditions discussed above , there might be regions for which there is more than one solution for each value of t and @xmath100 . to choose the stable solution among all of them , we require it to be an overall minimum of the thermodynamic potential . the set of parameters we use in the present work is the following ( those in ref . @xcite but without t hooft interactions ) , @xmath104 = 5.5 mev , @xmath105 = 112.0 mev , @xmath31 = 602.3 mev and @xmath106 = 4.638 . moreover , we considered the ratio @xmath107 obtained from fierz transformations of the one - gluon exchange interactions . as we impose vanishing pressure at vanishing temperature and chemical potentials for fermi momentum @xmath108 ( when @xmath109 ) @xcite , thus , @xmath110 is the same as in the bulk case . for the set of parameters we used , we found @xmath111 mev/@xmath112 ) . however , we emphasize that the values of the surface tension @xmath113 and the curvature energy @xmath114 are not changed by the choice of @xmath115 . as we previously mentioned , the value of @xmath51 is the largest root when solving @xmath116 with respect to @xmath53 , depending on @xmath117 and r. we find that these solutions can be fitted through the following rule , @xmath118 with @xmath119 in fm and @xmath51 in mev . the coefficients for @xmath120 mev are @xmath121 and @xmath122 . for @xmath123 mev we have @xmath124 and @xmath125 . in fig . [ fig1 ] we show our results for the surface tension @xmath113 and curvature energy @xmath114 of the color superconducting droplets as a function of the quark chemical potential , for two different temperatures and four different radii . the temperatures 30 mev and 5 mev are representative , respectively , of the conditions prevailing at the beginning and at the end of the cooling / deleptonization phase of protoneutron star evolution . we have checked that the results for @xmath126 mev are almost indistinguishable of those for zero temperature , thus , in practice they also represent old and cold neutron stars . we show results for drops with radii ranging from very small values of 5 fm , which have a large energy cost due to surface and curvature effects , to the bulk limit of @xmath127 . for given values of @xmath119 , @xmath18 and @xmath100 there may exist more than one solution of the equations . if more than one solution is found , the one that minimizes the thermodynamic potential is chosen . as explained in the figure caption , the left branches correspond to the chiral symmetry broken phase and the right curves after the discontinuity to the 2sc phase . for the curves presenting negative pressures we have introduced a dot indicating the zero pressure point . the part of the curve to the left of the dot corresponds to @xmath128 . note that , as previously mentioned , we subtracted @xmath46 to have vanishing pressure at vanishing @xmath18 and @xmath100 for @xmath127 . however , @xmath46 can be taken as a free parameter as in ref . @xcite . in this case , the dot in the figures can move along the curves . however , it is clear from eqs . ( [ surfacetension ] ) and ( [ curvatureenergy ] ) that a non - standard choice for @xmath46 does nt change the numerical values of @xmath113 and @xmath114 for given @xmath119 , @xmath18 and @xmath100 . our results show that the surface tension is in the range of @xmath129 mev/@xmath0 and the curvature energy is in the range of @xmath130 mev / fm . the large values of the surface and curvature energies are due to the linear term in the expression for @xmath32 in eq . ( [ omeguinha ] ) , which is not present in the thermodynamic potential of e.g. the mit bag model . for a given @xmath100 , the surface tension is an increasing function of @xmath119 . the curvature energy behaves differently at constant @xmath100 : for very small radii ( @xmath131 fm ) it increases with @xmath119 , but in the range from @xmath132 to @xmath133 it is a decreasing function of @xmath119 . of course , both the surface and curvature contributions to the free energy per unit volume @xmath134 tend to zero as @xmath109 , as can be checked from eq . ( [ eq17 ] ) . the different behavior with @xmath119 for @xmath113 and @xmath114 is a consequence of the functional form of the integrand together with the r - dependence of the ir cutoff . the integral for the surface tension has a fixed upper limit ( the ultraviolet cutoff @xmath31 ) , but the lower limit ( @xmath51 ) decreases with r. then , as @xmath119 increases , @xmath51 decreases , the area under the curve increases and as a result , the surface tension increases . for the curvature energy the effect is different . the integrand is positive for large @xmath53 but negative for small @xmath53 . for small @xmath119 , @xmath51 falls in the positive region of the integrand . thus , as @xmath51 decreases , the positive area under the curve increases , and the curvature energy increases . for larger @xmath119 , @xmath51 falls in the negative regionof the integrand . consequently , as @xmath51 decreases , the negative part of the area under the curve increases , and the curvature energy decreases . the results in fig . [ fig1 ] show that the temperature dependence of @xmath113 and @xmath114 is very weak for the values of @xmath18 that are relevant for protoneutron stars and cold neutron stars . note also that these results are of the same order of the obtained within the njl model for just deconfined quark matter _ out of chemical equilibrium _ under weak interactions @xcite . the present equation of state can support a two solar mass neutron star such as the pulsars psr j1614 - 2230 @xcite and psr j0348 - 0432 @xcite if a sufficiently stiff hadronic equation of state is employed for the outer layers of the star ( see for example fig 4 of @xcite ) . adding a vector interaction to the njl model , it is possible to stiffen the eos and obtain larger stellar masses . the vector term affects the surface tension in a non - trivial way . the chemical potentials gain an extra term @xmath135 that shift chemical equilibrium and the pressure gains a term proportional to the square of the density . the combined effect is difficult to estimate without a full calculation that will be addressed in future work . according to recent work @xcite , within a geometric approach to the surface tension evaluation , the surface tension can be lowered by the presence of a repulsive vector term and for magnetized quark matter the value will be further lowered @xcite . however , within the mre formalism the behavior may be different , and deserves further study . the large values of @xmath113 and @xmath114 have strong consequences for the physics of neutron star interiors , because the energy cost of forming quark drops within the mixed phase of hybrid stars would be very large . according to @xcite , beyond a limiting value of @xmath136 mev/@xmath0 the structure of the mixed phase becomes mechanically unstable and local charge neutrality is recovered . therefore , our results indicate that the hadron - quark interphase within a hybrid star should be a sharp discontinuity . the consequences of the large values of @xmath113 and @xmath114 for the triggering of the deconfinement transition in neutron and protoneutron stars have been studied in @xcite . the main difference with the present study of the quark - hadron interphase is the condition of chemical equilibrium . the nucleation ( deconfinement ) of the first quark matter drop that triggers the conversion of the core of a hadronic star is driven by strong interactions and consequently it happens out of chemical equilibrium under weak interactions . as shown in ref . @xcite the nucleation of quark matter is possible during the protoneutron star phase even for large values of the surface tension , because large drops ( with a size of hundreds of fm ) may have a huge nucleation rate . these large drops are charge neutral because flavor is conserved during the deconfinement transition @xcite and therefore they can be considerably larger than the debye screening length @xmath137 of the stellar plasma which is typically @xmath138 fm . since these drops can be very large ( @xmath139 fm @xcite ) , surface and curvature effects tend to vanish . this is not the case for droplets of quark matter in the hypothetical mixed phase of a hybrid star . since they are electrically charged their size can not exceed @xmath140 and therefore surface and curvature have a significant energy cost , inhibiting the formation of the mixed phase . r. balian and c. bloch , annals of physics 60 , 401 ( 1970 ) j. madsen , phys . d * 50 * , 3328 ( 1994 ) . o. kiriyama and a. hosaka , phys . rev . d * 67 * , 085010 ( 2003 ) . o. kiriyama , phys . d * 72 * , 054009 ( 2005 ) .
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in this paper we study the surface tension and the curvature energy of three - flavor quark matter in equilibrium under weak interactions within the nambu - jona - lasinio model .
we include the effect of color superconductivity and describe finite size effects within the multiple reflection expansion ( mre ) framework .
our calculations result in large values of the surface tension which disfavor the formation of mixed phases at the hadron - quark inter - phase inside a hybrid star .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in principle qcd is the fundamental theory that describes hadronic systems , but its non - perturbative character makes a direct application to hadrons difficult , with the exception of the high energy and momentum transfer regime . fortunately , at low @xmath2 effective field theories based on effective hadronic degrees of freedom have been applied with success . examples are chiral perturbation theory @xcite , small scale expansion @xcite , soliton models and constituent quark models @xcite . constituent quark models provided consistent descriptions encompassing the low - energy baryon spectrum , elastic and inelastic form factors , charge and magnetic radii , magnetic moments , axial and pseudoscalar form factors @xcite . although all those different frameworks may include the essential features of qcd , which are confinement and chiral symmetry , they are only partial simulations of the true underlying theory . recently , significant progress has occurred in calculations of qcd in the lattice , which evaluate the important qcd non - perturbative contributions at low @xmath2 directly from the underlying theory . up to now the applications of lattice qcd are restricted still to large pion masses ( @xmath3 mev ) corresponding to heavy quarks , and lattice spacings that are one order of magnitude smaller than the size of the nucleon ( @xmath4 fm ) . to extract information on the real world the results must be extrapolated both to the continuum limit ( @xmath5 ) and to the physical pion mass regime @xcite . however , except for the limit to the physical region , there is no simple way of interpreting the lattice qcd data . in this work we invert this procedure . we take here the challenge of describing the lattice data using a quark model . we start with the constituent quark model obtained from the covariant spectator theory . our model was presented in refs . @xcite , where it was adjusted to the experimental data for the four nucleon elastic form factors and the dominant form factor of the @xmath1 transition . in this work we extend this model to the unphysical region of the lattice data . although the constituent quark models were originally thought for and applied to the physical limit , a constituent quark model that is simultaneously consistent with the experimental data and lattice qcd data is valuable , because it includes indirectly the constraints of qcd , and therefore satisfies properties of the underlying theory . a similar procedure was considered in refs . @xcite using a larger number of parameters . our model does not include explicit pion cloud effects . however , as the electromagnetic interaction with the quarks is parametrized according to vector meson dominance ( vmd ) , part of the pion cloud effects are indirectly taken into account . therefore the model can be applied in the regions where the valence quark degrees of freedom are dominant , and it gives a good description of the nucleon and @xmath6 systems @xcite . for the nucleon , pion cloud effects are fundamental in the time - like region @xcite , although they are not so important in the region of small positive @xmath2 values @xcite , which is involved in the study of the electromagnetic transition . for the @xmath1 transition , as it happens also to other constituent quark models , without explicit pion cloud effects , the predictions for the magnetic moment , which dominates the transition , deviate necessarily from the data for low @xmath2 , showing that an effective pion cloud within the vmd current is not sufficient . because of the opening of the @xmath7 channel , an explicit pion cloud contribution must be considered @xcite . therefore our results in the physical low @xmath2 region give only the constituent quark core contributions @xcite , usually labeled as `` bare '' contributions . our results for these contributions are consistent with independent calculations of dynamical reaction models @xcite based on hadronic degrees of freedom . as for the e2 and c2 multipoles , in the large @xmath8 limit @xcite they represent second order corrections and are not considered here . as pion cloud effects in lattice qcd are expected to be small for @xmath9 0.40 gev @xcite , the description of the lattice qcd data appears as the ideal testing ground for our constituent quark model , where pion cloud effects are not explicitly included . this defines the main goal of this work . at present , lattice calculations are still restricted to high pion masses @xmath10 350 mev , although technical improvements are increasingly allowing to reach nearer and nearer the physical region @xcite . in this work we took the quenched lattice qcd data from refs . @xcite . in the unquenched calculations the effects of the sea quarks are explicitly considered . although unquenched lattice qcd data is already available , it is not expected that the quenched and unquenched data differ substantially in the region @xmath9 0.40 gev @xcite . in addition , there are some differences between unquenched data for the @xmath1 transition based on two different unquenched methods at similar pion masses @xcite that still have to be clarified in the future . for these reasons we took the conservative option of using quenched data exclusively . our new model presented here is based on three specific features : i ) the electromagnetic interaction with the constituent quarks is described within the impulse approximation , and considering vmd ; ii ) the wavefunctions of the quark - diquark system are parametrized by simple monopole factors reduced to the hulthen form , with one or two effective range parameters that balance the details of the short range and the long range behavior of the system ; iii ) the constituent quark magnetic anomalous moment scales with the inverse of its mass , which we write , following @xcite as a function of the current quarks mass . a current based on vmd is suitable to describe the interaction of the photon with the constituent quark , for the quark - antiquark spin-1 vertex . depending on the isospin , the intermediate meson pole corresponds to the @xmath11 or the @xmath12 meson , at low @xmath2 , or , in the large @xmath2 regime , to some other effective heavy meson pole @xmath13 . the photon interaction is then described as proceeding through the production of an intermediate meson state which annihilates subsequently into a quark - antiquark pair . vmd is successful in the description of the electromagnetic interaction with nucleons . the nucleon s - state wavefunction includes the correct spin structure for the quark - diquark spin 0 and 1 components , associated to isospin 0 and 1 states , respectively . for the @xmath6 , since total isospin is @xmath14 , the s - state wavefunction reduces to the diquark spin 1 with isospin 1 structure . as for the scalar wavefunction , it takes the phenomenological form _ b ( p , k ) = , [ eqpsis ] where @xmath15 , @xmath16 , with @xmath17 the baryon mass ( for the nucleon or @xmath6 ) , @xmath18 is the diquark mass and @xmath19 is a normalization constant . [ in ref . @xcite we considered @xmath20 for the nucleon and @xmath21 for the @xmath6 ] . the parameters @xmath23 can be interpreted as yukawa mass or range coefficients that distinguish between two different regimes for the momentum range . the parametrization of the momentum dependence in terms of @xmath24 absorbs the dependence on the baryon masses @xmath17 . the range parameters @xmath23 for the nucleon and the @xmath6 were fixed by the nucleon and @xmath1 form factor experimental data @xcite . in the spectator quark model the transition amplitude between a initial ( momentum @xmath25 ) nucleon ( @xmath0 ) and a final ( momentum @xmath26 ) baryon @xmath15 can be written , in impulse approximation @xcite as j^= 3 _ _ k complete baryon wavefunction ( including spin , isospin and momentum dependence ) , @xmath27 is the diquark polarization and @xmath28 the quark current operator dependent of the hadron isospin @xmath29 . the baryon polarizations are suppressed for simplicity . the symbol @xmath30 represents the invariant integral in the on - shell diquark moment @xmath31 , with @xmath32 as the diquark energy . the factor 3 takes account for the isospin symmetrization . the quark current @xcite takes the general form j_i^&= & ( f_1+(q^2 ) + f_1- ( q^2)_3 ) ^+ + & & ( f_2+(q^2 ) + f_2- ( q^2 ) _ 3 ) . the constituent quark form factors @xmath33 ( @xmath34 ) are parametrized using a vmd structure , and are normalized according to @xmath35 and @xmath36 , where @xmath37 are defined in terms of the quark anomalous magnetic moments @xmath38 and @xmath39 according with @xmath40 and @xmath41 . see refs . @xcite for details . in this first calculation the wavefunctions are reduced to s - states for the quark - diquark system . although it is well known that angular momentum components besides the s - states are essential for the description of the nucleon in deep inelastic scattering , their effects are not so evident in the elastic form factors , particularly for low @xmath2 @xcite . as for the @xmath6 , calculations of the valence quark contribution associated with higher angular momentum states ( d - states ) suggest a small effect @xcite . in fact , even with only s - wave components in the wavefunctions for both the nucleon and the @xmath6 , the model generates the dominant contribution for the @xmath1 transition , the dipole magnetic moment form factor . the non - zero contributions for the other form factors appear only when d states are included , or when pion cloud effects are taken in consideration @xcite . at the end , we will estimate the effect in the @xmath1 quadrupole transition form factors , in the lattice regime , from adding the d - states to the present model . as shown in @xcite the diquark mass scales out from the formulas obtained for the electromagnetic form factors . this allowed us to ignore any explicit dependence on the quark mass . this mass dependence was present in the nucleon , @xmath6 and vector meson masses in an implicit way only . however , in order to grant a comparison of our results with the lattice data , we extend here the model to include a dependence of the quark anomalous moment @xmath38 and @xmath39 on the quark mass . this dependence was not explicitly considered in refs . @xcite because they dealt only with the physical data . inspired by ref . @xcite , which combines chiral symmetry with conventional quark models , and applies an analytic continuation of the chiral expansion to the simple su(6 ) model , we then use the smooth variation of the hadronic properties with the current quark masses above 60 mev . therefore , in the spirit of the constituent quark models , we assume here that @xmath42 ( @xmath43 ) scales with @xmath44 , where @xmath45 is the constituent quark mass . labeling the quark anomalous moment at the physical point by @xmath46 , we can write @xmath42 , for an arbitrary constituent quark mass @xmath45 , as _ q= _ q^0 , [ eqkq ] where @xmath47 is the constituent quark mass at the physical point . to include an explicit dependence on the quark mass we consider the parametrization due to cloet _ et . @xcite m_q = m_+ cm_q = m_+ c m_q^phy , [ eqmq ] where @xmath48 ( @xmath49 ) is the ( physical ) current quark mass , @xmath50 is a new parameter corresponding to the constituent quark mass in the chiral limit ( @xmath51 ) , and @xmath52 is a coefficient of the order of the unity . in the same equation @xmath53 stands for the pion mass in the model , a parameter to be fixed by the lattice data , and @xmath54 for the physical mass . following ref . @xcite we considered @xmath55 mev . the parametrization in ( [ eqmq ] ) is most sensitive to @xmath50 . reference @xcite fixes @xmath56 gev . different descriptions using quark models and different lattice sizes lead to different values @xcite . for this reason we use @xmath50 as the only free parameter allowed to vary in the calculation presented here . it is needed to introduce an explicit dependence of the nucleon magnetic moment on the pion mass , and to enable the connection of the constituent quark model to the lattice qcd calculations @xcite . we consider three different cases . first we considered the case @xmath57 , corresponding to the limit where @xmath42 has no dependence on @xmath53 . we tested also the original parametrization @xmath58 0.42 gev @xcite . finally , although we did not performed a systematic fit , we tested several other values of @xmath50 . furthermore , to extend the spectator model to the region of the quenched lattice qcd data we still need to consider the nucleon and @xmath6 masses determined by the lattice simulations @xcite . this brings to the calculation an implicit dependence on the pion mass . as for the vectorial mesons included in the vmd , quark current picture , we use the parametrization @xcite m_= c_0 + c_1 m_^2 , [ eqmrho ] where @xmath59 gev and @xmath60 gev@xmath61 . the simple parametrization ( [ eqmrho ] ) describes well the quenched lattice qcd data and is consistent with finite volume corrections @xcite . in this applications we consider this parametrization together with model ii of the refs . @xcite , where the heavy vectorial meson mass is @xmath62 , with @xmath63 the nucleon mass of the lattice calculations . in fig . [ fignucleonff ] we compare the predictions of our model for the nucleon form factor with the lattice data from ref . @xcite , corresponding to the three values of @xmath50 . we consider in particular the isovector nucleon form factor because the contributions of the disconnected diagrams vanishes in lattice calculations , if the flavor su(2 ) symmetry is assumed as in ref . we considered the lowest pion masses and the smallest lattice spacings . in particular , we take the data corresponding to @xmath64 504 mev , 649 mev ( @xmath65 fm ) and @xmath66 mev ( @xmath67 fm ) . for larger lattice spacings a dependence on @xmath68 is observed @xcite . from the figure we conclude that @xmath56 gev and 0.80 gev gives an excellent description of the nucleon data . still , when the pion mass increases there is a systematic deviation of our model from the lattice data of ref . this deviation may be a consequence of the fact that the wavefunction parametrization in terms of a low momentum scale ( @xmath69 ) ( long range behavior ) and high momentum scale ( @xmath70 ) was kept unchanged . as the pion masses vary , at least the long range parameter may vary , and may vary more rapidly for larger pion masses . as for the short range parameter ( @xmath70 ) it sets the scale below which the short range physics becomes important . as observed in applications of the finite - range regularization effective field theory , the scale associated to the momentum cut - off ( and consequently short range effects ) can be expressed by an universal regulator which describes simultaneously the large pion mass regime and the physical regime @xcite . in our calculation @xmath70 is related to that universal regulator , and it is expected not to depend crucially on the pion mass value . for a finer analysis of our results , we compare our predictions for the isovector nucleon magnetic moment in physical nucleon magnetons , with the lattice qcd data and the chiral result from ref . @xcite . to convert @xmath71 given by the lattice data in units @xmath72 ( @xmath63 nucleon mass in lattice ) to `` physical '' units @xmath73 , we need to use the transformation @xmath74 . the results are presented in the fig . [ figgmv ] for @xmath56 gev and @xmath75 gev , as function of @xmath53 . the agreement of our results with the chiral expression from ref . @xcite shows the consistency of our calculations with chiral calculations . finally , in fig . [ figndeltaff ] we compare the quenched lattice data for the @xmath1 transition from ref . @xcite with the spectator quark model corresponding to @xmath76 gev , 0.80 gev and @xmath77 . all cases shown correspond to a lattice spacing of @xmath78 fm . the contribution of the quark core extracted from @xcite at the physical point is also included . to be consistent , we considered the parametrization of eq . ( [ eqmrho ] ) for @xmath79 , although the original ref . @xcite gives slightly different results . as for @xmath63 and @xmath80 , we use the values derived directly from the lattice data @xcite . for the larger pion masses we observe an almost perfect agreement between the predictions of @xmath81 and the data , although @xmath82 gev is also close . for @xmath83 gev we have also a good agreement , except for a slight deviation from the lattice data for @xmath84 gev@xmath85 2 gev@xmath86 . this deviation may result from the effect of the pion cloud for light pions , predicted to be important for @xmath87 gev @xcite . as for the physical pion mass case , our model is coherent with the constituent quark core data labeled bare form factor , which is extracted indirectly from experiment @xcite . in this case all lines coincide . as mentioned already , the d - states in the @xmath6 wavefunction induce contributions for the subleading electric and coulomb quadrupole transition form factors @xmath88 and @xmath89 . the exact contributions depend on the specific parametrization , in particular on the admixture coefficients for the two d - waves . in ref . @xcite it was shown how a percentage of d - states of @xmath90 can provide an excellent description of the quadrupole lattice data , without significantly changing the description of the dominant magnetic dipole form factor . that application corresponds to the @xmath91 limit and estimates the effect of the valence quark ( or bare ) contributions as only @xmath92 of the total quadrupole for both @xmath88 and @xmath89 , at the physical point ( the remaining contribution being the pion cloud ) . on the other hand , in that particular parametrization , and in the region of study @xmath93 gev@xmath86 , the effect of the constituent quark mass dependence described by eq . ( [ eqkq ] ) corresponds to a correction of less than 20% for both form factors , a correction smaller than the typical lattice errorbands ( @xmath94 ) . chiral perturbation methods for lattice qcd extrapolations are useful for small @xmath2 ( like @xmath95 gev@xmath86 ) , but are not adequate for the high @xmath2 region . constituent quark models can supply an alternative guidance for lattice qcd extrapolations . in this work we consider a covariant constituent quark model of the nucleon and the @xmath6 based on the spectator formalism for the quark - diquark system ( covariance is an important issue for the description of the high momentum transfer processes ) . the model contains no explicit pion cloud , besides the effects included in the @xmath11 term of the vdm picture for the electromagnetic current , and is therefore reduced to the bare quark hadron structure . since pion cloud effects are expected to be negligible for large values of the pion mass , the comparison of the model to the quenched data for pion masses larger than 450 mev is justified a priori . to accomplish that comparison the initial covariant quark model was extended to the lattice data region , @xmath96 mev . in light of the work in ref . @xcite , this extension was done by introducing the constituent quark mass in the chiral limit parameter , @xmath50 , and the nucleon @xmath11 , @xmath6 masses used in the lattice calculations . the main conclusion is that the covariant constituent quark model which was previously calibrated by means of a quantitative description of the nucleon , @xmath1 and @xmath6 form factor data in the physical region , as shown in previous works @xcite , after a simple extension involving one parameter only , describes also quantitatively well the lattice results for the nucleon isovector form factor and the @xmath1 `` bare '' magnetic form factor . with @xmath56 gev , consistently with the range @xmath97 gev suggested by several constituent quark models @xcite , we obtain a very good description of the lattice nucleon form factors data , but underestimate the @xmath1 magnetic moment form factor data by less than 9% at low @xmath2 for @xmath98 mev . note that the original parametrization @xmath56 gev @xcite was a result of a phenomenological fit and was not derived from first principles . an optimal description of the lattice data for both nucleon and @xmath1 transition form factors is achieved , once the scale of the constituent quark mass in the chiral limit is fixed as @xmath75 gev . [ the values @xmath56 and 0.80 gev are better for the nucleon data ; @xmath75 gev and @xmath99 are better for the @xmath1 data ] . the parameter @xmath75 gev is relatively large when compared with alternative constituent quark models , which may indicate that some of the effective quark - antiquark configurations contained in our model through the vmd mechanism in the electromagnetic current , for instance , do not correspond to the lattice calculations in the quenched approximation . still , the possibility of adjusting other parameters is very promising . a refinement of the description can be pursued , by increasing the number of adjustable parameters , or simply by introducing an extra dependence on the pion mass . for instance , in the future we certainly plan to check the assumption that the @xmath100 parameter which controls the long range regime does not depend on the pion mass value . in addition , we may consider as well an explicit dependence of the heavier meson mass pole @xmath13 , that regulates the short range effect in vmd mechanism , on the pion mass . alternatively , a pion mass dependence of the @xmath101 parameter can be introduced . furthermore , we want to study the quality of the description in the high pion masses region , not probed yet here . the authors are particularly thankful to gckeler for supply information about ref . g. r. would like to thank specially to ross young for the detailed explanations of the lattice proprieties and the extrapolations for the real world . g. r. also thank franz gross , ian cloet , michael pardon , anthony thomas and ping wang for helpful discussions . this work was partially supported by jefferson science associates , llc under u.s . doe contract no . de - ac05 - 06or23177 . g. r. was supported by the portuguese fundao para a cincia e tecnologia ( fct ) under grant no . sfrh / bpd/26886/2006 . this work has been supported in part by the european union ( hadronphysics2 project `` study of strongly interacting matter '' ) . b. julia - diaz , d. o. riska and f. coester , phys . c * 69 * , 035212 ( 2004 ) [ erratum - ibid . c * 75 * , 069902 ( 2007 ) ] [ arxiv : hep - ph/0312169 ] . s. boffi , l. y. glozman , w. klink , w. plessas , m. radici and r. f. wagenbrunn , eur . phys . j. a * 14 * , 17 ( 2002 ) [ arxiv : hep - ph/0108271 ] ; r. f. wagenbrunn , s. boffi , w. klink , w. plessas and m. radici , phys . lett . b * 511 * , 33 ( 2001 ) [ arxiv : nucl - th/0010048 ] . m. m. giannini , e. santopinto and a. vassallo , eur . j. a * 12 * , 447 ( 2001 ) [ arxiv : nucl - th/0111073 ] . e. pace , g. salme , f. cardarelli and s. simula , nucl . a * 666 * , 33 ( 2000 ) [ arxiv : nucl - th/9909025 ] ; f. cardarelli and s. simula , phys . c * 62 * , 065201 ( 2000 ) [ arxiv : nucl - th/0006023 ] . t. sato and t. s. h. lee , phys . c * 63 * , 055201 ( 2001 ) [ arxiv : nucl - th/0010025 ] ; s. s. kamalov , s. n. yang , d. drechsel , o. hanstein and l. tiator , phys . c * 64 * , 032201(r ) ( 2001 ) [ arxiv : nucl - th/0006068 ] .
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a covariant quark model , based both on the spectator formalism and on vector meson dominance , and previously calibrated by the physical data , is here extended to the unphysical region of the lattice data by means of one single extra adjustable parameter the constituent quark mass in the chiral limit .
we calculated the nucleon ( @xmath0 ) and the @xmath1 form factors in the universe of values for that parameter described by quenched lattice qcd .
a qualitative description of the nucleon and @xmath1 form factors lattice data is achieved for light pions .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
spectrum sensing plays an important role in cognitive radio ( cr ) systems , and cyclostationary feature detection is one of the main technologies for spectrum sensing in low snr cases [ 1 - 2 ] . the idea of cyclic detection is that one cr node samples the rf signals , transform the time domain signals into cyclic frequency domain , then decide primary user s occupancy of a target band regarding whether the cyclic spectrum on a significant cyclic frequency is above a certain threshold [ 2 ] . + although the simulation results of cyclostationary feature detection has been studied [ 3 - 5 ] , less attention has been paid to analyze the noise distribution on cyclic frequency domain . consequently , theoretical function between detection threshold @xmath1 and miss detection probability @xmath0 is not available , which makes it difficult for practical system design . for comparison , the theoretical function between detection threshold @xmath1 and miss detection probability @xmath0 of energy detection is available by using central and non - central chi - square distribution to model the distribution of the observation variable affected by time domain gaussian noise [ 6 ] , which makes energy detection to be a practical method for spectrum sensing . however , its performance in low snr cases is much poorer than cyclostationary feature detection [ 7 ] . in this paper , noise distribution on cyclic frequency domain has been analyzed and generalized extreme value distribution is found to be a precise match of the observation variable affected by time domain gaussian noise . a fast cyclic frequency domain feature detection algorithm [ 7 ] has been introduced to evaluate the coincidence between theoretical roc curve and the simulated roc curve , which proves the reliability of theoretical distribution model and feasibility of practical system design . + the rest part of the paper is organized as follows : section ii describes the system model of cyclostationary feature detection . noise distribution on cyclic frequency domain is analyzed in section iii . a fast cyclic frequency domain feature detection algorithm has been introduced in section iv . simulation results are given in section v. finally , conclusions are drawn in section vi . the spectrum sensing problem can be modeled as hypothesis testing . it is equivalent to distinguishing between the following two hypotheses : @xmath2 @xmath3 , @xmath4 and @xmath5 denote the received signal , the primary user s transmit signal and the gaussian noise , respectively . @xmath6 and @xmath7 represent the hypothesis that the primary user is active or inactive . due to the existence of noise , a certain threshold @xmath1 should be set to decide whether a primary user is active or not . probability of detection ( @xmath8 ) and false alarm ( @xmath0 ) are defined to evaluate the detection performance : @xmath9 the goal of detection is to maximize @xmath8 while maintain a given @xmath0 . + when feature detection is applied , the detection model ( 1 ) changes into : @xmath10 @xmath11 is the spectrum correlation density ( scd ) of the received signal @xmath3 , @xmath12 and @xmath13 is the scd of @xmath4 and @xmath5 , respectively [ 8 ] . theoretically , gaussian noise @xmath5 is not a cyclostationary statistic process , then @xmath14 when @xmath15 [ 8 ] . as for cyclostationary signal @xmath4 , there is a significant frequency set \{@xmath16 } , on which @xmath17 . due to the ideal non - cyclostationary characteristic of gaussian noise , any pre - set threshold @xmath1 on a significant frequency will lead to @xmath18 and @xmath19 . in practice , scd is calculated for limited length signals , therefore , @xmath20 when @xmath15 [ 9 ] . as shown in fig.1 , the background noise is obvious on @xmath21 square when calculating scd of a noise interfered am modulated signal . in order to analyze the background noise distribution , a limited - length gaussian noise sequence @xmath22 is considered : @xmath23 where @xmath24 is the length of the analysis window , @xmath25 is the index of analysis window . noise data in each analysis window are transformed into cyclic frequency domain , and then mapped from @xmath21 square to @xmath26 axis through the following expression : @xmath27 for each cycle frequency @xmath16 , the cyclic spectrum value is aligned to a set \{@xmath28 } , @xmath29 . according to the extreme definition of @xmath30 in ( 5 ) , generalized extreme value ( gev ) distribution is adopted to model the cyclic frequency domain noise [ 10 ] . the density function is : + when @xmath31 @xmath32 when @xmath33 @xmath34 where @xmath35 is the shape parameter , @xmath36 is the position parameter , @xmath37 is the scale parameter . and the parameters @xmath35 , @xmath36 , @xmath38 can be estimated by maximum likelihood estimation based on the noise sequence \{@xmath39}. for most cases , @xmath40 , then the likelihood function is defined as follows : @xmath41 let @xmath42,@xmath43 then : @xmath44 where @xmath45 and @xmath46 are the estimated value of @xmath36 and @xmath38 . by solving ( 9 ) , @xmath45 and @xmath46 are obtained . + after that , the likelihood function for @xmath35 is defined as : @xmath47\nonumber\\ & & -\sum_{i=1}^n[1+\kappa(\frac{n_i-\hat{u}}{\hat{\sigma}})]^{-\frac{1}{\kappa}}\end{aligned}\ ] ] let @xmath48 , the estimation of @xmath49 can be obtained by solving ( 10 ) . fig.2 shows the block diagram of a cyclic frequency domain feature detector [ 7 ] : in this detector , time domain signals are transformed to cyclic frequency domain , and then mapped to @xmath26 axis and get extreme value for each @xmath26 , finally , compared with a pre - set threshold @xmath1 to determine the occupancy of primary user . to decrease the computational complexity of feature detection , only one cycle frequency @xmath16 for a modulated signal is calculated for spectrum sensing [ 7 ] . the probability of detection ( @xmath8 ) and false alarm ( @xmath0 ) is defined by : @xmath50 where @xmath51 is cdf of generalized extreme value distribution . substitution of ( 6 ) and ( 7 ) into ( 11 ) yields : @xmath52 for a pre - set @xmath0 , the threshold can be estimated as : @xmath53 where @xmath54 . in this section , monte - carlo simulation results are presented to prove the reliability of the upper analytical results between @xmath0 and the threshold @xmath1 . + frequency smoothing method in [ 11 - 12 ] is applied to estimate scd of a time domain noise signal . simulation parameters are listed in table 1 . length of the analysis window is set to be 4096 and totally 10000 cyclic spectrum values on cyclic frequency @xmath55 , aligned by analysis window index , are considered to evaluate the theoretical curve . .simulation parameters list [ cols="^,^",options="header " , ] the histogram of the aligned cyclic spectrum values is shown in fig 3 , with compared to gev distribution . it is proved that the gev distribution precisely match the cyclic frequency domain noise data . + for further proof of the proposed model , the curves of _ receiver operating characteristics ( roc ) _ , which are theoretically derived from gev distribution , are plotted to compare with those derived from monte - carlo simulation . as to the theoretical curve , @xmath0 is pre - set according to system requirement . by using ( 13 ) , a theoretical threshold @xmath1 is obtained . finally , received signals under hypothesis @xmath6 are compared with the threshold to obtain the statistics result of @xmath8 . as to the simulated curve , threshold are chosen to be the same as the theoretical threshold \{@xmath1 } , after that , received signals under hypothesis @xmath6 and @xmath7 are compared with each @xmath1 to obtain the statistics results of @xmath8 and @xmath0 . finally , plot these ( @xmath8,@xmath0 ) points to form a continuous curve . + experiments results are shown in fig.4 , we can see that the theoretical roc curve ( red highlighted ) precisely match the simulated roc curve ( green highlighted ) for different received signal power levels . it is proved that the generalized extreme value distribution is efficient to model the noise distribution on cyclic frequency domain . in this paper , noise distribution on cyclic frequency domain is studied and generalized extreme value ( gev ) distribution is found to be an efficient method to model the cyclic frequency domain noise . maximum likelihood estimation is applied to estimate the parameters of gev . sensing threshold is consequently derived from system requirements ( a pre - set @xmath0 ) and theoretical cdf of gev distribution . monte carlo simulation has been carried out to prove that the simulated roc curve is precisely coincided with the theoretical roc curve . the project is founded by the corporation research department of huawei technology , the national 863 project of china , no . 2007aa01z237 , and the fund of ministry of science and technology of china , no . 2008dfa11950 . william a. gardner , and chad m. spooner , `` detection and source location of weak cyclostationary signals : simplifications of the maximum - likelihood receiver '' , ieee trans . vol-41 , no . 6 , june 1993 . paul d. sutton , keith e. nolan , and linda e. doyle , `` cyclostationary signatures in practical cognitive radio applications '' , selected areas in commun , ieee journal vol.26 , issue 1 , pp . 13 - 24 , jan . 2008 . jun ma , guodong zhao , and ye li , `` soft combination and detection for cooperative spectrum sensing in cognitive radio networks , '' ieee transactions on wireless communications , vol . 7 , no . 11 , pp . 4502 - 4507 , november 2008 . gan xiaoying , xu hao , xu youyun , qian liang , liu jing , `` noise analysis for limited length cyclostationary detection in cognitive radio systems '' , journal of pla university of science and technology,2008 , vol . 9 , no.6 , pp:633 - 636 .
