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You are an expert at summarizing long articles. Proceed to summarize the following text:
nearby dark clouds like taurus and perseus contain dozens of dense molecular cores where stars like our sun are currently forming or have done so in the recent past ( myers 1995 ) . their large number , together with their proximity and simple structure , make cores unique targets to study the complex physics involved in the formation of a star . dense cores that have not yet formed stars , the so called starless or pre - stellar cores , inform us of the initial conditions of star formation , and their study can help us elucidate the process by which pockets of cloud material condense and become gravitationally unstable . cores with deeply embedded young stellar objects ( `` protostellar cores '' ) are unique targets to study the complex motions that occur during the period of accretion , when a combination of infall , outflow , and rotation is necessary to assemble the star and redistribute the gas angular momentum . finally , evolved cores are primary targets to study the interaction between the newly born star and its environment . these feedback effects are responsible for the transition of the protostar from embedded to visible , and may be important determining the final mass of the star and stabilizing the nearby gas via turbulence generation . the observational study of dense cores has advanced enormously over the last decade thanks to the increase in resolution provided by the new millimeter and submillimeter interferometers , and also due to the systematic combination of observations of dust and molecular tracers ( e.g. , bergin & tafalla 2007 ) . this brief review summarizes some new results from dense cores studies and presents a number of current issues that will greatly benefit from alma observations . the limited space of this article makes any attempt to review the field necessarily incomplete , and the reader is referred for further information to the other contributions on star formation in these proceedings , in particular to those by van dishoeck , andr , shepherd , aikawa , wilner , johnstone , and crutcher . despite significant recent progress , our understanding of the structure and evolution of dense cores is still incomplete due in part to limitations in the resolution and sensitivity of the available observations . even the highest resolution data of nearby dense cores can not discern details finer than about 100 au , which is still insufficient to disentangle the complex kinematics of infall and outflow motions in the vicinity of a protostar . probably more important , the low temperatures of the gas and the dust in cores ( @xmath0 k ) make the emission of any core tracer intrinsically weak , so any increase in the resolution needs to be accompanied by a parallel increase in the sensitivity , or the observations will not achieve enough s / n to provide useful information . this is particularly important when using weak , optically thin tracers to sample the innermost gas in the core . these tracers , in addition , often present extended emission , which poses a problem to the current generation of interferometers that cover sparsely the @xmath1 plane and therefore suffer systematically from missing flux . the high resolution and collecting area afforded by alma , combined with its great sensitivity to extended emission , promises to revolutionize the field of dense cores studies . on the one hand , alma will allow studying the dense cores of nearby clouds with the greatest detail , achieving subarcsecond resolution with high sensitivity . on the other hand , alma will permit the systematic study of dense cores in more distant clouds , enlarging the sample of available targets from the current set of the nearest clouds to cores at distances of at least 1 kpc . the earliest phase of a core , the so - called starless or pre - stellar stage , is characterized by the lack of a point - like object at its center ( e.g. , di francesco et al . 2007 ) . this characterization is of course dependent on the current sensitivity limits of the observations , and is therefore susceptible of misclassifying a core with an embedded source of very low luminosity ( see the case of vellos below ) . still , the significant number of dense cores with no pointlike source detected even after deep spitzer space telescope observations suggests that a population of truly starless cores exists in nearby clouds like taurus ( werner et al . 2006 ) . starless cores present systematically a close to constant density of @xmath2-@xmath3 @xmath4 over the central 5000 - 10000 au followed by an almost power - law drop at large distances . this central flattening of the density profile has been observed in a number of cores using different observational techniques , like millimeter dust continuum emission ( ward - thompson et al . 1999 ) , mir absorption ( bacmann et al . 2000 ) , and nir extinction ( alves et al . 2001 ) , and therefore constitutes a robust result of recent core studies . the presence of a density flattening provides further evidence that starless cores have not yet developed a central singularity , and that they are of pre - stellar nature . the physical origin of the flattening , however , is still a matter of debate , as a number of interpretations are consistent with it . the most natural one is that the profile results from an equilibrium configuration in which the pressure of an isothermal gas balances its gravitational attraction , the so called bonnor - ebert profile ( e.g. , alves et al . 2001 ) . indeed , the gas temperature in a core is typically close to constant ( @xmath0 k ) , and the associated thermal pressure dominates the turbulent component by a factor of several ( e.g. , tafalla et al . the bonnor - ebert interpretation , however , seems in conflict with the non - spherical shape of most cores ( typical axial ratio is 2:1 , myers et al . 1991 ) , and with the fact that the density contrast observed in cores often exceeds the factor of 14 limit for stability of the bonnor - ebert analysis ( bacmann et al . 2000 ) . additional magnetic field support could be responsible for these deviations from the theoretical expectation , but unfortunately , the observation of this magnetic component is extremely hard to make ( see contribution from crutcher in this volume ) . even the apparently `` simple '' structure of the cores still eludes our understanding . when the density distribution of a core , as inferred from dust measurements , is compared with the observed emission from most molecular tracers , it is commonly found that they disagree significantly . as illustrated in fig . 1 for l1498 in taurus , the dust emission of a core often appears centrally concentrated ( with of course a relative flattening at the center ) , while all molecular species but nh@xmath5 and n@xmath6h@xmath7 present ring - like distributions around the continuum peak . radiative transfer analysis of the molecular emission indicates that the abundance of most species drops by at least a factor of 10 towards the high density peak of the molecular core ( caselli et al . 1999 , bergin et al . 2002 , tafalla et al . 2002 ) . such strong abundance decrease is suffered by all the c - bearing molecules as well as other species ( like so ) , while it does not affect significantly nh@xmath5 or n@xmath6h@xmath7 ( see di francesco et al . 2007 and bergin & tafalla 2007 for reviews ) . nh@xmath5 seems in fact to be enhanced toward the center of most cores ( tafalla et al . 2002 ) , while the n@xmath6h@xmath7 abundance tends to have a constant value or may drop at the very center of some cores ( bergin et al . 2002 , pagani et al . 2005 ) . cores therefore have a differentiated ( onion - like ) molecular composition , with a center rich in nh@xmath5 and n@xmath6h@xmath7 and a series of outer layers containing c - bearing species . the inhomogeneous composition of the starless dense cores most likely results from the freeze out of the main molecular species onto the cold dust grains at the center ( bergin & langer 1997 , aikawa et al . 2005 ) . the high densities and low temperatures typical of dense core centers make the freeze out time ( @xmath8 yr ) become much shorter than the core dynamical scale ( @xmath9 1 myr ) , and as a consequence , species like co disappear rapidly from the gas phase . other molecular species suffer the same fate as co , but more importantly , the original chemical balance , characterized by a relative large co abundance ( @xmath10 ) , is changed dramatically by freeze out . a new chemical balance emerges , and it is characterized by the enhancement of certain n - bearing species , like n@xmath6h@xmath7 , which are daughter products of n@xmath6 and whose abundance is controlled by the amount of co in the gas phase ( co is the main destroyer of n@xmath6h@xmath7 ) . even as n@xmath6 freezes out on the dust grains with a similar binding energy as co ( berg et al . 2005 ) , the n@xmath6h@xmath7 abundance can increase relatively from its value in the diffuse cloud ( where co is undepleted ) and give rise to the relatively `` high '' abundances ( few 10@xmath11 ) typical of dense cores . nh@xmath5 can then form from n@xmath6h@xmath7 via dissociative recombination ( geppert et al . 2004 ) , giving rise to the observed central enhancement ( aikawa et al . 2005 ) . another effect of the co depletion in cores is the enhancement of deuterated species . deuteration at the low ( 10 k ) temperature of dense cores occurs via the enhancement of h@xmath6d@xmath7 , which then passes the deuterium atom to other species via ion - molecule reactions ( dalgarno & lepp 1984 ) . as h@xmath6d@xmath7 is mainly destroyed by co , the depletion of co further enhances the h@xmath6d@xmath7 abundance , which in turn enriches in deuterium a number of additional species . high abundance of h@xmath6d@xmath7 has in fact been observed in the heavily co - depleted dense core l1544 ( caselli et al . 2003 ) , and a correlation of co depletion and high deuteration has been reported by bacmann et al . ( 2003 ) and crapsi et al . this deuteration in the cold and dense pre - stellar phase is responsible for the extreme deuteration values of species like h@xmath6co , ch@xmath5oh , and nh@xmath5 seen toward protostellar cores ( ceccarelli et al . 1998 , roueff et al . 2000 , van der tak et al . as cores evolve , they are expected to become more and more centrally concentrated until they reach the point of gravitational instability . one of the most pressing issues in star formation studies is to understand whether this process of concentration is driven by the loss of magnetic field support via ambipolar diffusion ( e.g. , shu et al . 1987 , mouschovias & ciolek 1999 ) or by the dissipation of turbulence via shocks ( e.g. , maclow & klessen 2004 ) . observations of dense cores can not yet distinguish between these scenarios , but do show a systematic correlation between central concentration and other indicators of evolution , like co depletion and deuterium fractionation ( crapsi et al . 2005 ) . evidence for inward motions also seems correlated with central concentration , and this suggests that some cores that we see now as starless have already begun collapsing to form stars . one of the best candidates for such a collapsing system is the l1544 core in taurus , whose pattern of inward motions has been studied in a number of molecules ( tafalla et al . 1998 , williams et al . 1999 , caselli et al . the l1544 dense core is characterized by a high central density and concentration ( ward - thompson et al . 1999 , tafalla et al . 2002 ) , a high degree of co depletion and deuterium fractionation ( caselli et al . 1999 , 2002 ) , and seems starless despite deep spitzer space telescope observations in the ir ( bourke , private communication ) . clearly this core , an similar objects , will be prime targets for alma observations . cores more evolved than l1544 are expected to contain already a luminous object surrounded by an envelope of accreting material . the little observable difference between the pre and proto - stellar phases of a core is illustrated by the case of l1521f , a core initially thought from molecular data to be an almost twin of l1544 ( crapsi et al . 2004 ) and later found with spitzer observations to have a luminous central star ( bourke et al . the central object in l1521f has a luminosity close to 0.1 l@xmath12 , and is characteristic of a new group of objects identified by the spitzer telescope and usually referred as vellos ( very low luminosity objects ) . these vellos seem associated with very weak nir nebulosity and low velocity bipolar outflows ( bourke et al . 2005 ) , and their status in the evolutionary sequence of protostars is still unclear . although some vellos could represent precursors of substellar objects ( proto brown dwarfs ) , it seems more likely that in the case of l1521f we are witnessing the very first moments of accretion , when the central source has an extremely low mass . the proto brown dwarf alternative is unlikely in this case because the dense core has about 5 m@xmath12 of mass ( crapsi et al . 2004 ) , and no clear perturbation seems stopping the accretion ( the outflow has too little mechanical power ) . the pristine nature of vellos makes them ideal candidates to study star - forming infall motions . the study of these motions has a long and rich tradition , and is plagued by difficulties as illustrated by the case of b335 . this dense core harbors a very young ( class 0 ) object whose inward motions were first characterized by zhou et al . these authors found that the spectral signatures from this core are in good agreement with the expectation from the inside - out collapse model of shu ( 1977 ) . high resolution observations with the plateau de bure interferometer by wilner et al . ( 2000 ) , however , have shown that some of the signatures of `` infall '' ( like the high velocity wings in the cs lines ) arise in fact from outflow acceleration , and not from an increase in velocity of the infalling material as it approaches the central object . a revisit of b335 ( and similar objects ) making use of alma s high angular resolution and selecting appropriate ( i.e. , depletion resistant ) tracers is therefore needed to clarify the still confusing picture of star - forming infall motions . the clean appearance of some vellos , together with their weaker outflow emission , offers an interesting alternative to the more evolved ( and massive ) objects like b335 , that have fully developed outflows . because of their lower mass , vellos may present weaker signatures of infall and may be tracing the very first moments of collapse . the combined study of vellos and more luminous class 0 and class i sources should therefore allow us to reconstruct the sequence of star - forming accretion as a function of time . the presence of a protostar at the center of a core affects not only the gas kinematics but its chemistry . the newly born star heats up the nearby gas and dust introducing a temperature gradient in its vicinity . in the @xmath13 au region where the dust temperature exceeds the co evaporation temperature ( @xmath14 - 30 k ) , this molecule returns to the gas phase and undoes part of the chemical processing that occurred during the pre - stellar phase ( jrgensen et al . 2004 , jrgensen 2004 ) . closer to the protostar ( @xmath15 au ) , the dust temperature reaches the 90 - 100 k value at which water evaporates from the grains , further enriching the chemistry . observations of some very young protostellar objects , like iras 16293 - 2422 , show that these very small regions have extreme abundance of a number of complex molecules like hcooh , hcooch@xmath5 , and ch@xmath5och@xmath5 ( cazaux et al . 2003 , bottinelli et al . 2004 ) . the chemical richness of these regions rivals that of the hot cores around massive protostars , justifying their common denomination as `` hot corinos '' ( ceccarelli et al . the exact origin of the complex molecules in these regions , however , is still not fully understood . one possibility is that they result from direct evaporation of species trapped in the water ice , while an alternative is that they result from the processing of simpler evaporated molecules . even the geometry of hot corinos remains unknown , with the innermost part of the envelope or a more stable disk - like distribution as the most likely locations . despite these temporary uncertainties , hot corinos offer a unique opportunity to study the innermost vicinity of low - mass protostars . their distinctive chemical composition makes them highly selective tracers of the most complex and interesting region of the protostar , where inflow , outflow , and rotation motions play comparable roles , and angular momentum is transfered between different gas components . hot corino studies with alma will surely constitute some of the first scientific projects of the instrument . at the same time that protostars accrete material , they eject powerful bipolar outflows of supersonic speed . co observations of these outflows reveal masses that are too large to originate directly from the central protostar , and indicate that most of the moving gas is core ambient material accelerated by a collimated stellar wind ( lada 1985 ) . the lobes of bipolar outflows , in addition , commonly coincide with evacuated cavities seen via scattered light from the protostar , further illustrating how the outflow phenomenon represents a major disruption in the core internal structure ( padgett et al . 1999 ) . despite more than two decades of intense outflow research , a number of outstanding problems remain , and alma observations represent our current best hope to solve them ( see also contribution by d. shepherd in this volume ) . the properties of the underlying wind , for example , are not yet understood , and several alternative models have been proposed over the years . the two main types of models that attempt to fit the observations are the jet - driven outflow and the wind - driven shell , each of them with a number of flavors ( see bachiller 1996 for a review ) . despite significant successes , however , neither type of model can reproduce the rich variety of kinematic properties found by observations , so each of of them is necessarily incomplete ( lee et al . 2002 ) . in the jet driven model , a highly collimated agent shocks and sweeps cloud material along an almost straight line . this model succeeds in explaining the highly collimated co outflows often found toward class 0 objects , but fails to reproduce observations of less collimated flows ( usually powered by class i sources ) , where the co emission arises from gas along limb - brightened shells ( like l1551 , see moriarty - schieven et al . 1987 ) . to fit these less collimated systems , the jet models need to broaden the outflow path , and this has been done by either invoking jet precession/wandering ( masson & chernin 1993 ) or large - scale bow shocks ( raga & cabrit 1993 ) . none of these elements however seems consistent with observations ( see arce et al . 2007 for more details ) , and this leaves the jet models limited to fitting the youngest , and admittedly more spectacular , bipolar outflows . wind - driven models , on the other hand , naturally produce shell - like structures thanks to a wide - angle agent that sweeps ambient material ( shu et al . these models , unfortunately , do not reproduce the appearance of the highly collimated outflows or the mass - velocity distribution commonly observed even in the poorly collimated flows ( masson & chernin 1992 ) . a combination of high resolution observations and new developments in outflow modeling are starting to show a possible solution to the current impasse . interferometer mapping of the outflow powered by the very young source iras 04166 + 2706 in taurus shows both jet and shell features simultaneously ( see fig . 2 and poster contribution by santiago - garca et al . the jet - like feature in this outflow , seen in both co and sio emission , is extremely rectilinear , appears only at the highest velocities ( between 30 and 50 km s@xmath16 ) , and shows no evidence for precession or wandering . the shell - like part appears at low velocities ( 2 to 10 km s@xmath16 ) and seems to delineate two opposed cavities with the iras source at their vertex . this cavity interpretation is supported by the fact that the blue outflow shell coincides with the walls of a nir scattering nebula seen in spitzer images , as expected from its more favorable projection . in addition , the high velocity jet runs along the axis of the two cavities showing a remarkable degree of symmetry ( see poster contribution for further details ) . the data from iras 04166 + 2706 , therefore , leads to the inevitable conclusion that , at least in some cases , both highly collimated and wide - angle components coexist in the outflow driving agent , and that a model that considers both components simultaneously is needed to explain the observations . interestingly enough , recent realistic modeling of the interaction between the x - wind of shu et al . ( 1994 ) and a toroidal core shows that both jet and shell components should be observed simultaneously in very young outflows ( shang et al . this so - called `` unified '' model of bipolar flows shows in fact a remarkable likeness with the iras 04166 + 2706 observations , both in geometry and kinematics ( compare fig . 2 and the models in shang et al . 2006 ) . the unified outflow model not only unifies the jet and wide - angle aspects of the outflows , but also brings together the evolution of flows and the dense cores , two elements often treated separately . evidence for outflow - core interaction has been reported in a number of systems ( e.g. , tafalla & myers 1997 , arce & sargent 2006 ) , but no unified framework of how this interaction happens or how outflows and cores evolve in parallel exists yet . the beautiful simulations of shang et al . ( 2006 ) illustrate how the most important elements of this interaction occur inside the central 1000 au region , which corresponds to less than @xmath17 even towards the most nearby clouds . high angular resolution observations with alma are clearly needed to sample the complex geometry and kinematics inside this critical region , and thus compare real outflows with their simulated counterparts . producing a unified picture of the different and interacting processes occurring during the formation of a solar - type star can be one of most significant achievements of alma .
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dense cores are the simplest star - forming sites that we know , but despite their simplicity , they still hold a number of mysteries that limit our understanding of how solar - type stars form . alma promises to revolutionize our knowledge of every stage in the life of a core , from the pre - stellar phase to the final disruption by the newly born star .
this contribution presents a brief review of the evolution of dense cores and illustrates particular questions that will greatly benefit from the increase in resolution and sensitivity expected from alma .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
since the launch in march 2009 , the _ kepler _ mission has discovered a few thousand planetary candidates , called kepler objects of interests ( kois ) , by detecting the flux deficit as a planet transits in front of its star @xcite . while some of the stars are observed to have one transiting planet ( called tranet from now on , following * ? ? ? * ) , others show up to 6 @xcite . a natural question to ask is , do all of these systems share the same intrinsic orbital structures ? for observing transiting planets , the two most relevant orbital parameters are the dispersion in orbital inclinations , and the typical spacing between adjacent planets . a number of groups have studied the inclination dispersion of kepler planets and reached the common conclusion that this must be small and is of order a few degrees @xcite . however , it has been pointed out that models with a single inclination dispersion falls short in explaining the number of single tranets relative to higher multiples , by a factor of three or more @xcite . this suggests that all kepler planets are not the same , and motivates models where the inclination dispersion itself is broadly distributed ( rayleigh of rayleigh , * ? ? ? * ; * ? ? ? * ) . however , the relative occurrences of different kepler multiples ( denoted here as 1p , 2p , 3p ... by the number of tranets seen in a system ) are sensitive to both the inclination dispersion and the intrinsic planet spacing . larger spacing between adjacent planets will raise the relative number of single tranet systems , as so will larger inclination dispersion . it is difficult to disentangle the two without the aid of further information . therefore we turn to a new measure , the ttv fraction . if a tranet is accompanied by another planet , its transit times deviate from strict periodicity ( transit - time - variation , ttv , * ? ? ? * ; * ? ? ? many studies have used ttv to confirm the planetary nature of kepler candidates . furthermore , it is realized that when the companion is near a mean - motion resonance ( mmr ) with the tranet , the ttv is particularly strong and exhibits a characteristic sinusoidal form @xcite . the amplitude and phase of this sinusoid have been simply related to the perturber s mass , as well as the orbital eccentricities @xcite , thereby allowing us to infer the interior composition and orbital parameters of these objects @xcite . just as the ttv signal can be used to infer the presence of unseen ( non - transiting ) companions around specific candidates ( e.g. * ? ? ? * ; * ? ? ? * ) , the number of tranets that exhibit sinusoidal ttvs provides constraints on near - mmr companions . since the period ratios of adjacent kepler pairs do not much prefer mmrs @xcite , these near - mmr companions can be taken as a proxy for companions that lie close to and inward of the 2:1 mmr . to be quantitative , we shall define the `` intrinsic ttv fraction '' as half the probability that a planet induces a sufficiently large ttv amplitude for detection in another planet in the system . the reason for the factor of a half is that when one planet has a large ttv , then typically so does its ttv partner , and we do not wish to double - count such a pair . when trying to measure this quantity observationally , we shall first count the number of observed tranets with measured ttv s , and then subtract one each time two tranets are ttv partners . dividing by the total number of tranets yields the `` measured ttv fraction , '' which is our estimate for the intrinsic fraction . our ability to measure ttv is affected by the noise level in the transit signals , which is in turn determined by a range of parameters including stellar brightness , the size of the planet relative to its host star , the orbital period and the transit duration . however , if we split the planet candidates into different groups , and if these groups share the same noise properties , then one can argue that the relative ttv fractions measured for different groups represent the relative differences in their intrinsic ttv fractions . in the following , we proceed to measure the relative ttv fractions among 1p , 2p , 3p and 4p+ systems , where 4p+ stands for systems that have four or more transiting planets . we carry out the analysis for all kois that have suitable light - curves , which include more than 2600 kois . we interpret the significance of our results in [ sec : discussion ] . we use the publicly available q0-q12 long cadence ( lc , pdc ) data for 2740 kois ( kepler objects of interest , * ? ? ? * ) . out of these , 134 kois have fewer than 7 transit time measurements , either because the transit periods are very long , or the signal - to - noise ratios ( snr ) are too small . these spread evenly across all multiplicities . this leaves us with 2606 kois , out of which there are , 1488 , 571 , 320 , 227 systems that are designated as 1p , 2p , 3p and 4p+ , respectively . we refer to this sample of 2606 kois as the ` full ' sample . we have also selected a ` reduced ' sample by excluding those kois for which timing measurements are less accurate . these include those with large noise ( @xmath0 ) , and short transit duration ( less than an hour ) . we include only kois with intermediate planet sizes ( @xmath1 ) as they likely have the lowest false positive rate @xcite . this reduced sample contains a total of 1989 kois , with 1097 , 446 , 253 , and 193 systems designated as 1p , 2p , 3p , and 4p+ , respectively . the pipeline used to measure the transit times has been developed and described in @xcite . we compare our transit time measurements to the published ones @xcite , and found good consistency ( see , e.g. , fig.[fig : ttv ] ) . our measurements for the ttv candidates are publicly available at http://www.astro.utoronto.ca/@xmath2jwxie/ttv . from the above transit time measurements , we derive ttv , which are the residuals after a best linear fit . we then search for a sinusoidal signal by obtaining a lomb - scargle ( ls ) periodogram @xcite on these residuals and identify the highest peak that has a period longer than twice the orbital period , as well as @xmath3 days . the former threshold comes about because twice the orbital period is the nyquist frequency for sampling ttv . the latter threshold is enforced because ttv at shorter periods can be significantly polluted by noise from chromospheric activities , as stellar rotation periods fall typically in the range from a few to a few tens of days @xcite . sinusoidal ttv caused by a perturber near a mmr has a `` super - period '' @xcite , @xmath4 where @xmath5 and @xmath6 are the orbital periods of the two planets that are near a first - order @xmath7 mmr by a fractional distance @xmath8 . for a planet pair with period @xmath9 days , @xmath10 , @xmath11 , we find @xmath12 days . a planet pair with a larger @xmath13 will have a shorter super - period , however their ttvs also become increasingly difficult to detect as the ttv amplitudes scale inversely with @xmath13 @xcite . all reported cases @xcite have @xmath13 falling between @xmath14 . this consideration , coupled with the above concern for chromospheric noise , leads us to discard sinusoids short - ward of 100 days . we also adopt an upper limit of @xmath15 days . this comes about because the data ( q0-q12 ) stretch only @xmath16 days . however , some ttv systems show strong , identifiable sinusoids even before a full ttv cycle is observed . this constraint is later relaxed and is found not to impact the conclusion . many of the sinusoids thus identified are false , caused by random alignment of noisy data . it is an important task to exclude these . we adopt the following strategy from @xcite , originally applied to detect planets from radial velocity data . for each koi , we scramble the time stamps of the original ttv data for @xmath17 times , perform a ls periodogram analysis on each set of data , obtaining the amplitude and frequency on the highest peak . the fap ( false alarm probability ) of the original ttv peak is estimated as the fraction of permutations that have higher sinusoids than the original ttv . we assign an fap value of @xmath18 if not a single random realization exceeds the observed sinusoid amplitude . our fap estimates compare well with those from @xcite . fig.[fig : pttv ] shows the fap and @xmath19 for each of the kois in our full sample , as well as those using a scrambled time series for the same kois . the latter set is equivalent to random noise and so acts as a control sample . the true data show a significant excess of objects at very low faps , when compared to those of the scrambled data . we adopt the following ` standard ' criterion ( the region labelled as ` s ' in fig . [ fig : pttv ] ) for identifying our ttv candidates : * \(1 ) @xmath20 and * \(2 ) ttv period between 100 and 1000 d. objects that have fap @xmath21 exhibit ttv amplitudes that range from one to hundreds of minutes , with ttv sensitivity higher for larger snr objects . our above fap criterion is on the conservative side : for an fap of @xmath22 , there should only be @xmath23 false positives among our ttv candidates , much fewer than the actual number of candidates ( @xmath24 ) . we experiment by relaxing the above criterion , either by raising the fap threshold to @xmath25 ( adding the ` f ' region in fig . [ fig : pttv ] ) , or by removing the 1000 day upper limit ( adding the ` p ' region ) . we report our results below . [ cols="^,^,^,^,^,^,^,^,^,^,^ " , ] @xmath26 see [ sec : method ] for definitions of various samples and ttv thresholds . the ` reduced ' sample is selected based on snr , transit duration , and luminosity ; the s , p , and f criteria are illustrated in fig . [ fig : pttv ] . + @xmath27 number of identified ttv candidates , versus numbers of kois in that category . the numbers in parentheses are the raw ttv candidates , while the the corrected ones after parentheses result from removing one of the two ttv candidates from the count whenever a ttv pair is seen . + @xmath28 the measured ttv fraction using the corrected ttv count . we observe a remarkable rise of the ttv fraction with transit multiplicity . in table 1 , we list the number of ttv candidates for different transiting multiplicities , for four different combinations of sample and ttv selection criteria . for ease of comparison against theory , we list the `` measured '' ttv fractions , obtained by removing one candidate from the raw count whenever both it and its ttv partner have observed ttvs . such a method is justified in [ sec : discussion ] . from now on , we focus on these `` measured '' fractions ( illustrated in fig . [ fig : frac ] ) in fact , we focus on the relative `` measured '' fractions , the ttv fractions normalized by that in 4p+ systems . the choice for the normalization is arbitrary . however , since the errorbars for these relative fractions are taken to be quadratic sums of the individual errorbars , which type of system one normalizes against does not affect the statistical conclusion . these results are presented in fig . [ fig : frac ] . except for case 3 , all other combinations give very similar results for the relative ttv fractions : 1p systems have about five times lower ( values from case 1 : @xmath29 ) ttv fraction than 4p+ , and 2p and 3p systems are about twice lower ( @xmath30 , and @xmath31 ) . results from case 3 are less reliable as the ttv selection criterion is too relaxed and allows for too many false positives . we have also used the ttv data from @xcite , published while we are editing our final draft , to confirm the above results ( fig . [ fig : frac ] ) . , we plot ttv fractions from their catalog ( their table 3 ) after applying our ` s ' selection criterion . the good agreement is encouraging , as we extract ttvs using a different method . ] we first discuss what potential bias may affect the absolute ttv fractions that we obtain , then move on to discuss biases that may affect the relative ttv fractions among groups of different transit multiplicity . it becomes clear that by focusing only on the relative ttv fractions , we can eliminate many , if not all , observational bias . we compare properties of the set of ttv candidates against the koi sample . the top panels of fig . [ fig_dis_ttv ] display four transit properties : signal - to - noise ratio , planet radius , orbital period and transit duration . the ttv sample have in general larger transit snr , larger planet radii ( disfavouring small planets ) and slightly longer transit durations than the average kois . in addition , they are also concentrated around orbital periods @xmath32 days . these characteristics allow for optimum ttv detections , as is demonstrated in recent ttv studies by @xcite . for instance , longer orbital periods generally lead to larger ttv amplitudes @xcite , yet too long orbital periods permit only a small number of transits to be observed . as such , we expect that the intrinsic ttv fraction , quantified as half the fraction of planets that have comparable ttv amplitudes as the ones detected here , should be higher than our reported values ( table 1 ) . on the other hand , we find no significant difference between the ttv and koi samples in terms of stellar mass , effective temperature , metallicity and stellar brightness ( stellar parameters from * ? ? ? ks tests performed to compare these two populations always return p - values greater than @xmath33 . this suggests that ttv candidates live in all possible systems . however , this deserves further study as currently there are large uncertainties in stellar parameters , and our ttv sample is relatively small . while the ttv sample as a whole are a biased representation of the koi sample , we find that the different sub - samples , separated by their transiting multiplicity , share similar distributions in both the transit parameters ( fig.[fig_dis_ttv ] ) and the stellar parameters ( not shown here ) . the large p - values returned from ks tests ( fig.[fig_dis_ttv ] ) do not support the hypothesis that the different subgroups experience different selection effects . moreover , we have confirmed that our reduced koi samples , when separated into groups of different transit multiplicities , are statistically similar in their transit and stellar parameters . since the ability to detect ttv above a certain threshold amplitude , only depends on these transit and stellar parameters , these two results then argue that the relative measured ttv fractions reflect the relative intrinsic ttv fractions . in other words , the significant correlation between ttv fraction and transit multiplicity that we observe ( fig.3 ) is unlikely to be caused by systematic biases on stellar / transit parameters . another potential bias could arise during transit detection transiting planets with significant ttvs can be systematically missed , or cataloged as false positives by the kepler pipeline @xcite . to remove this bias , @xcite designed an algorithm ( qats ) that can simultaneously detect transits and measure their ttvs . searches using qats have only found a handful of new planetary candidates ( private communication j. a. carter ) , which might indicate that the kepler catalogue is not significant impacted by this bias . nevertheless , we caution that there could be another possibility , namely , the qats could not fully remove the bias , which deserves further study but is out of the scope of this paper . last but not least , we note that the transit multiplicity of a given system is evolving as the catalog updates . for example , a 1p system may become a 2p system when a new transit candidate is found , either due to accumulation of new data and/or improvement of the pipeline / algorithm for planet detection . to see how these factors affect our results , we took an older version koi catalog from @xcite and performed the same analysis as done in the standard case ( case 1 in table 1 ) . we obtained similar results for the relative ttv fractions : ( @xmath34 , ( @xmath35 , ( @xmath36 and ( @xmath37 for the 1p , 2p , 3p and 4p systems , respectively . this suggests that the correlation between ttv fraction and transit multiplicity observed in fig . 3 may remain unaffected as more improved catalogues are published . , satisfies a rayleigh distribution @xmath38 , and where the planets mutual inclinations are described by an independent rayleigh distribution with @xmath39 . the results presented here are insensitive to the value of @xmath40 . we identify ttv planets as those that transit their host stars and have a companion within a fractional distance of @xmath41 from a first - order mmr . the solid lines indicate the measured fractions , while the dotted lines are the raw fractions ( i.e. , the fraction before removing doubly - counted ttv pairs ) , plotted as functions of @xmath42 , for different multiplicity groups . the raw fractions in the 1p systems can drop to half of that in the 4p systems , because one is likely to observe both planets in the same ttv pair in the latter case . in contrast , in the corrected form ( our so - called measured ttv fraction ) , the relative ttv fractions remain close to unity , and are largely independent of either transit multiplicity or model parameters . in contrast , the observed relative fractions ( case 1 in fig . [ fig : frac ] , marked here as arrows ) fall much below unity . , scaledwidth=45.0% ] we discuss the significance of our results by contrasting them against predictions from a simple toy model . assume all kois , independent of transit multiplicity , are drawn from the same intrinsic distribution , with similar dispersions in mutual inclinations and planet spacing ( with no preference for mmrs ) . in this case , single systems are the ones where the viewing angles are less favourable and we miss most of the planets in the system , while the higher multiples are ones where more planets are caught . one can estimate the ttv fraction for the theoretical population as half the fraction of planets that both transit and have companions within a certain distance from a first - order mmr . as one naively expects and as is confirmed by monte carlo simulations ( fig . [ fig : ttv_toymodel ] ) , the relative ttv fractions cluster around @xmath43 , and are largely independent of the model parameters and transit multiplicities . the lower ttv fraction observed for singles is unlikely to be completely explained by the higher false positive rates in koi singles . the reported false positive rate is of order @xmath44 @xcite . more importantly , our reduced sample , which is expected to have a lower false positive rate than the full sample , yields the same relative ttv fractions . moreover , since ttv amplitudes are strongly boosted by eccentricities as small as a few percent @xcite , the lower ttv fraction can be explained if higher multiple systems have higher eccentricities . however , this is likely excluded by the tight spacing observed among high multiples . a simple explanation for our results is that the basic assumption in our toy model is not true , namely , all kois can not be treated as the same intrinsic population @xcite . for example , there could be at least two distinct populations of kepler planets , different in their intrinsic frequencies of close companions . the high multiples ( 4p+ ) are dominated by a population that has a higher companion frequency , while the 1p systems may be dominated by a population that have a lower frequency of close companions . in other words , there are at least two populations of kepler planets , one that are closely spaced , and one that is sparsely spaced . in an upcoming publication , we will use ttv fractions obtained in this paper , together with a variety of other observational facts , to constrain the properties of these two populations of kepler planets . this will yield important constraints on the process of planet formation . , c. j. , bryson , s. , christiansen , j. , mullally , f. , rowe , j. , science office , k. , & kepler science team . 2013 , in american astronomical society meeting abstracts , vol . 221 , american astronomical society meeting abstracts , 216.02
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a transiting planet exhibits sinusoidal transit - time - variations ( ttvs ) if perturbed by a companion near a mean - motion - resonance ( mmr ) .
we search for sinusoidal ttvs in more than 2600 kepler candidates , using the publicly available kepler light - curves ( q0-q12 ) .
we find that the ttv fractions rise strikingly with the transit multiplicity .
systems where four or more planets transit enjoy four roughly five times higher ttv fraction than those where a single planet transits , and about twice higher than those for doubles and triples .
in contrast , models in which all transiting planets arise from similar dynamical configurations predict comparable ttv fractions among these different systems .
one simple explanation for our results is that there are at least two different classes of kepler systems , one closely packed and one more sparsely populated .
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recent observations of gamma - ray bursts ( grbs ) suggest that long duration grbs and type ib / c supernova ( sn ) explosions are tightly connected . for example , sn1998bw was observed in the positional error box of grb980425 @xcite . in this case the grb / sn association was based on the spatial and temporal coincidence of both events . the most remarkable example of long grb / sn link came in 2003 , when the spectra of both the grb030329 afterglow and of the sn2003dh were measured , since the burst happened closeby and it was quite bright . the supernova spectrum , which includes many complex lines , gradually appeared from the decaying afterglow spectrum after a few tens of days from the burst . the spectrum of sn2003dh after about a month from the explosion is quite similar with that of sn1998bw at the same stage . both sn1998bw and sn2003dh are type ic supernovae @xcite whose progenitor had lost the hydrogen and the helium envelopes during the pre - supernova stage . they are also categorized within a special class of supernova explosions , so - called hypernovae , whose explosion energy is about ten times higher , i.e. , @xmath6 , than that of ordinary supernova . indeed , our common view is that most long - lasting grbs are produced by core - collapse supernovae akin to sn1998bw . grbs are not exclusively linked to hypernovae . for instance , the long lasting burst ( @xmath7 s ) grb060218 ( or xrf060218 ) was associated with the type ic , which is not a hypernova . the explosion energy falls within the regular range for type ic events @xcite . @xcite argued that the progenitor star may not form a black hole but a neutron star , since the estimated mass of the progenitor during the main sequence is @xmath8 . on the other hand , there are recent examples of cases of long - lasting grbs ( and ) where no supernova signature was observed at all even if , considering the distance , the observational trace of a supernova should have been detected @xcite . though a variety of long duration grbs which have a strong connection with sns are observed , we still lack of a complete picture of the processes by which a sizable fraction of the energy involved in a type ib / c sn explosion is tapped in a relatively narrow channel , and produces a grb at a large distance from the original site of generation . @xcite proposed that the death of massive stars can be an origin of grbs . @xcite introduced the collapsar model to account for the progenitor system of long grbs . according to this model , a non - spherical outflow could be formed from the deep inside of the progenitor where a black hole or a proto - neutron star is born as a result of the collapse of the iron core @xcite . if the specific angular momentum of the iron core is sufficiently large , an accretion torus may develop around the central compact object . the system formed by a central object and an accretion disk has the potential to launch a bipolar outflow . the mechanisms proposed to extract energy out of such central engines are basically two ( e.g. , * ? ? ? * ) : thermal or hydromagnetic . thermal mechanisms rely on depositing a considerable amount of thermal energy in the vicinity of the rotation axis of the system , just above the poles of the central compact object , where a low density funnel has develop in the course of the evolution . the accretion energy of the hot torus is converted into a copious flux of neutrinos @xmath9 and anti - neutrinos @xmath10 . from the @xmath11-annihilation a hot @xmath12-plasma results . in its turn , @xmath12-pairs annihilate yielding a _ fireball _ of high energy photons . the conversion of the thermal energy of the fireball into kinetic energy partly determines the subsequent evolution of the plasma . it accelerates to ultrarelativistic speeds ( reaching lorentz factors of @xmath13 @xcite ) while , at the same time , interacts with the progenitor system . alternatively , mhd process may tap a fraction of the rotational energy of the bh or of the accretion disk to form an outflow @xcite . a number of works have dealt with the hydrodynamic properties of the outflows generated in collapsar progenitors as they propagate through the progenitor system ( and in some cases beyond the surface of the progenitor star ) . the problem is addressed by means of numerical relativistic hydrodynamic simulations with different degrees of complexity ( @xcite ) , which assume that a quasi - steady momentum flux has been produced at a certain distance from the region where the energy is released ( independent of which is the actual energy extraction mechanism -mhd or thermal- ) . therefore , such numerical works assume the existence of a nozzle through which the injection of a supersonic jet is produced , and put their focus on the modification of the morphology and of the dynamics of collimated outflows as they travel through the progenitor star . depending on the exact inflow conditions , a variety of different outflows result . for instance , @xcite finds a whole spectrum of outflows ranging from collimated , relativistic jets to poorly collimated expanding winds . thus , @xcite argue that such a variety of resulting outflows supports the idea that the same collapsar scenario can yield a number of different phenomena ( in agreement with the previous ideas of unification of high energy transients , e.g. , @xcite ) , such as , grbs , x - ray rich grbs , x - ray flashes , and normal supernovae . in the prompt grb phase we observe the emission of high energy photons from an ultrarelativistic outflow , which is generated at a distance @xmath14 cm , very far from the central engine . due to the large optical thickness of the outflow at scales comparable to that of the progenitor , we have not observed any electromagnetic emission directly from the progenitor so far , except , perhaps , some precursor activity @xcite . since the direct detection of progenitors of grbs is nowadays impossible ( if they happen at cosmological distances ) , the observation of the association between grbs and sne probably provides the best clue to understand the progenitors of the grbs . though more than 100 grbs per year are identified , most of them occur far away . thus , it is technically impossible to identify their accompanying supernovae . another clue regarding the nature of the grb progenitors comes from the environment and the host galaxies in which the burst takes place . both clues indicate that long - duration grbs are associated with the death of the most massive stars @xcite . the typical hosts of long grbs are star - forming , low metallicity galaxies ( with an star formation rate @xmath15y@xmath3 ; @xcite ) but bluer than typical starburst galaxies , with little dust @xcite , and lower masses than current ellipticals , i.e. , they correspond to the typical environments of formation of massive stars . @xcite also conclude , that the host galaxies of the grbs are significantly fainter and more irregular than the hosts of core - collapse supernovae . a very interesting question is weather it is possible to get any information on the nature of a grb progenitor from the observation of the afterglow emission . one possibility is to look at the angular distribution of integrated energy per unit of solid angle , as observed in the afterglow phase of the burst . @xcite estimated theoretically such an angular distribution assuming , that the kinetic energy of the jet is converted to thermal energy in the cocoon , till the head of the jet reaches the progenitor surface . the cocoon originates from jet material which crosses through the terminal strong shock of the collimated outflow and moves away from the center of the progenitor surrounding the beam of the jet . a fat cocoon develops for light jets , i.e. , jets whose rest - mass density is much lower than that of the ambient gas into which the jet propagates . when the jet breaks out the progenitor surface , the thermal energy is released to a low - density inter stellar medium ( ism ) . @xcite concluded , that the energy distribution per solid angle ( @xmath16 ) of the jet displays a @xmath17 dependence with the viewing angle after its eruption through the progenitor surface . recently , @xcite have used hydrodynamic simulations to test the theoretical prediction of @xcite , and found that their numerical models do not follow the inferred theoretical angular energy distribution . in this paper we also try to verify the analytic relation for the angular dependence of the energy with the polar angle that was proposed by @xcite . we explore a parameter space different from that of @xcite in order to compute the dependence of the angular energy distribution on the structure of the progenitor . the progenitor models are built upon the pre - supernova models of @xcite . along the way , we also characterize the hydrodynamic properties of relativistic jets propagating through different progenitor stars . the paper is organized as follows . we describe our physical model , the choice of stellar progenitors , and relevant numerical details in sect . [ model ] . in the appendix , we provide a study of a selected sample of models in order to justify our choice of numerical resolution and the effects it has on our conclussions . the dynamics of the injected bipolar outflows , and the extrapolated angular energy distribution in the afterglow phase is considered in sec . [ results ] . finally , we discuss our results and write down the conclusions of this work in sect . [ sec : conclusions ] . we will investigate the dependence of the properties of relativistic jets , injected in a pre - supernova stellar model at a certain distance from the center , using relativistic hydrodynamic simulations . in sect . [ sec : progenitor ] we show the different stellar progenitors used in this study . we provide the technical details of the numerical simulations in sect . [ sec : numerics ] . finally , in sect . [ sec : jet ] we specify the physical conditions used to inject relativistic jets in the pre - supernova progenitors described above . in the last years , some detailed calculations of stellar evolution of massive stars have been done including the effects of initial angular momentum , dynamo , metallicity , and mass loss rate @xcite . according to these studies , the metallicity of the progenitor strongly affects the evolution of the angular momentum distribution at the pre - supernova stage , in such a way , that low metallicity is preferred to obtain a large angular momentum in the core of the progenitor . for the purposes of this work , we employ some of the pre - supernova models computed by @xcite . we stick to the same naming convention than the former authors , and consider several sets of models ( tab . [ tab : progenitor ] ) . the first group corresponds to the he16-series of 16 models of @xcite , which include progenitor stars for which @xmath18 bare helium cores are evolved , that have solar metallicity , and different amounts of initial angular momentum , dynamo effects , mass - loss rates , etc . the last three models of tab . [ tab : progenitor ] , 16oc , 16 tb , and 16tc form the second group of progenitors . they correspond to stars with the same initial mass as those of the first group but with a smaller metallicity ( @xmath19 , @xmath20 ) . the second set of low - metallicity models has been chosen among the many other possibilities available because their radius , at the pre - supernova stage , are the smallest among all other low - metallicity progenitors ( in all cases , their stellar radii are @xmath21 cm ) . figure [ fig:1dradialmass ] shows the radial mass profiles of the models he16c , he16l , and he16n . he16c is representative of progenitors whose pre - supernova mass is small @xmath22 ( due to the vigorous mass loss rate in the late phases of its evolution ; tab . [ tab : progenitor ] ) . other members of this group of low pre - supernova mass are he16b , he16j , and he16k . we will refer to this group as type - l . the model he16n belongs to the group of more massive progenitors ( @xmath23 ) , to which we will refer as type - h models . finally , the model he16l ( also he16d ) falls in the middle of these two groups ( members of this group will be called type - m models ) . its total mass is about @xmath24 . figure [ fig:1dradialmass2 ] shows radial mass density profiles of the low metallicity models 16 tb , 16tc , and 16oc . though the mass of the models is similar ( @xmath25 ) , the density profiles are slightly different . in total , 19 models are considered in this study ( tab . [ tab : progenitor ] ) . we neglect for the progenitors any deviation from spherical symmetry arising from the rotation of the models . therefore , for the study we present here , each progenitor differs from the other mainly in its total mass , radius , and mass density profile at the pre - supernova stage . though a nonspherical structure is expected around the black hole due to the rapid rotation of the progenitor , it is reasonable to assume spherical symmetry for the envelopes of the progenitor for radial distances @xmath26 cm , which is where we put the innermost radial boundary in the numerical simulations of this study . thus , we only take from the models of @xcite the radial density and the radial velocity profiles which result by the end of the pre - supernova evolution . we assume that the pressure of the progenitor is very low , or , equivalently , that the initial specific internal energy ( @xmath27 ) is set to be very low ( @xmath28 , where @xmath29 is the speed of light ) . both the gravitational force produced by the central compact object , and the progenitor self gravity are ignored , since the timescale for the outflows to cross the progenitor and to break out from the surface of the progenitor , @xmath30s , is much shorter than free fall timescale for the stellar envelopes . we extend the radial mass density profile to the outside of the progenitor up to the outer computational boundary located at @xmath31 cm . for the models which barely loose mass during the latest stages of their evolution , the rest - mass density is assumed to be uniform @xmath32 and much smaller than that at the progenitor surface . if the progenitor star has a non - null mass loss rate ( i.e. , if the parameter @xmath33 , according to the nomenclature of @xcite ) , we take a @xmath34 dependence in radial mass - density profiles from the surface of the progenitor ( see fig . [ fig:1dradialmass ] ) . we map the spherically symmetric progenitor models of sect . [ sec : numerics ] into a two - dimensional grid in spherical coordinates ( @xmath35 ) . we assume that our models are axial and equatorially symmetric and , therefore , specify reflection boundary conditions at the polar axis ( @xmath36 ) and at the equator ( @xmath37 ) . the radial grid consists of @xmath38 points , uniformly spaced in @xmath39 , which extends from @xmath40 cm to @xmath41 cm . the smallest radial grid spacing , besides @xmath42 , is @xmath43 cm , while the largest one , besides @xmath44 , is @xmath45 cm . the resolution we have choosen here represents a trade - off between accuracy and feasibility of the numerical simulations , as we discuss in the appendix . free outflow ( i.e. , zero gradient ) boundary conditions are set at @xmath46 and @xmath44 . the polar grid has @xmath47 grid points uniformly spaced in the range @xmath48 , ( @xmath49 ) . we use the same 2d special relativistic hydrodynamic code of @xcite to perform our simulations . the code provides 3rd order accuracy in both space and time , by applying a ppm intra - cell interpolation and a tvd - runge kutta time integration . for the sake of simplicity , we employ an ideal gas equation of state ( @xmath50 ) with uniform adiabatic index @xmath51 , where @xmath52 , @xmath53 and @xmath27 are the pressure and the rest - mass density , respectively . we assume that a jet has been generated by the central engine , and that at a certain distance , quasi - steady injection conditions are settled through a well defined circular nozzle . thus , we inject plasma , in the radial direction , through the innermost radial boundary at @xmath54 in a cone of half - opening angle @xmath55 . the jet injection proceeds for a period @xmath56s . we parametrize the outflowing plasma by assuming that it is hot ( we set @xmath57 ) and moderately relativistic ( the lorentz factor being @xmath58 ) . we adopt the convention that the parameters of the outflow at the injection point are named with a subscript j. because of the conversion of thermal - to - kinetic energy , the injected flows have the potential to accelerate to bulk lorentz factors larger than 100 @xcite . during the first 3s , the power of the injected outflow is @xmath59 , where @xmath60 is the area of the injection surface , @xmath61 is the specific enthalpy , and @xmath62 is the radial component of the 3-velocity . the density and pressure of the injected outflow are obtained by setting @xmath63 , @xmath64 , @xmath65 , @xmath66 , and @xmath42 . we fix @xmath67 , which is higher than that adopted in previous studies @xcite . the total injected energy is several times @xmath68erg . since the main purpose of this study is to see the jet propagation and expansion of the cocoon into the interstellar medium after the shock breakout , we adopt this power to obtain a rapid propagation of the jet in the progenitor . this fast propagation is necessary to be consistent with the fact that we neglect the self - gravity of the star . if the jet crosses the progenitor much faster than the typical hydrodynamic timescale in the system , the progenitor remains roughly unchanged during the complete jet propagation through it and , therefore , we do not need to care about the progenitor evolution during such short timescales . after the initial phase of constant kinetic power injection , both the kinetic power and the injection lorentz factor are linearly decreased according to the laws @xmath69 , and @xmath70 , respectively , for @xmath71 . in this period of decaying injection power , the specific energy is kept fixed to the same value as it had at @xmath72 , and the density and the pressure are obtained from the other parameters ( as in the constant injection power phase ) . after @xmath73s , the flow injection ceases . with the parametrization considered above , the rest - mass density ( @xmath74 ) during the constant power phase is 154g@xmath75 . since the rest - mass density of the progenitor around the inner computational boundary is @xmath76g@xmath75 , the injected outflow is initially much lighter than medium in which it is injected . thus it is expected that the jet propagation velocity across the progenitor is smaller than the speed of light , and one naturally expects to generate relatively thick cocoons surrounding the beam of the jet . the dynamical evolution of our jet models can be split in two phases . the first one happens during the period in which the jet drills its way through the progenitor star . the second one shows up latter , when the jet breaks out of the stellar surface . the dynamics of our models during this two phases if roughly similar to that outlined by some previous works ( e.g. , @xcite ) and , therefore , we limit ourselves here to provide a shallow description of the most salient features . figure [ fig : contour ] shows a snapshot of the evolution of the density of the model he16n at @xmath77s , when the head of the jet is still in the progenitor . the left panel of the fig . [ fig : contour1_2 ] shows the lorentz factor contour at the same time shown in fig . [ fig : contour ] . the jet is well collimated both inside of the progenitor and as it travels through the ism . the bow shock develops close to the head of the jet and rises the pressure and the temperature of the envelope region it sweeps up ( in agreement with the findings of @xcite ) . it takes about 3.2s for the jet to cross the progenitor , hence , the average propagation velocity is @xmath78 . the right panel of fig . [ fig : contour1_2 ] shows the lorentz factor of the model he16c at @xmath79s , showing that the head of the jet in model he16c propagates faster than model he16n due to the lower density in model he16c . s ) of the rest - mass density of model he16n . the left and right panels are shown with different color scales in order to outline the most salient features of the cocoon and the cavity drilled by the jet ( left ) , and to show more clearly the structure of the beam ( right ) . both , the vertical and the horizontal axis are scaled by @xmath80 cm . a strong bow shock surrounds the jet , the cocoon , and the shocked progenitor gas . a vigorous back flow from the head of the jet can be seen . some vortices caused by kelvin - helmholtz instabilities are also indicated in the figure . [ fig : contour ] ] s , and right he16c at t@xmath81s ) . both , the vertical and the horizontal axis are scaled by @xmath80 cm . the largest lorentz factors are reached close to the polar axis where the first reconfinement shock appears . some weak reconfinement shocks can also be seen further downstream . [ fig : contour1_2 ] ] although the outflow has a finite initial opening angle , the the beam of the jet is almost parallel to the polar axis . the kinetic energy of the beam is dissipated when it crosses the reverse shock ( i.e , the mach disc ) at the head of the jet . after the beam plasma is decelerated at the mach disc and its pressure is risen to a much higher value than in the beam , it expands and flows back in a thick cocoon . the high pressure of the cocoon is the responsible for the beam collimation during the initial phase of propagation inside of the progenitor star . also during this early stage of the evolution , a strong backflow can be seen flanking the beam of the jet . some vortices develop between the jet and the backflow caused by the growth of kelvin - helmholtz modes . an schematic view of this process can be seen in fig . 5a of @xcite . the propagation of the jet inside the progenitor also drives a cavity limited by a shroud whose density and pressure is larger than in the cocoon . the shroud is swept up by a reverse shock that results from the interaction between the cavity and the progenitor envelope . however , this reverse shock is not strong enough to rise the temperature above the threshold in which nuclear reactions can take place . after the jet breaks out the progenitor surface , it proceeds to the ism , which is assumed to be rarefied for model he16n . in this phase , cocoon is almost freely released into the ism , because of the negligible pressure of the external medium ( fig . [ fig : contour2 ] ) . in spite of the fact that the inertial confinement provided by the stellar progenitor is lost in the ism , by the time the jet reaches the surface of the star , the beam has accelerated to @xmath82 ( see sect . [ sec : acceleration ] ) and , thus , it has entered into a ballistic regime , where lateral expansion is strongly suppressed . thereby , the jet remains well collimated as it propagates through the ism , and the half - opening angle of the beam reaches only a few degrees . these collimation properties have been confirmed by means of numerical models with better resolution in the @xmath83-direction ( see appendix ) . s. both , the vertical and the horizontal axis are scaled by @xmath80 cm . by this time , the head of the jet has already broken out of the progenitor surface , and the bow shock begins to expand into the ism . [ fig : contour2 ] ] in this section we focus on the dynamics of the acceleration of the flow to large lorentz factors @xmath84 , as requested by the the standard fireball model ( e.g. , * ? ? ? within such a theoretical model , an initial release of thermal energy is later converted into kinetic energy of the flow , as it expands into a dilute and cold environment . on the basis of this model , a number of numerical results have assessed that a jet , injected through a pre - established nozzle , is able to drill its way through a collapsar can reach an ultrarelativistic regime under likely inflow conditions ( * ? ? ? * ; * ? ? ? * ; * ? ? ? * and this paper ) . we focus here on the details of the dynamical phase in which a kinematically mildly relativistic jet ( @xmath85 ) speeds up to the ultrarelativistic regime @xmath84 converting its initial ( thermal ) energy @xmath86 into kinetic energy . to drive the discussion , we take as a prototype case that of model he16n . the acceleration process of other models is very similar . figures [ fig:1dprofiles]a-[fig:1dprofiles]c show one dimensional profiles of density , pressure , and lorentz factor along the polar axis , at different times . these profiles are qualitatively similar to the ones shown in previous papers ( e.g. , * ? ? ? * ; * ? ? ? * ) . during the injection the lorentz factor increases linearly ( fig . [ fig:1dprofiles]c ) , whereas both the density and pressure decrease as @xmath87 and @xmath88 ( fig . [ fig:1dprofiles]a , b ) , respectively . this is not unexpected , since the fluid expands radially ( almost freely ) in that region , where there are no shocks . note that this result differs a bit from models where the generation of the outflow is considered @xcite . if jet injection conditions are set through a nozzle at a certain distance to the center , the variability imprinted by the highly dynamical generation of the jet is erased . clearly , this minimizes the number of internal shocks in the outflowing jet . @xcite show in their fig . 2 that the outflow can accelerate to lorentz factors which are smaller than those attained in this work and in others where jet injection conditions are assumed . this is because of the modulation in the growth of the lorentz factor imposed by the development of kelvin - helmholtz modes in the course of the very early outflow evolution @xcite . nevertheless , models which consistently include the outflow generation ( by thermal energy deposition ) , can still accelerate to ultrarelativistic lorentz factors at large distance from the source . in such models the flow is kept hotter than in the present ones because of the occurrence of internal shocks already inside of the progenitor . however , since the outflow is optically thick , the thermal energy is not radiated away , but converted ( later ) into kinetic energy @xcite . ( d ) along the polar axis at times @xmath891.0 , 3.0 , 3.5 , 4.5 and 5.0s ( see legends ) , corresponding to model he16n . [ fig:1dprofiles ] ] one can notice that the maximum lorentz factor is reached behind the first recollimation shock in the jet ( figs . [ fig : contour1_2 ] and [ fig:1dprofiles]c ) coinciding with the location of the minimum density in the beam of the jet ( figs . [ fig : contour ] and [ fig:1dprofiles]a ) . inside of the progenitor star the jet reaches a maximum lorentz factor @xmath90 ( see dashed line in fig . [ fig:1dprofiles]c corresponding to @xmath91s ) . we note that the quick acceleration to large lorentz factor happening in the present models is a direct result of the fact that the flow is injected into a finite opening angle . differently , @xcite , who injected the jet parallel to the polar axis of the progenitor , did not find such a fast acceleration . in the later case , the occurrence of recollimation shocks much closer to the injection nozzle prevents the development a freely expanding , unshocked jet extending so far away as in the models of this work . indeed , such internal , recollimation shocks are the responsible of the confinement of the jet @xcite . nonetheless , the head propagation speed is very similar in @xcite as it is found here and , hence , also the crossing time of the progenitor by the jet . in @xcite an acceleration of the jet to @xmath84 at larger distances from the source happens ( by thermal - to - kinetic energy conversion ) , but it still takes place inside of the progenitor star . this resembles more the acceleration process of outflows generated self - consistently from the inner engine as pointed out above . the absolute maximum lorentz factor ( @xmath92 ) is attained by the time we begin to reduce the luminosity of the inflow ( at @xmath93s , see [ sec : numerics ] ) . by this time , the head of the jet has already broken up the surface of the progenitor . this numerical value is smaller than the maximum one which could be potentially reached , provided the conditions of the injected outflow , according to which @xmath94 , which is an indication of the maximum potentially reachable lorentz factor , along a flow line , if all thermal energy were converted to kinetic energy . after the energy deposition is switched off ( @xmath91s ) , the region previously occupied by the unshocked beam of the jet ( up to the first recollimation shock of the beam ; fig . [ fig:1dprofiles]c ) begins to shrink and to incorporate mass from its lateral boundaries ( c.f . , rest - mass density profiles between times 3.5s and 5.0s in fig . [ fig:1dprofiles]a up to @xmath95 cm ) . this fact reduces progressively both the beam lorentz factor ( fig . [ fig:1dprofiles]c ) and the ability of the beam fluid to reach large asymptotic values of @xmath5 ( fig . [ fig:1dprofiles]d ) . thereby , if the jet is injected with a finite half - opening angle , the ultrarelativistic part of the outflow ( i.e. , that where @xmath84 ) may persist only if the activity of the engine does not cease . the evolution of our models has been followed until the head of the jet reaches the outermost radial computational boundary at @xmath96 cm . at such distances , the outflow is still optically thick and , therefore , radiation can not escape freely . the jet and the surrounding cocoon , past a transient phase , that happens after the jet breaks out of the stellar surface , enter into a quasi self - similar phase , where the properties of the outflow roughly scale with the distance of propagation and become almost time independent . this fact allows us to extrapolate the properties of the jet from distances @xmath97 cm to the typical distances where the afterglow will take place ( namely , @xmath98 cm ) . thus , we can roughly estimate the angular energy distribution per unit solid angle ( @xmath16 ) that can be potentially emitted at the afterglow phase . in order to derive @xmath16 we have to make several assumptions . first , we assume that a fix fraction ( the same everywhere ) of the total energy ( internal plus kinetic ) will be converted into electromagnetic radiation . basically , this assumption is equivalent to state that the angular profile of the observed non - thermal radiation is simply a scaled version of the total energy angular profile . certainly , this is a rough approximation , since the non - thermal radiation from @xmath99-rays to radio frequencies will be produced by synchrotron ( and , perhaps , inverse compton ) processes of particles accelerated at shocks ( or , maybe , along the jet boundary layer ; e.g. , @xcite ) . obviously , there are shocks of very different properties in the ultrarelativistic beam and in the cocoon and , thus , we may expect somewhat different conversion efficiencies of the outflow energy into radiation in the beam and in the cocoon . finally , we assume that the angular energy distribution is _ frozen - in _ by the time when the head of the jet reaches the outer computational boundary . as commented above , our models evolve almost self - similarly a bit after they break out of the stellar surface , and therefore , we expect only a minor time evolution of the angular profiles of @xmath16 . we point out that the procedure we use to estimate @xmath16 differs from that of @xcite , who derived their @xmath16 profiles from the time integration of the energy flux trough a certain radius . under the hypothesis of self - similar evolution , this is equivalent to integrate , along the radial direction , the energy density of our models ( by the time they reach @xmath44 ) as follows , @xmath100 where the subscripts @xmath101 and @xmath102 are associated to the spherical grid coordinates @xmath103 , and @xmath104 , respectively , and @xmath105 and @xmath83 are the radial velocity in units of @xmath29 and the observer s viewing angle ( measured from the jet axis ) , respectively . the expression ( [ eq : dedomega ] ) includes the radiation contributions coming from regions outside of the line of sight ( see @xcite ) . the summation in the radial direction runs from the surface of the progenitor , located at @xmath106 or , equivalently , @xmath107 , to the outermost boundary . the summation in the azimuthal angle runs from @xmath108 to @xmath109 ( note that due to the assumed axial symmetry , we can copy the computed data of the quadrant @xmath110 to the quadrant @xmath111 ) . in order to avoid accounting for subrelativistic regions , which will not contribute to the afterglow energetics , we exclude the contributions of numerical cells where @xmath112 and @xmath113 in the expression ( [ eq : dedomega ] ) . have the potential to accelerate , at most , to @xmath114 . in actuality , the asymptotic lorentz factor of such parcels of fluid , will be much smaller , since they will decelerate as they incorporate mass from the external medium . ] the absolute value of the observed @xmath115 along every radial direction forming an angle @xmath83 with the polar axis depends , among other things , on two parameters whose exact value is not well constrained , neither by observations nor by the present day theory . these are ( i ) the efficiency of energy conversion to radiation , and ( ii ) the total energy injected . therefore , we will show only the angular profiles of @xmath115 normalized to the maximum value @xmath116 found for each model . figure [ fig : distri1 ] shows the normalized angular energy distributions corresponding to models he16c , he16l and he16n , which are prototypes of the types l , m and h , respectively . in the same figure we overplot fits to the normalized @xmath115 profiles . the fitting function is a smoothly broken power law ( sbp ) of the form , @xmath117^{1/n},\end{aligned}\ ] ] where @xmath118 is the value of the function @xmath119 at @xmath120 , @xmath121 is the angular location of the break point between the prebreak and postbreak power - laws , whose slopes are @xmath122 and @xmath123 , respectively , and @xmath124 is a numerical factor that controls the sharpness of the break . note that the maximum value of @xmath119 occurs at @xmath125 } , \label{eq : theta_max}\ ] ] when @xmath126 . otherwise , if @xmath127 and @xmath128 , the function diverges as @xmath129 . by inspection of fig . [ fig : distri1 ] , the angular energy distributions are remarkably well fitted by the function of eq . ( [ eq : fitting ] ) in the interval @xmath130 , i.e. , in the angular region occupied by the beam of the jet . at smaller latitudes ( @xmath131 ) the model data separates from the fitting function and presents systematically larger values than the latter . indeed , the data in such an interval can be well fitted by a simple power law , with a slope in the range @xmath132 $ ] ( see the inset in fig . [ fig : distri1 ] ) . the deviation from the spb function in this angular range is due to the contribution of the expanding , mildly relativistic cocoon . this cocoon contribution shows up more clearly ( the energy distribution tends to flatten in the range @xmath133 ) if we lower the thresholds on the values of @xmath134 and @xmath62 used to compute the angular energy distribution of each model . however , as we argued before , lowering these thresholds too much will catch up contributions from numerical cells whose asymptotic lorentz factor is too small to account for typical afterglow . low metallicity models 16 tb , 16tc , and 16oc , are also well fit by the function of eq . ( [ eq : fitting ] ) in roughly the same interval as the solar metallicity models ( fig . [ fig : distri2 ] ) . the values of the fit parameters are comparable to those of type - h models , with which they share a very similar progenitor mass ( @xmath135 ) . however , in these models , the cocoon contribution , which can be fit by a simple power - law with a slope @xmath136 between @xmath137 ( fig . [ fig : distri2 ] inset ) shows a faster decay of @xmath16 than type - h models for @xmath138 . the reason for this difference is the much deeper density drop of low - metallicity models close to the star surface ( fig . [ fig:1dradialmass2 ] ) compared with type - h models ( fig . [ fig:1dradialmass ] ) . the density of low - metallicity models in the region @xmath139@xmath140 cm is @xmath141 times smaller than in type - h models . hence , the beam of jets in such low - metallicity progenitors becomes much more ballistic than the corresponding beams of jets in the type - h group . since more ballistic beams reduce the sideways expansion of their cocoons , this explains that the angular energy distribution in low - metallicity stars is more narrowly concentrated than in solar - metallicity , type - h progenitors . in order to show more clearly the existence of correlations between the properties of the progenitor star and the @xmath16 distribution , we show in fig . [ fig : mass_al ] the dependence of the postbreak slope @xmath123 and on the stellar progenitor mass @xmath142 . there exists a correlation between @xmath123 and @xmath142 , such that the slope of lighter progenitors is steeper than that of heavier ones . there is roughly a linear dependence of @xmath123 on @xmath142 , which displays a relatively large dispersion . the reason for the dispersion being that for very similar values of the total progenitor mass , the rest - mass density radial profiles can be appreciably different ( see , e.g. , figs . [ fig:1dradialmass ] and [ fig:1dradialmass2 ] ) . this is particularly true in heavy progenitor models ( including type - h and low metallicity models ) . for the prebreak slope @xmath122 we find no obvious correlation with the progenitor mas , but in all the models considered here is very small ( @xmath143 ; tab . [ tab : progenitor ] ) . on the total mass of the progenitor . we identify the models by the last letter in the model name , e.g. , the label a stands for model he16a . the labels tb , tc , and oc stand for the models 16 tb , 16tc , and 16oc , respectively . [ fig : mass_al ] ] on the mass loss rate in units of the _ typical _ mass loss rate considered by @xcite . the naming convention is the same as in fig . [ fig : mass_al ] . [ fig : massloss_al ] ] we have also investigated the dependence of the slope @xmath123 and @xmath122 on the mass loss rate @xmath144 assumed in models of @xcite ( see tab . [ tab : progenitor ] ) . figure [ fig : massloss_al ] shows that there exist a good correlation between @xmath123 and @xmath144 , while @xmath122 seems to be independent of @xmath144 . the @xmath145 correlation tells us that , models with a larger mass loss rate posses a steeper slopes . this is not surprising considering the previously found correlation between @xmath123 and @xmath142 , since the stellar progenitor mass is mostly determined by the amount of mass lost in the form of winds during the latest stages of its evolution . we have not found any other good correlation between the fit parameters ( other than @xmath123 ) and the gross properties of the progenitors ( radius , average density , total angular momentum , rotation period , mass of the iron core , etc . ) . we shall note that the angular distribution @xmath16 is not directly observable . instead , the isotropic equivalent angular energy per solid angle @xmath146 can be detected . figure [ fig : distri3 ] shows the equivalent isotropic angular energy distribution for prototype models belonging to the types l , m and h , normalized to the value of the distribution at @xmath147 . as can be seen from fig . [ fig : distri3 ] ( see also tab . [ tab : progenitor ] ) , the values of @xmath148 are also negative for the @xmath146 , while the corresponding values @xmath122 are close to zerofor @xmath16 ( approximately , @xmath148@xmath149 ) . this happens because of the small value of the solid angles close to the symmetry axis , which makes systematically larger the higher latitude values of @xmath146 than those of @xmath16 . the values of @xmath150 are systematically smaller than the respective @xmath123 values ( roughly , @xmath150@xmath151 in our standard resolution runs ) . given the tight relations of the slopes in the @xmath16 and @xmath146 distributions , it is not surprising to find that there exists a good correlation between @xmath150 and @xmath142 ( fig . [ fig : mass_al2 ] ) which follows the same qualitative trend as the correlation between @xmath123 and @xmath142 . but for the equivalent isotropic energy distribution . [ fig : mass_al2 ] ] to sum up our findings so far , we realize that the values of the parameters of the fit function ( eq.[eq : fitting ] ) are chiefly correlated with the mass of the progenitor . in type - l progenitors , the velocity of propagation of the jet is larger than in more massive models , which results in jets developing more massive and hotter cocoons in type - m and type - h models than in light progenitors . this fact is evident by looking at fig . [ fig : comparison ] , where the pressure in the cocoon of the model he16n is higher than that in the model he16c in the course of the propagation of the jet up to the progenitor surface . the differences in the propagation velocity of the injected jets are set , to a large extend , by the differences in the average density between progenitors of different mass . the less massive progenitors of our sample ( type - l models ) tend to have the smaller average densities ( see , e.g. , fig . [ fig:1dradialmass ] , where model he16c displays a smaller density than model he16n at every radial point ) . consistently , jets propagating in type - l progenitors are faster as can be observed in fig . [ fig : average_velo ] . however , the time needed to reach the progenitor surface presents a dependence with the progenitor mass with a much larger scatter ( fig . [ fig : time_cross ] ) . particularly , models he16a , he16f , he16 g and he16h , all of which belong to the type - h group , display a progenitor crossing time comparable to that of the jets in the type - l group ( although its average propagation velocity is comparable to that of models of type - h ) . that s the reason why these models appear as _ outliers _ in the correlations between the slope @xmath152 and the progenitor mass ( figs . [ fig : mass_al ] and with the mass - loss rate ( fig . [ fig : massloss_al ] ) . since more massive progenitor stars yield hotter cocoons , once the cocoon is erupted through the stellar surface , it undergoes a larger lateral expansion ( compare models he16c and he16n in the lower panels of fig . [ fig:1dradialmass ] ) , i.e. , the energy carried by the jet spreads towards lower latitudes . this explains why the energy per solid angle is more concentrated towards the axis when the jet crosses a low mass progenitor . s ) and he16n ( right : at @xmath153s ) , when the head of the jet reaches progenitor surface in each model , i.e. , at shock break out . both , the vertical and the horizontal axis are scaled by @xmath80 cm . the radii of the progenitors are @xmath154 cm and @xmath155 cm for the models he16c and he16n , respectively . at the same distance @xmath156 from the injection nozzle , the pressure in both the jet and in the cocoon is higher for model he16n than for model he16c . this fact explains the larger lateral expansion after shock break out in model he16n than in model he16c . [ fig : comparison ] ] if we assume that the detectability of an event , for an observer looking such event at a certain viewing angle ( @xmath83 ) , is proportional to @xmath157 ( see , e.g. , @xcite ) , it turns out that @xmath83 should be rather small ( @xmath158 ; fig . [ fig : detecta ] ) to observe and event produced in collapsar progenitors ( note that the flanks of @xmath146 distribution are quite steep and , thus , it is very unlikely to detect events which are not directly pointing towards the observer ) . jets produced in type - l progenitors exhibit narrower observability profiles than those injected in more massive starts ( c.f . , compare the profiles of models he16c and he16n in fig . [ fig : detecta ] ) . therefore , it is more unlikely to detect off - axis events produced in light progenitors than in more massive ones . alternatively , we may state the the lower degree of collimation of relativistic jets in high - mass stars results in a higher probability of observing an event from a high - mass progenitor than a low - mass progenitor . we also find that the metallicity has little influence on the observability , because for similar progenitor masses , solar - metallicity models and low metallicity ones display almost identical observability angular profiles ( fig . [ fig : detecta ] ) . we also notice the very different observability profiles of jets produced in collapsars and jets produced in remnants of mergers of compact objects . in the latter case , a non - negligible observability is obtained in the viewing angle interval @xmath159 , because of the larger half - opening angles of the outflows originated in progenitors of short grbs ( see fig . 2 of @xcite ) . another relevant difference is that collapsar - jets show a detectability with a fast rise close to @xmath160 , followed by a shallower decay beyond the most probable detection angle @xmath161 . in contrast , most jets produced in merger remnants tend to decay very abruptly beyond the most likely observing angle , and have an observability rise much more moderate than that of collapsar - jets . the reason for this difference is the much shallower decay of the isotropic equivalent energy at the flanks of the beam in collapsar - jets than in jets from merger remnants . in this paper we explore the relation of the dynamical properties of ultrarelativistic jets generated in collapsars with the properties of the progenitor stars in which such jets propagate . we have particularly focused on the correlations that exist between the angular profile of the energy per solid angle ( as seen in the afterglow phase ) and the properties of the progenitors . along the way , we have pointed out which is the relevance of the fact that our numerical models are set up with a finite injection half - opening angle . using a non - zero injection angle affects the way in which the conversion of internal - to - kinetic energy takes place . if the flow is injected parallel to the polar axis , the development of reconfinement shocks happens closer to the injection nozzle than if the flow is injected radially within a cone of finite half - opening angle . when the recollimation shock occurs far away from the nozzle , the unshocked beam flow accelerates along a larger distance in a rarefaction that precedes such shock . thus , the beam reaches there larger lorentz factors than if the jet is injected parallel to the polar axis . the dissipation in cross shocks acts by simply recycling part of the kinetic energy of the outflow into thermal energy . the thermal energy is not lost , since the jet propagation is roughly adiabatic inside of the progenitor ( radiation losses are negligible there ) . instead , this thermal energy can be further converted into kinetic energy at larger distances . this process may happen several times before the outflow becomes transparent and radiation can freely escape . this explains why , in spite of the differences in the beam dynamics , all models ( independent of the injection half - opening angle ; c.f . , @xcite ) develop a roughly similar propagation speed and , by the time they reach the head of the jet , the gross properties of the outflow are similar . we have estimated the angular distribution of energy per solid angle in the afterglow phase by extrapolation of the state of our models when they reach a distance of @xmath162 cm . this extrapolation relies on the fact that the jets develop a rough self - similar behavior soon after they emerge from the progenitor surface . our results show that the equivalent isotropic energy per solid angle @xmath163 is only partly consistent with that of @xcite . however , the results in this paper do appear to be consistent with previous numerical simulations such as those of @xcite and @xcite . @xcite obtain that the angular energy distribution displays a relatively flat core which is flanked by a region where @xmath164 . our results show that the core of the distribution ( close to @xmath36 ) is not flat ( but decays as @xmath165 ) and that the energy per solid angle decays much faster than @xmath17 ( it does it as @xmath166 , with a value of @xmath167 depending on the mass of the progenitor ; see below ) . we can fit the @xmath146-data with sbp functions up to @xmath168 . at smaller latitudes , a simple power - law with a slope close to @xmath169 fits better the data . in this region ( @xmath170 ) , the cocoon contribution is the dominant ( at smaller values of @xmath83 , the beam of the jet dominates the energetics ) , and we find there the best consistency with the model of @xcite . we have correlated the properties of the angular @xmath163 distribution of the jet with the fundamental parameters of the progenitor star in which the jet has propagated . we find that the shape of the distribution is mostly influenced by the mass of the progenitor . when the mass of the progenitor is small ( @xmath171 ) , because of the occurrence of large mass losses due to winds in the latest stages of the star s evolution , then @xmath146 decays faster with @xmath83 than if the mass of the progenitor is large ( @xmath172 ) . we find that the reason for this behavior is that the average density of the progenitors tends to grow ( approximately ) with the mass . this means that the average jet propagation speed inside of the star is smaller , the larger is the mass . a smaller jet propagation speed results into thicker and hotter cocoons , since we fix the same mass , momentum and energy fluxes at the nozzle for all models . also the beam of the jet in the more massive progenitors is wider . this is the reason why low mass progenitors develop narrower @xmath146 profiles than high mass ones . the difference in the collimation of the energy distributions resulting from low and high mass progenitors has a direct influence on the number of observed events . we expect to see more events produced in heavy progenitors than in low mass ones . one could question whether the correlation that we have found between the mass of the progenitor and the width of the @xmath163 is an artifact of our numerical set up . we are fixing the luminosity of the jet to be the same independent of the progenitor mass . however , progenitors of different mass may develop central engines which release different power . a good proxy of the power of the central engine is the mass of the iron core of the progenitor . one may expect that the collapse of more massive iron cores results into larger central compact objects . if the power released by the central engine is dominated by the size of the central compact object , then models with more massive iron cores could release a larger power than models with low mass cores . then , how do we justify our numerical assumption that the power injected in the jet is roughly independent of the progenitor s mass ? . in support of our point we argue that , first , according to @xcite data , there is not a one to one correlation between the mass of the iron core and the mass of the progenitor and , second , the iron mass varies by less than a 30% in all the models considered here , while the total mass can be different by a factor of 3 . thus , within the simplifications we do in our models , the assumption of a common luminosity independent of the mass of the progenitor is justified . irrespective of these two arguments , if heavier progenitors would result into more luminous central engines , we shall point out that this trend will also result into wider jets and cocoons . this is a result early pointed out by @xcite , were it was shown that , taking the same progenitor , but increasing the luminosity of the central engine by a factor of 10 , results into thicker cocoons than in cases in which the luminosity is more moderate . the reason being that the release of a large power triggers large amplitude kelvin - helmholtz instabilities at the basis of the outflow , which transfer a large fraction of the momentum of the beam to a thicker shear layer between the beam and the cocoon . effectively , this process widens the cross sectional area of the beam , reducing its propagation speed and , consistently inflating larger cocoons . we have also found , that comparing progenitors of similar mass , the metallicity of the star has a small impact on the extrapolated @xmath146 profiles . the collimation of the jet is similar regardless of the stellar metallicity . however , the cocoon is more narrowly collimated in low - metallicity stars , because of the large density drop close to their surfaces . this reduced density makes the jets in low - metallicity stars much more ballistic , once they break out the stellar surface , than in solar - metallicity progenitors . unfortunately , this difference might not be observable , since it happens in regions where the energy per solid angle is much smaller than at the jet core ( unless orphan afterglows could be detected ; see , e.g. , @xcite ) . finally , we have found significant differences between the @xmath146 profiles of collapsar - jets and those of jets produced in merger remnants ( i.e. , between the angular energy distribution of jets associated to long and to short grbs ) . collapsar - jets are more narrowly collimated than jets of merger remnants , and the decay of @xmath146 beyond the central flat core is much steeper in the latter than in the former . this intrinsic difference manifest itself as a larger chance of detectability of jets from merger remnants at viewing angles up to @xmath173 , while , on the other hand , collapsar - jets could hardly be seen off - axis . we would like to thank a. heger for his kindness to allow us to use his progenitor models for this study . we would like to thank dr . hanawa for his helpful comments . we also thank the referee for his / her useful comments that helped to improve this paper . this work is partly supported by the grants - in - aid of the japanese ministry of education , science , culture , and sport ( 19540236 , 20041002 , 20340040 a.m. ) , the grants of the spanish ministry of science and innovation csd2007 - 00050 and aya2007 - 67626-c03 - 01 . m. a. a. is a ramn y cajal fellow of the spanish ministry of education and science . the calculations were carried out on a nec sx8 , at the cybermedia center , osaka university , on a nec sx8 , at yukawa institute for theoretical physics , kyoto university , on the space science simulator ( nec sx6 ) , at the japan aerospace exploration agency , and on a nex sx9 , at the center for computational astrophysics , national astronomical observatory of japan . the main purpose of this appendix is to justify the choice of numerical resolution that we have used in the main body of this paper . we have picked up three models , he16c , he16l and he16n ( representative of the models of the respective types - l , -m and h ) and performed a resolution study by progressively increasing the numerical resolution . our standard models have a working resolution @xmath174 zones . the standard computational grid is uniform in both @xmath39- and @xmath83-coordinates . in addition to the standard resolution , two higher resolutions have been considered . first , we compute models he16c - m , he16l - m , and he16n - m ( tab . [ tab : progenitor ] ) , which have an intermediate resolution of @xmath175 zones . in these models the radial grid is uniform in @xmath39 and the smallest radial grid spacing is @xmath176 cm , i.e. , 3/4 times smaller than that of our standard resolution cases . the polar grid possesses a uniform region close to the symmetry axis ( @xmath177 ) , where @xmath178 , followed by a uniformly spaced region in @xmath179 ( @xmath180 ) . the reason to consider two different regions in the @xmath83-spacing is that all our jet models develop cocoons whose angular extension is @xmath181 , i.e. , all the dynamics develops in a wedge covered by a finer mesh in our computational grid . thus , an increased resolution in the abovementioned wedge surrounding the polar axis yields an effective increase of the numerical resolution in the whole computational grid . even higher resolution models ( he16c - h , he16l - h , and he16n - h ) have also been computed . in this case , the grid consists of @xmath182 zones uniformly spaced in @xmath39 , with a minimum radial grid spacing @xmath183 cm , i.e. , one half of the width of smallest radial spacing of the standard resolution case . we take in this case @xmath47 zones , also split in two regions : a uniform grid in the interval @xmath184 ( @xmath185 ) , followed by a uniformly spaced region in @xmath179 ( @xmath180 ) . going to even larger resolutions in the @xmath83-direction increases the total computational time up to prohibitive limits due to the increased number of time steps associated to the fulfillment of the courant condition . we note that the jet dynamics is rather independent of the resolution , and also the gross morphological features are converged at the standard resolution , though , of course , finer details show up both in the cocoon and in the beam ( see fig . [ fig : pressure_reso ] ) . therefore , when we take radial averages to compute the angular energy distribution of our models ( eq . [ eq : dedomega ] ) , the differences are relatively small ( see below ) , and our standard resolution models can be considered to be sufficiently resolved to account for such global energetic properties . the time - scales to cross the progenitor and/or to reach the outer computational boundary ( at @xmath186 cm ) are slightly different from each other . since the jet propagates at slightly different speed depending on the resolution , the three snapshots correspond to slightly different evolutionary times . both , the vertical and the horizontal axis are scaled by @xmath80 cm . [ fig : pressure_reso ] ] figure [ fig : energy_distri_reso ] shows the isotropic equivalent angular energy distributions of models computed with three different resolutions . in all cases , a sbp function ( eq . [ eq : fitting ] ) fits properly the data up to @xmath187 . at larger latitude , the contribution of the expanding cocoon component dominates . since each distribution is normalized independently to its absolute maximum , the distributions corresponding to different resolutions do not overlap , but they show the same shape . the fitting parameters for the higher resolution cases are also listed in tab . [ tab : progenitor ] . there , we can see that the same correlations found in models with the standard resolution are reproduced at higher resolutions , namely , the correlation between @xmath188 or @xmath150with the progenitor mass ( sect . [ sec : dedomega ] ) . however , at higher resolution , we could guess additional correlations which are not obvious in models run with the standard resolution . particularly , the parameters @xmath121 and @xmath189 become smaller as the progenitor mass increases . this trend is however , an artifact of the models chosen as prototypes of each mass type . they are such , that the radius of the progenitor roughly grows with the mass . as pointed out by @xcite , progenitors with larger radii provide a larger inertial confinement which prevents the lateral expansion of the jet . hence , for the models chosen in this resolution study , a larger progenitor mass yields better collimated jets . taking the whole sample of models , but run at higher resolution , the correlations between @xmath121 or @xmath189 would not exist . cccc|ccccc|ccccc model & mass & total & radius & @xmath122 & @xmath123 & @xmath190 & @xmath191 & @xmath124 & @xmath192 & @xmath193 & @xmath194 & @xmath195 & @xmath196 + & loss & mass /@xmath197 & /@xmath198 cm & /@xmath199 & & ergs & & & & & ergs & & + he16a & 0 & 15.70 & 3.86 & -1.22 & -3.27 & 8.65 & 2.37 & -0.686 & -1.01 & -4.24 & 2.45 & 2.35 & -0.696 + he16b & 1.0 & 5.10 & 4.91 & -1.72 & -3.68 & 6.02 & 2.12 & -0.632 & -1.02 & -4.45 & 2.14 & 2.02 & -0.694 + he16c & 1.0 & 5.15 & 4.75 & -2.99 & -3.58 & 6.61 & 2.03 & -0.706 & -1.03 & -4.38 & 2.40 & 1.95 & -0.765 + he16d & 0.3 & 9.53 & 4.42 & -1.22 & -3.05 & 8.84 & 2.21 & -0.746 & -1.01 & -3.99 & 2.72 & 2.18 & -0.768 + he16e & 0.1 & 12.86 & 5.71 & -2.60 & -2.73 & 9.64 & 1.84 & -0.896 & -1.03 & -3.64 & 3.64&1.80 & -0.942 + he16f & 0.03 & 14.80 & 3.52 & -1.33 & -3.15 & 9.28 & 2.41 & -0.726 & -1.01 & -4.09 & 2.63&2.37 & -0.746 + he16 g & 0.01 & 15.56 & 3.31 & -1.27 & -3.26 & 8.55 & 2.44 & -0.719 & -1.01 & -4.30 & 2.24&2.47 & -0.705 + he16h & 0 & 15.68 & 3.31 & -1.71 & -3.41 & 8.96 & 2.34 & -0.690 & -1.02 & -4.37 & 2.58&2.32 & -0.701 + he16i & 0 & 15.88 & 4.54 & -1.21 & -3.17 & 8.12 & 2.33 & -0.714 & -1.01 & -4.14 & 2.32&2.32 & -0.722 + he16j & 1.0 & 5.13 & 4.81 & -3.64 & -3.47 & 7.23 & 1.89 & -0.764 & -1.04 & -4.29 & 2.77&1.82 & -0.824 + he16k & 1.0 & 5.16 & 4.81 & -3.49 & -3.57 & 6.67 & 2.02 & -0.741 & -1.04 & -4.38 & 2.39&1.95 & -0.798 + he16l & 0.3 & 9.58 & 4.18 & -1.01 & -3.19 & 8.51 & 2.27 & -0.707 & -1.01 & -4.15 & 2.53&2.24 & -0.721 + he16 m & 0.1 & 13.04 & 6.29 & -2.29 & -2.80 & 8.89 & 1.93 & -0.851 & -1.03 & -3.69 & 3.24&1.88 & -0.903 + he16n & 0.03 & 14.95 & 6.17 & -2.91 & -2.73 & 9.92 & 1.79 & -0.914 & -1.03 & -3.64 & 3.86&1.75 & -0.965 + he16o & 0.01 & 15.62 & 5.96 & -1.88 & -2.74 & 9.11 & 1.99 & -0.855 & -1.02 & -3.67 & 3.16&1.94 & -0.891 + he16p & 0 & 15.88 & 4.63 & -1.40 & -2.99 & 9.78 & 2.18 & -0.762 & -1.01 & -3.93 & 3.05&2.15 & -0.781 + 16 tb & 0.1 & 15.29 & 4.45 & -1.42 & -3.14 & 8.38 & 2.19 & -0.733 & -1.01 & -4.11 & 2.55&2.18 & -0.741 + 16tc & 0.1 & 15.23 & 4.87 & -1.46 & -3.14 & 7.51 & 2.30 & -0.735 & -1.02 & -4.08 & 2.22&2.27 & -0.754 + 16oc & 0.1 & 14.26 & 4.42 & -1.62 & -2.95 & 9.05 & 2.09 & -0.798 & -1.02 & -3.94 & 2.85&2.09 & -0.798 + he16c - m & 1.0 & 5.15 & 4.75 & -0.72 & -5.77 & 6.71 & 2.70 & -0.486 & -1.01 & -9.81 & 0.310&3.43 & -0.241 + he16l - m & 0.3 & 9.58 & 4.18 & -1.16 & -4.45 & 11.8 & 2.40 & -0.572 & -1.01 & -5.04 & 2.97&2.29 & -0.569 + he16n - m & 0.03 & 14.95 & 6.17 & -1.27 & -3.33 & 17.9 & 1.65 & -0.685 & -1.01 & -4.41 & 3.64&2.06 & -0.645 + he16c - h & 1.0 & 5.15 & 4.75 & -1.99 & -6.18 & 4.40 & 2.64 & -0.377 & -1.00 & -7.76 & 1.08&2.98 & -0.397 + he16l - h & 0.3 & 9.58 & 4.18 & -1.04 & -3.90 & 11.0 & 2.22 & -0.600 & -1.01 & -6.36 & 2.04&2.74 & -0.446 + he16n - h & 0.03 & 14.95 & 6.17 & -1.18 & -3.22 & 12.4 & 1.96 & -0.704 & -1.01 & -4.62 & 5.74&1.79 & -0.596 +
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collapsars are fast - spinning , massive stars , whose core collapse liberates an energy , that can be channeled in the form of ultrarelativistic jets .
these jets transport the energy from the collapsed core to large distances , where it is dissipated in the form of long - duration gamma - ray bursts . in this paper
we study the dynamics of ultrarelativistic jets produced in collapsars .
also we extrapolate our results to infer the angular energy distribution of the produced outflows in the afterglow phase .
our main focus is to look for global energetical properties which can be imprinted by the different structure of different progenitor stars .
thus , we employ a number of pre - supernova , stellar models ( with distinct masses and metallicities ) , and inject in all of them jets with fixed initial conditions .
we assume that at the injection nozzle , the jet is mildly relativistic ( lorentz factor @xmath0 ) , has a finite half - opening angle ( @xmath1 ) , and carries a power of @xmath2ergs@xmath3 . in all cases ,
well collimated jets propagate through the progenitor , blowing a high pressure and high temperature cocoon .
these jets arrive intact to the stellar surface and break out of it .
a large lorentz factor region @xmath4 develops well before the jet reaches the surface of the star , in the unshocked part of the beam , located between the injection nozzle and the first recollimation shock .
these high values of @xmath5 are possible because the finite opening angle of the jet allows for free expansion towards the radial direction .
we find a strong correlation between the angular energy distribution of the jet , after its eruption from the progenitor surface , and the mass of the progenitors .
the angular energy distribution of the jets from light progenitor models is steeper than that of the jets injected in more massive progenitor stars .
this trend is also imprinted in the angular distribution of isotropic equivalent energy .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
measurements of @xmath4 in semileptonic decays , @xmath5 in @xmath0@xmath1 mixing , and @xmath6 from cp violation in @xmath2@xmath3 and @xmath0@xmath1 mixing have firmly established the existence of a cp - violating phase in the ckm matrix . the present situation , often referred to as the `` standard analysis '' of the unitarity triangle , is summarized in figure [ fig : utfit ] . three comments are in order concerning this analysis : 1 . the measurements of cp asymmetries in kaon physics ( @xmath7 and @xmath8 ) and @xmath0@xmath1 mixing ( @xmath9 ) probe the imaginary part of @xmath10 and so establish cp violation in the top sector of the ckm matrix . the standard model predicts that the imaginary part of @xmath10 is related , by three - generation unitarity , to the imaginary part of @xmath11 , and that those two elements are ( to an excellent approximation ) the only sources of cp violation in flavor - changing processes . in order to test this prediction one must explore the phase @xmath12 in the bottom sector of the ckm matrix . 2 . with the exception of the @xmath9 measurement the standard analysis is limited by large theoretical uncertainties , which dominate the widths of the various bands in the figure . these uncertainties enter via the calculation of hadronic matrix elements . below i will discuss some novel methods to constrain the unitarity triangle using charmless hadronic @xmath0 decays , which are afflicted by smaller hadronic uncertainties and hence provide powerful new tests of the standard model , which can complement the standard analysis . 3 . with the exception of the measurement of @xmath4 the standard constraints are sensitive to meson antimeson mixing . mixing amplitudes are of second order in weak interactions and hence might be most susceptible to effects from physics beyond the standard model . the new constraints on @xmath13 discussed below allow a construction of the unitarity triangle that is over - constrained and independent of @xmath0@xmath1 and @xmath2@xmath3 mixing . it is in this sense complementary to the standard analysis . the phase @xmath14 can be probed via tree penguin interference in decays such as @xmath15 . experiment shows that amplitude interference is sizable in these decays . information about @xmath14 can be obtained from measurements of direct cp asymmetries ( @xmath16 ) , but also from the study of cp - averaged branching fractions ( @xmath17 ) . the challenge is , of course , to gain theoretical control over the hadronic physics entering the tree - to - penguin ratios in the various decays . recently , much progress has been made toward achieving that goal . hadronic weak decays simplify greatly in the heavy - quark limit @xmath18 . the underlying physics is that a fast - moving light meson produced by a point - like source ( the effective weak hamiltonian ) decouples from soft qcd interactions @xcite . a systematic implementation of this color transparency argument is provided by the qcd factorization approach @xcite , which makes rigorous predictions for hadronic @xmath0-decay amplitudes in the heavy - quark limit . one can hardly overemphasize the importance of controlling nonleptonic decay amplitudes in the heavy - quark limit . while a few years ago reliable calculations of such amplitudes appeared to be out of reach , we are now in a situation where hadronic uncertainties enter only at the level of power corrections suppressed by the heavy @xmath19-quark mass . in recent work , qcd factorization has been applied to the entire set of the 96 decays of @xmath0 and @xmath20 mesons into @xmath21 or @xmath22 final states ( @xmath23pseudoscalar meson , @xmath24vector meson ) @xcite . it has been demonstrated that the approach correctly reproduces the main features seen in the data , such as the magnitudes of the various tree and penguin amplitudes , and the fact that they have small relative strong - interaction phases . in the future , when more data become available , this will allow us to extract much useful information about the flavor sector of the standard model either from global fits or from analysis of certain classes of decay modes such as @xmath25 , @xmath26 , and @xmath27 . detailed comparison with the data may also reveal limitations of the heavy - quark expansion in certain modes , perhaps hinting at the significance of some power corrections in @xmath28 . despite of the success of qcd factorization in describing the data , there is an interest in analyzing ckm parameters using methods that rely as little as possible on an underlying theoretical framework . in this talk i discuss a method for constructing the unitarity triangle from @xmath0 physics using measurements whose theoretical interpretation is `` clean '' in the sense that it only relies on assumptions that can be tested using experimental data . i call this construction the cp-@xmath19 triangle , because it probes the existence of a cp - violating phase in the @xmath19 sector of the ckm matrix . the cp-@xmath19 triangle is over - determined and can be constructed using already existing data . most importantly , this construction is insensitive to potential new physics effects in @xmath0@xmath1 or @xmath2@xmath3 mixing . the present analysis is an update of @xcite using the most recent data as of summer 2003 . the first ingredient is the ratio @xmath29 extracted from semileptonic @xmath0 decays , whose current value is @xmath30 . several strategies have been proposed to determine @xmath4 with an accuracy of about 10% @xcite , which would be a significant improvement . the first plot in figure [ fig : cpt ] shows the corresponding constraint in the @xmath13 plane . here and below the narrow , dark - colored band shows the theoretical uncertainty , while the lighter band gives the current experimental value . the second ingredient is a constraint derived from the ratio of the cp - averaged branching fractions for the decays @xmath31 and @xmath32 , using a generalization of the method suggested in @xcite . the experimental inputs to this analysis are a certain tree - to - penguin ratio @xmath33 and the ratio @xmath34 } = 0.804\pm 0.085\ ] ] of two cp - averaged @xmath25 branching fractions @xcite . without any recourse to qcd factorization this method provides a bound on @xmath35 , which can be turned into a determination of @xmath35 ( for fixed value of @xmath36 ) when information on the relevant strong - interaction phase @xmath37 is available . the phase @xmath37 is bound by experimental data ( and very general theoretical arguments ) to be small , of order @xmath38 @xcite . ( in the future , this phase can be determined directly from the direct cp asymmetry in @xmath32 decays . ) it is thus conservative to assume that @xmath39 , corresponding to @xmath40 . with this assumption the corresponding allowed region in the @xmath13 plane was analyzed in @xcite . the resulting constraint is shown in the second plot in figure [ fig : cpt ] . the third constraint comes from a measurement of the time - dependent cp asymmetry @xmath41 in @xmath42 decays . the present experimental situation is still unclear , since the measurements by babar ( @xmath43 ) and belle ( @xmath44 ) are not in good agreement with each other @xcite . the naive average of these results gives @xmath45 . ( inflating the error according to the pdg prescription would yield @xmath46 , but for some reason the experimenters usually use the naive error without rescaling , and i will follow their example . ) the theoretical expression for the asymmetry is @xmath47 here @xmath48 is the cp - violating phase of the @xmath49@xmath50 mixing amplitude , which in the standard model equals @xmath51 . usually it is argued that for small @xmath52 ratio the quantity @xmath53 is approximately given by @xmath54 , and so apart from a `` penguin pollution '' the asymmetry @xmath55 . in order to become insensitive to possible new physics contributions to the mixing amplitude i adopt a different strategy @xcite . i use the measurement @xmath56 @xcite and write @xmath57 , with a sign ambiguity in the real part . ( the plus sign is suggested by the standard fit of the unitarity triangle . ) a measurement of @xmath58 can then be translated into a constraint on @xmath14 ( or @xmath59 and @xmath60 ) , which remains valid even if the @xmath61 measurement is affected by new physics . the result obtained with the current experimental values and assuming @xmath62 is shown in the third plot in figure [ fig : cpt ] . the resulting bands for @xmath63 are obtained by a reflection about the @xmath59 axis . this follows because the expression for @xmath58 is invariant under the simultaneous replacements @xmath64 and @xmath65 . each of the three constraints in figure [ fig : cpt ] are , at present , limited by rather large experimental errors , while comparison with figure [ fig : utfit ] shows that the theoretical limitations are smaller than for the standard analysis . yet , even at the present level of accuracy it is interesting to combine the three constraints and construct the resulting allowed regions for the apex of the unitarity triangle . the result is shown in the left - hand plot in figure [ fig : summary ] . note that the lines corresponding to the new constraints intersect the circles representing the @xmath4 constraint at large angles , indicating that the three measurements used in the construction of the cp-@xmath19 triangle provide highly complementary information on @xmath59 and @xmath60 . there are six ( partially overlapping ) allowed regions , three corresponding to @xmath62 ( dark shading ) and three to @xmath63 ( light shading ) . if we use the information that the measured value of @xmath7 requires a positive value of @xmath60 , then only the solutions in the upper half - plane remain . comparison with figure [ fig : utfit ] shows that one of these regions ( corresponding to @xmath62 ) is in perfect agreement with the standard fit . this is highly nontrivial , since with the exception of @xmath4 none of the standard constraints are used in this construction . interestingly , there is a second allowed region ( corresponding to @xmath63 ) which would be consistent with the constraint from @xmath7 but inconsistent with the constraints derived from @xmath9 and @xmath66 . such a solution would require a significant new physics contribution to @xmath0@xmath1 mixing . j. d. bjorken , nucl . suppl . * 11 * , 325 ( 1989 ) . h. d. politzer and m. b. wise , phys . b * 257 * , 399 ( 1991 ) . m. beneke , g. buchalla , m. neubert and c. t. sachrajda , phys . rev . lett . * 83 * , 1914 ( 1999 ) ; nucl . b * 591 * , 313 ( 2000 ) . m. beneke , g. buchalla , m. neubert and c. t. sachrajda , nucl . b * 606 * , 245 ( 2001 ) . m. beneke and m. neubert , nucl . phys . b * 651 * , 225 ( 2003 ) ; preprint hep - ph/0308039 . m. neubert , preprint hep - ph/0207327 . m. neubert , phys . rev . d * 49 * , 4623 ( 1994 ) ; phys . b * 543 * , 269 ( 2002 ) . r. d. dikeman and n. g. uraltsev , nucl . b * 509 * , 378 ( 1998 ) ; + i. i. bigi , r. d. dikeman and n. uraltsev , eur . j. c * 4 * , 453 ( 1998 ) . a. f. falk , z. ligeti and m. b. wise , phys . b * 406 * , 225 ( 1997 ) . c. w. bauer , z. ligeti and m. e. luke , phys . b * 479 * , 395 ( 2000 ) ; phys . d * 64 * , 113004 ( 2001 ) . m. neubert , jhep * 0007 * , 022 ( 2000 ) ; + m. neubert and t. becher , phys . b * 535 * , 127 ( 2002 ) . m. neubert and j. l. rosner , phys . b * 441 * , 403 ( 1998 ) ; phys . lett . * 81 * , 5076 ( 1998 ) ; m. neubert , jhep * 9902 * , 014 ( 1999 ) . i use the average of babar , belle , and cleo data as compiled in the second paper in @xcite .