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in cognitive radio systems , cyclostationary feature detection plays an important role in spectrum sensing , especially in low snr cases . to configure the detection threshold under a certain noise level and a pre - set miss detection probability @xmath0 , it s important to derive the theoretical distribution of the observation variable . in this paper , noise distribution in cyclic frequency domain has been studied and generalized extreme value ( gev ) distribution is found to be a precise match .
maximum likelihood estimation is applied to estimate the parameters of gev .
monte carlo simulation has been carried out to show that the simulated roc curve is coincided with the theoretical roc curve , which proves the efficiency of the theoretical distribution model . cognitive radio , spectrum sensing , cyclic feature detection , noise distribution on cyclic frequency domain
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many real systems can be viewed as a combination of slow and fast motions which leads to complicated double scale equations . already in the 19th century in applications to celestial mechanics it was well understood ( though without rigorous justification ) that a good approximation of the slow motion can be obtained by averaging its parameters in fast variables . later , averaging methods were applied in signal processing and , rather recently , to model climate weather interactions ( see @xcite , @xcite , @xcite and @xcite ) . the classical setup of averaging justified rigorously in @xcite presumes that the fast motion does not depend on the slow one and most of the work on averaging treats this case only . on the other hand , in real systems both slow and fast motions depend on each other which leads to the more difficult fully coupled case which we study here . this setup emerges , in particular , in perturbations of hamiltonian systems which leads to fast motions on manifolds of constant energy and slow motions across them . in this work we consider a system of differential equations for @xmath6 and @xmath7 @xmath8 with initial conditions @xmath9 on the product @xmath10 where @xmath11 is a compact @xmath12-dimensional @xmath13 riemannian manifold and @xmath14 are smooth in @xmath15 families of bounded vector fields on @xmath16 and on @xmath17 respectively , so that @xmath18 serves as a parameter for @xmath19 and @xmath20 for @xmath21 . the solutions of ( [ 1.1.1 ] ) determine the flow of diffeomorphisms @xmath22 on @xmath10 acting by @xmath23 . taking @xmath24 we arrive at the flow @xmath25 acting by @xmath26 where @xmath27 is another family of flows given by @xmath28 with @xmath29 being the solution of @xmath30 it is natural to view the flow @xmath31 as describing an idealised physical system where parameters @xmath32 are assumed to be constants ( integrals ) of motion while the perturbed flow @xmath22 is regarded as describing a real system where evolution of these parameters is also taken into consideration . essentially , the proofs of this paper work also in the slightly more general case when @xmath19 and @xmath21 in ( [ 1.1.1 ] ) together with their derivatives depend lipschitz continuously on @xmath33 ( cf . @xcite ) but in order to simplify notations and estimates we do not consider this generalisation here . assume that the limit @xmath34 exists and it is the same for `` many '' @xmath35s . for instance , suppose that @xmath36 is an ergodic invariant measure of the flow @xmath27 then the limit ( [ 1.1.3 ] ) exists for @xmath37almost all @xmath18 and is equal to @xmath38 if @xmath39 does not , in fact , depend on @xmath20 then @xmath40 and @xmath41 are also independent of @xmath20 and we arrive at the classical uncoupled setup . in this case lipschitz continuity of @xmath19 implies already that @xmath42 is also lipshitz continuous in @xmath20 , and so there exists a unique solution @xmath43 of the averaged equation @xmath44 in this case the standard averaging principle says ( see @xcite ) that for @xmath45-almost all @xmath18 , @xmath46 as the main motivation for the study of averaging is the setup of perturbations described above we have to deal in real problems with the fully coupled system ( [ 1.1.1 ] ) which only in very special situations can be reduced by some change of variables to a much easier uncoupled case where the fast motion does not depend on the slow one . observe that in the general case ( [ 1.1.1 ] ) the averaged vector field @xmath47 in ( [ 1.1.3 ] ) may even not be continuous in @xmath20 , let alone lipschitz , and so ( [ 1.1.4 ] ) may have many solutions or none at all . moreover , there may exist no natural well dependent on @xmath48 family of invariant measures @xmath36 since dynamical systems @xmath27 may have rather different properties for different @xmath20 s . even when all measures @xmath36 are the same the averaging principle often does not hold true in the form ( [ 1.1.5 ] ) , for instance , in the presence of resonances ( see @xcite and @xcite ) . thus even basic results on approximation of the slow motion by the averaged one in the fully coupled case can not be taken for granted and they should be formulated in a different way requiring usually stronger and more specific assumptions . if convergence in ( [ 1.1.3 ] ) is uniform in @xmath20 and @xmath18 then ( see , for instance , @xcite ) any limit point @xmath49 as @xmath50 of @xmath51 is a solution of the averaged equation @xmath52 it is known that the limit in ( [ 1.1.3 ] ) is uniform in @xmath18 if and only if the flow @xmath27 on @xmath11 is uniquely ergodic , i.e. it possesses a unique invariant measure , which occurs rather rarely . thus , the uniform convergence in ( [ 1.1.3 ] ) assumption is too restrictive and excludes many interesting cases . probably , the first relatively general result on fully coupled averaging is due to anosov @xcite ( see also @xcite and @xcite ) . relying on the liouville theorem he showed that if each flow @xmath27 preserves a probability measure @xmath36 on @xmath11 having a @xmath53 dependent on @xmath20 density with respect to the riemannian volume @xmath54 on @xmath11 and @xmath36 is ergodic for lebesgue almost all @xmath20 then for any @xmath55 @xmath56 where mes is the product of @xmath54 and the lebesgue measure in a relatively compact domain @xmath57 . an example in appendix to @xcite shows that , in general , this convergence in measure can not be strengthened to the convergence for almost all initial conditions and , moreover , in this example the convergence ( [ 1.1.5 ] ) does not hold true for any initial condition from a large open domain . such examples exist due to the presence of resonances , more specifically to the `` capture into resonance '' phenomenon , which is rather well understood in perturbations of integrable hamiltonian systems . resonances lead there to the wealth of ergodic invariant measures and to different time and space averaging . it turns out ( see @xcite ) that wealth of ergodic invariant measures with nice properties ( such as gibbs measures ) for axiom a and expanding dynamical systems also yields in the fully coupled averaging setup with the latter fast motions examples of nonconvergence as @xmath50 for large sets of initial conditions ( see remark [ rem1.2.12 ] ) . in hamiltonian systems , which are a classical object for applications of averaging methods , the whole space is fibered into manifolds of constant energy . for some mechanical systems these manifolds have negative curvature with respect to the natural metric and their motion is described by geodesic flows there . hyperbolic hamiltonian systems were discussed , for instance , in @xcite and a specific example of a particle in a magnetic field leading to such systems was considered recently in @xcite . of course , these lead to hamiltonian systems which are far from integrable . such situations fall in our framework and they are among main motivations for this work . this suggests to consider the equation ( [ 1.1.1 ] ) on a ( locally trivial ) fiber bundle @xmath58 with a base @xmath59 being an open subset in a riemannian manifold @xmath60 and fibers @xmath61 being diffeomorphic compact riemannian manifolds ( see @xcite ) . on the other hand , @xmath62 has a local product structure and if @xmath63 is bounded then the slow motion stays in one chart during time intervals of order @xmath64 with @xmath65 small enough . hence , studying behavior of solutions of ( [ 1.1.1 ] ) on each such time interval separately we come back to the product space @xmath10 setup and will only have to piece results together to see the picture on a larger time interval of length @xmath66 we assume in the first part of this work that @xmath39 is @xmath13 in @xmath20 and @xmath18 and that for each @xmath20 in a closure of a relatively compact domain @xmath67 the flow @xmath27 is anosov or , more generally , axiom a in a neighborhood of an attractor @xmath68 let @xmath69 be the sinai - ruelle - bowen ( srb ) invariant measure of @xmath27 on @xmath70 and set @xmath71 it is known ( see @xcite ) that the vector field @xmath47 is lipschitz continuous in @xmath72 and so the averaged equations ( [ 1.1.4 ] ) and ( [ 1.1.6 ] ) have unique solutions @xmath73 and @xmath74 still , in general , the measures @xmath69 are singular with respect to the riemannian volume on @xmath11 , and so the method of @xcite can not be applied here . we proved in @xcite that , nevertheless , ( [ 1.1.7 ] ) still holds true in this case , as well , and , moreover , the measure in ( [ 1.1.7 ] ) can be estimated by @xmath75 with some @xmath76 . the convergence ( [ 1.1.7 ] ) itself without an exponential estimate can be proved by another method ( see @xcite ) which can be applied also to some partially hyperbolic fast motions . an extension of the averaging principle in the sense of convergence of young measures is discussed in section [ sec1.11 ] . once the convergence of @xmath51 to @xmath77 as @xmath50 is established it is interesting to study the asymptotic behavior of the normalized error @xmath78.\ ] ] namely , in our situation it is natural to study the distributions @xmath79 as @xmath50 where @xmath54 is the normalized riemannian volume on @xmath11 and @xmath80 is a borel subset in the space @xmath81 of continuous paths @xmath82 $ ] on @xmath83 we will obtain in this work large deviations bounds for @xmath84 which will give , in particular , the result from @xcite saying that @xmath85 exponentially fast in @xmath5 where @xmath86 is the uniform norm on @xmath87 however , the main goal of this work is not to provide another derivation of ( [ 1.1.9 ] ) but to obtain precise upper and lower large deviations bounds which not only estimate measure of sets of initial conditions for which the slow motion @xmath88 exhibits substantially different behavior than the averaged one @xmath89 but also enable us to go further and to investigate much longer exponential in @xmath5 time behavior of @xmath88 . namely , we will be able to study exits of the slow motion from a neighborhood of an attractor of the averaged one and transitions of @xmath88 between basins of attractors of @xmath89 . such evolution , which becomes visible only on much longer than @xmath5 time scales , is usually called adiabatic in the framework of averaging . in the simpler case when the fast motion does not depend on the slow one such results were discussed in @xcite . still , even in this uncoupled situation descriptions of transitions of the slow motion between attractors of the averaged one were not justified rigorously both in the markov processes case of @xcite and in the dynamical systems case of @xcite . extending these technique to three scale equations may exhibit stochastic resonance type phenomena producing a nearly periodic motion of the slowest motion which is described in section [ sec1.10 ] below . these problems seem to be important in the study of climate weather interactions and they were discussed in @xcite and @xcite in the framework of a model describing transitions between steady climatic states with weather evolving as a fast chaotic system and climate playing the role of the slow motion . such `` very long '' time description of the slow motion is usually impossible in the traditional averaging setup which deals with perturbations of integrable hamiltonian systems . in the fully coupled situation we can not work just with one hyperbolic flow but have to consider continuously changing fast motions which requires a special technique . in particular , the full flow @xmath22 on @xmath10 defined above and viewed as a small perturbation of the partially hyperbolic system @xmath31 plays an important role in our considerations . it is somewhat surprising that the `` very long time '' behavior of the slow motion which requires certain `` markov property type '' arguments still can be described in the fully coupled setup which involves continuously changing fast hyperbolic motions . it turns out that the perturbed system still possesses semi - invariant expanding cones and foliations and a certain volume lemma type result on expanding leaves plays an important role in our argument for transition from small time were perturbation techniques still works to the long and `` very long '' time estimates . it is plausible that moderate deviations type results can be proved for @xmath90 when @xmath91 and that the distribution of @xmath92 in @xmath18 converges to the distribution of a gaussian diffusion process in @xmath16 . still , this requires somewhat different methods and it will not be discussed here . in this regard we can mention limit theorems obtained in @xcite for a system of two heavy and light particles which leads to an averaging setup for a billiard flow . for the simpler case when @xmath21 does not depend on @xmath72 i.e. when all flows @xmath93 are the same , the moderate deviations and gaussian approximations results were obtained previously in @xcite . related results in this uncoupled situation concerning hasselmann s nonlinear ( strong ) diffusion approximation of the slow motion @xmath2 were obtained in @xcite . we consider also the discrete time case where ( [ 1.1.1 ] ) is replaced by difference equations for sequences @xmath94 and @xmath95 so that @xmath96 where @xmath97 is lipschitz in both variables and the maps @xmath98 are smooth and depend smoothly on the parameter @xmath48 . introducing the map @xmath99 we can also view this setup as a perturbation of the map @xmath100 describing an ideal system where parameters @xmath48 do not change . assuming that @xmath101 are @xmath13 depending on @xmath20 families of either @xmath13 expanding transformations or @xmath13 axiom a diffeomorphisms in a neighborhood of an attractor @xmath70 we will derive large deviations estimates for the difference @xmath102 where @xmath103 solves the equation @xmath104 where @xmath105 and @xmath69 is the corresponding srb invariant measure of @xmath106 on @xmath70 . the discrete time results are obtained , essentially , by simplifications of the corresponding arguments in the continuous time case which enable us to describe `` very long '' time behavior of the slow motion also in the discrete time case . since our methods work not only for fast motions being axiom a diffeomorphisms but also when they are expanding transformations we can construct simple examples satisfying conditions of our theorems and exhibiting corresponding effects . in particular , we produce in section [ sec1.9 ] computational examples which demonstrate transitions of the slow motion between neighborhoods of attractors of the averaged system . a series of related results for the case when ordinary differential equations in ( [ 1.1.1 ] ) are replaced by fully coupled stochastic differential equations appeared in @xcite , @xcite@xcite , @xcite , and @xcite . hasselmann s nonlinear ( strong ) diffusion approximation of the slow motion in the fully coupled stochastic differential equations setup was justified in @xcite . when the fast process does not depend on the slow one such results were obtained in @xcite , @xcite , and @xcite . especially relevant for our results here is @xcite and we employ some elements of the probabilistic strategy from this paper . still , the methods there are quite different from ours and they are based heavily , first , on the markov property of processes emerging there and , secondly , on uniformity and nondegeneracy of the fast diffusion term assumptions which can not be satisfied in our circumstances as our deterministic fast motions are very degenerate from this point of view . note that the proof in @xcite contains a vicious cycle and substantial gaps which recently were essentially fixed in @xcite . some of the dynamical systems technique here resembles @xcite but the dependence of the fast motion on the slow one complicates the analysis substantially and requires additional machinery . a series of results on cramer s type asymptotics for fully coupled averaging with axiom a diffeomorphisms as fast motions appeared recently in @xcite@xcite . observe that the methods there do not work for continuous time axiom a dynamical systems considered here , they can not lead , in principle , to the standard large deviations estimates of our work and they deal with deviations of @xmath2 from the averaged motion only at the last moment and not of its whole path . various limit theorems for the difference equations setup ( [ 1.1.10 ] ) with partially hyperbolic fast motions were obtained recently in @xcite and @xcite . the study of deviations from the averaged motion in the fully coupled case seems to be quite important for applications , especially , from phenomenological point of view . in addition to perturbations of hamiltonian systems mentioned above there are many non hamiltonian systems which are naturally to consider from the beginning as a combination of fast and slow motions . for instance , hasselmann @xcite based his model of weather climate interaction on the assumption that weather is a fast chaotic motion depending on climate as a slow motion which differs from the corresponding averaged motion mainly by a diffusion term . though , as shown in @xcite , @xcite and @xcite , this diffusion error term does not help in the study of large deviations which are responsible for rare transitions of the slow motion between attractors of the averaged one , the latter phenomenon can be described in our framework and it seems to be important in certain models of climate fluctuations ( see @xcite and @xcite ) . very slow nearly periodic motions appearing in the stochastic resonance framework considered in section [ sec1.10 ] may also fit into this subject in the discussion on `` ice ages '' . of course , it is hard to believe that real world chaotic systems can be described precisely by an anosov or axiom a flow but one may take comfort in the chaotic hypothesis @xcite : `` a chaotic mechanical system can be regarded for practical purposes as a topologically mixing anosov system '' . many real systems can be viewed as a combination of slow and fast motions which leads to complicated double scale equations . already in the 19th century in applications to celestial mechanics it was well understood ( though without rigorous justification ) that a good approximation of the slow motion can be obtained by averaging its parameters in fast variables . later , averaging methods were applied in signal processing and , rather recently , to model climate weather interactions ( see @xcite , @xcite , @xcite and @xcite ) . the classical setup of averaging justified rigorously in @xcite presumes that the fast motion does not depend on the slow one and most of the work on averaging treats this case only . on the other hand , in real systems both slow and fast motions depend on each other which leads to the more difficult fully coupled case which we study here . this setup emerges , in particular , in perturbations of hamiltonian systems which leads to fast motions on manifolds of constant energy and slow motions across them . it is natural to view double scale models as describing physical systems considered as perturbations of an idealized one which depends on parameters @xmath2466 assumed to be constants ( integrals ) of motion . in part [ part2 ] we suppose that the evolution of this idealized sistem is described by certain family of markov processes @xmath2467 on a separable metric space @xmath11 . in the perturbed system parameters start changing slowly in time and we assume that the corresponding slow motion @xmath2468 is described by an ordinary differential equations in @xmath16 having the form @xmath2469 where @xmath2470 is lipschitz continuous and the fast motion @xmath2471 evolves on @xmath11 , it depends , in general , on the slow one and tends to @xmath2472 as @xmath50 . usually , @xmath2473 is determined by certain equations , in general , coupled with ( [ 2.1.1 ] ) which means that their coefficients depend on the slow motion @xmath2474 . assume that a nonrandom limit @xmath2475 exists in some sense , it `` essentially '' does not depend on @xmath18 and it depends lipschitz continuously on @xmath20 . then there exists a unique solution @xmath2476 of the averaged equation @xmath2477 the averaging principle suggests that often @xmath2478 in some sense . if unperturbed motions @xmath2479 do not depend on the slow variables @xmath20 and @xmath2479 then the averaged principle holds true under quite general circumstances but when the fast motion depends on the slow one ( coupled case ) the situation becomes more complicated and approximation of @xmath2480 by @xmath2481 in the weak or the average sense was justified under some conditions in@xcite and @xcite . an extension of the averaging principle in the sense of convergence of young measures is discussed in section [ sec2.9 ] below . in this work we are interested in large deviations bounds for probabilities that the time changed slow motion @xmath2482 belongs to various sets of curves which leads , in particular , to exponential bounds of the form @xmath2483 where @xmath2393 satisfies @xmath2484 when the fast motion do not depend on the slow one such results were obtained in @xcite and @xcite but the coupled case was dealt with much later in @xcite though ( as we indicated this to the author ) the proof there contained a vicious circle and substantial gaps which , essentially , were fixed recently in @xcite . still , @xcite is rather difficult to follow and we find it useful to provide a precise and consistent exposition of this important result which also deals with a more general case including fast motions being random evolutions whose extreme partial cases are diffusions and finite markov chains with continuous time . moreover , we go beyond bounded time large deviations and describe the adiabatic behaviour of the slow motion @xmath88 on exponentially large in @xmath5 time intervals such as its exits from a domain of attraction and transitions between attractors of the averaged system ( [ 2.1.6 ] ) . we observe that essentially the same proof yields the same results for a bit more general case when both @xmath19 in [ 2.1.1 ] and the coefficients of the random evolutions in the next section depend also lipschitz continuously on @xmath33 . we consider also the discrete time case where ( [ 2.1.1 ] ) is replaced by a difference equation of the form @xmath2485 where @xmath368 is the same as in ( [ 2.1.1 ] ) and the fast motion @xmath2486 is a perturbation of a family @xmath2487 of markov chains parametrized by @xmath48 . for somewhat less general discrete time situation large diviations bounds were obtained in @xcite by a simpler approach but in our more general situation we can rely only on methods similar to the continuous time case . moreover , unlike @xcite we go farther and study also very long time `` adiabatic '' behaviour of the slow motion similar to the continuous time case and illustrate some of the results by computer simulations for simple models . the strategy and many of arguments in part [ part2 ] are rather similar to part [ part1 ] where deterministic chaotic fast motions such as anosov and axiom a systems were considered . still , in view of the heavy dynamical systems background and machinery part [ part1 ] is hardly accessible for most of probabilists . by this reason we give full proofs here refering to part [ part1 ] only for proofs of some general results on large deviations , rate functionals and some others which do not rely on the specific dynamical systems setup . let @xmath107 be a @xmath13 flow on a compact riemannian manifold @xmath11 given by a differential equation @xmath108 a compact @xmath109invariant set @xmath110 is called hyperbolic if there exists @xmath111 and the splitting @xmath112 into the continuous subbundles @xmath113 of the tangent bundle @xmath114 restricted to @xmath115 the splitting is invariant with respect to the differential @xmath116 of @xmath117 @xmath118 is the one dimensional subbundle generated by the vector field @xmath21 , and there is @xmath119 such that for all @xmath120 and @xmath121 @xmath122 a hyperbolic set @xmath123 is said to be basic hyperbolic if the periodic orbits of @xmath124 are dense in @xmath115 @xmath124 is topologically transitive , and there exists an open set @xmath125 with @xmath126 such a @xmath123 is called a basic hyperbolic attractor if for some open set @xmath59 and @xmath127 @xmath128 where @xmath129 denotes the closure of @xmath130 if @xmath131 then @xmath107 is called an anosov flow . [ ass1.2.1 ] the family @xmath132 in ( [ 1.1.2 ] ) consists of @xmath13 vector fields on a compact @xmath12-dimensional riemannian manifold @xmath11 with uniform @xmath13 dependence on the parameter @xmath20 belonging to a neighborhood of the closure @xmath133 of a relatively compact open connected set @xmath57 . each flow @xmath134 on @xmath11 given by @xmath135 possesses a basic hyperbolic attractor @xmath70 with a splitting @xmath136 satisfying ( [ 1.2.2 ] ) with the same @xmath111 and there exists an open set @xmath137 and @xmath119 such that @xmath138 let @xmath139 be the absolute value of the jacobian of the linear map @xmath140 with respect to the riemannian inner products and set @xmath141 the function @xmath142 is known to be h " older continuous in @xmath143 since the subbundles @xmath144 are h " older continuous ( see @xcite and @xcite ) , and @xmath142 is @xmath53 in @xmath20 ( see @xcite ) . let @xmath145 satisfy ( [ 1.2.4 ] ) and set @xmath146\}.$ ] a set @xmath147 is called @xmath148separated for the flow @xmath106 if @xmath149 , @xmath150 imply @xmath151 for some @xmath152 $ ] , where @xmath153 is the distance function on @xmath154 for each continuous function @xmath155 on @xmath145 set @xmath156 @xmath157 if @xmath158 and @xmath159 the latter is monotone in @xmath160 and so the limit @xmath161 exists and it is called the topological pressure of @xmath155 for the flow @xmath162 let @xmath163 denotes the space of @xmath164invariant probability measures on @xmath70 then ( see , for instance , @xcite ) the following variational principle @xmath165 holds true where @xmath166 is the kolmogorov sinai entropy of the time - one map @xmath167 with respect to @xmath168 if @xmath169 is a h " older continuous function on @xmath70 then there exists a unique @xmath164invariant measure @xmath170 on @xmath171 called the equilibrium state for @xmath172 such that @xmath173 we denote @xmath174 by @xmath69 since it is usually called the sinai ruelle bowen ( srb ) measure . since @xmath70 are attractors we have that @xmath175 ( see @xcite ) . for any probability measure @xmath176 on @xmath177 define @xmath178 then @xmath179 observe that by the ruelle inequality ( see , for instance , @xcite , theorem s.2.13 ) , @xmath180 , and so in view of assumption [ ass1.2.1 ] for any @xmath181 , @xmath182 it is known that @xmath183 is upper semicontinuous in @xmath176 since hyperbolic flows are entropy expansive ( see @xcite ) . thus @xmath184 is a lower semicontinuous functional in @xmath176 and it is also convex ( and affine on @xmath163 ) since entropy @xmath185 is affine in @xmath176 ( see , for instance , @xcite ) . hence , by the duality theorem ( see @xcite , p.201 ) , @xmath186 observe that this formula can be proved more directly . namely , if we define @xmath184 by it in place of ( [ 1.2.8 ] ) then ( [ 1.2.8 ] ) follows for @xmath181 from theorem 9.12 in @xcite and it is easy to show directly that @xmath184 defined in this way equals @xmath187 for any finite signed measure @xmath176 which is not @xmath106-invariant . since we assume that the vector field @xmath19 is @xmath53 in both arguments ( here only continuity in @xmath18 is needed ) then for any @xmath188 and @xmath189 we can define @xmath190 and @xmath191 . then @xmath192 where @xmath193 if @xmath181 satisfying the condition in brackets exists and @xmath194 , otherwise . since , @xmath195 is convex and continuous the duality theorem ( see @xcite , p.201 ) yields that @xmath196 provided there exists a probability measure @xmath181 such that @xmath197 and @xmath194 , otherwise . clearly , @xmath198 is convex and lower semicontinuous in all arguments and , in particular , it is measurable . we set also @xmath199 . denote by @xmath81 the space of continuous curves @xmath200 $ ] in @xmath67 which is the space of continuous maps of @xmath201 $ ] into @xmath202 for each absolutely continuous @xmath203 its velocity @xmath204 can be obtained as the almost everywhere limit of continuous functions @xmath205 when @xmath206 . hence @xmath204 is measurable in @xmath207 , and so we can set @xmath208 where @xmath209 provided for lebesgue almost all @xmath210 $ ] there exists @xmath211 for which @xmath212 and @xmath213 otherwise . it follows from @xcite and @xcite that @xmath214 where @xmath215 is the unique solution of the equation @xmath216 where @xmath217 and the equality @xmath218 holds true if and only if @xmath219 define the uniform metric on @xmath81 by @xmath220 for any @xmath221 set @xmath222 since @xmath223 is lower semicontinuous and convex in @xmath224 and , in addition , @xmath225 if @xmath226 we conclude that the conditions of theorem 3 in ch.9 of @xcite are satisfied as we can choose a fast growing minorant of @xmath223 required there to be zero in a sufficiently large ball and to be equal , say , @xmath227 outside of it . as a result , it follows that @xmath228 is lower semicontinuous functional on @xmath81 with respect to the metric @xmath229 , and so @xmath230 is a closed set which plays a crucial role in the large deviations arguments below . we suppose that the coefficients of ( [ 1.1.1 ] ) satisfy the following [ ass1.2.2 ] there exists @xmath231 such that @xmath232 where @xmath233 is the @xmath234 norm of the corresponding vector fields on @xmath235 set @xmath236 and @xmath237 for all @xmath238,\,{{\varepsilon}}>0\}.$ ] clearly , @xmath239 the following is one of the main results of this paper . [ thm1.2.3 ] suppose that @xmath240 and @xmath241 , @xmath242 are solutions of ( [ 1.1.1 ] ) with coefficients satisfying assumptions [ ass1.2.1 ] and [ ass1.2.2 ] . set @xmath51 then for any @xmath243 and every @xmath244 there exists @xmath245 such that for @xmath246 @xmath247 and @xmath248 where , recall , @xmath54 is the normalized riemannian volume on @xmath154 the functional @xmath249 for @xmath203 is finite if and only if @xmath250 for @xmath211 and lebesgue almost all @xmath210.$ ] furthermore , @xmath249 achieves its minimum 0 only on @xmath251 satisfying ( [ 1.2.14 ] ) for all @xmath252.$ ] finally , for any @xmath253 there exist @xmath254 and @xmath255 such that for all @xmath246 @xmath256 where @xmath257 is the unique solution of ( [ 1.2.14 ] ) . observe that ( [ 1.2.18 ] ) ( which was proved already in @xcite by a less precise large deviations argument ) follows from ( [ 1.2.17 ] ) and the lower semicontinuity of the functional @xmath228 and it says , in particular , that @xmath258 converges to @xmath259 in measure on the space @xmath260 with respect to the metric @xmath229 . it is naturally to ask whether we have here also the convergence for @xmath54-almost all @xmath261 . an example due to a.neishtadt discussed in @xcite shows that in the classical situation of perturbations of integrable hamiltonian systems , in general , the averaging principle holds true only in the sense of convergence in measure on the space of intitial conditions but not in the sense of the almost everywhere convergence . this example concerns the simple system @xmath262 with the one dimensional slow motion @xmath263 and the fast motion @xmath264 evolving on the circle while the corresponding averaged motion @xmath265 satisfies the equation @xmath266 . the resonance occurs here only when @xmath267 but it suffices already to create troubles in the averaging principle . namely , it turns out that for any initial condition @xmath268 with @xmath269 there exists a sequence @xmath270 such that @xmath271 though , of course , convergence in measure holds true here ( see @xcite ) . recently ( see @xcite and remark [ rem1.2.12 ] ) , such nonconvergence examples were constructed for the difference equations averaging setup ( [ 1.1.10 ] ) with expanding fast motions and there is no doubt that such examples exist also in the continuous time setup ( [ 1.1.1 ] ) when fast motions are axiom a flows as in this paper . observe also that ( [ 1.2.16 ] ) and ( [ 1.2.17 ] ) remain true ( with the same proof ) if we replace @xmath54 there by @xmath69 but as an example in @xcite shows we can not , in general , replace @xmath54 there by an arbitrary gibbs measure @xmath36 of @xmath27 . next , let @xmath272 be a connected open set and put @xmath273 where we take @xmath274 if @xmath275 for all @xmath276 the following result follows directly from theorem [ thm1.2.3 ] . [ cor1.2.4 ] under the conditions of theorem [ thm1.2.3 ] for any @xmath277 and @xmath278 @xmath279,\,{{\gamma}}_0=x,\ , { { \gamma}}_t\not\in v\right\}\nonumber . \end{aligned}\ ] ] precise large deviations bounds such as ( [ 1.2.16 ] ) and ( [ 1.2.17 ] ) of theorem [ thm1.2.3 ] ( which will be needed uniformly on certain unstable discs ) are crucial in our study in sections [ sec1.7 ] and [ sec1.8 ] of the `` very long '' , i.e. exponential in @xmath5 , time `` adiabatic '' behaviour of the slow motion which can not be described usually in the traditional theory of averaging where only perturbations of integrable hamiltonian systems are considered . namely , we will describe such long time behavior of @xmath88 in terms of the function @xmath280 under various assumptions on the averaged motion @xmath281 observe that @xmath282 satisfies the triangle inequality @xmath283 for any @xmath284 and it determines a semi metric on @xmath67 which measures `` the difficulty '' for the slow motion to move from point to point in terms of the functional @xmath285 . introduce the averaged flow @xmath286 on @xmath287 by @xmath288 where , recall , @xmath289 and @xmath290 for any probability measure @xmath176 on @xmath11 . call a @xmath286-invariant compact set @xmath291 an @xmath285-compact if for any @xmath292 there exist @xmath293 and an open set @xmath294 such that whenever @xmath295 and @xmath296 we can pick up @xmath297 $ ] and @xmath298 satisfying @xmath299 it is clear from this definition that @xmath300 for any pair points @xmath301 of an @xmath285-compact @xmath302 and by the above triangle inequality for @xmath282 we see that @xmath303 takes on the same value when @xmath304 is fixed and @xmath20 runs over @xmath302 . we say that the vector field @xmath19 on @xmath305 is complete at @xmath306 if the convex set of vectors @xmath307 , \,\nu\in{{\mathcal m}}_x\}$ ] contains an open neigborhood of the origin in @xmath16 . in lemma [ lem1.6.2 ] we will show that if @xmath291 is a compact @xmath286-invariant set such that @xmath19 is complete at each @xmath295 and either @xmath302 contains a dense orbit of the flow @xmath286 ( i.e. @xmath286 is topologically transitive on @xmath302 ) or @xmath300 for any @xmath308 then @xmath302 is an @xmath285-compact . moreover , to ensure that @xmath302 is an @xmath285-compact it suffices to assume that @xmath19 is complete only at some point of @xmath302 and the flow @xmath286 on @xmath302 is minimal , i.e. the @xmath286-orbits of all points are dense in @xmath302 or , equivalently , for any @xmath292 there exists @xmath309 such that the orbit @xmath310\}$ ] of length @xmath311 of each point @xmath295 forms an @xmath312-net in @xmath302 . the latter condition obviously holds true when @xmath302 is a fixed point or a periodic orbit of @xmath286 but also , more generally , when @xmath286 on @xmath302 is uniquely ergodic ( see @xcite , @xcite , @xcite ) . among well known examples of uniquely ergodic flows we can mention irrational translations of tori and horocycle flows on surfaces of negative curvature . a compact @xmath286-invariant set @xmath291 is called an attractor ( for the flow @xmath286 ) if there is an open set @xmath313 and @xmath314 such that @xmath315 for an attractor @xmath302 the set @xmath316 , which is clearly open , is called the basin ( domain of attraction ) of @xmath302 . an attractor which is also an @xmath285-compact will be called an @xmath285-attractor . in what follows we will speak about connected open sets @xmath317 with piecewise smooth boundaries @xmath318 . the latter can be introduced in various ways but it will be convenient here to adopt the definition from @xcite saying that @xmath318 is the closure of a finite union of disjoint , connected , codimension one , extendible @xmath53 ( open or closed ) submanifolds of @xmath16 which are called faces of the boundary . the extendibility condition means that the closure of each face is a part of a larger submanifold of the same dimension which coincides with the face itself if the latter is a compact submanifold . this enables us to extend fields of normal vectors to the boundary of faces and to speak about minimal angles between adjacent faces which we assume to be uniformly bounded away from zero or , in other words , angles between exterior normals to adjacent faces at a point of intersection of their closures are uniformly bounded away from @xmath319 and @xmath320 . the following result which will be proved in section [ sec1.7 ] describes exits of the slow motion from neighborhoods of attractors of the averaged motion . [ thm1.2.5 ] let @xmath291 be an @xmath285-attractor of the flow @xmath286 whose basin contains the closure @xmath321 of a connected open set @xmath317 with a piecewise smooth boundary @xmath318 such that @xmath322 and assume that for each @xmath323 there exists @xmath324 and an @xmath325invariant probability measure @xmath326 on @xmath327 such that @xmath328,\ ] ] i.e. @xmath329 and the former vector points out into the interior while the latter into the exterior of @xmath317 . set @xmath330 and @xmath331 . then @xmath332 takes on the same value @xmath333 and @xmath334 coincides with the same compact nonempty set @xmath335 for all @xmath336 while @xmath337 for all @xmath338 . furthermore , for any @xmath338 , @xmath339 and for each @xmath340 there exists @xmath341 such that for all small @xmath342 , @xmath343 next , set @xmath344 where @xmath345 dist@xmath346 and @xmath347 if @xmath348 and @xmath349 , otherwise . then for any @xmath338 and @xmath253 there exists @xmath350 such that for all small @xmath342 , @xmath351 finally , for every @xmath338 and @xmath253 , @xmath352 provided @xmath353 and the latter holds true if and only if for some @xmath277 there exists @xmath354 such that @xmath355 for lebesgue almost all @xmath252 $ ] with @xmath211 then @xmath353 . theorem [ thm1.2.5 ] asserts , in particular , that typically the slow motion @xmath88 performs rare ( adiabatic ) fluctuations in the vicinity of an @xmath285-attractor @xmath302 since it exists from any domain @xmath313 with @xmath356 for the time much smaller than @xmath357 ( as the corresponding number @xmath358 will be smaller ) and by ( [ 1.2.23 ] ) it can spend in @xmath359 only small proportion of time which implies that @xmath88 exits from @xmath59 and returns to @xmath360 ( exponentially in @xmath5 ) many times before it finally exits @xmath317 . we observe that in the much simpler uncoupled setup corresponding results in the case of @xmath302 being an attracting point were obtained for a continuous time markov chain and an axiom a flow as fast motions in @xcite and @xcite , respectively , but the proofs there rely on the lower semicontinuity of the function @xmath282 which does not hold true in general , and so extra conditions like @xmath285-compactness of @xmath302 or , more specifically , the completness of @xmath19 at @xmath302 should be assumed there , as well . it is important to observe that the intuition based on diffusion type small random perturbations of dynamical systems should be applied with caution to problems of large deviations in averaging since the @xmath285-functional of theorem [ thm1.2.3 ] describing them is more complex and have rather different properties than the corresponding functional emerging in diffusion type random perturbations of dynamical systems ( see @xcite ) . the reason for this is the deterministic nature of the slow motion @xmath88 which unlike a diffusion can move only with a bounded speed , and moreover , even in order to ensure its `` diffusive like '' local behaviour ( i.e. to let it go in many directions ) some extra nondegeneracy type conditions on the vector field @xmath19 are required . our next result describes rare ( adiabatic ) transitions of the slow motion @xmath88 between basins of attractors of the averaged flow @xmath286 which we consider now in the whole @xmath16 and impose certain conditions on the structure of its @xmath361-limit set . [ ass1.2.6 ] assumptions [ ass1.2.1 ] and [ ass1.2.2 ] hold true for @xmath362 , the family @xmath363 is a compact set of diffeomorphisms in the @xmath13 topology , @xmath364 for some @xmath231 independent of @xmath365 and there exists @xmath366 such that @xmath367 the condition ( [ 1.2.26 ] ) means that outside of some ball all vectors @xmath368 have a bounded away from zero projection on the radial direction which points out to the origin . this condition can be weakened , for instance , it suffices that @xmath369 but , anyway , we have to make some assumption which ensure that the slow motion stays ( at least , for `` most '' initial points @xmath261 ) in a compact region where really interesting dynamics takes place . next , suppose that the @xmath361-limit set of the averaged flow @xmath286 is compact and it consists of two parts , so that the first part is a finite number of @xmath285-attractors @xmath370 whose basins @xmath371 have piecewise smooth boundaries @xmath372 and the remaining part of the @xmath361-limit set is contained in @xmath373 . we assume also that for any @xmath374 there exist @xmath324 and an @xmath375-invariant measures @xmath376 such that @xmath377\,\ , \mbox{and}\,\ , i=1, ... ,k,\ ] ] i.e. @xmath378 and it points out into the interior of @xmath379 which means that from any boundary point it is possible to go to any adjacent basin along a curve with an arbitrarily small @xmath285-functional . let @xmath253 be so small that the @xmath380-neighborhood @xmath381 of each @xmath382 is contained with its closure in the corresponding basin @xmath383 . for any @xmath384 set @xmath385 in section [ sec1.8 ] we will derive the following result . [ thm1.2.7 ] the function @xmath386 takes on the same value @xmath387 for all @xmath388 . let @xmath389 . then for any @xmath384 , @xmath390 and for any @xmath340 there exists @xmath341 such that for all small @xmath342 , @xmath391 next , set @xmath392 then for any @xmath384 and @xmath253 there exists @xmath393 such that for all small @xmath342 , @xmath394 now , suppose that the vector field @xmath19 is complete on @xmath395 for some @xmath396 ( which strengthens ( [ 1.2.27 ] ) there ) and the restriction of the @xmath361-limit set of @xmath286 to @xmath395 consists of a finite number of @xmath285-compacts . assume also that there is a unique index @xmath397 such that @xmath398 . then for some @xmath399 and all small @xmath342 , @xmath400 finally , suppose that the above conditions hold true for all @xmath401 . define @xmath402 , @xmath403 and recursively , @xmath404 where @xmath405 , @xmath406 if @xmath407 , and set @xmath408 . then for any @xmath384 and @xmath340 there exists @xmath341 such that for all @xmath409 and sufficiently small @xmath342 , @xmath410 and for some @xmath399 , @xmath411 generically there exists only one index @xmath412 such that @xmath413 and in this case theorem [ thm1.2.7 ] asserts that @xmath414 arrives ( for `` most '' @xmath261 ) at @xmath415 after it leaves @xmath383 . if @xmath416 contains more than one index then the method of the proof of theorem [ thm1.2.7 ] enables us to conclude that in this case @xmath417 arrives ( for `` most '' @xmath261 ) at @xmath418 after leaving @xmath383 but now we can not specify the unique basin of attraction of one of @xmath419 s where @xmath420 exits from @xmath383 . if the succession function @xmath421 is uniquely defined then it determines an order of transitions of the slow motion @xmath88 between basins of attractors of @xmath89 and because of their finite number @xmath88 passes them in certain cyclic order going around such cycle exponentially many in @xmath5 times while spending the total time in a basin @xmath383 which is approximately proportional to @xmath422 . if there exist several cycles of indices @xmath423 where @xmath424 and @xmath425 then transitions between different cycles may also be possible . in the uncoupled case with fast motions being continuous time markov chains a description of such transitions via certain hierarchy of cycles appeared in @xcite and @xcite without detailed proofs but relying on some heuristic arguments . in our fully coupled deterministic setup a rigorous justification of the corresponding description seems to be difficult in a more or less general situation though for some specific simple examples ( as , for instance , those which are considered in section [ sec1.9 ] ) this looks feasible while it is not clear whether it is possible to describe in our situation a limiting as @xmath426 behaviour of the slow motion @xmath427 when @xmath33 is small but fixed . the proof of theorems [ thm1.2.5 ] , [ thm1.2.7 ] and to certain extent also of theorem [ thm1.2.3 ] rely , in particular , on certain `` markov property type '' arguments which enable us to extend estimates on relatively short time intervals to very long time intervals by , essentially , iterating them where the crucial role is played by a volume lemma type result of section [ sec1.3 ] together with the technique of @xmath428-separated sets and bowen s @xmath428-balls on unstable leaves of the perturbed flow @xmath22 . moreover , the proof of ( [ 1.2.32 ] ) and ( [ 1.2.33 ] ) require certain rough strong markov property type arguments which enable us to study the slow motion at subsequent hitting times @xmath429 of small neighborhoods of attractors of the averaged motion . in order to produce a wide class of systems satisfying the conditions of theorem [ thm1.2.7 ] we can choose , for instance , a vector field @xmath430 on @xmath16 whose @xmath361-limit set satisfies the conditions stated above for the averaged system together with a family of vector fields @xmath431 on @xmath16 ( parametrized by @xmath432 ) such that @xmath433 and then set @xmath434 . as a specific example we can take the flows @xmath435 to be geodesic flows on the manifold @xmath11 with ( changing ) constant negative curvature @xmath20 , @xmath436 to be a one dimensional vector field on @xmath437 and @xmath431 can be just a function @xmath438 on @xmath11 with zero integral with respect to the lebesgue measure there . in section [ sec1.9 ] we will derive similar results for the discrete time case where differential equations ( [ 1.1.1 ] ) are replaced by difference equations ( [ 1.1.10 ] ) . namely , recall that a compact subset @xmath123 of a compact riemannian manifold @xmath11 is called hyperbolic if it is @xmath439-invariant and there exists @xmath111 and the splitting @xmath440 into the continuous subbundles @xmath441 of the tangent bundle @xmath114 restricted to @xmath115 the splitting is invariant with respect to the differential @xmath442 of @xmath443 and there is @xmath444 such that for all @xmath120 and @xmath445 the inequalities ( [ 1.2.2 ] ) with @xmath207 replaced by @xmath446 hold true . a hyperbolic set @xmath123 is said to be basic hyperbolic if the periodic orbits of @xmath447 are dense in @xmath115 @xmath447 is topologically transitive , and there exists an open set @xmath125 with @xmath448 such a @xmath123 is called a basic hyperbolic attractor if for some open set @xmath59 and @xmath449 @xmath450 where @xmath129 denotes the closure of @xmath130 if @xmath131 then @xmath107 is called an anosov flow . if @xmath439 is a @xmath13 endomorphism of @xmath11 and there exists @xmath111 such that @xmath451 for all @xmath452 then @xmath439 is called an expanding map ( or transformation ) of @xmath11 . it will be convenient for our exposition to use the notation of the expanding subbundle @xmath453 also in the case of expanding maps where , of course , @xmath454 . we replace now assumption [ ass1.2.1 ] by the following one . [ ass1.2.8 ] the family @xmath455 in ( [ 1.1.10 ] ) consists of @xmath13-diffeomorphisms or endomorphisms of a compact @xmath12-dimensional riemannian manifold @xmath11 with uniform @xmath13 dependence on the parameter @xmath20 belonging to a neighborhood of the closure @xmath133 of a relatively compact open connected set @xmath57 . all @xmath456 are either expanding maps of @xmath11 or diffeomorphisms possessesing basic hyperbolic attractors @xmath70 with hyperbolic splittings satisfying ( [ 1.2.2 ] ) with the same @xmath111 and there exists an open set @xmath137 and @xmath444 satisfying ( [ 1.2.4 ] ) with @xmath446 in place of @xmath207 . let @xmath457 be the absolute value of the jacobian of the linear map @xmath458 with respect to the riemannian inner products and set @xmath459 the function @xmath142 is known to be h " older continuous in @xmath143 since the subbundles @xmath144 are h " older continuous ( see @xcite ) , and @xmath142 is @xmath53 in @xmath20 ( see @xcite ) . the topological pressure @xmath460 of a function @xmath155 for @xmath439 is defined similarly to the continuous time ( flow ) case above but now time should run only over integers and the integral @xmath461 should be replaced by the sum @xmath462 ( see @xcite ) . again the variational principle ( [ 1.2.6 ] ) holds true and if @xmath169 is a h " older continuous function on @xmath70 there exists a unique @xmath463invariant measure @xmath170 on @xmath171 called the equilibrium state for @xmath464 which satisfies ( [ 1.2.7 ] ) . in particular , @xmath465 is usually called the sinai ruelle bowen ( srb ) measure . since @xmath70 are attractors we have that @xmath175 ( see @xcite ) and the same holds true in the expanding case , as well . next , we define @xmath184 , @xmath195 , @xmath466 , @xmath198 , @xmath223 , @xmath249 , and @xmath251 as in ( [ 1.2.8 ] ) and ( [ 1.2.10])([1.2.14 ] ) . in place of assumption [ ass1.2.2 ] we will rely now on the similar one concerning the equation ( [ 1.1.10 ] ) . [ ass1.2.9 ] there exists @xmath231 such that @xmath467 where the first @xmath468 is the @xmath53 norm of the corresponding vector fields on @xmath305 and the second expression is the @xmath13 norm ( with respect to the corresponding riemannian metrics ) of the map @xmath469 acting by @xmath470 . [ thm1.2.10 ] assume that assumptions [ ass1.2.8 ] and [ ass1.2.9 ] are satisfied and that @xmath471 is obtained by ( [ 1.1.10 ] ) . for @xmath472 $ ] define @xmath473 and set @xmath51 . then theorem [ thm1.2.3 ] and corollary [ cor1.2.4 ] hold true with the corresponding functionals @xmath474 . theorems [ thm1.2.5 ] and [ thm1.2.7 ] hold true , as well , under the corresponding assumptions about the family @xmath475 ( with @xmath362 in the case of theorem [ thm1.2.7 ] ) and about the averaged system ( [ 1.1.11 ] ) ( in particular , about its attractors ) in place of the system ( [ 1.1.6 ] ) . in section [ sec1.9 ] we exhibit computations which demonstrate the phenomenon of theorem [ thm1.2.7 ] in the discrete time case for two simple examples where @xmath476 are one dimensional maps @xmath477 ( mod 1 ) and the averaged equation has three attracting fixed points . in the last section [ sec1.10 ] we discuss a stochastic resonance type phenomenon which can be exhibited in three scale systems where fast motions are hyperbolic flows ( hyperbolic diffeomorphisms , expanding transformations ) as above depending on the intermediate and slow motions while the intermediate motion performs rare transitions between attracting fixed points of corresponding averaged systems which under certain conditions creates a nearly periodic motion of the slow one dimensional motion . [ rem1.2.11 ] computation or even estimates of functionals @xmath249 seem to be quite difficult already for simple discrete ( and , of course , more for continuous ) time examples since this leads to complicated nonclassical variational problems . this is crucial in order to estimate numbers @xmath387 which according to theorem [ thm1.2.7 ] are responsible for transitions of the slow motion between basins of attractors of the averaged system . [ rem1.2.12 ] the estimate ( [ 1.2.18 ] ) shows that @xmath420 tends as @xmath50 to @xmath259 uniformly on @xmath201 $ ] in the sense of convergence in measure @xmath54 considered on the space of initial conditions @xmath432 . a natural question to ask is whether the convergence for almost all ( fixed ) initial conditions also takes place in our circumstances . in @xcite we give a negative answer to this question , in paricular , for the following simple discrete time example @xmath478 identifying 0 and 1 we view @xmath18 variable as belonging to the circle in order to fit into our setup where the fast motion runs on a compact manifold . the averaged equation ( [ 1.1.11 ] ) has here zero in the right hand side so the averaged motion stays forever at the initial point . the discrete time version of ( [ 1.2.18 ] ) asserted by theorem [ thm1.2.10 ] implies that @xmath479 in the sense of convergence in ( the lebesgue ) measure on the circle but we show in @xcite that for each @xmath20 there is a set @xmath480 of full lebesgue measure on the circle such that if @xmath481 then @xmath482 as @xmath50 of the left hand side in ( [ 1.2.36 ] ) is positive , i.e. there is no convergence for lebesgue almost all @xmath18 there . namely , it turns out that for almost all initial conditions there exists a sequence @xmath483 such that the fast motion @xmath484 stays for a time of order @xmath485 close to an orbit @xmath486 ( mod @xmath487 of the doubling map with @xmath488 being a generic point with respect to a gibbs invariant measure @xmath45 of this map satisfying @xmath489 which prevents ( [ 1.2.36 ] ) . for readers convenience we exhibit , first , in this section the setup and necessary technical results from @xcite and though their proofs can can be found in @xcite we provide for completness and readers convenience their slightly modified and corrected version also here . any vector @xmath490 can be uniquely written as @xmath491 where @xmath492 and @xmath493 and it has the riemannian norm @xmath494 where @xmath495 is the usual euclidean norm on @xmath16 and @xmath496 is the riemannian norm on @xmath114 . the corresponding metrics on @xmath11 and on @xmath10 will be denoted by @xmath497 and @xmath498 , respectively , so that if @xmath499 then @xmath500 it is known ( see @xcite ) that the hyperbolic splitting @xmath501 over @xmath70 can be continuously extended to the splitting @xmath502 over @xmath145 which is forward invariant with respect to @xmath503 and satisfies exponential estimates with a uniform in @xmath306 positive exponent which we denote again by @xmath504 i.e. we assume now that @xmath505 provided @xmath506 , @xmath507 , @xmath508 , and @xmath509 moreover , by @xcite ( see also @xcite ) we can choose these extensions so that @xmath510 and @xmath511 will be h " older continuous in @xmath512 and @xmath53 in @xmath20 in the corresponding grassmann bundle . actually , since @xmath145 is contained in the basin of each attractor @xmath70 , any point @xmath513 belongs to the stable manifold @xmath514 of some point @xmath515 ( see @xcite ) , and so we choose naturally @xmath510 to be the tangent space to @xmath516 at @xmath517 now each vector @xmath518 can be represented uniquely in the form @xmath519 with @xmath520 , @xmath521 , @xmath522 and @xmath523 . we denote also @xmath524 and @xmath525 . for each small @xmath526 set @xmath527 and @xmath528 which are unstable cones around @xmath453 and @xmath529 respectively . similarly , we define @xmath530 and @xmath531 which are stable cones around @xmath532 and @xmath533 respectively . put @xmath534\}$ ] , where , recall , @xmath22 is the flow determined by the equations ( [ 1.1.1 ] ) . [ lem1.3.1 ] . there exist @xmath535 such that if @xmath536 and @xmath537 then @xmath538 and for any @xmath539 @xmath540 @xmath541 let @xmath542 @xmath543 @xmath544 and @xmath545 then @xmath546 and @xmath547 @xmath548 @xmath549 @xmath550 for some @xmath551 independent of @xmath552 and @xmath553 if @xmath554 hence , for @xmath121 @xmath555 and @xmath556 if @xmath557 then @xmath558 and @xmath559 hence , by the above , @xmath560 set @xmath561 choose @xmath562 and @xmath563 . then we obtain that @xmath564 for all @xmath565 $ ] , and so by continuity of the splitting @xmath566 and by perturbation arguments it follows that @xmath567 for all @xmath568 $ ] provided @xmath33 is small enough . repeating this argument for @xmath569 $ ] , @xmath570 we conclude the proof of the first part of ( [ 1.3.2 ] ) and its second part follows in the same way . next , for @xmath557 and @xmath121 @xmath571 choose @xmath572 so small ( for instance , @xmath573 ) that for all @xmath574 and @xmath575 , @xmath576 for all @xmath577.$ ] then , @xmath578 for all such @xmath207 , and so if @xmath33 small enough we have also @xmath579 . using ( [ 1.3.2 ] ) and repeating this argument for @xmath580 in place of @xmath552 we derive ( ii ) for all @xmath581 . the proof for stable cones @xmath582 is similar . for any linear subspace @xmath583 of @xmath584 denote by @xmath585 absolute value of the jacobian of the linear map @xmath586 with respect to inner products induced by the riemannian metric . for each @xmath587 set also @xmath588 let @xmath589 be the dimension of @xmath590 which does not depend on @xmath20 and @xmath512 by continuity considerations . if @xmath583 is an @xmath591dimensional subspace of @xmath592 , @xmath593 and @xmath594 then it follows easily from assumption [ ass1.2.2 ] and lemma [ lem1.3.1 ] that there exists a constant @xmath595 independent of @xmath596 and of a small @xmath33 such that for any @xmath597 @xmath598 recall , that an embedded @xmath599 @xmath600dimensional disc @xmath601 in @xmath10 , @xmath602 is the image of an @xmath603-dimensional disc ( ball ) @xmath604 in @xmath605 centered at 0 under a @xmath606 diffeomorphism of a neighborhood of 0 in @xmath605 into @xmath10 and we define the boundary @xmath607 of @xmath601 as the image of the boundary @xmath608 of @xmath604 considered in the corresponding @xmath603-dimensional euclidean subspace of @xmath605 . denote by @xmath609 the ball in @xmath610 centered at @xmath611 and let @xmath612 , @xmath551 be the set of all @xmath53 embedded @xmath613dimensional closed discs @xmath614 such that @xmath615 , @xmath616 and if @xmath617 then @xmath618 where @xmath619 is the tangent bundle over @xmath601 and @xmath620 is the interior metric on @xmath621 each disc @xmath622 will be called unstable or expanding and , clearly , @xmath623 and if @xmath624 and @xmath625 are small enough and @xmath626 then dist@xmath627 for any @xmath628 let @xmath629 and @xmath630 . set @xmath631\}$ ] and let @xmath632 and @xmath633 be natural projections on the first and second factors , respectively . [ lem1.3.2 ] let @xmath634 , @xmath635 be as in lemma [ lem1.3.1 ] and @xmath277 . there exist @xmath636 such that if @xmath637 , @xmath615 , @xmath638 , @xmath639 , @xmath640 and @xmath641 then \(i ) @xmath642 for any @xmath643 and @xmath644,$ ] where @xmath645 is the interior distance on @xmath59 ; \(ii ) @xmath646 and @xmath647 provided @xmath648 ; \(iii ) for all @xmath643 and @xmath649 , @xmath650 ( iv ) @xmath651 provided @xmath652 , where @xmath653 is the induced ( not normalized ) riemannian volume on @xmath621 \(i ) let @xmath654 be a smooth curve on @xmath655 connecting @xmath656 and @xmath657 then @xmath658 is a smooth curve on @xmath659 connecting @xmath660 and @xmath661 since @xmath662 then by ( [ 1.3.3 ] ) , length@xmath663length@xmath664 if @xmath665 then for such @xmath207 and @xmath666 @xmath667 observe that ( i ) is nontrivial only for large @xmath668 , so minimizing in @xmath654 in the above inequality we derive the assertion ( i ) . next , we derive ( ii ) . its first part follows from ( [ 1.3.2 ] ) . by the definition of @xmath317 , @xmath669 for any @xmath670 . it remains to show that @xmath671 for any @xmath672 indeed , suppose @xmath673 set @xmath674 next , we conclude via perturbation arguments that @xmath675 provided @xmath33 is small enough . let @xmath676 it follows from lemma [ lem1.3.1 ] that @xmath677 for all @xmath152.$ ] hence , if @xmath678 then @xmath679 and so @xmath680 in order to derive ( iii ) take an arbitrary smooth curve @xmath654 on @xmath659 connecting @xmath681 and @xmath660 . then @xmath682 it follows that if @xmath683 with @xmath684 and @xmath685 then @xmath686 , @xmath687 @xmath688 and so @xmath689 hence @xmath690 and so @xmath691 minimizing the right hand side here over such @xmath654 we obtain ( iii ) using ( i ) . finally , ( iv ) follows from ( [ 1.3.5 ] ) , ( i ) , ( ii ) , and the h " older continuity of @xmath692 ( as a function on @xmath693 for each @xmath694 and @xmath695 small enough set @xmath696 and @xmath697 which are local stable and unstable manifolds for @xmath106 at @xmath698 according to @xcite and @xcite these families can be included into continuous families of @xmath699 and @xmath591dimensional stable and unstable @xmath53 discs @xmath700 and @xmath701 respectively , defined for all @xmath261 and such that @xmath700 is tangent to @xmath702 , @xmath703 is tangent to @xmath144 , @xmath704 , and @xmath705 actually , as we noted it above if @xmath261 then @xmath18 belongs to a stable manifold @xmath706 of some @xmath707 and we choose @xmath708 to be the subset of @xmath709 [ lem1.3.3 ] for any @xmath710 small enough and a continuous function @xmath711 on @xmath712 uniformly in @xmath622 , @xmath713 and @xmath714 $ ] , @xmath715 where @xmath653 is the induced riemannian volume on @xmath621 set @xmath716 @xmath717 and @xmath718 by standard transversality considerations we can define a one - to - one map @xmath719 by @xmath720 provided @xmath721 are small and @xmath722 is sufficiently large . by the absolute continuity of the stable foliation arguments ( see , for instance , @xcite , section 3.3 ) we conclude that @xmath723 and its inverse have bounded jacobians . it follows that it suffices to establish ( [ 1.3.6 ] ) for @xmath724 where @xmath725 uniformly in @xmath726,\,{{\gamma}}_0>0.$ ] set @xmath727 @xmath728 and @xmath729 then @xmath730 and so @xmath731 where @xmath496 is the supremum norm on @xmath732 since @xmath733 by @xcite then given @xmath292 there exist @xmath734 @xmath735 and @xmath736 such that for any @xmath737 we can define a one - to - one map @xmath738 by @xmath739 by standard absolute continuity of the stable foliation considerations ( see @xcite , section 3.3 ) it follows that @xmath740 and its inverse have bounded jacobians which together with the above arguments yield that it suffices to prove lemma [ lem1.3.3 ] only when @xmath741 and so ( see @xcite ) , @xmath742 we observe also that without loss of generality we can assume @xmath654 to be sufficiently small since we can always cover @xmath743 by @xmath744 with @xmath745 @xmath746 and small @xmath747 so proving lemma [ lem1.3.3 ] for all such @xmath748 will imply it for @xmath743 itself . so now assume that @xmath749 and we claim that for any @xmath292 there exists @xmath750 such that @xmath751 forms an @xmath752net in @xmath70 for any @xmath753 indeed , by topological transitivity there exists @xmath515 whose orbit is dense in @xmath171 and so by standard ergodicity considerations with respect to any ergodic invariant measure with full support on @xmath70 ( take , for instance , the srb measure ) we conclude that for any @xmath754 already @xmath755 is dense in @xmath70 . then by transversality of @xmath756 and @xmath757 there exists @xmath758 such that @xmath759 for some @xmath760 and so the forward orbit of @xmath512 is dense in @xmath70 , whence our claim holds true . by compactness and structural stability considerations it follows that we can choose the same @xmath761 for all @xmath261 and @xmath762 for any set @xmath763 put @xmath764\}.$ ] recall , that a finite set @xmath765 is called @xmath766separated for the flow @xmath27 if @xmath767 implies @xmath768 a set @xmath765 is called @xmath766spanning if for any @xmath694 there is @xmath769 such that @xmath770 . let @xmath751 be an @xmath752net in @xmath70 and @xmath771 be a maximal @xmath766separated subset of @xmath751 . then @xmath772 are disjoint sets . by transversality of @xmath751 and @xmath756 there exists @xmath595 such that for any @xmath694 we can find @xmath773 such that @xmath774 and so for some @xmath775 @xmath776 with some @xmath777 large enough but independent of @xmath778 hence , @xmath771 is @xmath779spanning , and so @xmath780 . assume , first , that @xmath781 in ( [ 1.3.6 ] ) is h " older continuous in @xmath782 then by standard volume lemma arguments ( see @xcite ) we obtain for @xmath783 and @xmath784 that @xmath785 where @xmath786 does not depend on @xmath787 now ( [ 1.3.6 ] ) follows from the above integral estimates and the uniform in @xmath788separated and @xmath788spanning sets approximation of the topological pressure ( see , for instance , @xcite and @xcite ) . for a general continuous @xmath711 approximate it uniformly by h " older continuous functions and ( [ 1.3.6 ] ) will follow in this case again . the limit ( [ 1.3.6 ] ) is uniform in @xmath789 and in @xmath611 since @xmath790 uniformly continuous in @xmath789 ( easy to see ) and it is uniformly continuous in @xmath611 ( see @xcite ) and , furthermore , it follows from lemma 5.1 from @xcite that the family @xmath791 is equicontinuous in @xmath611 . [ prop1.3.4 ] for any @xmath792 with @xmath793 large and @xmath794 small enough there exists a positive function @xmath795 such that @xmath796 and for any @xmath797 @xmath798 , @xmath799 , @xmath800 @xmath615 and @xmath801 satisfying @xmath802 we have @xmath803 where @xmath804 is the inner product , @xmath805 @xmath806 , and @xmath653 is the induced riemannian volume on @xmath621 by ( [ 1.1.1 ] ) and ( [ 1.2.15 ] ) for any @xmath807 @xmath808 where , @xmath809 and , recall , @xmath810 then , by gronwall s inequality @xmath811 hence , @xmath812 recall , that by lemma [ lem1.3.2](iii ) for any @xmath813 , @xmath814 set @xmath815 then @xmath816 where @xmath817 , @xmath818 is the tangent space to @xmath819 at @xmath512 and @xmath820 is the induced riemannian volume on @xmath821 by ( [ 1.3.4 ] ) , ( [ 1.3.5 ] ) , lemma [ lem1.3.2](i ) , and the h " older continuity of the function @xmath822 @xmath823 for some @xmath777 independent of @xmath824 and @xmath825 since @xmath826 by lemma [ lem1.3.2](ii ) , it follows from ( [ 1.2.15 ] ) and the above estimates that @xmath827 where @xmath828 @xmath829 is a constant independent of @xmath830 and @xmath831 with @xmath832 . next , choose @xmath806 and @xmath833 $ ] so that @xmath834 is an integer . if @xmath835 then ( [ 1.3.8 ] ) follows from ( [ 1.3.9])([1.3.11 ] ) and lemma [ lem1.3.3 ] . now , let @xmath836 and @xmath837 . then by ( [ 1.2.15 ] ) and lemma [ lem1.3.2](i ) for any @xmath838 @xmath839 and @xmath840 where @xmath841 . integrating the inequalities above we obtain @xmath842 from the estimates ( [ 1.3.9])([1.3.11 ] ) together with lemma [ lem1.3.3 ] we conclude that for some @xmath843 independent of @xmath844 and @xmath789 , @xmath845 where @xmath846 and @xmath847 , @xmath848 as @xmath849 observe that by ( [ 1.1.1 ] ) , ( [ 1.2.15 ] ) and lemma [ lem1.3.2](iii ) , @xmath850 and so setting @xmath851 we obtain by ( [ 1.2.15 ] ) and @xcite ( see also @xcite and @xcite ) that @xmath852 for some @xmath853 independent of @xmath854 and @xmath831 with @xmath832 provided , say , @xmath855 which we can assume without loss of generality . a finite set @xmath856 will be called @xmath857separated if @xmath858 @xmath859 implies that @xmath860 let @xmath861 be a maximal @xmath862separated set in @xmath601 and define @xmath863 then for @xmath864 , @xmath865 where @xmath866 and the left hand side of ( [ 1.3.16 ] ) follows from lemma [ lem1.3.2](i ) assuming that @xmath33 is small enough . observe also that @xmath867 are disjoint for different @xmath868 for @xmath869 set @xmath870 and @xmath871 with @xmath872 then by ( [ 1.3.12])([1.3.16 ] ) and lemma [ lem1.3.2](iii ) , @xmath873 similarly , we obtain @xmath874 emloying these estimates recursively for @xmath875 and estimating @xmath876 by ( [ 1.3.14 ] ) with @xmath877 and with @xmath878 in place of @xmath879 we derive ( [ 1.3.8 ] ) with @xmath880 provided @xmath881 next , under assumption [ ass1.2.6 ] we derive a volume lemma type assertion ( see @xcite ) which will be needed in sections [ sec1.7 ] and [ sec1.8 ] and which will hold true on any time intervals and not just on time intervals of order @xmath5 as in lemma [ lem1.3.2](iv ) . in order to do so we will consider a subset of embedded @xmath13 discs from @xmath882 taking special care of their @xmath13 bounds . namely , let exp@xmath883 be the exponential map which is a diffeomorphism of @xmath884 onto the open @xmath885neighborhood @xmath886 of @xmath18 provided @xmath253 is small enough . given @xmath887 and @xmath888 set @xmath889 which is a diffeomorphism of @xmath890 onto @xmath891 . let @xmath892 , @xmath893 , @xmath261 be an embedded @xmath13 disc . assuming that @xmath894 we can define @xmath895 which is a @xmath13 hypersurface in @xmath896 . if @xmath380 is sufficiently small then the tangent subbubndle @xmath897 over @xmath898 still stays close to @xmath899 , and so we can represent @xmath898 as a parametric set @xmath900 where @xmath901 , @xmath902 and @xmath903 . we will write that @xmath904 if the parametric representation of the corresponding @xmath898 as above satisfies @xmath905 [ lem1.3.5 ] there exists @xmath906 such that for any @xmath581 we can choose @xmath253 small enough and @xmath907 large enough so that if @xmath908 and @xmath894 then @xmath909 since the differential @xmath910exp@xmath911 of the exponential map at zero is the identity map it follows from the definition of @xmath612 that @xmath912 where @xmath913 ( uniformly in all @xmath601 as above ) as @xmath914 . let @xmath893 and set @xmath915 which for each fixed @xmath916 and a sufficiently small @xmath253 ( depending on @xmath207 ) defines a diffeomorphism of @xmath890 onto its image . by ( [ 1.3.2 ] ) the tangent subbundle over @xmath917 is contained in @xmath918 , and so for small @xmath253 the tangent subbundle @xmath919 over @xmath920 stays close to @xmath921 where @xmath922 and @xmath923 . hence , we can represent @xmath924 in a parametric form @xmath925 where @xmath926 and @xmath927 . fix some @xmath928 so that ( [ 1.3.3 ] ) holds true . write @xmath929 , so that , in particular , @xmath930 then @xmath931 where @xmath901 , @xmath932 , @xmath48 , @xmath80 is an @xmath933matrix , @xmath19 is an @xmath934matrix with @xmath935 and @xmath936 for all @xmath937 and @xmath938 where @xmath913 as @xmath914 . by ( [ 1.2.15 ] ) , ( [ 1.3.1 ] ) and assumption [ ass1.2.6 ] it follows that there exists a constant @xmath939 such that for any @xmath940 , @xmath48 , @xmath432 , @xmath941 by ( [ 1.3.1 ] ) we can choose @xmath916 large enough and then @xmath33 and @xmath380 small enough so that for all @xmath942 , @xmath943 now @xmath207 is fixed and we can choose @xmath33 and @xmath380 so small that ( [ 1.3.19 ] ) implies that , @xmath944 in order to shorten notations for every vector function @xmath945 , @xmath946 we denote by @xmath947 the jacobi matrix @xmath948 and by @xmath949 we denote the collection @xmath950 . we set also @xmath951 observe that by assumptions [ ass1.2.1 ] , [ ass1.2.2 ] and [ ass1.2.6 ] for any @xmath916 there exists @xmath952 such that @xmath953 and @xmath954 it follows by ( [ 1.3.17])([1.3.23 ] ) ( with natural product notations ) that @xmath955 and @xmath956 similarly , @xmath957 choosing @xmath958 we obtain that if @xmath959 then @xmath960 and the assertion of lemma [ lem1.3.5 ] follows . the main purpose of the previous result is to derive the following volume lemma type assertion which plays an essential role in section [ sec1.6 ] . [ lem1.3.6 ] for any @xmath961 there exists @xmath962 such that if @xmath963 and @xmath964 is large enough then for any @xmath916 and @xmath965 satisfying @xmath966 , @xmath967 set @xmath968 and @xmath969 . similarly to lemma [ lem1.3.2](ii ) , @xmath970 , and so by uniformity considerations there exists @xmath971 independent of @xmath972 and @xmath601 as above such that @xmath973 choose @xmath974 so that @xmath975 and set @xmath976 . then for any @xmath977 , @xmath978 and by lemma [ lem1.3.2](i ) , @xmath979 by ( [ 1.3.5 ] ) , ( [ 1.3.22 ] ) , ( [ 1.3.26 ] ) , and ( [ 1.3.27 ] ) together with lemma [ lem1.3.5 ] we conclude that there exists a constant @xmath722 such that @xmath980 now ( [ 1.3.24 ] ) follows from ( [ 1.3.25 ] ) , ( [ 1.3.26 ] ) , and ( [ 1.3.28 ] ) with @xmath981 we will need the following version of general large deviations bounds when usual assumptions hold true with errors . an upper bound similar to ( [ 1.4.3 ] ) below appeared previously in @xcite . for simplicity we will formulate the result for @xmath982valued random vectors though the same arguments work for random variables with values in a banach space . the proof is a strightforward modification of the standard one ( cf . @xcite ) but still we exhibit it here for readers convenience . [ lem1.4.1 ] let @xmath983 , @xmath984 be uniformly bounded on compact sets functions on @xmath16 and @xmath985 be a family of @xmath982valued random vectors on a probability space @xmath986 such that @xmath987 with probability one for some constant @xmath793 and all @xmath855 . for any @xmath988 and @xmath989 set @xmath990 ( i ) for any @xmath991 there exists @xmath992 such that whenever for some @xmath993 , @xmath994 and each @xmath831 with @xmath995 , @xmath996 then for any compact set @xmath997 , @xmath998 where @xmath999 ( ii ) suppose that @xmath1000 , @xmath1001 and there exists @xmath994 such that @xmath1002 and @xmath1003 if ( [ 1.4.2 ] ) holds true then for any @xmath253 , @xmath1004 ( iii ) assume that @xmath1005 satisfy ( [ 1.4.5 ] ) . for any @xmath991 there exists @xmath1006 such that whenever for some @xmath1007 and each @xmath831 with @xmath1008 the inequality ( [ 1.4.2 ] ) holds true together with @xmath1009 then for any @xmath1010 , @xmath1011 where @xmath1012 @xmath1013 @xmath1014 , @xmath1015 @xmath1016 and @xmath1017 denotes the closure of @xmath59 . \(i ) by ( [ 1.4.1 ] ) for any @xmath1018 and @xmath1019 there exists @xmath1020 such that @xmath1021 set @xmath1022 and cover the compact set @xmath1023 by open balls @xmath1024 let @xmath1025 be a finite subcover with a minimal number @xmath446 of elements . observe that @xmath446 does not exceed the maximal number of points in @xmath1026 with pairwise distances at least @xmath1027 and the latter number depends only on @xmath1028 and @xmath1029 . by ( [ 1.4.2 ] ) and ( [ 1.4.9 ] ) for each @xmath1030 , @xmath1031 since @xmath1032 and @xmath1033 then summing these inequalities in @xmath1030 we obtain @xmath1034 since @xmath446 is bounded by a number depending only on @xmath1035 and @xmath793 we can choose @xmath1036 so that @xmath1037 which together with ( [ 1.4.10 ] ) yield ( [ 1.4.3 ] ) . \(ii ) by ( [ 1.4.2 ] ) and ( [ 1.4.5 ] ) , @xmath1038 and ( [ 1.4.6 ] ) follows . \(iii ) by ( [ 1.4.5 ] ) and ( [ 1.4.7 ] ) for any @xmath1039 , @xmath1040 where @xmath1041 is the expectation with respect to the probability measure @xmath1042 on @xmath1043 such that @xmath1044 now by ( [ 1.4.2 ] ) and ( [ 1.4.5 ] ) for any @xmath831 with @xmath995 we obtain that @xmath1045 where @xmath1046 and @xmath1047 . observe that @xmath1048 thus , applying ( i ) on the probability space @xmath1049 we derive that @xmath1050 provided @xmath1007 for a sufficiently large @xmath1036 . this together with ( [ 1.4.11 ] ) yield ( [ 1.4.8 ] ) . [ lem1.4.2 ] let @xmath1051 be a nondecreasing sequence of lower semicontinuous functions on a metric space @xmath1052 and let @xmath1053 assume that @xmath285 is also lower semicontinuous and for any compact set @xmath1054 denote @xmath1055 then @xmath1056 by the lower semicontinuity of @xmath1057 and @xmath285 and by compactness of @xmath1058 it follows that there exist @xmath1059 such that @xmath1060 and @xmath1061 . passing if needed to a subsequence assume