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the study of charmless hadronic two - body decays of @xmath0 mesons is one of the most fascinating topics in @xmath0 physics .
a construction of the unitarity triangle based on such decays is presented , which is independent of @xmath0@xmath1 and @xmath2@xmath3 mixing .
it provides stringent tests of the standard model with small theoretical uncertainties .
clns 03/1841 address = f.r .
newman laboratory for elementary - particle physics + cornell university , ithaca , ny 14853 , usa
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the merging of two galaxies will produce a binary black hole at the center of the newly formed galaxy . if the two black holes do not stall , they will ultimately merge due to emission of gravitational wave radiation . the gravitational waves carry away linear momentum , causing the centre of mass of the coalescing bh system to recoil in the opposite direction ( peres 1962 , bekenstein 1973 ) . early analytical calculations predicted that mergers of non - spinning black holes can attain kicks with velocities of up to a few hundred kms@xmath0 ( e.g. , fitchett & detweiler 1984 , favata et al . 2004 , blanchet et al . 2005 , damour & gopakumar 2006 ) , recently confirmed by numerical simulations ( e.g. , baker et al . 2006 , herrmann et al . 2007a , gonzlez et al . these velocities are above the escape velocity of dwarf galaxies , low - mass spirals , and high - redshift dark matter halos . if many bhs were removed from their hosts in the early history of the universe , this would have profound consequences for galaxy assembly and bh growth in the early universe , and would give rise to a population of interstellar and intergalactic bhs ( e.g. , madau et al . 2004 , merritt et al . 2004 , madau & quataert 2004 , haiman 2004 , yoo & miralda - escud 2004 , volonteri & perna 2005 , volonteri & rees 2006 , libeskind et al . 2006 ) . recent numerical relativity simulations of certain configurations of merging , _ spinning _ bhs have produced much higher recoil velocities , up to several thousand kms@xmath0 ( campanelli et al . 2007a , b , gonzlez et al . 2007b , tichy & marronetti 2007 , herrmann et al . 2007b , dain et al . 2008 , schnittman et al . 2008 ) , scaling to an expected maximum around 4000 kms@xmath0 ( campanelli et al . 20007a , b , baker et al . 2008 ) for maximally spinning equal - mass binaries with anti - aligned spins in the orbital plane . these kick velocities exceed the escape velocities of even massive elliptical galaxies ( fig . 2 of merritt et al . 2004 ) and therefore the new results reinforce and enhance consequences studied earlier for the smaller kicks , with potentially far - reaching implications for the early phases of bh growth from early stellar - mass precursors or later intermediate - mass precursors ( schnittman 2007 , volonteri 2007 ) and consequently for the frequency of gravitational wave signals detectable with _ lisa _ ( sesana 2007 ) , for the scatter in the @xmath1 relation ( libeskind et al . 2006 ) , and for the offsets and oscillations of recoiling bhs in galaxy cores ( gualandris & merritt 2008 ) . the recoiling black holes will carry a fraction of nuclear gas and stars with them ( merritt et al . 2004 , 2006 , madau & quataert 2004 , loeb 2007 ) . they would be detectable spatially in the form of seyfert or quasar activity offset from the galaxy core ( madau & quataert 2004 ) , or in the form of broad emission lines kinematically offset from the narrow emission lines ( bonning et al . 2007 , komossa et al . 2008 ) . because of the broad astrophysical implications , the search for and actual identification of such recoiling black holes is of great interest , and will place important constraints on bh growth during the epoch of structure formation , on predictions of maximum recoil velocity , and on arguments suggesting that the bh spin configurations leading to maximal recoil velocities should be rare in gas - rich mergers ( bogdanovi et al . 2007 ) . bonning et al . ( 2007 ) searched for recoiled smbhs in the sloan digital sky survey ( sdss ) database , looking for systematic kinematic offsets between broad - line gas attached to the recoiling bh , and narrow - line gas left behind . they did not find any promising candidate , and concluded that smbh recoil with large kick velocities is relatively rare . here , we present the best candidate to date for a recoiling smbh , the quasar + sdssj092712.65 + 294344.0 ( sdssj0927 + 2943 hereafter ) . its unusual emission - line spectrum matches key predictions from the recoiled - smbh scenario . we use a cosmology with @xmath2=70 kms@xmath0mpc@xmath0 , @xmath3=0.3 and @xmath4=0.7 throughout this letter . sdssj0927 + 2943 at redshift @xmath5=0.713 is a luminous quasar , observed in the course of the sdss ( adelman - mccarthy et al . 2007 ) , and was found by us in a systematic search for active galactic nuclei ( agn ) with high [ oiii ] velocity shifts . the sdss spectrum , corrected for the galactic reddening of e(b - v ) = 0.021 mag , is displayed in fig . the underlying continuum spectral energy distribution ( sed ) was modeled as a powerlaw with a best - fit slope of @xmath6 ( where @xmath7 ) . each emission line was fit by a single gaussian except the feii multiplets , which were modeled by templates built from i zw 1 ( @xmath8 , vron - cetty et al . 2004 ; @xmath9 , tsuzuki et al . the redshifts of the feii lines were tied either to mgii ( the uv multiplets ) or to broad h@xmath10 ( the optical multiplets ) . two systems of strong emission lines can be identified in the spectrum , which we refer to as the `` red '' ( r ) and `` blue '' ( b ) systems . the red system consists of very narrow emission lines ( red nels , r - nels hereafter ) of [ oiii]5007 , [ oii]3727 , [ neiii]3869 , faint [ nev]3426 and balmer lines , all of them almost unresolved ( fwhm , obs([oiii ] ) = 230 kms@xmath0 ; the widths of the narrow lines are all very similar , and were therefore all fixed to the same value in order to derive fluxes ) . the blue system shows classical broad balmer and mgii2798 emission lines ( bels ) , plus unusually broad nels ( blue nels , b - nels hereafter ) . all lines of the blue system are blueshifted by about 2650 kms@xmath0 relative to the r - nels(see tab . 1 for redshifts ; the value of 2650 kms@xmath0 is the shift between broad h@xmath10 and r-[oiii ] ) . the b - nels show broad [ nev ] with a width of fwhm([nev])=2080 kms@xmath0 , and broad [ neiii ] with fwhm([neiii])=1020 kms@xmath0 . [ oiii ] and [ oii ] are present , too , with widths of 460 kms@xmath0 . the bels appear in balmer lines and in mgii with fwhm(h@xmath10)=5740 kms@xmath0 and fwhm(mgii)=3530 kms@xmath0 . line ratios indicate agn - like excitation in both systems , r - nels and b - nels . emission - line properties are summarized in tab . 1 . in order to see whether any ( faint ) broad - line emission is also accompanying the r - nels , we have performed the following test . we have first subtracted the best - fit continuum , feii multiplets , and nels from the observed spectrum , and then fit the h@xmath10 regime with two broad gaussians . the redshift of the second gaussian was fixed to that of the r - nels , and its width constrained to be in the range @xmath11 km s@xmath0 . no successful fit could be obtained if a contribution of the second gaussian is enforced ; its contribution is always consistent with zero . we therefore conclude that only the b - nels are accompanied by broad - line gas at the same redshift . the broadband sed of sdssj0927 + 2943 is rather blue with sdss magnitudes of @xmath12 , @xmath13 , @xmath14 , @xmath15 , @xmath16 , and _ galex _ ( martin et al . 2005 ) magnitudes of @xmath17 and @xmath18 . assuming that the standard broad - line region ( blr ) scaling relations ( kaspi et al . 2005 ) hold , we expect a blr radius of @xmath190.1 pc and estimate a smbh mass of sdssj0927 + 2943 of @xmath20 m@xmath21 from the width of h@xmath10 and the luminosity at 5100 . we have searched the x - ray archives for observations of sdssj0927 + 2943 . the quasar is serendipitously located close to the edge of two rosat hri images . the deeper exposure , of 19 ks duration , was performed in april - may 1995 . x - ray emission from sdssj0927 + 2943 is detected with a countrate of 0.0037@xmath22 cts / s , which translates into a soft x - ray luminosity of @xmath23 erg / s ( assuming an x - ray spectrum with no intrinsic absorption , and with photon index @xmath24 ) . the second hri image was taken in november 1994 with a duration of 10 ks . sdssj0927 + 2943 is detected with a countrate of 0.0047@xmath25 cts / s , consistent with the other measurement . the x - ray detection demonstrates the presence of an inner accretion disk ( relevant for the discussion in sect . lllcc r - nel & [ oiii]5007 & 0.71279 & 170 & 10.1 + & h@xmath10 & 0.71279 & 170 & 1.0 + & h@xmath26 & 0.71279 & 170 & 0.4 + & [ neiii]3869 & 0.71279 & 170 & 0.7 + & [ oii]3727 & 0.71279 & 170 & 2.6 + & [ nev]3426 & 0.71279 & 170 & 0.3 + b - nel & [ oiii]5007 & 0.69713 & 460 & 6.7 + & h@xmath10 & 0.69713 & 460 & 1.0 + & [ neiii]3869 & 0.69678 & 1020 & 1.8 + & [ oii]3727 & 0.69801 & 460 & 1.5 + & [ nev]3426 & 0.69709 & 2080 & 4.0 + bel & h@xmath10 & 0.69770 & 5740 & 30.7 + & h@xmath26 & 0.69970 & 3880 & 10.7 + & h@xmath27 & 0.69996 & 3260 & 3.8 + & mgii 2798 & 0.69832 & 3530 & 29.5 several pairs of quasars have been detected at projected separations of 3 - 10@xmath28 . while some of them can be explained by lensing , others very likely represent real pairs ( e.g. , kochanek et al . 1999 ; review by komossa 2003 ) . is sdssj0927 + 2943 actually a binary quasar ? we temporarily refer to this hypothetical system as sdssj0927 + 2943a , b . the difference in velocity of the two sets of emission lines exceeds the peculiar velocities observed in galaxy clusters , and is too large for the two galaxies to form a bound merger . their redshift difference corresponds to @xmath19100 mpc , if their redshifts are cosmological . consequently , we would then have a very unlikely projection effect , including not just one , but two intrinsically extremely unusual agn : sdssj0927 + 2943a with exceptionally broad neon lines of [ neiii ] and [ nev ] . and sdssj0927 + 2943b as one of the rare type-2 quasars with , in addition , exceptionally narrow emission lines [ their observed width of @xmath19230 kms@xmath0 is below that typically observed in quasar narrow - line regions ( fwhm @xmath29 kms@xmath0 ; zakamska et al . 2003 ) ] . this rare source would have to be projected by chance behind the other unusual agn . we are therefore let to consider the alternative hypothesis that we actually see only one agn , its smbh and bound emission - line region having been ejected from the core with a line - of - sight speed of 2650 kms@xmath0 , while the bulk of the narrow - line region ( nlr ) and other interstellar medium ( ism ) was left behind and shines in narrow emission lines . this is the scenario actually predicted in the context of the recent black hole recoil simulations discussed in sect . 1 , and it is the observational signature bonning et al . ( 2007 ) searched for , and komossa et al . ( 2008 ) paid attention to , but did not detect . in this picture , sdssj0927 + 2943 underwent a merger in the past . upon merging , its central smbh recoiled taking with it the blr and perhaps other very high ionization gas , leaving behind the nlr . gas with a velocity larger than the recoil velocity will remain bound to the smbh , and a trail of partially bound gas may form . accretion activity may have switched off temporarily in the course of the binary smbh merging , once the orbital decay time due to emission of gravitational waves had become smaller than the viscous timescale of the disk ( e.g. , liu et al . 2003 , milosavljevi & phinney 2005 ) . after coalescence , the inner disk will re - form quickly ( loeb 2007 ) and the accreting smbh will then illuminate the bound gas , a trail of partially bound gas and swept - up ism , the surrounding ism / halo and the left - behind bulk of the nlr gas . the nlr gas left behind will retain memory of the original ionization from the pre - merger accretion phase for only limited time , given by the light - travel time , and the hydrogen recombination timescale , @xmath30 yr , where @xmath31 is the gas temperature in 10@xmath32 k , and @xmath33 is the gas density in units of 10@xmath34 @xmath35 . [ oiii ] will fade away more quickly than hydrogen lines ( binette & robinson 1987 ) . however , once the recoiling smbh re - forms its inner accretion disk , it will re - illuminate parts of the nlr left behind ( from larger distance than before ) , and any surrounding ism . emission - line ratios would depend on the average density which would be higher in the classical nlr than in other ism / halo gas , and on metal abundances which will be lower in the halo . which aspects of this scenario do we observe in sdssj0927 + 2943 ? the blue system of emission - lines , the bels and b - nels , represents gas bound to the recoiling smbh , seen in broad balmer lines , broad mgii and broad forbidden lines , especially [ nev ] . after coalescence and recoil , matter orbiting the smbh with a velocity much larger than the recoil velocity will remain bound to the smbh ( merritt et al . 2006 ) ; i.e. matter within a region whose size is given by @xmath36 . in our case , @xmath37 cm , which is about a factor of 3 larger than the blr of sdssj0927 + 2943 . the width of [ neiii ] ( of the b - nel system ) of 1020 kms@xmath0 is too narrow for this gas to be originally bound to the smbh ( except in case of projection effects ) , but as the disk keeps accreting onto the recoiling smbh , it will spread radially , and will also drive some outflows . the semi - broad [ oii ] and [ oiii ] most likely have this same origin , and/or are from swept - up ism in front of the path of the recoiling smbh emission from the gas bound to the recoiling smbh ; even though shielding geometries might be constructed in which this actually becomes possible . ] . the red system of emission lines , the r - nels , represents the nlr gas left behind , and/or ism surrounding the recoiling smbh , not bound to it , but illuminated by its accretion disk . all r - nels have the same profiles and are very narrow , and the emission - line ratios imply an agn ionizing continuum . can we distinguish between nlr and other ism / halo gas , in terms of line ratios and line profiles ? ratios _ would depend on density and distance . as the recoiling smbh moves away from the galaxy core , illumination of the nlr from an , on average larger , distance would decrease the degree of ionization , while illumination of low - density ism surrounding the recoiling smbh would increase the degree of ionization of the line - emitting gas . it is possible that we see a mix of these two processes . we do not expect to see pure very low - density halo gas . its high ionization parameter would lead to a much higher degree of ionization than we actually observe , as we have also verified by test calculations with ferland s photoionization code _ cloudy _ ( s. komossa et al . , in preparation ) . the density - sensitive [ sii ] doublet is outside the observed wave band , but will be accessible with nir spectroscopy . independent evidence for the origin of the r - nels comes from their _ profiles _ which are very narrow . their width is below the stellar velocity dispersion of @xmath38=260 kms@xmath0 predicted from the @xmath39 relation ( ferrarese & ford 2005 ) . this indicates that the r - nel gas illuminated by the recoiling smbh either does not feel the full bulge potential , or the local velocity field of the illuminated gas does not reflect the full velocity dispersion . this latter situation is expected if the recoiling smbh was in a disk where the velocity pattern is predominantly rotational , so that we only see a fraction of the full pattern . the whole parameter space of bh recoil velocities is still being explored ( e.g. , baker et al . 2006 , 2007 , campanelli et al . 2007a , b , choi et al . 2007 , gonzlez et al . 2007a , b , herrmann et al . 2007a , b , pollney et al . 2007 , schnittman et al . 2008 , and references therein ; see pretorius 2007 for a review ) . several recent calculations focused on ( almost ) equal mass bhs with specific spin configurations such that kick velocities are maximized . the line - of - sight velocity we measure is comparable to the recoil velocities predicted by the runs of gonzles et al . ( 2007b ; @xmath40 kms@xmath0 ) and by dain et al . ( 2008 ; 3300 kms@xmath0 ) . the scaling formulae of campanelli et al . ( 2007a ) and of baker et al . ( 2008 ) predict maximal recoil velocities of @xmath193800 kms@xmath0 . in order to reach a high kick velocity , the pre - merger bhs of sdssj0927 + 2943 must have been rapidly spinning and of nearly equal mass ; i.e. the galaxy hosting sdssj0927 + 2943 must have undergone a major merger . assuming that the recoiling smbh carried with it an amount of mass which is not larger than its own mass , and that most of that gas is ultimately available for accretion , we can estimate an upper limit on the duration of the quasar activity , @xmath41 . we further assume that the gas continues to accrete at its current rate of @xmath42 , where @xmath43 is the eddington luminosity , and we use a radiative efficiency of @xmath44=0.37 which is appropriate for rapidly spinning bhs . this implies an upper limit on the lifetime of quasar activity of @xmath45 yr . if the space velocity of the recoiling smbh was close to the maximum possible velocity , it would reach a projected separation from the core on the order of a few kiloparsec within @xmath1910@xmath46 yr . future _ hst _ imaging of sdssj0927 + 2943 will be valuable to distinguish whether the host galaxy shows a distorted morphology as would be expected if the merger was recent . with its spatial resolution of 0.1@xmath28 , _ hst _ would allow resolution of a spatial scale of @xmath191 kpc in the galaxy and measurement of a corresponding offset of the accreting smbh from the galaxy core . the x - ray detection of sdssj0927 + 2943 holds promise for a deep _ chandra _ study of smbh and host galaxy . if the recoiling smbh is a radio emitter , high - resolution radio observations would provide even more precise measurement of its location . in summary , sdssj0927 + 2943 is the best candidate to date for a recoiling supermassive black hole . its spectrum shows the characteristic signature of two separate emission - line systems , kinematically offset by @xmath192650 kms@xmath0 broad lines from gas bound to the recoiling hole , and narrow lines from gas left behind . further study of this source and detection of similar ones will provide important information on bh recoil , relevant timescales , and the frequency of bhs removed from their host galaxies , and therefore on simulations of galaxy formation and evolution , and on the question how many bhs grew early by merging .
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we present sdssj092712.65 + 294344.0 as the best candidate to date for a recoiling supermassive black hole ( smbh ) .
sdssj0927 + 2943 shows an exceptional optical emission - line spectrum with two sets of emission lines : one set of very narrow emission lines , and a second set of broad balmer and broad high - ionization forbidden lines which are blueshifted by 2650 kms@xmath0 relative to the set of narrow emission lines .
this observation is most naturally explained if the smbh was ejected from the core of the galaxy , carrying with it the broad - line gas while leaving behind the bulk of the narrow - line gas .
we show that the observed properties of sdssj0927 + 2943 are consistent with predictions and expectations from recent numerical relativity simulations which demonstrate that smbhs can receive kicks up to several thousand kms@xmath0 due to anisotropic emission of gravitational waves during the coalescence of a binary .
our detection of a strong candidate for a rapidly recoiling smbh implies that kicks large enough to remove smbhs completely from their host galaxies do occur , with important implications for models of black hole and galaxy assembly at the epoch of structure formation , and for recoil models .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
it is well known that metallic nanoparticles can sustain localized surface plasmon ( lsp ) oscillations , whose resonance frequencies in the quasi - static limit depend solely on the geometry of the nanoparticle , the permittivity of the metal and the surrounding permittivity . the dependency of the lsp resonance ( lspr ) on the surrounding medium makes metallic particles extremely good sensors , progressing towards the detection of single molecules @xcite . however , the weak effect of retardation on the lsp resonance in nanosized metal particles leaves only one parameter to truly engineer : the geometry . by modifying the structure of the metal nanoparticle to have a dielectric core with a metal shell , an increased tunability is achieved due to the plasmon hybridization of the inner and outer surfaces of the metal @xcite . especially the spherical core - shell structure has received a considerable amount of attention in recent years @xcite due to its excellent and tunable sensing properties , which show great promise in biological studies such as cancer therapy @xcite . the plasmon hybridization allows one to position the lsp resonance of the nanoshell as desired by simply varying the core size @xmath0 and/or outer radius @xmath1 appropriately @xcite . the hybridization of the inner and outer surface plasmons increases when the metal shell becomes thinner @xcite , which gives rise to significantly altered lsp resonances compared to usual homogeneous metal nanoparticles . studies of the hybridization between two spherical @xcite or cylindrical @xcite metal nanoparticles in few - nm proximity reveal that effects of nonlocal response increase with increasing hybridization . furthermore , nanosized metal particles @xcite and metal films @xcite are also strongly affected by nonlocal effects . the core - shell particle thus calls for a nonlocal description , since it features an ultra - thin metallic shell with resulting strong plasmon hybridization . the use of arrays of nanotubes with high aspect ratio for biosensing @xcite and hydrogen sensing @xcite has yielded impressive results , yet only few theoretical studies have been performed on the nanotube @xcite . _ investigate the plasmonic modes and dispersion relations of the nanotube @xcite , while zhu _ et al . _ perform calculations using the discrete dipole approximation to discuss the changes of the resonance wavelength of the nanotube due to variations of the aspect ratio @xcite . thus , to our knowledge no systematic study has yet been performed that addresses which parameters determine the lspr refractive - index sensitivity of a nanotube - based sensor . in this paper , we fill this gap with a systematic study of the sensing and scattering properties of a single infinitely long cylindrical core - shell nanowire ( see inset of fig . [ fig : fig1 ] ) , which is a good description of dilute arrays of non - interacting nanotubes with high aspect ratio . on the basis of this study , we propose how to optimize a nanotube - based sensor to achieve the utmost sensitivity for the refractive - index sensing of both gases and liquids . the outline of this paper is as follows . in sec . [ sec : theory ] we discuss the physical principles of local and nonlocal response , and introduce the sensitivity and figure - of - merit ( fom ) as quantitative measures of the performance of a lspr - based sensor . section [ sec : results ] is dedicated to the study of a nanotube with a silica core and gold shell . we determine the dependency of the sensitivity and fom on the shape and size of the nanotube , using both local and nonlocal theory to model the response of the gold shell . our conclusions and outlook on nanotube - based sensors is given in sec . [ sec : con ] , and details on the analytical calculations in the appendix . the ability of lspr - based sensors to detect changes in the refractive index of their surrounding medium is usually quantified by the sensitivity and fom @xcite . the sensitivity @xmath2 is determined as the shift in wavelength of the considered lsp resonance in the extinction spectrum of the sensor , when varying the background refractive index @xmath3 , while the fom is given as @xmath4 where @xmath5 is the resonance linewidth , calculated as the fwhm of the considered lsp resonance in the extinction spectrum . thus , to determine the performance of the nanotube as a lspr sensor , we must calculate its extinction cross section , as this quantifies the extinction spectrum and therefore allows us to determine the sensitivity and fom . predictions for the extinction cross section depend on how the optical response of electrons in the metal is modeled . the common approach to describe the response of metals is by making the local approximation which assumes that the response field at a certain position is proportional to the driving field at that position , with the proportionality function being a position- and frequency - dependent dielectric function . this approach has the rather unphysical consequence that all surface charges reside on an infinitely thin layer on the boundaries of the metal , thereby neglecting the actual extent ( or wave nature ) of the electrons . while the local approximation is justified as long as the metal boundaries are far apart such that the interaction between electrons due to their extent can be neglected ( i.e. large metallic structures ) , it can not be safely assumed for nanosized metal particles where the wavelength of the electron becomes comparable in size to the metal particle . describing the metal using the semiclassical hydrodynamic drude model @xcite , we relax the local approximation by allowing the existence of local inhomogeneity in the density of the electron gas , which gives rise to pressure waves . the electron - gas pressure waves provide a means to transport energy in the metal in addition to the electromagnetic waves , which gives rise to nonlocal response : the response of the metal at a certain spatial point can depend on the driving field at other nearby points ( on the length scale of the fermi wavelength ) in the metal . in the appendix , we provide an analytical expression for the extinction cross section in the cases of both nonlocal and local response , for a normally incident tm - polarized wave , see the inset of fig . [ fig : fig1 ] . we have checked the analytical expression with our numerical implementation of the hydrodynamic drude model @xcite , which showed perfect agreement ( not shown in this paper ) . we consider the specific core - shell structure , where the core is silica ( @xmath6 ) with dielectric constant @xmath7 and the shell is gold ( au ) modeled with the data by raki _ _ @xcite . to clearly show the difference in extinction cross section in local and nonlocal response , we start by examining the case where interband effects in au are neglected . figure [ fig : fig1 ] depicts the extinction cross section for a @xmath8 silica - au cylinder in vacuum comparing the local and nonlocal model . the local approximation shows three distinct peaks , two at low frequencies ( dipole and quadrupole peaks ) and one at a high frequency ( near 7ev ) . these are due to the interaction between the localized plasmons at the inner and outer surface of the nanoshell , or equivalently , the interaction between a cavity mode and a cylinder mode @xcite . the nonlocal description allows the same classification of peaks as the local approximation @xcite , although the high - frequency peak is blueshifted compared to the local model . since sensing depends on peak shifts , it is important to take possible nonlocal blueshifts into account however , the low - frequency resonances show no noticeable blueshift , because the strength of the nonlocal blueshift does not only increase with decreasing thickness of the metal layer @xcite but it also depends on the frequency , with a decreasing blueshift for lower frequencies . thus we find that there is an intricate interplay between plasmon hybridization and nonlocal response : since a thinner metal shell produces stronger plasmon hybridization , the dipole and quadrupole peaks are pushed to such low frequencies that the nonlocal blueshift effect due to nanosized metallic features is counteracted by the low frequency of the resonances . the panel on the right of fig . [ fig : fig1 ] shows the nonlocal normalized intensity distribution in the metal at the dipole and quadrupole resonance frequencies , illustrating the expected dipole and quadrupole nature of the resonances . above the plasma energy @xmath9 we see the characteristic additional resonances in the nonlocal model due to the excitation of longitudinal modes , as previously reported for different metal nanoparticles @xcite . the difference between the red and green curves in fig . [ fig : fig1 ] shows the importance of taking into account interband transitions in the response of the metal shell . the implications on the dipole and quadrupole resonances are that they are redshifted and damped due to interband transitions , with greatest impact on the quadrupole peak . in the remaining part of this paper , we will always use measured values for the dielectric function @xcite , i.e. we take interband transitions into account . we will concentrate on the dipole resonance , since this peak is the strongest , is close to visible and infrared frequencies and can be affected by the shape and size of the cylinder and the background permittivity . there are two geometrical properties that can be modified in the nanotube structure : the first is the shape defined by the @xmath10 ratio and the second is the overall size , that is , varying the outer radius @xmath1 but keeping @xmath10 constant . in fig . [ fig : fig2 ] we show the effect of shape variations of the nanotube on its sensing abilities , which is quantified through the change in the dipole resonance wavelength when the background refractive index is increased . we see that regardless of the shape , the dependency is always approximately linear . however , as shown in fig . [ fig : fig2](i ) there is no significant dependency on the background refractive index for low @xmath10 ratios , indicating the lack of ability to sense . only when the shell becomes thin ( @xmath11 ) does the resonance wavelength shift with the refractive index . the thinner the shell , the greater is the average slope of the curves . relaxing the nonlocal description to a local one does not change this trend , because the dipole resonances occur at too low energies for the nonlocal blueshift to kick in . furthermore , the resonance wavelength shifts to higher wavelengths when the shell becomes thinner , because the coupling between the cavity and cylinder modes increases . thus , even though fig . [ fig : fig2](iv ) represents a nanotube with a @xmath12 thin metal shell , where nonlocal blueshifts are expected to be very prominent , the local approximation predicts sensitivities that are almost identical to the nonlocal description . so , as in fig . [ fig : fig1 ] , here in fig . [ fig : fig2 ] we see that for ultra - thin nanotubes the usual observation of larger nonlocal blueshifts for smaller structures does not occur . the nonlocal blueshift cancels out with the decrease of the resonance energy due to increased hybridization . .sensitivity and figure - of - merit calculations eq . ( [ eq : fom ] ) in the nonlocal description at the refractive index of air @xmath13 ( for gas sensing ) and water @xmath14 ( for liquid sensing ) for the four different shapes of fig . [ fig : fig2 ] . [ cols="^,^,^,^,^ " , ] [ tab : tab1 ] for a more quantitative description of the sensitivity of the nanotube , we present sensitivity and fom calculations of the nanotube structures shown in fig . [ fig : fig2 ] at the refractive index of air and water in table [ tab : tab1 ] . as in fig . [ fig : fig2 ] , it is again clear from table [ tab : tab1 ] that increased sensitivity can be achieved for thinner metal shells . comparing the sensitivity of the nanotube with other lspr sensors based on different nanoparticle geometries @xcite , where the sensitivity is in the range @xmath15 nm per refractive index unit ( riu ) , shows that the nanotube is comparable in sensitivity for ratios @xmath16 , while it is superior for very high @xmath10 ratios . comparison of the fom with other nanoparticle lsp sensors also shows equally good performance by the nanotube , although the fom is mainly dependent on the properties of au and not easily improved by changing the geometry @xcite . the sensitivity values in table [ tab : tab1 ] also reveal that the nanotube has a high sensitivity at both the refractive index of air and water , which shows the versatility of a nanotube - based sensor and its applicability as both a gas and liquid sensor . besides shape variations , we also varied the size @xmath1 of the nanotube , while keeping @xmath10 constant . figure [ fig : fig3 ] depicts the dipole resonance wavelength as a function of the background refractive index for three different sizes with @xmath17 . the sensing ability of the nanotube is not as dependent on size as it is on shape , which can be seen by the three almost parallel lines in fig . [ fig : fig3 ] . even though the sensitivity does not change much with increasing size , there is still an optimum size which occurs at @xmath18 and @xmath19 for liquid and gas sensing , respectively , see the inset of fig . [ fig : fig3 ] . the fact that it is neither the smallest nor the biggest nanotube size that gives the highest sensitivity can be explained by a trade - off between the total structure size and the shell thickness . if the size of the structure is too small , then we have a weak lsp excitation and thereby poor sensing ability , but if the structure size is too big ( with the shape kept constant ) the absolute shell thickness increases , which also decreases the sensing ability , as we saw in fig . [ fig : fig2 ] . therefore , for a larger @xmath10 value the optimum size will also be larger . in fig . [ fig : fig3 ] we also show the calculations using the local approximation . as seen , effects are surprisingly well accounted for even with a local description , despite the fact that we actually consider very thin metallic shells , for instance a 3 nm shell in fig . [ fig : fig3](iii ) , with concomitant strong plasmon hybridization . the strong hybridization in ultra - thin metal shells shifts the dipole resonance to very low energies , where the nonlocal blueshift is weak . the sensitivity and consequently the fom are therefore weakly influenced by nonlocal response . although it is hardly visible in fig . [ fig : fig3 ] , the local resonances do in fact occur at slightly longer wavelengths than in the nonlocal description , revealing a small nonlocal blueshift . we have examined the infinite single dielectric - metal nanotube structure as an approximation for a dilute array of nanotubes with high aspect ratio . we calculate the extinction properties of a silica - gold nanotube analytically for both local and nonlocal response by extending the mie theory for nanowires to nanotube geometries . our investigation reveals that in contrast to the spherical nanoshell @xcite , the sensing ability of the nanotube is highly dependent on the shape of the structure , where few - nm thin shells produce extreme sensitivities . the sensitivity is shown to be less dependent on the overall structure size . the sensitivity at the refractive index of air and water of ultra - thin nanotubes are superior to other nanoparticle geometries , making nanotubes very promising for both gas and liquid sensing . our results also show unexpectedly that nonlocal response has negligible influence on the extinction and sensing properties of the nanotube , even though the metal shell is ultra - thin ( a few nm ) , because the hybridization in the nanotube is so strong that the dipole resonance is pushed to very low energies . the strength of the nonlocal blueshift is an interplay between the metal thickness and the resonance energy , where a thinner shell produces a stronger blueshift while a lower energy produces a weaker blueshift . this interplay is surprisingly well - balanced in the nanotube structure , because a thinner shell gives rise to lower resonance energies . with the high sensitivity and good fom of the nanotube geometry , we propose a sensor based on ultra - thin nanotubes . the robustness of the sensitivity of the nanotube to size variations provides desirable advantages , since fluctuations in size due to imperfect fabrication will have less impact . in the special case of gas sensing , the sensitivity may be further improved by a factor of two by designing the nanotube to have a hollow core . with a hollow core , the inner surface of the metal shell is also exposed to the surrounding medium , which significantly improves the sensitivity . however , mechanical stability is sacrificed with a hollow core if for instance the nanotubes are to stand vertically on a substrate . the nonlocal optical properties of the nanotube are determined by solving maxwell s wave equation coupled to the hydrodynamic equation for the current @xcite . we solve the coupled set of equations by extending the mie theory for wires of ref . @xcite to core - shell structures . by expanding the electromagnetic fields in the dielectric core , metal shell and surrounding medium in cylindrical bessel functions , we can most easily take into account maxwell s boundary conditions along with the additional boundary condition of a vanishing normal component of the current in the nonlocal case @xcite . although quantum tunneling is not taken into account with this treatment , we do not expect any such effects to be important in this structure @xcite . to determine the extinction property of the infinite cylindrical nanotube we calculate the extinction cross section @xcite @xmath20 where @xmath21 is the background wave vector , @xmath22 is the background permittivity and @xmath23 is a cylindrical bessel - function expansion coefficient for the scattered electromagnetic field . we consider a normally incident electric - field polarization perpendicular to the cylinder axis ( tm ) , as sketched in the inset of fig . [ fig : fig1 ] . the nonlocal - response scattering coefficient is calculated analytically as @xmath24 - \sqrt{\epsilon } j_n^\prime(k_0 r_2 ) \left[j_n p_n - h_n q_n\right ] } { \sqrt{\epsilon_\text{b } } h_n(k_0 r_2 ) \left[c_n+ j_n^\prime p_n - h_n^\prime q_n\right ] - \sqrt{\epsilon } h_n^\prime(k_0 r_2 ) \left[j_n p_n - h_n q_n\right]}. \label{eq : annl}\ ] ] here , @xmath25 and @xmath26 are the bessel and hankel functions of the first kind , @xmath27 and @xmath28)$ ] is the drude local - response function that includes interband effects through @xmath29 . the argument of the bessel and hankel functions are @xmath30 unless written explicitly otherwise . the coefficients @xmath31 , @xmath32 and @xmath33 are given by @xmath34 , \label{eq : pn } \\ q_n & = q_n \alpha_n + j_n(k_\text{c}r_1 ) \left[j_n(k_\text{t}r_1 ) \delta_n + j_n \tau_n \right ] , \label{eq : qn } \\ c_n & = \frac{i n}{k_0 r_2 } \left [ h_n(k_\text{l}r_2 ) c_n - j_n(k_\text{l}r_2 ) d_n \right ] , \label{eq : cn}\end{aligned}\ ] ] where @xmath35 and @xmath36 is the dielectric constant of the core . furthermore , @xmath37 and @xmath38 with @xmath39 being the fermi velocity of the metal shell . the coefficients @xmath40 , @xmath41 , @xmath42 , @xmath43 and @xmath44 of eqs . ( [ eq : pn]-[eq : qn ] ) are given as @xmath45 , \label{eq : alphan } \\ \delta_n & = -\frac{k_\text{l}n^2 \sqrt{\epsilon_\text{c } } \epsilon_\text{other } ( \epsilon-\epsilon_\text{other})}{k_\text{t}k_0 ^ 2 r_1 ^ 2 } \left [ j_n^\prime(k_\text{l}r_2 ) h_n(k_\text{l}r_1)- h_n^\prime(k_\text{l}r_2 ) j_n(k_\text{l } r_1 ) \right ] , \label{eq : deltan } \\ \tau_n & = -\frac{k_\text{l}n^2 \sqrt{\epsilon_\text{c } } \epsilon_\text{other } ( \epsilon-\epsilon_\text{other})}{k_\text{t}k_0 ^ 2 r_1 r_2 } \left [ h_n^\prime(k_\text{l}r_1 ) j_n(k_\text{l}r_1)- j_n^\prime(k_\text{l}r_1 ) h_n(k_\text{l } r_1 ) \right ] , \label{eq : taun}\end{aligned}\ ] ] while the coefficients @xmath46 and @xmath47 of eq . ( [ eq : cn ] ) are given as @xmath48 + j_n^\prime(k_\text{l}r_1 ) g_n \left [ j_n p_n - h_n q_n \right ] , \label{eq : cn } \\ d_n & = f_n \left [ h_n^\prime(k_\text{l}r_2 ) \eta_n + h_n(k_\text{l } r_1 ) \kappa_n \right ] + h_n^\prime(k_\text{l}r_1 ) g_n \left [ j_n p_n - h_n q_n \right ] , \label{eq : dn}\end{aligned}\ ] ] where @xmath49 , \label{eq : etan } \\ \kappa_n & = \frac{n^2 ( \epsilon-\epsilon_\text{other})}{k_\text{t } r_2 r_1 } \left [ j_n(k_\text{t}r_1 ) h_n - h_n(k_\text{t}r_1 ) j_n \right ] . \label{eq : kappan}\end{aligned}\ ] ] the local - response result can be retrieved in the limit of a vanishing fermi velocity for which @xmath50 , @xmath51 and @xmath52 .
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we study the refractive - index sensing properties of plasmonic nanotubes with a dielectric core and ultra - thin metal shell .
the few - nm thin metal shell is described by both the usual drude model and the nonlocal hydrodynamic model to investigate the effects of nonlocality .
we derive an analytical expression for the extinction cross section and show how sensing of the refractive index of the surrounding medium and the figure - of - merit are affected by the shape and size of the nanotubes .
comparison with other localized surface plasmon resonance sensors reveals that the nanotube exhibits superior sensitivity and comparable figure - of - merit .