that @xmath1062 as @xmath206 . since @xmath1063 then @xmath1064 assume now that @xmath1065 . since @xmath1066 then for any @xmath342 there exists @xmath1067 such that @xmath1068 by the lower semicontinuity of @xmath1069 it follows that for @xmath1070 large enough @xmath1071 where we use also that @xmath1072 is a nondecreasing sequence . since ( [ 1.4.19 ] ) holds true for any @xmath1070 large enough and for each @xmath342 we can pass there to the limit so that , first , @xmath1073 and then @xmath50 yielding that @xmath1074 which together with ( [ 1.4.17 ] ) give ( [ 1.4.15 ] ) under the condition @xmath1065 . if @xmath1075 then @xmath1076 and for any @xmath1077 there exists @xmath1078 such that @xmath1079 for any @xmath1080 . by the lower semicontinuity of @xmath1057 we conclude that @xmath1081 for @xmath1082 large enough which implies that @xmath1083 for all sufficiently large @xmath54 . hence @xmath1084 and since @xmath80 is arbitrary the left hand side of ( [ 1.4.20 ] ) equals infinity , i.e. again ( [ 1.4.15 ] ) holds trues with both parts of it being equal @xmath187 . in the next section we will employ the following general result which will enable us to subdivide time into small intervals freezing the slow variable on each of them so that the estimate ( [ 1.3.8 ] ) of proposition [ prop1.3.4 ] becomes sufficiently precise and , on the other hand , we will not change much the corresponding functionals @xmath228 appearing in required large deviations estimates . this result is certainly not new , it is cited in @xcite as a folklore fact and a version of it can be found in @xcite , p.67 but for readers convenience we give its proof here . [ lem1.4.3 ] let @xmath1085 be a measurable function on @xmath437 equal zero outside of @xmath201 $ ] and such that @xmath1086 . for each positive integer @xmath54 and @xmath1087 $ ] define @xmath1088{{\delta}}-c)$ ] where @xmath1089 and @xmath1090 $ ] denotes the integral part . then there exists a sequence @xmath1091 such that for lebesgue almost all @xmath1087 $ ] , @xmath1092 for each @xmath253 there exists a @xmath53 function @xmath711 on @xmath437 equal zero outside of @xmath201 $ ] and such that @xmath1093 define @xmath1094 as above with @xmath711 in place of @xmath1095 . then @xmath1096 we have also @xmath1097 since @xmath1098 it follows from ( [ 1.4.22])([1.4.24 ] ) that @xmath1099 this together with the chebyshev inequality and the borel cantelli lemma yield ( [ 1.4.21 ] ) for some sequence @xmath1091 and lebesgue almost all @xmath1087 $ ] . [ lem1.5.1 ] let @xmath1100 @xmath1101 @xmath1102 @xmath1103 , @xmath1104 @xmath1105 , @xmath1106 , @xmath1107 and for @xmath252 $ ] , @xmath1108 then @xmath1109 @xmath1110 and @xmath1111 where , recall , @xmath1112 and @xmath1113 if @xmath1114 . by ( [ 1.2.12 ] ) , @xmath1115 and ( [ 1.5.2 ] ) follows . observe , that @xmath1116 and ( [ 1.5.3 ] ) follows in view of ( [ 1.2.15 ] ) . next , by ( [ 1.2.15 ] ) , @xmath1117 and ( [ 1.5.4 ] ) follows from the gronwall inequality . for any @xmath1118 and @xmath1119 set @xmath1120 and @xmath1121 with @xmath1122 given by ( [ 1.2.10 ] ) . [ prop1.5.2 ] let @xmath1123 and @xmath1124 be the same as in lemma [ lem1.5.1 ] and assume that @xmath1125 fix also @xmath1126 so that proposition [ prop1.3.4 ] holds true . \(i ) there exist @xmath1127 and @xmath1128 independent of @xmath1129 such that if @xmath1130 and @xmath1131 then for any @xmath1132 , @xmath1133 where @xmath1134 , @xmath1135 does not depend on @xmath1129 and @xmath1136 in particular , if for each @xmath1137 there exists @xmath1138 such that @xmath1139 and @xmath1140 then ( [ 1.5.6 ] ) holds true with @xmath1141 in place of @xmath1142 , @xmath1137 . \(ii ) for any @xmath1143 there exist @xmath1144 and @xmath1145 , the latter depending also on @xmath1146 , such that if @xmath1147 and @xmath1148 satisfy ( [ 1.5.8 ] ) and ( [ 1.5.9 ] ) , @xmath1149 , @xmath1150 and @xmath1151 then @xmath1152 with some @xmath1128 depending only on @xmath21 and @xmath1153 . \(i ) assuming that @xmath625 is small and @xmath1154 is large so that @xmath1155 is still small , we consider for each @xmath240 and @xmath261 closed discs @xmath1156 and @xmath629 with @xmath1157 . for each small @xmath1158 set @xmath1159 @xmath1160 then @xmath1161 provided @xmath1162 . for any pair of compact sets @xmath1163 and @xmath695 a finite set @xmath1164 will be called @xmath1165-separated if @xmath1166 @xmath859 implies that @xmath1167 choose a maximal @xmath1168-separated set @xmath1169 in @xmath1170 ( where maximal means that the set can not be enlarged still remaining @xmath1171-separated ) . then @xmath1172 and , by lemma [ lem1.3.2](i ) for small @xmath33 , @xmath1173 , and @xmath1174 , @xmath1175 set @xmath1176 and @xmath1177 assuming that @xmath1178 . then for @xmath1179 , @xmath1180 , @xmath1181 by lemma [ lem1.3.2](i ) if @xmath1173 and @xmath33 is small enough then @xmath1182 for any @xmath1183 and using , in addition , assumption [ ass1.2.2 ] and the inequality ( [ 1.5.3 ] ) we obtain that for any @xmath1184 , @xmath1185 hence , if @xmath1186 then for @xmath1187 and @xmath1188 , @xmath1189 provided @xmath33 is small enough , and so , by ( [ 1.5.3 ] ) and ( [ 1.5.4 ] ) , @xmath1190 where we set @xmath1191 . since @xmath1122 is ( lipschitz ) continuous in @xmath1192 there exists @xmath1193 such that @xmath1194 let @xmath1195 and @xmath1196 since @xmath1122 is lipschitz continuous ( and even @xmath53 ) in @xmath789 and @xmath1197 ( see @xcite ) it follows from ( [ 1.5.14 ] ) that @xmath1198 where @xmath1199 depends only on @xmath1200 . since @xmath1201 provided @xmath1202 we derive from lemma [ lem1.3.2](iv ) , proposition [ prop1.3.4 ] , and lemma [ lem1.4.1](i ) that for such @xmath1203 and any @xmath988 , @xmath1204 where @xmath1205 as , first , @xmath50 and then @xmath1206 . since @xmath1207 are disjoint for different @xmath1208 we obtain from ( [ 1.5.13 ] ) and lemma [ lem1.3.2](iv ) that @xmath1209 employing ( [ 1.5.11 ] ) , ( [ 1.5.17 ] ) and ( [ 1.5.18 ] ) for @xmath1210 with @xmath1211 and @xmath1212 , respectively , and using only ( [ 1.5.17 ] ) for @xmath1213 we derive that @xmath1214 provided @xmath1215 and @xmath1216 with @xmath1217 satisfying ( [ 1.5.7 ] ) and with the same @xmath1218 as in ( [ 1.5.6 ] ) . let @xmath1219 be a ball on @xmath1220 centered at @xmath512 and having radius @xmath1221 , @xmath1222 in the interior metric on @xmath1220 ( which , recall , is a semi - invariant extension of the family of local unstable manifolds on @xmath70 see section 3 and @xcite ) . then @xmath1223 if @xmath1154 . recall , that if @xmath625 is small enough then the extended local unstable and stable discs @xmath1224 and @xmath1225 are defined for all @xmath513 and , in fact , by ( [ 1.2.4 ] ) , the compactness arguments and by @xcite such discs can be defined for all @xmath512 from a small neighborhood @xmath59 of @xmath1226 which is still contained in the basin of attraction of each @xmath327 . for each @xmath1227 set @xmath1228 and assume that @xmath625 is small enough so that @xmath1229 . then ( [ 1.5.19 ] ) together with the fubini theorem yield ( [ 1.5.6 ] ) with the box @xmath1230 in place of the whole @xmath145 . relying on the transversality of unstable and weakly stable submanifolds together with compactness arguments we conclude that there exist an integer @xmath1231 depending only on @xmath625 such that @xmath145 can be covered by @xmath1231 boxes @xmath1232 which yields now ( [ 1.5.6 ] ) in the required form . \(ii ) we start proving ( [ 1.5.10 ] ) by using ( [ 1.5.12 ] ) in order to conclude similarly to ( [ 1.5.13 ] ) that if @xmath1173 , @xmath1233 , and @xmath1234 then @xmath1235 . hence , @xmath1236 where the last inequality holds true since @xmath1237 are disjoint for different @xmath1238 . using ( [ 1.5.16 ] ) , lemma [ lem1.3.2](iv ) , proposition [ prop1.3.4 ] , and lemma [ lem1.4.1](iii ) we obtain that for any @xmath1233 , @xmath1239 , @xmath1240 and @xmath1241 , @xmath1242 where @xmath1243 @xmath1244 @xmath1245 and @xmath1246 as , first , @xmath50 and then @xmath1206 . by lemma [ lem1.3.2](iv ) and the definitions of @xmath1247 and @xmath1248 , @xmath1249 employing ( [ 1.5.20])([1.5.22 ] ) for @xmath1250 with @xmath1251 , @xmath1252 ... ,@xmath1253 and @xmath1254 , respectively , and using only ( [ 1.5.21 ] ) for @xmath1213 we derive that @xmath1255 for some @xmath1256 provided , say , @xmath1257 and @xmath1258 . since @xmath466 is differentiable in @xmath1192 ( see @xcite ) then @xmath1259 for any @xmath1260 ( see theorems 23.5 and 25.1 in @xcite ) , and so by the lower semicontinuity of @xmath223 in @xmath224 ( and , in fact , also in @xmath20 ) , @xmath1261 this together with lemma [ lem1.4.2 ] yield that @xmath1262 appearing in the definition of @xmath1263 is positive provided @xmath21 is sufficiently large . in fact , it follows from the lower semicontinuity of @xmath223 that @xmath1262 is bounded away from zero by a positive constant independent of @xmath1264 and @xmath1147 , @xmath1137 if these points vary over fixed compact sets and ( [ 1.5.8 ] ) together with ( [ 1.5.9 ] ) hold true . now , given @xmath1019 choose , first , sufficiently large @xmath21 as needed and then subsequently choosing small @xmath1265 and @xmath1266 , then small @xmath65 , and , finally , small enough @xmath33 we end up with an estimate of the form @xmath1267 where @xmath1128 and @xmath1268 satisfies ( [ 1.5.7 ] ) . finally , ( [ 1.5.10 ] ) follows from ( [ 1.5.23 ] ) , ( [ 1.5.24 ] ) and the fubini theorem ( similarly to ( i ) ) . next , we pass directly to the proof of theorem [ thm1.2.3 ] starting with the lower bound . some of the details below are borrowed from @xcite but we believe that our exposition and the way of proof are more precise , complete and easier to follow . assume that @xmath1269 , and so that @xmath654 is absolutely continuous , since there is nothing to prove otherwise . then by ( [ 1.2.13 ] ) , @xmath1270 for lebesgue almost all @xmath1271.$ ] by ( [ 1.2.15 ] ) and ( [ 1.3.6 ] ) , @xmath1272 and so if @xmath1270 it follows from ( [ 1.2.12 ] ) that @xmath1273 . suppose that @xmath1274 and let ri@xmath1275 be the interior of @xmath1275 in its affine hull ( see @xcite ) . then either ri@xmath1276 or @xmath1275 ( by its convexity ) consists of one point and recall that @xmath1277 for lebesgue almost all @xmath1271 $ ] . by ( [ 1.2.10 ] ) and ( [ 1.5.25 ] ) , @xmath1278 this together with the nonnegativity and lower semi - continuity of @xmath1279 yield that there exists @xmath1280 such that @xmath1281 and by a version of the measurable selection ( of the implicit function ) theorem ( see @xcite , theorem iii.38 ) , @xmath1280 can be chosen to depend measurably in @xmath1271 $ ] . of course , if ri@xmath1282 then @xmath1275 contains only @xmath1280 and in this case @xmath1283 for lebesgue almost all @xmath1271 $ ] . taking @xmath1284 and @xmath1285 we obtain @xmath1286 observe that @xmath1287 is measurable as a function of @xmath1288 and @xmath224 since it is obtained via ( [ 1.2.12 ] ) as a supremum in one argument of a family of continuous functions , and so this supremum can be taken there over a countable dense set of @xmath1192 s . hence , the set @xmath1289,\,{{\alpha}}\in{{\mathcal d}}(l_s)\}=\ell^{-1}[0,\infty)$ ] is measurable , and so the set @xmath1290\}$ ] is measurable , as well . its projection @xmath1291:\ , ( s,{{\alpha}})\in b\,\mbox{for some}\,{{\alpha}}\in{{\mathbb r}}^d\}$ ] on the first component of the product space is also measurable and @xmath317 is the set of @xmath1271 $ ] such that @xmath1275 contains more than one point . employing theorem iii.22 from @xcite we select @xmath1292 measurably in @xmath1293 and such that @xmath1294 . by convexity and lower semicontinuity of @xmath1279 it follows from corollary 7.5.1 in @xcite that @xmath1295 for each @xmath253 set @xmath1296 then , clearly , @xmath1297 is a measurable function of @xmath1288 , and so @xmath1298 and @xmath1299 are measurable in @xmath1288 , as well . by theorems 23.4 and 23.5 from @xcite for each @xmath1300 there exists @xmath1301 such that ( [ 1.5.27 ] ) holds true . given @xmath1302 take @xmath1303 and for @xmath1271\setminus v$ ] set @xmath1284 . then @xmath1304 for each @xmath1132 set @xmath1305 if the corresponding @xmath1306 in ( [ 1.5.27 ] ) satisfies @xmath1307 and @xmath1308 , otherwise . note , that ( [ 1.5.27 ] ) remains true with @xmath1309 in place of @xmath1310 with @xmath1285 if @xmath1308 . as observed above @xmath1311 whenever @xmath1312 , and so @xmath1313 for lebesgue almost all @xmath1271 $ ] . we recall also that @xmath1314 and @xmath1315 for lebesgue almost all @xmath1271 $ ] . since @xmath1269 , @xmath1316 , and @xmath1317 as @xmath1318 for lebesgue almost all @xmath1271 $ ] , we conclude from ( [ 1.5.29 ] ) and the above observations that for @xmath21 large enough @xmath1319 next , we apply lemma [ lem1.4.3 ] to conclude that there exists a sequence @xmath1320 such that for each @xmath1321 and lebesgue almost all @xmath1322 , @xmath1323 where @xmath1324{{\delta}}_j - c$ ] , @xmath1090 $ ] denotes the integral part and we assume @xmath1325 and @xmath1326 if @xmath1327 . choose @xmath1328 $ ] and set @xmath1329 , @xmath1330 where @xmath1331 if @xmath1332 , @xmath1333 , @xmath1334 @xmath1335 and @xmath1336 , @xmath1337 for @xmath1338 and @xmath1339 for @xmath1340 where @xmath1341 . since @xmath1342 for lebesgue almost all @xmath1271 $ ] then @xmath1343 and , in addition , @xmath1344 by ( [ 1.5.29])([1.5.31 ] ) . this together with ( [ 1.5.3 ] ) and ( [ 1.5.4 ] ) yield that for @xmath1345 , @xmath1346 provided @xmath1347 where @xmath1348 and @xmath1349 are the same as in lemma [ lem1.5.1 ] , the latter is defined with @xmath1350 , @xmath1338 and @xmath1351 . choose @xmath1352 so small and @xmath1353 so large that @xmath1354 then by ( [ 1.5.32 ] ) , @xmath1355 by ( [ 1.5.29])([1.5.31 ] ) , @xmath1356 and by the construction above the conditions of the assertion ( ii ) of proposition [ prop1.5.2 ] hold true , so choosing @xmath1353 sufficiently large we derive ( [ 1.2.16 ] ) ( with @xmath1357 in place of @xmath1029 ) from ( [ 1.5.10 ] ) , ( [ 1.5.33 ] ) and ( [ 1.5.34 ] ) provided @xmath33 is small enough . next , we pass to the proof of the upper bound ( [ 1.2.17 ] ) . assume that ( [ 1.2.17 ] ) is not true , i.e. there exist @xmath1358 and @xmath240 such that for some sequence @xmath1359 as @xmath1360 , @xmath1361 since @xmath1362 by ( [ 1.2.15 ] ) all paths of @xmath1363 $ ] and of @xmath1364 $ ] given by ( [ 1.5.1 ] ) ( the latter for any measurable @xmath155 ) belong to a compact set @xmath1365 which consists of curves starting at @xmath20 and satisfying the lipschitz condition with the constant @xmath604 . let @xmath1366 denotes the open @xmath625-neighborhood of the compact set @xmath230 and @xmath1367 . for any small @xmath1368 choose a @xmath1352-net @xmath1369 in @xmath1370 where @xmath1371 . since @xmath1372 then there exists @xmath1373 and a subsequence of @xmath1374 , for which we use the same notation , such that @xmath1375 denote such @xmath1376 by @xmath1377 , choose a sequence @xmath1378 and set @xmath1379 . since @xmath1370 is compact there exists a subsequence @xmath1380 converging in @xmath81 to @xmath1381 which together with ( [ 1.5.36 ] ) yield @xmath1382 for all @xmath1368 . we claim that ( [ 1.5.37 ] ) contradicts ( [ 1.5.2 ] ) and the assertion ( i ) of proposition [ prop1.5.2 ] . indeed , set @xmath1383 by the monotone convergence theorem @xmath1384 similarly to our remark ( before assumption [ ass1.2.2 ] ) in section [ sec1.2 ] it follows from the results of section 9.1 of @xcite that the functionals @xmath1385 and @xmath1386 are lower semicontinuous in @xmath155 and @xmath654 ( see also section 7.5 in @xcite ) . this together with ( [ 1.5.38 ] ) enable us to apply lemma [ lem1.4.2 ] in order to conclude that @xmath1387 where @xmath1388 . the last inequality in ( [ 1.5.39 ] ) follows from the lower semicontinuity of @xmath228 . thus we can and do choose @xmath1132 such that @xmath1389 by the lower semicontinuity of @xmath1390 in @xmath155 there exists a function @xmath1391 on @xmath1392 such that for each @xmath1393 , @xmath1394 next , we restrict the set of functions @xmath155 to make it compact . namely , we allow from now on only functions @xmath155 for which there exists @xmath1393 such that either @xmath1395 or @xmath1396 for @xmath1397 , @xmath1398 and @xmath1399 where @xmath54 is a positive integer . it is easy to see that the set of such functions @xmath155 is compact with respect to the uniform convergence topology in @xmath81 and it follows that @xmath1400 in ( [ 1.5.41 ] ) constructed with such @xmath155 in mind is lower semicontinuous in @xmath654 . hence @xmath1401 now take @xmath1402 satisfying ( [ 1.5.37 ] ) and for any integer @xmath1403 set @xmath1404 , @xmath1405 , @xmath1406 and @xmath1407 @xmath1408 . define a piecewise linear @xmath1409 and a piecewise constant @xmath1410 by @xmath1411 and @xmath1398 with @xmath1412 . since @xmath1402 is lipschitz continuous with the constant @xmath604 then @xmath1413 if @xmath54 is large enough and @xmath342 is sufficiently small then @xmath1414 where @xmath1415 is the same as in ( [ 1.5.6 ] ) . since @xmath1416 it follows from ( [ 1.5.44 ] ) and ( [ 1.5.45 ] ) that @xmath1417 and by ( [ 1.5.41 ] ) and the first inequality in ( [ 1.5.45 ] ) we obtain that @xmath1418 hence , by ( [ 1.5.6 ] ) and the second inequality in ( [ 1.5.45 ] ) for all @xmath33 small enough , @xmath1419 provided @xmath1420 ( taking into account that @xmath1421 ) . by ( [ 1.5.2 ] ) and the definition of vectors @xmath1422 for any @xmath1423 , @xmath1424 therefore , @xmath1425 choosing , first , @xmath54 large enough so that @xmath65 satisfies ( [ 1.5.45 ] ) with all sufficiently small @xmath33 and also that @xmath1426 , and then choosing @xmath1352 so small that @xmath1427 , we conclude that ( [ 1.5.47 ] ) together with ( [ 1.5.49 ] ) contradicts ( [ 1.5.37 ] ) , and so the upper bound ( [ 1.2.17 ] ) holds true . since @xmath218 if and only if @xmath1428 satisfying ( [ 1.2.14 ] ) the estimate ( [ 1.2.18 ] ) follows from ( [ 1.2.17 ] ) and the lower semicontinuity of the functional @xmath228 , completing the proof of theorem [ thm1.2.3 ] . in this section we study essential properties of the functionals @xmath228 which will be needed in the proofs of theorems [ thm1.2.5 ] and [ thm1.2.7 ] in the next sections . we will start with the following general fact which do not require specific conditions of theorems [ thm1.2.5 ] and [ thm1.2.7 ] . [ lem1.6.1 ] there exists @xmath758 such that if @xmath1429 then any @xmath1430 from the space @xmath163 of @xmath27-invariant probability measures on @xmath70 can be included into a weakly continuous in @xmath611 family @xmath1431 ( considered in the space of probability measures on @xmath1432 ) for which @xmath1433 is @xmath53 in @xmath611 and the entropy @xmath1434 is continuous in @xmath611 as @xmath1435 . furthermore , there exists @xmath1436 such that @xmath1437 and for any @xmath340 there exists @xmath1438 such that if @xmath1439 is small enough the structural stability theorem for axiom a flows obtained in @xcite can be applied in order to compare @xmath167 and @xmath1440 but here we will need its more recent form derived in @xcite , @xcite , and @xcite which yields a homeomorphism @xmath1441 and a continuous function @xmath1442 on @xmath327 both with @xmath53 dependence on @xmath611 and such that the conjugate flow @xmath1443 satisfies @xmath1444 where @xmath1445 is the identity map on @xmath70 and @xmath1446 . by the standard direct verification we see that @xmath1447 is an @xmath1448-invariant probability measure . it is known ( see , for instance , @xcite , theorem 4.2 ) that then the probability measure @xmath1449 on @xmath327 defined by its radon nikodim derivative @xmath1450 is @xmath375-invariant . in our case this can be seen easily since for any @xmath53 function @xmath169 on @xmath327 , @xmath1451 where the last equality holds true by @xmath1452-invariance of @xmath45 . now @xmath1453 this together with ( [ 1.2.15 ] ) yield the differentiability of @xmath1454 in @xmath611 taking into account that @xmath1442 and @xmath1455 are @xmath53 in @xmath611 ( see @xcite ) and since the proof of this fact relies on a version of the implicit function theorem ( see @xcite ) which provides derivatives in @xmath611 uniformly in @xmath306 whenever @xmath1435 and @xmath1456 is small enough we derive also ( [ 1.6.1 ] ) . next , clearly , @xmath1457 . if we knew that @xmath1430 were ergodic then , of course , @xmath45 would be ergodic , as well , and it would follow from theorem 10.1 in @xcite that @xmath1458 which would yield the differentiability of @xmath1434 in @xmath611 . in the general case we obtain from @xcite that @xmath1459 and so @xmath1460 since by ruelle s inequality ( see , for instance @xcite ) , @xmath1461 we derive both the continuity of @xmath1434 in @xmath611 and the first part of ( [ 1.6.2 ] ) . the second part of ( [ 1.6.2 ] ) follows from its first part in view of ( [ 1.2.8 ] ) taking into account that the function @xmath1462 defined by ( [ 1.2.5 ] ) is h " older continuous in @xmath18 and uniformly lipschitz continuous ( even @xmath53 ) in @xmath20 ( see @xcite ) and that @xmath368 is lipschitz continuous in both variables ( see ( [ 1.2.15 ] ) ) . the following result gives , in particular , sufficient conditions for a set to be an @xmath285-compact . [ lem1.6.2 ] ( i ) there exists @xmath1436 and for each @xmath306 where the vector field @xmath19 is complete there exists @xmath1463 such that if @xmath1464 and @xmath1465 then we can construct @xmath298 with @xmath1466 satisfying @xmath1467 it follows that @xmath1468 and @xmath1469 are locally lipschitz continuous in @xmath611 belonging to the open @xmath1456-neighborhood of @xmath20 when @xmath1470 is fixed . \(ii ) let @xmath291 be a compact @xmath286-invariant set which either contains a dense in @xmath302 orbit of @xmath286 or @xmath300 for any pair @xmath308 . suppose that @xmath19 is complete at each point of @xmath302 . then @xmath302 is an @xmath285-compact . \(iii ) assume that for any @xmath292 there exists @xmath309 such that for each @xmath295 its orbit @xmath310\}$ ] of length @xmath311 forms an @xmath312-net in @xmath302 or , equivalently , that @xmath286 is a minimal flow on @xmath302 . suppose that @xmath19 is complete at a point of @xmath302 . then @xmath302 is an @xmath285-compact . \(i ) fix some @xmath306 . in view of the ergodic decomposition ( see , for instance , @xcite ) any @xmath1471 can be represented as an integral over the space of ergodic measures from @xmath163 . using the specification ( see @xcite and @xcite ) any ergodic @xmath1471 can be approximated ( in the weak sense ) by @xmath27-invariant measures sitting on its periodic orbits , i.e. by measures of the form @xmath1472 where @xmath1473 , @xmath1474 is a periodic orbit of @xmath27 with a period @xmath1475 . this is done in a standard way by choosing a generic point of an ergodic measure @xmath45 , i.e. a point @xmath512 which satisfies @xmath1476 for any continuous function @xmath711 on @xmath70 , and then approximating the orbit of @xmath512 by periodic orbits of @xmath27 using the specification theorem ( see theorem 3.8 in @xcite ) . it is well known ( see @xcite ) that there are countably many periodic orbits of @xmath27 which together with the above discussion yield that the closed convex hull @xmath1477 of the set @xmath1478 is a periodic orbit of @xmath1479 coincides with @xmath1480 . now assume that @xmath19 is complete at @xmath20 . then @xmath1481\}= \{{{\alpha}}{{\gamma}}_x^{(0)},\,{{\alpha}}\in[0,1]\}$ ] contains an open neighborhood of 0 in @xmath16 . but then we can find a simplex @xmath1482 with vertices in @xmath1477 such that @xmath1483\}$ ] contains an open neighborhood of 0 in @xmath16 and for some periodic orbits @xmath1484 of @xmath1485 , @xmath1486 where we denote @xmath1487 . by compactness of @xmath1482 it follows also that @xmath1488 now , set @xmath1489 , @xmath1490 and include each @xmath1491 into the weakly continuous in @xmath611 families @xmath1492 constructed in lemma [ lem1.6.1 ] for @xmath611 in some neighborhood of @xmath20 . if @xmath1493 and @xmath1494 is small enough each simplex @xmath1495 intersects and not at 0 with any ray emanating from @xmath1496 or , in other words , @xmath1497\}$ ] contains an open neighborhood of 0 in @xmath16 and , moreover , @xmath1498 since all @xmath1499 are @xmath375-invariant probability measures provided @xmath1500 we conclude that for any @xmath611 in the @xmath1494-neighborhood of @xmath20 and any vector @xmath552 there exists an @xmath375-invariant probability measure @xmath1501 such that @xmath1502 has the same direction as @xmath552 and @xmath1503 where @xmath604 is the same as in ( [ 1.2.15 ] ) . it follows that any two points @xmath1504 and @xmath1505 from the open @xmath1494-neighborhood of @xmath20 can be connected by a curve @xmath654 lying on the interval connecting @xmath1504 and @xmath1505 with @xmath1506 , i.e. @xmath1507 with some @xmath1508 $ ] and by ( [ 1.2.9 ] ) , @xmath1509 in view of the triangle inequality for @xmath282 what we have proved yields the continuity of @xmath1468 and @xmath1469 in @xmath611 belonging to the open @xmath1494-neighborhood of @xmath20 when @xmath1470 is fixed . covering @xmath133 by @xmath1494-neighborhoods of points @xmath1429 and choosing a finite subcover we obtain ( i ) with the same constant for all @xmath133 . next , we derive the sufficient conditions of ( ii ) for the @xmath285-compactness . first , observe that both assumptions there imply that for any @xmath292 there exist @xmath1510 and @xmath1511 such that @xmath1512 form an @xmath1513-net in @xmath302 and @xmath1514 where @xmath793 is the same as in ( i ) . indeed , if there exists a dense orbit of @xmath286 in @xmath302 then a sufficiently long piece of this orbit will work as such @xmath654 with its @xmath285-functional equal 0 . if @xmath300 for any @xmath308 then we can choose an @xmath1513-net @xmath1515 in @xmath302 and then construct curves @xmath1516 such that @xmath1517 with @xmath1518 . taking @xmath1519 for @xmath1520 $ ] we obtain the required curve . now , for each @xmath295 let @xmath1521 be the open @xmath1494-neighborhood of @xmath20 in @xmath16 where the construction of the part ( i ) can be implemented . since @xmath302 is compact we can choose from the cover @xmath1522 of @xmath302 a finite subcover @xmath1523 of @xmath302 . for any positive @xmath312 such that @xmath1513 is less than the lebesgue number ( see @xcite ) of @xmath1524 we construct @xmath1512 as above and then for any @xmath336 there is @xmath1525 and @xmath1526 such that @xmath1527 , and so by the assertion ( i ) we can connect @xmath611 and @xmath1470 by a curve @xmath1528 with @xmath1529 and @xmath1530 . it follows that any two points @xmath308 can be connected by a curve @xmath298 with @xmath1531 $ ] and @xmath1532 . now set @xmath1533 and suppose that @xmath1534 . let @xmath1535 be smaller than the lebesgue number of the cover @xmath1536 of @xmath1537 and set @xmath1538dist@xmath1539 . then for any @xmath296 there exists @xmath295 with @xmath1540 , and so @xmath1541 for some @xmath1525 . hence , by ( i ) there exists a curve @xmath1542 connecting @xmath20 with @xmath611 and such that @xmath1543 and @xmath1544 . by above we can connect any @xmath1545 with @xmath20 by a curve @xmath298 with @xmath1531 $ ] and @xmath1532 and then using @xmath1546 we arrive at a combined curve connecting @xmath1547 with @xmath611 and satisfying the conditions required to ensure that @xmath302 is an @xmath285-compact by taking @xmath1548 . \(iii ) now assume that for any @xmath292 and each @xmath295 its piece of the @xmath286-orbit of length @xmath311 forms an @xmath312-net in @xmath302 and suppose that @xmath19 is complete at @xmath1549 . set @xmath1550 where @xmath1551 is the differential of @xmath286 at @xmath611 . let @xmath295 and @xmath304 with dist@xmath1552 where @xmath1553 . then for some @xmath1554 , @xmath1555 and @xmath1556 , @xmath1557 for some @xmath1558 $ ] , and so @xmath1559 . in addition , for any @xmath1560 there exists @xmath1561 so that @xmath1562 with @xmath1563 $ ] . now , by the assertion ( i ) we can connect @xmath1564 with @xmath1565 by a curve @xmath1566 with @xmath1567 and @xmath1568 , then connect @xmath1565 with @xmath1569 by a curve @xmath1570 with @xmath1571 and @xmath1572 . finally , we can connect @xmath20 with @xmath611 by the curve @xmath1573 with @xmath1574 and such that @xmath1575 for @xmath1576 $ ] , @xmath1577 for @xmath1578 $ ] , @xmath1579 for @xmath1580 $ ] , and @xmath1581 for @xmath1582 $ ] yielding that @xmath302 is an @xmath285-compact . the following assertion which relies on lemma [ lem1.6.1 ] will be also useful in our analysis . [ lem1.6.3 ] for any @xmath292 and @xmath277 there exists @xmath1583 such that if @xmath1584 , @xmath1269 , @xmath1585 , and @xmath1586 then we can find @xmath1587 , @xmath1588 with @xmath1589 satisfying @xmath1590 by ( [ 1.2.13 ] ) and the lower semicontinuity of the functionals @xmath1591 there exist measures @xmath1592 $ ] such that @xmath1593 for lebesgue almost all @xmath252 $ ] and @xmath1594 for lebesgue almost all @xmath252 $ ] . recall also that @xmath204 is measurable in @xmath207 . introduce the ( measurable ) map @xmath1595\times{{\mathcal p}}(\bar{{\mathcal w}})\to{{\mathbb r}}\cup\{\infty\}\times{{\mathbb r}}^d$ ] defined by @xmath1596 . recall that @xmath204 is measurable in @xmath207 , and so another map @xmath1597\to { { \mathbb r}}\cup\{\infty\}\times{{\mathbb r}}^d$ ] defined by @xmath1598 is also measurable in @xmath252 $ ] . then @xmath1599 and it follows from the measurable selection in the implicit function theorem ( see @xcite , theorem iii.38 ) that measures @xmath1600 satisfying this condition can be chosen to depend measurably on @xmath252 $ ] . now , given @xmath292 we pick up a small @xmath1583 which will be specified later on and employ lemma [ lem1.4.3 ] in the same way as in ( [ 1.5.31 ] ) together with ( [ 1.2.9 ] ) , ( [ 1.2.11 ] ) , and ( [ 1.2.13 ] ) in order to conclude that for all @xmath409 large enough there exists @xmath1601 such that if @xmath1602 then @xmath1603 set @xmath1604 for @xmath1605 and @xmath1606 for @xmath1607 . then @xmath1608 $ ] defines a polygonal line such that @xmath1609 next , set @xmath1610 for all @xmath1611 $ ] and continue the construction of @xmath1546 in the following recursive way . suppose that @xmath1612 is already defined for all @xmath1613 $ ] and some @xmath1614 . denote @xmath1615 , @xmath1616 , @xmath1617 and suppose that @xmath1618 where @xmath604 is the same as in ( [ 1.2.15 ] ) and @xmath1456 comes from lemma [ 1.6.1 ] . for @xmath1619 $ ] define @xmath1612 as the integral curve starting at @xmath1620 of the vector field @xmath1621 with @xmath1622 obtained in lemma [ lem1.6.1 ] for @xmath1623 , i.e. @xmath1612 is the solution of the equation @xmath1624 this definition is legitimate since in view of ( [ 1.2.15 ] ) and our assumption on @xmath1620 the curve @xmath1625 $ ] does not exit the @xmath1456-neighborhood of @xmath1626 . by ( [ 1.6.1 ] ) and the above for all @xmath1619 $ ] , @xmath1627 and so @xmath1628}| \tilde{{\gamma}}_t-\psi_t^{(n)}|\leq|z_j - y_j|(1+ctn^{-1})+\zeta t / n+ckt^2n^{-2}.\ ] ] assuming that @xmath1586 with @xmath1629 small enough and since @xmath1630 we obtain successively from here that for all @xmath1631 , @xmath1632 which enables us to continue our construction recursively for @xmath1631 if @xmath1629 and @xmath1633 are small enough yielding also that @xmath1634 hence , the first part of ( [ 1.6.3 ] ) follows provided @xmath1629 and @xmath1633 are sufficiently small . next , observe that @xmath1635}|\tilde{{\gamma}}_t - x_j|\leq |z_j - y_j| |y_j - x_j|+ktn^{-1}\leq ktn^{-1}(1+e^{ct})+\zeta(1 + 2e^{ct})\ ] ] and the right hand side here can be made as small as we wish choosing @xmath1629 small and @xmath446 large . hence , by lemma [ lem1.6.1 ] we can make @xmath1636}\big\vert i_{\tilde{{\gamma}}_t}(\mu_{x_j\tilde{{\gamma}}_t})-i_{x_j}(\nu_{t_j^{(n)}})\big\vert < \eta/2t\ ] ] which together with ( [ 1.6.4 ] ) yield the second part of ( [ 1.6.3 ] ) . the following result will enable us to control the time which the slow motion can spend away from the @xmath361-limit set of the averaged motion . [ lem1.6.4 ] let @xmath1637 be a compact set not containing entirely any forward semi - orbit of the flow @xmath286 . then there exist positive constants @xmath1638 and @xmath1639 such that for any @xmath1640 and @xmath641 , @xmath1641\big\}\geq a[t / t]\ ] ] where @xmath1642 $ ] denotes the integral part of @xmath1643 . for each @xmath1640 set @xmath1644 . by the assumption of the lemma @xmath1645 for each @xmath1640 and it follows from continuous dependence of solutions of ( [ 1.1.6 ] ) on initial conditions that @xmath1646 is upper semicontinuous . hence , @xmath1647 . set @xmath1648 and @xmath1649 for all @xmath1271\}$ ] . since no @xmath1650 can be a solution of the equation ( [ 1.2.14 ] ) then @xmath1651 for any @xmath1650 . the set @xmath1652 is closed with respect to the uniform convergence and since the functional @xmath228 is lower semicontinuous we obtain that @xmath1653 this together with ( [ 1.2.13 ] ) yield the assertion of lemma [ lem1.6.4 ] . untill now we have not used specific assumptions of theorem [ thm1.2.5 ] but some of them will be needed for the following auxiliary result . [ lem1.6.5 ] let @xmath317 be a connected open set with a piecewise smooth boundary and assume that ( [ 1.2.20 ] ) holds true . then the function @xmath1654 is upper semicontinuous at any @xmath1655 for which @xmath1656 . let @xmath1657 be an @xmath285-compact . \(i ) then for each @xmath1658 the function @xmath303 takes on the same value @xmath1659 for all @xmath295 , and so @xmath1654 takes on the same value @xmath333 for all @xmath295 and the set @xmath1660 coincides with the same ( may be empty ) set @xmath335 for all @xmath295 . furthermore , for each @xmath253 there exists @xmath1661 such that for any @xmath295 we can construct @xmath1662 with @xmath1663 $ ] satisfying @xmath1664 \(ii ) suppose that @xmath1665 and dist@xmath1666 for some @xmath338 and @xmath1667 as @xmath426 . then @xmath337 and for any @xmath253 there exist @xmath1668 ( depending only on @xmath380 and the function @xmath1218 but not on @xmath20 ) and @xmath1669 with @xmath1670 $ ] satisfying @xmath1671 in particular , if @xmath353 then @xmath1672 and if @xmath302 is an @xmath285-attractor of the flow @xmath286 then @xmath1672 for all @xmath338 . \(iii ) suppose that for any open set @xmath313 the compact set @xmath1673 does not contain entirely any forward semi - orbit of the flow @xmath286 . then the function @xmath1659 is lower semicontinuous in @xmath1658 , @xmath1674 as dist@xmath1675 , and @xmath335 is a nonempty compact set . let @xmath1656 for some @xmath1655 . then for any @xmath340 there exist @xmath277 and @xmath1676 such that @xmath1677 and @xmath1678 . by lemma [ lem1.6.3 ] for any @xmath292 we can choose @xmath1583 so that if @xmath1679 then there exists @xmath1587 such that @xmath1680 and @xmath1681 . let @xmath1682 be measures obtained in lemma [ lem1.6.1 ] for @xmath1683 with @xmath176 satisfying the second part of ( [ 1.2.20 ] ) . since the boundary @xmath318 is piecewise smooth it follows from the continuous dependence of solutions of ordinary differential equations on initial conditions that for all small @xmath292 there exists @xmath1684 as @xmath1685 such that if @xmath1686 $ ] is an integral curve of the vector field @xmath1687 with @xmath1688 then @xmath1689 . since @xmath1690 we can define @xmath1691 for @xmath1692 $ ] . now , @xmath1693 and by ( [ 1.2.9 ] ) , @xmath1694 thus we can choose @xmath312 so small that @xmath1695 and the upper semicontinuity of @xmath1654 at @xmath1565 follows . from now on till the end of the proof of this lemma we assume that @xmath302 is an @xmath285-compact and prove , first , the assertion ( i ) . it follows from the definition of an @xmath285-compact that @xmath1696 for any pair @xmath1697 , and so @xmath1698 for any such @xmath1699 and each @xmath1658 . it follows that @xmath1654 takes on the same value @xmath333 for all @xmath295 and all sets @xmath1700 coincide with some , may be empty , set @xmath335 . fix @xmath1549 . then for each @xmath253 there exists @xmath1701 and @xmath1702 such that @xmath1703 by the definition of an @xmath285-compact there exists @xmath1704 such that for any @xmath336 we can construct @xmath1705 with @xmath1706 $ ] satisfying @xmath1707 defining @xmath1708 by @xmath1709 for @xmath1710 $ ] and @xmath1711 for @xmath1712 $ ] we obtain a curve satisfying ( [ 1.6.5 ] ) with @xmath1713 . next , we prove ( ii ) assuming that @xmath353 and that dist@xmath1714 for some @xmath338 with @xmath1667 as @xmath426 . by ( i ) , for any @xmath292 there exists @xmath1715 such that for any @xmath336 we can construct @xmath1716 with @xmath1717 $ ] and @xmath1718 satisfying ( [ 1.6.5 ] ) with @xmath1719 . for such @xmath312 and @xmath1720 choose @xmath1629 by lemma [ lem1.6.3 ] so that if @xmath1721 and @xmath336 then in the same way as at the beginning of the proof of this lemma we can construct @xmath1722 with @xmath1684 as @xmath1685 such that @xmath1723 pick up @xmath1724 so that @xmath1725 . then @xmath1726 for @xmath1727 and some @xmath336 . now construct as above @xmath1546 for such @xmath611 and define @xmath1669 with @xmath1728 setting @xmath1729\,\,\mbox{and}\,\ , \hat{{\gamma}}^x_t=\tilde{{\gamma}}(t-\tilde t)\,\,\mbox{for}\,\ , t\in[\tilde t , \tilde t+t_z+t(\eta)].