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the electron - electron energy - relaxation time determines a number of parameters of nonequilibrium superconductors such as the relaxation times for the amplitude and the phase of the order parameter @xcite . it is also important for nonequilibrium superconducting radiation detectors based on the resistive and inductive responses . the electron - electron relaxation time is responsible for the quasiparticle multiplication coefficient which in turn determines the responsivity and detectivity of the detector and its noise characteristics @xcite . the energy relaxation time also serves as a pair - breaking parameter in the superconducting density of states which is measured in the tunneling experiment @xcite . recently the energy relaxation time was measured in cuprate superconductors by studying an electronic instability at high vortex velocities in the mixed state @xcite . the electron - electron relaxation time of clean superconductors was calculated for the three - dimensional case in ref . @xcite and for the two - dimensional case in ref . @xcite . in the last work all channels of the electron - electron interaction not only the coulomb interaction was taken into account by using the matrix classification of the interaction channels developed earlier in refs . @xcite and @xcite . the importance of considering all channels of interaction was realized long ago for the problem of gauge invariance in superconductors @xcite . it was also emphasized in ref . @xcite that interference between different channels of interaction cancels divergences in the interaction correction to the superconducting order parameter . it is known that in normal impure and low dimensional metals the diffusive motion of electrons leads to enhancement of the electron - electron relaxation @xcite , @xcite . the purpose of the present paper is to calculate the electron - electron relaxation time in impure two - dimensional superconductors using the formalism of refs . @xcite and @xcite . earlier attempt to study electron - electron relaxation in impure two - dimensional superconductors @xcite took into account only the coulomb electron - electron interaction and therefore ignored the other relevant interaction channels . as we will show in the present paper including all channels of the interaction is very important for the electron - electron relaxation , and leads to results qualitatively different from that of ref . another difference is that we found important contributions to the scattering relaxation time from gapless collective excitations which where mised in ref . note also that we derived the quasiparticle energy relaxation time from the quantum kinetic equation , not as an imaginary part of the self - energy as in ref . @xcite . then we study the superconductor - normal metal two - layer system . such a system was already studied in ref . @xcite for a clean system . we consider a disordered case in the present work and show that recombination relaxation rate is strongly enhanced due to inter - layer electron - electron interaction . we use the keldysh diagram technique for nonequilibrium processes in which the electron green s functions , along with the electron - electron interaction potential , the electron self energy and the polarization operator are represented by supermatrices @xmath0 the matrix electron green s function in an impure superconductor in the nambu representation has the form @xmath1^*= { -\xi_p\hat\tau_3-\epsilon^r\hat\tau_0+\delta^r\hat\tau_1 \over \xi_p^2-(e^r)^2},\cr p=({\bf p},\epsilon),\ \ \ \xi_p={p^2-p_f^2\over 2 m } , \end{aligned}\ ] ] where @xmath2 are the pauli matrices , @xmath3 is the electron mass , and @xmath4 where @xmath5 is the energy gap , @xmath6 is the electron - impurity relaxation time , and @xmath7 in a spatially uniform system the kinetic components @xmath8 and @xmath9 are satisfied the equations @xmath10,\cr \hat\sigma(p)=s(\epsilon)[\hat \sigma^a(p)-\hat \sigma^r(p ) ] , \end{aligned}\ ] ] where @xmath11 , and @xmath12 is the fermi distribution function . similar relations hold for matrix interaction potentials and polarization operators : @xmath13,\cr \hat\pi^c(q)=(2n(\omega)+1)[\hat\pi^r(q)-\hat\pi^a(q)],\ \ \end{aligned}\ ] ] where @xmath14 and @xmath15 is the bose distribution function . for averaging over impurity position it is convenient to introduce the following expressions @xmath16 where @xmath17 is the two - spin electron density of states . calculations give e.g. @xmath18 , \cr \eta_3^{aa}={\zeta_+\over 2}[(1-a)\hat\tau_3+bi\hat\tau_2 ] , \end{aligned}\ ] ] where @xmath19 @xmath20 values of @xmath21 for different matrices @xmath22 are presented in tab . 1 . note also that the following identities hold : @xmath23 and @xmath24 . treating the electron - electron interaction in superconductors we use the matrix formalism developed in refs . the bare vertices for the electron - electron interaction are classified in terms of the pauli matrices . physical meaning of the corresponding operators @xmath25 is the following . matrix @xmath26 corresponds to the order parameter amplitude @xmath5 , matrix @xmath27 corresponds to the order parameter phase @xmath28 , matrix @xmath29 corresponds to the electron density , and the vector matrix @xmath30 corresponds to the electric current , the later will not be considered in the present paper(see ref . note also that each impurity vertex carries the matrix @xmath29 . therefore each interaction vertex operates in both keldysh and nambu spaces and has the form @xmath31 , where @xmath22 indicates the component in the nambu space and tensor @xmath32 stands for the keldysh space , @xmath33 is a boson index and @xmath3 and @xmath34 are the electron indices . in the representation corresponding to eq . 1 the nonzero components of tensor @xmath32 are @xmath35 we will omit coefficient @xmath36 in intermediate equations for the impurity renormalized vertices and restore it in final equations for the polarization operators and the electron self energies . impurity averaging leads to the ladder equation for the scalar vertex @xmath37 shown in fig . 1 . such an equation should be written for each bare matrix @xmath22 in the nambu space . the solution of these equations for the vertex @xmath37 in the keldysh - nambu space is obtained following ref . we start with the equation for the vertex @xmath38 , @xmath39 the solution of this equation is @xmath40 note that renormalized vertex @xmath38 has components proportional not only to to matrix @xmath29 but also to matrix @xmath27 . the other vertices @xmath41 and @xmath42 are presented in tab . the vertices with the other keldysh indices are obtained from the equations @xmath43,\cr \hat \gamma^1_{21}(\hat\tau_i)=-s(\epsilon+\omega ) [ \hat \gamma^1_{22}(\hat\tau_i)- \hat \gamma^2_{21}(\hat\tau_i)],\cr \hat \gamma^2_{11}(\hat\tau_i)=s(\epsilon)\hat \gamma^2_{21}(\hat\tau_i ) -s(\epsilon+\omega)\hat \gamma^2_{12}(\hat\tau_i)\cr -[s(\epsilon)-s(\epsilon+\omega)](\hat\gamma^1_{22}(\hat\tau_i))^*. \end{aligned}\ ] ] note also that index structure of the renormalized vertices in the keldysh space is different from the index structure of the bare vertex described by eq . it is important that the complex conjugate in eq . ( 14 ) operates only in the keldysh space , thus any @xmath44 in tab . 2 originating from @xmath22 matrix algebra are not affected by the complex conjugate , e. g. ( compare with eq . ( 13 ) ) @xmath45 the effective screened electron - electron interaction in a superconductor according to refs . 6 - 8 is shown in fig . the solution of this equation in @xmath46 matrix form is @xmath47/\cal d & \pi_{23}/\cal d\cr 0 & \pi_{32}/\cal d & -(2/\lambda+\pi_{22})/\cal d } , \end{aligned}\ ] ] where @xmath48- \pi_{23}\pi_{32 } , \end{aligned}\ ] ] and @xmath49 is the bcs coupling constant ( @xmath50 ) , @xmath51 is the nonscreened two - dimensional coulomb potential . the polarization operators renormalized by impurities are expressed through the vertex @xmath52 by the equation @xmath53 using eq . ( 14 ) for @xmath54 and tab . 2 we find the polarization operators , @xmath55\biggr ) , \end{aligned}\ ] ] @xmath56\biggr ) , \end{aligned}\ ] ] @xmath57\biggr ) , \end{aligned}\ ] ] @xmath58 to calculate the electron relaxation we need the imaginary part of the potentials ( the polarization operators ) in the quasiparticle representation . making a transformation from the electronic representation to the quasiparticle representation we use eq . ( 4 ) . as a result we separate out the processes of scattering and recombination of quasiparticles in eqs . ( 18)-(21 ) . for the imaginary part of the polarization operators we have @xmath59c_{ii}(q,\epsilon,\omega ) , \end{aligned}\ ] ] @xmath60c_{ii}(q,\epsilon,-\omega ) , \end{aligned}\ ] ] where @xmath61 are @xmath62 @xmath63 @xmath64 for the off - diagonal polarization operator we have @xmath65 , \end{aligned}\ ] ] @xmath66 . \end{aligned}\ ] ] as we will see in the next chapter , calculating the electron relaxation time we need the imaginary part of the propagators @xmath67 . the imaginary part of the propagators may originate from the poles of the propagators which correspond to collective excitations or from the imaginary part of the polarization operators @xmath68 , which correspond to real processes of scattering and recombination of quasiparticles . we will restrict our calculations to low temperatures @xmath69 , where large frequencies @xmath70 , are important for both recombination and scattering processes and small frequencies @xmath71 are important only for the scattering processes . the imaginary part of diagonal polarization operators for in these regions are presented in tab . real parts of the polarization operators are analyzed in appendix a. now we study in detail each of the matrix elements of @xmath72 . following eq . ( a17 ) the imaginary part of the potential @xmath73 for scattering processes and for small arguments @xmath74 and @xmath75 is @xmath76 this approximation is justified because there is no singularity in the order parameter amplitude propagator @xmath73 at small frequency and momentum for any finite @xmath5 , which means that fluctuations of the amplitude of the order - parameter are massive . for the recombination processes , large frequencies @xmath77 are important , and according to tab . 3 @xmath78 may be neglected . as was shown in appendix a , the propagators @xmath79 and @xmath80 for @xmath74 and @xmath75 have the form @xmath81 @xmath82 thus the imaginary part of the propagators @xmath79 and @xmath80 for small arguments comes from the pole corresponding to the phase mode , not from the imaginary part of the polarization operators . for large frequencies @xmath83 we use the following approximation , @xmath84 @xmath85 where @xmath86 is the statically screened coulomb potential in the normal state , which was presented above for the two - dimensional case , @xmath87 is the screening momentum . such an approximation is justified due to absence of collective excitation in this frequency region . the kinetic equation for nonequilibrium distribution function in a spatially uniform system is @xmath88[\hat\sigma^c(p)-s(\epsilon)(\hat\sigma^a(p ) -\hat\sigma^r(p ) ) ] . \end{aligned}\ ] ] the electron energy relaxation time @xmath89 is determined from the equation @xmath90 the electron self - energy is shown in fig . 3 . using the results of section 2 , we have @xmath91\cr \times{\delta\over \delta s(\epsilon ) } \lbrace{\rm im}v^a_{ii}(q){\rm re}[{\rm pr}_{\hat\tau_i}(\hat\gamma^2_{11 } ( \hat\tau_i))]-2{\rm im}v^a_{23}{\rm im}[{\rm pr}_{\hat\tau_3 } ( \hat\gamma^2_{11}(\hat\tau_2))]\rbrace , \end{aligned}\ ] ] where @xmath92 means the component proportional to matrix @xmath22 ( projection on @xmath22 ) . in eq . ( 37 ) summation on repeated indices is implied . using tab . 2 and relation @xmath93 we present eq . ( 37 ) in the form @xmath94\cr \times\biggl[{\rm im}v_{11}^a(q ) \biggl(\biggl({\xi_\epsilon\over \epsilon } + { \epsilon(\epsilon+\omega)+\delta^2\over \epsilon\xi_{\epsilon+\omega}}\biggr ) { dq^2\over ( \xi_{\epsilon+\omega}+\xi_\epsilon)^2+(dq^2)^2}\cr -\biggl({\xi_\epsilon\over \epsilon } -{\epsilon(\epsilon+\omega)+\delta^2\over \epsilon\xi_{\epsilon+\omega}}\biggr ) { dq^2\over ( \xi_{\epsilon+\omega}-\xi_\epsilon)^2+(dq^2)^2}\biggr)\cr + { \rm im}v_{22}^a(q ) \biggl(\biggl({\xi_\epsilon\over \epsilon } + { \epsilon(\epsilon+\omega)-\delta^2\over \epsilon\xi_{\epsilon+\omega}}\biggr ) { dq^2\over ( \xi_{\epsilon+\omega}+\xi_\epsilon)^2+(dq^2)^2}\cr -\biggl({\xi_\epsilon\over \epsilon } -{\epsilon(\epsilon+\omega)-\delta^2\over \epsilon\xi_{\epsilon+\omega}}\biggr ) { dq^2\over ( \xi_{\epsilon+\omega}-\xi_\epsilon)^2+(dq^2)^2}\biggr)\cr + { \rm im}v_{33}^a(q ) \biggl(\biggl({\xi_\epsilon\over \epsilon } -{\epsilon(\epsilon+\omega)-\delta^2\over \epsilon\xi_{\epsilon+\omega}}\biggr ) { dq^2\over ( \xi_{\epsilon+\omega}+\xi_\epsilon)^2+(dq^2)^2}\cr -\biggl({\xi_\epsilon\over \epsilon } + { \epsilon(\epsilon+\omega)-\delta^2\over \epsilon\xi_{\epsilon+\omega}}\biggr ) { dq^2\over ( \xi_{\epsilon+\omega}-\xi_\epsilon)^2+(dq^2)^2}\biggr)\cr + 2i{\rm im}v^a_{23}{\omega\delta\over\epsilon\xi_{\epsilon+\omega } } \biggl({dq^2\over ( \xi_{\epsilon+\omega}+\xi_\epsilon)^2+(dq^2)^2 } + { dq^2\over ( \xi_{\epsilon+\omega}-\xi_\epsilon)^2+(dq^2)^2}\biggr)\biggr].\end{aligned}\ ] ] in order to separate out the processes of scattering and recombination of quasiparticles we need to make a transformation from the electronic representation to the quasiparticle representation in eq . note that the presence of imaginary factor @xmath44 in the last term in eq . ( 38 ) means that for the contribution of the nondiagonal channels of interaction requires the states under the gap to be taken into account according to the equation : @xmath95^{1/2},\ \ |\epsilon+\omega|<\delta$ ] . such states should also be included in equations for the polarization operator @xmath96 , in eqs . ( 28 ) and ( 29 ) only the states above the gap were included . the analysis similar to that presented in ref . 6 for the clean case shows that contribution from the nondiagonal channels of interaction may be neglected . for electrons on the fermi surface , @xmath97 , @xmath98 { dq^2\over \omega^2-\delta\omega + ( dq^2)^2}\cr \times\biggl[[{\rm im}v_{22}^a(q)_{recom } + { \rm im}v_{33}^a(q)_{recom } ] \biggl({\delta\over 2\omega}\biggl)^{1/2}\biggr]\cr + { 2\over \pi^3}\int_0^\infty d\omega\int_0^\infty dqq[n(\omega)+n(\omega+\delta ) ] { dq^2\over 2\delta\omega+\omega^2 + ( dq^2)^2}\cr \times\biggl[{\rm im}v_{11}^a(q)_{scatt } \biggl({2\delta+\omega\over \omega}\biggr)^{1/2}+[{\rm im}v_{22}^a(q)_{scatt}+ { \rm im}v_{33}^a(q)_{scatt } ] \biggl({\delta\over 2\omega}\biggl)^{1/2}\biggr],\end{aligned}\ ] ] @xmath99 { dq^2\over \omega^2-\delta\omega + ( dq^2)^2}\cr \times\biggl[{\rm im}v_{11}^a(q)_{scatt } \biggl({\omega-2\delta\over \omega}\biggr)^{1/2}+[{\rm im}v_{22}^a(q)_{scatt } + { \rm im}v_{33}^a(q)_{scatt } ] \biggl({\delta\over 2\omega}\biggl)^{1/2}\biggr]\cr + { 2\over \pi^3}\int_{2\delta}^\infty d\omega\int_0^\infty dqq[n(\omega)+n(\omega+\delta ) ] { dq^2\over 2\delta\omega+\omega^2 + ( dq^2)^2}\cr \times\biggl[[{\rm im}v_{22}^a(q)_{recom}+ { \rm im}v_{33}^a(q)_{recom } ] \biggl({\delta\over 2\omega}\biggl)^{1/2}\biggr ] . \end{aligned}\ ] ] eqs . ( 39 ) and ( 40 ) describe processes of scattering `` two into two '' and `` three into one '' quasiparticles correspondingly . further calculations will be performed for low temperatures @xmath69 . it may be shown that the most important contribution to the recombination time originates from terms @xmath100 and @xmath101 in eq . ( 40 ) , @xmath102 calculating the scattering relaxation time from terms @xmath100 and @xmath101 in eq . ( 38 ) we use the approximation of eqs . ( 33 ) and ( 34 ) and we use tab . 3 for the imaginary parts of the polarization operators . as a result we get @xmath103 as for the contribution to the scattering relaxation time from terms @xmath104 and @xmath105 in eq . ( 38 ) we note that for small energy transfers , @xmath71 the imaginary part of the propagators @xmath79 and @xmath80 originates from the poles corresponding to the phase mode as seen in eqs . ( 31 ) and ( 32 ) ) . integrating these poles over the momentum @xmath106 we get @xmath107 note that the main contribution comes from the propagator @xmath80 corresponding to the fluctuation of the phase of the order parameter . the term @xmath108 does not have poles corresponding to the collective mode , thus using eq . ( 30 ) we find that the scattering relaxation time has a power law divergence , @xmath109 this divergence is similar to the logarithmic divergence of the phase or energy relaxation times in the normal impure two - dimensional case@xmath110 . according to ref . 14 the cutoff frequency @xmath111 is defined by the relaxation time @xmath89 , which physically means that the kinetic equation can not be applied for energy transfers less than @xmath112 , thus the self - consistent solution of eq . ( 44 ) is @xmath113 the low - frequency singularity in the scattering relaxation time mentioned above is for electrons exactly at the fermi surface , @xmath97 . we note that for the electrons above the fermi surface , @xmath114 the singularity in the scattering relaxation time associated with the potential @xmath108 is weaker but it does not disappear . more accurately such a divergence must be regularized directly in the physically measurable quantity e.g. the tunneling conductance . however it is not necessary because the contribution to the scattering relaxation time from the phase collective mode , eq . ( 43 ) , is more important because it does not have a small exponential factor such as that presented in eqs . ( 42 ) and ( 45 ) . the appearance of the nonexponential scattering relaxation at low temperature is a direct consequence of the gapless phase mode in two dimensions . in three dimensions the phase mode have a gap and the main contribution to the scattering relaxation comes from the potential @xmath73 . @xmath115 where @xmath116 is the three - dimensional density of states . again after regularization of singularity in eq . ( 46 ) we have @xmath117 the recombination relaxation time is obtained similar to eq . ( 41 ) , @xmath118 we consider a system of two disordered electron layers with different density of states , @xmath119 , elastic scattering times , @xmath120 , mean free paths , @xmath121 , and diffusion coefficients @xmath122 . the layers are coupled by the coulomb potentials , there is no superconducting coupling between the layers . first we consider the screened coulomb potentials in the normal state . the nonscreened coulomb potentials within the layer , @xmath123 , and between electrons in different planes , @xmath124 , are @xmath125 where @xmath126 and @xmath127 are the dielectric constants of the electron layer and the inter - layer media , @xmath128 is the distance between layers . we assume that @xmath129 and we absorbed @xmath126 into @xmath130 in all further calculations small momentum transfers are important , thus we assume @xmath131 , @xmath132 in this chapter the lower indices of the potentials and the polarization operators refer to the layer , e.g. @xmath73 means the coulomb potential between electrons in the layer 1 , etc . the screened potentials are satisfied the equations @xmath133 we will use the definitions : @xmath134 , @xmath135 , and @xmath136 . the solution of eq . ( 3 ) is @xmath137 the polarization operators in each layers for @xmath138 and @xmath139 are chosen in the form corresponding to a normal state , because for recombination processes large frequencies @xmath77 are important , while collective excitations in a superconductor exist only for @xmath74 . @xmath140 the potentials are @xmath141 @xmath142 where @xmath143 we assume that layer 1 is in the superconducting state and layer 2 is in the normal state . we will calculate the recombination relaxation time in the superconducting layer due to inter - layer electron - electron interaction @xmath144 . from eq . ( 40 ) we have @xmath145 { d_1q^2\over \omega^2-\delta\omega + ( d_1q^2)^2}\cr { \rm im}u^a(q)\biggl({\delta\over 2\omega}\biggl)^{1/2}. \end{aligned}\ ] ] for the imaginary part of the iner - layer interaction we use the approximation similar to eq . ( 33 ) , @xmath146 corresponds to the electron scattering in the normal layer and thus does not have a small exponential factor typical for a superconductor . as a result @xmath147 we see that relaxation rate in a superconductor - normal metal two - layer system is increased by an exponential factor in comparison with the recombination rate in a single layer , see eq . we derived the kinetic equation describing the electron relaxation in two - dimensional impure superconductors . for electrons at the fermi surface , @xmath97 , the recombination and scattering relaxation time were calculated for low temperatures , @xmath69 . we took into account all channels of the electron - electron interaction in the superconductor . we found that the recombination relaxation rate comes from the quasiparticle scattering ( recombination processes ) associated with the fluctuations of the electron density and the phase of the order parameter , the propagators @xmath79 and @xmath80 . the recombination relaxation rate has double exponential smallness at low temperatures ( see eq . ( 41 ) ) , associated with exponentially small number of available quasiparticles . the scattering relaxation rate has a power law temperature dependence ( see eq . ( 43 ) ) due to singularity in the propagators @xmath80 and @xmath79 associated with the gapless collective mode , the phase mode . the contribution to the scattering relaxation rate from the fluctuations of the amplitude of the order parameter , @xmath73 has an infrared divergence similar to the phase relaxation time in the two - dimensional normal metal @xcite , however after regularization the corresponding contribution to the scattering relaxation rate has a small exponential factor and therefore is less important that the contribution from the collective excitations . we also shown that in the superconductor - normal metal two - layer system the recombination relaxation rate in the superconducting layer due to the inter - layer coulomb interaction is strongly increased at low temperatures @xmath69 by an exponential factor @xmath148 in comparison with a single superconducting layer . this fact may be important for constructing superconducting radiation detectors @xcite . in this appendix we obtain equations for the polarization operators for some limiting cases and prove some identities for them . though some of the results were already presented in ref . 8 , for matsubara frequencies and in ref . 15 for @xmath149 the analysis for continious frequencies has some advantages and helps us to estimate the imaginary parts of the interaction propagators . we start with the polarization operators for @xmath150 . @xmath151 may be taken directly from eq . ( 21 ) , @xmath152 where @xmath153 then we take @xmath154 from eq . ( 19 ) , @xmath155 , \end{aligned}\ ] ] and transform it using the identities @xmath156 @xmath157 to the form @xmath158 now recalling the bcs selfconsistancy equation @xmath159 we see that @xmath160 to transform @xmath161 , @xmath162 , \end{aligned}\ ] ] we use the identities @xmath163 @xmath164 and get @xmath165 now we recall another identity , @xmath166 and get @xmath167 we see from eqs . ( a5 ) , ( a8 ) , and ( a13 ) that the following identity holds @xmath168 at low temperatures , @xmath69 and for small arguments @xmath74 and @xmath75 , the polarization operators are @xmath169,\end{aligned}\ ] ] @xmath170,\end{aligned}\ ] ] @xmath171 . \end{aligned}\ ] ] thus for small arguments the following relation holds @xcite @xmath172 using eq . ( a16 ) we have for the propagator @xmath73 @xmath173 we see that @xmath73 is not singular which means that fluctuations of the amplitude of the order parameter are massive , thus the imaginary part of the propagator @xmath73 originates from @xmath174 . the screened coulomb potential is presented in the form @xmath175 where @xmath176 the propagator @xmath80 may be written in a similar way @xmath177 the poles in the propagators @xmath79 and @xmath80 correspond to the collective excitation , the phase mode , which in two dimensions is @xcite given by equation @xmath178 . in quasi - one - dimensional superconductors the nonscreened coulomb potential is @xmath179 , @xmath180 , where @xmath181 is a cross - sectional size , and the density of states is @xmath182 , thus the spectrum of the pase mode is @xcite @xmath183 it is interesting to see how the spectrum of the phase mode changes in different two - layer systems . first we consider a system of two identical impure superconducting planes coupled by the coulomb interaction , but no josephson coupling between the planes , thus the order parameters are independent in each planes . for the screened coulomb potential in each layer we have from eq . ( 52 ) @xmath184\tilde\pi(q)\over [ 1-v_0(q)\tilde\pi(q)]^2-[u_0(q)\tilde\pi(q)]^2}. \end{aligned}\ ] ] to avoid confusion we dropped lower indices in the coulomb potential @xmath185 . the spectrum of the phase modes is defined by equations @xmath186 and @xmath187 . the solution of these equations for @xmath188 is @xmath189 and @xmath190 . these new phase modes are similar to in - phase and out - of - phase plasmons in symmetric two - layer clean normal metal system , see @xcite and @xcite . now we consider a two - layer superconductor - normal metal disordered system . the polarization operators in the superconducting and normal layers according to eqs . ( a22 ) and ( 53 ) are @xmath192 @xmath193 the spectrum of collective excitations is determined from the equation @xmath194 for small momenta @xmath195 , @xmath196 eq . ( b4 ) leads to @xmath197 the solution of eq . ( b5 ) is a phase mode with small damping @xmath198 the last inequality is satisfied provided @xmath199 and @xmath200 . this result was independently obtained in @xcite . if the opposite inequality is valid , the solution of eq . ( b6 ) is a diffusion mode , @xmath201.\end{aligned}\ ] ] , ed . by d. n. langenberg and a. i. larkin , elsevier ( 1986 ) . a. v. sergeev and m. reizer , int . b * 10 * , 635 ( 1996 ) . d. s. pyun and t. r. lemberger , phys . b * 43 * , 3732 ( 1991 ) . s. g. doettinger , s. kittenberger , r. p. huebener , and c. c. tsuei , phys . b * 56 * , 14157 ( 1997 ) . reizer , phys . b * 39 * , 1602 ( 1989 ) . reizer , phys . b. * 57 * , 1147 ( 1998 ) . i. o. kulik , o. entin - wohlman , and r. orbach , j. low temp . phys . * 43 * , 591 ( 1981 ) . r. a. smith , m. reizer , j. w. wilkins , phys . rev . b * 51 * , 6470 ( 1995 ) . j. r. shrieffer , _ theory of superconductivity _ , chapter 8 , addison - wesley , redwood city , 1988 . a. schmid , z. phys . * 271 * , 251 ( 1974 ) . b. l. altshuler and a. g. aronov , pisma zh . * 30 * , 514 ( 1979 ) ; [ sov . jetp lett . * 30 * , 482 ( 1979 ) ] . for a review see b. l. altshuler and a. g. aronov , _ electron - electron interaction in disordered systems _ , edited by a. l. efros and m. polak north - holland , amsterdam , 1985 . t. p. devereaux and d. belitz , phys . 44 * , 4587 ( 1991 ) ; j. low temp . phys . * 77 * , 319 ( 1989 ) . m. yu . reizer and a. v. sergeev , zh . fiz . * 90 * , 1056 ( 1986 ) [ sov . zetp * 63 * , 616 ( 1986 ) ] . b. l. altshuler , a. g. aronov , and d. e. khmelnitskii , j. phys . c * 15 * , 7367 ( 1982 ) . u. eckern and f. pelzer , j. low temp . phys . * 73 * , 433 ( 1988 ) . y. takada , j. phys . . jpn . * 43 * , 1627 ( 1977 ) ; s. das sarma and a. madhukar , phys . rev . b * 23 * , 805 ( 1981 ) . j. e. mooij and g. schn , phys . * 55 * , 114 ( 1985 ) . b. n. narozhny , i. l. aleiner , and b. l. altshuler , phys . b ( 1999 ) .
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the electron - electron relaxation in impure two - dimensional superconductors is studied .
all channels of the electron - electron interaction classified in the nambu representation are taken into account .
it is shown that the recombination relaxation rate originates from quasipartical processes associated with fluctuations of the electron density and the phase of the order parameter . at low temperatures
the recombination relaxation rate has a double exponential temperature dependence .
the scattering relaxation rate at low temperatures has a power law temperature dependence due to contributions from gapless collective excitations , the phase modes .
two - layer superconductor - normal metal system is also considered .
it is shown that the recombination relaxation rate in the superconducting layer has a single exponential factor at low temperatures in comparison with a one layer superconducting system .
this increase in the recombination relaxation rate originates from the inter - layer coulomb interaction and may be used in constructing of superconducting radiation detectors .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
among the alternative approaches to describe the dark cosmological sector , consisting of dark matter ( dm ) and dark energy ( de ) , so called holographic de models have received considerable attention @xcite . the underlying holographic principle states that the number of degrees of freedom in a bounded system should be finite and related to the area of its boundary @xcite . on this basis , a field theoretical relation between a short distance ( ultraviolet ) cutoff and a long distance ( infrared ) cutoff was established @xcite . such relation ensures that the energy in a box of size @xmath3 does not exceed the energy of a black hole of the same size . if applied to the dynamics of the universe , @xmath3 has to be a cosmological length scale . different choices of this cutoff scale result in different de models . for the most obvious choice , the hubble scale , only models in which dm and de are interacting with each other also nongravitationally , give rise to a suitable dynamics @xcite . following ref . , there has been a considerable number of investigations based on the future event horizon as cutoff scale . all models with a cutoff at the future event horizon , however , suffer from the serious drawback that they can not describe a transition from decelerated to accelerated expansion . a future event horizon does not exist during the period of decelerated expansion . we are focussing here on a further option that has received attention more recently , a model based on a cutoff length proportional to the ricci scale . a distance proportional to the ricci scale has been identified as a causal connection scale for perturbations @xcite . as a cutoff length in de models it was first used in ref . . subsequent investigations include ref . and , for the perturbation dynamics , refs . + here , we reconsider the dynamics of a two - component system of pressureless dm and ricci - type de both in the homogeneous and isotropic background and on the perturbative level@xcite . whereas most dynamic de scenarios start with an assumption for the equation - of - state ( eos ) parameter for the de , the starting point of holographic models is an expression for the de energy density from which the eos is then derived . as was pointed out in ref . , the mere definition of the holographic de density , independently of the choice of the specific cutoff length , implies an interaction with the dm component . requiring this interaction to vanish is equivalent to impose an additional condition on the dynamics . in the case of ricci - type de this condition establishes a simple relation between the matter fraction and the necessarily time - dependent eos parameter . of course , a time - varying eos parameter is not compatible with a cosmological constant . our main aim here is to perform a gauge - invariant perturbation analysis for this model . it will turn out that the general perturbation dynamics suffers from instabilities . there exists just one single configuration without instabilities at finite values of the scale factor @xmath4 @xcite . we also update previous tests of the homogeneous and isotropic background dynamics using recent results for the differential age of old objects based on the @xmath5 dependence , data from snia and from bao . the cosmic medium is assumed to consist of pressureless dm with energy density @xmath6 and a holographic de component with energy density @xmath7 . in the spatially flat case friedmann s equation is @xmath8 where @xmath9 is the hubble rate and @xmath4 is the scale factor of the robertson - walker metric . in general , both components are not separately conserved but obey the balance equations @xmath10 with a source ( or loss ) term @xmath11 , such that the total energy @xmath12 is conserved . here , @xmath13 is the eos parameter of the de and @xmath14 is the pressure associated with the holographic component . in terms of the scale factor as independent variable , the acceleration equation can be written @xmath15 where @xmath16 is the ratio of the energy densities . total _ effective eos of the cosmic medium is @xmath17 according to the balance equations ( [ cons2 ] ) , the ratio @xmath18 changes as @xmath19\ . \label{dr2}\ ] ] following refs . , we write the holographic energy density as @xmath20 the quantity @xmath3 is the infrared cutoff scale and @xmath21 is the reduced planck mass . the numerical constant @xmath22 determines the degree of saturation of the condition @xmath23 which is at the heart of any holographic de model . it states that the energy in a box of size @xmath3 should not exceed the energy of a black hole of the same size @xcite . differentiation of the expression ( [ ans ] ) for the holographic de density and use of the energy balances ( [ cons2 ] ) yields @xmath24 in general , there is no reason for @xmath11 to vanish . assuming @xmath25 provides us with a specific relationship between @xmath26 and the ratio of the rates @xmath27 and @xmath28 . any nonvanishing @xmath11 will modify this relationship . with @xmath11 from ( [ ql ] ) , the general dynamics ( [ dr2 ] ) of the energy - density ratio @xmath18 becomes @xmath29\ , . \label{drl}\ ] ] the case without interaction is characterized by [ cf . ( [ dr2 ] ) ] @xmath30 with a generally time - dependent @xmath26 . different choices of the cutoff scale @xmath3 give rise to different expressions for the total effective eos parameter in eq . ( [ w ] ) and to different relations between @xmath26 and @xmath18 . we shall briefly sketch the situations for the hubble radius and for the future event horizon as cutoff lengths before considering in detail the ricci scale . for @xmath31 the holographic de density is @xmath32 for the deceleration parameter one has @xmath33 in the interaction - free case we recover the einstein - de sitter value @xmath34 . the condition for accelerated expansion is @xmath35 . to describe a transition from decelerated to accelerated expansion , @xmath11 has to change from @xmath36 to @xmath35 . a viable scenario can be realized , e.g. , by a choice @xcite @xmath37 where @xmath38 is an interaction constant . the resulting dynamics is that of a generalized chaplygin gas with a hubble rate @xmath39^{1/n}\ , , \label{hq}\ ] ] where @xmath38 is related to the present value @xmath40 of the deceleration parameter @xmath41 by @xmath42 for @xmath43 one reproduces the @xmath2cdm dynamics . with @xmath44 , where @xmath45 is the future event horizon , the holographic de density ( [ ans ] ) is @xmath46 the de balance ( [ cons2 ] ) can be written as @xmath47 with an effective eos ( the superscript e denotes the event horizon ) @xmath48 this effective eos does not directly depend on @xmath26 . however , the ratio @xmath18 that enters @xmath49 is determined by @xmath26 via eq . ( [ drl ] ) . notice also , that this effective eos for the de component is different from the total effective eos of the cosmic medium which is @xmath50 . in the previous hubble - scale - cutoff case both these quantities were identical . different from the previous hubble - scale cutoff , there exists a non - interacting limit with accelerated expansion in the present case . in this special situation @xmath51\ \mathrm{and } \ r_{e}h = - \frac{2}{1 + 3 \omega } \label{q0}\ ] ] are valid . explicitly , the relation between @xmath26 and @xmath18 is @xmath52 it is obvious , that for any @xmath53 , the parameter @xmath26 remains always smaller than @xmath54 , demonstrating the impossibility of a matter - dominated period in this context . our interest in the present paper will be the ricci - scale cutoff . the role of a distance proportional to the ricci scale as a causal connection scale for perturbations was noticed in ref . . in ref . it was used for the first time as a de cutoff scale . the ricci scalar is @xmath55 . for the corresponding cutoff scale one has @xmath56 , i.e. , @xmath57 where @xmath58 . upon using ( [ dh ] ) we obtain @xmath59 for the holographic de density . notice that the ( not yet known ) eos parameter explicitly enters @xmath60 . use of friedmann s equation provides us with @xmath61 which coincides with the result in ref . . obviously , a constant value of @xmath26 necessarily implies a constant @xmath18 and vice versa . for the source term we have @xcite @xmath62\rho_{h } \ , . \label{q=}\ ] ] it is obvious that a constant eos parameter @xmath26 is compatible with @xmath25 only for @xmath63 , i.e. , if @xmath7 behaves as dust . for a time - varying eos parameter @xmath64 , however , there exists a non trivial case @xmath25 : @xmath65 in the remainder of the paper we shall consider this case , for which the eos parameter is explicitly given by @xmath66 at high redshift we have @xmath67 the property that noninteracting ricci - de behaves as dust at high redshift was first pointed out in ref . . this model naturally reproduces an early matter - dominated era . for @xmath68 and @xmath69 , the ratio @xmath18 approaches @xmath70 for @xmath71 . this value is only roughly ten times larger than the present value @xmath72 . for the @xmath2cdm model the corresponding difference is about nine orders of magnitude . in this sense , the coincidence problem is considerably alleviated for our ricci cdm model for which the hubble rate becomes @xmath73 } { 3\omega_{0}- r_{0}\left[1+r_{0}-3\omega_{0}\right ] } } \ , . \label{solhq0}\ ] ] generally , the two - component model is described by an energy - momentum tensor @xmath74 with @xmath75 and @xmath76 . the quantity @xmath77 denotes the total four - velocity of the cosmic substratum . latin indices run from @xmath78 to @xmath79 . the total @xmath80 splits into a matter component and a holographic de component , @xmath81 with ( @xmath82 ) @xmath83 for separately conserved fluids we have @xmath84 and @xmath85 . in general , each component has its own four - velocity , with @xmath86 . for the homogeneous and isotropic background we assume @xmath87 . indicating first - order perturbations about the homogeneous and isotropic background by a hat symbol , the perturbed time components of the four - velocities are @xmath88 restricting ourselves to scalar perturbations , we define the ( three- ) scalar quantities @xmath89 , @xmath90 and @xmath91 by @xmath92 introducing fractional energy - density perturbations @xmath93 and changing to gauge - invariant variables according to @xmath94 energy and momentum conservations for the cosmic medium as a whole reduce to @xcite @xmath95 the superscript c indicates that the corresponding variables are defined with respect to a comoving observer . the perturbation @xmath96 has to be determined from the perturbed raychaudhuri equation for @xmath97 . in our context one finds at linear order @xmath98 with @xmath99 where @xmath100 is the three - dimensional laplacian . combing eqs . ( [ balcomb ] ) and ( [ dthetacfin ] ) and specifying the pressure perturbations will result in a second - order equation for @xmath101 . on the other hand , with @xmath102 and @xmath103 it will be useful to consider the combination @xmath104 , where @xmath105 . our aim is to obtain an equation also for @xmath106 . to this purpose we have to introduce a further ingredient . so far the pressure perturbations are not sufficiently specified . in general , pressure perturbations in two - component systems are nonadiabatic . firstly , because of the two - component nature itself , secondly because each of the components may be nonadiabatic on its own . the relevant combination for our de component is @xmath107\,,\ ] ] which is a gauge - invariant expression . now an assumption for the perturbed eos parameter @xmath108 is necessary to proceed . we shall restrict ourselves here to adiabatic internal perturbations of the de component , equivalent to a vanishing of the combination ( [ hatph-1 ] ) : @xmath109 this assumption of an adiabatic de component allows us to relate the otherwise undetermined perturbation @xmath108 of the eos parameter to the de energy perturbation @xmath110 . we emphasize that the total perturbation dynamics remains nonadiabatic due to the two - component nature of the medium . the resulting coupled equations for @xmath101 and @xmath106 in the @xmath111 space then are @xcite ( the prime denotes a derivative with respect to @xmath4 ) @xmath112 \frac{\delta^{c\prime}}{a}\nonumber\\ & - & \left[\frac{3}{2 } + 12\frac{p}{\rho } - \frac{9}{2}\frac{p^{2}}{\rho^{2 } } - 9\frac{p^{\prime}}{\rho^{\prime } } - \frac{k^{2}}{a^{2}h^{2}}\frac{p}{\rho^{\prime}}\right]\frac{\delta^{c}}{a^{2}}\nonumber\\ & & \qquad\qquad\qquad\qquad = \frac{k^{2}}{a^{2}h^{2}}\frac{p^{\prime}}{\rho^{\prime}}\frac{\rho_{m}}{\rho}\frac{s_{mh}}{a^{2 } } \ \label{prprdeltas}\end{aligned}\ ] ] and @xmath113\frac{s^{\prime}_{mh}}{a}\nonumber\\ & + & \frac{r}{1+\omega}\frac{p}{\rho}\frac{k^{2}}{a^{2}h^{2}}\frac{s_{mh}}{a^{2 } } = \frac{1+r}{1+\omega}\frac{p}{\rho}\frac{k^{2}}{a^{2}h^{2}}\frac{\delta^{c}}{a^{2}}\,,\end{aligned}\ ] ] respectively , where we have to exclude the case @xmath114 . to obtain the matter - energy perturbations , we decompose the total energy - density perturbation @xmath101 according to @xmath115 combination with @xmath116 leads to @xmath117\,,\ ] ] which describes the matter - energy perturbations as a combination of @xmath101 and @xmath106 . to obtain its dynamics one has to solve the coupled system of equations ( [ prprdeltas ] ) and ( [ prprs ] ) . the matter density perturbation @xmath118 in relation ( [ deltamc ] ) is defined with respect to the _ total _ comoving gauge . to obtain the matter density perturbation , comoving with the matter velocity , @xmath119 , we have to consider @xmath120 since @xcite @xmath121 the quantity of interest is @xmath122 obviously , @xmath118 and @xmath123 differ by the last term in relation ( [ deltamcm ] ) . because of the factor @xmath124 ( assuming @xmath125 ) one expects that on scales smaller than the hubble scale the differences between @xmath118 and @xmath123 are small . ( dotted line ) , @xmath101 ( dash - dotted line ) , @xmath126 ( dashed line ) and @xmath127 ( thin solid line ) for @xmath128 , @xmath129 and @xmath130 . for comparison the @xmath2cdm result ( thick solid line ) is also included . ] in fig . [ fig1 ] we show the behavior of the quantities @xmath106 , @xmath101 , @xmath118 and @xmath131 for @xmath132 with @xmath133 . while this figure confirms that differences between @xmath118 and @xmath123 are indeed small on the chosen scale , there appear oscillations of all the perturbation quantities very close to the present time . this reminds of a similar feature in ( generalized ) chaplygin - gas models which apparently have jeopardized these models @xcite . still more serious is the existence of instabilities at future values @xmath134 of the scale factor , related to a crossing of the phantom divide @xmath114 . instabilities occur if the denominator @xmath135 in ( [ prprs ] ) vanishes , i.e. , if @xmath26 approaches @xmath136 . from eq . ( [ wsol ] ) one finds the condition for @xmath137 , @xmath138 which determines the value @xmath139 of the scale factor at which the instability occurs . solving for @xmath139 yields @xmath140 now we assume @xmath141 and consider the cases @xmath142 and @xmath143 separately . for @xmath144 and @xmath145 we have @xmath146 for @xmath147 we find @xmath148 , i.e. , the instability sets in at a finite value of the scale factor in the future . for @xmath149 , i.e. for a phantom eos , there appears an instability in the past at @xmath150 . since such kind of instability has not been observed , a present phantom eos is definitely excluded in the context of our model . the limit between the two regimes is just @xmath151 where we have @xmath152 , i.e. , an instability at the present epoch . the only case without instabilities at finite values of the scale factor is a fixed relation @xmath153 between the initially independent values of @xmath72 and @xmath1 . since @xmath154 necessarily , this implies @xmath155 . consequently , the only physically acceptable case is @xmath156 the parameters @xmath1 and @xmath72 are necessarily related to each other and can not be chosen independently . in a sense , @xmath72 quantifies the deviation of @xmath1 from @xmath157 . under this condition we have @xmath158 . this is exactly the result found by karwan and thitapura in their study of instabilities through nonadiabatic perturbations in a system of matter and ricci de @xcite . the solutions for @xmath26 and @xmath18 then simplify to @xmath159 respectively . combination of both solutions has the important consequence @xmath160 this makes all the coupling terms ( and some others ) in the coupled system ( [ prprdeltas ] ) and ( [ prprs ] ) vanish . also the pressure perturbations @xmath161 vanish . the square of the hubble rate turns out to be @xmath162 notice that we have the same number of free parameters as in the @xmath2cdm model , but there is no @xmath2cdm limit of ( [ hstab ] ) . the behavior of the perturbation quantities on the basis of ( [ wsol+ ] ) and ( [ dotp0 ] ) is visualized in fig . [ fig2 ] . this figure confirms that for the chosen configuration there are neither oscillations nor instabilities . from this point of view the model appears acceptable . ( dotted line ) , @xmath101 ( dash - dotted line ) , @xmath126 ( dashed line ) and @xmath127 ( thin solid line ) for @xmath129 and @xmath130 on the basis of ( [ wsol+ ] ) and ( [ dotp0 ] ) . the @xmath2cdm result is represented by the thick solid line . the relative density perturbations @xmath106 are negligible during the entire evolution . the results for @xmath126 and @xmath127 are almost identical . ] we have tested this model by the differential age of old objects based on the @xmath5 dependence @xcite as well as by the data from snia @xcite and from bao @xcite . the details of the analysis are given in ref . . the results are presented in fig . [ fig3 ] which shows the two - dimensional probability - distribution function ( pdf ) at @xmath163 ( @xmath164 of confidence level ) , @xmath165 ( @xmath166 of confidence level ) and @xmath167 ( @xmath168 of confidence level ) . the estimation for @xmath169 , based on a combination of the three tests at @xmath165 , is @xmath170 , while for @xmath171 we find @xmath172 . the straight line represents the combination @xmath173 which is singled out by the stability analysis of the perturbation dynamics . the tension to the results for the background dynamics is obvious , an agreement is possible only at the @xmath167 level . and @xmath171 resulting from a combination of the three tests . the straight line represents the instability - avoiding configuration @xmath173 . ] noninteracting ricci - type de is characterized by a necessarily time - dependent eos parameter . this makes it an observationally testable alternative to the @xmath2cdm model . there exists a a relationship between this eos parameter and the matter content of the universe . the ratio of the energy densities of dm and de varies considerably less than for the @xmath2cdm model . since the time of radiation decoupling it has changed by about one order of magnitude compared with roughly nine orders of magnitude for the @xmath2cdm model this amounts to a remarkable alleviation of the coincidence problem . ricci - type de behaves almost as dust at high redshift . our statistical analysis , based on recent observational data from snia , bao and @xmath5 , results in a preferred value of @xmath174 for the ricci - de parameter which confirms earlier studies in the literature @xcite . within a gauge - invariant analysis we calculated the matter perturbations as a combination of the total energy perturbations of the cosmic medium and the relative perturbations of the components . the perturbation dynamics suffers from instabilities that exclude a present phantom - type eos . it is only for a specific relation between the values @xmath0 of the present matter density and the present eos parameter @xmath1 that the dynamics remains stable for any finite scale - factor value . this relation corresponds to a ricci - de parameter @xmath175 @xcite . holographic ricci - type de represents a theoretically appealing scenario which does not need additional parameters except @xmath176 and @xmath0 . despite of its attractive features , the stable configuration is only marginally consistent with the observationally preferred background values of @xmath0 and @xmath1 . this work was supported by the comisin nacional de ciencias y tecnologa " ( chile ) through the fondecyt grants no . 1110230 and no . 1130628 ( r.h . and j.c.f and w.z acknowledge support by fondecyt - concurso incentivo a la cooperacin internacional " no . 1130628 as well as by cnpq ( brazil ) and fapes ( brazil ) . we thank alonso romero for carefully checking our calculations .
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we consider the dynamics of a cosmological substratum of pressureless matter and holographic dark energy with a cutoff length proportional to the ricci scale .
stability requirements for the matter perturbations are shown to single out a model with a fixed relation between the present matter fraction @xmath0 and the present value @xmath1 of the equation - of - state parameter of the dark energy .
this model has the same number of free parameters as the @xmath2cdm model but it has no @xmath2cdm limit .
we discuss the consistency between background observations and the mentioned stability - guaranteeing parameter combination .
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the @xmath0 is a new hadronic resonance discovered in 2003 by the belle collaboration @xcite and subsequently confirmed by the cdf , babar , and d0 collaborations @xcite . all the properties of the @xmath0 that have been measured thus far are compatible with its identification as a weakly - bound molecule whose constituents are a superposition of the charm meson pairs @xmath1 and @xmath2 @xcite . because it is so weakly bound , the @xmath0 has many features in common with the deuteron , which is a weakly - bound baryonic molecule consisting of a proton and a neutron . their binding energies are both small compared to the natural energy scale associated with pion exchange : @xmath3 , where @xmath4 is the reduced mass of the two constituents . the binding energy 2.2 mev of the deuteron is small compared to the natural scale of about 20 mev . after taking into account a recent precision measurement of the @xmath5 mass by the cleo collaboration @xcite , the difference between the measured mass of the @xmath0 and the @xmath1 threshold is @xmath6 this is small compared to the natural energy scale of about 10 mev . the belle collaboration has set an upper bound on the width of the @xmath7 @xcite : @xmath8 this is also small compared to the natural energy scale . the small width is most easily understood if the @xmath7 is below the @xmath1 threshold . because the binding energies of the deuteron and the @xmath0 are small compared to the natural energy scales , they have universal properties that are determined by the large scattering length @xmath9 of the constituents @xcite . the universality of few - body systems with a large scattering length has many applications in atomic , nuclear , and particle physics @xcite . the universal features of the @xmath0 were first exploited by voloshin to describe its decays into @xmath10 and @xmath11 , which can proceed through decay of the constituent @xmath12 or @xmath13 @xcite . universality has also been applied to the production process @xmath14 @xcite , to the line shape of the @xmath7 @xcite , and to decays of @xmath7 into @xmath15 and pions @xcite . these applications rely on factorization formulas that separate the length scale @xmath9 from all the shorter distance scales of qcd @xcite . the factorization formulas can be derived using the operator product expansion for a low - energy effective field theory @xcite . there is one feature of the @xmath0 that is very different from the deuteron . the mass difference between the constituents @xmath12 and @xmath16 , @xmath17 mev , is very close to the @xmath18 mass : @xmath19 mev . thus the @xmath12 mass is very close to the @xmath20 threshold : @xmath21 mev . this is important because the pion is the lightest meson that can be exchanged between the @xmath12 and @xmath16 . the small energy gap between the @xmath12 mass and the @xmath20 threshold makes it easy for the @xmath12 to make a transition to @xmath20 . an important consequence is that the @xmath7 could have a substantial 3-body component @xmath10 . suzuki has argued that the near equality of @xmath22 and @xmath23 implies that the pion - exchange interaction is too weak to bind the @xmath7 @xcite , but this conclusion has been criticized @xcite . an analysis of the effect of the near equality @xmath24 in the case of the @xmath0 is complicated by other effects that may be comparable in importance . there are other nearby two - body and three - body thresholds . the thresholds for the charged mesons @xmath25 and @xmath26 are higher than the @xmath1 threshold by @xmath27 mev . the @xmath28 and @xmath29 thresholds are higher than the @xmath1 threshold by only @xmath30 mev . there are several other features that complicate the analysis of the effects of pions on the @xmath7 . the constituent @xmath31 s are spin-1 particles . the charge conjugation @xmath32 requires the molecule to be a superposition of charm mesons : @xmath33 . the interaction of the charm mesons with pions depends on the pion momentum . a quantitative analysis of the effects of pions on the @xmath7 might require all these effects to be taken into account . since the new feature of the @xmath0 is the near equality between the mass difference @xmath22 of the constituents and the mass @xmath23 of a meson that can be exchanged between them , it is worthwhile to analyze this feature in a simple model that avoids all the other complications of the @xmath7 . the simplest such model is one in which the primary constituents @xmath34 and @xmath35 are spin-0 mesons with a large positive scattering length and a momentum - independent coupling to a lighter spin-0 meson @xmath36 . we will calculate the effects of the exchanged meson in this model to next - to - leading order in the @xmath37 coupling constant @xmath38 . in sec . [ sec : model ] , we write down the lagrangian for the model and discuss its parameters . in sec . [ sec : zeroth ] , we summarize the results of the model at 0@xmath39 order in @xmath38 that were derived previously in refs . the results include the short - distance production rate for @xmath7 and the line shape of @xmath7 in a short - distance decay channel . the ultraviolet divergences can be removed by renormalization of the coupling constant for the @xmath40 contact interaction and by renormalization of short - distance coefficients in the operator product expansion . in sec . [ sec : second ] , we calculate the observables in sec . [ sec : zeroth ] to next - to - leading order in @xmath38 . we also calculate the short - distance production rate for @xmath41 and the decay rate of @xmath7 into @xmath41 . there are new ultraviolet divergences that can be removed by renormalization of the @xmath35 mass . we show that after summing a geometric series of next - to - leading order corrections , all remaining ultraviolet divergences can again be removed by renormalization of the coupling constant for the @xmath40 contact interaction and by renormalization of short - distance coefficients in the operator product expansion . a further resummation is required to eliminate an infrared divergence at the @xmath40 threshold in the next - to - leading order corrections . in sec . [ sec : summary ] , we summarize our results . the two - loop feynman diagrams that arise at next - to - leading order in @xmath38 are calculated in an appendix . we consider the simplest possible model with a bound state @xmath7 whose mass is close to a 3-body threshold . the primary constituents @xmath34 and @xmath35 of the bound state are scalar mesons whose masses @xmath42 and @xmath43 satisfy @xmath44 . there is also a scalar meson @xmath36 whose mass @xmath23 is close to the mass difference @xmath45 . we will refer to this model as the _ scalar meson model_. it is defined by a nonrelativistic quantum field theory with three complex scalar fields @xmath34 , @xmath35 , and @xmath36 . the free terms in the lagrangian on a field indicates its complex conjugate . ] are @xmath46 the interaction terms for the scalar meson model are @xmath47 the interaction vertices are illustrated in fig . [ fig : vertex ] . the coupling constants @xmath48 and @xmath38 have mass dimensions @xmath49 and @xmath50 , respectively . we assume that there is a fine - tuning that makes the @xmath40 scattering length large , which implies that the @xmath40 contact interaction with coupling constant @xmath48 must be treated nonperturbatively . the subscript on @xmath48 emphasizes that it is a bare coupling constant that requires renormalization . we assume that the @xmath37 interaction can be treated perturbatively and we calculate its effects through order @xmath51 . at this order , no renormalization of @xmath38 is required . however , we will see that nonperturbative renormalization of @xmath48 requires summing a geometric series to all orders in @xmath51 . there are also ultraviolet divergent corrections to the mass of the @xmath35 which are cancelled by the mass counterterm @xmath52 in eq . ( [ lint ] ) . contact interaction , ( b ) the @xmath53 interaction , and ( c ) the @xmath35 mass counterterm . [ fig : vertex],width=604 ] it is convenient to introduce concise notations for various combinations of the masses . the sum and difference of the @xmath34 and @xmath35 masses are @xmath54 the reduced masses of @xmath40 and of @xmath55 are @xmath56 [ mred ] if @xmath23 is much smaller than @xmath42 , the natural energy and 3-momentum scales associated with exchange of the meson @xmath36 between the mesons @xmath34 and @xmath35 are @xmath57 and @xmath23 , respectively . we assume that the energy gap between the @xmath35 mass and the @xmath55 threshold is small compared to the natural energy scale : @xmath58 the total number of heavy mesons @xmath34 and @xmath35 is conserved by the interactions in eq . ( [ lint ] ) . we are interested in the sector of the theory with two heavy mesons and , more specifically , in the threshold region where the invariant mass @xmath59 of all the particles is very close to @xmath60 : @xmath61 this restricts the possible scattering states to @xmath40 and @xmath41 . for numerical illustrations , we will use masses that correspond to the @xmath62 system . the meson masses are @xmath63 mev , @xmath64 mev , and @xmath65 mev . thus @xmath66 mev and @xmath67 mev . the natural energy scale is @xmath68 mev . the condition in eq . ( [ dm - small ] ) is only marginally satisfied : @xmath69 . we take the binding energy of @xmath7 to be @xmath70 mev , which is near the central value of the measurement of @xmath71 in eq . ( [ mx - cleo ] ) . if we express this binding energy as @xmath72 , the binding momentum is @xmath73 mev . we impose an ultraviolet cutoff on the momenta @xmath74 of particles in the center - of - momentum frame that restricts them to the region @xmath75 . this guarantees that all the particles remain nonrelativistic . it was shown in ref . @xcite that at leading order @xmath76 , all ultraviolet divergences can be absorbed into @xmath48 and into short - distance coefficients in the operator product expansion . at order @xmath51 , there are new ultraviolet divergences that can be absorbed into the @xmath35 mass . we will find that the remaining ultraviolet divergences can again be absorbed into @xmath48 and into short - distance coefficients in the operator product expansion . the coupling constant @xmath48 must be tuned to near the critical value at which the @xmath40 scattering length diverges in order for the bound state @xmath7 to have a small binding energy @xmath77 . in order to compare results at order @xmath76 and order @xmath51 , we must have a well - defined renormalization prescription for @xmath48 or , equivalently , a clear definition of the binding energy @xmath78 . we will find that all the amplitudes for processes near the @xmath40 threshold have a resonant term with a common factor @xmath79 that depends on the energy @xmath80 in the rest frame . for example , the line shape in a short - distance decay channel of @xmath7 produced by a short - distance process has a long - distance factor @xmath81 , where @xmath59 is the invariant mass of the decay products . the square of the resonant factor can be expressed in the form @xmath82 ^ 2 + [ { \rm i m } \ , \mathcal{a}^{-1}(e)]^2 } . \ ] ] as a function of the real energy @xmath80 , this resonance factor has a peak when @xmath80 is close to @xmath83 . the peak arises because @xmath84 vanishes near that value of @xmath80 and @xmath85 is much smaller than @xmath84 except near that value of @xmath80 . as our renormalization prescription for @xmath48 , we demand that @xmath84 vanishes at an energy determined by a real variable @xmath86 : @xmath87 the dependence of an observable on the bare parameter @xmath48 can be eliminated in favor of the renormalization parameter @xmath86 . we assume that @xmath86 is small compared to the natural momentum scale : @xmath88 . the amplitude @xmath79 has a pole at a complex energy @xmath89 near the real energy at which @xmath90 has its maximum . we choose to define the mass @xmath71 and the width @xmath91 of @xmath7 by expressing that complex energy in the form @xmath92 we also choose to define the binding energy @xmath78 by @xmath93 we assume that the binding energy and the width of @xmath7 are both small compared to the natural energy scale : @xmath94 . in ref . @xcite , the authors proposed a renormalization scheme in which @xmath48 was tuned to give a prescribed value for the complex parameter @xmath89 . the binding energy @xmath78 and the width @xmath91 were taken as the input parameters that determine @xmath89 . this renormalization scheme requires @xmath48 to be a complex coupling constant . the imaginary part of @xmath48 takes into account inelastic scattering channels for @xmath95 that are outside the energy range described by the effective field theory . in this paper , we assume that any such inelastic channels do not give a significant contribution to the width of @xmath7 . the coupling constant @xmath48 is therefore real valued . we could use the binding energy @xmath78 as the input parameter and eliminate @xmath48 in favor of @xmath78 , and then @xmath91 would be calculable . we find it more convenient to eliminate @xmath48 in favor of the variable @xmath86 introduced in eq . ( [ reainv : zero ] ) . in this renormalization scheme , @xmath78 and @xmath91 are both calculable in terms of @xmath86 , @xmath38 , and the masses @xmath42 , @xmath43 , and @xmath23 . the most convenient input for determining the coupling constant @xmath38 is the width @xmath96 of the heavy meson @xmath35 . since @xmath97 , the @xmath53 vertex in fig . [ fig : vertex](b ) allows @xmath35 to decay into @xmath55 . calculating the width using nonrelativistic phase space , we obtain @xmath98^{1/2 } , \label{gam2-nr}\ ] ] where @xmath4 and @xmath99 are the reduced masses defined in eqs . ( [ mred ] ) . the value of the coupling constant @xmath38 can be determined from @xmath96 and the masses . we assume that the width of @xmath35 is small compared to the natural energy scale : @xmath100 . we also assume @xmath96 is small enough to allow perturbation theory in the coupling constant @xmath38 . for numerical illustrations , we will use values of the parameters that correspond to the @xmath62 system . we take the width of @xmath35 to be @xmath101 mev , which is approximately the total width for @xmath12 @xcite . this is much smaller than the natural energy scale @xmath57 and also rather small compared to @xmath102 : @xmath103 . using eq . ( [ gam2-nr ] ) , we determine the numerical value of the @xmath53 coupling constant to be @xmath104 mev@xmath105 . when considering the dependence of observables on @xmath38 , it will be more convenient to regard them as functions of @xmath96 . we now list all the energies that are assumed to be small compared to the natural scale @xmath57 : * the energy @xmath106 of a @xmath40 or @xmath41 system relative to the @xmath40 threshold , * the energy difference @xmath107 between the @xmath35 mass and the @xmath55 threshold , * the decay width @xmath96 of @xmath35 , which is given in eq . ( [ gam2-nr ] ) , * the binding energy @xmath78 of the molecule , which is approximately equal to @xmath108 , where @xmath86 is a renormalization parameter . we will carry out our calculations under the assumption that these energies are all comparable and that the masses @xmath43 , @xmath42 , and @xmath23 are also comparable . the only approximations we will make are those that can be justified by a nonrelativistic approximation , such as @xmath109 . at order @xmath51 , renormalization of the @xmath35 mass is necessary . in the _ on - shell renormalization scheme _ , the parameter @xmath43 in the free lagrangian in eq . ( [ lfree ] ) is chosen to be the physical @xmath35 mass . the physical mass @xmath43 and the width @xmath96 of @xmath35 can be defined by specifying that the exact @xmath35 propagator at 0 momentum has a pole in the energy at @xmath111 . the pole in the energy @xmath112 at 0 momentum is the solution to the equation @xmath113 where @xmath114 is the @xmath35 self - energy . the exact @xmath35 self - energy is the sum of the one - loop diagram in fig . [ fig : self](a ) and the mass counterterm in fig . [ fig : self](b ) : @xmath115 there are no other diagrams for the self - energy at higher order in @xmath38 . the one - loop diagram is calculated in section [ sec : self - energy ] of the appendix . the analytic expression for the self - energy diagram using dimensional regularization is given in eq . ( [ sigma2-dimreg0 ] ) . the result in a general regularization scheme is @xmath116 \right)^{1/2 } . \label{d2self}\end{aligned}\ ] ] the term @xmath117 is real valued and includes a linear ultraviolet divergence . the divergence is cancelled by a divergent term in the mass counterterm @xmath52 in eq . ( [ sigma : exact ] ) . in the on - shell renormalization scheme , the finite terms in @xmath52 are chosen so that the solution to eq . ( [ pole - eq ] ) satisfies @xmath118 . the wavefunction normalization factor for a @xmath35 with momentum @xmath119 is @xmath120 self - energy . [ fig : self],width=529 ] since eq . ( [ pole - eq ] ) can not be solved analytically , the on - shell renormalization scheme is not the most convenient choice . a simpler renormalization scheme for the @xmath35 mass is to choose the mass counterterm so the self - energy @xmath121 vanishes at the @xmath40 threshold : @xmath122 . the self - energy is then @xmath123 \right)^{1/2 } . \label{d2self : ren}\end{aligned}\ ] ] with this renormalization prescription , the parameter @xmath43 differs at order @xmath124 from the physical @xmath35 mass , which is given by the real part of the solution to eq . ( [ pole - eq ] ) . the width @xmath96 , which is given by the imaginary part of the solution to eq . ( [ pole - eq ] ) , reduces at leading order in @xmath38 to @xmath125 . this agrees with the explicit expression for the decay rate of @xmath35 at order @xmath51 obtained in eq . ( [ gam2-nr ] ) . to order @xmath51 , the wavefunction normalization factor in eq . ( [ z2-def ] ) is @xmath126 \big)^{-1/2 } .\ ] ] since the momentum @xmath119 is restricted to the nonrelativistic region , the term @xmath127 can always be neglected compared to 1 . the resulting wavefunction normalization factor is independent of @xmath119 : @xmath128^{-1/2}. \label{z2-nr}\ ] ] the correction term is pure imaginary . the @xmath37 interaction allows a virtual @xmath35 whose energy is greater than @xmath129 to decay into @xmath130 . in the amplitude for such a process at leading order in @xmath38 , the propagator of the virtual @xmath35 diverges . near such a point in momentum space , the corrections to the @xmath35 propagator from higher orders in @xmath38 are not suppressed . this problem can be solved by summing the @xmath35 propagator corrections to all orders . the effect is to add @xmath131 to the denominator @xmath132 of the @xmath35 propagator . the imaginary part of the self - energy @xmath114 eliminates any divergence in the @xmath35 propagator for real values of the energy . one drawback of this solution is that it leads to unnecessarily complicated expressions for the integrals in loop diagrams with @xmath35 propagators . summing the @xmath35 self - energy corrections to all orders is essential only near the pole in the propagator . in other regions of the energy @xmath112 , @xmath114 is a perturbative correction to the denominator of the propagator that is suppressed by a power of @xmath51 . in those regions , summing the @xmath35 propagator corrections to all orders is optional . at the pole in @xmath112 , the self - energy @xmath114 reduces to @xmath133 . an alternative resummation that eliminates the divergence in the propagator for real values of the energy without changing the order in @xmath38 of the truncation error is to sum only the @xmath133 term in the self - energy to all orders and treat the remainder of the self - energy as a perturbation . an advantage of this partial resummation is that it leads to simpler expressions for the integrals in loop diagrams with @xmath35 propagators . a systematic method for implementing this partial resummation is the _ complex mass scheme _ @xcite . this method has been used in standard model calculations at next - to - leading order @xcite . the application of this method to the scalar meson model involves adding cancelling terms to the free and interaction terms in the lagrangian : @xmath134 where @xmath96 is the @xmath35 width and @xmath135 is the @xmath35 wavefunction normalization factor . to order @xmath51 , @xmath96 and @xmath135 are given by eqs . ( [ gam2-nr ] ) and ( [ z2-nr ] ) . since @xmath136 is pure imaginary and @xmath135 is complex , the free lagrangian consisting of the sum of eqs . ( [ lfree ] ) and ( [ dlfree ] ) corresponds to a nonhermitian hamiltonian . the effects of the nonhermiticity are cancelled exactly if the interaction terms in eq . ( [ dlint ] ) are calculated to all orders . at any finite order in perturbation theory , the complex mass scheme gives amplitudes that are uniformly accurate to the appropriate order in @xmath38 , even if there is a virtual @xmath35 that decays into @xmath55 . the effect of adding eq . ( [ dlfree ] ) to the free lagrangian in eq . ( [ lfree ] ) is to change the @xmath35 propagator : @xmath137 the effect of adding eq . ( [ dlint ] ) to the interaction lagrangian in eq . ( [ lint ] ) is to change the counterterm vertex in fig . [ fig : vertex ] : @xmath138 . \label{newcounterterm}\ ] ] it is convenient to also rescale the fields @xmath139 and @xmath140 by the same complex factor @xmath141 . this eliminates the factor of @xmath135 from the numerator of the @xmath35 propagator in eq . ( [ newpropagator ] ) . it also multiplies all the interaction terms in eqs . ( [ lint ] ) and ( [ dlint ] ) by @xmath135 or @xmath141 . since we will use nonperturbative renormalization for the coupling constant @xmath142 for the @xmath40 contact interaction , the factor of @xmath135 only affects the value of the unphysical bare coupling constant @xmath48 . since @xmath143 , the other factors of @xmath135 or @xmath141 in the interaction vertices contribute only at order @xmath124 and higher . we will be calculating only to order @xmath51 , so we will ignore these factors of @xmath135 and @xmath141 . the perturbation expansion in the complex mass scheme corresponds to ordinary perturbation theory in @xmath38 followed by a resummation to all orders of a subset of terms that are higher order in @xmath38 . we will refer to calculations to order @xmath76 in this perturbation expansion as _ leading order ( lo ) _ in the complex mass scheme . we will refer to calculations through order @xmath51 in this perturbation expansion as _ next - to - leading order ( nlo ) _ in the complex mass scheme . we can use the scalar meson model to describe processes with asymptotic states that include not only the particles @xmath34 and @xmath36 , but also the bound state @xmath7 . the local composite operator @xmath145 has a nonzero amplitude to create @xmath7 from the vacuum . thus @xmath146 can be used as an interpolating field for @xmath7 . it is more convenient to use the operator @xmath147 , because this is a renormalized operator whose matrix elements do not depend on the ultraviolet cutoff @xcite . the corresponding propagator for @xmath7 is @xmath148 the momentum 4-vector in the fourier transform is @xmath149 , where @xmath80 and @xmath150 are the energy and momentum of @xmath7 . at @xmath151 , this propagator has a pole at the complex energy @xmath89 . near the pole in the energy , the behavior of the propagator at @xmath151 is @xmath152 the energy @xmath89 determines the binding energy @xmath78 and the width @xmath91 of @xmath7 through eq . ( [ epole ] ) . the residue of the pole defines the wavefunction normalization factor @xmath153 , which is complex valued . because the propagator @xmath154 has mass dimension @xmath49 , @xmath153 has mass dimension @xmath155 . strictly speaking , since @xmath7 has a nonzero width @xmath91 , there are no t - matrix elements for processes involving @xmath7 in the initial or final state , because the decay of the @xmath7 prevents it from being a truly asymptotic state . however if @xmath91 is sufficiently small , the @xmath7 can propagate over a long time interval before decaying , and it can therefore be treated as a quasi - asymptotic state . we can use the lehmann - symanzik - zimmermann ( lsz ) formalism to define t - matrix elements for the @xmath7 that are closely related to t - matrix elements for decay products of the @xmath7 whose total energy is tuned to the peak of the @xmath7 resonance . to define the t - matrix element @xmath156 for a process with @xmath7 in the initial or final state , we start with a green s functions for the operator @xmath157 or @xmath158 . the connected green s function at 4-momentum @xmath159 is amputated by multiplying by the inverse propagator @xmath160 , it is normalized by multiplying by @xmath161 , and then it is evaluated at the real energy @xmath162 . this gives the t - matrix element multiplied by @xmath163 for a state @xmath7 with the standard nonrelativistic normalization . the t - matrix element @xmath156 for a state @xmath7 with the standard relativistic normalization is obtained by multiplying by @xmath164 . to justify this prescription , we consider a process @xmath165 , where @xmath166 and @xmath167 both represent one or more particles and @xmath168 denotes a set of particles that can be decay products of @xmath7 . the t - matrix for this process will have a resonant enhancement when the invariant mass @xmath59 of the particles in @xmath168 is near the mass @xmath71 . the resonant contribution to the t - matrix element for @xmath165 in the rest frame of @xmath168 can be approximated by @xmath169 \approx i { \cal t}[a \to b + x ] \ , \frac{i}{2 m_x(m - m_x + i \gamma_x/2 ) } \ , i { \cal t}[x \to c ] . \label{tabc : res}\ ] ] our prescription for t - matrix elements such as @xmath170 $ ] and @xmath171 $ ] involves evaluating the amputated connected green s function at the real energy @xmath172 . this ensures that the expression in eq . ( [ tabc : res ] ) is as good an approximation as possible to the t - matrix element @xmath173 $ ] near the peak of the resonance . as a simple illustration of the prescription for t - matrix elements for processes involving @xmath7 , we consider the forward scattering process @xmath174 . the relevant green s function in the rest frame of @xmath7 is the negative of the propagator @xmath175 . this must be multiplied by two factors of @xmath176 , one for the @xmath7 in the inital state and one for the @xmath7 in the final state . the t - matrix element @xmath177 $ ] is then obtained by evaluating this at @xmath178 and multiplying it by @xmath179 : @xmath180 = - \frac{2 m_x z_x}{\delta_x(m_x,0 ) } . \label{txx}\ ] ] the scalar meson model defined by the lagrangian in eqs . ( [ lfree ] ) and ( [ lint ] ) can be a low - energy approximation to a more fundamental quantum field theory . the fundamental theory may include high energy processes that can create @xmath40 or @xmath41 with invariant mass near @xmath60 . if _ long - distance _ effects involving momenta much smaller than @xmath23 can be separated from _ short - distance _ effects involving momenta of order @xmath23 and larger , we can use the scalar meson model to calculate the long - distance effects . the basic tool required to separate long - distance effects from short - distance effects is the _ operator product expansion _ applications of the operator product expansion to the scalar meson model at 0@xmath39 order in @xmath38 were described in ref . below , we give the leading terms in the operator product expansions of the t - matrix elements for some of the processes considered in ref . @xcite . the general production process for @xmath7 has the form @xmath181 , where @xmath166 and @xmath167 each represent one or more particles . there can be analogous production processes for @xmath41 . in the case of @xmath41 , we assume that the invariant mass @xmath59 satisfies the condition in eq . ( [ e - small ] ) . this implies that the relative momenta of the mesons , which we denote generically by @xmath182 , are small compared to the natural momentum scale @xmath23 . we call the production process a short - distance process if all the particles in @xmath166 and @xmath167 have momenta in the rest frame of @xmath7 or @xmath41 that are of order @xmath23 or larger . the t - matrix element for a short - distance production process can be expanded in powers of the small energy differences @xmath106 and @xmath107 divided by @xmath57 and higher energy scales and in powers of the small relative momenta @xmath74 divided by @xmath23 and higher momentum scales . the operator product expansion can be used to organize the expansions of the t - matrix elements into sums of products of short - distance coefficients and matrix elements of local operators between the vacuum state @xmath183 and the appropriate final state . the leading terms in the expansions are those with the lowest dimension operator , which is @xmath145 . the corresponding renormalized operator is @xmath157 . in feynman diagrams , the local operator @xmath157 is represented by a dot from which a @xmath34 line and a @xmath35 line emerge , as illustrated in fig . [ fig : operators ] . if we keep only the leading terms , the factorization formulas for the t - matrix elements reduce to and @xmath184 differ from those in ref . @xcite by a factor of @xmath48 . this choice simplifies many equations . ] @xmath185 & = & \sqrt{2m_x } \ ; { \cal c}_a^{b,12 } \ , \langle x| \lambda_0 d_1^\dagger d_2^\dagger(0 ) | \emptyset \rangle , \label{taxpb } \\ { \cal t}[a \to b + d_1 d_1 \phi ] & = & \sqrt{8m_1 ^ 2 m } \ ; { \cal c}_a^{b,12 } \ , \langle d_1 d_1 \phi | \lambda_0 d_1^\dagger d_2^\dagger(0 ) | \emptyset \rangle . \label{taddphib0}\end{aligned}\ ] ] [ taddx ] the wilson coefficient @xmath186 is a function of high energy scales , such as @xmath23 and @xmath4 , and of the total momentum of the @xmath7 or @xmath41 . the only dependence on whether the final state contains @xmath7 or @xmath41 is in the matrix elements of the local operator @xmath187 . short - distance and long - distance effects are separated in eqs . ( [ taddx ] ) . and @xmath188 operators . [ fig : operators],width=94 ] the decay of @xmath7 into @xmath41 can be described within the scalar meson model . the fundamental theory may also allow the decay of @xmath7 into other final states @xmath168 . we call such a process a short - distance decay if all the particles in @xmath168 have momenta in the rest frame that are of order @xmath23 or larger . the t - matrix element for such a process can be expanded in powers of the small energy differences @xmath189 and @xmath107 divided by @xmath57 and higher energy scales . the operator product expansion can be used to organize the expansion of the t - matrix element into sums of products of short - distance coefficients and matrix elements of local operators between the state @xmath190 and the vacuum state . the leading term in the expansion is the one with the lowest dimension operator , which is @xmath146 . the corresponding renormalized operator is @xmath147 . if we keep only this term , the factorization formula for the t - matrix element reduces to@xmath191 = \sqrt{2m_x } \ ; { \cal c}_{12}^{c } \ , \langle \emptyset | \lambda_0 d_1 d_2(0 ) | x \rangle . \label{txcm}\ ] ] the wilson coefficient @xmath184 is a function of high energy scales , such as @xmath23 and @xmath4 . short - distance and long - distance effects are separated in eqs . ( [ txcm ] ) . if the fundamental theory includes processes that allow the production of @xmath7 via @xmath181 and the decay of @xmath7 via @xmath192 , it also allows the process @xmath165 , where @xmath168 represents the same particles but with a variable invariant mass @xmath59 instead of @xmath71 . this process has a resonant enhancement when @xmath59 is near the @xmath40 threshold as specified by eq . ( [ e - small ] ) . we call it a _ short - distance _ process if each of the particles in @xmath166 and @xmath167 has momentum large compared to @xmath23 in the rest frame of @xmath168 and if the relative momentum between each particle in @xmath168 and each particle in @xmath166 or @xmath167 is large compared to @xmath23 . the t - matrix element for such a process can be described within the scalar meson model by a double operator product expansion . the leading terms in this expansion are @xmath193 & = & { \cal c}_a^{b,\,c } + { \cal c}_a^{b,12 } { \cal c}_{12}^c \int d^4x \ , e^{i p \cdot x } \langle \emptyset | \lambda_0 d_1 d_2(x ) \lambda_0 d_1^\dagger d_2^\dagger(0 ) | \emptyset \rangle , \label{t : abcm}\end{aligned}\ ] ] where the 4-vector is @xmath194 . the wilson coefficients @xmath186 and @xmath195 are the same ones that appear in the operator product expansions in eqs . ( [ taddx ] ) and ( [ txcm ] ) . the first term @xmath196 on the right side of eq . ( [ t : abcm ] ) takes into account the direct production of @xmath168 at short distances . this term can be expanded in powers of the small energy differences @xmath106 and @xmath102 divided by energy scales of order @xmath57 or higher . the leading term in this expansion is simply a constant . according to eq . ( [ x - prop ] ) , the fourier transform in eq . ( [ t : abcm ] ) is just the @xmath7 propagator evaluated at @xmath197 . short - distance and long - distance effects are not yet separated in eq . ( [ t : abcm ] ) , because the @xmath7 propagator requires an additive renormalization @xcite . at @xmath199 order in @xmath38 , the only interaction in the scalar meson model is the contact interaction between @xmath34 and @xmath35 with coupling constant @xmath48 . the @xmath41 states remain noninteracting at this order in @xmath38 . in this section , we summarize some of the results at @xmath199 order in @xmath38 that were obtained in ref . @xcite and we generalize those results to the complex mass scheme . for the amplitude for the propagation of @xmath95 between contact interactions . the open dots indicate that the vertex factors @xmath201 are omitted . [ fig : l - lo],width=188 ] at 0@xmath39 order in @xmath38 , all the observables for processes near the @xmath40 threshold are related in a simple way to the connected green s function for @xmath202 . the basic building block for this green s function is the amplitude @xmath203 for the propagation of the @xmath40 pair between successive contact interactions , which is given by the feynman diagram in fig . [ fig : l - lo ] . the amplitude @xmath204 is calculated in section [ sec : propcon0 ] of the appendix . the analytic expression using dimensional regularization is given in eq . ( [ l0 - 3d ] ) . the result in a general regularization scheme is @xmath205 where @xmath206 is the energy variable defined by @xmath207 this variable vanishes at the @xmath95 threshold @xmath208 . it is real and positive if @xmath209 and it is pure imaginary with a negative imaginary part if @xmath210 . the term @xmath211 in eq . ( [ l0-sub ] ) is real valued and includes a linear ultraviolet divergence . the amputated connected green s function @xmath212 for @xmath202 can be calculated nonperturbatively by summing the geometric series represented by fig . [ fig : alo ] to all orders in @xmath48 : @xmath213 the renormalization prescription for @xmath48 specified by eq . ( [ reainv : zero ] ) is that @xmath214 must vanish at @xmath215 . this requires the inverse bare coupling constant to be @xmath216 using this expression for @xmath217 and the expression for @xmath204 in eq . ( [ l0 - 3d ] ) , the amplitude in eq . ( [ a - bare ] ) reduces to @xmath218 if we were to use the complex mass scheme , the amplitude @xmath204 would be given by eq . ( [ l0-sub ] ) with @xmath60 replaced by @xmath219 . if we again demand that @xmath214 must vanish at @xmath215 , the renormalized expression for the amplitude would be @xmath220 where @xmath221 is a complex energy variable defined by @xmath222 and @xmath223 is its value at @xmath224 : @xmath225 the variable @xmath221 vanishes at the complex threshold @xmath226 and it has a negative imaginary part for all real values of @xmath80 . for @xmath202 at @xmath227 order in @xmath38 can be obtained by summing a geometric series of one - loop diagrams . [ fig : alo],width=604 ] if the local composite operator @xmath147 is used as an interpolating field for @xmath7 , the propagator for @xmath7 is given in eq . ( [ x - prop ] ) . the diagrams for the propagator of @xmath7 at 0@xmath39 order in @xmath38 are shown in fig . [ fig : xprop ] . these diagrams form a geometric series whose sum in the rest frame @xmath228 is @xmath229 using the expression for @xmath230 in eq . ( [ a - bare ] ) , we can express the propagator for @xmath7 as @xmath231 . \label{deltax : ren}\ ] ] to obtain a renormalized propagator for @xmath7 that does not depend on the ultraviolet cutoff , an additive renormalization is necessary . after adding the constant @xmath232 , we obtain the renormalized propagator @xmath233 . propagator at 0@xmath39 order in @xmath38 . the interpolating field for the @xmath7 is @xmath234 . [ fig : xprop],width=604 ] the term @xmath235 in the propagator for @xmath7 in eq . ( [ deltax : ren ] ) has a pole at the complex energy @xmath89 given in eq . ( [ epole ] ) . using the renormalized expression for @xmath230 in eq . ( [ a0-ren ] ) , we find that the binding energy and width of @xmath7 at 0@xmath39 order in @xmath38 are @xmath236 [ egamx:00 ] the wavefunction normalization factor for @xmath7 is @xmath237 if we were to use the complex mass scheme , the amplitude @xmath230 would be given by eq . ( [ a0-cms ] ) . this amplitude has a pole at a complex energy @xmath89 that determines the binding energy and width of @xmath7 . the complex parameter @xmath89 satisfies @xmath238 where @xmath221 and @xmath223 are defined in eqs . ( [ gamma - e ] ) and ( [ gammax ] ) . comparing with eq . ( [ epole ] ) , we find that at lo in the complex mass scheme , the binding energy and width of @xmath7 are @xmath239 [ egamx:0 ] if we treat @xmath96 as being of order @xmath51 , the binding energy of @xmath7 in eq . ( [ ex:0 ] ) differs from @xmath108 only at order @xmath124 . the wavefunction normalization constant for @xmath7 at lo in the complex mass scheme is @xmath240 the operator product expansion of the t - matrix element for a short - distance production process @xmath181 is given in eq . ( [ taxpb ] ) . the vacuum to@xmath7 matrix element can be calculated via the lsz formalism using @xmath158 as an interpolating field for @xmath7 @xcite : @xmath241 at 0@xmath39 order in @xmath38 , the normalization constant @xmath153 is given in eq . ( [ zx ] ) . at lo in the complex mass scheme , @xmath153 is given in eq . ( [ zx : cms ] ) . the t - matrix element in eq . ( [ taxpb ] ) can be expressed as the product of a short - distance factor and a long - distance factor : @xmath242 = \sqrt{2m_x } \ , { \cal c}_a^{b,12 } \ , z_x^{1/2}. \label{taxb - fact}\ ] ] the rate for producing @xmath7 is obtained by squaring the t - matrix element in eq . ( [ taxb - fact ] ) and integrating over the appropriate phase space . if @xmath166 consists of a single particle , its decay rate into @xmath243 can be expressed in the factored form @xmath244 = \gamma_a^b \ , m_{12 } |z_x| . \label{gamabx}\ ] ] we have followed ref . @xcite in choosing the long - distance factor in eq . ( [ gamabx ] ) to be @xmath245 , which is dimensionless . the short - distance factor @xmath246 is @xmath247 where the 4-vector @xmath248 satisfies @xmath249 . the operator product expansion of the t - matrix element for a short - distance decay process @xmath192 is given in eq . ( [ txcm ] ) . the @xmath7to vacuum matrix element can be calculated via the lsz formalism using @xmath157 as an interpolating field for @xmath7 @xcite : ) is the complex conjugate of the matrix element in eq . ( [ xdd0 ] ) . however @xmath190 in eq . ( [ 0ddx ] ) is an in state , while @xmath250 in eq . ( [ xdd0 ] ) is the hermitian conjugate of an out state . these two states are related by the s - matrix : @xmath251 . ] @xmath252 at 0@xmath39 order in @xmath38 , the normalization constant @xmath153 is given in eq . ( [ zx ] ) . at lo in the complex mass scheme , @xmath153 is given by eq . ( [ zx : cms ] ) . the t - matrix element in eq . ( [ txcm ] ) can be expressed as the product of a short - distance factor and a long - distance factor : @xmath191 = \sqrt{2m_x } \ , { \cal c}_{12}^{c } \ , z_x^{1/2 } . \label{txcm - fact}\ ] ] the decay rate of @xmath7 into the particles represented by @xmath168 is obtained by squaring the t - matrix element in eq . ( [ txcm - fact ] ) and integrating over the phase space of those particles . it can be expressed in the factored form @xmath253 = \gamma^c \ , m_{12 } | z_x | . \label{gamxcm - fact}\ ] ] we have followed ref . @xcite in choosing the long - distance factor to be @xmath254 , which is dimensionless . the short - distance factor @xmath255 is @xmath256 where the 4-vector @xmath248 satisfies @xmath249 . if @xmath7 can be produced via the short - distance process @xmath181 and if it can decay via the short - distance process @xmath192 , then there can be resonant enhancement of the process @xmath165 , where @xmath168 represents the same particles but with a variable invariant mass @xmath59 instead of @xmath71 . the t - matrix element for this process can be expressed as the double operator product expansion in eq . ( [ t : abcm ] ) . the fourier transform of the matrix element is just the @xmath7 propagator evaluated at @xmath257 . the t - matrix element therefore reduces to @xmath193 & = & { \cal c}_a^{b , c } + { \cal c}_a^{b,12 } { \cal c}_{12}^c \frac{i \lambda_0 ^ 2 l_0(m)}{1-\lambda_0 l_0(m)}. \label{t : abc - bare}\end{aligned}\ ] ] the t - matrix element can be expressed in a form in which the short - distance effects and long - distance effects are separated : @xmath193 & = & -i{\cal c}_a^{b,12 } { \cal c}_{12}^c \left [ \mathcal{a}_0(m ) - ( 2 \pi / m_{12 } ) c_a^{b , c } \right ] , \label{t : abc - simple}\end{aligned}\ ] ] where @xmath230 is given in eq . ( [ a0-ren ] ) and @xmath258 is a complex constant with dimensions of length that is completely determined by short - distance factors : @xmath259 the nonresonant term @xmath258 in eq . ( [ t : abc - simple ] ) is a combination of short - distance factors , so the natural scale for @xmath258 is @xmath260 . the condition @xmath88 and the condition on @xmath59 in eq . ( [ e - small ] ) imply that the nonresonant term in eq . ( [ t : abc - simple ] ) is small compared to the resonant term in the threhsold region secified by eq . ( [ e - small ] ) . we therefore set @xmath261 . the invariant mass distribution of the particles in @xmath168 is obtained by squaring the t - matrix element and integrating over the momenta of all the particles in the final state . if @xmath166 consists of a single heavy particle , the invariant - mass distribution of the particles in @xmath168 has the form @xmath262 & = & \big ( \gamma_a^b \gamma^c \big ) \ , \frac{m_{12}^2}{2 \pi } \left| \mathcal{a}_0(m ) \right|^2 . \label{dgamabc}\end{aligned}\ ] ] the short - distance factors @xmath246 and @xmath255 are the same as in eqs . ( [ gamabx ] ) and ( [ gamxcm - fact ] ) . at 0@xmath39 order in @xmath38 , @xmath230 is given in eq . ( [ a0-ren ] ) . at lo in the complex mass scheme , @xmath230 is given in eq . ( [ a0-cms ] ) . note that if the invariant mass distribution in eq . ( [ dgamabc ] ) is divided by the product of the decay rates @xmath263 $ ] and @xmath264 $ ] in eqs . ( [ gamabx ] ) and ( [ gamxcm - fact ] ) , the short - distance factors cancel . this combination of observables has only a long - distance factor given by @xmath265 . of the @xmath7 resonance in a short - distance decay channel at leading order in @xmath38 . the dotted line is at order @xmath76 in ordinary perturbation theory ( using @xmath230 in eq . ( [ a0-ren ] ) ) . the dashed lines are at lo in the complex mass scheme ( using @xmath230 in eq . ( [ a0-cms ] ) ) . the line shapes are shown for @xmath73 mev and for both @xmath101 mev ( upper solid line ) and @xmath266 mev ( lower solid line ) . the units on the vertical axis are @xmath267 . [ fig : lineshapelo],width=453 ] the line shape of @xmath7 as a function of the invariant mass @xmath80 of the particles in @xmath168 is given by the factor @xmath268 in eq . ( [ dgamabc ] ) . in fig . [ fig : lineshapelo ] , we compare the line shapes at 0@xmath39 order in @xmath38 and at lo in the complex mass scheme . at 0@xmath39 order in @xmath38 , the line shape diverges at @xmath269 . in the complex mass scheme , the line shape is a resonance whose maximum is near @xmath269 and whose width is determined by @xmath96 . if @xmath270 , the line shape for @xmath271 is a nonrelativistic breit - wigner resonance whose full width at half - maximum is @xmath96 : @xmath272 ^ 2 + \gamma_2 ^ 2/4}.\end{aligned}\ ] ] however , a breit - wigner resonance has tails that fall off like @xmath273 and it is integrable . in contrast , the line shape @xmath268 with @xmath230 given by eq . ( [ a0-cms ] ) has tails that fall off only like @xmath274 and it is therefore not integrable . thus the area under the resonance is not as well - defined as for a breit - wigner resonance . in this section , we calculate the results in section [ sec : zeroth ] to next - to - leading order in @xmath38 . since the @xmath275 interaction allows transitions to @xmath41 states , we also calculate the rates for processes whose final state includes @xmath41 . one of the basic building blocks for the amplitudes for processes near the @xmath95 threshold is the amplitude for the propagation of @xmath95 between contact interactions . the term in this amplitude of order @xmath76 , @xmath203 , is given in eq . ( [ l0-sub ] ) . the term of order @xmath51 , @xmath276 , is the sum of the three feynman diagrams in fig . [ fig : l - nlo ] . these diagrams are calculated in section [ sec : propcon2 ] of the appendix . the final expression for @xmath277 is the sum of the amplitudes @xmath278 and @xmath279 given by eqs . ( [ l3-integral ] ) and ( [ l2-analytic ] ) : @xmath280 where @xmath206 is the energy variable defined in eq . ( [ kappa - e ] ) , the function @xmath281 is @xmath282 , \label{f - kappa}\end{aligned}\ ] ] and @xmath283 is the value of @xmath206 at the @xmath41 threshold @xmath284 : @xmath285^{1/2}. \label{kappa1}\end{aligned}\ ] ] the function @xmath286 in the integrand in eq . ( [ f - kappa ] ) is a rational function of @xmath287 that increases from 0 to 1 as @xmath287 increases from 0 to 1 : @xmath288 the term @xmath289 in eq . ( [ l2-bare ] ) is real valued and includes logarithmic ultraviolet divergences . the function @xmath281 vanishes at the @xmath41 threshold @xmath290 by definition . the first few terms in the laurent expansion of @xmath281 around the @xmath40 threshold at @xmath291 can be obtained from eqs . ( [ l2a - laurent ] ) and ( [ l2b - laurent ] ) . the leading term is @xmath292 for the amplitude for the propagation of @xmath95 between contact interactions . the open dots indicate that the vertex factors @xmath201 are omitted . [ fig : l - nlo],width=453 ] at 0@xmath39 order in @xmath38 , the ultraviolet divergence in @xmath204 was eliminated by summing a geometric series in @xmath293 and then renormalizing the coupling constant @xmath48 . the ultraviolet divergence in @xmath277 can be eliminated in a similar way . the geometric series of diagrams that must be summed are those that can be obtained by replacing each one - loop subdiagram in fig . [ fig : alo ] by the sum of the one - loop subdiagram and the three diagrams for @xmath277 in fig . [ fig : l - nlo ] . the sum of that geometric series gives an amplitude @xmath294 that can be obtained by making the substitution @xmath295 in eq . ( [ a - bare ] ) : @xmath296}. \label{a2-bare}\ ] ] our renormalization prescription for @xmath48 in eq . ( [ reainv : zero ] ) requires @xmath297 to vanish at @xmath269 . using this prescription , the amplitude in eq . ( [ a2-bare ] ) can be expressed as @xmath298 } . \label{a2-nlo}\end{aligned}\ ] ] of the @xmath7 resonance in a short - distance decay channel . the dotted lines are at order @xmath51 in ordinary perturbation theory ( using @xmath299 in eq . ( [ a2-nlo ] ) ) . the dashed lines are at lo in the complex mass scheme ( using @xmath230 in eq . ( [ a0-cms ] ) ) . the line shapes are shown for @xmath73 mev and for both @xmath101 mev ( the two narrower resonances that are almost indistinguishable ) and for @xmath266 mev ( the two wider resonances ) . the line shapes are normalized so their maximum values are 1 . the inset shows their behavior near the @xmath40 threshold . [ fig : lineshapenlocmslo],width=453 ] the quantity @xmath300 with @xmath299 given in eq . ( [ a2-nlo ] ) is the line shape of the @xmath7 resonance in a short - distance decay channel to order @xmath51 in ordinary perturbation theory . in fig . [ fig : lineshapenlocmslo ] , we compare this line shape with that at lo in the complex mass scheme , which is @xmath301 with @xmath230 given by eq . ( [ a0-cms ] ) . they are shown for two values of the coupling constant @xmath38 that correspond to the width of @xmath35 being @xmath101 mev and @xmath302 mev . the two line shapes have the same qualitative behavior below the resonance and near the peak of the resonance . they have qualitatively different behavior near the @xmath40 threshold @xmath303 . unlike @xmath301 , the function @xmath300 vanishes at the @xmath40 threshold and has a sharp peak just above the threshold . the inset in fig . [ fig : lineshapenlocmslo ] shows the behavior near the @xmath40 threshold , which is clearly unphysical . the reason @xmath300 vanishes at the @xmath40 threshold can be seen from the laurent expansion of @xmath281 around @xmath291 . the leading term , which is given in eq . ( [ f - laurent ] ) , diverges like @xmath304 as @xmath305 . no matter how small the coupling constant @xmath38 , the term proportional to @xmath306 can not be treated as a perturbation in the region near @xmath291 . the resolution of the problem can be found by noting that the diverging term in eq . ( [ f - laurent ] ) can be expressed as @xmath307 where @xmath96 is the width of the @xmath35 from its decay into @xmath55 , which is given in eq . ( [ gam2-nr ] ) . thus the divergence at @xmath291 implies that near the @xmath40 threshold , the imaginary part of the @xmath35 self - energy must be resummed to all orders . that resummation shifts the branch point in the leading order amplitude from the real threshold @xmath208 to the complex threshold @xmath308 . making this change and then imposing our renormalization scheme , the amplitude in eq . ( [ a2-nlo ] ) becomes @xmath309 } , \label{a2-resum}\end{aligned}\ ] ] where @xmath221 and @xmath223 are defined in eqs . ( [ gamma - e ] ) and ( [ gammax ] ) . this amplitude is a smooth function of @xmath80 at @xmath291 . a convenient way to implement the resummation in eq . ( [ a2-resum ] ) that can be extended staightforwardly to higher orders in @xmath38 is to use the complex mass scheme . this automatically gives an amplitude @xmath299 that has smooth behavior at the @xmath95 threshold and is correct through relative order @xmath51 . in the complex mass scheme , the contribution to @xmath277 from the diagrams in figs . [ fig : l - nlo](a ) and [ fig : l - nlo](b ) are obtained by replacing the energy variable @xmath206 by @xmath221 defined in eq . ( [ gamma - e ] ) and by replacing @xmath283 by @xmath310 , which is the value of @xmath221 at the @xmath41 threshold @xmath311 : @xmath312^{1/2 } . \label{gamma1}\end{aligned}\ ] ] in the complex mass scheme , there are additional contributions to the function @xmath277 coming from the one - loop diagram in fig . [ fig : l - nlo](c ) with the counterterm vertex in eq . ( [ dlint ] ) . the additional contributions are given in eq . ( [ l2c ] ) . adding those terms is equivalent to making the substitution @xmath313 using the expressions to order @xmath51 for @xmath96 in eq . ( [ gam2-nr ] ) and @xmath314 in eq . ( [ z2-nr ] ) , we can see that the additional terms in eq . ( [ fsub ] ) cancel the @xmath315 and @xmath316 terms coming from the laurent expansions of @xmath279 in eq . ( [ l2b - laurent ] ) . our renormalization prescription for @xmath48 in eq . ( [ reainv : zero ] ) requires @xmath297 to have a zero at @xmath269 . using this prescription , the amplitude can be expressed as @xmath317 } , \label{a2-ren : resum}\end{aligned}\ ] ] where @xmath221 and @xmath223 are given by eqs . ( [ gamma - e ] ) and ( [ gammax ] ) and the function @xmath318 is @xmath319 . \label{fcms}\end{aligned}\ ] ] the function @xmath318 can be obtained from @xmath281 in eq . ( [ f - kappa ] ) by subtracting several terms in the laurent expansion in @xmath316 and then replacing @xmath206 and @xmath283 by @xmath221 and @xmath310 . the terms that are subtracted are the imaginary term of order @xmath315 , the real terms of order @xmath320 , and the imaginary term of order @xmath316 in the second of the two terms on the right side of eq . ( [ f - kappa ] ) . the real terms of order @xmath320 are subtracted because they would cancel in the denominator in eq . ( [ a2-ren : resum ] ) anyway . the leading terms in the laurent expansion of @xmath318 around the complex @xmath40 threshold at @xmath321 are @xmath322.\end{aligned}\ ] ] for real values of @xmath80 , there is an ambiguity in the function @xmath323 in eq . ( [ a2-ren : resum ] ) for @xmath324 because of the branch point at @xmath325 in eq . ( [ fcms ] ) . this ambiguity can be resolved by the substitution @xmath326 . of the @xmath7 resonance in a short - distance decay channel using the complex mass scheme . the dashed lines are at lo in the complex mass scheme ( using @xmath230 in eq . ( [ a0-cms ] ) ) . the solid lines are at nlo in the complex mass scheme ( using @xmath299 in eq . ( [ a2-ren : resum ] ) ) . the line shapes are shown for @xmath73 mev and for both @xmath101 mev ( the two narrower resonances that are almost indistiguishable ) and for @xmath266 mev ( the two wider resonances ) . the line shapes are normalized so their maximum values are 1 . [ fig : lineshapenlocmsnlo],width=453 ] in fig . [ fig : lineshapenlocmsnlo ] , we compare the line shapes of @xmath7 in a short - distance decay channel at nlo in the complex mass scheme , which is @xmath300 with @xmath299 given by eq . ( [ a2-ren : resum ] ) , and that at lo in the complex mass scheme , which is @xmath301 with @xmath230 given by eq . ( [ a0-cms ] ) . they are shown for two values of the coupling constant @xmath38 that correspond to the width of @xmath35 being @xmath101 mev and @xmath302 mev . the curves are normalized so their maximum values are 1 . the line shapes at lo and nlo have the same qualitative behavior . as @xmath38 decreases , the quantitative differences between the line shapes at lo and nlo decrease . thus the complex mass scheme seems to provide a good solution to the infrared problem near the @xmath40 threshold that arises in ordinary perturbation theory . we will therefore use the complex mass scheme in most of the calculations in the remainder of this section . if the local composite operator @xmath147 is used as the interpolating field for @xmath7 , the propagator for @xmath7 is defined in eq . ( [ x - prop ] ) . the expression for the propagator at 0@xmath39 in @xmath38 is given in eq . ( [ deltax-0 ] ) . a renormalizable expression for the @xmath7 propagator that is accurate to order @xmath51 can be obtained simply by making the substitution @xmath327 in eq . ( [ deltax-0 ] ) : @xmath328 } { 1 - \lambda_0 [ l_0(e ) + l_2(e ) ] } . \label{deltax-2}\ ] ] comparing with the expression for the amplitude @xmath299 in eq . ( [ a2-bare ] ) , we see that the propagator in eq . ( [ deltax-2 ] ) can be expressed as @xmath329 . \label{deltax-2ren}\ ] ] the expression for @xmath299 at nlo in the complex mass scheme is given in eq . ( [ a2-ren : resum ] ) . this amplitude has a pole at a complex energy @xmath89 that determines the binding energy and the width of @xmath7 . the energy @xmath89 is the solution to @xmath330 = 0 , \label{gamma - pole : nlo}\ ] ] where @xmath221 and @xmath223 are defined in eqs . ( [ gamma - e ] ) and ( [ gammax ] ) . the wavefunction normalization constant for @xmath7 is @xmath331 if we treat @xmath96 as order @xmath51 , the solution to eq . ( [ gamma - pole : nlo ] ) for the complex variable @xmath89 through order @xmath51 is @xmath332 . \ ] ] using the expression for @xmath318 in eq . ( [ fcms ] ) , we find that the binding energy and width of the @xmath7 to order @xmath51 are @xmath333 , \label{gamx : nlo}\end{aligned}\ ] ] [ egamx : nlo ] where the upper endpoint @xmath334 of the integral is @xmath335 for @xmath73 mev , the second term on the right side of eq . ( [ gamx : nlo ] ) is @xmath336 . thus the width of the constituent @xmath35 accounts for most of the width of @xmath7 at order @xmath51 . we can obtain an analytic expression for the width @xmath91 if @xmath337 . in this limit , the upper limit @xmath334 on the integral approaches 1 and the width reduces to @xmath338 . \label{gamma - x : nlolim}\ ] ] as @xmath339 varies from 0 to @xmath340 , the factor in square brackets ranges from @xmath341 to @xmath342 . note that the expression for @xmath91 in eq . ( [ gamma - x : nlolim ] ) can be smaller than @xmath96 for some values of the parameters . the operator product expansion of the t - matrix element for the short - distance production process @xmath181 is given in eq . ( [ taxpb ] ) . at nlo in the complex mass scheme , the operator matrix element is given by eq . ( [ xdd0 ] ) , where @xmath153 is now the wavefunction normalization constant for @xmath7 in eq . ( [ zx - nlo ] ) . the factored expression for the t - matrix element is given by eq . ( [ taxb - fact ] ) . if @xmath166 consists of a single particle , its decay rate into @xmath343 can be expressed in the factored form in eq . ( [ gamabx ] ) , where @xmath153 is given in eq . ( [ zx - nlo ] ) . the operator product expansion for the t - matrix element for the short - distance decay process @xmath192 is given in eq . ( [ txcm ] ) . at nlo in the complex mass scheme , the operator matrix element is given in eq . ( [ 0ddx ] ) , where @xmath153 is the wavefunction normalization constant for @xmath7 in eq . ( [ zx - nlo ] ) . the factored expression for the t - matrix element is given by eq . ( [ txcm - fact ] ) . the decay rate of @xmath7 into the particles represented by @xmath168 can be expressed in the factored form in eq . ( [ gamxcm - fact ] ) , where @xmath153 is given in eq . ( [ zx - nlo ] ) . the t - matrix element for the short - distance process @xmath165 , where @xmath168 is a set of particles with invariant mass @xmath59 close to the @xmath40 threshold , can be expressed as the double operator product expansion in eq . ( [ t : abcm ] ) . the t - matrix element at order @xmath76 is given in eq . ( [ t : abc - bare ] ) . an expression for this t - matrix element that is accurate through order @xmath51 can be obtained by substituting @xmath327 : @xmath193 & = & { \cal c}_a^{b , c } + { \cal c}_a^{b,12 } { \cal c}_{12}^c \frac{i \lambda_0 ^ 2 [ l_0(m ) + l_2(m ) ] } { 1 - \lambda_0 [ l_0(m ) + l_2(m ) ] } . \label{t : abc - bare - nlo}\end{aligned}\ ] ] the expression for the t - matrix element in which short - distance and long - distance factors are separated is given by eq . ( [ t : abc - simple ] ) with @xmath344 replaced by @xmath345 . the invariant mass distribution for the short - distance decay channel is @xmath262 & = & \big ( \gamma_a^b \gamma^c \big ) \ , \frac{m_{12}^2}{2 \pi } \left| \mathcal{a}_2(m ) \right|^2 , \label{dgamabc : nlo}\end{aligned}\ ] ] where @xmath246 and @xmath255 are the short - distance factors defined in eqs . ( [ gamaddb ] ) and ( [ gam - c ] ) . the expression for @xmath345 at nlo in the complex mass scheme is given in eq . ( [ a2-ren : resum ] ) . the line shapes at lo and nlo in the complex mass scheme are compared in fig . [ fig : lineshapenlocmsnlo ] . if the invariant mass distribution in eq . ( [ dgamabc : nlo ] ) is divided by the product of the decay rates @xmath263 $ ] and @xmath264 $ ] in eqs . ( [ gamabx ] ) and ( [ gamxcm - fact ] ) , the short - distance factors cancel , leaving only the long - distance factors @xmath346 , where @xmath299 and @xmath153 are given in eqs . ( [ a2-ren : resum ] ) and ( [ zx - nlo ] ) . the corresponding quantity at lo in the complex mass scheme is @xmath347 , where @xmath230 and @xmath153 are given in eqs . ( [ a0-cms ] ) and ( [ zx : cms ] ) . the dependence of this long - distance factor on the invariant mass @xmath59 is illustrated in fig . [ fig : lineshapenlocmsnlo ] , where the curves are all normalized to maximum value 1 . the peak value of @xmath348 is also completely determined by long distances . for @xmath73 mev and @xmath101 mev , the peak value decreases from 130.1 mev@xmath349 at lo to 119.2 mev@xmath349 at nlo . for @xmath73 mev and @xmath266 mev , the peak value decreases from 8.34 mev@xmath349 at lo to 6.58 mev@xmath349 at nlo . the decrease in the difference between the peak values at nlo and lo from about 27% at @xmath266 mev to about 9% at @xmath101 mev supports the assumption that the perturbative expansion is well - behaved in the complex mass scheme . the fundamental theory may have short - distance production processes @xmath351 , where @xmath166 and @xmath167 both represent one or more particles whose momenta in the @xmath41 rest frame are of order @xmath23 or larger . the @xmath41 invariant mass @xmath59 is assumed to be close to @xmath60 , as specified by the condition in eq . ( [ e - small ] ) . the operator product expansion of the t - matrix element for this process is given in eq . ( [ taddphib0 ] ) . for the vacuum to@xmath352 matrix element of the operator @xmath188 . [ fig : ddphi],width=453 ] we first calculate the rate for @xmath351 to order @xmath51 using ordinary perturbation theory . at order @xmath38 , the operator matrix element in eq . ( [ taddphib0 ] ) is given by the series of feynman diagrams in fig . [ fig : ddphi ] summed over the two permutations of the external @xmath34 lines . the sum of the geometric series of diagrams is @xmath353 multiplied by the first diagram in the series . according to the optical theorem , the square of the first diagram integrated over the 3-body phase space for @xmath41 is given by the sum of the 3-body cuts through the feynman diagrams for @xmath277 in fig . [ fig : l - nlo ] . if @xmath354 , the sum of the 3-body cuts gives the imaginary part of @xmath277 . thus the square of the t - matrix element in eq . ( [ taddphib0 ] ) integrated over the 3-body phase space of @xmath41 is @xmath355 \right|^2 \nonumber \\ & & \hspace{7 cm } = |{\cal c}_a^{b,12 } |^2 |\mathcal{a}_0(m)|^2 ( -2 ) \ , { \rm i m } \ , l_2(m ) , \label{tddphi : g2}\end{aligned}\ ] ] where @xmath356 , @xmath357 , and @xmath358 are 4-momenta for @xmath34 , @xmath34 , and @xmath36 , respectively , and @xmath359 . if @xmath166 consists of a single particle , the invariant mass distribution for @xmath360 at order @xmath51 can be expressed as @xmath361 & = & \gamma_a^b \ , \frac{m_{12}}{2\pi } \left| \mathcal{a}_0 ( m)\right|^2 ( -2 ) \ , { \rm i m } \ , l_2(m ) , \label{ddphiimd : g2}\end{aligned}\ ] ] where @xmath246 is the short - distance factor defined in eq . ( [ gamaddb ] ) and we have replaced a multiplicative factor of @xmath59 by @xmath60 . if there is a short - distance process @xmath351 , then the process @xmath362 is also allowed . the invariant mass distribution for @xmath362 at order @xmath76 is given by the expression on the right side of eq . ( [ ddphiimd : g2 ] ) with @xmath363 replaced by @xmath364 . the threshold for this process is @xmath365 . since @xmath35 ultimately decays into @xmath55 , a decay into @xmath40 can also be regarded as a contribution to the inclusive decay into @xmath366 . the result in eq . ( [ ddphiimd : g2 ] ) is an order @xmath51 correction to the inclusive invariant mass distribution for @xmath41 . it is nonzero for all @xmath367 . for @xmath368 , there are additional corrections of order @xmath51 that can be obtained by replacing any of the one - loop subdiagrams in fig . [ fig : ddphi ] by one of the two - loop diagrams for @xmath277 in fig . [ fig : l - nlo ] . nonperturbative renormalization of the coupling constant @xmath48 then requires that a geometric series of these corrections be summed to all orders . the net effect is to replace @xmath344 in eq . ( [ ddphiimd : g2 ] ) by @xmath345 . thus the complete result for the inclusive invariant mass distribution for @xmath41 through order @xmath51 is @xmath361 & = & \gamma_a^b \ , \frac{m_{12}}{2\pi } \left| \mathcal{a}_2 ( m)\right|^2 ( -2 ) \ , { \rm i m } \ , [ l_0(m ) + l_2(m ) ] , \label{dgamddphi : g2}\end{aligned}\ ] ] where the functions @xmath369 , @xmath370 , and @xmath371 are given in eqs . ( [ a2-nlo ] ) , ( [ l0-sub ] ) , and ( [ l2-bare ] ) . note that taking the imaginary parts of @xmath370 and @xmath371 eliminates the additive ultraviolet - divergent terms . a more compact expression for the distribution in eq . ( [ dgamddphi : g2 ] ) is @xmath361 & = & \gamma_a^b \ , \frac{m_{12}}{\pi } \ , { \rm i m } \ , \mathcal{a}_2(m ) . \label{ddphiimd : nlocms}\end{aligned}\ ] ] the inclusive invariant mass distribution in eq . ( [ dgamddphi : g2 ] ) when calculated using ordinary perturbation theory has unphysical behavior at the threshold @xmath372 . the factor @xmath373 has a zero at @xmath372 and a sharp peak just above that threshold as illustrated in fig . [ fig : lineshapenlocmslo ] . the term @xmath374 also has unphysical behavior at this threshold . the imaginary part of @xmath277 in the region @xmath375 is given by the sum of eqs . ( [ iml2a ] ) and ( [ iml2b ] ) . the resulting expression for @xmath376 is @xmath377 , \label{iml2ab}\end{aligned}\ ] ] where the upper limit @xmath334 of the integral is @xmath378 the function in eq . ( [ iml2ab ] ) diverges like @xmath379 as @xmath305 . as a consequence , near the @xmath40 threshold , corrections of higher order in @xmath51 that come from the @xmath35 self energy are not suppressed . this is the same problem that prompted us to introduce the complex mass scheme in section [ sec : xamp2 ] . we next consider the inclusive rate for @xmath351 in the complex mass scheme . at lo in the complex mass scheme , the invariant mass distribution for @xmath41 can be obtained from eq . ( [ ddphiimd : nlocms ] ) by replacing @xmath345 by @xmath344 in eq . ( [ a0-cms ] ) . this can be written more explicitly as @xmath361 & = & \gamma_a^b \ , \frac{-2 \ , { \rm i m } \ , \gamma(m)}{|\gamma(m ) - { \rm re } \ , \gamma_x|^2 } , \label{ddphiimd : locms}\end{aligned}\ ] ] where @xmath221 and @xmath223 are defined in eqs . ( [ gamma - e ] ) and ( [ gammax ] ) . this expression has a nonzero imaginary part at all energies @xmath80 , even below the @xmath41 theshold . for @xmath380 , the imaginary part is small and it is cancelled by higher orders in the complex mass scheme . at nlo in the complex mass scheme , the invariant mass distribution for @xmath41 is obtained from eq . ( [ ddphiimd : nlocms ] ) by using the expression for @xmath345 in eq . ( [ a2-ren : resum ] ) . resonance at nlo in the complex mass scheme . the solid lines are for a short - distance decay channel ( @xmath300 using @xmath299 in eq . ( [ a2-ren : resum ] ) ) . the dash - dotted lines are for the @xmath41 channel ( @xmath381 using @xmath299 in eq . ( [ a2-ren : resum ] ) ) . the line shapes are shown for @xmath73 mev and for both @xmath101 mev ( narrower resonances ) and @xmath266 mev ( wider resonances ) . the line shapes are normalized so their maximum values are 1 . [ fig : lineshapeaw],width=453 ] in fig . [ fig : lineshapeaw ] , we compare the line shape of @xmath7 in the @xmath41 channel , which is proportional to @xmath381 , with the line shape of @xmath7 in a short - distance decay channel , which is proportional to @xmath300 . they are shown for two values of the coupling constant @xmath38 that correspond to the width of @xmath35 being @xmath101 mev and @xmath302 mev . the curves are normalized so their maximum values are 1 . the line shapes have the same qualitative behavior . the line shape for @xmath41 is shifted towards higher energy by an amount that decreases as @xmath38 decreases . the belle collaboration has observed an enhancement in the production of @xmath10 near threshold in the decay @xmath382 @xcite . the enhancement peaks at the invariant mass @xmath383 mev , where we have combined the errors in quadrature . if we use the recent precision measurement of the @xmath5 mass by the cleo collaboration @xcite , the peak is above the @xmath1 threshold by @xmath384 mev . in contrast , the peak observed in the short - distance decay channel @xmath385 is below the @xmath1 threshold by @xmath386 mev . the difference is @xmath387 mev , which differs from 0 by two standard deviations . part of the discrepancy may be due to a shift in the invariant mass distribution analogous to the one illustrated in fig . [ fig : lineshapeaw ] . the width @xmath91 of the @xmath7 resonance was defined in eq . ( [ epole ] ) in terms of the energy @xmath89 of the pole in the @xmath7 propagator : @xmath389 . at lo in the complex mass scheme , @xmath91 is simply @xmath96 . at nlo in the complex mass scheme , @xmath91 is obtained by solving eq . ( [ gamma - pole : nlo ] ) for @xmath89 . if the width of the @xmath7 resonance is sufficiently small , its width can also be interpreted as its decay rate . a direct calculation of that decay rate will give a result that is closely related to but not identical to @xmath91 . we will denote the result of the direct calculation of the inclusive decay rate by @xmath390 $ ] . we can use the optical theorem to calculate the inclusive decay rate for @xmath391 : @xmath392 = \frac{1}{m_x } \ , { \rm i m } \ , { \cal t}[x \to x ] . \label{opticalthm}\end{aligned}\ ] ] the forward t - matrix element for @xmath174 can be defined using the lsz formalism and it is given in eq . ( [ txx ] ) . at nlo in the complex mass scheme , the normalization constant @xmath153 and the propagator @xmath175 are given in eqs . ( [ zx - nlo ] ) and ( [ deltax-2 ] ) . the resulting expression for the decay rate of @xmath7 in eq . ( [ opticalthm ] ) is @xmath392 = -2 \ , { \rm i m } \frac{z_x{\cal a}_2^{-1}(m_x)}{\lambda_0 [ l_0(m_x ) + l_2(m_x)]}. \label{imtxx}\end{aligned}\ ] ] we have used eq . ( [ a2-bare ] ) to express this in a form with a factor of @xmath393 , because @xmath394 does not depend on the ultraviolet cutoff @xmath395 . the decay rate in eq . ( [ imtxx ] ) depends on @xmath395 through the bare coupling constant @xmath48 and through the ultraviolet - divergent additive constants in @xmath396 and @xmath397 . from the renormalized expression for @xmath394 