\ ] ] then @xmath1730 and @xmath1731 $ ] . choosing @xmath312 so small that @xmath1732 and then taking @xmath1733 we conclude that @xmath1734 satisfies ( [ 1.6.6 ] ) . since @xmath312 is arbitrary we obtain that @xmath337 . if @xmath302 is an @xmath285-attractor whose basin contains @xmath321 then we can choose @xmath1667 as @xmath1735 which in view of the continuous dependence of @xmath1736 on @xmath20 will be the same for all @xmath1737 ( though for this lemma @xmath1738 as above depending on @xmath20 would suffice , as well ) , so our conditions are satisfied now for all @xmath338 . hence , in this case @xmath1654 is finite in the whole @xmath317 , completing the proof of ( ii ) . finally , we prove ( iii ) . recall , that by the definition of an @xmath285-compact @xmath302 it follows that @xmath300 whenever @xmath308 . for all @xmath292 let @xmath294 be open sets appearing in the definition of an @xmath285-compact . if @xmath295 and @xmath296 then @xmath1739 . hence , if dist@xmath1740 as @xmath206 then @xmath1741 . now , let @xmath1742 and @xmath1743 as @xmath206 . for each @xmath1744 set @xmath1745dist@xmath1746 and let @xmath1747 . without loss of generality we will assume that @xmath1748 for some @xmath1749 and all @xmath1750 . fix @xmath295 . by the definition of the function @xmath282 for any @xmath1583 we can choose @xmath1751 and @xmath1752 , @xmath1753 such that @xmath1754 for each @xmath1755 set @xmath1756 consider @xmath1757 defined by @xmath1758 for @xmath1759 $ ] which stays in @xmath1760int@xmath1761 ( where int@xmath1762 means the interior of a set @xmath1762 ) , and so by lemma [ lem1.6.4 ] we conclude that @xmath1763 provided , say , @xmath1764 where @xmath1765 depends only on @xmath312 . in order to verify the lower semicontinuity of @xmath303 at @xmath1766 we have only to consider the case @xmath1767 and so we can assume that @xmath1768 for all @xmath1769 . passing to a subsequence and denoting its members by the same letters we can assume also that @xmath1770 the curves @xmath1771 are lipschitz continuous with a constant @xmath604 from ( [ 1.2.15 ] ) , and so this sequence is relatively compact . hence , we can choose a uniformly converging subsequence and denoting , again , its members by the same letters we obtain now that @xmath1772 where @xmath1773 with @xmath1774,\,$ ] dist@xmath1775 and @xmath1776 . each curve @xmath1777 can be extended to a curve in @xmath1778 with @xmath1779 and the same @xmath285-functional by adding to one of its ends a piece of the orbit of the flow @xmath286 . hence , we can rely on the lower semicontinuity of the functional @xmath228 in order to derive from ( [ 1.6.7 ] ) that @xmath1780 by the definition of an @xmath285-compact there exists @xmath1781 with @xmath1782 $ ] such that @xmath1783 and @xmath1784 . it follows that @xmath1785 and since @xmath312 and @xmath1629 can be chosen arbitrarily small we conclude that @xmath1786 obtaining the lower semicontinuity of @xmath303 at @xmath1766 . finally , the lower semicontinuity of @xmath303 in @xmath323 for a fixed @xmath295 implies that @xmath1787 is nonempty and compact and since @xmath1787 is the same for all @xmath295 by ( i ) , the proof of lemma [ lem1.6.5 ] is complete . in this section we study essential properties of the functionals @xmath228 which will be needed in the proofs of theorems [ thm2.2.5 ] and [ thm2.2.7 ] in the next sections . the following result which follows from @xcite is a basic step in our analysis of functionals @xmath2235 and our thanks go to r. pinsky who quickly produced on our request @xcite deriving some properties of functionals @xmath2791 needed here . [ lem2.4.1 ] for each @xmath1429 and any vector measure @xmath2543 on @xmath1052 with @xmath2792 , @xmath2793 if and only if each @xmath2794 has density @xmath2795 with respect to the riemannian volume @xmath54 on @xmath1052 such that @xmath2796 where @xmath2797 is the riemannian gradient and @xmath496 is a corresponding norm . furthermore , there exists @xmath1436 such that for any @xmath1429 and each @xmath45 as above for which ( [ 2.4.1 ] ) holds true , @xmath2798 where @xmath2799 , and if @xmath2800 is another point then @xmath2801 next , using lemma [ lem2.4.1 ] we are able to show that each point where @xmath19 is complete can be connected with close points by curves with small @xmath285-functionals which , in particular , enables us to obtain important examples of @xmath285-compacts . [ lem2.4.2 ] ( i ) there exists @xmath1436 and for each @xmath1429 where the vector field @xmath19 is complete there exists @xmath1463 such that if @xmath1464 and @xmath1465 then we can construct @xmath298 with @xmath1466 satisfying @xmath1467 it follows that @xmath1468 and @xmath1469 are locally lipschitz continuous in @xmath611 belonging to the open @xmath1456-neighborhood of @xmath20 when @xmath1470 is fixed . \(ii ) let @xmath291 be a compact @xmath286-invariant set which either contains a dense in @xmath302 orbit of @xmath286 or @xmath300 for any pair @xmath308 . suppose that @xmath19 is complete at each point of @xmath302 . then @xmath302 is an @xmath285-compact . \(iii ) assume that for any @xmath292 there exists @xmath309 such that for each @xmath295 its orbit @xmath310\}$ ] of length @xmath311 forms an @xmath312-net in @xmath302 and suppose that @xmath19 is complete at a point of @xmath302 . then @xmath302 is an @xmath285-compact . \(i ) fix some @xmath1429 and assume that @xmath19 is complete at @xmath20 . then we can find a simplex @xmath1482 with vertices in @xmath2802 such that @xmath1483\}$ ] contains an open neighborhood of 0 in @xmath16 and @xmath1486 for some @xmath2803 with @xmath2804 . by compactness of @xmath1482 it follows that @xmath1488 by ( [ 2.2.1 ] ) there exists a small @xmath2805 such that if @xmath1493 then each simplex @xmath2806 intersects and not at 0 with any ray emanating from @xmath1496 or , in other words , @xmath1497\}$ ] contains an open neighborhood of 0 in @xmath16 and , moreover , @xmath1498 it follows that for any @xmath611 in the @xmath1494-neighborhood of @xmath20 and any vector @xmath552 there exist @xmath2807 with @xmath2808 such that @xmath2809 observe that by ( [ 2.4.2 ] ) and convexity of @xmath2810 , @xmath2811 for some @xmath722 . hence , any two points @xmath1504 and @xmath1505 from the open @xmath1494-neighborhood of @xmath20 can be connected by a curve @xmath654 lying on the interval connecting @xmath1504 and @xmath1505 with @xmath1506 , i.e. @xmath1507 with some @xmath1508 $ ] and by ( [ 2.2.9 ] ) , @xmath2812 in view of the triangle inequality for @xmath282 what we have proved yields the continuity of @xmath1468 and @xmath1469 in @xmath611 belonging to the open @xmath1494-neighborhood of @xmath20 when @xmath1470 is fixed . covering @xmath133 by @xmath2813neighborhoods of points @xmath1429 and choosing a finite subcover we obtain ( i ) with the same constant @xmath1436 for all points in @xmath133 . [ lem2.4.3 ] for any @xmath292 and @xmath277 there exists @xmath1583 such that if @xmath1584 , @xmath1269 , @xmath1585 , and @xmath1586 then we can find @xmath1587 , @xmath1588 with @xmath1589 satisfying @xmath2814 by ( [ 2.2.9 ] ) , ( [ 2.2.19 ] ) and the lower semicontinuity of the functionals @xmath1591 there exist measures @xmath1592 $ ] such that @xmath1593 for lebesgue almost all @xmath252 $ ] and @xmath1594 for lebesgue almost all @xmath252 $ ] . recall also that @xmath204 is measurable in @xmath207 . introduce the ( measurable ) map @xmath1595\times{{\mathcal p}}(\bar{{\mathcal w}})\to{{\mathbb r}}\cup\{\infty\}\times{{\mathbb r}}^d$ ] defined by @xmath1596 . recall that @xmath204 is measurable in @xmath207 , and so another map @xmath1597\to { { \mathbb r}}\cup\{\infty\}\times{{\mathbb r}}^d$ ] defined by @xmath1598 is also measurable in @xmath252 $ ] . then @xmath1599 and it follows from the measurable selection in the implicit function theorem ( see @xcite , theorem iii.38 ) that measures @xmath1600 satisfying this condition can be chosen to depend measurably on @xmath252 $ ] . since @xmath1269 and the @xmath263-functionals are nonnegative then @xmath2815 for lebesgue almost all @xmath252 $ ] ( and , actually , without loss of generality we can assume that @xmath2816 is finite for all @xmath252 $ ] ) . now let @xmath2817,\ ] ] which in view of ( [ 2.2.1 ] ) determines @xmath1587 . then by ( [ 2.2.1 ] ) , @xmath2818 and by gronwall s inequality @xmath2819 this together with ( [ 2.4.2 ] ) and ( [ 2.4.3 ] ) yields that @xmath2820 for some @xmath722 independent of @xmath1629 and @xmath654 . exchanging @xmath654 and @xmath1546 , applying the same argument and using the inequality @xmath2821 we conclude that @xmath2822 choosing @xmath1629 small enough we arrive at ( [ 2.4.4 ] ) . [ lem2.4.4 ] let @xmath1637 be a compact set not containing entirely any forward semi - orbit of the flow @xmath286 . then there exist positive constants @xmath1638 and @xmath1639 such that for any @xmath1640 and @xmath641 , @xmath1641\big\}\geq a[t / t]\ ] ] where @xmath1642 $ ] denotes the integral part of @xmath1643 . the result is a simple consequence of lower semicontinuity of functionals @xmath474 and the fact that @xmath218 if and only if @xmath654 is a part of an orbit of the flow @xmath286 . further details of the argument can be found in lemma [ lem1.6.4 ] in part [ part1 ] and in lemma 2.2(a ) , chapter 4 of @xcite . [ lem2.4.5 ] let @xmath317 be a connected open set with a piecewise smooth boundary and assume that ( [ 2.2.24 ] ) holds true . then the function @xmath1654 is upper semicontinuous at any @xmath1655 for which @xmath1656 . let @xmath1657 be an @xmath285-compact . \(i ) then for each @xmath1658 the function @xmath303 takes on the same value @xmath1659 for all @xmath295 , and so @xmath1654 takes on the same value @xmath333 for all @xmath295 and the set @xmath1660 coincides with the same ( may be empty ) set @xmath335 for all @xmath295 . furthermore , for each @xmath253 there exists @xmath1661 such that for any @xmath295 we can construct @xmath1662 with @xmath1663 $ ] satisfying @xmath2823 \(ii ) suppose that @xmath1665 and dist@xmath1666 for some @xmath338 and @xmath1667 as @xmath426 . then @xmath337 and for any @xmath253 there exist @xmath1668 ( depending only on @xmath380 and the function @xmath1218 but not on @xmath20 ) and @xmath1669 with @xmath1670 $ ] satisfying @xmath2824 in particular , if @xmath353 then @xmath1672 and if @xmath302 is an @xmath285-attractor of the flow @xmath286 then @xmath1672 for all @xmath338 . \(iii ) suppose that for any open set @xmath313 the compact set @xmath1673 does not contain entirely any forward semi - orbit of the flow @xmath286 . then the function @xmath1659 is lower semicontinuous in @xmath1658 , @xmath1674 as dist@xmath1675 , and @xmath335 is a nonempty compact set . in this section we derive theorems [ thm2.2.5 ] relying on certain `` markov property type '' arguments which are substantial modifications of the corresponding arguments from sections 4 and 5 of @xcite . in this and the following section in order to simplify notations we will write @xmath1788 for @xmath1789 ( both introduced in section [ sec1.3 ] ) with some large @xmath964 so that appropriate discs on ( extended ) unstable leaves @xmath1790 and all their @xmath1791-iterates belong to this set . we start with the following result which will not only yield theorem [ thm1.2.5 ] but also will play an important role in the proof of theorem [ thm1.2.7 ] in the next section . [ prop1.7.1 ] let @xmath317 be a connected open set with a piecewise smooth boundary @xmath318 such that @xmath1792 . assume that for each @xmath323 there exist @xmath1793 and an @xmath107-invariant probability measure @xmath176 on @xmath327 so that @xmath1794,\ ] ] i.e. @xmath1795 and it points out into the exterior of @xmath321 . \(i ) suppose that for some @xmath1796 and any @xmath1658 there exists @xmath1797 such that for some @xmath1798 $ ] , @xmath1799 then for each @xmath338 , @xmath1800 and for any @xmath340 there exists @xmath341 such that for all small @xmath342 , @xmath1801 \(ii ) assume that there exists an open set @xmath1762 such that @xmath317 contains its closure @xmath1802 and the intersection of @xmath1803 with the @xmath361-limit set of the flow @xmath286 is empty . let @xmath1652 be a compact subset of @xmath318 such that @xmath1804 for some @xmath1805 . then for some @xmath277 and any @xmath1438 there exists @xmath1806 such that for each @xmath338 and any small @xmath342 , @xmath1807 suppose that for some @xmath338 , @xmath1808 then @xmath1809 and for each @xmath277 there exists @xmath1810 such that for all small @xmath342 , @xmath1811 and if the set @xmath1652 from ( [ 1.7.5 ] ) coincides with the whole @xmath318 then @xmath1812 the corresponding to ( [ 1.7.3 ] ) , ( [ 1.7.4 ] ) , ( [ 1.7.8 ] ) and ( [ 1.7.9 ] ) assertions hold true also when @xmath145 and @xmath54 in these estimates are replaced by a disc @xmath622 with @xmath1813 and by @xmath653 , respectively ( ( see ( [ 1.7.21 ] ) and[1.7.22 ] ) , ( [ 1.7.34 ] ) , ( [ 1.7.36 ] ) and ( [ 1.7.37 ] ) below ) . observe that applying to ( [ 1.5.19 ] ) and ( [ 1.5.23 ] ) the arguments which were used in order to derive theorem [ thm1.2.3 ] from proposition [ prop1.5.2 ] and the latter from proposition [ prop1.3.4 ] and lemma [ lem1.4.1 ] we obtain that ( [ 1.2.16 ] ) and ( [ 1.2.17 ] ) can be written for any disc @xmath1814 in place of the whole @xmath145 , namely , for any @xmath203 with @xmath1815 , @xmath1816 , and @xmath33 small enough @xmath1817 and @xmath1818 which holds true in the same sense as ( [ 1.2.16])([1.2.17 ] ) and ( [ 1.7.10])([1.7.11 ] ) are uniform in @xmath601 as above . in order to prove ( i ) we observe , first , that the assumption ( [ 1.7.1 ] ) above together with lemma [ lem1.6.2](i ) and the compactness of @xmath318 considerations enable us to extend any @xmath1819 slightly so that it will exit some fixed neighborhood of @xmath317 with only slight increase in its @xmath285-functional . hence , from the beginning we assume that for each @xmath1438 there exists @xmath1820 such that for any @xmath1821 we can find @xmath277 , @xmath1797 and @xmath1798 $ ] satisfying @xmath1822 where @xmath1823dist@xmath1824 . it follows that for any @xmath1825 , and @xmath1826 , @xmath1827\big\}\\ & = \big\ { v\in d:\,\tau^{{\varepsilon}}_{\phi_{{\varepsilon}}^{kt/{{\varepsilon}}}v}(v)>t,\,\ , \forall\ , k=0,1, ... ,n-1\big\}\subset g^{{\varepsilon}}_{n,{{\delta}}}\nonumber\\ & \stackrel{\mbox{def}}{=}\big\ { v\in d:\,\phi_{{\varepsilon}}^{kt/{{\varepsilon}}}v \not\in\bigcup_{z\in v}a_{{{\delta}}}^{{{\varepsilon}},t}({{\varphi}}^z ) , \forall\ , k=0,1, ... ,n-1\big\}\nonumber \end{aligned}\ ] ] where for any subset @xmath1828 and @xmath1829 , @xmath1830 for @xmath1831 define @xmath1832 and @xmath1833 which are , clearly , compact sets satisfying @xmath1834 let @xmath861 be a maximal @xmath1835- separated set in @xmath1836 . then @xmath1837 and the left hand side of ( [ 1.7.13 ] ) is a disjoint union . this together with lemma [ lem1.3.6 ] give @xmath1838 by lemma [ lem1.3.2](ii ) , @xmath1839 clearly , for any @xmath1840 , @xmath1841 in view of ( [ 1.7.15 ] ) we can apply ( [ 1.7.10 ] ) which together with the choice of curves @xmath1842 yield that for any @xmath1019 and @xmath33 small enough , @xmath1843 where @xmath1844 . by lemma [ lem1.3.6 ] it follows that @xmath1845 for some @xmath254 . since @xmath1846 are disjoint for different @xmath1840 we derive from ( [ 1.7.14 ] ) , ( [ 1.7.16 ] ) and ( [ 1.7.18 ] ) that @xmath1847 where @xmath1848 . applying ( [ 1.7.19 ] ) for @xmath869 we obtain that @xmath1849 this together with ( [ 1.7.12 ] ) yield that for any @xmath1438 there exists @xmath1850 such that for all small @xmath342 , @xmath1851 observe that by ( [ 1.7.12 ] ) and ( [ 1.7.20 ] ) , @xmath1852 in the same way as at the end of the proof of proposition [ prop1.5.2](i ) we fix now an initial point @xmath1853 and choose discs @xmath601 to be small balls on the ( extended ) local unstable manifolds @xmath1854 which by means of the fubini theorem and compactness arguments enable us to extend ( [ 1.7.21 ] ) and ( [ 1.7.22 ] ) to the case when @xmath653 is replaced by @xmath54 and @xmath601 by @xmath145 yielding ( [ 1.7.3 ] ) and ( [ 1.7.4 ] ) since @xmath1192 and @xmath1029 in ( [ 1.7.22 ] ) can be chosen arbitrarily small as @xmath50 . next , we derive the assertion ( ii ) . let @xmath916 and @xmath446 be the integral part of @xmath1855 where @xmath277 will be chosen later . let , again , @xmath1826 and @xmath338 . then @xmath1856 let @xmath604 be the intersection of the @xmath361-limit set of the flow @xmath286 with @xmath321 . then @xmath604 is a compact set and by our assumption @xmath1857 . hence , @xmath1858 and if we set @xmath1859 then @xmath1860 . now suppose that @xmath1861 for some @xmath864 and @xmath1862 with @xmath338 and @xmath513 . then either there is @xmath1863 $ ] such that @xmath1864 for all @xmath1865 $ ] or there exist @xmath1866 such that @xmath1867 and @xmath1868 while @xmath1869 . set @xmath1870 and either there is @xmath1871 $ ] so that @xmath1872 for all @xmath1865 $ ] or @xmath1873 and @xmath1874 for some @xmath1875 . then for any @xmath864 , @xmath1876 where @xmath1844 and @xmath1877 . let @xmath1878 ( later both discs will be small balls on @xmath1879 ) assuming that @xmath625 is small and @xmath1154 is large so that @xmath1155 is still small . choose a maximal @xmath1880-separated set @xmath1881 in @xmath910 and let @xmath1882 then for @xmath33 small enough , @xmath1883 and for any @xmath1884 , @xmath1885 if @xmath1886 and @xmath1887 then by lemma [ lem1.3.2](iii ) , @xmath1888 is of order @xmath33 , and so for each @xmath292 if @xmath33 is small enough then @xmath1889 . for each @xmath1890 set @xmath1891 and suppose that for some @xmath292 there is @xmath1892 so that @xmath1893 then @xmath1894 , where @xmath1895 is the same as in theorem [ thm1.2.3 ] , and so @xmath1896 hence , @xmath1897 by lemma [ lem1.3.2](ii ) , @xmath1898 and so applying ( [ 1.7.11 ] ) to @xmath1899 we obtain from ( [ 1.7.27]) ( [ 1.7.29 ] ) that for any @xmath1438 and sufficiently small @xmath33 uniformly in discs @xmath1899 as above , @xmath1900 this together with lemma [ lem1.3.6 ] yield that for each @xmath1901 , @xmath1902 for some @xmath722 depending only on @xmath794 . combining this with ( [ 1.7.24])([1.7.26 ] ) and lemma [ lem1.3.6 ] we obtain that for any @xmath864 , @xmath1903 for some @xmath1904 depending only on @xmath794 . next , we will specify @xmath1905 in ( [ 1.7.27 ] ) choosing @xmath1906 . for each @xmath1821 we can write @xmath1907 where @xmath1908 and @xmath1909 for some @xmath1875 with @xmath1910 dist@xmath1911 and @xmath1912 and there is @xmath1871 $ ] so that @xmath1913 for all @xmath1865\}$ ] . by ( [ 1.7.5 ] ) and the lower semicontinuity of the functional @xmath1914 it follows that for any @xmath1583 we can choose @xmath292 small enough so that @xmath1915 since @xmath1916 is disjoint with the @xmath361-limit set of the flow @xmath286 and the latter is closed then if @xmath312 is sufficiently small @xmath1917 is also disjoint with this @xmath361-limit set and , in particular , it does not contain any forward semi - orbit of @xmath286 . hence we can apply lemma [ lem1.6.4 ] which in view of ( [ 1.2.13 ] ) implies that there exists @xmath988 such that for all small @xmath292 , @xmath1918 which is not less than @xmath1919 if we take @xmath1920 . now , ( [ 1.7.32 ] ) and ( [ 1.7.33 ] ) produce ( [ 1.7.27 ] ) with @xmath1921 , and so ( [ 1.7.30 ] ) follows with such @xmath1905 . this together with ( [ 1.7.23 ] ) yield that for any @xmath1438 we can choose sufficiently small @xmath1922 and then @xmath292 so that for all @xmath33 small enough @xmath1923 now assume that ( [ 1.7.7 ] ) holds true for some @xmath338 . recall , that @xmath218 implies that @xmath654 is a piece of an orbit of the flow @xmath286 . since no @xmath203 satisfying @xmath1924}\,\mbox{dist } ( { { \gamma}}_t,\partial v)\leq a(x)/2\ ] ] can be such piece of an orbit we conclude by the lower semicontinuity of @xmath228 that @xmath1925 whenever ( [ 1.7.35 ] ) holds true for some @xmath1926 independent of @xmath654 ( but depending on @xmath20 ) . hence , by ( [ 1.7.11 ] ) , @xmath1927 provided @xmath33 is small enough and ( [ 1.7.8 ] ) follows . observe also that any @xmath298 with @xmath1928 and @xmath1929 should contain a piece which either belongs to some @xmath1930 or @xmath1931 , as above , or to satify ( [ 1.7.35 ] ) . by ( [ 1.7.32 ] ) , ( [ 1.7.33 ] ) , and the above remarks it follows that @xmath1932 for such @xmath654 where @xmath1933 depends only on @xmath20 , and so @xmath1934 . finally , similarly to proposition [ prop1.5.2 ] we fix @xmath1935 , choose discs @xmath601 and @xmath910 to be small balls on the ( extended ) local unstable manifolds @xmath1936 and using the fubini theorem we extend ( [ 1.7.34 ] ) and ( [ 1.7.36 ] ) to the case when @xmath601 and @xmath653 are replaced by @xmath145 and @xmath54 , respectively , yielding ( [ 1.7.6 ] ) . if @xmath1937 then by ( [ 1.7.6 ] ) and ( [ 1.7.8 ] ) , @xmath1938 and , since @xmath1438 is arbitrary , ( [ 1.7.9 ] ) follows completing the proof of proposition [ prop1.7.1 ] . now we will derive theorem [ thm1.2.5 ] from proposition [ prop1.7.1 ] . assume , first , that @xmath353 . then by lemma [ 1.6.4 ] , @xmath1654 is finite and continuous in the whole @xmath317 . moreover , since @xmath302 is an @xmath285-attractor the conditions of lemma [ lem1.6.5 ] are satisfied with some @xmath1667 as @xmath426 the same for all points of @xmath317 which yields the conditions of proposition [ prop1.7.1](i ) with @xmath1939 for any @xmath253 . hence , ( [ 1.7.3 ] ) and ( [ 1.7.4 ] ) hold true with @xmath1940 . since @xmath302 is an @xmath285-attractor of the flow @xmath286 and its basin contains @xmath321 then the intersection of @xmath1941 with the @xmath361-limit set of @xmath286 is empty . by the definition of an @xmath285-attractor for any @xmath292 there exists an open set @xmath294 such that @xmath1739 whenever @xmath295 and @xmath296 . hence , by the triangle inequality for the function @xmath282 and lemma [ lem1.6.5 ] for any set @xmath1942 , @xmath1943 if @xmath1937 then by lemma [ lem1.6.5 ] the right hand side of ( [ 1.7.38 ] ) equals @xmath1944 . assuming that @xmath1665 we can apply proposition [ prop1.7.1](ii ) with such @xmath1919 yielding ( [ 1.7.6 ] ) , ( [ 1.7.8 ] ) and since @xmath292 is arbitrary ( [ 1.2.21 ] ) and ( [ 1.2.22 ] ) follow in this case . if @xmath1945 then ( [ 1.2.22 ] ) is trivial and by ( [ 1.7.38 ] ) , @xmath1946 for any @xmath1947 and @xmath1948 , and so we can apply proposition [ prop1.7.1](ii ) with any @xmath1919 which sais that the left hand side in ( [ 1.7.9 ] ) equals @xmath187 , and so ( [ 1.2.21 ] ) holds true in this case , as well . next , we establish ( [ 1.2.23 ] ) . for small @xmath1949 and large @xmath277 which will be specified later on set @xmath1950 , @xmath1951 and @xmath1952 . then @xmath1953 if @xmath380 is sufficiently small then @xmath1954 is still contained in the basin of @xmath302 with respect to the flow @xmath286 , and so we can choose @xmath1153 ( depending only on @xmath380 ) so that @xmath1955 then for some @xmath988 , @xmath1956 and so if @xmath1957 and @xmath1958 then dist@xmath1959 for any @xmath1960 . relying on ( [ 1.7.11 ] ) we obtain that for any @xmath1961 with @xmath1962 , @xmath1963 provided @xmath33 is small enough . next , the same arguments which yield ( [ 1.7.34 ] ) and ( [ 1.7.36 ] ) enable us to conclude that if @xmath1438 is small enough then for any @xmath1964 with @xmath1965 , @xmath1966 now let @xmath622 , @xmath893 , @xmath1967 and @xmath1968 with @xmath625 small and @xmath793 large so that @xmath1155 is still small . let @xmath1969 and @xmath1970 be maximal @xmath1971 and @xmath1972separated sets , respectively . then @xmath1973 and since the last union is contained in a small neighborhood of @xmath601 and @xmath1974 are disjoint for different @xmath1975 we obtain using lemma [ lem1.3.6 ] that @xmath1976 similarly , @xmath1977 and @xmath1978 since by lemma [ lem1.3.2](ii ) , @xmath1979 we can apply ( [ 1.7.39])([1.7.41 ] ) together with lemma [ lem1.3.6 ] ( similarly to the proof of ( [ 1.7.30 ] ) ) in order to conclude that for sufficiently small @xmath1192 and any much smaller @xmath33 , @xmath1980 choosing discs @xmath601 to be small balls on the ( extended ) local unstable manifolds @xmath1981 together with the fubini theorem we extend this estimate to @xmath1982 for some @xmath722 depending on @xmath380 but independent of @xmath446 and @xmath33 . finally , ( [ 1.2.22 ] ) and ( [ 1.7.42 ] ) together with the chebyshev inequality yield that for @xmath1983 $ ] , each @xmath338 , a small @xmath1438 and any much smaller @xmath342 , @xmath1984 since @xmath1985 and we can choose @xmath1192 to be arbitrarily small , ( [ 1.7.43 ] ) yields ( [ 1.2.23 ] ) . in order to complete the proof of theorem [ thm1.2.5 ] we have to derive ( [ 1.2.24 ] ) . if @xmath1986 then there is nothing to prove , so we assume that @xmath335 is a proper subset of @xmath318 and in this case , clearly , @xmath1665 . since @xmath1987 is compact and disjoint with @xmath335 which is also compact then by the lower semicontinuity of @xmath1659 established in lemma [ lem1.6.5](iii ) it follows that @xmath1988 for some @xmath1438 and all @xmath348 . then by ( [ 1.7.38 ] ) , @xmath1989 for any @xmath1990 and @xmath1991 . hence , applying proposition [ prop1.7.1 ] we obtain that @xmath1992 and @xmath1993 for some @xmath1019 and all @xmath33 small enough yielding ( [ 1.2.24 ] ) and completing the proof of theorem [ thm1.2.5 ] . [ prop2.5.1 ] let @xmath317 be a connected open set with a piecewise smooth boundary @xmath318 such that @xmath1792 . assume that for each @xmath323 there exist @xmath1793 and a probability measure @xmath45 with @xmath2825 so that @xmath2826,\ ] ] i.e. @xmath2827 and it points out into the exterior of @xmath321 . \(i ) suppose that for some @xmath1796 and any @xmath1658 there exists @xmath1797 such that for some @xmath1798 $ ] , @xmath2828 then for any @xmath338 uniformly in @xmath432 , @xmath2829 and for any @xmath340 there exists @xmath341 such that uniformly in @xmath432 for all small @xmath342 , @xmath2830 \(ii ) assume that there exists an open set @xmath1762 such that @xmath317 contains its closure @xmath1802 and the intersection of @xmath1803 with the @xmath361-limit set of the flow @xmath286 is empty . let @xmath1652 be a compact subset of @xmath318 such that @xmath2831 for some @xmath1805 . then for some @xmath277 and any @xmath1438 there exists @xmath1806 such that uniformly in @xmath432 for each @xmath338 and any small @xmath342 , @xmath2832 suppose that for some @xmath338 , @xmath2833 then @xmath1809 and for each @xmath277 there exists @xmath1810 such that uniformly in @xmath432 for all small @xmath342 , @xmath2834 and if the set @xmath1652 from ( [ 2.5.5 ] ) coincides with the whole @xmath318 then for all @xmath338 uniformly in @xmath432 , @xmath2835 in order to prove ( i ) we observe , first , that the assumption ( [ 2.5.1 ] ) above together with lemma [ lem2.4.2](i ) and the compactness of @xmath318 considerations enable us to extend any @xmath1819 slightly so that it will exit some fixed neighborhood of @xmath317 with only slight increase in its @xmath285-functional . hence , from the beginning we assume that for each @xmath1438 there exists @xmath1820 such that for any @xmath1821 we can find @xmath277 , @xmath1797 and @xmath1798 $ ] satisfying @xmath1822 where @xmath1823dist@xmath1824 . employing the markov property we obtain that for any @xmath2836 , @xmath2837\big\}\\ & = p\big\{\tau^{{\varepsilon}}_{z^{{\varepsilon}}_{x , y}(kt)}(v)>t,\,\ , \forall\ , k=0,1, ... ,n-1\big\}\leq\big(\sup_{w\in v\times{{\bf m } } } p\big\{\tau^{{\varepsilon}}_{w}(v)>t\big\}\big)^n . \nonumber \end{aligned}\ ] ] from ( [ 2.2.10 ] ) and ( [ 2.5.2 ] ) it follows that @xmath2838 by ( [ 2.5.10 ] ) and ( [ 2.5.11 ] ) , @xmath2839 and @xmath2840 yielding ( [ 2.5.3 ] ) and ( [ 2.5.4 ] ) since @xmath1192 and @xmath1029 in ( [ 2.5.13 ] ) can be chosen arbitrarily small as @xmath2841 . next , we derive the assertion ( ii ) . let @xmath916 and @xmath446 be the integral part of @xmath1855 where @xmath277 will be chosen later . let , again , @xmath2842 with @xmath338 and @xmath432 . then @xmath2843 let @xmath604 be the intersection of the @xmath361-limit set of the flow @xmath286 with @xmath321 . then @xmath604 is a compact set and by our assumption @xmath1857 . hence , @xmath1858 and if we set @xmath1859 then @xmath1860 . now suppose that @xmath2844 for some @xmath864 and @xmath2845 with @xmath338 and @xmath2846 . then either there is @xmath1863 $ ] such that @xmath2847 for all @xmath1865 $ ] or there exist @xmath1866 such that @xmath1867 and @xmath2848 while @xmath2849 . set @xmath1870 and either there is @xmath1871 $ ] so that @xmath1872 for all @xmath1865 $ ] or @xmath1873 and @xmath1874 for some @xmath1875 . then for any @xmath864 , @xmath2850 for each @xmath1890 set @xmath1891 and suppose that for some @xmath292 there is @xmath1892 so that @xmath2851 then @xmath1894 , where @xmath1895 is the same as in theorem [ thm2.2.2 ] , and so @xmath2852 from ( [ 2.2.10 ] ) and ( [ 2.5.15])([2.5.17 ] ) we obtain that for any @xmath1438 and all sufficiently small @xmath33 , @xmath2853 for some @xmath1904 . next , we will specify @xmath1905 in ( [ 2.5.16 ] ) choosing @xmath1906 . for each @xmath1821 we can write @xmath2854 where @xmath1908 and @xmath1909 for some @xmath1875 with @xmath1910 dist@xmath1911 and @xmath1912 and there is @xmath1871 $ ] so that @xmath1913 for all @xmath1865\}$ ] . by ( [ 2.5.5 ] ) and the lower semicontinuity of the functional @xmath1914 it follows that for any @xmath1583 we can choose @xmath292 small enough so that @xmath2855 since @xmath1916 is disjoint with the @xmath361-limit set of the flow @xmath286 and the latter is closed then if @xmath312 is sufficiently small @xmath1917 is also disjoint with this @xmath361-limit set and , in particular , it does not contain any forward semi - orbit of @xmath286 . hence we can apply lemma [ lem2.4.4 ] which in view of ( [ 2.2.9 ] ) implies that there exists @xmath988 such that for all small @xmath292 , @xmath2856 which is not less than @xmath1919 if we take @xmath1920 . now , ( [ 2.5.20 ] ) and ( [ 2.5.21 ] ) produce ( [ 2.5.16 ] ) with @xmath1921 , and so ( [ 2.5.18 ] ) follows with such @xmath1905 . this together with ( [ 2.5.14 ] ) yield that for any @xmath1438 we can choose sufficiently small @xmath1922 and then @xmath292 so that for all @xmath33 small enough @xmath2857 and ( [ 2.5.6 ] ) follows . now assume that ( [ 2.5.7 ] ) holds true for some @xmath338 . recall , that @xmath218 implies that @xmath654 is a piece of an orbit of the flow @xmath286 . since no @xmath203 satisfying @xmath2858}\,\mbox{dist } ( { { \gamma}}_t,\partial v)\leq a(x)/2\ ] ] can be such piece of an orbit we conclude by the lower semicontinuity of @xmath228 that @xmath1925 whenever ( [ 2.5.23 ] ) holds true for some @xmath1926 independent of @xmath654 ( but depending on @xmath20 ) . hence , by ( [ 2.2.11 ] ) , @xmath2859 provided @xmath33 is small enough and ( [ 2.5.8 ] ) follows . observe also that any @xmath298 with @xmath1928 and @xmath1929 should contain a piece which either belongs to some @xmath1930 or to @xmath1931 , as above , or to satify ( [ 2.5.23 ] ) . by ( [ 2.5.20 ] ) , ( [ 2.5.21 ] ) , and the above remarks it follows that @xmath1932 for such @xmath654 where @xmath1933 depends only on @xmath20 , and so @xmath1934 . if @xmath1937 then by ( [ 2.5.6 ] ) and ( [ 2.5.8 ] ) , @xmath2860 and , since @xmath1438 is arbitrary , ( [ 2.5.9 ] ) follows completing the proof of proposition [ prop2.5.1 ] . now we will derive theorem [ thm2.2.5 ] from proposition [ prop2.5.1 ] . assume , first , that @xmath353 . then by lemma [ lem2.4.5 ] , @xmath1654 is finite in the whole @xmath317 . moreover , since @xmath302 is an @xmath285-attractor the conditions of lemma [ lem2.4.5 ] are satisfied with some @xmath1667 as @xmath426 the same for all points of @xmath317 which yields the conditions of proposition [ prop2.5.1](i ) with @xmath1939 for any @xmath253 . hence , ( [ 2.5.3 ] ) and ( [ 2.5.4 ] ) hold true with @xmath1940 . since @xmath302 is an @xmath285-attractor of the flow @xmath286 and its basin contains @xmath321 then the intersection of @xmath1941 with the @xmath361-limit set of @xmath286 is empty . by the definition of an @xmath285-attractor for any @xmath1583 there exists an open set @xmath2861 such that @xmath2862 whenever @xmath295 and @xmath1947 . hence , by the triangle inequality for the function @xmath282 and lemma [ lem2.4.5 ] for any set @xmath1942 , @xmath2863 if @xmath1937 then by lemma [ lem2.4.5 ] the right hand side of ( [ 2.5.26 ] ) equals @xmath2864 . assuming that @xmath1665 we can apply proposition [ prop2.5.1](ii ) with such @xmath1919 yielding ( [ 2.5.6 ] ) , ( [ 2.5.8 ] ) and since @xmath1583 is arbitrary ( [ 2.2.25 ] ) and ( [ 2.2.26 ] ) follow in this case . if @xmath1945 then ( [ 2.2.26 ] ) is trivial and by ( [ 2.5.26 ] ) , @xmath1946 for any @xmath1947 and @xmath1948 , and so we can apply proposition [ prop2.5.1](ii ) with any @xmath1919 which sais that the left hand side in ( [ 2.5.9 ] ) equals @xmath187 , and so ( [ 2.2.25 ] ) holds true in this case , as well . next , we establish ( [ 2.2.27 ] ) . for small @xmath1949 and large @xmath277 which will be specified later on set @xmath2865 and define the event @xmath2866 then @xmath2867 if @xmath380 is sufficiently small then @xmath1954 is still contained in the basin of @xmath302 with respect to the flow @xmath286 , and so we can choose @xmath1153 ( depending only on @xmath380 ) so that @xmath1955 then for some @xmath988 , @xmath1956 and so if @xmath1957 and @xmath1958 then dist@xmath1959 for any @xmath1960 . relying on ( [ 2.2.11 ] ) and the markov property we obtain that for any @xmath2868 with @xmath1821 , @xmath2869 provided @xmath33 is small enough . next , the same arguments which yield ( [ 2.5.22 ] ) and ( [ 2.5.24 ] ) together with the markov property enable us to conclude that if @xmath1438 is small enough then for any @xmath2868 with @xmath1821 , @xmath2870 applying ( [ 2.5.27])([2.5.29 ] ) we conclude that for sufficiently small @xmath1192 and any much smaller @xmath33 , @xmath2871 finally , ( [ 2.2.26 ] ) and ( [ 2.5.30 ] ) together with the chebyshev inequality yield that for @xmath1983 $ ] , each @xmath338 , a small @xmath1438 and any much smaller @xmath342 , @xmath2872 since @xmath1985 and we can choose @xmath1192 to be arbitrarily small , ( [ 2.5.31 ] ) yields ( [ 2.2.27 ] ) . in order to complete the proof of theorem [ thm2.2.5 ] we have to derive ( [ 2.2.28 ] ) . if @xmath1986 then there is nothing to prove , so we assume that @xmath335 is a proper subset of @xmath318 and in this case , clearly , @xmath1665 . since @xmath1987 is compact and disjoint with @xmath335 which is also compact then by the lower semicontinuity of @xmath1659 established in lemma [ lem2.4.5](iii ) it follows that @xmath1988 for some @xmath1438 and all @xmath348 . then by ( [ 2.5.26 ] ) , @xmath1989 for any @xmath1990 and @xmath1991 hence , applying proposition [ prop2.5.1 ] we obtain that @xmath2873 and @xmath2874 for some @xmath1019 and all @xmath33 small enough yielding ( [ 2.2.28 ] ) and completing the proof of theorem [ thm2.2.5 ] . in this section we will prove theorem [ thm1.2.7 ] relying , again , on proposition [ prop1.7.1 ] together with `` markov property type '' arguments and at the end of the proof we will apply even some rough `` strong markov property type '' arguments in order to deal with subsequent transitions between basins of attractors . in view of ( [ 1.2.27 ] ) and lemma [ lem1.6.2]i any curve @xmath298 starting at @xmath1994 and ending at @xmath1995 can be extended into each @xmath1996 with arbitrarily small increase in its @xmath285-functional . hence , @xmath1997 where @xmath1998 . let @xmath1999 be an open ball of radius at least @xmath2000 centered at the origin of @xmath16 . by assumption [ ass1.2.6 ] the slow motion @xmath420 can not exit @xmath1999 provided @xmath2001 and @xmath261 . furthermore , it is clear that @xmath1999 contains the @xmath361-limit set of the averaged flow @xmath286 . assumption [ ass1.2.6 ] enables us to deal only with restricted basins @xmath2002 and though the boundaries @xmath2003 of @xmath2004 may include now parts of the boundary @xmath2005 of @xmath1999 it makes no difference since @xmath88 can not reach @xmath2005 if it starts in @xmath1999 . set @xmath2006 where @xmath253 is small enough . we claim that in view of ( [ 1.2.27 ] ) each @xmath383 satisfies conditions of proposition [ prop1.7.1](i ) for any @xmath1438 with @xmath2007 and some @xmath2008 depending on @xmath1192 . indeed , set @xmath2009 in view of ( [ 1.2.9 ] ) and ( [ 1.2.27 ] ) there exists @xmath907 such that if @xmath312 is small enough and @xmath2010 we can construct a curve @xmath2011 with @xmath2012 , @xmath2013 for some @xmath2014 $ ] and @xmath2015 where @xmath2016 . since @xmath2017 is the basin of @xmath419 there exists @xmath2018 such that @xmath2019 and extending @xmath2020 by the piece of the orbit of @xmath286 we obtain a curve @xmath2021 starting at @xmath611 , entering @xmath2022 and satisfying @xmath2023 . hence , for @xmath2010 the condition ( [ 1.7.2 ] ) holds true with @xmath2024 and @xmath2025 . since the @xmath361-limit set of the flow @xmath286 is contained in @xmath2026 it follows from assumption [ ass1.2.6 ] and compactness considerations that there exists @xmath2027 such that for any @xmath2028 we can find @xmath2029 $ ] with @xmath2030 . if @xmath2031 then we take @xmath2032 $ ] to satisfy ( [ 1.7.2 ] ) for @xmath2024 and @xmath2033 . if @xmath2034 then we extend the curve @xmath2035 $ ] as in the above argument which yields a curve @xmath2036 starting at @xmath611 , ending in some @xmath2037 and having its @xmath285-functional not exceeding @xmath2038 . finally , in the same way as in the proof of theorem [ thm1.2.5 ] for any @xmath1438 there exists @xmath2039 such that whenever @xmath2040 we can construct @xmath2041 such that ( [ 1.7.2 ] ) holds true with @xmath2042 and @xmath2043 and , moreover , dist@xmath2044 for some @xmath2045 and @xmath2046 with @xmath2047 . then in the same way as above we can extend @xmath2020 to some @xmath2048 so that @xmath2049 for some @xmath1373 as above , @xmath2050 and @xmath2051 which gives ( [ 1.7.2 ] ) for all @xmath2052 with @xmath2053 provided @xmath312 is small enough . hence , proposition [ prop1.7.1](i ) yields the estimates ( [ 1.7.3 ] ) and ( [ 1.7.4 ] ) for @xmath2054 in place of @xmath2055 with @xmath2056 . in order to obtain the corresponding bounds in the other direction observe that in view of ( [ 1.2.27 ] ) , @xmath2057 since @xmath2058 is contained in the basin of @xmath382 we can apply to @xmath2059 the same estimates as in theorem [ thm1.2.5 ] which together with ( [ 1.8.2 ] ) and the fact that the exit time of @xmath88 from @xmath2059 is smaller than its exit time from @xmath383 provide the remaining bounds yielding ( [ 1.2.28 ] ) and ( [ 1.2.29 ] ) . next , we derive ( [ 1.2.30 ] ) similarly to ( [ 1.2.23 ] ) but taking into account that @xmath373 may contain parts of the @xmath361-limit set of the flow @xmath286 which allows the slow motion @xmath88 to stay long time near these boundaries . still , set @xmath2060 using the same arguments as above we conclude that for any @xmath292 there exists @xmath2018 such that whenever @xmath2061 we can construct @xmath1797 with @xmath2062 and @xmath2063 . this together with ( [ 1.7.21 ] ) and assumption [ ass1.2.6 ] yield that for any disc @xmath2064 , @xmath2065 for some @xmath2066 and all small @xmath33 . set @xmath2067 @xmath2068 and @xmath2069 where @xmath312 is much smaller than @xmath1192 . then proceeding similarly to the proof of ( [ 1.2.23 ] ) as in ( [ 1.7.40])([1.7.43 ] ) above we arrive at ( [ 1.2.30 ] ) . next , we obtain ( [ 1.2.31 ] ) relying on additional assumptions specified in the statement of theorem [ thm1.2.7 ] . let @xmath2070 be the same as above and @xmath2071 . since @xmath382 is an @xmath285-attractor it follows from lemma [ lem1.6.5](i ) that @xmath303 and @xmath2072 coincide with the same function @xmath2073 and the same ( in general , may be empty ) set @xmath2074 , respectively , for all @xmath2075 . by lemma [ lem1.6.2](i ) , our assumption that @xmath19 is complete on @xmath395 implies that @xmath2076 is continuous in a neighborhood of @xmath395 , and so @xmath2074 is a nonempty compact set . since we assume that @xmath2077 is the unique index @xmath1373 for which @xmath2078 then by ( [ 1.2.27 ] ) , @xmath2079 observe that if @xmath2080 is an @xmath285-compact then either @xmath2081 or @xmath2082 . denote by @xmath2083 the @xmath361-limit set of the averaged flow @xmath286 . since @xmath2084 consists of a finite number of @xmath285-compacts it follows that @xmath2085 by the continuity of @xmath2076 in @xmath2086 there exists @xmath988 such that @xmath2087 these considerations enable us to construct a connected open set @xmath1762 with a piecewise smooth boundary @xmath2088 such that @xmath2089 and for @xmath2090 and some @xmath2091 , @xmath2092 provided @xmath2093 . the idea of this construction is that if @xmath2094 then the slow motion should exit @xmath1762 through the part @xmath1652 of its boundary . somewhat similarly to the proof of proposition [ prop1.7.1](ii ) we will show that for `` most '' initial conditions @xmath18 this can only occur after the time @xmath2095 and , on the other hand , we conclude from ( [ 1.2.29 ] ) that for `` most '' initial conditions @xmath18 the exit time @xmath2054 does not exceed @xmath2096 . let @xmath2097 be a sufficiently small open neighborhood of @xmath2074 so that , in particular , @xmath2098 and set @xmath2099 for each disc @xmath2100 we can write @xmath2101 where @xmath2102 for some small @xmath1438 , @xmath2103 $ ] , @xmath2104 \big\}$ ] for a sufficiently small @xmath292 , @xmath2105 , @xmath2106\big\}$ ] , @xmath2107 , and @xmath2108 . observe that @xmath2109 satisfies conditions of proposition [ prop1.7.1](i ) with arbitrarily small @xmath2110 , so similarly to ( [ 1.7.21 ] ) ( and taking into account lemma [ lem1.3.6 ] ) we can estimate @xmath2111 similarly to the proof of proposition [ prop1.7.1](ii ) we obtain also that @xmath2112 where we , first , choose @xmath312 small and then @xmath1153 large enough . next , we estimate @xmath2113 for @xmath2114 by the following markov property type argument . let @xmath2115 and choose a maximal @xmath2116-separated set @xmath771 in @xmath601 . let @xmath2117 , @xmath2118 and @xmath2119 . assume that dist@xmath2120 for any @xmath2121 . by lemma [ lem1.6.2](i ) , @xmath303 is continuous in @xmath611 when @xmath611 belong to a sufficiently small neighborhood of @xmath395 which together with the definition of @xmath285-compacts yields that @xmath2122 for any @xmath2123 , provided @xmath312 is small enough . observe that @xmath2124 for any @xmath2125 , provided @xmath33 is sufficiently small . these together with the arguments similar to the proof of proposition [ prop1.7.1](ii ) yield the estimate @xmath2126 for all @xmath33 small enough . since @xmath2127 are disjoint for different @xmath2128 we obtain by lemma [ lem1.3.6 ] , @xmath2129 where @xmath2130 . in a similar way we obtain that for each disc @xmath2131 , @xmath2132 provided @xmath33 is small enough . by ( [ 1.8.7])([1.8.9 ] ) together with lemma [ lem1.3.6 ] , @xmath2133 provided @xmath2114 and @xmath33 is small enough . summing in @xmath2134 and @xmath446 we obtain from ( [ 1.8.4])([1.8.6 ] ) and ( [ 1.8.10 ] ) that for a small @xmath1192 and all sufficiently small @xmath33 , @xmath2135 taking discs @xmath601 to be small balls on the ( extended ) local unstable manifolds @xmath2136 and using the fubini theorem as before we obtain ( [ 1.8.11 ] ) for @xmath54 and @xmath145 in place of @xmath653 and @xmath601 , respectively . on the other hand , employing proposition [ prop1.7.1](i ) we derive that @xmath2137 for some @xmath1019 and all @xmath33 small enough which together with ( [ 1.8.11 ] ) considered for @xmath54 and @xmath145 in place of @xmath653 and @xmath601 yield ( [ 1.2.31 ] ) . in order to complete the proof of theorem [ thm1.2.7 ] it remains to derive ( [ 1.2.32 ] ) and ( [ 1.2.33 ] ) . both statements hold true for @xmath1213 in view of ( [ 1.2.29 ] ) and ( [ 1.2.31 ] ) but , in fact , we will use them as the induction base with @xmath653 and @xmath601 in place of @xmath54 and @xmath145 where @xmath622 and @xmath2138 which holds true in view of ( [ 1.8.4])([1.8.11 ] ) together with the corresponding form of proposition [ prop1.7.1 ] . for such @xmath601 and @xmath409 set @xmath2139 and @xmath2140 as the induction hypotesis we assume that for any @xmath340 there exist @xmath2141 and @xmath1019 such that for all small @xmath33 , @xmath2142 set @xmath2143 where @xmath604 is the same as in ( [ 1.2.15 ] ) so that if @xmath2144 then @xmath2145.