in eq . ( [ a2-bare ] ) , we can see that the divergent terms in @xmath398 must be cancelled by @xmath217 as @xmath399 . the leading divergence is a linear divergence in @xmath211 that is linear in @xmath395 . thus @xmath217 must include a cancelling linear divergence . this implies that @xmath400 $ ] must approach 1 as @xmath399 . upon taking the limit @xmath399 , eq . ( [ imtxx ] ) reduces to @xmath392 = -2 \ , { \rm i m } \left ( z_x \mathcal{a}_2^{-1}(m_x ) \right ) . \label{imtxx - finite}\end{aligned}\ ] ] an explicit expression for the inclusive decay rate of @xmath7 can be obtained from eq . ( [ imtxx - finite ] ) by inserting the renormalized expression for @xmath394 in eq . ( [ a2-ren : resum ] ) : @xmath392 = \frac{m_{12}}{2 \pi } \ , ( -2 ) \ , { \rm i m } \left [ z_x \big ( \gamma(m_x ) + f_{\rm cms}(\gamma(m_x ) ) - { \rm re } [ \gamma_x + f_{\rm cms}(\gamma_x ) ] \big ) \right ] , \nonumber \\ \label{gamddphi}\end{aligned}\ ] ] where @xmath153 , @xmath221 , @xmath223 , and @xmath318 are given in eqs . ( [ zx - nlo ] ) , ( [ gamma - e ] ) , ( [ gammax ] ) , and ( [ fcms ] ) . if we treat @xmath96 as order @xmath51 , the decay rate in eq . ( [ gamddphi ] ) reduces at order @xmath51 to the corresponding result for the width @xmath91 in eq . ( [ gamx : nlo ] ) . of the @xmath7 resonance ( solid line ) and the decay rate @xmath401 $ ] ( dashed line ) at nlo in the complex mass scheme as functions of @xmath96 . the vertical axis is in units of @xmath96 . the intercept on the vertical axis is approximately @xmath402 . [ fig : widths],width=453 ] in fig . [ fig : widths ] , we compare the width @xmath91 obtained by solving eq . ( [ gamma - pole : nlo ] ) and the decay rate @xmath401 $ ] in eq . ( [ gamddphi ] ) as functions of @xmath96 . they agree for small @xmath96 , because the difference between @xmath91 and @xmath401 $ ] is of order @xmath124 . they differ by less than 1% if @xmath403 mev . we have examined the effects of a weakly - bound hadronic molecule on a nearby 3-body threshold , where the three bodies consist of a primary constituent of the molecule and the decay products of the other constituent . we studied these effects in the simplest possible model : the scalar meson model , with spin-0 constituents @xmath34 , @xmath35 , and @xmath36 and momentum - independent interactions . the primary constitutents of the molecule @xmath7 are @xmath40 and the 3-body threshold is that for @xmath41 . the @xmath40 contact interaction with coupling constant @xmath48 was treated nonperturbatively . the @xmath37 interaction with coupling constant @xmath38 was treated perturbatively . several observables were calculated to all orders in @xmath48 and to next - to - leading order in @xmath38 . both ultraviolet problems and infrared problems were encountered in these calculations . we found that at next - to - leading order in @xmath38 , the ultraviolet divergences can be removed by a perturbative renormalization of the @xmath35 mass , a nonperturbative renormalization of @xmath48 , and a nonperturbative renormalization of short - distance coefficients in the operator product expansion . the nonperturbative renormalizations required the summing of a geometric series of order-@xmath51 subdiagrams to all orders . the subdiagrams represent order-@xmath51 contributions to the amplitude for the propagation of @xmath40 between contact interactions . thus the renormalized amplitudes at next - to - leading order include all terms of order @xmath51 together with subsets of terms from all higher orders in @xmath38 . we found that the next - to - leading order amplitudes suffer from an infrared problem at the @xmath40 threshold . the problem is illustrated dramatically in fig . [ fig : lineshapenlocmslo ] , which shows the line shape @xmath90 in a short - distance decay channel as a function of the energy @xmath80 . the pathological behavior at @xmath303 arises because the @xmath37 interaction shifts the @xmath40 threshold from the real value @xmath60 to the complex value @xmath219 . since @xmath96 is of order @xmath51 , a strict expansion in powers of @xmath38 leads to singularities at @xmath303 . one possible solution to this problem is to sum the geometric series of self - energy corrections to the @xmath35 propagator to all orders , but this leads to very complicated loop integrals . a simpler solution is to use the complex mass scheme , in which the width @xmath96 is included in the feynman rule for the @xmath35 propagator and its effects are systematically compensated at higher orders in @xmath38 through counterterms . this partial resummation of the @xmath35 propagator corrections leads to smooth behavior of the line shape @xmath90 at the @xmath40 threshold as illustrated in fig . [ fig : lineshapenlocmsnlo ] . having solved the infrared problems , we calculated several observables to nlo in the complex mass scheme . they include the line shapes of the @xmath7 resonance in both a short - distance decay channel and in the @xmath41 channel . the @xmath0 is a weakly - bound hadronic molecule whose primary constitutents are a superposition of charm mesons : @xmath33 . its mass is near the 3-body thresholds for @xmath10 , @xmath28 , and @xmath29 . the methods we applied to the scalar meson model can be extended straightforwardly to this multichannel problem . a conventional perturbation expansion in the @xmath404 coupling constant will have infrared problems at the @xmath1 threshold which are associated with the decay @xmath405 . these infrared problems can be avoided by using the complex mass scheme . the ultraviolet problems can be expected to be more severe in this system , because the @xmath406 interaction is proportional to the 3-momentum of the pion . thus renormalization may not be as simple as in the scalar meson model . if the renormalization problem can be solved , it will be straightforward to calculate observables to nlo in the complex mass scheme . one of the most interesting applications will be to predict the difference between the line shapes of the @xmath0 resonance in a short - distance decay channel , such as @xmath385 , and in the @xmath10 channel . eb thanks s. fleming , m. kusunoki , and t. mehen for valuable discussions . jl thanks the physics department at the ohio state university for its hospitality while some of this work was carried out . this research was supported in part by the department of energy under grant de - fg02 - 91-er4069 and by the korea research foundation under moehrd basic research promotion grant krf-2006 - 311-c00020 . in this appendix , we calculate the loop integrals that appear at order @xmath76 and at order @xmath51 in the scalar meson model . many of the loop integrals we need to evaluate are ultraviolet divergent . they can be regularized by using dimensional regularization of the integrals over the 3-momenta . we analytically continue the 3-dimensional integrals to @xmath407 dimensions using the prescription @xmath408 where @xmath409 is the renormalization scale and @xmath410 is the pochhammer symbol : @xmath411 the function of @xmath412 in the prefactor is designed to simplify the analytic expressions for dimensionally regularized integrals by cancelling the effects of the analytic continuation of angular integrals . thus the integral of a scalar function of @xmath413 is @xmath414 loop integrals are evaluated by first inserting the appropriate nonrelativistic propagators : @xmath415 where @xmath416 and @xmath417 . integrals over the loop energy are then evaluated using the residue theorem . after using the feynman parameter trick to combine denominators , the dimensionally regularized integrals over the 3-momenta can be evaluated . the final steps are the evaluation of the feynman parameter integrals and the analytic continuation to @xmath418 . it will be convenient to express the results in terms of the variables the one - loop diagram for the self - energy @xmath114 of @xmath35 is shown in fig . [ fig : self](a ) . the expression for the diagram is @xmath420 the residue theorem can be used to evaluate the energy integral : @xmath421 if we implement dimensional regularization using the prescription in eq . ( [ dimreg ] ) , the analytic result is @xmath422 - i \varepsilon \right)^{\frac{1}{2 } - \epsilon } . \label{sigma2-dimreg}\ ] ] the self - energy in dimensional regularization is given by the analytic continuation of eq . ( [ sigma2-dimreg ] ) to @xmath418 : @xmath423 - i \varepsilon \right)^{\frac{1}{2 } } . \label{sigma2-dimreg0}\ ] ] the momentum integral in eq . ( [ sigma2a ] ) has a linear ultraviolet divergence that is set to zero by dimensional regularization . the self - energy in a general regularization scheme is given in eq . ( [ d2self ] ) . the linear ultraviolet divergence is contained in the extra term @xmath117 , which is real valued . the amplitude for the propagation of @xmath40 between contact interactions is given at order @xmath76 by the one - loop diagram in fig . [ fig : l - lo ] . in the center - of - momentum frame where the 4-momentum is @xmath424 , this amplitude is @xmath425 the residue theorem can be used to evaluate the energy integral : @xmath426 if we implement dimensional regularization using the prescription in eq . ( [ dimreg ] ) , the analytic result is @xmath427 where @xmath316 is defined in eq . ( [ kappa - def ] ) . the amplitude in dimensional regularization is given by analytic continuation to @xmath418 : @xmath428 the momentum integral in eq . ( [ l0-e ] ) has a linear ultraviolet divergence that is set to zero by dimensional regularization . the amplitude in a general regularization scheme is given in eq . ( [ l0-sub ] ) . the linear ultraviolet divergence is contained in the extra term @xmath211 , which is real valued . the amplitude for the propagation of @xmath40 between contact interactions has contribution of order @xmath51 from the two - loop diagrams in fig . [ fig : l - nlo ] . in the center - of - momentum frame where the 4-momentum is @xmath429 , the order-@xmath51 amplitude @xmath276 is a function of @xmath80 only . we proceed to calculate the contributions from each of the three diagrams . the contribution to the amplitude @xmath276 from the two - loop diagram in fig . [ fig : l - nlo](a ) in which @xmath36 is exchanged between the @xmath34 and @xmath35 is @xmath431 the integrals over the loop energies @xmath432 and @xmath433 can be evaluated by closing the contours in the upper half - planes . after introducing feynman parameters and evaluating the integrals over the loop momenta , the amplitude reduces to @xmath434^{-3/2 + \epsilon } \nonumber \\ & & \hspace{2 cm } \times \left [ ( x+y ) \kappa^2 + ( 1-x - y)(m_1/m_{12})(\kappa^2-\kappa_1 ^ 2)\right]^{-2 \epsilon } , \end{aligned}\ ] ] where @xmath316 and @xmath283 are defined in eqs . ( [ kappakappa1 ] ) . one of the feynman parameter integrals can be evaluated analytically by changing variables to @xmath435 and @xmath436 . the integral over @xmath437 gives a hypergeometric function : @xmath438^{-2 \epsilon } , \label{l3-eps}\end{aligned}\ ] ] where @xmath286 is the rational function of @xmath287 given in eq . ( [ t - z ] ) . the pole at @xmath418 in eq . ( [ l3-eps ] ) is a logarithmic ultraviolet divergence . the amplitude @xmath278 in dimensional regularization is defined by the laurent expansion of the expression in eq . ( [ l3-eps ] ) to order @xmath439 . the expansion in powers of @xmath412 is facilitated by using a transformation formula to replace the hypergeometric function in eq . ( [ l3-eps ] ) by one of the form @xmath440 : @xmath441 the expansion of @xmath440 to order @xmath412 is simple : @xmath442 expanding the amplitude @xmath278 in eq . ( [ l3-eps ] ) to 0@xmath39 order in @xmath412 , it reduces to @xmath443}{t(z ) } + \frac{2t(z)}{3 } \ , { } _ 2f_1 \left ( 1,1,\mbox{$\frac{5}{2}$};t(z ) \right ) \nonumber \\ & & \hspace{5 cm } -2 \ln \frac{z\kappa^2+(1-z)(m_1/m_{12})(\kappa^2-\kappa_1 ^ 2)}{\mu^2 } \bigg ] \bigg ) . \label{l3-dimreg}\end{aligned}\ ] ] we have used the integral @xmath444 this can be evaluated by changing variables to @xmath445 $ ] . most of the terms in eq . ( [ l3-dimreg ] ) are independent of @xmath80 . the expression can be simplified by expressing it in the form @xmath446 the amplitude @xmath447 at the @xmath41 threshold is real - valued and includes the logarithmic divergence . the amplitude @xmath278 has an imaginary part for @xmath448 . for energies in the range @xmath375 , the imaginary part is @xmath449 where the upper limit @xmath334 of the integral is @xmath450 the first few terms in the laurent expansion of @xmath278 in powers of @xmath316 are @xmath451 . \label{l2a - laurent}\end{aligned}\ ] ] the contribution to @xmath276 from the two - loop diagram in fig . [ fig : l - nlo](b ) with a self - energy correction to the @xmath35 line is @xmath452 where @xmath453 is the contribution to the @xmath35 self - energy from the one - loop diagram in fig . [ fig : self](a ) . the analytic result for @xmath453 in dimensional regularization is given in eq . ( [ sigma2-dimreg ] ) . since @xmath453 has a branch cut for @xmath432 in the lower half - plane , it is convenient to close the @xmath432 contour in the upper half - plane . after introducing feynman parameters , the integral over @xmath454 can be evaluated . if we implement dimensional regularization using the prescription in eq . ( [ dimreg ] ) , the result is @xmath455^{-2\epsilon } , \label{l2-eps}\end{aligned}\ ] ] where @xmath316 and @xmath283 are defined in eqs . ( [ kappakappa1 ] ) and @xmath456 is given by @xmath457 the pole at @xmath418 in eq . ( [ l2-eps ] ) is a logarithmic ultraviolet divergence . the amplitude @xmath279 in dimensional regularization is defined by the laurent expansion of the expression in eq . ( [ l2-eps ] ) to order @xmath439 . expanding to 0@xmath39 order in @xmath412 , the expression reduces to @xmath458 . \label{l2-dimreg}\end{aligned}\ ] ] the integral over @xmath459 can be evaluated analytically . we can simplify the expression by replacing @xmath460 by 1 , because the difference is suppressed by a factor of @xmath461 . most of the terms in eq . ( [ l2-dimreg ] ) are independent of @xmath80 . the expression can be simplified by expressing it in the form @xmath462 the amplitude @xmath463 at the @xmath41 threshold is real - valued and includes the logarithmic divergence . the amplitude @xmath279 has an imaginary part for @xmath448 . for energies in the range @xmath375 , the imaginary part of @xmath279 is @xmath464 note that this diverges at the @xmath40 threshold @xmath291 . the first few terms in the laurent expansion of @xmath278 in powers of @xmath316 are @xmath465 . \label{l2b - laurent}\end{aligned}\ ] ] in a general regularization scheme , the one - loop self - energy subdiagram @xmath453 in eq . ( [ l2b ] ) has a linear ultraviolet divergence , which is included in the term @xmath117 in eq . ( [ d2self ] ) . the resulting divergence in @xmath279 is cancelled by the one - loop diagram with a @xmath35 mass counterterm in fig . [ fig : l - nlo](c ) . the counterterm vertex is @xmath466 , with @xmath467 . in the complex mass scheme , the @xmath35 propagator counterterm has the more complicated form given in eq . ( [ newcounterterm ] ) . with this counterterm , the contribution to @xmath277 of order @xmath51 from the one - loop diagram in fig . [ fig : l - nlo](c ) is @xmath468 \big).\end{aligned}\ ] ] in the term with the factor @xmath469 , the integral is proportional to @xmath204 . this amplitude in a general regularization scheme is given in eq . ( [ l0-sub ] ) . in the term with the factor @xmath470 , the integral is convergent . the complete result for the diagram is @xmath471
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the @xmath0 seems to be a loosely - bound hadronic molecule whose constituents are two charm mesons .
a novel feature of this molecule is that the mass difference of the constituents is close to the mass of a lighter meson that can be exchanged between them , namely the pion .
we analyze this feature in a simple model with spin-0 mesons only .
various observables are calculated to next - to - leading order in the interaction strength of the exchanged meson .
renormalization requires summing a geometric series of next - to - leading order corrections .
the dependence of observables on the ultraviolet cutoff can be removed by renormalizations of the mass of the heaviest meson , the coupling constant for the contact interaction between the heavy mesons , and short - distance coefficients in the operator product expansion .
the next - to - leading order correction has an unphysical infrared divergence at the threshold of the two heavier mesons that can be eliminated by a further resummation that takes into account the nonzero width of the heaviest meson .
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there are two important phenomena observed in evolutionary dynamical systems of any kind : _ self - organization _ and _ emergence_. both phenomena are the exclusive result of endogenous interactions of the individual elements of an evolutionary dynamical system . emergence characterizes the patterns that are situated at a higher macro level and that arise from interactions taking place at the lower micro level of the system . self - organization , besides departing from the individual micro interactions , implies an increase in order of the system , being usually associated to the promotion of a specific functionality and to the generation of patterns . typically , complex patterns emerge in a system of interacting individuals that participate in a self - organizing process . self - organization is more frequently related to the process itself , while emergence is usually associated to an outcome of the process . although less frequently mentioned , the emergence of patterns from self - organizing processes may be strongly dependent on _ locality_. emergence and self - organization are not enough to distinguish between two important and quite different circumstances : the presence of an influence that impacts the system globally and , conversely , the absence of any global influence and the lack of information about any global property of the system . in the latter case , the system itself is the exclusive result of local interactions . such a global influence ( entity or property ) is often associated with the concept of _ environment_. noteworthy , the latter circumstance may be considered a case of the former : when that global entity does not exist , the environment for each agent is just the set of all the other agents . conversely , when the global entity exists , it is considered part of the environment and may have an inhomogeneous impact on the individual dynamics . regardless of the environmental type , economical , ecological and social environments share as a common feature the fact that the agents operating in these environments usually try to improve some kind of utility , related either to profit , to food , to reproduction or to comfort and power . a general concept that is attached to this improvement attempt is the idea of _ adaptation_. in the economy , adaptation may be concerned with the development of new products to capture a higher market share or with the improvement of the production processes to increase profits : that is , innovation . in ecology , adaptation concerns better ways to achieve security or food intake or reproduction chance and , in the social context , some of the above economical and biological drives plus a few other less survival - oriented needs . in all cases , adaptation aims at finding strategies to better deal with the surrounding environment ( @xcite ) . natural selection through fitness landscapes or geographic barriers are good examples how global influences are considered when modeling adaptation in an evolutionary process . on the other hand , adaptation also operates in many structure generating mechanisms that can be found in both physical and social sciences but that are built on the exclusive occurrence of local interactions . in biology , the ultimate domain of evolution and natural selection , we are confronted with tremendous organic diversity virtually infinite forms and shapes none of which found twice but the distribution is well structured in a way that allows us to order this diversity and to speak of species , families , orders etc . a quite illustrative description is given by the evolutionary geneticist theodusius dobzhanski ( @xcite : p.21 ) : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ suppose that we make a fairly large collection , say some 10,000 specimens , of birds or butterflies or flowering plants in a small territory , perhaps 100 square kilometers . no two individuals will be exactly alike . let us , however , consider the entire collection . the variations that we find in size , in color , or in other traits among our specimens do not form continuous distributions . instead , arrays of discrete distributions are found . the distributions are separated by gaps , that is , by the absence of specimens with intermediate characteristics . we soon learn to distinguish the arrays of specimens to which the vernacular names english sparrow , chickadee , bluejay , blackbird , cardinal , and the like , are applied . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ if we had to make a visual representation of this description of intra and interspecies variations it would perhaps look like the multi - modal distribution shown in figure [ fig : distribution01 ] . what we call a species , is in fact some norm or mean characteristics of a cluster of individuals . evolutionary theory is ultimately a theory about the history which led to such a pattern . and if the organic diversity we observe nowadays evolved in a way that is characterized by some kind of > > tree of live < < , then there must be events that may lead to the split of a connected set of individuals ( protospecies ) into ( at least ) two sets that are not connected any longer ( see figure [ fig : speciation ] ) . in biology , this is called _ speciation_. as we will see in this article , though , the generation of such a split with simple but well known evolutionary models in which `` natural selection impels and directs evolutionary changes '' ( ibid . p.2 ) is not straightforward . it so happens that constraints on the interaction behavior are required . the phenotype of living beings is not the only domain where patterns of structured diversity as illustrated in figure [ fig : distribution01 ] are observed . phenomena include certain phases of structure formation in physical cosmology , distribution of cultural behavior , languages and dialects , herd behavior in finance , among others . especially for the latter examples in the field of socio - cultural dynamics a variety of models has been proposed which do not rely on the evolutionary concept of ( natural ) selection . they are rather based on the idea of exclusively _ local interactions ( li ) _ implemented in form of a system of agents that interact locally according to simple rules like assimilation or conformity . in these systems , finding strategies to better deal with the surrounding environment ( and thus improving fitness ) is not constrained by any global property . it may , however , be constrained by local ( individual ) rules . as we shall see later in this paper , constraints on the mechanisms of selection , interaction and replacement and the way they are combined in the modeling of an evolutionary process have an important bearing on both adaptation and emergence of speciation . locality operating in each of these mechanisms seems to be the fundamental modeling principle by which emergence of a multi - modal distribution as shown in figure [ fig : distribution01 ] can be explained . on the basis of these observations about the > > modelability < < of speciation with evolutionary and self - organisatory models , we study in this paper the conditions and mechanisms required for speciation and the emergence of a multi - modal distribution . in this analysis , we use computational ( section [ sec : computation ] ) as well as mathematical ( section [ sec : mathematic ] ) arguments . our models simulate how a population of individuals evolves in time in an abstract attribute space @xmath0 that represent phenetic traits , attitudes , verbal behavior , etcetera . modeling agents as points in an attribute space of this kind is of course a highly artificial abstraction from the complexity and multi dimensionality of real agents . for the purposes of this paper , let us conceptualize an _ interaction event _ , defining the system evolution from one time step to the other , by the following three components : 1 . selection of agents , 2 . application of interaction rules , 3 . replacement of agents . any interaction event ( e.g. , mating , communication , ... ) that takes place in the course of a simulation of the model consists of the sequential application of these three steps . the reason to dissect the interaction events in this way is two fold : 1 . we want to look at the dynamical and structural effects of constraints applied to each of the three components independently ; 2 . the scheduling of interaction events may have a crucial effect on the model behavior , and with the distinction between selection and interaction on the one hand , and replacement on the other , we are able to make this effect explicit . the way interaction events are scheduled in the implementation of the models is not always given much importance in existing simulation studies . in the presence of constraints on the selection and interaction mechanisms , however , the outcome as well as the dynamical properties depend in a crucial way on the different choices . on the other hand , there are studies that do analyze the differences between synchronous and asynchronous update ( see , for instance , @xcite ) as well as studies on non overlapping ( nolg ) and respectively overlapping generations ( olg ) in biology and economics ( for instance , @xcite ) . here we show that especially when the interaction is constrained ( as in the case of assortative mating ) there emerges an important qualitative difference between olg and nolg models . namely , speciation is observed in the former , but not in the latter case , whereas adaptation is favored by the latter and hindered by the former . however , by the distinction of selection , interaction and replacement we are able to show that in fact the difference between local and non - local replacement plays the determinant role ( and not the distinction between olg and nolg ) . even though locality also impacts selection and interaction mechanisms , it is on the replacement mode where relies the fundamental difference with respect to the conditions required for either adaptiveness or speciation . this paper is organized as follows : section 2 addresses the main issues of both the fitness landscape and the self - organizing models from a computer simulation framework . in both cases , microscopic implementation rules are tested against their capability of reproducing adaptiveness and speciation . in section 3 , the emergence of speciation is analytically shown to be dependent on the choice of different replacement modes . this is accomplished through a probabilistic description of a minimal model of just three phenetic traits where the transition probabilities between traits follow a markov chain . section 4 is targeted at presenting concluding remarks and a framework that relates interaction events to the emergence of collective structures in adaptive and self - organizing complex systems . in biology , and population genetics in particular , adaptive walks on fitness landscapes have been studied intensively . the main questions addressed by fitness landscapes approaches are related to the possible structure of the landscapes ( e.g. , @xcite ) , to how populations climb an adaptive peak in the landscape ( e.g. , @xcite ) , and to the circumstances under which a population might wander from one peak to another by crossing adaptive valleys ( e.g. @xcite ) . one of the best known models for populations on fitness landscapes is the wright - fisher model with non overlapping generations ( sometimes called wright - fisher sampling and shortened in the sequel by wf model , see @xcite and also @xcite ) . consider a population of @xmath1 individuals which is said to constitute the original generation ( @xmath2 ) . we consider only the case of sexual reproduction in this paper , in which the genotype of a new born individual is obtained by the recombination of the genoms of two randomly chosen parent individuals . as noted above , the choice of two parents and the application of a recombination rule is referred to as interaction ( or mating ) event . in the wf model , @xmath1 such mating events are performed until a new generation of @xmath1 individuals is complete . as soon as it is complete , the parent generation is canceled and the process is repeated taking the new generation as parents . therefore , in the wf model the population size is always maintained at @xmath1 . we will denote the generation number by @xmath3 . we implemented this simple model and performed simulations on different toy fitness landscapes . the microscopic rules involved into the creation of a new individual , that is , the mating event , are as follows : 1 . selection of two individuals with a probability proportional to their fitness , 2 . application of recombination and mutation rules , 3 . replacement of an agent from the parent generation . in this toy model , we consider only one phenetic trait ( locus ) that takes discrete values ( from 0 to 99 ) . we denote the traits of the two chosen parent individuals @xmath4 and @xmath5 as @xmath6 and @xmath7 respectively and model recombination by taking the average of the two , @xmath8 . to model mutations we add a random value to @xmath9 . in the wf model , @xmath9 is stored at an arbitrary place in the children array and one of the main objectives of this paper is clarify that this has important consequences for the model dynamics . an adaptive landscape is introduced into the model by assigning a fitness value to each of the 100 traits . for the first analysis shown in figure [ fig : onepeak.wf ] , a single peaked fitness function with a peak at trait 75 is used and the fitness assigned to trait @xmath10 is given by @xmath11 we have used the normal distribution with @xmath12 and @xmath13 in the construction of the fitness landscape ( solid line in figure [ fig : onepeak.wf ] ) . in the iteration process , individuals are chosen as parents with a probability proportional to @xmath14 , @xmath10 being the trait of the respective individual . for the illustrative model realizations in this section , we set @xmath15 . initially , the 500 individuals are distributed in this space according to a normal distribution with mean @xmath16 and @xmath17 ( see first image of figure [ fig : onepeak.wf ] ) . this section is mainly thought as an illustration of the different behaviors and patterns generated by certain constraints on the interaction mechanism . as the qualitative effects of different assumptions become evident and comprehensible in single simulations of the model , there is no need for a rigorous statistical analysis of suites of simulations with varying initial conditions . moreover , a mathematical analysis of the model dynamics is presented in the second part of this paper ( section [ sec : mathematic ] ) . [ cols="^,^,^ " , ] the framework presented in table 1 schematically shows the consequences of adopting ( un)constrained mechanisms to the emergent outcome of a so process . it helps to emphasize that the emergence of some specific patterns may be strongly dependent on the way constraints dictate limitations on the selection , interaction and replacement mechanisms . more specifically , it shows that differently ( un)constraining the replacement mechanism of an so process provides the conditions required for either speciation ( the emergence of multi - modal distributions ) or adaptation , since these features appear as two opposing phenomena , not achieved by one and the same model . in the same way that random interbreeding leads to conservative dynamics , random replacement is also an opposing force to speciation since newcomers may take the place of former - distant agents . in so doing , at the macro level , random replacement sets aside the effect of bounded confidence and - like undirected genetic drift - may lead to the merging of subpopulations . even though we show in this paper that natural selection , operating as an external , environmental mechanism , is neither necessary nor sufficient for the creation of clustered populations , we do not want to argue against natural selection as an important mechanism in the biological domain and a substantive driving force in the speciation process . to the contrary , the concept of ( natural ) selection operating at a global level may provide us with plausible interpretations of the model results , even in disciplines where such interpretations are still lacking . in the words of t. dobzhanski ( @xcite , p.5 - 6 ) : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ [ ... ] in biology nothing makes sense except in the light of evolution . it is possible to describe living beings without asking questions about their origins . the descriptions acquire meaning and coherence , however , only when viewed in the perspective of evolutionary development . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ financial support of the german federal ministry of education and research ( bmbf ) through the project _ linguistic networks _ is gratefully acknowledged ( http://project.linguistic-networks.net ) . this work has also benefited from financial support from the fundao para a cincia e a tecnologia ( fct ) , under the _ 13 multi - annual funding project of uece , iseg , technical university of lisbon_.
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in this paper , we inspect well known population genetics and social dynamics models . in these models , interacting individuals , while participating in a self - organizing process , give rise to the emergence of complex behaviors and patterns .
while one main focus in population genetics is on the adaptive behavior of a population , social dynamics is more often concerned with the splitting of a connected array of individuals into a state of global polarization , that is , the emergence of speciation . applying computational and mathematical tools
we show that the way the mechanisms of selection , interaction and replacement are constrained and combined in the modeling have an important bearing on both adaptation and the emergence of speciation . differently
( un)constraining the mechanism of individual replacement provides the conditions required for either speciation or adaptation , since these features appear as two opposing phenomena , not achieved by one and the same model .
even though natural selection , operating as an external , environmental mechanism , is neither necessary nor sufficient for the creation of speciation , our modeling exercises highlight the important role played by natural selection in the interplay of the evolutionary and the self organization modeling methodologies .
_ keywords _ : * emergence*,*self - organization*,*agent based models * , * speciation * , * markov chains*. _ msc : _
37l60 , 37n25 , 05c69 .
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a conventional boundary layer theory of fluid flow used for free convective description assumes zero velocity at leading edge of a heated plate . more advanced theories of self - similarity also accept this same boundary condition @xcite , @xcite , @xcite . however experimental visualization definitely shows that in the vicinity of edge the fluid motion exists sb , @xcite , @xcite . it is obvious from the point of view of the mass conservation law . in the mentioned convection descriptions the continuity equation is not taken into account that diminishes the number of necessary variables . for example the pressure is excluded by cross differentiation of navier - stokes equation component . the consequence of zero value of boundary layer thickness at the leading edge of the plate yields in infinite value of heat transfer coefficient which is in contradiction with the physical fact that the plate do not transfer a heat at the starting point of the phenomenon . the whole picture of the phenomenon is well known : the profiles of velocity and temperature in normal direction to a vertical plate is reproduced by theoretical concepts of prandtl and self-similarity.while the evolution of profiles along tangent coordinate do not look as given by visualisation of isotherms ( see e.g. gdp ) . it is obvious that isotherms dependance on vertical coordinate @xmath1 significantly differs from power low depandance @xmath3 of boundary layer theories . in this article we develop the model of convective heat transfer taking into account nonzero fluid motion at the vicinity of the starting edge . our model is based on explicit form of solution of the basic fundamental equations ( navier - stokes and fourier - kirchhoff ) as a power series in dependant variables . the mass conservation law in integral form is used to formulate a boundary condition that links initial and final edges of the fluid flow . we consider a two - dimensional free convective fluid flow in @xmath4 plane generated by vertical isothermal plate of height @xmath2 placed in udisturbed surrounding . the algorithm of solution construction is following . first we expand the basic fields , velocity and temperature in power serious of horizontal variable @xmath0 , it substitution into the basic system gives a system of ordinary differential equations in @xmath1 variable . such system is generally infinite therefore we should cut the expansion at some power . the form of such cutting defines a model . the minimal number of term in the modeling is determined by the physical conditions of velocity and temperature profiles . from the scale analysis of the equations we neglect the horizontal ( normal to the surface of the plate ) component velocity . the minimum number of therms is chosen as three : the parabolic part guarantee a maximum of velocity existence while the third therm account gives us change of sign of the velocity derivative . the temperature behavior in the same order of approximation is defined by the basic system of equations . the first term in such expansion is linear in @xmath0 , that account boundary condition on the plate ( isothermic one ) . the coefficient , noted as @xmath5 satisfy an ordinary differential equation of the fourth order . it means that we need four boundary condition in @xmath1 variable . the differential links of other coefficients with @xmath6 add two constants of integrations hence a necessity of two extra conditions . these conditions are derived from conservation laws in integral form . the solution of the basic system , however , need one more constant choice . this constant characterize linear term of velocity expansion and evaluated by means of extra boundary condition . in the second section we present basic system in dimensional and dimensionless forms . by means of cross - differentiation we eliminate the pressure therm and next neglect the horizontal velocity that results in two partial differential equations for temperature and vertical component of velocity . in the third section we expand both velocity and temperature fields into taylor series in @xmath0 and derive ordinary differential equations for the coefficients by direct substitution into basic system . the minimal ( cubic ) version is obtained disconnecting the infinite system of equations by the special constraint . the fourth and fives sections are devoted to boundary condition formulations and its explicit form in therms of the coefficient functions of basic fields . it is important to stress that the set of boundary conditions and conservation laws determine all necessary parameters including the grasshof anf rayleigh numbers in the stationary regime under consideration . the last section contains the solution @xmath5 in explicit form and results of its numerical analysis . the solution parameters values as the function of the plate height @xmath2 and parameters whivh enter the grasshof number @xmath7 estimation are given in the table form , which allows to fix a narrow domain of the scale parameter @xmath8 being the characteristic linear dimension of the flow at the starting level . let us consider a two dimensional stationary flow of incompressible fluid in the gravity field . the flow is generated by a convective heat transfer from solid plate to the fluid . the plate is isothermal and vertical . in the cartesian coordinates @xmath0 ( horizontal and orthogonal to the palte)@xmath9 ( vertical and tangent to the palte ) the navier - stokes ( ns ) system of equations have the form @xcite.:@xmath10@xmath11 in the above equations the pressure terms are divided in two parts @xmath12 . the first of them is the hydrostatic one that is equal to mass force @xmath13 , where : @xmath14 is the density of a liquid at the nondisturbed area where the temperature is @xmath15 . the second one is the extra pressure denoted by @xmath16the part of gravity force @xmath17 arises from dependence of the extra density on temperature , @xmath18 is a coefficient of thermal expansion of the fluid . in the case of gases @xmath19the last terms of the above equations represents the friction forces with the kinematic coefficient of viscosity @xmath20 the mass continuity equation in the conditions of natural convection of incompressible fluid in the steady state @xcite has the form : @xmath21 the temperature dynamics is described by the stationary fourier - kirchhoff ( fk ) equation : @xmath22 where @xmath23 and @xmath24 are the components of the fluid velocity @xmath25 , @xmath26 - temperature and @xmath27 - pressure disturbances correspondingly and @xmath28 is the thermal diffusivity . from the point of clarity of further transformations we use the same scale @xmath29 along both variables @xmath0 and @xmath1 . we will return to the eventual difference between characteristic scales in different directions while the solution analysis to be provided . after introducing variables : @xmath30@xmath31 we obtain in boussinesq approximation ( in all terms besides of buoyancy one we put @xmath32 ) .@xmath33@xmath34 and fk equation is written as @xmath35 where @xmath36 @xmath8 is a characteristic linear dimension and @xmath37 is characteristic velocity : @xmath38 then @xmath39 , @xmath40 and @xmath41is the grashof number , which after plugging ( @xmath42 takes the form:@xmath43 after cross differentiation of equations ( @xmath44 and ( @xmath45 ) we have : @xmath46 = \ ] ] @xmath47 \label{ns1 + 2}\ ] ] the fk equation rescales as @xmath48 and @xmath49 where @xmath50 next we would formulate the problem of free convection around the heated vertical isothermal plate @xmath51 @xmath52 , dropping the primes . in this case we assume the angle between the plate and a stream line is small that means a possibility to neglect the horizontal component of velocity of fluid , denoting the vertical component as @xmath53 . in this paper we restrict ourselves by the assumption that @xmath54 and @xmath55 , that yields @xmath56 = 0 , \label{ns - a}\ ] ] @xmath57 the aim of this paper is the theory application to the standard example of a finite vertical plate . having only two basic functions we consider the power series expansions of the velocity and temperature in cartesian coordinates : @xmath58 @xmath59 according to standard boundary conditions on the plate we assume that the both functions tend to zero when @xmath60 , so we choose for a calculation the variable that has the zero value for nondimentional temperature ( @xmath61 ) . it means that the value of @xmath62 outside of the convective flow tends to @xmath63 substituting expressions ( @xmath64 and ( @xmath65 into the equations ( @xmath66 we take into account the linear independance of monomials @xmath67 that gives a system of coupled nonlinear equations for the coefficients @xmath68 @xmath69 , .... and @xmath70 , @xmath71 , @xmath72 such system is infinite hence for a practical use we need to choose appropriate scheme of closed formulation for finite number of variables . the formulation should be based on physical assumptions for a concrete conditions . we would like to restrict ourselves by the fourth order approximation for both variables that means we neglect higher order terms starting from fifth one . the area of the approximations validity is defined by the comparison of terms in expantions ( @xmath64 and ( @xmath73 as it will be clear from further analysis we should consider the functions : @xmath74 @xmath75 @xmath70 and @xmath76 as variables of the first order , while @xmath77 and @xmath78 to be the second one . from the relations that appear after substitution of ( @xmath64 and ( @xmath79 into ( @xmath80 and ( @xmath81 it follows that @xmath82 and @xmath83 finally from both equations ( @xmath84 ( @xmath81 we obtain the system of equations for the coefficients @xmath85 , @xmath86 , @xmath69 and @xmath87@xmath88 @xmath89 @xmath90 @xmath91 the first two ( @xmath92 ( @xmath93 arise from fk equation and the rest of them are from the ns one . the system of equations is closed if @xmath94 . it means that the number of equations and the number of unknown functions is the same . in the first approximation the velocity and temperature are expressed as : @xmath95 from ( @xmath96 ) one has @xmath97 from ( @xmath98 ) it follows that @xmath99 hence ( @xmath100 ) goes to:@xmath101@xmath102 the equation ( @xmath103 ) reads : @xmath104 this results in @xmath105 the form of the equation ( @xmath106 ) indicates that for unique solution one needs four boundary conditions for given parameters @xmath107 and @xmath108 . apart from such conditions we should also have values for @xmath109 and @xmath110 . so for expicit determination of @xmath53 and @xmath111 we need eight conditions . looking for the boundary conditions let us apply conservation laws of mass , momentum and energy , applying the laws to a control volume @xmath112 ( see fog.1 ) . the first one is the conservation of mass in two dimensions that in steady state looks as : @xmath113 where : @xmath114 is the sum of all lateral surfaces @xmath115 ( fig.1 ) . the mass conservation law in the integral form ( @xmath116 ) is formulated by a division of the surface @xmath117 to two the lower @xmath118 and upper @xmath119 boundaries only . according to our main assumption about two - dimensionality of the stream we neglect a dependence of variables on @xmath120 coordinate and @xmath121 $ ] . hence the condition of total mass conservation looks as follows : @xmath122 where the flow from below @xmath123 is approximately the product of density at temperature @xmath124 and velocity of the incoming flow in the interval @xmath125.$ ] we follow the idea of the velocity field continuity at @xmath126 , hence @xmath127 . for the left side in approximations mentioned above one has ( @xmath128 @xmath129 @xmath130 ) : @xmath131 @xmath132 and outcoming flow @xmath119 is expressed similarily : @xmath133 @xmath134 the mass conservation law yields @xmath135 the next condition is connected with the conservation of energy in a control volume @xmath112 ( area @xmath114 with unit width see fig.1 ) arises from fk equation ( @xmath136 ) by integration over the volume.@xmath137 the left side of the energy conservation equation ( @xmath138 ) is transformed similar applying the identity @xmath139(@xmath140 @xmath141 and ( @xmath142 ) . according to our assumptions we left only flows accross @xmath143 and basing on homogenity of the problem with respect to the coordinate @xmath120 we have : @xmath144 to link the incoming fluid temperature @xmath124 @xmath145 with the solution at @xmath126 and the outgoing fluid ( see @xmath146 ) we put @xmath147 that results in : @xmath148 @xmath149 where @xmath150 . the equation ( @xmath151 is the ordinary differential equation of the fourth order , therefore its solution needs four constants of integration . these constants depend on two parameters @xmath107 and @xmath108 , which enter the coefficients of the eq.(@xmath151 . the function @xmath5 defines the rest functions @xmath152 and @xmath85 via above relations . it means that we have six constants determining the solution of problem and we need also six corresponding boundary conditions . the temperature values in the vicinity of the boundary edge point and taken as value -1 ( temperature of incoming from the bottom flow ) . in dimensional form the interval of consideration has the characteristic length @xmath8 which we identify with a parameter we used when dimensional variables where itroduced ( @xmath61 ) . let us remind that scale @xmath8 is connected with special ( local , horizontal ) grashof number @xmath153 ( @xmath154).the total height of the plate is denoted @xmath2 for a stationary process an edge conditions may be considered as initial one for a cauchy problem . having a power series approximation of such conditions we choose the coefficients of the series using weierstrass theorem . it means that we equalize the coefficients to scalar product of intial conditions and orthonormal polynomials on the interval @xmath155 .$ ] in our case the temperature profile @xmath156 represents this condition , while the function is constant @xmath157 on the interval @xmath158 $ ] in nondimensional variables . in the approximation of the thirdpower orthogonal polynomials we have : @xmath159 because nondimentional temperature of the fluid at the lower half plane , according to above , is @xmath160where the polynomialas are defined as : @xmath161 @xmath162 the normalization for @xmath163 , @xmath164and orthogonality condition @xmath165 give the link between constants : @xmath166 . , which plugging into @xmath167 results in @xmath168 = @xmath169finally @xmath170 substituting the result into @xmath171 gives two equations @xmath172 , @xmath173 , which solving and projecting @xmath174 , @xmath175 @xmath176.yield boundary conditions for the coefficients for temperature expansion : @xmath177 plotting the temperature approximation at the level @xmath178 @xmath179 approximation ] is given by the fig.2 . let us recall that @xmath180 ( see eq . ( @xmath181))@xmath182 therefore @xmath183 ( @xmath184 ) we will consider as boundary condition for @xmath185 the temperature gradient values @xmath186 on the plate decrease when @xmath1 grows.at the leading edge we pose the condition @xmath187 because the plate lose the contact with the fluid . it gives third boundary condition ( @xmath146 ) @xmath188 the phenomenon of free convective heat transfer from isothermal vertical plate ( @xmath189 ) imply that temperature gradient on the plate is negative ( @xmath190 ) and decrease along @xmath1 ( @xmath191 ) . it is also known that velocity profile has maximum at the distance @xmath192 @xmath193 . the extrema for the curve is defined by derivative of @xmath53 as a function of @xmath194 hence the relation @xmath195 indicates that for @xmath196 , @xmath197 and @xmath198 we have two extremal points @xmath199{\frac{\alpha^{2}}{9\beta^{2}}-\frac{% \gamma}{3\beta}}\text { \ \ \ \ and \ \ \ \ } x_{0}(y)=-\frac{\alpha } { 3\beta}+% \sqrt[2]{\frac{\alpha^{2}}{9\beta^{2}}-\frac{\gamma}{3\beta } } \label{xm0}\ ] ] if @xmath200 notations are chosen to mark maximum position point as @xmath192 while the minimum one is @xmath201 . in the exeptional case of @xmath202the expression simplifies @xmath203 which is positive for @xmath204the second extremum do not exist now ( see fig.3 ) . [ fig : fig-3 ] there is a possibility to choose the value @xmath205 considering the @xmath206 as a conditional boundary of the upward stream.we define hence @xmath207 . at the starting horizontal edge of the vertical plate the vertical velocity component of incoming flow ( @xmath146 ) varies slow so we assume that @xmath208 hence @xmath209 the extrema of the velocity profile ( @xmath210 ) after account of ( @xmath211 ) and ( @xmath130 ) is transformed as , for maximum : @xmath212{\frac{c_{2}{}^{2}}{\left ( \frac{g_{r}}{2}% c\left ( y\right ) \right ) ^{2}}+\frac{2\gamma}{g_{r}c\left ( y\right ) } } $ ] and minimum one : @xmath213{\frac{% c_{2}{}^{2}}{\left ( \frac{g_{r}}{2}c\left ( y\right ) \right ) ^{2}}+\frac{% 2\gamma}{g_{r}c\left ( y\right ) } } $ ] . the following identity @xmath214 holds for : @xmath215 suppose there exists a level @xmath216 at which @xmath217 where @xmath218 denotes the boundary layer thickness analog . the equation ( @xmath219 ) is solved with respect to @xmath220 that gives : @xmath221 as function of the problem parameters . then plugging ( @xmath222 ) for the expression for the @xmath223 yields @xmath224 let us return to the expression for the temperature ( @xmath146 ) with neglecting the last term in temperature ( the possibility of such assumption will be explained below ) on the level @xmath225 and substitute ( @xmath222 ) and ( @xmath226 ) into it equalizing to the temperature of surraunding @xmath227 . @xmath228 we have : @xmath229 where : @xmath230 from the equation ( @xmath106 ) after plugging @xmath108 ( @xmath231 ) and taking into account @xmath232 ( @xmath233 ) we have @xmath234 the equation was studied recently @xcite where the solution was given by@xmath235 @xmath236+\exp[-\frac{sy}{2}](b_{1}\cos[\frac{\sqrt{3}}{2}sy]% + b_{2}\sin[\frac{\sqrt{3}}{2}sy ] ) , \label{c(y)}\ ] ] where @xmath237{2\pr g_{r } } \label{s}\ ] ] is expressed via @xmath238 = @xmath239we have also boundary conditions : ( @xmath240 ) @xmath241 solution of the system results in a rather big expression for @xmath242as function of @xmath243which we skip in theis text , going to following approximtion . the explicit form of the equation ( @xmath244 shows that the three last terms have exponential behaviour as function of @xmath245it means that there are three different domains of the fluid flow structure . the first is the starting one where all terms are significant . the leading edge is characterized by two first terms and the medium domain is described by the only first one . we choose the parameter @xmath216 such that it belongs to that medium range . in such conditions @xmath246 @xmath247 where @xmath248 plugging @xmath243 in the form of ( @xmath249 into the table of boundary conditions gives @xmath250 a_1=--b_1+,@xmath251b_1=-b_2 + -,@xmath252b_2=.@xmath253 let us consider the natural approxmation @xmath254 . after substitution of the expression for @xmath255 into @xmath256and next @xmath256 into @xmath257 we have approximate formulas : @xmath250a_0=-,@xmath251 a_1=,@xmath251b_1=-,@xmath251b_2@xmath258-++.@xmath258@xmath259 it defines the expression for @xmath70 as the function of parameters @xmath260 ( @xmath261 , the plate height @xmath2 and the new one ( @xmath262 @xmath263 the velocity profile at the level @xmath225 is defined by ( @xmath264 and the parameters values ( @xmath265 : @xmath266 the mass conservation equation ( @xmath267 ) after substitution of @xmath268 @xmath269 @xmath270 @xmath271 , @xmath272 and denoting @xmath273 has the form : @xmath274 the only real solution of the equation ( @xmath275 ) value that have physical sense is @xmath276 now we can return to the energy conservation equation ( @xmath277 plugging the boundary conditions for the domain restricted by the plate on interval ( @xmath278 ) . it simpifies the expression for the integral along the plate surface ( heat transfer from the plate on this interval ) . consequently we change @xmath279 to @xmath223 and neglect the integrant oscilations at vicinity of @xmath126 . @xmath280 we estimate the heat flux integral from the plate as @xmath281 and take into account the expressions for parameters @xmath282that yields : @xmath283 . as further considerations show , the value of @xmath225 mayby chosen as close to the plate height @xmath2 . chosing @xmath285 and plugging the values of the parameters @xmath286 @xmath287and ( @xmath288 ) @xmath289{2ra}=\allowbreak 4.\,\allowbreak5782 $ ] into the table of the function @xmath5 coefficients gives [ cols= " < " , ] substitution of the table values into ( @xmath290 ) we have the expression which allows to plot the function @xmath5 . in the same approximation the typical velocity profile @xmath291 ( @xmath292 ) at the the stability interval ( y @xmath293 : $ ] ) @xmath294 . substitution of the grashof number @xmath295= @xmath296 gives @xmath297 that is represented by the plot . . ] in the same condition the temperature profile @xmath298 is defined by the expression ( @xmath146 ) and results in the plot ] to understand the phenomenon it is useful to return to dimensional picture . as a main space scale it is choosen the parameter @xmath8 ( @xmath61 ) which is connected with the grashof number by ( @xmath154 ) @xmath299{\frac{1}{% bg\phi } \nu^{2}g_{r}}$ ] . where @xmath300 . for the air example and the temperature @xmath301 @xmath302 , @xmath303 , @xmath304 the viscosity coeficient @xmath305 = @xmath306 the coeficient of thermal expansion @xmath307 and for conditions of our model ( @xmath285 ) @xmath308 we estimate @xmath8 as : @xmath309{\frac{1}{\frac{1}{303}10\cdot20}\left ( 16\cdot10^{-6}\right ) ^{2}\cdot68.\,\allowbreak54}=\allowbreak2.\,\allowbreak985\times 10^{-3}m\thickapprox3mm$ ] first of all we would stress again that the model we present here have the engieering character of approximations , but include direct possibilities for a development by simple taking next terms of expansions into account . a modification of boundary conditions which would improve the transient regimes at both ends of the y - dependence is also possible . newertheless in this simple modeling we observe some important characteristic features of real convection phenomenon as almost parallel streamlines and isotherms in the stability region ( as , for example in visualizations of interferometric study from @xcite ) . it follows from functional parameter @xmath5 behaviour inside the domain and small contribution of cubic therm in the expresion for temperature ( @xmath146 ) . our explicit solution form and parameter values estimation allows to conclude that : \1 . the streamlines and isotherms of the flow are almost paralel to the vertical heating plate surface in the domain of stability , \2 . velocity values of the fluid flow at starting edge of the plate are nonzero , \3 . the set of boundary conditions yields in the complete set of the solution parameter including the local grashof number and hence , the characteristic linear dimension length @xmath8 in normal to plate direction @xmath0 , \4 . the sesults allow to descibed the natural heat transfer phenomenon for given fluid in therms only the temperature difference @xmath310 and the plate heigth @xmath311 which are novel in comparison with former theories . 99 y.jaluria.:natural convection heat and mass transfer ; pergamon press , oxford , 1980 . e .. schmidt and w.beckmann , das temperatur- und geschwindigkeitfeld vor einer wrme abgebenden senkrechten plate bei natrlicher konvektion , tech mech . u. thermodynamik , bd.1 , nr . 10 , okt.1930 , pp.341 - 349 and bd . 1 , nr.11 , nov . 1930 , pp.391 - 406 . h.c.li and g. p. peterson , experimental studies of natural convection heat transfer of al2o3/diwater nanoparticle suspensions ( nanofluids ) , hindawi publishing corporation , advances in mechanical engineering , volume 2010 , article i d 742739
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the model under consideration is based on approximate analytical solution of two dimensional stationary navier - stokes and fourier - kirchhoff equations .
approximations are based on the typical for natural convection assumptions : the fluid noncompressibility and bousinesq approximation .
we also assume that ortogonal to the plate component ( @xmath0 ) of velocity is neglectible small .
the solution of the boundary problem is represented as a taylor series in @xmath0 coordinate for velocity and temperature which introduces functions of vertical coordinate ( @xmath1 ) , as coefficients of the expansion .
the correspondent boundary problem formulation depends on parameters specific for the problem : grashoff number , the plate height ( @xmath2 ) and gravity constant .
the main result of the paper is the set of equations for the coefficient functions for example choice of expansion terms number .
the nonzero velocity at the starting point of a flow appears in such approach as a development of convecntional boundary layer theory formulation .