\ ] ] choose also @xmath2146 so that for any @xmath2147 , @xmath2148 let @xmath2149 be a maximal @xmath2150-separated set where @xmath2151 as before . set @xmath2152 then for @xmath33 small enough , @xmath2153 we claim that there exists @xmath1438 such that if @xmath2154 then for all small @xmath33 , @xmath2155 indeed , let @xmath2156\,\,\mbox{and}\,\,{{\gamma}}_t\not\in u_{{{\delta}}/2 } ( { { \mathcal o}}_{{{\iota}}_n(i)})\,\,\mbox{for all}\,\ , t\in[0,na]\big\}$ ] . then by ( [ 1.8.14 ] ) and the lower semicontinuiti of the functional @xmath2157 we obtain that @xmath2158 since by lemma [ lem1.3.2](ii ) and ( iii ) for any @xmath2159 the distance @xmath2160 has the order of @xmath33 we conclude from ( [ 1.8.13 ] ) and ( [ 1.8.16 ] ) that for each @xmath2161 , @xmath2162 hence , by ( [ 1.7.11 ] ) and lemma [ lem1.3.6 ] it follows that for any @xmath2163 and all @xmath33 small enough , @xmath2164 and since @xmath2165 are disjoint for different @xmath2166 we apply lemma [ 1.3.6 ] once more and obtain ( [ 1.8.15 ] ) . set @xmath2167 . applying ( [ 1.8.12 ] ) with @xmath1213 to each @xmath2168 and using lemma [ lem1.3.6 ] we derive also that @xmath2169 for some @xmath1438 and all small @xmath33 . by ( [ 1.8.17 ] ) we can write @xmath2170 observe that for any finite measure @xmath45 , measurable sets @xmath2171 and integers @xmath2172 , @xmath2173 which follows applying @xmath2174 to @xmath2175 and @xmath2176 for @xmath2177 . applying ( [ 1.8.19 ] ) for @xmath2178 and @xmath2179 it follows from ( [ 1.8.15 ] ) that @xmath2180 i.e. for any @xmath2181 , @xmath2182 this together with ( [ 1.8.12 ] ) and ( [ 1.8.18 ] ) complete the induction step and proves ( [ 1.2.32 ] ) and ( [ 1.2.33 ] ) for @xmath601 and @xmath653 in place of @xmath145 and @xmath54 . finally , as before we complete the proof of theorem [ thm1.2.7 ] by choosing discs @xmath601 to be small balls on the ( extended ) local unstable manifolds @xmath2183 which together with the fubini theorem enables us to extend the estimates to @xmath145 and @xmath54 as required in ( [ 1.2.32 ] ) and ( [ 1.2.33 ] ) . in this section we will prove theorem [ thm2.2.7 ] relying , again , on proposition [ prop2.5.1 ] together with markov and strong markov property of the markov process @xmath2875 . in view of ( [ 2.2.31 ] ) and lemma [ lem2.4.2](i ) any curve @xmath298 starting at @xmath1994 and ending at @xmath1995 can be extended into each @xmath1996 with arbitrarily small increase in its @xmath285-functional . hence , @xmath2876 where @xmath1998 . let @xmath1999 be an open ball of radius at least @xmath2000 centered at the origin of @xmath16 . by assumption [ ass2.2.6 ] the slow motion @xmath420 can not exit @xmath1999 provided @xmath2001 and @xmath261 . furthermore , it is clear that @xmath1999 contains the @xmath361-limit set of the averaged flow @xmath286 . assumption [ ass2.2.6 ] enables us to deal only with restricted basins @xmath2002 and though the boundaries @xmath2003 of @xmath2004 may include now parts of the boundary @xmath2005 of @xmath1999 it makes no difference since @xmath88 can not reach @xmath2005 if it starts in @xmath1999 . set @xmath2006 where @xmath253 is small enough . we claim that in view of ( [ 2.2.31 ] ) each @xmath383 satisfies conditions of proposition [ prop2.5.1](i ) for any @xmath1438 with @xmath2007 and some @xmath2008 depending on @xmath1192 . indeed , set @xmath2009 in view of ( [ 2.2.31 ] ) and lemma [ lem2.4.1 ] there exists @xmath907 such that if @xmath312 is small enough and @xmath2010 we can construct a curve @xmath2011 with @xmath2877 , @xmath2878 for some @xmath2014 $ ] and @xmath2015 . since @xmath2017 is the basin of @xmath419 there exists @xmath2018 such that @xmath2019 and extending @xmath2020 by the piece of the orbit of @xmath286 we obtain a curve @xmath2021 starting at @xmath611 , entering @xmath2022 and satisfying @xmath2879 . hence , for @xmath2010 the condition ( [ 2.5.2 ] ) holds true with @xmath2024 and @xmath2880 . since the @xmath361-limit set of the flow @xmath286 is contained in @xmath2026 it follows from assumption [ ass2.2.6 ] and compactness considerations that there exists @xmath2027 such that for any @xmath2028 we can find @xmath2029 $ ] with @xmath2030 . if @xmath2031 then we take @xmath2032 $ ] to satisfy ( [ 2.5.2 ] ) for @xmath2024 and @xmath2033 . if @xmath2034 then we extend the curve @xmath2035 $ ] as in the above argument which yields a curve @xmath2036 starting at @xmath611 , ending in some @xmath2037 and having its @xmath285-functional not exceeding @xmath2881 . finally , in the same way as in the proof of theorem [ thm2.2.5 ] for any @xmath1438 there exists @xmath2039 such that whenever @xmath2040 we can construct @xmath2041 such that ( [ 2.5.2 ] ) holds true with @xmath2042 and @xmath2043 and , moreover , dist@xmath2044 for some @xmath2045 and @xmath2046 with @xmath2047 . then in the same way as above we can extend @xmath2020 to some @xmath2048 so that @xmath2049 for some @xmath1373 as above , @xmath2050 and @xmath2882 which gives ( [ 2.5.2 ] ) for all @xmath2052 with @xmath2053 provided @xmath312 is small enough . hence , proposition [ prop2.5.1](i ) yields the estimates ( [ 2.5.3 ] ) and ( [ 2.5.4 ] ) for @xmath2054 in place of @xmath2055 with @xmath2056 . in order to obtain the corresponding bounds in the other direction observe that in view of ( [ 2.2.31 ] ) , @xmath2883 since @xmath2058 is contained in the basin of @xmath382 we can apply to @xmath2059 the same estimates as in theorem [ thm2.2.5 ] which together with ( [ 2.6.2 ] ) and the fact that the exit time of @xmath88 from @xmath2059 is smaller than its exit time from @xmath383 provide the remaining bounds yielding ( [ 2.2.32 ] ) and ( [ 2.2.33 ] ) . next , we derive ( [ 2.2.34 ] ) similarly to ( [ 2.2.27 ] ) but taking into account that @xmath373 may contain parts of the @xmath361-limit set of the flow @xmath286 which allows the slow motion @xmath88 to stay long time near these boundaries . still , set @xmath2060 using the same arguments as above we conclude that for any @xmath292 there exists @xmath2018 such that whenever @xmath2061 we can construct @xmath1797 with @xmath2062 and @xmath2063 . this together with ( [ 2.5.12 ] ) and assumption [ ass2.2.6 ] yield that @xmath2884 for some @xmath2066 and all small @xmath33 . set @xmath2885 @xmath2886 and @xmath2069 where @xmath312 is much smaller than @xmath1192 . then proceeding similarly to the proof of ( [ 2.2.27 ] ) as in ( [ 2.5.28])([2.5.31 ] ) above we arrive at ( [ 2.2.34 ] ) . next , we obtain ( [ 2.2.35 ] ) relying on additional assumptions specified in the statement of theorem [ thm2.2.7 ] . let @xmath2070 be the same as above and @xmath2071 . since @xmath382 is an @xmath285-attractor it follows from lemma [ lem2.4.5](i ) that @xmath303 and @xmath2072 coincide with the same function @xmath2073 and the same ( in general , may be empty ) set @xmath2074 , respectively , for all @xmath2075 . by lemma [ lem2.4.2](i ) , our assumption that @xmath19 is complete on @xmath395 implies that @xmath2076 is continuous in a neighborhood of @xmath395 , and so @xmath2074 is a nonempty compact set . since we assume that @xmath2077 is the unique index @xmath1373 for which @xmath2078 then by ( [ 2.2.31 ] ) , @xmath2079 observe that if @xmath2080 is an @xmath285-compact then either @xmath2081 or @xmath2082 . denote by @xmath2083 the @xmath361-limit set of the averaged flow @xmath286 . since @xmath2084 consists of a finite number of @xmath285-compacts it follows that @xmath2085 by the continuity of @xmath2076 in @xmath2086 there exists @xmath988 such that @xmath2087 these considerations enable us to construct a connected open set @xmath1762 with a piecewise smooth boundary @xmath2088 such that @xmath2089 and for @xmath2090 and some @xmath2091 , @xmath2887 provided @xmath2093 . the idea of this construction is that if @xmath2094 then the slow motion should exit @xmath1762 through the part @xmath1652 of its boundary . somewhat similarly to the proof of proposition [ prop2.5.1](ii ) we will show that `` most likely '' this can only occur after the time @xmath2095 and , on the other hand , we conclude from ( [ 2.2.33 ] ) that except for small probability the exit time @xmath2054 does not exceed @xmath2096 . let @xmath2097 be a sufficiently small open neighborhood of @xmath2074 so that , in particular , @xmath2098 and set @xmath2099 then @xmath2888 where @xmath2102 for some small @xmath1438 , @xmath2103 $ ] , @xmath2889 \big\}$ ] for a sufficiently small @xmath292 , @xmath2890 , @xmath2891\big\}$ ] , @xmath2892 , and @xmath2893 . observe that @xmath2109 satisfies conditions of proposition [ prop2.5.1](i ) with arbitrarily small @xmath2110 , so similarly to ( [ 2.5.12 ] ) we can estimate @xmath2894 similarly to the proof of proposition [ prop2.5.1](ii ) we obtain also that @xmath2895 where we , first , choose @xmath312 small and then @xmath1153 large enough . next , relying on the markov property and the arguments similar to the proof of proposition [ prop2.5.1](ii ) we estimate @xmath2896 provided @xmath2114 and @xmath33 is small enough . summing in @xmath2134 and @xmath446 we obtain from ( [ 2.6.4])([2.6.7 ] ) that for a small @xmath1192 and all sufficiently small @xmath33 , @xmath2897 employing proposition [ prop2.5.1](i ) we derive that @xmath2898 for some @xmath1019 and all @xmath33 small enough which together with ( [ 2.6.8 ] ) yield ( [ 2.2.35 ] ) . in order to complete the proof of theorem [ thm2.2.7 ] it remains to derive ( [ 2.2.36 ] ) and ( [ 2.2.37 ] ) . both statements hold true for @xmath1213 in view of ( [ 2.2.33 ] ) and ( [ 2.2.35 ] ) and we proceed by induction . set @xmath2899 and @xmath2900 as the induction hypotesis we assume that for any @xmath340 there exist @xmath2141 and @xmath1019 such that for all small @xmath33 , @xmath2901 by ( [ 2.2.35 ] ) and the strong markov property @xmath2902 which implies ( [ 2.2.37 ] ) . similarly , by ( [ 2.2.33 ] ) and the strong markov property @xmath2903 proving ( [ 2.2.36 ] ) and completing the proof of theorem [ thm2.2.7 ] . finally , we prove theorem [ thm2.2.8 ] employing the arguments similar to 2 and 3 in ch . 6 of @xcite . namely , in order to obtain the upper bound in ( [ 2.2.38 ] ) observe that for any @xmath2904 there are @xmath2905 such that if @xmath2906 and a curve @xmath298 satisfies @xmath2907 and dist@xmath2908 then @xmath2909 . using lemma [ lem2.4.4 ] and the upper bound of large deviations ( [ 2.2.11 ] ) we can choose @xmath2910 such that for all small @xmath33 and any @xmath2911 , @xmath2912 any path of @xmath88 starting at a point of @xmath2913 and reaching @xmath2914 at time @xmath2915 either spends the time @xmath2916 without touching the set @xmath2917 or arrives at @xmath2918 during the time @xmath2916 . in the latter case @xmath2919 and by ( [ 2.2.11 ] ) and ( [ 2.6.12 ] ) for any @xmath1345 with @xmath2920 , all @xmath33 small enough and @xmath2046 , @xmath2921 for some @xmath2922 independent of @xmath33 . any path of @xmath2923 starting at @xmath2924 and reaching @xmath2914 at the time @xmath2915 must first hit at time @xmath2925 the set @xmath2913 , and so ( [ 2.6.13 ] ) together with the markov property yields the upper bound in ( [ 2.2.38 ] ) . in order to derive the lower bound in ( [ 2.2.38 ] ) observe that using the definition of @xmath285-attractors and lemma [ lem2.4.3 ] ( similarly to the proof of lemma [ lem1.6.5](ii ) in part [ part1 ] and see also 2 in ch . 6 of @xcite ) we conclude that for any @xmath2904 there exists @xmath2926 such that if @xmath2606 then for any @xmath2927 there exists a curve @xmath298 such that @xmath2928 for @xmath2929 $ ] , @xmath2930 for @xmath2931 , @xmath2932 and , finally , @xmath2933 . then by ( [ 2.2.10 ] ) for all small @xmath342 , @xmath2934 for some @xmath2922 independent of @xmath33 which together with ( [ 2.6.13 ] ) yields ( [ 2.2.38 ] ) . for readers convenience we start this section with the setup and necessary technical results from @xcite refering there for the corresponding proofs . these results are similar to section [ sec1.3 ] and we refer the reader also to @xcite where more details of proofs can be found than in @xcite and though @xcite deals only with the continuous time case the corresponding discrete time proofs can be obtained , essentially , by simplification . we will discuss below mainly the axiom a case since the corresponding proofs for expanding transformations can be obtained , essentially , by simplification of the same arguments , roughly speaking , by ignoring the stable direction . as in section [ sec1.3 ] we will use the representations @xmath491 of vectors @xmath490 , the norms @xmath2184 and the distances @xmath497 and @xmath153 on @xmath11 and on @xmath10 , respectively . it is known ( see @xcite ) that the hyperbolic splitting @xmath2185 over @xmath70 can be continuously extended to the splitting @xmath2186 over @xmath145 which is forward invariant with respect to @xmath2187 and satisfies exponential estimates ( [ 1.3.1 ] ) with a uniform in @xmath306 exponent @xmath2188 moreover , by @xcite ( see also @xcite ) we can choose these extensions so that @xmath510 and @xmath511 will be h " older continuous in @xmath512 and @xmath53 in @xmath20 in the corresponding grassmann bundle . actually , since @xmath145 is contained in the basin of each attractor @xmath70 , any point @xmath513 belongs to the stable manifold @xmath514 of some point @xmath515 ( see @xcite ) , and so we choose naturally @xmath510 to be the tangent space to @xmath516 at @xmath517 now each vector @xmath518 can be represented uniquely in the form @xmath2189 with @xmath2190 , @xmath521 , and @xmath523 . for each small @xmath526 set @xmath2191 and @xmath528 which are cones around @xmath453 and @xmath529 respectively . similarly , we define @xmath2192 and @xmath528 which are cones around @xmath532 and @xmath533 respectively . the corresponding version of lemma [ lem1.3.1 ] is proved in @xcite and the discrete time versions of lemmas [ lem1.3.2 ] and [ lem1.3.3 ] follow in the same way as in @xcite . let , again , @xmath612 be the set of all @xmath53 embedded @xmath613dimensional closed discs @xmath614 such that @xmath615 , @xmath616 and if @xmath617 then @xmath618 . for @xmath622 and @xmath2193 set @xmath2194 and let @xmath632 and @xmath633 be natural projections on the first and second factors , respectively . the same proof as in @xcite yields the following discrete time version of proposition [ prop1.3.4 ] . [ prop1.9.1 ] for any @xmath792 with @xmath793 large and @xmath794 small enough there exists a positive function @xmath795 satisfying ( [ 1.3.7 ] ) such that for any @xmath2195 @xmath2196 , @xmath799 , @xmath2197 @xmath615 and @xmath2198 we have @xmath2199 where @xmath2200 and @xmath806 . next , observe that the results of section [ sec1.4 ] above are so general that they work both for the continuous and the discrete time case . now , we will discuss the discrete time version of lemma [ 1.5.1 ] . [ lem1.9.1 ] let @xmath1100 @xmath1101 @xmath1102 @xmath1103 , @xmath1104 @xmath1105 and @xmath2201-[{{\varepsilon}}^{-1}t_{j-1}])^{-1 } \sum_{[{{\varepsilon}}^{-1}t_{j-1}]\leq k\leq [ { { \varepsilon}}^{-1}t_j]}b(x , y^{{\varepsilon}}_v(k)).\ ] ] set for @xmath2202,\ , n\in{{\mathbb n}}$ ] , @xmath2203 and for @xmath2204 @xmath2205 then @xmath2206 @xmath2207 and @xmath2208 where , recall , @xmath1112 and @xmath1113 if @xmath1114 . the proof of ( [ 1.9.5 ] ) and ( [ 1.9.6 ] ) is strightforward using the definitions ( [ 1.9.2])([1.9.4 ] ) in the same way as the proof of ( [ 1.5.2 ] ) and ( [ 1.5.3 ] ) only the integrals in the latter case should be replaced by the corresponding sums in the former one . the estimate ( [ 1.9.7 ] ) follows in the same way as ( [ 1.5.4 ] ) only the use of the standard gronwall inequality in the latter proof should be replaced by the discrete time version of the gronwall inequality as in lemma 4.20 of @xcite . now the proof of the discrete time version of proposition [ prop1.5.2 ] and of the remaining part of the proof of large deviations bounds ( [ 1.2.16 ] ) and ( [ 1.2.17 ] ) for the discrete time case proceeds almost verbatim as the corresponding continuous time proofs in section [ sec1.5 ] . observe that in the discrete time case the functionals @xmath228 are given again by ( [ 1.2.13 ] ) with @xmath184 defined by ( [ 1.2.8 ] ) where @xmath2209 and @xmath2210 is given by ( [ 1.2.34 ] ) . the property of @xmath263-functionals described in lemma [ lem1.6.1 ] follows directly in the discrete time case via conjugation since we do not have to deal with the time change here . other auxiliary results of section [ sec1.6 ] are derived in the discrete time case exactly in the same way as there . the proof of the discrete time versions of theorems [ thm1.2.5 ] and [ thm1.2.7 ] under the corresponding assumptions goes through exactly in the same way as its continuous time counterpart in section [ sec1.6 ] yielding the assertion of theorem [ thm1.2.10 ] . next , we exhibit computations demonstrating a discrete time version of theorem [ thm1.2.7 ] for simple examples . the maps @xmath106 in both examples have the form @xmath2211 where @xmath2212 and @xmath2213 $ ] but by identifying the end points of the unit interval we view @xmath106 as expanding maps of the circle @xmath2214 . the function @xmath19 from ( [ 1.1.10 ] ) is given in the first example by @xmath2215 hence , we are dealing here with the maps @xmath2216 defined by @xmath2217 all maps @xmath106 preserve the normalized lebesgue measure @xmath2218 on @xmath2214 and it is the srb measure @xmath69 for each @xmath106 in this simple case . the averaged equation ( [ 1.1.11 ] ) for @xmath2219 has here the form @xmath2220 where @xmath2221 . the one dimensional vector field @xmath47 has three attracting fixed points @xmath2222 and two repelling fixed points 1 and @xmath2223 . in order to apply the discrete time version of theorem [ thm1.2.7 ] ( i.e. theorem [ thm1.2.10 ] ) to this example we have to verify that @xmath19 is complete at the fixed points @xmath2224 of the averaged system . since at these points @xmath106 coincides with the map @xmath2225(mod 1 ) we can take the periodic orbits @xmath2226 and @xmath2227 of the latter and notice that the average of @xmath2228 along the former is @xmath2229 and along the latter @xmath2230 which yields completness of @xmath19 at zeros of @xmath2231 . according to the corresponding part of theorem [ thm1.2.10 ] which is a discrete time version of theorem [ thm1.2.7 ] the transitions between @xmath2232 and @xmath2233 are determined by @xmath2234 which are obtained via the functionals @xmath2235 given by ( [ 1.2.13 ] ) . even here these functionals are not easy to compute though their main ingredients the functionals @xmath184 from ( [ 1.2.8 ] ) are given now by the simple formula @xmath2236 and the set of @xmath106-invariant measures can be reasonably described since all @xmath106 s are conjugate to the simple map @xmath2237 . we plot below the histogram of a single orbit of the slow motion @xmath2238 with @xmath2239 and the initial values @xmath2240 . the histogram shows that most of the points of the orbit stay near the attractors @xmath2241 and @xmath2233 and @xmath2242 hops between basins of attraction of these points . the form of the histogram indicates ( according to theorem [ thm1.2.7 ] ) the equality @xmath2243 and in this case theorem [ thm1.2.7 ] ( or its discrete time version ) can not specify whether the slow motion exits from the basin of @xmath2244 to the basin of @xmath2245 or to the basin of @xmath2233 . observe that theorem [ thm1.2.7 ] is an asymptotical as @xmath50 result and it takes an exponential in @xmath5 time for a typical orbit to exit from the basin of one attractor and to hop to the basin of another one . hence , the computations should be done for small @xmath33 and exponentially long in @xmath5 orbits which is time consuming , so we put a big coefficient in front of @xmath2246 which makes this exponent smaller . of course , it is hard to be absolutely sure that @xmath33 in our computations is small enough and the number of iterates is large enough to demonstrate faithfully the real situation in this case but we found that our histograms are rather robust , for instance , their shapes have the same form for @xmath2239 when the number of iterates ranges from @xmath2247 to , at least , @xmath2248 and various initial conditions were checked , as well . our second example differs from the first one only in @xmath19 which is given now by @xmath2249 here the averaged system has the same attracting fixed points @xmath2250 but one of two repelling fixed points moves from @xmath2223 to @xmath2251 . this makes the basin of attraction of @xmath2252 smaller while the left interval of the basin of attraction of @xmath2253 becomes larger . the latter leads to the inequality @xmath2254 which according to the discrete time version of theorem [ thm1.2.7 ] makes it more difficult for the slow motion to exit to the left from the basin of @xmath2244 than to the right . as in the first example in order to apply the latter result we have to check that @xmath19 is complete at all zeros of @xmath2231 but since we did this already for all integer points it remains to verify completness only for @xmath2255 which follows since @xmath2228 equals 1 and @xmath2223 at two fixed points @xmath2256 and @xmath2257 of @xmath2258 , respectively . in the histogram here we plot @xmath2238 with @xmath2239 and the initial values @xmath2259 . in compliance with the discrete time version of theorem [ thm1.2.7 ] the histogram demonstrates that the slow motion leaves the basin of @xmath2233 and after arriving at the basin of @xmath2244 it exits mostly to the basin of @xmath2245 , and so the slow motion hops mostly between basins of @xmath2245 and @xmath2244 staying most of the time in small neighborhoods of these points . theorem [ thm2.2.10 ] follows by a slight modification ( essentially , by simplification ) of the proof of theorems [ thm2.2.2 ] , in particular , the standard gronwall inequality required in the proof of lemma [ lem2.3.4 ] should be replaced by its discrete time version from @xcite . we have also to check that ( [ 2.2.41 ] ) holds true here which is easier to do than in the continuous time case . indeed , @xmath2935 where @xmath2936 , @xmath2937 , @xmath2938 . by ( [ 2.2.1 ] ) , @xmath2939 since all @xmath2940 which is compact and @xmath2941 , i.e. all @xmath1626 stay also in a compact set , we obtain from our assumptions on transition densities that for all @xmath2942 and @xmath2943 , @xmath2944 for some @xmath1436 independent of @xmath2945 and @xmath20 staying in a compact set . hence , @xmath2946 where @xmath2947 is obtained from @xmath2948 by replacing @xmath242 in the latter by @xmath2949 . it follows from standard facts on principal eigenvalues of positive operators ( see , for instance , @xcite and @xcite ) that uniformly in @xmath432 and @xmath2538 the limit @xmath2950 exists and it satisfies the conditions of assumption [ ass2.2.9 ] , and so taking the logarithm in the ineguality above and dividing by @xmath2539 we arrive at ( [ 2.2.41 ] ) . theorem [ thm2.2.12 ] also follows by a slight modification of proofs of theorems [ thm2.2.5 ] and [ thm2.2.7 ] , only we have to derive a result which replaces lemma [ lem2.4.1 ] providing required properties of @xmath263-functionals given by ( [ 2.2.44 ] ) . since , without loss of generality , we can assume that @xmath2951 for some @xmath1436 and by ( [ 2.2.44 ] ) , @xmath2952 where the supremum is taken over positive continuous functions @xmath2545 . then @xmath2953 it is easy to see from here that @xmath2793 if and only if @xmath2954 and the density @xmath711 is bounded . hence , in this case , @xmath2955 since @xmath2956 for some @xmath2957 , we obtain @xmath2958 two last inequalities provide all properties of @xmath263-fuctionals which are needed in order to replace lemma [ 2.4.1 ] and to proceed with arguments of sections [ sec2.4][sec2.6 ] in the discrete time case . theorem [ thm2.2.10 ] provides , in particular , an approximation of the slow motion by the averaged one in probability but , in general , we do not have convergence in ( [ 2.1.4 ] ) also with probability one ( see @xcite ) . sometimes , we can derive this almost sure convergence from the upper large deviations bound estimating the derivative in @xmath33 of the slow motion as in the following example . let @xmath368 be a bounded 1-periodic in @xmath18 function on @xmath2959 with bounded derivatives and let @xmath2960 be a sequence of independent identically distributed ( i.i.d . ) random variables . define recursively @xmath2961 where @xmath2962 and @xmath2963 , @xmath2964 . then @xmath2965 set @xmath2966 which are sequences of bounded matrices and vectors . taking into account the equalities @xmath2967 we obtain from ( [ 2.7.2 ] ) by induction ( with the agreement @xmath2968 ) that @xmath2969 since @xmath2970 and @xmath2971 are bounded we obtain that @xmath2972 for some @xmath1436 independent of @xmath446 and @xmath33 . since @xmath368 is 1-periodic in @xmath18 we can replace the second equality in ( [ 2.7.1 ] ) by @xmath2973 i.e. we consider now @xmath2974 evolving on the interval @xmath2975 $ ] with 0 and 1 identified which makes it the circle of radius @xmath2976 . suppose that the distribution of @xmath2977 has a @xmath53 density @xmath2978 with respect to the lebesgue measure which is positive on @xmath2975 $ ] . now we have the family of markov chains @xmath2979 with transition probabilities @xmath2980 thus we are in the framework of our main model satisfying assumption [ ass2.2.9 ] , and so the assertion of theorem [ thm2.2.10 ] holds true . let @xmath2635 be the invariant measure of the markov chain @xmath2549 ( which is unique since the doeblin condition is satisfied here ) and assume that @xmath2981 which is , essentially , not a restriction since we always can consider @xmath2722 in place of @xmath368 . this means that @xmath2982 and we derive from theorem [ thm2.2.10 ] that for any @xmath253 there exists @xmath2560 such that for all small @xmath33 , @xmath2983 set @xmath2723 then @xmath2725 and by the borel cantelli lemma we obtain that there exists @xmath2726 finite with probability one so that for all @xmath2727 , @xmath2984 by ( [ 2.7.3 ] ) for @xmath2729 and @xmath2730 , @xmath2985 it follows that with probability one , @xmath2986 the conditions above can be relaxed a bit but this method will not already work if , for instance , the second equality in ( [ 2.7.1 ] ) is replaced by @xmath2987 since in this case the derivative @xmath2988 may grow exponentially in @xmath446 and , indeed , we show in @xcite that for the latter example there is no convergence with probability one in ( [ 2.7.10 ] ) provided @xmath253 is small enough . next , we exhibit two examples of computations which demonstrate adiabatic transitions between attractors of the averaged system via the statistics of proportions of time the slow motion spends in basins of different attractors . the fast motions @xmath2989 in both examples are given by the second equation in ( [ 2.7.1 ] ) where @xmath2960 are i.i.d . random variables with the uniform distribution on @xmath2975 $ ] . the slow motion @xmath2990 is given by the first equation in ( [ 2.7.1 ] ) where in the first example @xmath2991 and in the second example @xmath2992 the markov chains @xmath2549 preserve here the lebesgue measure on @xmath2975 $ ] which is the unique invariant measure for them , and so the averaged equation ( [ 2.1.6 ] ) for @xmath2993 has the right hand side @xmath2994 in the first case and , @xmath2995 in the second case . the one dimensional vector field @xmath47 has three attracting fixed points @xmath2222 and two repelling fixed points 1 and @xmath2223 , while @xmath2996 has the same attracting fixed points but one repelling fixed point moves now from @xmath2223 to @xmath2251 making the basin of @xmath2252 smaller which makes it easier for the slow motion to escape from there . it is easy to see that @xmath2997 and @xmath2998 are complete at the fixed points of the averaged system , and so theorem [ thm2.2.12 ] is applicable in this situation . according to the corresponding part of theorem [ thm2.2.12 ] the transitions between @xmath2232 and @xmath2233 are determined by @xmath2234 which are obtained via the functionals @xmath2235 given by ( [ 2.2.9 ] ) but even here these functionals are not easy to compute . the functionals @xmath2235 yield non classical variational problems and the effective ways of their computation remain for further research . in the first example we plot above the histogram with @xmath2999 intervals of a single orbit of the slow motion @xmath3000 with @xmath2239 and the initial values @xmath3001 . the histogram shows that most of the points of the orbit stay near the attractors @xmath2241 and @xmath2233 and @xmath2242 hops between basins of attraction of these points . the form of the histogram indicates the equality @xmath2243 , which follows also by the symmetry considerations , but in this case theorem [ thm2.2.12 ] can not specify whether the slow motion mostly exits from the basin of @xmath2244 to the basin of @xmath2245 or to the basin of @xmath2233 . in the second example the basin of attraction of @xmath2252 becomes smaller while the left interval of the basin of attraction of @xmath2253 becomes larger . the latter leads to the inequality @xmath2254 which according to theorem [ thm2.2.12 ] makes it more difficult for the slow motion to exit to the left from the basin of @xmath2244 than to the right . in the histogram below ( which has again @xmath2999 intervals ) we plot @xmath3000 with @xmath2239 and the initial values @xmath3002 . in compliance with theorem [ thm2.2.12 ] the histogram demonstrates that the slow motion leaves the basin of @xmath2233 and after arriving at the basin of @xmath2244 it exits mostly to the basin of @xmath2245 , and so the slow motion hops mostly between basins of @xmath2245 and @xmath2244 staying most of the time in small neighborhoods of these points . still , a complete rigorous explanation of these histograms even for our simple examples requires nontrivial additional arguments . it is interesting to observe that these histograms have the same form as in section [ sec1.9 ] of part [ part1 ] where randomness is generated by the expanding ( chaotic ) map @xmath3003 instead of adding uniformly distributed random variables as we do it here . the scheme for the stochastic resonance type phenomenon described below is a slight modification of the model suggested by m.freidlin ( cf . @xcite ) and it can be demonstrated in the setup of three scale systems @xmath2260 @xmath2261 , @xmath2262 , @xmath2263 with initial conditions @xmath2264 , @xmath2265 and @xmath2266 . we assume that @xmath2267 , @xmath2268 while @xmath2269 evolves on a compact @xmath12-dimensional @xmath13 riemannian manifold @xmath11 and the coefficients @xmath80 , @xmath19 , @xmath21 are bounded smooth vector fields on @xmath2270 , @xmath16 and @xmath11 , respectively , depending on other variables as parameters . the solution of ( [ 1.10.1 ] ) determines the flow of diffeomorphisms @xmath2271 on @xmath2272 acting by @xmath2273 . taking @xmath2274 we arrive at the ( unperturbed ) flow @xmath2275 acting by @xmath2276 where @xmath2277 is another family of flows given by @xmath2278 with @xmath2279 which are solutions of @xmath2280 it is natural to view the flow @xmath31 as describing an idealized physical system where parameters @xmath2281 , @xmath32 are assumed to be constants of motion while the perturbed flow @xmath2282 is regarded as describing a real system where evolution of these parameters is also taken into consideration but unlike the averaging setup ( [ 1.1.1 ] ) we have now two sets of parameters moving with very different speeds . set @xmath2283 @xmath2284 @xmath2285 and pass from ( [ 1.10.1 ] ) to the equations in the new time @xmath2286 assume that the equation ( [ 1.10.1 ] ) satisfy the assumptions similar to assumptions [ ass1.2.1 ] , [ ass1.2.2 ] , [ ass1.2.6 ] together with other corresponding conditions appearing in the setup of theorem [ thm1.2.7 ] ( with @xmath2287 in place of @xmath16 ) , in particular , that @xmath2288 form a compact set of flows in the @xmath13 topology with @xmath13 dependence on @xmath2289 and for all @xmath2289 they are axiom a flows in a neighborhood @xmath145 which contains a basic hyperbolic attractor @xmath2290 for @xmath2277 and @xmath145 itself is contained in the basin of each @xmath2290 . set @xmath2291 where @xmath2292 is the srb measure for @xmath2277 and let @xmath2293 be the solution of the averaged equation @xmath2294 first , we apply averaging and large deviations estimates in averaging from the previous section to two last equations in ( [ 1.10.3 ] ) freezing the slowest variable @xmath512 ( i.e. taking for a moment @xmath2295 ) . namely , set @xmath2296 and @xmath2297 so that @xmath2298 suppose for simplicity that @xmath2299 ( i.e. both @xmath2300 and @xmath2301 are one dimensional ) and that the solution @xmath2302 of ( [ 1.10.5 ] ) has the limit set consisting of two attracting points @xmath2245 and @xmath2244 , which for simplicity we assume to be independent of @xmath512 , and a repelling fixed point @xmath2303 depending on @xmath512 and separating their basins . as an example of @xmath2304 we may have in mind @xmath2305 let @xmath2306 be the large deviations rate functional for the system ( [ 1.10.6 ] ) defined in ( [ 1.2.13 ] ) and set for @xmath2307 , @xmath2308 ( cf . with @xmath387 in theorem [ thm1.2.7 ] ) . set @xmath2309 and assume that for all @xmath512 , @xmath2310 which means in view of the averaging principle ( see theorem [ thm1.2.3 ] and the following it discussion ) that @xmath2311 decreases ( increases ) while @xmath2312 stays close to @xmath2245 ( to @xmath2244 ) for `` most '' @xmath18 s with respect to the riemannian volume on @xmath11 restricted to @xmath145 . the following statement suggests a `` nearly '' periodic behavior of the slowest motion . [ conj1.10.1 ] suppose that there exist strictly increasing and decreasing functions @xmath2313 and @xmath2314 , respectively , so that @xmath2315 and @xmath2316 for some @xmath1019 while @xmath2317 for @xmath2318 . assume that @xmath2319 and @xmath50 in such a way that @xmath2320 then for any @xmath2289 there exists @xmath119 so that the slowest motion @xmath2321 converges weakly ( as @xmath914 so that ( [ 1.10.10 ] ) holds true ) as a random process on the probability space @xmath2322 ( where @xmath2323 is the normalized riemannian volume on @xmath145 ) to a periodic function @xmath2324 , @xmath2325 with @xmath2326 the argument supporting this conjecture goes as follows . set @xmath2327 , @xmath2328 and @xmath2329 which satisfy @xmath2330 since @xmath2331 moves much slower than @xmath2332 we can freeze the former and in place of ( [ 1.10.11 ] ) we can study ( [ 1.10.6 ] ) . applying the arguments of theorem [ thm1.2.7 ] to the pair @xmath2333 from ( [ 1.10.6 ] ) we conclude by ( [ 1.2.30 ] ) that the intermediate motion @xmath2334 most of the time stays very close to either @xmath2245 or @xmath2244 before it exits from the corresponding basin , and so in view of an appropriate averaging principle ( which follows , for instance , from theorem [ 1.2.3 ] ) on bounded time intervals the slowest motion @xmath2335 mostly stays close to the corresponding averaged motion determined by the vector fields @xmath2336 and @xmath2337 given by ( [ 1.10.4 ] ) . when @xmath2334 is close to @xmath2245 the slowest motion @xmath2335 decreases until @xmath2338 where @xmath2339 . in view of ( [ 1.2.29 ] ) and the scaling ( [ 1.10.10 ] ) between @xmath33 and @xmath380 , a moment later @xmath2340 becomes less than @xmath625 and @xmath2341 jumps immediately close to @xmath2244 . there @xmath2342 , and so @xmath2335 starts to grow until it reaches @xmath2343 where @xmath2344 . a moment later @xmath2345 becomes smaller than @xmath625 and in view of ( [ 1.2.29 ] ) the intermediate motion @xmath2334 jumps immediately close to @xmath2245 . this leads to a nearly periodic behavior of @xmath2335 . in order to make these arguments precise we have to deal here with an additional difficulty in comparison with the two scale setup considered in previous sections since now the large deviations @xmath285-functionals from theorem [ thm1.2.3 ] and the @xmath282-functions describing adiabatic fluctuations and transitions of theorems [ thm1.2.5 ] and [ thm1.2.7 ] depend on another very slowly changing parameter . still , the technique of sections [ sec1.7 ] and [ sec1.8 ] above applied on time intervals where changes in the @xmath512-variable can be neglected should work here but the details of this approach have not been worked out yet . on the other hand , when the fast motion @xmath2269 does not depend on the slow motions , i.e. when the coefficient @xmath21 in ( [ 1.10.1 ] ) depend only on the coordinate @xmath18 ( but not on @xmath512 and @xmath20 ) , then the above arguments can be made precise without much effort . indeed , we can obtain estimates for transition times @xmath2346 and @xmath2347 of @xmath2348 between neighborhoods of @xmath2245 and @xmath2244 as in theorem [ thm1.2.7 ] applying the latter to @xmath2349 and @xmath2350 from ( [ 1.10.6 ] ) with freezed @xmath512-variable . this is possible since the method of proposition [ prop1.7.1 ] requires us to make large deviations estimates , essentially , only for probabilities @xmath2351 , i.e. on bounded time intervals , and then combine them with the markov property type arguments . during such times the slowest motion @xmath2300 can move only a distance of order @xmath2352 . thus freezing @xmath512 and using the gronwall inequality for the equation of @xmath2 in order to estimate the resulting error we see that the latter is small enough for our purposes . observe , that it would be much more difficult to justify freezing @xmath512 in the coefficient @xmath21 of @xmath1 , if we allow the latter to depend on @xmath512 , since a strightforward application of the gronwall inequality there would yield an error estimate of an exponential in @xmath5 order which is comparable with @xmath2353 . still , it may be possible to take care about the general case using methods of sections [ sec1.4 ] and [ sec1.5 ] since we produce large deviations estimates there by gluing large deviations estimates on smaller time intervals where the @xmath20-variable ( and so , of course , @xmath512-variable ) can be freezed . next , set @xmath2354 where now @xmath2355 does not depend on @xmath33 and @xmath380 . then by ( [ 1.10.1 ] ) together with the gronwall inequality we obtain that @xmath2356 where @xmath964 is the lipschitz constant of @xmath80 . if @xmath20 belongs to the basin @xmath382 then according to theorem [ thm1.2.7 ] @xmath2349 , and so also @xmath2301 , stays most of the time near @xmath382 up to its exit from the basin of the latter which yields according to the above inequality that @xmath2300 stays close to @xmath2357 during this time . but now we can employ the averaging principle for the pair @xmath2358 which sais that @xmath2359 stays close on the time intervals of order @xmath2360 to the averaged motion @xmath2361 defined by @xmath2362 and in view of ( [ 1.10.9 ] ) , @xmath2363 decreases while @xmath2364 increases which leads to the behavior described in conjecture [ conj1.10.1 ] . a similar conjecture can be made under the corresponding conditions for the discrete time case determined by a three scale difference system of equations of the form @xmath2365 where @xmath80 and @xmath19 are smooth vector functions and @xmath2366 is a smooth map ( a diffeomorphism or an endomorphism ) . we obtain an example where discrete time versions of conditions of conjecture [ conj1.10.1 ] hold true setting , for instance , @xmath2367 , @xmath2368 and @xmath2369 ( mod 1 ) . the scheme for the stochastic resonance type phenomenon described below is a slight modification of the model suggested by m.freidlin ( cf . @xcite ) and it can be demonstrated in the setup of three scale systems @xmath3004 @xmath3005 , @xmath3006 , @xmath3007 with initial conditions @xmath3008 , @xmath2265 and @xmath2266 and the last equation in ( [ 2.8.1 ] ) is a stochastic differential equation coupled with first two ordinary differential equations though together they should be considered as a system of stochastic differential equations ( with a degeneration in the first two ) . we assume that @xmath3009 , @xmath2268 while @xmath2269 evolves on a compact @xmath12-dimensional @xmath13 riemannian manifold @xmath11 and the coefficients @xmath80 , @xmath19 , @xmath21 are bounded smooth vector fields on @xmath2270 , @xmath16 and @xmath11 , respectively , depending on other variables as parameters . we suppose also that @xmath3010 is a uniformly positive definite smooth matrix field on @xmath11 . in the same way as in section [ sec2.2 ] we can generalize the setup taking @xmath2269 to be random evolutions but in order to simplify the notations we restrict ourselves to fast motions @xmath2269 being diffusions . the solution of ( [ 2.8.1 ] ) determines a markov diffusion process which the triple @xmath3011 . taking @xmath2274 we arrive at the ( unperturbed ) process @xmath3012 where @xmath3013 solves the unperturbed stochastic differential equation @xmath3014 it is natural to view the diffusion @xmath3015 as describing an idealized physical system where parameters @xmath3016 and @xmath32 are assumed to be constants of motion while the perturbed process @xmath3011 is regarded as describing a real system where evolution of these parameters is also taken into consideration but unlike the averaging setup ( [ 2.1.1 ] ) we have now two sets of parameters moving with very different speeds . let @xmath3017 be the unique invariant measure of the diffusion @xmath3018 . set @xmath3019 and let @xmath3020 be the solution of the averaged equation @xmath3021 first , we apply averaging and large deviations estimates in averaging from the previous sections to two last equations in ( [ 2.8.1 ] ) freezing the slowest variable @xmath488 ( i.e. taking for a moment @xmath2295 ) . namely , set @xmath3022 and @xmath3023 so that @xmath3024 suppose for simplicity that @xmath2299 ( i.e. both @xmath3025 and @xmath2301 are one dimensional ) and that the solution @xmath3026 of ( [ 2.8.4 ] ) has the limit set consisting of two attracting points @xmath2245 and @xmath2244 , which for simplicity we assume to be independent of @xmath488 , and a repelling fixed point @xmath3027 depending on @xmath488 and separating their basins . as an example of @xmath2304 we may have in mind @xmath3028 . let @xmath3029 be the large deviations rate functional for the system of last two equations in ( [ 2.8.1 ] ) defined in ( [ 2.2.9 ] ) and set for @xmath2307 , @xmath3030 ( cf . with @xmath387 in theorem [ thm2.2.7 ] ) . set @xmath3031 and assume that for all @xmath488 , @xmath3032 which means in view of the averaging principle ( see theorem [ thm2.2.2 ] and the following it discussion ) that @xmath3033 decreases ( increases ) with high probability while @xmath3034 stays close to @xmath2245 ( to @xmath2244 ) . [ conj2.8.1 ] suppose that there exist strictly increasing and decreasing functions @xmath3035 and @xmath3036 , respectively , so that @xmath3037 and @xmath3038 for some @xmath1019 while @xmath3039 for @xmath2318 . assume that @xmath2319 and @xmath50 in such a way that @xmath3040 then for any @xmath3041 there exists @xmath119 so that the slowest motion @xmath3042 converges in distribution ( as @xmath914 so that ( [ 2.8.9 ] ) holds true ) to a periodic function @xmath2324 , @xmath2325 with @xmath3043 the argument supporting this conjecture goes as follows . since @xmath3025 moves much slower than @xmath2301 we can freeze the former and in place of ( [ 2.8.1 ] ) we can study , first , ( [ 2.8.5 ] ) . applying the arguments of theorem [ thm2.2.7 ] to the pair @xmath2333 from ( [ 2.8.5 ] ) we conclude from ( [ 2.2.34 ] ) that the intermediate motion @xmath2301 most of the time stays very close to either @xmath2245 or @xmath2244 before it exits from the corresponding basin , and so in view of an appropriate averaging principle ( which follows , for instance , from theorem [ 2.2.2 ] ) on bounded time intervals the slowest motion @xmath3025 mostly stays close to the corresponding averaged motion determined by the vector fields @xmath2336 and @xmath2337 given by ( [ 2.8.7 ] ) . when @xmath2301 is close to @xmath2245 the slowest motion @xmath3025 decreases until @xmath3044 where @xmath3045 . in view of ( [ 2.2.33 ] ) and the scaling ( [ 2.8.9 ] ) between @xmath33 and @xmath380 , a moment later @xmath3046 becomes less than @xmath625 and @xmath3047 jumps immediately close to @xmath2244 . there @xmath3048 , and so @xmath3025 starts to grow until it reaches @xmath3049 where @xmath3050 . a moment later @xmath3051 becomes smaller than @xmath625 and in view of ( [ 2.2.33 ] ) the intermediate motion @xmath2301 jumps immediately close to @xmath2245 . this leads to a nearly periodic behavior of @xmath3025 . in order to make these arguments precise we have to deal here with an additional difficulty in comparison with the two scale setup considered in previous sections since now the large deviations @xmath285-functionals from theorem [ thm2.2.2 ] and the @xmath282-functions describing adiabatic fluctuations and transitions of theorems [ thm2.2.5 ] and [ thm2.2.7 ] depend on another very slowly changing parameter . still , we can use the technique of sections [ sec2.5 ] and [ sec2.6 ] above applied on time intervals where changes in the @xmath488-variable can be neglected should work here but the details of this approach have not been worked out yet . on the other hand , when the fast motion @xmath2269 does not depend on the slow motions , i.e. when the coefficients @xmath1265 and @xmath21 in ( [ 2.8.1 ] ) depend only on the coordinate @xmath18 ( but not on @xmath488 and @xmath20 ) , then the above arguments can be made precise without much effort . indeed , we can obtain estimates for transition times @xmath2346 and @xmath2347 of @xmath2348 between neighborhoods of @xmath2245 and @xmath2244 as in theorem [ thm2.2.7 ] applying the latter to @xmath2349 and @xmath2350 from ( [ 2.8.5 ] ) with freezed @xmath488-variable . this is possible since the method of proposition [ prop2.5.1 ] requires us to make large deviations estimates , essentially , only for probabilities @xmath3052 , i.e. on bounded time intervals , and then combine them with the markov property arguments . during such times the slowest motion @xmath3025 can move only a distance of order @xmath2352 . thus freezing @xmath488 and using the gronwall inequality for the equation of @xmath2 in order to estimate the resulting error we see that the latter is small enough for our purposes . observe , that it would be much more difficult to justify freezing @xmath488 in the coefficients @xmath1265 and @xmath21 of @xmath1 , if we allow the latter to depend on @xmath488 , since a strightforward application of the gronwall inequality there would yield an error estimate of an exponential in @xmath5 order which is comparable with @xmath2353 . still , it may be possible to take care about the general case using methods of section [ sec2.3 ] since we produce large deviations estimates there by gluing large deviations estimates on smaller time intervals where the @xmath20-variable ( and so , of course , @xmath488-variable ) can be freezed . next , set @xmath3053 where now @xmath2355 does not depend on @xmath33 and @xmath380 . then by ( [ 2.8.1 ] ) together with the gronwall inequality we obtain that @xmath3054 where @xmath604 is the lipschitz constant of @xmath80 . if @xmath20 belongs to the basin @xmath382 then according to theorem [ thm2.2.7 ] @xmath2349 , and so also @xmath2301 , stays most of the time near @xmath382 up to its exit from the basin of the latter which yields according to the above inequality that @xmath3025 stays close to @xmath3055 during this time . but now we can employ the averaging principle for the pair @xmath3056 which sais that @xmath3057 stays close on the time intervals of order @xmath2360 to the averaged motion @xmath3058 defined by @xmath3059 and in view of ( [ 2.8.9 ] ) , @xmath3060 decreases while @xmath3061 increases which leads to the behavior described in conjecture [ conj2.8.1 ] . a similar conjecture can be made under the corresponding conditions for the discrete time case determined by a three scale difference system of equations of the form @xmath3062 where @xmath80 and @xmath19 are smooth vector functions and @xmath3063 are coupled with @xmath3064 and @xmath3065 perturbations of a parametrized by @xmath488 and @xmath20 appropriate family of markov chains having smooth transition densities similar to those considered in theorem [ thm2.2.12 ] . this section deals with the averaging principle and a bit with the corresponding large deviations in the sense of convergence of young measures and i thank k.gelfert for asking me about young measures applications in averaging and for indicating to me the paper @xcite . let @xmath45 belongs to the space @xmath2370 of probability measures on @xmath10 and consider the young measure ( which is a map from a measure space to a space of measures , see @xcite ) @xmath2371 from @xmath2372\times{{\mathbb r}}^d\times{{\bf m}},\ell_t\times\mu)$ ] to @xmath2370 defined by @xmath2373 where @xmath2374 is the lebesgue measure on @xmath201 $ ] , @xmath2375 is the unit mass at @xmath512 , and @xmath2376 are solutions of @xmath2377 on the product @xmath2370 . we assume that for all @xmath2378 and @xmath2379 the coefficients @xmath21 and @xmath19 satisfy @xmath2380 for some @xmath907 independent of @xmath2381 . of course , we could require the lipschitz continuity and the boundedness conditions ( [ 1.11.1 ] ) only in some open domain as in section [ sec1.2 ] but we can always extend these vector fields to the whole @xmath16 keeping these properties intact . suppose that @xmath2382 has a disintegration @xmath2383 such that for each lipschitz continuous function @xmath711 on @xmath11 and any @xmath2378 , @xmath2384 depending only on @xmath964 where @xmath2385 is both a lipschitz constant of @xmath711 and it also bounds @xmath2386 . set @xmath2387 then by ( [ 1.11.1 ] ) and ( [ 1.11.3 ] ) , @xmath2231 is bounded and lipschitz continuous , and so there exists a unique solution @xmath2388 of ( [ 1.1.3 ] ) . for any bounded continuous function @xmath711 on @xmath10 define @xmath2389 where @xmath2390 . by the definition ( see @xcite ) , the young measures @xmath2371 converge as @xmath50 to the young measure @xmath2391 defined by @xmath2392 @xmath2393 , if for any bounded continuous function @xmath1095 on @xmath201\times{{\mathbb r}}^d\times{{\bf m}}$ ] , @xmath2394 the following result provides a verifiable ( in some interesting cases ) criterion for even stronger convergence . [ thm1.11.1 ] let @xmath2382 has the disintegration ( [ 1.11.2 ] ) satisfying ( [ 1.11.3 ] ) . then @xmath2395 for any bounded continuous function @xmath2396 on @xmath201\times{{\mathbb r}}^d\times { { \bf m}}$ ] where @xmath2397 if and only if for each @xmath2398 and any finite collection @xmath2399 of bounded lipschitz continuous functions on @xmath10 there exists an integer valued function @xmath2400 as @xmath50 such that for any @xmath253 and @xmath2401 , @xmath2402 where @xmath2403 and , recall , @xmath2404 . first , we prove that ( [ 1.11.5 ] ) implies ( [ 1.11.6 ] ) . let @xmath2399 be bounded lipschitz continuous functions on @xmath2405 and set @xmath2406 if @xmath2407 then by ( [ 1.11.5 ] ) for each @xmath2401 , @xmath2408 choose an integer valued function @xmath2409 as @xmath50 so that @xmath2410 and let @xmath2411 . set @xmath2412 , @xmath2413 and @xmath2414 , @xmath2415 . then by ( [ 1.11.7 ] ) , @xmath2416 by ( [ 1.11.1 ] ) , @xmath2417 where @xmath2418 is the lipschitz constant of @xmath2419 . similarly , by ( [ 1.11.1 ] ) and ( [ 1.11.3 ] ) , @xmath2420 and @xmath2421 it follows from ( [ 1.11.10])([1.11.13 ] ) that @xmath2422 given @xmath253 choose @xmath2423 such that for all @xmath2424 and @xmath2401 , @xmath2425 then by ( [ 1.11.14 ] ) , @xmath2426 by chebyshev s inequality @xmath2427 by ( [ 1.11.9 ] ) the right hand side of ( [ 1.11.15 ] ) tends to 0 as @xmath50 yielding ( [ 1.11.6 ] ) . next , we derive ( [ 1.11.5 ] ) from ( [ 1.11.6 ] ) . since @xmath1095 in ( [ 1.11.5 ] ) is a bounded function and @xmath1029 is a probability measure it is easy to see that it suffices to prove ( [ 1.11.5 ] ) when the integration in @xmath20 there is restricted to compact subsets of @xmath16 . but if we integrate in ( [ 1.11.5 ] ) in @xmath20 running over a compact set @xmath2428 then by ( [ 1.1.1 ] ) and ( [ 1.11.1 ] ) , @xmath2429 i.e. both @xmath2430 and @xmath2431 belong to the @xmath2432neighborhood @xmath2433 of the set @xmath1762 when @xmath1640 and @xmath1271 $ ] . on @xmath201\times g_{kt}\times{{\bf m}}$ ] we can approximate @xmath1095 uniformly by lipschitz continuous functions . thus , in place of ( [ 1.11.5 ] ) it suffices to show that for any compact set @xmath2428 and a bounded lipschitz continuous function @xmath1095 on @xmath201\times g_{kt}\times{{\bf m}}$ ] with a lipschitz constant @xmath2434 in all variables , @xmath2435 by ( [ 1.11.2 ] ) , ( [ 1.11.3 ] ) and ( [ 1.11.4 ] ) , @xmath2436 since ( [ 1.11.6 ] ) holds true also for @xmath2437 , it follows from theorem 2.1 of @xcite that @xmath2438 and so we have only to deal with the first absolute value in the right hand side of ( [ 1.11.18 ] ) . as before set @xmath2412 , @xmath2413 , @xmath2414 and fix a large @xmath2398 . let @xmath2439=[jn / n({{\varepsilon}})]$ ] then by ( [ 1.11.1 ] ) , ( [ 1.11.2 ] ) and ( [ 1.11.3 ] ) , @xmath2440 and @xmath2441 now using ( [ 1.11.20 ] ) , ( [ 1.11.21 ] ) and assuming that @xmath2442 for some constant @xmath2443 we obtain @xmath2444 where @xmath2445 integrating against @xmath45 both parts of ( [ 1.11.22 ] ) over @xmath2446 we obtain @xmath2447 where @xmath2448 by the assumption there exists an integer valued function @xmath2409 as @xmath50 such that ( [ 1.11.6 ] ) holds true for all @xmath2449 and then @xmath2450 as @xmath50 . hence , letting first @xmath50 , then @xmath2319 and , finally , @xmath2451 we obtain ( [ 1.11.17 ] ) in view of ( [ 1.11.18 ] ) and ( [ 1.11.19 ] ) , completing the proof of theorem [ thm1.11.1 ] . observe that ( [ 1.11.5 ] ) holding true for all bounded continuous functions is , in principle , stronger than the averaging principle in the form ( [ 1.11.19 ] ) since the latter is equivalent to ( [ 1.11.5 ] ) with @xmath2452 . in fact , if we require ( [ 1.11.6 ] ) only for one function @xmath2437 then in the same way as in the proof of theorem [ thm1.11.1 ] above we conclude that ( [ 1.11.6 ] ) is equivalent to ( [ 1.11.19 ] ) if we consider the latter over all compacts @xmath2428 ( which was proved earlier in theorem 2.1 of @xcite ) . still , the main interesting classes of systems , we are aware of , for which ( [ 1.11.5 ] ) holds true are the same for which ( [ 1.11.19 ] ) is satisfied though it is easy to construct examples of ( somewhat degenerate ) right hand sides @xmath21 and @xmath19 in ( [ 1.1.1 ] ) for which ( [ 1.11.19 ] ) holds true but ( [ 1.11.5 ] ) fails ( since in the latter we require convergence for all functions @xmath1095 and in the former only for @xmath2452 ) . [ cor1.11.2 ] suppose that there exists an integer valued function @xmath2400 as @xmath50 such that @xmath2457 and for any @xmath253 and each bounded lipschitz continuous function @xmath2458 on @xmath10 , @xmath2459 then ( [ 1.11.6 ] ) is also satisfied , and so ( [ 1.11.5 ] ) holds true . in the same way as in @xcite we obtain that ( [ 1.11.24 ] ) holds true in the anosov theorem setup when @xmath36 is an @xmath27-invariant measure which is ergodic for @xmath1029-almost all @xmath20 , where @xmath1029 is the normalized lebesgue measure on a large compact in @xmath16 , and @xmath2460 with @xmath2461 differentiable in @xmath20 and @xmath18 . furthermore , in the same way as in theorem 2.4 of @xcite or similarly to theorem 2.4 of @xcite we conclude that ( [ 1.11.6 ] ) and ( [ 1.11.24 ] ) hold true under assumptions [ ass1.2.1 ] and [ ass1.2.2 ] . moreover , employing the method of @xcite the result can be extended to some partially hyperbolic systems . observe that under assumptions [ ass1.2.1 ] and [ ass1.2.2 ] we can obtain also large deviations bounds in the form ( [ 1.2.16 ] ) and ( [ 1.2.17 ] ) for @xmath2462 with the functional @xmath2463\big\},\,\,\,\bar f_\nu(s , x)= \int f(s , x , y)d\nu(y ) , \end{aligned}\ ] ] where @xmath1095 is a bounded lipschitz continuous vector function . this can be done deriving first an estimate similar to proposition [ prop1.3.4 ] for @xmath2464 in place of @xmath2465 there , which should follow in the same way as the proof of proposition 4.4 of @xcite , and proceeding similarly to sections [ sec1.4 ] and [ sec1.5 ] above . of course , analogous results can be obtained in the discrete time setup of difference equations ( [ 1.1.10 ] ) . in this section we derive the averaging principle and discuss corresponding large deviations in the sense of convergence of young measures adapted to our probabilistic setup . for more detailed information about young measures we refer the reader to @xcite and references there . let @xmath45 belongs to the space @xmath2370 of probability measures on @xmath10 . we consider a random young measure @xmath2371 from @xmath2372\times{{\mathbb r}}^d\times{{\bf m}},\ell_t\times\mu)$ ] to @xmath2370 which we define by @xmath2373 where @xmath2374 is the lebesgue measure on @xmath201 $ ] , @xmath2375 is the unit mass at @xmath512 , and @xmath2376 are the same as in ( [ 2.1.1 ] ) . suppose that @xmath2382 has a disintegration @xmath3066 such that for each lipschitz continuous function @xmath711 on @xmath11 and any @xmath2378 , @xmath3067 depending only on @xmath964 where @xmath2385 is both a lipschitz constant of @xmath711 and it also bounds @xmath2386 . set @xmath3068 and assume that ( [ 2.2.1 ] ) holds true which together with ( [ 2.9.2 ] ) yields that @xmath2231 is bounded and lipschitz continuous , and so there exists a unique solution @xmath3069 of ( [ 2.1.3 ] ) . for any bounded continuous function @xmath711 on @xmath10 define @xmath3070 where @xmath2390 . in the spirit of @xcite ) we say that the young measures @xmath2371 converge as @xmath50 to a young measure @xmath2391 defined by @xmath2392 @xmath2393 , if for any bounded continuous function @xmath3071 on @xmath2372\times{{\mathbb r}}^d\times{{\bf m}}$ ] , @xmath3072 the following result provides a verifiable ( in some interesting cases ) criterion for even stronger convergence . [ thm2.9.1 ] let @xmath2382 has the disintegration ( [ 2.9.1 ] ) satisfying ( [ 2.9.2 ] ) . then @xmath3073 for any bounded continuous function @xmath2396 on @xmath201\times{{\mathbb r}}^d\times { { \bf m}}$ ] where @xmath2397 if and only if for each @xmath2398 and any finite collection @xmath2399 of bounded lipschitz continuous functions on @xmath10 there exists an integer valued function @xmath2400 as @xmath50 such that for any @xmath253 and @xmath2401 , @xmath3074 where @xmath2403 . first , we prove that ( [ 2.9.4 ] ) implies ( [ 2.9.5 ] ) . let @xmath2399 be bounded lipschitz continuous functions on @xmath2405 and set @xmath3075 if @xmath2407 then by ( [ 2.9.4 ] ) for each @xmath2401 , @xmath3076 choose an integer valued function @xmath2409 as @xmath50 so that @xmath3077 and let @xmath2411 . set @xmath2412 , @xmath2413 and @xmath2414 , @xmath2415 . then by ( [ 2.9.6 ] ) , @xmath3078 where @xmath3079 and @xmath3080 . by ( [ 2.2.1 ] ) , @xmath3081 where @xmath2418 is the lipschitz constant of @xmath2419 . similarly , by ( [ 2.2.1 ] ) and ( [ 2.9.2 ] ) , @xmath3082 and @xmath3083 it follows from ( [ 2.9.9])([2.9.12 ] ) that @xmath3084 given @xmath253 choose @xmath2423 such that for all @xmath2424 and @xmath2401 , @xmath2425 then by ( 9.13 ) and the markov property , @xmath3085 where @xmath3086 is the conditional expectation . by chebyshev s inequality @xmath3087 by ( [ 2.9.8 ] ) the right hand side of ( [ 2.9.14 ] ) tends to 0 as @xmath50 yielding ( [ 2.9.5 ] ) . next , we derive ( [ 2.9.4 ] ) from ( [ 2.9.5 ] ) . since @xmath1095 in ( [ 2.9.4 ] ) is a bounded function and @xmath1029 is a probability measure it is easy to see that it suffices to prove ( [ 2.9.4 ] ) when the integration in @xmath20 there is restricted to compact subsets of @xmath16 . but if @xmath20 belongs to a compact set @xmath2428 in view of ( [ 2.1.1 ] ) and ( [ 2.2.1 ] ) the slow motion @xmath3088 , as well as the averaged one @xmath3089 , stays during the time @xmath3090 in a @xmath2432neighborhood @xmath2433 of @xmath1762 . but on @xmath201\times g_{kt}\times{{\bf m}}$ ] we can approximate @xmath1095 uniformly by lipschitz continuous functions . thus , in place of ( [ 2.9.4 ] ) it suffices to show that for any compact set @xmath2428 and a bounded lipschitz continuous function @xmath1095 on @xmath201\times g_{kt}\times{{\bf m}}$ ] with a lipschitz constant @xmath2434 in all variables , @xmath3091 by ( [ 2.2.1 ] ) , ( [ 2.9.2 ] ) and ( [ 2.9.3 ] ) , @xmath3092 by ( [ 2.1.1 ] ) , ( [ 2.2.1 ] ) , ( [ 2.9.2 ] ) and ( [ 2.9.3 ] ) , @xmath3093 this together with the gronwall inequality gives @xmath3094 now we see that the integral term in the right hand side of ( [ 2.9.17 ] ) is a particular case of the integral term in the right hand side of ( [ 2.9.16 ] ) with @xmath2452 , and so it suffices to estimate only the latter . set , again , @xmath2412 , @xmath2413 , @xmath2414 , @xmath2415 and fix a large @xmath2398 . let @xmath3095=[jn / n({{\varepsilon}})]$ ] then by ( [ 2.2.1 ] ) , ( [ 2.9.1 ] ) and ( [ 2.9.2 ] ) , @xmath3096 and @xmath3097 now using ( [ 2.9.18 ] ) , ( [ 2.9.19 ] ) together with the markov property and assuming that @xmath2442 for some constant @xmath2443 we obtain @xmath3098 where @xmath2445 integrating against @xmath45 both parts of ( [ 2.9.20 ] ) over @xmath2446 we obtain @xmath3099 where @xmath3100 by the assumption there exists an integer valued function @xmath2409 as @xmath50 such that ( [ 2.9.5 ] ) holds true for all @xmath3101 and then @xmath2450 as @xmath50 . hence , letting first @xmath50 , then @xmath2319 and , finally , @xmath2451 we obtain ( [ 2.9.15 ] ) in view of ( [ 2.9.16 ] ) and ( [ 2.9.17 ] ) , completing the proof of theorem [ thm2.9.1 ] . observe that ( [ 2.9.4 ] ) holding true for all bounded continuous functions is , in principle , stronger than the averaging principle in the form @xmath3102 since ( [ 2.9.22 ] ) is equivalent to ( [ 2.9.15 ] ) with @xmath2452 . in fact , if we require ( [ 2.9.5 ] ) only for one function @xmath2437 then it will be equivalent to ( [ 2.9.22 ] ) which follows in the same way as the proof of theorem [ thm2.9.1 ] above . still , the main interesting classes of systems , we are aware of , for which ( [ 2.9.4 ] ) holds true are the same for which ( [ 2.9.22 ] ) is satisfied though it is easy to construct examples of ( somewhat degenerate ) right hand sides @xmath19 in ( [ 2.1.1 ] ) for which ( [ 2.9.22 ] ) holds true but ( [ 2.9.4 ] ) fails ( since in the latter we require convergence for all functions @xmath1095 and in the former only for @xmath2452 ) . it follows from @xcite that the assumptions of theorem [ thm2.9.1 ] hold true when the unperturbed fast motions @xmath2472 are diffusion processes on @xmath11 so that @xmath36 is an invariant measure of @xmath2549 on @xmath11 ergodic for @xmath1029-almost all @xmath20 , where @xmath1029 is the normalized lebesgue measure on a large compact in @xmath16 , and @xmath2460 with @xmath2461 differentiable in @xmath20 and @xmath18 . this can be extended to random evolutions considered in previous sections . observe that under assumptions of theorem [ thm2.2.2 ] we can obtain also large deviations bounds in the form ( [ 2.2.10 ] ) and ( [ 2.2.11 ] ) for @xmath2462 with the functional @xmath2463\big\},\,\,\bar f_\nu(s , x)= \int f(s , x , y)d\nu(y ) , \end{aligned}\ ] ] where @xmath1095 is a bounded lipschitz continuous vector function . the proof can be carried out quite similarly to the proof of theorem [ thm2.2.2 ] . analogous results can be obtained in the discrete time setup of difference equations ( [ 2.1.7 ] ) . we will assume that right hand side of ( [ 2.1.1 ] ) is bounded and lipschitz continuous , i.e. for some @xmath231 , @xmath2488 where @xmath497 is the metric on @xmath11 . our large deviations estimates will be derived under the following general assumption on the fast motion which is satisfied , as we explain it below , for random evolutions which are markov processes with switching at random times between a finite number of diffusion processes . [ ass2.2.1 ] there exist a convex differentiable in @xmath1192 and lipschitz continuous in other variables function @xmath195 defined for all @xmath831 and for @xmath2489 from the closure @xmath133 of a relatively compact open connected set @xmath57 and a positive function @xmath2490 satisfying @xmath2491 such that for all @xmath916 , @xmath2492 and @xmath832 , @xmath2493 where @xmath806 and @xmath804 is the inner product . set @xmath2494 @xmath191 and @xmath2495 . since @xmath2496 then @xmath2497 in view of assumption [ ass2.2.1 ] and standard convex analysis duality results ( see @xcite and @xcite ) @xmath198 is ( strictly ) convex , lower semicontinuous and we have also that @xmath2498 it follows also from assumption [ ass2.2.1 ] that @xmath2499 since @xmath2496 by ( [ 2.2.6 ] ) and @xmath198 is lower semicontinuous then it follows from ( [ 2.2.5 ] ) that there exists a unique @xmath2500 such that @xmath2501 set @xmath2502 . if @xmath2503 depends lipschitz continuously in @xmath20 then we can define the averaged motion @xmath2504 in this general setup as the solution of the ordinary differential equation @xmath2505 denote by @xmath81 the space of continuous curves @xmath200 $ ] in @xmath67 which is the space of continuous maps of @xmath201 $ ] into @xmath202 for each absolutely continuous @xmath203 its velocity @xmath204 can be obtained as the almost everywhere limit of continuous functions @xmath205 when @xmath206 . hence @xmath204 is measurable in @xmath207 , and so we can set @xmath2506 define the uniform metric on @xmath81 by @xmath220 for any @xmath221 set @xmath2507 since @xmath223 is lower semicontinuous and convex in @xmath224 and , in addition , @xmath225 if @xmath2508 it follows that the conditions of theorem 3 in ch.9 of @xcite are satisfied as we can choose a fast growing minorant of @xmath223 required there to be zero in a sufficiently large ball and to be equal , say , @xmath227 outside of it . as a result we conclude that @xmath228 is lower semicontinuous functional on @xmath81 with respect to the metric @xmath229 , and so @xmath230 is a closed set which plays a crucial role in the large deviations arguments below . set @xmath2509 . [ thm2.2.2 ] suppose that ( [ 2.2.1 ] ) and assumption [ ass2.2.1 ] hold true . set @xmath51 and let @xmath240 . then for any @xmath243 and every @xmath244 there exists @xmath245 such that for @xmath1151 uniformly in @xmath432 , @xmath2510 and @xmath2511 the main class of markov processes satisfying our conditions which we have in mind consists of random evolutions on @xmath2513 where @xmath1052 is a compact @xmath446-dimensional @xmath13 riemannian manifold and the unperturbed parametric family of markov processes @xmath2472 is the pair @xmath2514 governed by the stochastic differential equations @xmath2515 where @xmath2516 and for all @xmath2517 , @xmath2518 we assume that @xmath2519 are bounded positive @xmath53 functions , @xmath2520 is a @xmath53 field of positively definite symmetric matrices on @xmath1052 , @xmath2521 is a @xmath53 vector field and all functions are defined and satisfy the above properties for @xmath2522 and @xmath20 belonging to an open neighborhood of @xmath133 . here @xmath2523 is the brownian motion and the equation ( [ 2.2.12 ] ) is written in local coordinats . observe that the existence and some properties of such markov processes are discussed in @xcite . the generator @xmath2524 of the markov process @xmath2525 is the operator acting on @xmath13 vector functions @xmath2526 on @xmath1052 by the formula @xmath2527 where @xmath2528 is the elliptic second order differential operator @xmath2529 now , the perturbed fast motion @xmath2530 satisfies @xmath2531 @xmath2532 and @xmath2533 where @xmath2 is given by ( [ 2.1.1 ] ) with @xmath2534 smoothly depending on @xmath20 and @xmath488 , so that the triple @xmath2535 is a markov processes . the following result which will be proved in section [ sec2.3 + ] claims , in particular , that random evolutions above satisfy assumption [ ass2.2.1 ] [ prop2.2.4 ] for the process @xmath2536 defined by ( [ 2.2.12 ] ) and ( [ 2.2.13 ] ) the limit @xmath2537 exists uniformly in @xmath2538 , @xmath432 and @xmath832 , it is strictly convex and differentiable in @xmath1192 and lipschitz continuous in other variables , and it does not depend under our conditions on @xmath488 and @xmath2539 . in this circumstances the function @xmath198 given by ( [ 2.2.4 ] ) can be represented in the explicit form @xmath2540 where @xmath2541 and the first infinum is taken over the set @xmath2542 of probability measures on @xmath11 , i.e. over the vector measures @xmath2543 with @xmath2544 , and the second one is taken over positive vector functions @xmath2545 on @xmath1052 belonging to the domain of the operator @xmath2524 . clearly , @xmath2546 and , furthermore , @xmath2547 if and only if @xmath45 is the invariant measure @xmath2548 of the markov process @xmath2549 which is unique in our circumstances since the doeblin condition ( see @xcite ) holds true here . the vector field @xmath2550 is @xmath53 in @xmath20 , and so we can define the averaged motion @xmath2476 by @xmath2551 hence , @xmath218 if and only if @xmath2552 for all @xmath252 $ ] . the processes @xmath1 given by ( [ 2.2.16 ] ) and ( [ 2.2.17 ] ) together with the function @xmath195 satisfy assumption [ ass2.2.1 ] . clearly , if @xmath2553 above then @xmath1 becomes a diffusion process and if all operators @xmath2528 are just zero then we arrive to the case of continuous time markov chains as fast motions which also satify all our assumptions . we observe also that both proposition [ prop2.2.4 ] and the results below can be extended to the case when @xmath2528 are hypoelliptic operators satisfying natural conditions so that we could rely , in particular , on results of section 6.3 from @xcite . suppose that the coefficients @xmath2554 and @xmath2555 in ( [ 2.2.12 ] ) and ( [ 2.2.13 ] ) do not depend on @xmath20 . then @xmath2556 is an ergodic markov process with the unique invariant measure @xmath45 and for any @xmath18 almost surely @xmath2557 and by standard general results on the uncoupled averaging ( see @xcite ) it follows that for any @xmath18 almost surely @xmath2558 in the fully coupled case ( i.e. when @xmath2559 depend on @xmath20 ) theorem [ thm2.2.2 ] implies in the case of fast motions given by ( [ 2.2.16 ] ) and ( [ 2.2.17 ] ) that for each @xmath253 there is @xmath2560 such that for all small @xmath33 , @xmath2561 which means , in particular , that in this case we have in ( [ 2.2.22 ] ) convergence in probability . examples from @xcite show that , in general , in the fully coupled setup we do not have convergence in ( [ 2.2.22 ] ) with probability one though in some cases such convergence can be derived from ( [ 2.2.22 + ] ) if the derivatives of @xmath2 and @xmath1 in @xmath33 grow subexponentially in @xmath5 on time intervals of order @xmath5 ( see remark [ rem2.3.6 ] ) . in the following assertions we assume always that the fast motions are obtained by means of ( [ 2.2.12 ] ) and ( [ 2.2.13 ] ) so that we could rely on ( [ 2.2.18])([2.2.20 ] ) though , in principle , it is possible to impose some general conditions on functions @xmath223 which would enable us to proceed with our arguments . precise large deviations bounds such as ( [ 2.2.10 ] ) and ( [ 2.2.11 ] ) of theorem [ thm2.2.2 ] are crucial in our study in sections [ sec2.5 ] and [ sec2.6 ] of the `` very long '' , i.e. exponential in @xmath5 , time `` adiabatic '' behaviour of the slow motion . namely , we will describe such long time behavior of @xmath88 in terms of the function @xmath280 under various assumptions on the averaged motion @xmath281 observe that @xmath282 satisfies the triangle inequality @xmath283 for any @xmath284 and it determines a semi metric on @xmath67 which measures `` the difficulty '' for the slow motion to move from point to point in terms of the functional @xmath285 . introduce the averaged flow @xmath286 on @xmath287 by @xmath2562 where @xmath2563 is the same as in ( [ 2.2.21 ] ) and set @xmath2564 for any probability measure @xmath2543 on @xmath2513 . call a @xmath286-invariant compact set @xmath291 an @xmath285-compact if for any @xmath292 there exist @xmath293 and an open set @xmath294 such that whenever @xmath295 and @xmath296 we can pick up @xmath297 $ ] and @xmath298 satisfying @xmath2565 it is clear from this definition that @xmath300 for any pair points @xmath301 of an @xmath285-compact @xmath302 and by the above triangle inequality for @xmath282 we see that @xmath303 takes on the same value when @xmath304 is fixed and @xmath20 runs over @xmath302 . we say that the vector field @xmath19 on @xmath305 is complete at @xmath306 if the convex set of vectors @xmath2566 , \mu\in{{\mathcal p}}({{\bf m}}),\ , i_x(\mu)<\infty\}$ ] contains an open neigborhood of the origin in @xmath16 . it follows by lemma [ lem1.6.2 ] in part [ part1 ] that if @xmath291 is a compact @xmath286-invariant set such that @xmath19 is complete at each @xmath295 and either @xmath302 contains a dense orbit of the flow @xmath286 ( i.e. @xmath286 is topologically transitive on @xmath302 ) or @xmath300 for any @xmath308 then @xmath302 is an @xmath285-compact . moreover , to ensure that @xmath302 is an @xmath285-compact it suffices to assume that @xmath19 is complete already at some point of @xmath302 and the flow @xmath286 on @xmath302 is minimal , i.e. the @xmath286-orbits of all points are dense in @xmath302 or , equivalently , for any @xmath292 there exists @xmath309 such that the orbit @xmath310\}$ ] of length @xmath311 of each point @xmath295 forms an @xmath312-net in @xmath302 which is equivalent to minimality of the flow @xmath286 on @xmath302 ( see @xcite ) . the latter condition obviously holds true when @xmath302 is a fixed point or a periodic orbit of @xmath286 but also , more generally , when @xmath286 on @xmath302 is uniquely ergodic ( see @xcite ) . a compact @xmath286-invariant set @xmath291 is called an attractor ( for the flow @xmath286 ) if there is an open set @xmath313 and @xmath314 such that @xmath315 for an attractor @xmath302 the set @xmath316 , which is clearly open , is called the basin ( domain of attraction ) of @xmath302 . an attractor which is also an @xmath285-compact will be called an @xmath285-attractor . in what follows we will speak about connected open sets @xmath317 with piecewise smooth boundaries @xmath318 . the latter can be introduced in various ways but it will be convenient here to adopt the definition from @xcite saying that @xmath318 is the closure of a finite union of disjoint , connected , codimension one , extendible @xmath53 ( open or closed ) submanifolds of @xmath16 which are called faces of the boundary . the extendibility condition means that the closure of each face is a part of a larger submanifold of the same dimension which coincides with the face itself if the latter is a compact submanifold . this enables us to extend fields of normal vectors to the boundary of faces and to speak about minimal angles between adjacent faces which we assume to be uniformly bounded away from zero or , in other words , angles between exterior normals to adjacent faces at a point of intersection of their closures are uniformly bounded away from @xmath319 and @xmath320 . the following result which will be proved in section [ sec2.5 ] describes exits of the slow motion from neighborhoods of attractors of the averaged motion . [ thm2.2.5 ] let @xmath291 be an @xmath285-attractor of the flow @xmath286 whose basin contains the closure @xmath321 of a connected open set @xmath317 with a piecewise smooth boundary @xmath318 such that @xmath322 and assume that for each @xmath323 there exists @xmath324 and a probability measure @xmath2567 with @xmath2568 such that @xmath2569,\ ] ] i.e. @xmath2570 and the former vector points out into the interior while the latter into the exterior of @xmath317 . set @xmath330 and @xmath331 . then @xmath332 takes on the same value @xmath333 and @xmath334 coincides with the same compact nonempty set @xmath335 for all @xmath336 while @xmath337 for all @xmath338 . furthermore , for any @xmath338 uniformly in @xmath432 , @xmath2571 and for each @xmath340 there exists @xmath341 such that uniformly in @xmath432 for all small @xmath342 , @xmath2572 next , set @xmath344 where @xmath345 dist@xmath346 and @xmath347 if @xmath348 and @xmath349 , otherwise . then for any @xmath338 and @xmath253 there exists @xmath350 such that uniformly in @xmath432 for all small @xmath342 , @xmath2573 finally , for every @xmath338 and @xmath253 , @xmath2574 provided @xmath353 and the latter holds true if and only if for some @xmath277 there exists @xmath354 such that @xmath355 for lebesgue almost all @xmath252 $ ] with @xmath211 then @xmath353 . theorem [ thm2.2.5 ] asserts , in particular , that typically the slow motion @xmath88 performs rare ( adiabatic ) fluctuations in the vicinity of an @xmath285-attractor @xmath302 since it exists from any domain @xmath313 with @xmath356 for the time much smaller than @xmath357 ( as the corresponding number @xmath358 will be smaller ) and by ( [ 2.2.27 ] ) it can spend in @xmath359 only small proportion of time which implies that @xmath88 exits from @xmath59 and returns to @xmath360 ( exponentially in @xmath5 ) many times before it finally exits @xmath317 . we observe that in the much simpler uncoupled setup corresponding results in the case of @xmath302 being an attracting point were obtained for a continuous time markov chain as a fast motion in @xcite but the proofs there rely on the lower semicontinuity of the function @xmath282 which does not hold true in general , and so extra conditions like @xmath285-compactness of @xmath302 or , more specifically , the completness of @xmath19 at @xmath302 should be assumed there , as well . it is important to observe that the intuition based on diffusion type small random perturbations of dynamical systems should be applied with caution to problems of large deviations in averaging since the @xmath285-functional of theorem [ thm2.2.2 ] describing them is more complex and have rather different properties than the corresponding functional emerging in diffusion type random perturbations of dynamical systems ( see @xcite ) . the reason for this is the deterministic nature of the slow motion @xmath88 which unlike a diffusion can move only with a bounded speed and , moreover , even in order to ensure its `` diffusive like '' local behaviour ( i.e. to let it go in many directions ) some extra nondegeneracy type conditions on the vector field @xmath19 are required . our next result describes rare ( adiabatic ) transitions of the slow motion @xmath88 between basins of attractors of the averaged flow @xmath286 which we consider now in the whole @xmath16 and impose certain conditions on the structure of its @xmath361-limit set . [ ass2.2.6 ] assumption [ ass2.2.1 ] holds true for @xmath362 , the families @xmath2575 and @xmath2576 of matrix and vector fields are compact sets in the @xmath53 topology , @xmath2577 for some @xmath231 independent of @xmath365 and there exists @xmath366 such that @xmath2578 the condition ( [ 2.2.30 ] ) means that outside of some ball all vectors @xmath368 have a bounded away from zero projection on the radial direction which points out to the origin . this condition can be weakened , for instance , it suffices that @xmath369 but , anyway , we have to make some assumption which ensure that the slow motion stays in a compact region where really interesting dynamics takes place . next , suppose that the @xmath361-limit set of the averaged flow @xmath286 is compact and it consists of two parts , so that the first part is a finite number of @xmath285-attractors @xmath370 whose basins @xmath371 have piecewise smooth boundaries @xmath372 and the remaining part of the @xmath361-limit set is contained in @xmath373 . we assume also that for any @xmath374 there exist @xmath324 and probability measures @xmath2579 such that @xmath2580 and @xmath2581\,\ , \mbox{and}\,\ , i=1, ... ,k,\ ] ] i.e. @xmath2582 and it points out into the interior of @xmath379 which means that from any boundary point it is possible to go to any adjacent basin along a curve with an arbitrarily small @xmath285-functional . let @xmath253 be so small that the @xmath380-neighborhood @xmath381 of each @xmath382 is contained with its closure in the corresponding basin @xmath383 . for any @xmath384 set @xmath2583 [ thm2.2.7 ] the function @xmath386 takes on the same value @xmath387 for all @xmath388 . let @xmath389 . then for any @xmath384 uniformly in @xmath432 , @xmath2584 and for any @xmath340 there exists @xmath341 such that for all small @xmath342 , @xmath2585 next , set @xmath392 then for any @xmath384 and @xmath253 there exists @xmath393 such that uniformly in @xmath432 for all small @xmath342 , @xmath2586 now , suppose that the vector field @xmath19 is complete on @xmath395 for some @xmath396 ( which strengthens ( [ 2.2.31 ] ) there ) and the restriction of the @xmath361-limit set of @xmath286 to @xmath395 consists of a finite number of @xmath285-compacts . assume also that there is a unique index @xmath397 such that @xmath398 . then for any @xmath384 there exists @xmath399 such that uniformly in @xmath432 for all small @xmath342 , @xmath2587 finally , suppose that the above conditions hold true for all @xmath401 . define @xmath402 , @xmath403 and recursively , @xmath404 where @xmath405 , @xmath406 if @xmath407 , and set @xmath408 . then for any @xmath384 and @xmath340 there exists @xmath341 such that uniformly in @xmath432 for all @xmath409 and sufficiently small @xmath342 , @xmath2588 and for some @xmath399 , @xmath2589 generically there exists only one index @xmath412 such that @xmath413 and in this case theorem [ thm2.2.7 ] asserts that @xmath414 arrives ( for `` most '' @xmath261 ) at @xmath415 after it leaves @xmath383 . if @xmath416 contains more than one index then the method of the proof of theorem [ thm2.2.7 ] enables us to conclude that in this case @xmath417 arrives ( for `` most '' @xmath261 ) at @xmath418 after leaving @xmath383 but now we can not specify the unique basin of attraction of one of @xmath419 s where @xmath420 exits from @xmath383 . if the succession function @xmath421 is uniquely defined then it determines an order of transitions of the slow motion @xmath88 between basins of attractors of @xmath89 and because of their finite number @xmath88 passes them in certain cyclic order going around such cycle exponentially many in @xmath5 times while spending the total time in a basin @xmath383 which is approximately proportional to @xmath422 . if there exist several cycles of indices @xmath423 where @xmath424 and @xmath425 then transitions between different cycles may also be possible . in the uncoupled case with fast motions being continuous time markov chains a description of such transitions via certain hierarchy of cycles appeared without a detailed proof in @xcite and @xcite . in our fully coupled setup the corresponding description does not seem to be different from the uncoupled situation since its justification relies only on the markov property arguments and estimates of probabilities of transitions of @xmath88 from @xmath2590 to @xmath2022 . set @xmath2591 . following @xcite we call a graph consisting of arrows @xmath2592 @xmath2593 an @xmath1525-graph if every point @xmath2594 is the origin of exactly one arrow and the graph has no circles . let @xmath2595 be the set of all @xmath1525-graphs . next , choose @xmath253 so small that @xmath2596 and define stopping times @xmath2597 and by induction for @xmath864 , @xmath2598 @xmath2599 define the markov chain @xmath2600 which evolves on the phase space @xmath2601 where @xmath2602 . [ thm2.2.8 ] let @xmath2603 be the transition probability of the markov chain @xmath2604 . then for any @xmath1438 there exist @xmath2605 such that if @xmath2606 and @xmath1151 then @xmath2607 whenever @xmath2608 . furthermore , if @xmath2609 is an invariant measure of @xmath2604 on @xmath2601 then @xmath2610 where @xmath2611 since total times spent by a markov process in different sets are asymptotically proportional to masses given to these sets by corresponding invariant measures then theorem [ thm2.2.8 ] ( together with theorem [ thm2.2.7 ] ) yields actually that the slow motion @xmath427 spends in a basin @xmath2017 of the attractor @xmath419 a percentage of total time approximately proportional to @xmath2612 which will be illustrated by computational examples in section [ sec2.7 ] . in fact , this description is effective only if there is a unique @xmath2613 and a graph @xmath2614 such that @xmath2615 is minimal possible among all such sums over all @xmath1373-graphs . in this case the slow motion spends in @xmath2616 a proportion of time close to one . [ ass2.2.9 ] there exist a convex differentiable in @xmath1192 and lipschitz continuous in other variables function @xmath195 defined for all @xmath831 and for @xmath2489 from the closure @xmath133 of a relatively compact open connected set @xmath57 and a positive function @xmath2490 satisfying ( [ 2.2.2 ] ) such that for all @xmath2617 , @xmath2492 and @xmath832 , @xmath2618 where @xmath806 and @xmath2619 appears in ( [ 2.1.7 ] ) . [ thm2.2.10 ] suppose that ( [ 2.2.1 ] ) and assumption [ ass2.2.9 ] are satisfied and that @xmath2620 are given by ( [ 2.1.7 ] ) . for @xmath2621 $ ] define @xmath2622 and set @xmath2623 . then theorem [ thm2.2.2 ] and corollary [ cor2.2.3 ] hold true with the corresponding functionals @xmath228 . the main model of markov chains serving as fast motions @xmath2624 , we have in mind , is obtained in the following way . we start with a parametric family of markov chains @xmath2625 on a compact @xmath13 riemannian manifold @xmath1052 with transition probabilities @xmath2626 having positive densities @xmath2627 with respect to the riemannian volume @xmath54 , so that @xmath2628 is @xmath53 in @xmath20 and continuous in other variables . next , we define @xmath2629 and @xmath2630 adding to ( [ 2.1.7 ] ) another equation @xmath2631 [ prop2.2.11 ] let @xmath2632 be as above . then the limit @xmath2633 exists uniformly in @xmath2489 running over a compact set and in @xmath432 and it satisfies conditions of assumption [ ass2.2.9 ] . in this circumstances the functionals @xmath228 appearing in the large deviations estimates ( [ 2.2.10 ] ) and ( [ 2.2.11 ] ) again have the form ( [ 2.2.9 ] ) with @xmath223 given by ( [ 2.2.19 ] ) where now @xmath2634 clearly , @xmath2546 and , furthermore , @xmath2547 if and only if @xmath45 is the invariant measure @xmath2635 of the markov chain @xmath2549 which is unique since the doeblin condition ( see @xcite ) holds true here . the vector field @xmath2636 is @xmath53 in @xmath20 , and so we can define uniquely the averaged motion @xmath2476 by ( [ 2.2.21 ] ) and , again , @xmath218 if and only if @xmath2637 for all @xmath252 $ ] . furthermore , @xmath2624 given by ( [ 2.2.42 ] ) satisfies ( [ 2.2.41 ] ) . the existence of the limit ( [ 2.2.43 ] ) and its properties in our circumstances are well known ( see @xcite , @xcite , @xcite , @xcite , @xcite ) and the fact that ( [ 2.2.41 ] ) holds true here will be explained at the beginning of section [ sec2.7 ] . [ thm2.2.12 ] let the fast motion @xmath2638 be constructed as above via ( [ 2.2.42 ] ) then with the corresponding definitions of @xmath285-compacts and under similar conditions the conclusions of theorems [ thm2.2.5 ] , [ thm2.2.7 ] and [ thm2.2.8 ] remain true for the corresponding slow motion @xmath427 defined in theorem [ thm2.2.10 ] . observe , that we can easily produce a wide class of systems satisfying the conditions of theorems [ thm2.2.5 ] , [ thm2.2.7 ] , and [ thm2.2.8 ] or theorem [ thm2.2.12 ] by setting @xmath434 so that @xmath2639 where @xmath2635 is the unique invariant measure of @xmath2549 and the vector field @xmath436 , which becomes now the averaged vector field @xmath2231 , has an @xmath2640-limit set satisfying conditions of the above theorems . simple examples of this construction will be exhibited in section [ sec2.7 ] for which we also compute historgrams indicating proportions of time the slow motion spends near different attracting points of the averaged motion . we observe that the functional @xmath228 , which plays a crucial role in the above theorems , seems to be quite difficult to compute since this leads to difficult nonclassical variational problems . we will need the following version of general large deviations bounds when usual assumptions hold true with errors . the proof is a strightforward modification of the standard one ( cf . @xcite ) and its details can be found in part [ part1 ] , lemma [ lem1.4.1 ] . [ lem2.3.1 ] let @xmath983 , @xmath984 be uniformly bounded on compact sets functions on @xmath16 and @xmath985 be a family of @xmath982valued random vectors on a probability space @xmath986 such that @xmath987 with probability one for some constant @xmath793 and all @xmath855 . for any @xmath988 and @xmath989 set @xmath2641 ( i ) for any @xmath991 there exists @xmath992 such that whenever for some @xmath993 , @xmath994 and each @xmath831 with @xmath995 , @xmath2642 then for any compact set @xmath997 , @xmath2643 where @xmath2644 ( ii ) suppose that @xmath1000 , @xmath1001 and there exists @xmath994 such that @xmath1002 and @xmath2645 if ( [ 2.3.2 ] ) holds true then for any @xmath253 , @xmath2646 ( iii ) assume that @xmath1005 satisfy ( [ 2.3.5 ] ) . for any @xmath991 there exists @xmath1006 such that whenever for some @xmath1007 and each @xmath831 with @xmath1008 the inequality ( [ 2.3.2 ] ) holds true together with @xmath2647 then for any @xmath1010 , @xmath2648 where @xmath1012 @xmath1013 @xmath1014 , @xmath1015 @xmath1016 and @xmath1017 denotes the closure of @xmath59 . [ lem2.3.2 ] let @xmath1051 be a nondecreasing sequence of lower semicontinuous functions on a metric space @xmath1052 and let @xmath2649 assume that @xmath285 is also lower semicontinuous and for any compact set @xmath1054 denote @xmath1055 then @xmath2650 we will need also the following general result which will enable us to subdivide time into small intervals freezing the slow variable on each of them so that the estimate ( [ 2.2.3 ] ) of assumption [ ass2.2.1 ] becomes sufficiently precise and , on the other hand , we will not change much the corresponding functionals @xmath228 appearing in required large deviations estimates . this result is certainly not new , it is cited in @xcite as a folklore fact and a version of it can be found in @xcite , p.67 while for a complete proof we refer the reader to part [ part1 ] , lemma [ lem1.4.3 ] . [ lem2.3.3 ] let @xmath1085 be a measurable function on @xmath437 equal zero outside of @xmath201 $ ] and such that @xmath1086 . for each positive integer @xmath54 and @xmath1087 $ ] define @xmath1088{{\delta}}-c)$ ] where @xmath1089 and @xmath1090 $ ] denotes the integral part . then there exists a sequence @xmath1091 such that for lebesgue almost all @xmath1087 $ ] , @xmath2651 [ lem2.3.4 ] let @xmath1100 @xmath1101 @xmath1102 @xmath1103 , @xmath1104 @xmath1105 , @xmath1106 , @xmath1107 and for @xmath252 $ ] , @xmath2652 then @xmath2653 @xmath2654 and @xmath2655 where , recall , @xmath1112 and @xmath1113 if @xmath1114 . \(i ) there exist @xmath1127 and @xmath1128 independent of @xmath2657 such that if @xmath1130 and @xmath1131 then for any @xmath1132 , @xmath2658 where @xmath1134 , @xmath1135 does not depend on @xmath1129 and @xmath2659 in particular , if for each @xmath1137 there exists @xmath1138 such that @xmath2660 and @xmath2661 then ( [ 2.3.16 ] ) holds true with @xmath1141 in place of @xmath1142 , @xmath1137 . \(ii ) for any @xmath1143 there exist @xmath1144 and @xmath1145 , the latter depending also on @xmath1146 , such that if @xmath1147 and @xmath1148 satisfy ( [ 2.3.18 ] ) and ( [ 2.3.19 ] ) , @xmath1149 , @xmath1150 and @xmath1151 then @xmath2662 with some @xmath1128 depending only on @xmath21 and @xmath1153 . \(i ) introduce the events @xmath2663 so that we have @xmath2664 now for @xmath1345 by the markov property @xmath2665 if @xmath2666 then @xmath2667 in view of ( [ 2.3.13 ] ) and ( [ 2.3.14 ] ) satisfies @xmath2668 since @xmath1122 is lipschitz continuous in @xmath789 and @xmath1197 it follows from ( [ 2.3.22 ] ) that @xmath2669 provided @xmath2666 where @xmath1199 depends only on @xmath1200 . in view of assumption [ ass2.2.1 ] we can estimate from above the probability in the right hand side of ( [ 2.3.22 ] ) by means of lemma [ lem2.3.1](i ) which together with ( [ 2.3.24 ] ) yield that @xmath2670 where @xmath1205 as , first , @xmath50 and then @xmath1206 . applying ( [ 2.3.25 ] ) for @xmath1250 and estimating @xmath2671 by means of lemma [ lem2.3.1](i ) we derive ( [ 2.3.16 ] ) in view of ( [ 2.3.21 ] ) . \(ii ) in order to obtain ( [ 2.3.20 ] ) we rely on assumption [ ass2.2.1 ] and lemma [ lem2.3.1](iii ) estimating from below the probability in the right hand side of ( [ 2.3.22 ] ) which together with ( [ 2.3.23 ] ) yield @xmath2672 where @xmath1243 @xmath1244 @xmath1245 and @xmath1246 as , first , @xmath50 and then @xmath1206 . employing ( [ 2.3.26 ] ) for @xmath1250 and estimating @xmath2671 by means of lemma [ lem2.3.1](iii ) we obtain from ( [ 2.3.21 ] ) that @xmath2673 for some @xmath1256 provided , say , @xmath1257 and @xmath1258 . since @xmath466 is differentiable in @xmath1192 then @xmath1259 for any @xmath1260 ( see theorems 23.5 and 25.1 in @xcite ) , and so by the lower semicontinuity of @xmath223 in @xmath224 ( and , in fact , also in @xmath20 ) , @xmath1261 this together with lemma [ lem2.3.2 ] yield that @xmath1262 appearing in the definition of @xmath1263 is positive provided @xmath21 is sufficiently large . in fact , it follows from the lower semicontinuity of @xmath223 that @xmath1262 is bounded away from zero by a positive constant independent of @xmath1264 and @xmath1147 , @xmath1137 if these points vary over fixed compact sets and ( [ 2.3.18 ] ) together with ( [ 2.3.19 ] ) hold true . now , given @xmath1019 choose , first , sufficiently large @xmath21 as needed and then subsequently choosing small @xmath1265 and @xmath1266 , then small @xmath65 , and , finally , small enough @xmath33 we end up with an estimate of the form @xmath2674 where @xmath1128 and @xmath1268 satisfies ( [ 2.3.17 ] ) . finally , ( [ 2.3.20 ] ) follows from ( [ 2.3.27 ] ) and ( [ 2.3.28 ] ) . the remaining part of the proof of theorem [ thm2.2.2 ] contains mostly some convex analysis arguments and it repeats almost verbatim the corresponding part of the proof of theorem [ thm1.2.3 ] in part [ part1 ] but for readers convenience we exhibit it also here . we remark that some of the details below are borrowed from @xcite but we believe that our exposition and the way of proof are more precise , complete and easier to follow . we start with the lower bound . assume that @xmath1269 , and so that @xmath654 is absolutely continuous , since there is nothing to prove otherwise . then by ( [ 2.2.9 ] ) , @xmath1270 for lebesgue almost all @xmath1271.$ ] by ( [ 2.2.1 ] ) and assumption [ ass2.2.1 ] , @xmath2675 for some @xmath2676 , and so if @xmath1270 it follows from ( [ 2.2.4 ] ) that @xmath2677 . suppose that @xmath2678 and let ri@xmath1275 be the interior of @xmath1275 in its affine hull ( see @xcite ) . then either ri@xmath1276 or @xmath1275 ( by its convexity ) consists of one point and recall that @xmath1277 for lebesgue almost all @xmath1271 $ ] . by ( [ 2.2.6 ] ) and ( [ 2.3.29 ] ) , @xmath2679 this together with the nonnegativity and lower semi - continuity of @xmath1279 yield that there exists @xmath1280 such that @xmath1281 and by a version of the measurable selection ( of the implicit function ) theorem ( see @xcite , theorem iii.38 ) , @xmath1280 can be chosen to depend measurably in @xmath1271 $ ] . of course , if ri@xmath1282 then @xmath1275 contains only @xmath1280 and in this case @xmath1283 for lebesgue almost all @xmath1271 $ ] . taking @xmath1284 and @xmath1285 we obtain @xmath2680 observe that @xmath1287 is measurable as a function of @xmath1288 and @xmath224 since it is obtained via ( [ 2.2.4 ] ) as a supremum in one argument of a family of continuous functions , and so this supremum can be taken there over a countable dense set of @xmath1192 s . hence , the set @xmath1289,\,{{\alpha}}\in{{\mathcal d}}(l_s)\}=\ell^{-1}[0,\infty)$ ] is measurable , and so the set @xmath1290\}$ ] is measurable , as well . its projection @xmath1291:\ , ( s,{{\alpha}})\in b\,\mbox{for some}\,{{\alpha}}\in{{\mathbb r}}^d\}$ ] on the first component of the product space is also measurable and @xmath317 is the set of @xmath1271 $ ] such that @xmath1275 contains more than one point . employing theorem iii.22 from @xcite we select @xmath1292 measurably in @xmath1293 and such that @xmath1294 . by convexity and lower semicontinuity of @xmath1279 it follows from corollary 7.5.1 in @xcite that @xmath2681 for each @xmath253 set @xmath1296 then , clearly , @xmath1297 is a measurable function of @xmath1288 , and so @xmath1298 and @xmath1299 are measurable in @xmath1288 , as well . by theorems 23.4 and 23.5 from @xcite for each @xmath1300 there exists @xmath1301 such that ( [ 2.3.31 ] ) holds true . given @xmath1302 take @xmath1303 and for @xmath1271\setminus v$ ] set @xmath1284 . then @xmath2682 for each @xmath1132 set @xmath1305 if the corresponding @xmath1306 in ( [ 2.3.31 ] ) satisfies @xmath1307 and @xmath1308 , otherwise . note , that ( [ 2.3.31 ] ) remains true with @xmath1309 in place of @xmath1310 with @xmath1285 if @xmath1308 . as observed above @xmath1311 whenever @xmath1312 , and so @xmath1313 for lebesgue almost all @xmath1271 $ ] . we recall also that @xmath1314 and @xmath1315 for lebesgue almost all @xmath1271 $ ] . since @xmath1269 , @xmath1316 , and @xmath1317 as @xmath1318 for lebesgue almost all @xmath1271 $ ] , we conclude from ( [ 2.3.33 ] ) and the above observations that for @xmath21 large enough @xmath2683 next , we apply lemma [ lem2.3.3 ] to conclude that there exists a sequence @xmath1320 such that for each @xmath1321 and lebesgue almost all @xmath1322 , @xmath2684 where @xmath1324{{\delta}}_j - c$ ] , @xmath1090 $ ] denotes the integral part and we assume @xmath1325 and @xmath1326 if @xmath1327 . choose @xmath1328 $ ] and set @xmath1329 , @xmath1330 where @xmath1331 if @xmath1332 , @xmath1333 , @xmath1334 @xmath1335 and @xmath1336 , @xmath1337 for @xmath1338 and @xmath1339 for @xmath1340 where @xmath1341 . since @xmath2685 for lebesgue almost all @xmath1271 $ ] then @xmath2686 and , in addition , @xmath2687 by ( [ 2.3.33])([2.3.35 ] ) . this together with ( [ 2.3.13 ] ) and ( [ 2.3.14 ] ) yield that for @xmath1345 , @xmath2688 provided @xmath1347 where @xmath1348 and @xmath1349 are the same as in lemma [ lem2.3.4 ] , the latter is defined with @xmath1350 , @xmath1338 and @xmath1351 . choose @xmath1352 so small and @xmath1353 so large that @xmath2689 then by ( [ 2.3.36 ] ) , @xmath2690 by ( [ 2.3.33])([2.3.35 ] ) , @xmath2691 and by the construction above the conditions of the assertion ( ii ) of proposition [ prop2.3.5 ] hold true , so choosing @xmath1353 sufficiently large we derive ( [ 2.2.6 ] ) ( with @xmath1357 in place of @xmath1029 ) from ( [ 2.3.20 ] ) , ( [ 2.3.37 ] ) and ( [ 2.3.38 ] ) provided @xmath33 is small enough . next , we pass to the proof of the upper bound ( [ 2.2.7 ] ) . assume that ( [ 2.2.7 ] ) is not true , i.e. there exist @xmath1358 and @xmath240 such that for some sequence @xmath1359 as @xmath1360 , @xmath2692 since @xmath2693 by ( [ 2.2.1 ] ) all paths of @xmath1363 $ ] and of @xmath1364 $ ] given by ( [ 2.3.11 ] ) ( the latter for any measurable @xmath155 ) belong to a compact set @xmath1365 which consists of curves starting at @xmath20 and satisfying the lipschitz condition with the constant @xmath604 . let @xmath1366 denotes the open @xmath625-neighborhood of the compact set @xmath230 and @xmath1367 . for any small @xmath1368 choose a @xmath1352-net @xmath1369 in @xmath1370 where @xmath1371 . since @xmath2694 then there exists @xmath1373 and a subsequence of @xmath1374 , for which we use the same notation , such that @xmath2695 denote such @xmath1376 by @xmath1377 , choose a sequence @xmath1378 and set @xmath1379 . since @xmath1370 is compact there exists a subsequence @xmath1380 converging in @xmath81 to @xmath1381 which together with ( [ 2.3.40 ] ) yield @xmath2696 for all @xmath1368 . we claim that ( [ 2.3.41 ] ) contradicts ( [ 2.3.12 ] ) and the assertion ( i ) of proposition [ prop2.3.5 ] . indeed , set @xmath1383 by the monotone convergence theorem @xmath2697 similarly to our remark in section [ sec2.2 ] it follows from the results of section 9.1 of @xcite that the functionals @xmath1385 and @xmath1386 are lower semicontinuous in @xmath155 and @xmath654 ( see also section 7.5 in @xcite ) . this together with ( [ 2.3.42 ] ) enable us to apply lemma [ lem2.3.2 ] in order to conclude that @xmath2698 where @xmath1388 . the last inequality in ( [ 2.3.43 ] ) follows from the lower semicontinuity of @xmath228 . thus we can and do choose @xmath1132 such that @xmath2699 by the lower semicontinuity of @xmath1390 in @xmath155 there exists a function @xmath2700 on @xmath1392 such that for each @xmath2701 , @xmath2702 next , we restrict the set of functions @xmath155 to make it compact . namely , we allow from now on only functions @xmath155 for which there exists @xmath1393 such that either @xmath1395 or @xmath1396 for @xmath1397 , @xmath1398 and @xmath1399 where @xmath54 is a positive integer . it is easy to see that the set of such functions @xmath155 is compact with respect to the uniform convergence topology in @xmath81 and it follows that @xmath1400 in ( [ 2.3.45 ] ) constructed with such @xmath155 in mind is lower semicontinuous in @xmath654 . hence @xmath2703 now take @xmath1402 satisfying ( [ 2.3.41 ] ) and for any integer @xmath1403 set @xmath1404 , @xmath1405 , @xmath1406 and @xmath1407 @xmath1408 . define a piecewise linear @xmath1409 and a piecewise constant @xmath1410 by @xmath2704 and @xmath1398 with @xmath1412 . since @xmath1402 is lipschitz continuous with the constant @xmath2705 then @xmath2706 if @xmath54 is large enough and @xmath342 is sufficiently small then @xmath2707 where @xmath1415 is the same as in ( [ 2.3.16 ] ) . since @xmath1416 it follows from ( [ 2.3.48 ] ) and ( [ 2.3.49 ] ) that @xmath1417 and by ( [ 2.3.45 ] ) and the first inequality in ( [ 2.3.49 ] ) we obtain that @xmath2708 hence , by ( [ 2.3.16 ] ) and the second inequality in ( [ 2.3.49 ] ) for all @xmath33 small enough , @xmath2709 provided @xmath1420 ( taking into account that @xmath1421 ) . by ( [ 2.3.12 ] ) and the definition of vectors @xmath1422 for any @xmath1423 , @xmath2710 therefore , @xmath2711 choosing , first , @xmath54 large enough so that @xmath65 satisfies ( [ 2.3.49 ] ) with all sufficiently small @xmath33 and also that @xmath2712 , and then choosing @xmath1352 so small that @xmath2713 , we conclude that ( [ 2.3.51 ] ) together with ( [ 2.3.53 ] ) contradicts ( [ 2.3.41 ] ) , and so the upper bound ( [ 2.2.7 ] ) holds true , completing the proof of theorem [ thm2.2.2 ] . [ rem2.3.6 ] in view of examples from @xcite in the fully coupled setup we should not expect convergence ( [ 2.2.22 ] ) in the averaging principle with probability one in spite of exponentially fast convergence in probability ( [ 2.2.22 + ] ) provided by the upper large deviations bound ( [ 2.2.11 ] ) . still , when derivatives of @xmath2 and @xmath1 in @xmath33 grow not too fast we can derive convergence with probability one from ( [ 2.2.22 + ] ) . indeed , consider , for instance , the following example @xmath2714 where @xmath2715 is a constant , @xmath2523 is the standard one dimensional brownian motion , @xmath368 satisfies ( [ 2.2.1 ] ) and it is @xmath2716-periodic in @xmath18 and @xmath21 has a bounded derivative in @xmath20 . set @xmath2717 then @xmath2718 the solution of this linear equation is easy to estimate which yields that for some constant @xmath1436 , @xmath2719 let @xmath2635 be the invariant measure of the diffusion @xmath2720 ( mod 1 ) ( which is unique since the doeblin condition is satisfied here ) and assume that @xmath2721 which does not harm the generality since we always can consider @xmath2722 in place of @xmath368 . set @xmath2723 where @xmath2724 is the same as in ( [ 2.2.22 + ] ) written for our specific situation . then @xmath2725 and by the borel cantelli lemma we obtain that there exists @xmath2726 , finite with probability one , so that for all @xmath2727 , @xmath2728 by ( [ 2.3.55 ] ) for @xmath2729 and @xmath2730 , @xmath2731 it follows that with probability one , @xmath2732 which is what we need since in our case @xmath2733 in view of ( [ 2.3.56 ] ) . in this section we will prove proposition [ prop2.2.4 ] . observe that @xmath195 obtained by ( [ 2.2.18 ] ) is the principal eigenvalue of the operator @xmath2734 acting on @xmath13 vector functions @xmath2526 on the manifold @xmath1052 by the formula ( see @xcite ) , @xmath2735 where @xmath2489 and @xmath1192 are considered as parameters . according to @xcite this operator satisfies the strong maximum principle . thus , the first part of proposition [ prop2.2.4 ] follows from the well known results on operators satisfying the maximum principle ( see @xcite , @xcite and @xcite ) and the results on the principle eigenvalue of positive operators ( see @xcite , @xcite and @xcite ) and of its smooth dependence on parameters which can be derived from the general perturbation theory of linear operators ( see @xcite ) . now we obtain from ( [ 2.2.18 ] ) that for @xmath2736 uniformly in @xmath2737 @xmath2522 and @xmath832 , @xmath2738 where @xmath2739 as @xmath2740 . next , we want to compare @xmath2741 in order to do this we introduce auxiliary random evolutions @xmath2742 and @xmath2743 governed by the stochastic differential equations @xmath2744 and @xmath2745 respectively , where @xmath2746 , @xmath2747 and for @xmath2748 , @xmath2749 according to @xcite ( which relies on theorem 2 in 6 , ch . vii of @xcite ) the distributions in the path space of the processes @xmath2750 and @xmath2751 are absolutely continuous with respect to the distributions in the path space of the processes @xmath2752 and @xmath2753 , respectively , with the densities @xmath2754 and @xmath2755 respectively , where @xmath2756 @xmath2757 and @xmath2758 . thus , we have to compare @xmath2759 and @xmath2760 observe that by ( [ 2.2.1 ] ) , @xmath2761 let @xmath2762 be both an upper bound for @xmath2763 and their lipschitz constant then we see from ( [ 2.2.1 ] ) and ( [ 2.3+.6])([2.3+.10 ] ) that @xmath2764 employing the witney theorem embed smoothly @xmath1052 as a compact submanifold in an euclidean space @xmath2765 of a sufficiently high dimension @xmath601 and extend the operator @xmath2734 from @xmath1052 to @xmath16 so that its coefficients remain @xmath13 and they vanish outside a relatively compact set containing @xmath1052 ( cf . now we can view ( [ 2.3+.2 ] ) and ( [ 2.3+.3 ] ) as stochastic differential equations in @xmath16 keeping the same notations for their coefficients and processes there . then using standard martingale moment estimates for stochastic integrals ( see , for instance , @xcite ) together with ( [ 2.3+.10 ] ) and the lipschitz continuity of coefficients in ( [ 2.3+.2 ] ) and ( [ 2.3+.3 ] ) we obtain @xmath2766 for some @xmath595 independent of @xmath2767 and @xmath33 . hence , by the gronwall inequality @xmath2768 observe also that the distribution of @xmath2769 can be written explicitly as ( see 55 in @xcite ) , @xmath2770 in order to estimate the last expression in the right hand side of ( [ 2.3+.11 ] ) we note that for any random variable @xmath552 , @xmath2771 and so by the cauchy schwarz and the chebyshev s inequalities @xmath2772 now by ( [ 2.3+.11])([2.3+.14 ] ) together with the cauchy schwarz inequality we obtain that for @xmath2736 uniformly in @xmath2538 , @xmath2522 and @xmath832 , @xmath2773 for another constant @xmath777 independent of @xmath2774 and @xmath33 . choose @xmath2775 and set @xmath2776 $ ] . by ( [ 2.2.21 ] ) , @xmath2777 if @xmath2778 and @xmath2779 then by ( [ 2.3+.1 ] ) , ( [ 2.3+.15 ] ) and the lipschitz continuity of @xmath2780 we obtain that @xmath2781 for some constant @xmath2782 independent of @xmath2783 , @xmath2522 , @xmath2784 and @xmath33 . observe that by the markov property , @xmath2785 now by ( [ 2.3+.10 ] ) and ( [ 2.3+.17 ] ) applying ( [ 2.3+.18 ] ) for @xmath2786 we obtain that @xmath2787 for some @xmath2788 independent of @xmath2538 , @xmath2789 , @xmath2790 and @xmath33 , which yields ( [ 2.2.3 ] ) completing the proof of proposition [ prop2.2.4 ] .
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the work treats dynamical systems given by ordinary differential equations in the form @xmath0 where fast motions @xmath1 depend on the slow motion @xmath2 ( coupled with it ) and they are either given by another differential equation @xmath3 or perturbations of an appropriate parametric family of markov processes with freezed slow variables . in the first case
we assume that the fast motions are hyperbolic for each freezed slow variable and in the second case we deal with markov processes such as random evolutions which are combinations of diffusions and continuous time markov chains .
first , we study large deviations of the slow motion @xmath2 from its averaged ( in fast variables @xmath1 ) approximation @xmath4 the upper large deviation bound justifies the averaging approximation on the time scale of order @xmath5 , called the averaging principle , in the sense of convergence in measure ( in the first case ) or in probability ( in the second case ) but our real goal is to obtain both the upper and the lower large deviations bounds which together with some markov property type arguments ( in the first case ) or with the real markov property ( in the second case ) enable us to study ( adiabatic ) behavior of the slow motion on the much longer exponential in @xmath5 time scale , in particular , to describe its fluctuations in a vicinity of an attractor of the averaged motion and its rare ( adiabatic ) transitions between neighborhoods of such attractors . when the fast motion @xmath1 does not depend on the slow one we arrive at a simpler averaging setup studied in numerous papers but the above fully coupled case , which better describes real phenomena ,
leads to much more complicated problems
. 0.1 cm [ part1 ]
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