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mutually unbiased basis sets are known to provide an optimal basis for quantum tomography @xcite , to play key roles in quantum cryptography @xcite , and to be instrumental in solving the mean king problem in prime power dimensions @xcite . the generalized pauli operators associated with mub s include the stabilizers of quantum error correcting codes @xcite , and serve as entanglement witnesses @xcite for the mub states . of interest for the foundations of quantum physics , the mub concept sharpens the concept of complementarity @xcite , and raises the question of existence in composite dimensions . an excellent comprehensive review of mubs has recently appeared @xcite . we deal here with hilbert spaces of prime power dimensions ( @xmath9 ) , where @xmath10 mubs are known to exist @xcite . this is both the largest possible number , and also the number required for a complete operator basis ( in representing the density matrix , for example ) . so , while each mub is a complete orthonormal basis in the hilbert space , the set of @xmath10 mubs is a complete ( nonorthogonal ) basis in the space of all operators , which has dimension @xmath11 . regarding terminology , to avoid reference to a `` complete set of complete sets , '' and prompted by the fact that different mubs ( or the observable sets associated with them ) are maximally complementary @xcite , i will use the term `` full complement , '' or sometimes just `` complement , '' to denote the set of all @xmath10 mubs . partial mub sets have been discussed in connection with composite dimensions and referred to as `` constellations '' @xcite . the natural systems to which mubs apply consist of @xmath1 @xmath4-state objects ( _ qupits _ ) . in such systems , while mub complements exhibit only a single entanglement type for @xmath12 ( and all @xmath4 ) , the number of distinct types proliferates with increasing @xmath1 . the variety is illustrated in a number of recent discussions , mostly on multiple qubit systems but also multiple qutrit systems @xcite . in particular , a systematic study by romero and collaborators @xcite illustrates a broad range of entanglement patterns that occur naturally in a construction scheme for full mub complements . such complements are catalogued for up to 4 qubits . wieniak and collaborators @xcite have developed a construction scheme aimed at experimental implementation and discussed the total entanglement content of full mub complements of bipartite systems . with the general mub problem in mind , our purpose here is to develop a general framework , independent of construction schemes , for exploring mub entanglement patterns for all @xmath4 and @xmath1 . the project begins by proving three general theorems ( the `` rules '' ) that underlie and lead quickly to an array of more specific results . many of the latter apply to all @xmath4 , but are @xmath1-specific , as each step in @xmath1 introduces further richness . all results refer to one of two levels - that of individual mubs and that of full complements . at the individual level , mub types are characterized by first specifying the _ separation pattern _ - how many , and how big , are the irreducible subsets of qupits defined by the factorization of the wavefunction ? - and next , by describing the _ entanglement pattern _ - what is the nature of the entanglement within each irreducible subset ? at the level of the full complement , we ask about the possible _ mub distributions _ - what combinations of mub types can coexist within full complements . at the first level , we will show that all conceivable separation patterns are possible , and we will show with examples how to describe the entanglement within the nonseparable factors . at the level of the full complement , we will show how to deduce constraints on the possible mub distributions . for @xmath12 and 3 , surprisingly , the general global constraints mentioned in the abstract suffice to determine all mub distributions for all @xmath4 . the @xmath6 case is considerably more complex and requires the derivation of more detailed constraint equations . let us begin with a review of basic concepts and notation in section ii . in section iii we prove the three general theorems . these rules are applied in section iv to obtain the entanglement patterns of individual mubs , and to deduce constraints on their possible distributions within full complements , taking the @xmath12 - 4 cases in turn . in section v we summarize results and comment on unresolved questions . in hilbert spaces of dimension @xmath13 , two orthonormal bases ( @xmath14 and @xmath15 ) are mutually unbiased if any state @xmath16 in basis @xmath14 has uniform probably of being found in any state @xmath17 in basis @xmath15 ; that is , if ||^2 = 1/d . [ mub ] thus , measurements in the two bases provide no redundant information . since measurements in any basis provide @xmath18 independent probabilities , and since @xmath19 real parameters are needed to determine an unknown quantum state ( its density matrix @xmath20 ) , it follows that @xmath10 mub s are required . in this way the mub projectors form a complete nonorthogonal basis in operator space this required number of mubs is ( only ) known to exist in power - of - prime dimensions . there is an intimate connection between mubs and generalized pauli operators ( hereafter called simply `` pauli operators '' ) which underlies several construction schemes ( provides a comprehensive listing @xcite ) . these operators are conventionally written in the form of a tensor product , _ n , m = x^nz^m x_1^n_1x_2^n_2 ... x_n^n_nz_1^m_1 ... z_n^m_n , [ pauli1 ] whose factors , acting on individual qupits , are powers of the generalized ( @xmath21 ) pauli matrices , z = _ k=0^p-1 ^k 1truecm 1truecm x = _ k=0^p-1 , [ pauli2 ] where @xmath22 , and @xmath23 is the raising operator of @xmath24 . the powers @xmath25 and @xmath26 are @xmath4-nary numbers , _ eg _ , @xmath27 , whose digits take the values 0,1, ... ,@xmath28 . thus , there are @xmath29 operators @xmath30 ( including the identity @xmath31 ) , which make up a complete and orthonormal basis in operator space ( with the trace operation as inner product ) . the desired connection with mub s is described in : the @xmath30 partition into @xmath2 internally - commuting subsets , each consisting of @xmath28 traceless operators ( excluding @xmath32 ) . the corresponding eigenbases then form a complete complement of mub s . the above are standard definitions and conventions . it will be useful to adopt a couple of more special conventions for use throughout this paper . first , the operator set @xmath30 does not form a group , because multiplication generates irreducible phase factors . however , for odd @xmath4 the set @xmath33 does form a group , of order @xmath34 , and for @xmath7 the analogous set @xmath35 forms a group of order @xmath36 . these are called discrete heisenberg - weyl , or generalized pauli groups @xcite . we shall not make direct use of them , but we shall take advantage of the freedom to redefine the phases of the @xmath30 in the original set : we choose phases so that the compatible subsets form groups , and we call these _ compatibility groups_. they are all isomorphic to those consisting of @xmath37 and @xmath38 , each of which is generated by the @xmath1 independent elements , @xmath39 and @xmath40 , respectively . thus , to construct another compatibility group , we may choose a generator set @xmath41 that consists of any @xmath1 elements in the original compatible subset that do not form a subgroup , and write the resulting group elements as g^n g_1^n_1g_2^n_2 ... g_n^n_n . [ ggroup ] thus , all of the compatibility groups are representations of the same group - the abelian group of order @xmath42 generated by @xmath1 elements . a simple example of a compatibility group so generated is y^n = ( x_1z_1)^n_1(x_2z_2)^n_2 ... (x_nz_n)^n_n . [ ygroup ] note that phase factors are introduced with respect to the original pauli operators because , , @xmath43 . the generator set @xmath44 , by itself , completely determines the states of the basis ( @xmath44 ) in the hilbert space , through the eigenvalue equations @xmath45 , where @xmath46 is a @xmath4-nary representation of the state index @xmath47 . the eigenvalues of a general group element are then given by g^n = ^n k , [ eigenvalues ] where @xmath48 , and the spectral representation of @xmath49 is therefore just the fourier transform @xcite g^n = _ k ^n k _ k ^n k ( g , k ) , [ specrep ] where @xmath50 is the projector onto state @xmath47 in basis @xmath44 . this mub projector is then given by the inverse transform , ( g , k ) = p^-n _ n ^-n k g^n . [ inverse ] the existence of these simple transform relationships between every compatibility group and its corresponding mub projector set is a consequence of defining the former to be a group . the only remaining arbitrary phases are those of the generators . in this section we establish the general rules that will form the basis for the rest of the work . for ease of reference and completeness i will state and prove these results as three separate numbered theorems . for transparency , here , in plain english , is what they will say about mub states : ( i ) a given qupit is perfectly pure or totally entangled , ( ii ) the distribution of one - qupit operator factors in the compatibility group correlates with this purity , ... , and ( iii ) in any full mub complement , every qupit appears pure @xmath2 times , and totally entangled @xmath3 times . these theorems and the results that follow from them rely on the assumption that mub states are eigenstates of pauli operators . while this is restrictive for individual mub pairs , it is not restrictive for known mub complements or known construction schemes @xcite , allowing for unitary equivalence . an example may help to illustrate . consider the standard basis in 4d , and another related to it by the unitary transformation @xmath51 ( where @xmath52 0,1,2,3 ) , which is not an eigenbasis of the pauli operators of . the two bases are mu , but a full complement can not be completed containing both of them . however , full complements can be found containing either basis without the other : starting with the well - known full complement containing the standard basis , one could apply @xmath53 to each of its bases to obtain another full complement . the latter are not eigenbases of the original pauli operators , but clearly they _ are _ eigenbases of transformed pauli operators , which may be thought of as corresponding to redefined parts ( and redefined quantization axes ) . the results of this paper then apply with reference to these redefined parts . regarding the existence of a mub complement outside of this equivalence - i believe that this question also remains unresolved @xcite . we will return to these points in the conclusions . as a brief preliminary , one - qupit states within the @xmath1-qupit system are defined by the reduced density matrices , _ i = , [ rhoi ] where tr@xmath54 denotes the partial trace over states of all but the @xmath55-th qupit . perfect purity means that @xmath56 is a projector , while total impurity means that @xmath57 . one can define the purity of the state @xmath58 as p_i = ( ptr _ i^2 - 1)/(p-1 ) , [ purity ] which takes its extremal values , 1 and 0 , in the respective cases . * theorem i : * if the system is in a pure eigenstate of pauli operators ( a generator set @xmath44 ) , then any individual qupit must exist in a state of either perfect purity , or total impurity , the same for all eigenstates of @xmath44 . * proof : * the generators produce a compatibility group , and the @xmath1-qupit density matrix representing a pure eigenstate , @xmath59 , may be expanded as in . considering now the pauli matrix factors that act on just the @xmath55th qupit , the generator set @xmath44 must fall into one of two categories : only one pauli matrix , say @xmath60 ( and possibly powers of it ) , appears in the generator set , or more than one appear ( including , say , @xmath61 and @xmath62 ) , that are not powers of one another . consider the latter case , which is simpler : let @xmath63 and @xmath64 be generators that contain the factors @xmath61 and @xmath62 . no operator of the form @xmath65 ( where @xmath66 is any one - qupit pauli matrix ) commutes with both @xmath63 and @xmath64 , and all such operators are thereby excluded from the compatibility group . as a result , the only operator with a nonvanishing partial trace tr@xmath54 is the global identity @xmath32 . since @xmath32 enters the summation ( [ inverse ] ) with the coefficient @xmath67 , and tr@xmath54 produces a factor of @xmath68 , the reduced density matrix for the @xmath55th qupit is _ i = p^-1 _ i , [ t1 ] indicating that the @xmath55th qupit is totally impure . now turn to the other case : if only @xmath60 ( and possibly powers ) appear in the generator set , then only @xmath60 and its powers can appear in the compatibility group ( again refering only to those factors that act on the @xmath55th qupit . since the `` one - body '' operators @xmath69 commute with all of these , they must belong to the compatibility group . these one - body operators are the only ones that survive the partial trace . since each of them enters the summation ( ) with coefficient @xmath70 , and since @xmath71 produces a factor of @xmath68 in each term , we find in this case that _ i = p^-1 _ n ^- n_ik_i z_i^n_i = . [ t2 ] this shows that @xmath58 is a projector onto the eigenstate of @xmath60 whose eigenvalue is @xmath72 , that is , _ i^2 = _ i 1.2truecm 1.2truecm z_i _ i = ^k_i _ i. [ t3 ] this proof is independent of the choice of the eigenstate @xmath73 ) in the basis @xmath44 , and so clearly the @xmath55th qubit is perfectly pure for all eigenstates in this basis . here is a related more detailed theorem on the distribution of one - qupit matrices associated with a single qupit . * theorem ii : * in any compatibility group of @xmath1-qupit pauli operators , the distribution of one - qupit factors acting on the @xmath55th qupit must be one of two types : ( i ) only a single pauli matrix and its powers occur , and each power occurs an equal number ( @xmath74 ) of times , or ( ii ) every pauli matrix occurs , and each occurs an equal number ( @xmath75 ) of times . * proof : * consider any set @xmath44 of @xmath1 generators of the compatibility group . this set must be one of the two types considered in the foregoing proof : suppose first that only one pauli matrix ( say @xmath60 ) , and possibly powers of @xmath60 appear . let @xmath63 be a generator containing @xmath60 as a factor , and let @xmath64 , @xmath76 , ... , @xmath77 be the rest . @xmath63 by itself generates a cyclic subgroup containing all powers of @xmath60 . then , @xmath63 and @xmath64 by themselves generate a subgroup of order @xmath78 in which , by virtue of the rearrangement theorem , every power of @xmath60 appears @xmath4 times ( no matter which power of @xmath60 is present in @xmath64 ) . one may repeat this argument , multiplying the order of the subgroup by @xmath4 at each stage , until the full compatibility group is generated , with each power of @xmath60 being produced @xmath74 times . in the other case , let @xmath63 and @xmath64 be generators containing the @xmath61 and @xmath62 factors , respectively . these two generators , by themselves , generate a subgroup of order @xmath78 in which every pauli matrix factor @xmath66 appears once and only once . ( to see this , note that @xmath61 and @xmath62 , by themselves , generate the one - qupit pauli group @xcite , but since @xmath63 and @xmath64 commute , the multiplicity of phase factors is absent . ) now , by including a third generator , @xmath76 , one generates a subgroup of order @xmath79 in which , by the rearrangement theorem , each pauli matrix factor appears @xmath4 times . repeating the process through @xmath77 , one generates the full compatibility group with each pauli matrix factor appearing @xmath75 times . the second result is particularly striking in light of the fact that the nature of the entanglement of the @xmath55th qupit may vary widely , in the sense that its entanglement may be shared with any number of other qupits in the system . nevertheless , only two kinds of pauli matrix distributions , with the correponding purities , are possible . we use both of the foregoing theorems to deduce the total entanglement content - as measured by the one - qupit purities - of a full complement of mub s . this total content is constrained by the requirement that the two types of one - qupit pauli matrix distributions be consisent with the set of all pauli operators , which must appear in the full complement . * theorem iii : * within any full complement of @xmath0 mub s , every qupit is perfectly pure in @xmath2 basis sets , and totally entangled in the remaining @xmath3 . * proof : * consider the @xmath55th qupit . recall that the total number of pauli operators ( excluding @xmath32 ) is @xmath80 , and that these exactly accommodate the @xmath0 compatibility groups containing @xmath81 traceless operators each . each pauli matrix factor @xmath66 appears in @xmath82 pauli operators , except for @xmath83 which appears in @xmath84 because we are not counting @xmath32 in the individual groups . this number must equal the sum of @xmath83 factors appearing in all of the compatibility groups . according to the previous theorem , there are @xmath85 such factors in compatibility groups in which the @xmath55th qupit is pure , and @xmath86 such factors in all other compatibility groups . if @xmath87 is the number of compatibility groups ( or basis sets ) in which it is pure , then , in order to account for all @xmath83 factors , we must have p^2n-2 -1 = _ s^ ( p^n-1 -1 ) + ( p^n+1-_s^)(p^n-2 -1 ) . [ accounting ] solving this equation , we find the number of basis sets in which the @xmath55th qupit is pure , _ s^ = p+1 , [ nusubs ] and consequently , the number of basis sets in which it is totally entangled , _ e^ = p^n - p. [ nusube ] the following corollary arises when all qupits take their pure states simultaneously : * corollary * : the maximum number of product mubs is @xmath2 , and in any mub complement where this number is realized , all of the remaining mubs ( @xmath3 ) must be totally entangled ( in the sense that every qupit is totally entangled ) @xcite . this is the standard distribution . note that the probability of finding the @xmath55th qupit pure in a mub state picked at random from any full complement is equal to the averaged purity ( ) , p_i _ comp^ = _ s^ _ s^+_e^ = p+1 p^n+1 , [ piave ] which vanishes exponentially with @xmath1 . we discuss the @xmath88 cases in turn . the first two are simpler , and we find that theorems i and iii are sufficient to determine all possible mub distributions , although ii provides useful insights . with @xmath6 , we require theorem ii in deriving more detailed constraints that apply to individual qupits . * bipartite systems * clearly , if one qupit is pure , then so must be the other . in light of theorem i , then , both purities must be unity , or both zero . because these purities coincide , the corollary of theorem iii applies : there are @xmath2 product bases and @xmath89 totally entangled bases - the standard distribution is inevitable . we shall refer to all of the entangled bases as generalized bell bases , because they share the common property that their compatibility groups consist solely of two - body operators , , those containing no @xmath90 factors @xcite . to see the consequences of this , write one of the two generators as @xmath91 . the most general eigenstates of @xmath63 may then be written as @xmath4-term expansions in the product basis of @xmath92 and @xmath93 , = 1 _ k c_k _ u _ v , [ bform ] where the eigenvalues of @xmath94 are @xmath95 and the coefficients @xmath96 are determined by the other generator , call it @xmath97 . commutativity demands that both @xmath98 and @xmath99 , so @xmath64 induces cyclic permutations ( of order @xmath4 ) in the product states @xmath100 . therefore the @xmath96 are unimodular , and the @xmath4 eigenvalues ( @xmath101 ) of @xmath64 are nondegenerate , like those of @xmath63 . this confirms explicitly what we know from theorem i - namely , that measurements of one - qupit properties ( , @xmath92 or @xmath93 ) must produce random distributions over all possible outcomes . the generalized bell states defined above are contained within a broader class definitions given elsewhere @xcite . the more restrictive definition given here - defining classes of states by the pauli operators of which they form eigenbases - applies nonetheless to all mubs that are compatible with known full complements , and we shall employ such definitions throughout this work as we proceed to larger n. we note for future reference that the precise form of the product state expansion ( [ bform ] ) depends on the choice of basis . a bad choice would require a @xmath78-term expansion , but even a good choice could look slightly different . for example , if eigenstates of @xmath102 were expanded in the same product basis used in , one would find sums of @xmath103 . as a final note on bell states , our working definition may be given in words alone : a generalized bell state is any totally entangled two - qupit eigenstate of pauli operators ( since total entanglement requires that the two pauli operators be of the form @xmath94 and @xmath104 ) . * tripartite systems * the standard mub complement has @xmath2 product bases and @xmath105 totally entangled bases . we shall refer to all of _ these _ totally entangled bases as generalized ghz , or @xmath44-bases , because they have common properties describable as follows : let us first illustrate with a specific example that generalizes a standard choice of generators for qubits @xcite , g ( g_1 , g_2 , g_3 ) = ( xxy , xyx , yxx ) , [ ghz1 ] to arbitrary @xmath4 . to identify an optimal product basis for an expansion , replace the latter two generators by @xmath106 and @xmath107 . recalling the usual definition @xmath108 on the @xmath55th qupit ( modulo possible phase factors ) , the result is g = ( xxy , izz^-1 , ziz^-1 ) . [ ghz2 ] clearly the most general joint eigenstates of @xmath109 and @xmath110 are @xmath4-term expansions in the standard basis , = 1 _ k c_k , [ ghz3 ] where @xmath101 and @xmath95 are the eigenvalues of @xmath109 and @xmath110 , respectively , and the @xmath96 are determined by @xmath63 . the @xmath96 are again unimodular because @xmath63 generates a cyclic group of order @xmath4 , of which the @xmath4 product states form a basis . this again illustrates the randomness of one - qupit properties in totally entangled states . to demonstrate the commonality of all totally entangled three - qupit bases , we note that at least one generator must be a three - body operator ( having no @xmath90 factors ) , which we write in complete generality as @xmath111 . now , according to theorem ii , the inverse of each factor occurs @xmath4 times in the compatibility group , once with the inverse of @xmath63 itself , and @xmath28 times in other three - body operators in which it is the only inverse ( footnote @xcite ) . choosing two from the latter category , one containing @xmath112 and the other containing @xmath113 , and multiplying @xmath63 by each in turn , we obtain the generator set g = ( uvw , ibc , aic ) , [ ghz4 ] where compatibility requires that @xmath114 is common to @xmath64 and @xmath76 as indicated . clearly , @xmath115 , @xmath116 , and @xmath114 define the product basis for the @xmath4-term expansions , = 1 _ k c_k _ a _ b _ c , [ ghz5 ] and each of @xmath115 , @xmath116 , and @xmath114 must differ from corresponding factors that appear in three - body operators of the compatibility group . in other words , every three - body operator in the compatibility group induces cyclic permutations of the states composing the product basis . the similarity of generator sets shows that all totally entangled three - qupit bases have @xmath4-term expansions in some special product basis , and that all of their compatibility groups ( of the same @xmath1 and @xmath4 ) have the same numbers of three - body and two - body operators . again , a purely verbal definition is possible : a generalized ghz state is any totally entangled 3-qupit eigenstate of pauli operators . a general statement for @xmath117 is possible but less categorical . the new aspect of mubs that enters with @xmath118 is the appearance of a third ( nonstandard ) mub type , and with it , the possibility of composing a full complement with varying combinations . the third type is biseparable , and thereby nonsymmetric with respect to qupits - one qupit separates , leaving the other two in a bell state . we shall refer to these as `` separable - bell '' bases , with the shorthand notation @xmath119 ( or @xmath120 if we wish to identify the pure qupit ) . such mub bases are known for @xmath7 and 3 ( ) , and to describe them for arbitrary @xmath4 , we consider a generator set s_1b = ( iia , uvi , sti ) , [ sb1 ] where @xmath94 and @xmath104 are commuting two - body operators acting on qupits 2 and 3 . the @xmath79 joint eigenstates of this set may be written as = , [ sb2 ] which describes qupit 1 in the @xmath47th eigenstate of @xmath115 , and qupits 2 and 3 in the bell state denoted by the eigenvalues @xmath121 and @xmath4 of @xmath94 and @xmath104 , respectively . similarly , the compatibility group of @xmath122 is a tensor product of that associated with qupit 1 ( @xmath123 ) and that of the bell basis of qupits 2 and 3 . the tensor product is a common characteristic of all separable mubs , and the eigenstates of a particular mub all have the same character - the separation pattern involves the same entangled subsets of qupits , and the nature of the entanglement within each subset is the same . the three mub types discussed above , including the three variations of the sb bases , exhaust all of the possibilities for three qupits . the remaining question now is , what combinations the three types of bases may appear in the full complement ? one can answer this question simply by conserving the number of pure qupits while conserving the number of basis sets . we then find that we can remove a single product basis ( @xmath124 ) while adding three @xmath119 and removing two @xmath44 bases : + 2 g 3sb [ stoich1 ] table i shows the possibilities for three particles with any @xmath4 . the cases of @xmath125 and 3 dramatize the role of totally entangled states with increasing dimension of the hilbert space . in fact , case ( a ) , dimension @xmath126 , is the only multiparticle mub dimension in which a complement can be found with no totally entangled bases . and more typically , a majority of mubs are totally entangled : in case ( b ) at least 4/7 of all bases are @xmath44 bases , and even for _ two _ qutrits , 6 of the 10 bases are bell bases . for @xmath118 and general @xmath4 , the minimum number of @xmath44 bases is given by @xmath127 , an ever - increasing fraction of the total number of bases as @xmath4 increases . @xmath128 @xmath129 it is noteworthy that @xmath119 bases can be introduced only in steps of three , reflecting the condition that the three variations @xmath130 must balance in the full complement , since the other mub types are symmetric with respect to permutations of qupits . this condition follows from the conservation of pure states for each qupit separately . * quadrapartite systems * the @xmath6 case is more complex in a number of respects . most importantly , new mub types enter with increasing @xmath4 . but even with @xmath7 , the number of distinct mub types exceeds the number of separation patterns . figure 1 shows the five separation patterns that characterize all @xmath4 , and lists seven mub types , six of which account for all @xmath7 options , and a seventh which represents , but is not exhaustive for @xmath131 . let us first discuss the mub types for general @xmath4 , and later specialize to particular cases for constraints and stoichiometries . the separable mubs compatibility groups are tensor products of those of their constituent mubs , and their generator sets may be constructed accordingly . with @xmath132 , a single generator is associated with the separating particle ( for example @xmath133 ) , while three generators ( each of the form @xmath134 or their alternatives ) are associated with the three particles forming @xmath135 states . there are four variations on this pattern corresponding to the choices of the separating particle . in the @xmath136 case , one could pick two generators of the form @xmath137 , and two of the form @xmath138 . there are three variations on this separation pattern , corresponding on the three ways of picking the two entangled pairs , as compared with six variations on the @xmath139 pattern from the six ways of picking a single entangled pair . let us discuss the nonseparable bases in somewhat more detail beginning with four - particle ghz bases ( @xmath140 ) . these are straightforward generalizations of the three - particle bases @xmath135 , and a standard generator set @xcite consists of the four operators g^(4 ) = ( xxxy , xxyx , xyxx , yxxx ) . [ geng ] from an alternative generator set , ( @xmath141 ) , it is apparent that eigenstates may again be written as superpositions of @xmath4 product states in the standard basis . a more general characterization of ghz states is provided in the appendix . cluster bases ( @xmath142 ) were introduced in connection with measurement - based , one - way quantum computation @xcite , and in fact both cluster and ghz states are special cases of a broad class of @xmath1-qubit states , called graph states , which form the basis of this @xcite . has shown that graph states may be classified in terms of curves in phase space , which provides a further connection with the mub problem . cluster bases are defined here , for all @xmath4 , by generator sets of which a standard example , introduced for the qubit case @xcite , is c^(4 ) = ( xzxi , zxix , xixz , ixzx ) . [ genc ] cluster states have stronger entanglement links between smaller groupings of particles , making their entanglement more robust against decoherence @xcite than ghz entanglement , which is shared equally among all particles . this is reflected in the fact that @xmath143 has only two 2-body operators in its compatibility group , as compared with three in the @xmath144 case . for this reason , its generator set can only be simplified to ( @xmath145 ) , and as a result , the eigenstate expansions can be reduced to no less than @xmath78 terms in the standard basis . a general characterization of @xmath142 accompanies that of @xmath144 in the appendix , which then goes on to show that these , together with the four separable bases , exhaust all mub possibilities for four qubits . as a final example , i have found that a new type of basis , one that has no counterpart for qubits , is necessary for the existence of full mub complements when @xmath146 , for reasons that will become apparent . a generator set giving rise to such a basis is p^(4 ) = ( zxyw , xzwy , wyxz , ywzx ) , [ genp ] where standard definitions @xmath147 and @xmath148 are followed . the essential point is that the generators are tensor products of four noncommuting one - body matrices , which rules out qubits , but makes possible the elimination of 2-body operators from the compatibility groups for @xmath149 . less essential is that the four generators are related by pairwise permutations of operators ( hence the notation @xmath150 ) . the eigenstates have bell correlations between all pairs of particles , not just the chosen pairs as in @xmath136 states . so , unlike cluster or @xmath136 states , the entanglement is shared equally among all four particles , but unlike ghz states , the entanglement is robust . one can perform measurements on any two particles , in any two different bases , and produce a bell state of the other two . to show that the @xmath150 basis does not exhaust the possibilities for @xmath149 , we mention another generator set involving cyclic permutations , ( @xmath151 ) , where the @xmath152 factors are inserted for compatibility . the corresponding basis could play a role similar to that of @xmath150 in filling mub complements for @xmath153 , although it turns out to be relatively inconsequential when @xmath154 . in any case , since it would needlessly complicate the discussion without changing our conclusions , we exclude this example from the analysis . it is interesting to note in passing , that despite the differences in appearance among the generator sets of the four ( five ) totally entangled bases , the total numbers of @xmath90 factors appearing in their compatibility groups must be the same , namely 4(@xmath155 ) , in accordance with theorem ii . this has consequences for stoichiometry , in particular for the standard distributions , and justifies classifying @xmath136 bases as totally entangled . let us now turn to questions of stoichiometry . while in previous cases we were able to deduce the allowed entanglement patterns from global constraints alone ( those involving total numbers of pure and entangled qupits in mub complements ) , with @xmath156 this is no longer the case . the existence of multiple totally entangled basis types requires that we consider more microscopic constraints associated with the distributions of @xmath90 factors , as was done for qubits in . to this end , we define a quantity that is capable of distinguishing among all mub types under consideration . the `` @xmath25-body profile '' of a particular mub is the distribution of @xmath25-body operators ( @xmath157 , 2 , ... , @xmath1 ) in its compatibility group , where ( as implied earlier ) @xmath25-body operators are those with @xmath158 identity factors , @xmath90 . this distribution is normalized to the total number of operators in the compatibility group , @xmath81 . examples of @xmath25-body profiles are given in table ii , where we include the @xmath12 and 3 cases both for comparison with @xmath6 , and also to show how global information is recovered . the number of operators in each category , summed over all mubs , must equal the numbers listed at the bottom of each column . the latter represent the @xmath25-body profile of the set of all pauli operators , and are thus independent of the particular mub choices . they are determined by generating all of the pauli operators as expansions in the tensor products , ( i_1,z_1,x_1, ... ,x_1z_1^(p-1 ) ) ... ( i_n , z_n , x_n, ... ,x_nz_n^(p-1 ) ) . [ pauliik ] thus , the total number of @xmath25-body operators is @xmath159 , as shown . the condition that the mub sums equal these bottom lines , column by column , provides ( @xmath160 ) independent constraint equations . ( in the exceptional case of @xmath12 , both equations are independent . ) @xmath161 @xmath162 it is immediately apparent from all of the first columns that the maximum number of @xmath124 bases is always given by @xmath2 , the number that defines the standard mub complement . part ( a ) confirms that this is the only choice for @xmath12 , and its second column then determines the number of bell bases ( @xmath89 ) , in accordance with the required total number of mubs . part ( b ) reproduces all @xmath118 results , as were summarized on table i. the three columns provide three equations , but only two are linearly independent : the first column determines all possible combinations of @xmath124 and @xmath119 bases , and the second column then determines the number of @xmath135 bases , which is again consistent with the required total number of mubs , @xmath163 . the third column provides no further constraint . proceeding to the case of @xmath6 , the calculation of the @xmath25-body profiles for the separable bases is straightforward , since their compatibility groups are tensor products of those whose profiles have already been calculated . the new nonseparable bases require more thought . we found that the more symmetrical generator sets listed in eqs . [ geng]-[genp ] were helpful in working out the profiles for general @xmath4 . one can see by inspection of table ii(c ) that there is a qualitative difference between @xmath6 and the other cases . consider just the first 6 mub types , which represent all possibilities for @xmath7 . looking at the 3-body factors in column ( iii ) , we can see that as @xmath4 increases , the number of @xmath144 and/or @xmath143 mubs would have to increase as @xmath164 in order to satisfy just eq . but then they could not satisfy eq . ( ii ) , for they would produce too many two - body operators . clearly , one eventually needs a basis which , like @xmath150 , has no two - body operators . this need makes itself felt already with @xmath165 , and becomes urgent with @xmath166 . with these differences in mind , let us consider the @xmath7 , 3 and 5 cases sequentially , to show how the general picture evolves with increasing @xmath4 . * four qubits * @xmath167 the @xmath25-body profiles for @xmath7 are shown on table iii . to explore stoichiometries , consider the three equations ( i , ii , and iii ) represented by the first three columns , respectively . equation ( i ) , by itself , determines all possible combinations of the first three mub types , 4n()+2n(s^2b)+n(sg^(3 ) ) = 12 . [ n4p21 ] next , notice that we can isolate the @xmath136 and @xmath140 mubs because of their simple profiles . indeed , by simply adding ( i ) and ( iii ) we obtain the sum of all _ other _ mubs , n ( ) + n(s^2b ) + n(sg^(3 ) ) + n(c^(4 ) ) = 15 . [ n4p22 ] since there are 17 mubs in total we know immediately that n(bb ) + n(g^(4 ) ) = 2 , [ n4p23 ] a result which also follows from 2(ii ) + ( iii ) - ( i ) , which reproduces the total number . there are 16 ways to satisfy , with @xmath168 ) determined in each case by . for each of these combinations , there are 3 ways to satisfy , for a total of 48 possible mub distributions . to illustrate the range , a standard and a nonstandard distribution are shown in the two leftmost columns of table iv . these examples are chosen to show the maximum and minimum numbers of the new ( nonseparable ) @xmath143 mubs , which make up a majority of mubs in 30 of the 48 possible distributions . the dominance of @xmath143 mubs is related to the large number of 3-body operators in their profile . similar complements were found in through an explicit construction , except that the @xmath140 mubs were not produced , so that 16 combinations were obtained with 2 @xmath136 mubs present in all of them . @xmath169 * four qutrits * the @xmath25-body profiles for the @xmath165 case are shown in table v. again , the first column restricts the combinations of the first three mub types , 4n ( ) + 2n(s^2b ) + n(sg^(3 ) ) = 16 . [ n4p31 ] the first and second columns together [ ( ii)@xmath1703(i ) ] restrict other combinations , 4n(bb ) + 3n(g^(4 ) ) + n(c^(4 ) ) = 72 , [ n4p32 ] and the inclusion of the third column [ ( iii)@xmath1712(ii)@xmath1706(i ) ] yields the total mub count , n ( ) + ... + n(p^(4 ) ) = 82 . [ n4p33 ] @xmath172 there are 25 combinations of the first three mub types that satisfy , as compared with 16 such combinations in the qubit case . but , in the absence of @xmath150 mubs , one can not solve both for all of these combinations , and we find a total of only 11 mub distributions . to trace the reasons , we subtract from [ n4p33 ] and solve for @xmath150 : n(p^(4 ) ) = 10 + 3n(bb ) + 2n(g^(4 ) ) - [ n ( ) + n(ssb ) + n(sg^(3 ) ) ] . [ n4p34 ] without @xmath150 mubs the left side vanishes , and there can be solutions only if the quantity in square brackets is 10 or larger . this condition fails for the standard distribution , for which this number is @xmath173 . in this case , the minimum number of @xmath150 mubs is 6 , as shown on table iv . the other entry maximizes the quantity in square brackets at @xmath174 . in both entries we then minimize the number of @xmath150 mubs by maximizing the number of @xmath143 mubs . one can increase the number of @xmath150 mubs over these minima by adding @xmath136 and/or @xmath144 and subtracting @xmath143 mubs . although its numbers can be small , the @xmath150 mubs play a critical role in maintaining the balance of 3-body operators ( table v ) at no cost in two - body operators . with @xmath150 mubs included , the multiplicity of each of the 25 solutions of is large ( we estimate more than 200 ) , for a total of probably more than 5000 solutions . we can not argue that all of these solutions represent realizable mub distributions , because we can not rule out the possibility of more subtle constraints . such concerns are beyond the scope of the present paper . * four ququints * again consulting table v for the @xmath166 case , it is striking to see how three simple equations can again emerge from appropriate combinations . the first column gives us directly 4n ( ) + 2n(s^2b ) + n(sg^(3 ) ) = 24 , [ n4p51 ] the combination [ ( iii)@xmath1718(ii)@xmath17064(i ) ] relates the other four quantities , 8n(bb ) + 5n(g^(4 ) ) + 3n(c^(4 ) ) + 2n(4 ) = 1600 , [ n4p52 ] and still another combination [ ( iii)@xmath1712(ii)@xmath17022(i ) ] yields the total mub count , n ( ) + ... + n(p^(4 ) ) = 626 . [ n4p53 ] there are 49 combinations of the first three mub types that satisfy , but in the absence of the @xmath150 mubs , _ none _ of these admits solutions of . to see how this situation arises , solve the latter two equations for @xmath175 while eliminating @xmath176 : n(p^(4 ) ) = 278 + 5n(bb ) + 2n(g^(4 ) ) - 3[n()+n(sg^(3))+n(ssb ) ] . [ n4p54 ] the quantity in square brackets has minimum and maximum values of 6 ( the standard distribution ) and 24 , as shown on table v , corresponding to lower bounds on @xmath175 of 260 and 206 , respectively . the latter is the absolute minimum number of @xmath150 mubs in any full complement . again , one can add @xmath150 mubs by removing @xmath143 and adding @xmath136 and/or @xmath144 mubs , so that @xmath150 can be the majority mub type in some complements . while @xmath150 is critical for both @xmath165 and 5 cases , it plays a considerably more dominant role here . the underlying reason is that the ratio of the numbers of 3-body to 2-body operators increases considerably in going from @xmath165 to 5 , as shown in table v. we estimate the total number of solutions of eqs . [ n4p51]-[n4p53 ] to be in excess of @xmath177 , but again , we can not argue that all such solutions represent realizable mub distributions , or provide a revised estimate , without a further study of possible constraints . the examples of this section have shown us that with every step in @xmath1 , and with some steps in @xmath4 , full complements require not only those mub types generated from smaller systems , but also new , nonseparable mub types that exhibit new entanglement characteristics inaccessible to smaller systems . in the step from @xmath12 to 3 , @xmath178 mubs are required for the standard distribution , although a nonstandard distribution ( @xmath119 only ) is possible with @xmath7 . with the step to @xmath165 , no mub complement exists without @xmath178 . in the step to @xmath6 , the @xmath143 mubs are indespensible to all mub distributions with @xmath7 . with the step to @xmath165 , the new @xmath150 mubs become possible , and they in turn make possible the standard distribution . at @xmath166 , the @xmath150 mubs become indispensable to all distributions . projecting to larger systems , the distinguishing feature of the @xmath150 generator set is that a different ( noncommuting ) pauli matrix factor is associated with each qupit . the number of such factors in general is @xmath2 , and when this is equal to the number of qupits , a new type of entanglement becomes possible . thus we predict that when @xmath1 is equal to any prime plus 1 , then that prime ( @xmath179 ) is a critical value for the emergence of new entangled states as @xmath4 is increased at fixed n. these states should play critical roles in filling mub complements for @xmath4 equal to or slightly greater than @xmath180 . we have exploited the connections between mubs and pauli operators to develop a general framework for investigating both the entanglement properties of individual mubs , and the combinations of such mubs that can be found in full complements . we began by proving general theorems regarding mubs as eigenbases of pauli operators : we showed that the purities of individual qupits in such eigenbases must be either 0 or 1 , that the purity alone dictates the distribution of pauli matrix factors ( including @xmath90 ) in the compatibility groups of these mubs , and that every qupit must adopt these special purities the same number of times within any mub complement : ( @xmath2 ) times pure , and ( @xmath3 ) times totally entangled . an immediate corollary is that one may have at most @xmath2 product mubs in a full complement , and when one does , all remaining mubs must be totally entangled . this defines the standard distribution . armed with these theorems and the general properties of pauli operators , one quickly obtains more specific results : when @xmath12 , only product and generalized bell bases are possible , for any @xmath4 , and the standard mub distribution is inevitable . with @xmath118 , the unique totally entangled bases are generalized ghz bases , but a third mub type becomes possible , namely separable - bell bases . this makes possible @xmath5 distinct mub distributions . with @xmath6 and @xmath7 there are six mub types , including two nonseparable bases and a third ( @xmath136 ) that is separable but totally entangled . there are 48 possible mub distributions , with cluster bases making up the majority of mubs in most of these . with @xmath6 and larger @xmath4 , further mub types exist , and at least one such mub type ( @xmath150 ) is essential to forming a standard mub complement with @xmath165 , and to forming _ any _ mub complement with @xmath166 . several results have emerged in the course of working the above examples , and it seems useful to synthesize these in one place : ( 1 ) a mub can exist in any separation pattern . ( 2 ) all states in a particular mub have common separation and entanglement patterns - the generator set contains all information about the nature of the entanglement , while the eigenvalues specify the states . ( 3 ) compatibility groups of separable bases are tensor products of those of the nonseparable constituent bases . it follows that ( 4 ) within nonseparable groupings of qupits , those with two qupits must be in generalized bell states , those with three qupits - generalized ghz states . those with 4 qupits have the same broader array of options available to 4-qupit systems . perhaps the most important lesson to be drawn from the present examples is that , although it is easy to construct mubs from those found at lower @xmath1 , either as tensor products , or as larger-@xmath1 counterparts such as @xmath181 , the more interesting challenge is to find the new nonseparable mub types , with no counterparts at smaller @xmath1 ( or sometimes @xmath4 ) , that make full complements possible . it may be a general feature that such mubs tend to dominate mub distributions near the @xmath1 and @xmath4 values where they first emerge , only to be superceeded by other mub types as the system size increases . in this sense , every @xmath1 is critical , but not every @xmath4 . we predict that when @xmath1 is a prime plus 1 ( , @xmath182 , 8 , ... ) , there will be a critical value , @xmath183 , for the introduction of new entangled states that will play critical roles in mub distributions . closing thoughts on the existence question : the intimate connection between mubs and entanglement for @xmath184 highlights the way in which all known mub complements take advantage of the symmetry associated with equivalent parts , unique to dimensions @xmath42 @xcite . theorem i , which makes no reference to dimension , can not hold in ( at least some ) composite dimensions : in the simplest counterexample , 6 dimensions , the qubit can be totally entangled , but the qutrit can not be . yet , the import of theorem i is that for all known mub complements , there exists a factorization into parts ( represented by some set of generalized pauli operators ) , in terms of which theorem i ( and the others ) hold . thus , in addition to the existence question in composite dimensions , there is also an existence question in @xmath42 dimensions - do mub complements exist that violate theorem i ? perhaps entanglement considerations such as the present ones will help in answering these persistent questions . i would like to thank bill wootters and winton brown for many stimulating discussions , and i would like to thank the james franck institute for its hospitality during the time when this work was completed . to generalization with maximum transparency , we follow the alternative forms written in the text and define @xmath144 bases by the generator set 4 = ( abii , aici , aiid , stuv ) , [ ageng ] where every four - body factor must differ from its two - body counterpart ( @xmath185 , etc . ) . the two - body operators provide the special product basis for the @xmath4-term expansion . the generalization to @xmath1-qupit ghz states is apparent . the @xmath143 bases are best defined in a similar way , although a bit less transparently because there are only two independent two - body operators , 4 = ( aici , ibid , sbui , itcv ) . [ agenc ] again , the two - body operators provide a product basis for the expansion , which in this case requires @xmath78 terms . individual factors in the three - body operators must differ from corresponding factors in the two - body operators , except where their equality is explicit . it should be noted that there are two variations on the @xmath143 generator set , corresponding to the other ways of pairing the two - body factors . one such variation is ( @xmath186 ) . the three alternatives are mathematically equivalent , although one can make a physical distinction based on entanglement links between pairs . the stronger entanglement links in a system represented by are between neighbors in the sequence ( 1 - 2 - 3 - 4 - 1 ) . in the variation given above , the sequence is ( 1 - 3 - 2 - 4 - 1 ) , and in the other possible variation it is ( 1 - 2 - 4 - 3 - 1 ) . the various possibilities are not unphysical , as one can imagine unlike particles with tetrahedral coordination . we now argue that the two bases defined above are the only nonseparable options for qubits . we first argue that both are the unique nonseparable representatives of their respective @xmath25-body profiles as shown on table iii . we then show that other profiles can not exist for qubits . it is straightforward to verify that the @xmath144 generators produce six two - body operators involving the same factors , @xmath115 - @xmath187 . these exhaust all 12 of the available @xmath90 factors ( theorem ii demands three @xmath90 factors per qubit ) , so that all remaining operators are 4-body operators , as shown in the table . the only other basis that can share this profile is @xmath136 , in which the two - body operators have factors that do not commute individually . but , by virtue of this fact , the @xmath136 basis has four independent two - body operators , so that these can compose the generator set . the separability of the basis is then obvious . turning to the @xmath143 case , it is straightforward to show that the generators of produce no further two - body operators beyond the two shown , so that remaining @xmath90 factors must appear with the 8 three - body operators . one might wonder whether a different basis could be found with the same profile by using four - body generators in place of the three - body generators . the answer is no - it is easy show that this would generate only @xmath144 or @xmath136 bases , depending upon whether one of the four - body generators shares factors with one of the two - body generators . the remaining point is to rule out other four - qubit profiles . it suffices to consider just the two - body operators , whose maximum number is six . we will show that the numbers 4 and 0 are impossible for qubits . the former case is very simple - it is impossible to find four commuting two - body operators that do not generate two more ( and these will immediately identify themselves as belonging to either a @xmath144 or @xmath136 compatibility group ) . as to the latter case , assume that there are no two - body operators . then all 12 of the @xmath90 factors must appear in one - body operators , making three appearances on each qubit ( theorem ii ) . consider any two of these operators that have their @xmath90 factors on the same qubit . commutativity demands that they have exactly one other factor in common , so that their product is a two - body operator . this forms a contradiction and shows that there is no profile without two - body operators . to briefly summarize the results of this appendix , the 6 mub types listed on table iii exhaust the possibilities for four qubits . the 5 corresponding @xmath1-body profiles are also exhaustive ; in particular , a @xmath150-like profile does not exist for qubits . 99 i. ivanovi , j. phys . a * 14 * , 3241 ( 1981 ) . w.k . wootters and b.d . fields , ann . phys . * 191 * , 363 ( 1989 ) ; and in _ bell s theorem , quantum theory and conceptions of the universe _ , edited by m. kafatos ( kluwer academic publishers , dordrecht , 1989 ) , pp . bennett and g. brassard , in _ proc . conf . on computers , systems , and signal processing _ , bangalore , india ( 1984 ) , p. 175 . d. bruss and c. macchiavello , phys . letters * 88 * , 127901 ( 2002 ) . ekert , phys . letters , * 67 * , 661 ( 1991 ) . s. grblacher et . al . , new j. phys . * 8 * , 75 ( 2006 ) . aravind , z. naturforsch . * 58a * , 85 ( 2003 ) . p. shor , phys . a * 52 * , r2493 ( 1995 ) . d. gottesmanm , phys . a * 54 * , 1862 ( 1996 ) . r. calderbank , e.m . rains , p.w . shor , and n.j.a.sloane , phys , rev . letters , * 78 * , 405 ( 1997 ) . p. hyllus , o. ghne , d. bruss , and m. lewenstein , phys . rev . * 72 * , 012321 ( 2005 ) . brukner and a. zeilinger , phys . a * 63 * , 022113 ( 2001 ) . j. lawrence , . brukner , and a. zeilinger , phys . a * 65 * , 032320 ( 2002 ) . t. durt , b .- englert , i. bengtsson , and k. zyczkowski , int * 8 * , 535 ( 2010 ) . s. brierley and s. weigert , phys . a * 78 * , 042312 ( 2008 ) . j. lawrence , phys . a * 70 * , 012302 ( 2004 ) . romero , g. bjrk , a.b . klimov , and l.l . snchez - soto , phys . a * 72 * , 062310 ( 2005 ) . m. wieniak , t. paterek , and a. zeilinger , e - print arxiv : quant - ph/1102.2080v2 . briegel and r. raussendorf , phys . letters , * 86 * , 910 ( 2001 ) . a.b . klimov , c. munoz , and l.l . snchez - soto , eprint quant - ph/1007.1751 ( 2010 ) . see p. 90 of . s. bandyopadhyay , p.o . boykin , v. roychowdhury , and f. vatan , algorithmica * 34 * , 512 ( 2002 ) ; or e - print quant - ph/0103162 ( 2001 ) . nielsen and i.l . chuang , _ quantum computation and quantum information _ ( cambridge univ . press , cambridge , england , 2000 ) , p. 454 . one may think of this as an @xmath1-dimensional discrete fourier transform with @xmath4 points in each dimension , or as a fourier transform over a galois field , with @xmath25 and @xmath47 as galois field variables . boykin , m. sitharam , p.h . tiep , and p. wojcan , quant . * 7 * , 371 ( 2007 ) . the one - qupit pauli groups may be written @xmath188 for odd @xmath4 , and @xmath189 for @xmath7 , where @xmath66 represents all of the @xmath78 pauli matrix factors . these are special cases of the discussion following . this result was derived in another way in . a one - body pauli operator transforms only a single qupit ( for example , @xmath190 in a 2-qupit system and @xmath191 in a 3-qupit system ) . such an operator can have no totally entangled eigenstates , because in any eigenstate the transformed qupit must be pure . a.b . klimov , d. sych , l.l . snchez - soto , and g. leuchs , phys . rev . a * 79 * , 052101 ( 2009 ) . t. durt , e - print arxiv : quant - ph/0401046 ; also see discussion on pp . 43 - 46 of . mermin , phys . letters , * 65 * , 1838 ( 1990 ) . briegel and r. raussendorf , phys . letters , * 86 * , 5188 ( 2001 ) . briegel , d.e . browne , w. dr , r. raussendorf , and m. van den nest , nature physics * 5 * , 19 ( 2009 ) . this symmetry is referred to in the concluding paragraph ( p. 91 ) of .
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a few simply - stated rules govern the entanglement patterns that can occur in mutually unbiased basis sets ( mubs ) , and constrain the combinations of such patterns that can coexist ( , the stoichiometry ) in full complements of ( @xmath0 ) mubs .
we consider hilbert spaces of prime power dimension ( as realized by systems of @xmath1 prime - state particles , or _ qupits _ ) , where full complements are known to exist , and we assume only that mubs are eigenbases of generalized pauli operators , without using a particular construction .
the general rules include the following : 1 ) in any mub , a particular qupit appears either in a pure state , or totally entangled , and 2 ) in any full mub complement , each qupit is pure in ( @xmath2 ) bases ( not necessarily the same ones ) , and totally entangled in the remaining ( @xmath3 ) .
it follows that the maximum number of product bases is @xmath2 , and when this number is realized , all remaining ( @xmath3 ) bases in the complement are characterized by the total entanglement of every qupit .
this `` standard distribution '' is inescapable for two qupits ( of any @xmath4 ) , where only product and generalized bell bases are admissible mub types .
this and the following results generalize previous results for qubits @xcite and qutrits @xcite , drawing particularly upon . with three qupits
there are three mub types , and a number of combinations ( @xmath5 ) are possible in full complements . with @xmath6 , there are 6 mub types for @xmath7 , but new mub types become possible with larger @xmath4 , and these are essential to realizing full complements . with this example , we argue that new mub types , showing new entanglement characteristics , should enter with every step in @xmath1 , and when @xmath1 is a prime plus 1 , also at critical @xmath4 values , @xmath8 .
such mubs should play critical roles in filling complements .
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being able to read news from other countries and written in other languages allows readers to be better informed . it allows them to detect national news bias and thus improves transparency and democracy . existing online translation systems such as _ google translate _ and _ _ bing translator _ _ are thus a great service , but the number of documents that can be submitted is restricted ( google will even entirely stop their service in 2012 ) and submitting documents means disclosing the users interests and their ( possibly sensitive ) data to the service - providing company . for these reasons , we have developed our in - house machine translation system onts . its translation results will be publicly accessible as part of the europe media monitor family of applications , @xcite , which gather and process about 100,000 news articles per day in about fifty languages . onts is based on the open source phrase - based statistical machine translation toolkit moses @xcite , trained mostly on freely available parallel corpora and optimised for the news domain , as stated above . the main objective of developing our in - house system is thus not to improve translation quality over the existing services ( this would be beyond our possibilities ) , but to offer our users a rough translation ( a `` gist '' ) that allows them to get an idea of the main contents of the article and to determine whether the news item at hand is relevant for their field of interest or not . a similar news - focused translation service is `` found in translation '' @xcite , which gathers articles in 23 languages and translates them into english . `` found in translation '' is also based on moses , but it categorises the news after translation and the translation process is not optimised for the news domain . europe media monitor ( emm ) gathers a daily average of 100,000 news articles in approximately 50 languages , from about 3,400 hand - selected web news sources , from a couple of hundred specialist and government websites , as well as from about twenty commercial news providers . it visits the news web sites up to every five minutes to search for the latest articles . when news sites offer rss feeds , it makes use of these , otherwise it extracts the news text from the often complex html pages . all news items are converted to unicode . they are processed in a pipeline structure , where each module adds additional information . independently of how files are written , the system uses utf-8-encoded rss format . inside the pipeline , different algorithms are implemented to produce monolingual and multilingual clusters and to extract various types of information such as named entities , quotations , categories and more . onts uses two modules of emm : the named entity recognition and the categorization parts . named entity recognition ( ner ) is performed using manually constructed language - independent rules that make use of language - specific lists of trigger words such as titles ( president ) , professions or occupations ( tennis player , playboy ) , references to countries , regions , ethnic or religious groups ( french , bavarian , berber , muslim ) , age expressions ( 57-year - old ) , verbal phrases ( deceased ) , modifiers ( former ) and more . these patterns can also occur in combination and patterns can be nested to capture more complex titles , @xcite . in order to be able to cover many different languages , no other dictionaries and no parsers or part - of - speech taggers are used . to identify which of the names newly found every day are new entities and which ones are merely variant spellings of entities already contained in the database , we apply a language - independent name similarity measure to decide which name variants should be automatically merged , for details see @xcite . this allows us to maintain a database containing over 1,15 million named entities and 200,000 variants . the major part of this resource can be downloaded from http://langtech.jrc.it/jrc-names.html all news items are categorized into hundreds of categories . category definitions are multilingual , created by humans and they include geographic regions such as each country of the world , organizations , themes such as natural disasters or security , and more specific classes such as earthquake , terrorism or tuberculosis , articles fall into a given category if they satisfy the category definition , which consists of boolean operators with optional vicinity operators and wild cards . alternatively , cumulative positive or negative weights and a threshold can be used . uppercase letters in the category definition only match uppercase words , while lowercase words in the definition match both uppercase and lowercase words . many categories are defined with input from the users themselves . this method to categorize the articles is rather simple and user - friendly , and it lends itself to dealing with many languages , @xcite . in this section , we describe our statistical machine translation ( smt ) service based on the open - source toolkit moses @xcite and its adaptation to translation of news items . * which is the most suitable smt system for our requirements ? * the main goal of our system is to help the user understand the content of an article . this means that a translated article is evaluated positively even if it is not perfect in the target language . dealing with such a large number of source languages and articles per day , our system should take into account the translation speed , and try to avoid using language - dependent tools such as part - of - speech taggers . inside the moses toolkit , three different statistical approaches have been implemented : _ phrase based statistical machine translation _ ( pbsmt ) @xcite , _ hierarchical phrase based statistical machine translation _ @xcite and _ syntax - based statistical machine translation _ @xcite . to identify the most suitable system for our requirements , we run a set of experiments training the three models with europarl v4 german - english @xcite and optimizing and testing on the news corpus @xcite . for all of them , we use their default configurations and they are run under the same condition on the same machine to better evaluate translation time . for the syntax model we use linguistic information only on the target side . according to our experiments , in terms of performance the hierarchical model performs better than pbsmt and syntax ( 18.31 , 18.09 , 17.62 bleu points ) , but in terms of translation speed pbsmt is better than hierarchical and syntax ( 1.02 , 4.5 , 49 second per sentence ) . although , the hierarchical model has the best bleu score , we prefer to use the pbsmt system in our translation service , because it is four times faster . * which training data can we use ? * it is known in statistical machine translation that more training data implies better translation . although , the number of parallel corpora has been is growing in the last years , the amounts of training data vary from language pair to language pair . to train our models we use the freely available corpora ( when possible ) : europarl @xcite , jrc - acquis @xcite , dgt - tm , opus @xcite , se - times @xcite , tehran english - persian parallel corpus @xcite , news corpus @xcite , un corpus @xcite , czeng0.9 @xcite , english - persian parallel corpus distributed by elra and two arabic - english datasets distributed by ldc . this results in some language pairs with a large coverage , ( more than 4 million sentences ) , and other with a very small coverage , ( less than 1 million ) . the language models are trained using 12 model sentences for the content model and 4.7 million for the title model . both sets are extracted from english news . for less resourced languages such as farsi and turkish , we tried to extend the available corpora . for farsi , we applied the methodology proposed by @xcite , where we used a large language model and an english - farsi smt model to produce new sentence pairs . for turkish we added the movie subtitles corpus @xcite , which allowed the smt system to increase its translation capability , but included several slang words and spoken phrases . * how to deal with named entities in translation ? * news articles are related to the most important events . these names need to be efficiently translated to correctly understand the content of an article . from an smt point of view , two main issues are related to named entity translation : ( 1 ) such a name is not in the training data or ( 2 ) part of the name is a common word in the target language and it is wrongly translated , e.g. the french name `` bruno le maire '' which risks to be translated into english as `` bruno mayor '' . to mitigate both the effects we use our multilingual named entity database . in the source language , each news item is analysed to identify possible entities ; if an entity is recognised , its correct translation into english is retrieved from the database , and suggested to the smt system enriching the source sentence using the xml markup option in moses . this approach allows us to complement the training data increasing the translation capability of our system . * how to deal with different language styles in the news ? news title writing style contains more gerund verbs , no or few linking verbs , prepositions and adverbs than normal sentences , while content sentences include more preposition , adverbs and different verbal tenses . starting from this assumption , we investigated if this phenomenon can affect the translation performance of our system . we trained two smt systems , @xmath0 and @xmath1 , using the europarl v4 german - english data as training corpus , and two different development sets : one made of content sentences , news commentaries @xcite , and the other made of news titles in the source language which were translated into english using a commercial translation system . with the same strategy we generated also a title test set . the @xmath1 used a language model created using only english news titles . the news and title test sets were translated by both the systems . although the performance obtained translating the news and title corpora are not comparable , we were interested in analysing how the same test set is translated by the two systems . we noticed that translating a test set with a system that was optimized with the same type of data resulted in almost 2 blue score improvements : title - testset : 0.3706 ( @xmath1 ) , 0.3511 ( @xmath0 ) ; news - testset : 0.1768 ( @xmath1 ) , 0.1945 ( @xmath0 ) . this behaviour was present also in different language pairs . according to these results we decided to use two different translation systems for each language pair , one optimized using title data and the other using normal content sentences . even though this implementation choice requires more computational power to run in memory two moses servers , it allows us to mitigate the workload of each single instance reducing translation time of each single article and to improve translation quality . to evaluate the translation performance of onts , we run a set of experiments where we translate a test set for each language pair using our system and google translate . lack of human translated parallel titles obliges us to test only the content based model . for german , spanish and czech we use the news test sets proposed in @xcite , for french and italian the news test sets presented in @xcite , for arabic , farsi and turkish , sets of 2,000 news sentences extracted from the arabic - english and english - persian datasets and the se - times corpus . for the other languages we use 2,000 sentences which are not news but a mixture of jrc - acquis , europarl and dgt - tm data . it is not guarantee that our test sets are not part of the training data of google translate . each test set is translated by google translate - translator toolkit , and by our system . bleu score is used to evaluate the performance of both systems . results , see table [ results ] , show that google translate produces better translation for those languages for which large amounts of data are available such as french , german , italian and spanish . surprisingly , for danish , portuguese and polish , onts has better performance , this depends on the choice of the test sets which are not made of news data but of data that is fairly homogeneous in terms of style and genre with the training sets . the impact of the named entity module is evident for arabic and farsi , where each english suggested entity results in a larger coverage of the source language and better translations . for highly inflected and agglutinative languages such as turkish , the output proposed by onts is poor . we are working on gathering more training data coming from the news domain and on the possibility of applying a linguistic pre - processing of the documents . .[results ] automatic evaluation . [ cols="<,^,^ " , ] the translation service is made of two components : the connection module and the moses server . the connection module is a servlet implemented in java . it receives the rss files , isolates each single news article , identifies each source language and pre - processes it . each news item is split into sentences , each sentence is tokenized , lowercased , passed through a statistical compound word splitter , @xcite , and the named entity annotator module . for language modelling we use the kenlm implementation , @xcite . according to the language , the correct moses servers , title and content , are fed in a multi - thread manner . we use the multi - thread version of moses @xcite . when all the sentences of each article are translated , the inverse process is run : they are detokenized , recased , and untranslated / unknown words are listed . the translated title and content of each article are uploaded into the rss file and it is passed to the next modules . the full system including the translation modules is running in a 2xquad - core with intel hyper - threading technology processors with 48 gb of memory . it is our intention to locate the moses servers on different machines . this is possible thanks to the high modularity and customization of the connection module . at the moment , the translation models are available for the following source languages : arabic , czech , danish , farsi , french , german , italian , polish , portuguese , spanish and turkish . our translation service is currently presented on a demo web site , see figure [ fig::demo ] , which is available at http://optima.jrc.it / translate/. news articles can be retrieved selecting one of the topics and the language . all the topics are assigned to each article using the methodology described in [ cat ] . these articles are shown in the left column of the interface . when the button `` translate '' is pressed , the translation process starts and the translated articles appear in the right column of the page . the translation system can be customized from the interface enabling or disabling the named entity , compound , recaser , detokenizer and unknown word modules . each translated article is enriched showing the translation time in milliseconds per character and , if enabled , the list of unknown words . the interface is linked to the connection module and data is transferred using rss structure . in this paper we present the optima news translation system and how it is connected to europe media monitor application . different strategies are applied to increase the translation performance taking advantage of the document structure and other resources available in our research group . we believe that the experiments described in this work can result very useful for the development of other similar systems . translations produced by our system will soon be available as part of the main emm applications . the performance of our system is encouraging , but not as good as the performance of web services such as google translate , mostly because we use less training data and we have reduced computational power . on the other hand , our in - house system can be fed with a large number of articles per day and sensitive data without including third parties in the translation process . performance and translation time vary according to the number and complexity of sentences and language pairs . the domain of news articles dynamically changes according to the main events in the world , while existing parallel data is static and usually associated to governmental domains . it is our intention to investigate how to adapt our translation system updating the language model with the english articles of the day . the authors thank the jrc s optima team for its support during the development of onts . c. callison - burch , and p. koehn and c. monz and k. peterson and m. przybocki and o. zaidan . 2009 . . proceedings of the joint fifth workshop on statistical machine translation and metricsmatr , pages 1753 . uppsala , sweden . p. koehn and f. j. och and d. marcu . proceedings of the 2003 conference of the north american chapter of the association for computational linguistics on human language technology , pages 4854 . edmonton , canada . p. koehn and h. hoang and a. birch and c. callison - burch and m. federico and n. bertoldi and b. cowan and w. shen and c. moran and r. zens and c. dyer and o. bojar and a. constantin and e. herbst 2007 . . proceedings of the annual meeting of the association for computational linguistics , demonstration session , pages 177180 . columbus , oh , usa . r. steinberger and b. pouliquen and a. widiger and c. ignat and t. erjavec and d. tufi and d. varga . proceedings of the 5th international conference on language resources and evaluation , pages 21422147 . genova , italy . m. turchi and i. flaounas and o. ali and t. debie and t. snowsill and n. cristianini . proceedings of the european conference on machine learning and knowledge discovery in databases , pages 746749 . bled , slovenia .
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we propose a real - time machine translation system that allows users to select a news category and to translate the related live news articles from arabic , czech , danish , farsi , french , german , italian , polish , portuguese , spanish and turkish into english .
the moses - based system was optimised for the news domain and differs from other available systems in four ways : ( 1 ) news items are automatically categorised on the source side , before translation ; ( 2 ) named entity translation is optimised by recognising and extracting them on the source side and by re - inserting their translation in the target language , making use of a separate entity repository ; ( 3 ) news titles are translated with a separate translation system which is optimised for the specific style of news titles ; ( 4 ) the system was optimised for speed in order to cope with the large volume of daily news articles .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
video compression is a major requirement in many of the recent applications like medical imaging , studio applications and broadcasting applications . compression ratio of the encoder completely depends on the underlying compression algorithms . the goal of compression techniques is to reduce the immense amount of visual information to a manageable size so that it can be efficiently stored , transmitted , and displayed . 3-d dwt based compressing system enables the compression in spatial as well as temporal direction which is more suitable for video compression . moreover , wavelet based compression provide the scalability with the levels of decomposition . due to continuous increase in size of the video frames ( hd to uhd ) , video processing through software coding tools is more complex . dedicated hardware only can give higher performance for high resolution video processing . in this scenario there is a strong requirement to implement a vlsi architecture for efficient 3-d dwt processor , which consumes less power , area efficient , memory efficient and should operate with a higher frequency to use in real - time applications . + from the last two decades , several hardware designs have been noted for implementation of 2-d dwt and 3-d dwt for different applications . majority of the designs are developed based on three categories , viz . ( i ) convolution based ( ii ) lifting - based and ( iii ) b - spline based . most of the existing architectures are facing the difficulty with larger memory requirement , lower throughput , and complex control circuit . in general the circuit complexity is denoted by two major components viz , arithmetic and memory component . arithmetic component includes adders and multipliers , whereas memory component consists of temporal memory and transpose memory . complexity of the arithmetic components is fully depends on the dwt filter length . in contrast size of the memory component is depends on dimensions of the image . as image resolutions are continuously increasing ( hd to uhd ) , image dimensions are very high compared to filter length of the dwt , as a result complexity of the memory component occupied major share in the overall complexity of dwt architecture . + convolution based implementations @xcite-@xcite provides the outputs within less time but require high amount of arithmetic resources , memory intensive and occupy larger area to implement . lifting based a implementations requires less memory , less arithmetic complex and possibility to implement in parallel . however it require long critical path , recently huge number of contributions are noted to reduce the critical path in lifting based implementations . for a general lifting based structure @xcite provides critical path of @xmath0 , by introducing 4 stage pipeline it cut down to @xmath1 . in @xcite huang et al . , introduced a flipping structure it further reduced the critical path to @xmath2 . though , it reduced the critical path delay in lifting based implementation , it requires to improve the memory efficiency . majority of the designs implement the 2-d dwt , first by applying 1-d dwt in row - wise and then apply 1-d dwt in column wise . it require huge amount of memory to store these intermediate coefficients . to reduce this memory requirements , several dwt architecture have been proposed by using line based scanning methods @xcite-@xcite . huang et al . , @xcite-@xcite give brief details of b - spline based 2-d idwt implementation and discussed the memory requirements for different scan techniques and also proposed a efficient overlapped strip - based scanning to reduce the internal memory size . several parallel architectures were proposed for lifting - based 2-d dwt @xcite-@xcite . y. hu et al . @xcite , proposed a modified strip based scanning and parallel architecture for 2-d dwt is the best memory - efficient design among the existing 2-d dwt architectures , it requires only 3n + 24p of on chip memory for a n@xmath3n image with @xmath4 parallel processing units ( pu ) . several lifting based 3-d dwt architectures are noted in the literature @xcite-@xcite to reduce the critical path of the 1-d dwt architecture and to decrease the memory requirement of the 3-d architecture . among the best existing designs of 3-d dwt , darji et al . @xcite produced best results by reducing the memory requirements and gives the throughput of 4 results / cycle . still it requires the large on - chip memory ( @xmath5 ) . in this paper , we propose a new parallel and memory efficient lifting based 3-d dwt architecture , requires only @xmath6 words of on - chip memory and produce 8 results / cycle . the proposed 3-d dwt architecture is built with two spatial 2-d dwt ( cdf 9/7 ) processors and four temporal 1-d dwt ( haar ) processors . proposed architecture for 3-d dwt replaced the multiplication operations by shift and add , it reduce the cpd from @xmath7 to @xmath8 . further reduction of cpd to @xmath9 is done by introducing pipeline in the processing elements . to eliminate the temporal memory and to reduce the latency , haar wavelet is incorporated in temporal processor . the resultant architecture has reduce the latency , on chip memory and to increase the speed of operation compared to existing 3-d dwt designs . the following sections provide the architectural details of proposed 3-d dwt through spatial and temporal processors . organization of the paper as follows . theoretical background for dwt is given in section ii . detailed description of the proposed architecture for 3-d dwt is provided in section iii . implementation results and performance comparison is given in section iv . finally , concluding remarks are given in section v. + lifting based wavelet transform designed by using a series of matrix decomposition specified by the daubechies and sweledens in @xcite . by applying the flipping @xcite to the lifting scheme , the multipliers in the longest delay path are eliminated , resulting in a shorter critical path . the original data on which dwt is applied is denoted by @xmath10 $ ] , and the 1-d dwt outputs are the detail coefficients @xmath11 $ ] and approximation coefficients @xmath12 $ ] . for the image ( 2-d ) above process is performed in rows and columns as well . eqns.(1)-(6 ) are the design equations for flipping based lifting ( 9/7 ) 1-d dwt @xcite and the same equations are used to implement the proposed row processor ( 1-d dwt ) and column processor ( 1-d dwt ) . [ lift_2d_eq ] @xmath13 & \leftarrow a'*x[2n-1]+\{x[2n]+x[2n-2]\ } \ldots p1\\ { l_1}[n]&\leftarrow b'*x[2n]+\{{h_1}[n]+{h_1}[n-1]\ } \ldots u1\\ { h_2}[n ] & \leftarrow c'*{h_1}[n]+\{{l_1}[n]+{l_1}[n-1]\}\ldots p2\\ { l_2}[n ] & \leftarrow d'*{l_1}[n]+\{{h_2}[n]+{h_2}[n-1]\}\ldots u2\\ h[n ] & \leftarrow k0 * \{{h_2}[n]\}\\ l[n ] & \leftarrow k1 * \{{l_2}[n]\}\end{aligned}\ ] ] where @xmath14 , @xmath15 , @xmath16 , @xmath17 , @xmath18 , and @xmath19 @xcite . the lifting step coefficients @xmath20 , @xmath21 , @xmath22 , @xmath23 and scaling coefficient @xmath24 are constants and its values @xmath25 , @xmath26 , @xmath27 , and @xmath28 , and @xmath29 lifting based wavelets are always memory efficient and easy to implement in hardware . the lifting scheme consists of three steps to decompose the samples , namely , splitting , predicting ( eqn . ( 1 ) and ( 3 ) ) , and updating ( eqn . ( 2 ) and ( 4 ) ) . haar wavelet transform is orthogonal and simple to construct and provide fast output . by considering the advantages of the haar wavelets , the proposed architecture uses the haar wavelet to perform the 1-d dwt in temporal direction ( between two adjacent frames ) . @xcite developed a lifting based haar wavelet . the equations of the lifting scheme for the haar wavelet transform is as shown in eqn.([eq2 ] ) @xmath30 = \left ( { \begin{array}{*{20}{c } } { \sqrt 2 } & 0\\ 0&{\frac{1}{{\sqrt 2 } } } \end{array } } \right)\left ( { \begin{array}{*{20}{c } } 1&{s(z)}\\ 0&1 \end{array } } \right)\left ( { \begin{array}{*{20}{c } } 1&0\\ { - p(z)}&1 \end{array } } \right)\left ( \begin{array}{l } { x_0}(z)\\ { x_1}(z ) \end{array } \right)\ ] ] @xmath31 eqn.([eq3 ] ) is extracted by substituting predict value @xmath32 as 1 and update step @xmath33 value as 1/2 in eqn.([eq2 ] ) , which is used to develop the temporal processor to apply 1-d dwt in temporal direction ( @xmath34 dimension ) . where l and h are the low and high frequency coefficients respectively . the proposed architecture for 3-d dwt comprising of two parallel spatial processors ( 2-d dwt ) and four temporal processors ( 1-d dwt ) , is depicted in fig . [ blockdia_1 ] . after applying 2-d dwt on two consecutive frames , each spatial processor ( sp ) produces 4 sub - bands , viz . ll , hl , lh and hh and are fed to the inputs of four temporal processors ( tps ) to perform the temporal transform . output of these tps is a low frequency frame ( l - frame ) and a high frequency frame ( h - frame ) . architectural details of the spatial processor and temporal processors are discussed in the following sections . in this section , we propose a new parallel and memory efficient lifting based 2-d dwt architecture denoted by spatial processor ( sp ) and it consists of row and column processors . the proposed sp is a revised version of the architecture developed by the y. hu et al.@xcite . the proposed architecture utilizes the strip based scanning @xcite to enable the trade - off between external memory and internal memory . to reduce the critical path in each stage flipping model @xcite-@xcite is used to develop the processing element ( pe ) . each pe has been developed with shift and add techniques in place of multiplier . lifting based ( 9/7 ) 1-d dwt process has been performed by the processing unit ( pu ) in the proposed architecture . as shown in fig . [ 3d_2 ] , the proposed pu is designed with five pes , and each pe ( except first pe ( shift@xmath35pe ) ) has been constructed with two pipeline stages for further reduction of cpd . this modified pu , reduces the cpd to @xmath36 ( adder delay ) . fig . [ blockdia_1 ] shows that the number of inputs to the spatial processor is equal to 2p+1 , which is also equal to the width of the strip . where p is the number of parallel processing units ( pus ) in the row processor as well as column processor . we have designed the proposed architecture with two parallel processing units ( p = 2 ) . the same structure can be extended to p = 4 , 8 , 16 or 32 depending on external bandwidth . whenever row processor produces the intermediate results , immediately column processor start to process on those intermediate results . row processor takes 9 clocks to produce the temporary results then after column processor takes 9 more clocks to to give the 2-d dwt output ; finally , temporal processor takes 3 more clocks after 2-d dwt results are available to produce 3-d dwt output . as a summary , proposed 2-d dwt and 3-d dwt architectures have constant latency of 18 and 21 clock cycles respectively , regardless of image size n and number of parallel pus ( p ) . details of the row processor and column processor are given in the following sub - sections . let @xmath37 be the image of size @xmath38 , extend this image by one column by using symmetric extension . now image size is @xmath39 . refer @xcite for the structure of strip based scanning method . the proposed architecture initiates the dwt process in row wise through row processor ( rp ) then process the column dwt by column processor ( cp ) . [ 3d_1](a ) . shows the generalized structure for a row processor with @xmath40 number of pus . @xmath41 has been considered for our proposed design . for the first clock cycle , rp get the pixels from @xmath42 to @xmath43 simultaneously . for the second clock rp gets the pixels from next row i.e. @xmath44 to @xmath45 , the same procedure continues for each clock till it reaches the bottom row i.e. , @xmath46 to @xmath47 . then it goes to the next strip and rp get the pixels from @xmath43 to @xmath48 and it continues this procedure for entire image . each pu consists of five pipeline stages and each pipeline stage is processed by one processing element ( pe ) as depicted in fig . [ 3d_2](b ) . first stage ( shift@xmath35pe ) provide the partial results which is required at @xmath49 stage ( pe@xmath35alpha ) , likewise processing elements pe@xmath35alpha to pe@xmath35delta ( @xmath49 stage to @xmath50 stage ) gives the partial results along with their original outputs . ( e.g. , consider the pe@xmath35alpha of pu-1 , it needs to provide output corresponding to eqn.(1 ) ( @xmath51 $ ] ) , along with @xmath52 $ ] , it also provides the partial output @xmath53 $ ] which is required for the pe@xmath35beta ) . structure of the pes are given in the fig . [ 3d_2](b ) , it shows that multiplication is replaced with the shift and add technique . the original multiplication factor and the value through the shift and add circuit are noted in table.[tab1 ] , it shows that variation between original and adopted one is extremely small . as shown in fig . [ 3d_2](b ) , time delay of shift@xmath35pe is one @xmath9 and remaining all pes are having delay of @xmath54 . to reduce the cpd of pu , pes from pe@xmath35alpha to pe@xmath35delta are divided in to two pipeline stages , and each pipeline stage has a delay of @xmath9 , as a result cpd of pu is reduced to @xmath9 and pipeline stages are increased to nine and is shown in fig . [ 3d_2](c ) . the outputs @xmath55 $ ] , @xmath56 $ ] , and @xmath57 $ ] corresponding to pe@xmath35alpha and pe@xmath35beta of last pu and pe@xmath35gama of last pu is saved in the memories memory@xmath35alpha , memory@xmath35beta and memory@xmath35gama respectively , shown in fig . [ 3d_1](a ) . those stored outputs are inputted for next subsequent columns of the same row . for a @xmath58 image rows is equivalent to @xmath59 . so the size of the each memory is @xmath60 words and total row memory to store these outputs is equals to @xmath61 . output of each pu are under gone through a process of scaling before it producing the outputs h and l. these outputs are fed to the transposing unit . the transpose unit has @xmath4 number of transpose registers ( one for each pu ) . [ 3d_3](a ) shows the structure of transpose register , and it gives the two h and two l data alternatively to the column processor . the structure of the column processor ( cp ) is shown in fig . [ 3d_1](b ) . to match with the throughput of rp , cp is also designed with two number of pus in our architecture . each transpose register produces a pair of h and l in an alternative order and are fed to the inputs of one pu of the cp . the partial results produced are consumed by the next pe after two clock cycles . as such , shift registers of length two are needed within the cp between each pipeline stages for caching the partial results ( except between @xmath62 and @xmath63 pipeline stages ) . at the output of the cp , four sub - bands are generated in an interleaved pattern , @xmath64 and so on . outputs of the cp are fed to the re - arrange unit . [ 3d_3](b ) shows the architecture for re - arrange unit , and it provides the outputs in sub - band order @xmath65 and @xmath66 simultaneously , by using @xmath4 registers and @xmath67 multiplexers . for multilevel decomposition , the same dwt core can be used in a folded architecture with an external frame buffer for the ll sub - band coefficients . .original and adopted values for multiplication [ cols= " < , < , < " , ] the proposed 3-d dwt architecture has been described in verilog hdl . a uniform word length of 14 bits has been maintained throughout the design . simulation results have been verified by using xilinx ise simulator . we have simulated the matlab model which is similar to the proposed 3-d dwt hardware architecture and verified the 3-d dwt coefficients . rtl simulation results have been found to exactly match the matlab simulation results . the verilog rtl code is synthesized using xilinx ise 14.2 tool and mapped to a xilinx programmable device ( fpga ) 7z020clg484 ( zynq board ) with speed grade of -3 . table [ fpga_results ] shows the device utilization summary of the proposed architecture and it operates with a maximum frequency of 265 mhz . the proposed architecture has also been synthesized using synopsys design compiler with 90-nm technology cmos standard cell library . it consumes 43.42 mw power and occupies an area equivalent to 231.45 k equivalent gate at frequency of 200 mhz . the performance comparison of the proposed 2-d and 3-d dwt architectures with other existing architectures is figure out in tables [ 2dcompare ] and [ 3dcompare ] respectively . the proposed 2-d processor requires zero multipliers , 34p ( pis number of parallel pus ) adders , 60p+3n internal memory . it has a critical path delay of @xmath9 with a throughput of four outputs per cycle with @xmath68/2p computation cycles to process an image with size @xmath38 . when compared to recent 2-d dwt architecture developed by the y.hu et al . @xcite , cpd reduced to @xmath9 from @xmath69 with the cost of small increase in hardware resources . table [ 3dcompare ] shows the comparison of proposed 3-d dwt architecture with existing 3-d dwt architecture . it is found that , the proposed design has less memory requirement , high throughput , less computation time and minimal latency compared to @xcite , @xcite , @xcite , and @xcite . though the proposed 3-d dwt architecture has small disadvantage in area and frequency , when compared to @xcite , the proposed one has a great advantage in remaining all aspects . table [ 3d_asic ] gives the comparison of synthesis results between the proposed 3-d dwt architecture and @xcite . it seems to be proposed one occupying more cell area , but it included total on chip memory also , where as in @xcite on chip memory is not included . power consumption of the proposed 3-d architecture is very less compared to @xcite . in this paper , we have proposed memory efficient and high throughput architecture for lifting based 3-d dwt . the proposed architecture is implemented on 7z020clg484 fpga target of zynq family , also synthesized on synopsys design vision for asic implementation . an efficient design of 2-d spatial processor and 1-d temporal processor reduces the internal memory , latency , cpd and complexity of a control unit , and increases the throughput . when compared with the existing architectures the proposed scheme shows higher performance at the cost of slight increase in area . the proposed 3-d dwt architecture is capable of computing 60 uhd ( 3840@xmath702160 ) frames in a second . 30 q. dai , x. chen , and c. lin,``a novel vlsi architecture for multidimensional discrete wavelet transform,''__ieee transactions on circuits and systems for video technology _ _ , vol . 14 , no . 8 , pp . 1105 - 1110 , aug . 2004 . c. cheng and k. k. parhi , `` high - speed vlsi implementation of 2-d discrete wavelet transform , '' _ ieee trans . signal process . 393 - 403 , jan . 2008 . b. k. mohanty and p. k. meher , `` memory - efficient high - speed convolution - based generic structure for multilevel 2-d dwt.''__ieee transactions on circuits and systems for video technology , _ _ vol . 353 - 363 , feb . 2013 . i. daubechies and w. sweledens , `` factoring wavelet transforms into lifting schemes , '' _ j. fourier anal . 247 - 269 , 1998 . huang , p.c . tseng , and l .- g . chen , `` flipping structure : an efficient vlsi architecture for lifting - based discrete wavelet transform , '' _ ieee trans . signal process . 1080 - 1089 , apr . 2004 . xiong , j .- w . tian , and j. liu , `` a note on flipping structure : an efficient vlsi architecture for lifting - based discrete wavelet transform , '' _ ieee transactions on signal processing , _ vol . 54 , no . 5,pp . 1910 - 1916 , may 2006 c .- t . huang , p .- c . tseng , and l .- g . chen , `` analysis and vlsi architecture for 1-d and 2-d discrete wavelet transform , '' _ ieee trans . signal process . 1575 - 1586 , apr . cheng , c .- t . huang , c .- y . ching , c .- j . chung , and l .- g . chen , `` on - chip memory optimization scheme for vlsi implementation of line based two - dimentional discrete wavelet transform , '' _ ieee transactions on circuits and systems for video technology , _ vol . 17 , no . 7 , pp . 814 - 822 , jul . liao , m. k. mandal , and b. f. cockburn , `` efficient architectures for 1-d and 2-d lifting - based wavelet transforms , '' _ ieee transactions on signal processing , _ vol . 5 , pp . 1315 - 1326 , may 2004 . b.f . wu and c.f . chung , `` a high - performance and memory - efficient pipeline architecture for the 5/3 and 9/7 discrete wavelet transform of jpeg2000 codec , '' _ ieee trans . circuits syst . video technol . , _ vol . 15 , no . 12 , pp . 1615 - 1628 , dec . xiong , j. tian , and j. liu , `` efficient architectures for two - dimensional discrete wavelet transform using lifting scheme , '' _ ieee transactions on image processing , _ vol . 607 - 614 , mar . w. zhang , z. jiang , z. gao , and y. liu , `` an efficient vlsi architecture for lifting - based discrete wavelet transform , '' _ ieee transactions on circuits and systems - ii : express briefs , _ vol . 158 - 162 , mar . b. k. mohanty and p. k. meher , `` memory efficient modular vlsi architecture for high throughput and low - latency implementation of multilevel lifting 2-d dwt , '' _ ieee transactions on signal processing , _ vol . 5 , pp . 2072 - 2084 , may 2011 . a.darji , s. agrawal , ankit oza , v. sinha , a.verma , s. n. merchant and a. n. chandorkar , `` dual - scan parallel flipping architecture for a lifting - based 2-d discrete wavelet transform,''__ieee transactions on circuits and systems - ii : express briefs , _ _ vol . 61 , no . 6 , pp . 433 - 437 , jun . 2014 . b. k. mohanty , a. mahajan , and p. k. meher , `` area and power efficient architecture for high - throughput implementation of lifting 2-d dwt , '' _ ieee trans . circuits syst . briefs , _ vol . 59 , no . 7 , pp . 434 - 438 , jul . 2012 . y. hu and c. c. jong,``a memory - efficient high - throughput architecture for lifting - based multi - level 2-d dwt,''__ieee transactions on signal processing , _ _ vol . 20 , pp.4975 - 4987 , oct . 15 , 2013 . y. hu and c. c. jong , `` a memory - efficient scalable architecture for lifting - based discrete wavelet transform,''__ieee transactions on circuits and systems - ii : express briefs , _ _ vol . 60 , no . 8 , pp . 502 - 506 , aug . j. xu , z.xiong , s. li , and ya - qin zhang , `` memory - constrained 3-d wavelet transform for video coding without boundary effects , '' _ ieee transactions on circuits and systems for video technology , _ vol . 812 - 818 , sep . 2002 . m. weeks and m. a. bayoumi , `` three - dimensional discrete wavelet transform architectures,''__ieee transactions on signal processing , _ _ vol . 50 , no . 8 , pp.2050 - 2063 , aug . z. taghavi and s. kasaei , `` a memory efficient algorithm for multidimensional wavelet transform based on lifting , '' _ in proc . acoust speech signal process . ( icassp ) _ , vol . 401 - 404 , 2003 . q. dai , x. chen , and c. lin , `` novel vlsi architecture for multidimensional discrete wavelet transform , '' _ ieee transactions on circuits and systems for video technology , _ vol . 14 , no . 8 , pp . 1105 - 1110 , a. das , a. hazra , and s. banerjee,``an efficient architecture for 3-d discrete wavelet transform,''__ieee transactions on circuits and systems for video technology , _ _ vol . 286 - 296 , feb . 2010 . b. k. mohanty and p. k. meher , `` memory - efficient architecture for 3-d dwt using overlapped grouping of frames,''__ieee transactions on signal processing _ _ , vol . 11 , pp.5605 - 5616 , nov . 2011 . a. darji , s. shukla , s. n. merchant and a. n. chandorkar , `` hardware efficient vlsi architecture for 3-d discrete wavelet transform , '' _ proc . of @xmath71 int . conf . on vlsi design and @xmath72 int . conf . on embedded systems _ pp . 348 - 352 , 5 - 9 jan . 2014 . w.sweldens , `` the lifting scheme : a construction of second generation of wavelets , '' _ siam journal on mathematical analysis , _ vol.29 no.2 , pp . 511 - 546 , 1998 .
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this paper presents a memory efficient , high throughput parallel lifting based running three dimensional discrete wavelet transform ( 3-d dwt ) architecture .
3-d dwt is constructed by combining the two spatial and four temporal processors .
spatial processor ( sp ) apply the two dimensional dwt on a frame , using lifting based 9/7 filter bank through the row rocessor ( rp ) in row direction and then apply in the colum direction through column processor ( cp ) . to reduce the temporal memory and the latency , the temporal processor ( tp ) has been designed with lifting based 1-d haar wavelet filter .
the proposed architecture replaced the multiplications by pipeline shift - add operations to reduce the cpd .
two spatial processors works simultaneously on two adjacent frames and provide 2-d dwt coefficients as inputs to the temporal processors .
tps apply the one dimensional dwt in temporal direction and provide eight 3-d dwt coefficients per clock ( throughput ) .
higher throughput reduces the computing cycles per frame and enable the lower power consumption .
implementation results shows that the proposed architecture has the advantage in reduced memory , low power consumption , low latency , and high throughput over the existing designs .
the rtl of the proposed architecture is described using verilog and synthesized using 90-nm technology cmos standard cell library and results show that it consumes 43.42 mw power and occupies an area equivalent to 231.45 k equivalent gate at frequency of 200 mhz .
the proposed architecture has also been synthesised for the xilinx zynq 7020 series field programmable gate array ( fpga ) .
index terms : discrete wavelet transform , 3-d dwt , lifting based dwt , vlsi architecture , flipping structure , strip - based scanning .
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in recent years , fifth generation ( 5 g ) wireless networks have attracted extensive research interest . according to the 3rd generation partnership project ( 3gpp ) @xcite , 5 g networks should support three major families of applications , including enhanced mobile broadband ( embb ) @xcite ; massive machine type communications ( mmtc ) @xcite ; and ultra - reliable and low - latency communications ( urllc ) @xcite . on top of this , enhanced vehicle - to - everything ( ev2x ) communications are also considered as an important service that should be supported by 5 g networks @xcite . these scenarios require massive connectivity with high system throughput and improved spectral efficiency ( se ) and impose significant challenges to the design of general 5 g networks . in order to meet these new requirements , new modulation and multiple access ( ma ) schemes are being explored . orthogonal frequency division multiplexing ( ofdm ) @xcite has been adopted in fourth generation ( 4 g ) networks . with an appropriate cyclic prefix ( cp ) , ofdm is able to combat the delay spread of wireless channels with simple detection methods , which makes it a popular solution for current broadband transmission . however , traditional ofdm is unable to meet many new demands required for 5 g networks . for example , in the mmtc scenario @xcite , sensor nodes usually transmit different types of data asynchronously in narrow bands while ofdm requires different users to be highly synchronized , otherwise there will be large interference among adjacent subbands . to address the new challenges that 5 g networks are expected to solve , various types of modulation have been proposed , such as filtering , pulse shaping , and precoding to reduce the out - of - band ( oob ) leakage of ofdm signals . filtering @xcite is the most straightforward approach to reduce the oob leakage and with a properly designed filter , the leakage over the stop - band can be greatly suppressed . pulse shaping @xcite can be regarded as a type of subcarrier - based filtering that reduces overlaps between subcarriers even inside the band of a single user , however , it usually has a long tail in time domain according to the heisenberg - gabor uncertainty principle @xcite . introducing precoding @xcite to transmit data before ofdm modulation is also an effective approach to reduce leakage . in addition to the aforementioned approaches to reduce the leakage of ofdm signals , some new types of modulations have also been proposed specifically for 5 g networks . for example , to deal with high doppler spread in ev2x scenarios , transmit data can be modulated in the delay - doppler domain @xcite . the above modulations can be used with orthogonal multiple access ( oma ) in 5 g networks . oma is core to all previous and current wireless networks ; time - division multiple access ( tdma ) and frequency - division multiple access ( fdma ) are used in the second generation ( 2 g ) systems , code - division multiple access ( cdma ) in the third generation ( 3 g ) systems , and orthogonal frequency division multiple access ( ofdma ) in the 4 g systems . for these systems , resource blocks are orthogonally divided in time , frequency , or code domains , and therefore there is minimal interference among adjacent blocks and makes signal detection relatively simple . however , oma can only support limited numbers of users due to limitations in the numbers of orthogonal resources blocks , which limits the se and the capacity of current networks . to support a massive number of and dramatically different classes of users and applications in 5 g networks , various noma schemes have been proposed . as an alternative to oma , noma introduces a new dimension by perform multiplexing within one of the classic time / frequency / code domains . in other words , noma can be regarded as an `` add - on '' , which has the potential to be harmoniously integrated with existing ma techniques . the core of noma is to utilize power and code domains in multiplexing to support more users in the same resource block . there are three major types of noma : power - domain noma , code - domain noma , and noma multiplexing in multiple domains . with noma , the limited spectrum resources can be fully utilized to support more users , therefore the capacity of 5 g networks can be improved significantly even though extra interference and additional complexity will be introduced at the receiver . to address the various challenges of 5 g networks , we can either develop novel modulation techniques to reduce multiple user interference for oma or directly use noma . the rest of this article is organized as follows . in section [ sec : waveform ] , novel modulation candidates for oma in 5 g networks are compared . in section [ sec : ma ] , various noma schemes are discussed . section [ sec : conclusion ] concludes the article . in this section , we will discuss new modulation techniques for 5 g networks . since ofdm is widely used in current wireless systems and standards , many potential modulation schemes for 5 g networks are delivered from ofdm for backward compatibility reasons . therefore , we will first introduce traditional ofdm . denote @xmath0 , for @xmath1 , to be the transmit complex symbols . then the baseband ofdm signal can be expressed as @xmath2 for @xmath3 , where @xmath4 , @xmath5 is the subcarrier bandwidth and @xmath6 is the symbol duration . to ensure that transmit symbols can be recovered without distortion , @xmath7 , which is also called the orthogonal condition . it can be easily shown that @xmath8 if the orthogonal condition holds . denote @xmath9 to be the sampled version of @xmath10 , where @xmath11 . it can be easily seen @xcite that @xmath12 is the inverse discrete fourier transform ( idft ) of @xmath13 , which can be implemented by fast fourier transform ( fft ) and significantly simplifies ofdm modulation and demodulation . to address the delay spread of wireless channels , a cp is usually used in ofdm . if the length of the cp is larger than the delay span ( the duration between the first and the last taps / paths of a channel ) , then the demodulated ofdm signal can be expressed as @xmath14 where @xmath15 is the frequency response of the wireless channel at @xmath16 and @xmath17 is the impact of additive channel noise . therefore , the channel distortion becomes a multiplication of channel frequency response in ofdm systems while it is convolution in single - carrier systems , which makes the detection of ofdm signal much easier . from the above discussion , ofdm can effectively deal with the delay spread of broadband wireless channels and fft can be used to significantly simplify its complexity , therefore it has been widely used in the current wireless communication systems and standards . however , as we can see from ( [ eq : ofdmsig ] ) , the ofdm signal is time - limited . therefore , its oob leakage is pretty high , especially when users are asynchronized as typical of 5 g networks . to address this issue , a guard band is usually inserted between the signals of two adjacent users in the frequency domain in addition to a cp or a guard interval in the time domain , which reduces the se of ofdm . this is even more severe for the users using a narrow frequency band . 5 g networks have to support not only a massive number of users but also dramatically different types of users that have different demands . traditional ofdm can no longer satisfy these requirements , and therefore novel modulation techniques with much lower oob leakage are required . the new modulation techniques for 5 g networks currently need to consider backward compatibility with traditional ofdm systems but should also have the following key features to address the new challenges . 1 . high se : new modulation techniques should be able to mitigate oob leakage among adjacent users so that the system se can be improved significantly by reducing the guard band / time resources . 2 . loose synchronization requirements : massive number of users are expected to be supported , especially for the internet of things ( iot ) , which makes synchronization difficult . therefore , new modulation techniques are expected to accept asynchronous scenarios . 3 . flexibility : the modulation parameters ( e.g. , subcarrier width and symbol period ) for each user should be configured independently and flexibly to support users with different data rate requirements . the modulation techniques for oma mainly include pulse shaping , subband filtering , precoding design , guard interval ( gi ) shortening , and modulation in the delay - doppler domain . in this section , we introduce those promising modulation techniques subsequently . pulse shaping , which is also regarded as subcarrier - based filtering , can effectively reduce oob leakage . according to the heisenberg - gabor uncertainty principle @xcite , the time and frequency widths of the pulses can not be reduced at the same time . therefore , the waveforms based on pulse shaping is usually non - orthogonal in both time and frequency domains to maintain high se . compared with traditional ofdm , the transceiver structure supporting pulse shaped modulation is more complex . here , we introduce two typical modulations based on pulse shaping , i.e. , filter bank multicarrier ( fbmc ) @xcite and generalized frequency division multiplexing ( gfdm ) @xcite . as shown in fig . [ fig : fbmc ] , fbmc @xcite consists of idft and dft , synthesis and analysis polyphase filter banks . the prototype filter in fbmc performs the pulse shaping . there are two types of typical pulses : the pulse based on the isotropic orthogonal transform algorithm ( iota ) @xcite and the pulse adopted in the phydyas project @xcite . the length of the pulse in the time domain is determined by the required performance and is usually several times the length of the symbol period . the bandwidth of the pulse , which is different from the pulse in the traditional ofdm that has a long tail , is limited within a few subbands . to achieve the best se , offset quadrature amplitude modulation ( oqam ) is usually applied to make fbmc real - domain orthogonal in time and frequency domains @xcite . therefore , the transmit signal over @xmath18 consecutive block periods can be expressed as @xmath19 where @xmath20 and @xmath21 are the numbers of subcarriers and symbols , respectively , @xmath22 is the transmit symbol at subcarrier @xmath23 and symbol @xmath24 , and @xmath25 is the prototype filter coefficient at the @xmath26-th time - domain sample . it is worth noting that the transmit symbols here refer to the pulse amplitude modulation ( pam ) symbols that are derived from the staggering of quadrature amplitude modulation ( qam ) symbols . thus the interval between two adjacent blocks is only half of the block period due to the offset in oqam . the parameter , @xmath27 in ( [ eq : fbmcsig ] ) , is defined as @xmath28 which is used to form the oqam structure . with a properly designed prototype filter such as iota and the oqam structure , the interference from the nearby overlapped symbols caused by a matched filter ( mf ) receiver becomes pure imaginary , which can be easily cancelled . [ fig : gfdm ] demonstrates the block diagram of gfdm . ofdm and single - carrier frequency division multiplexing ( sc - fdm ) can be regarded as two special cases of gfdm @xcite . the unique feature of gfdm is to use circular shifted filters , rather than linear filters that are used in fbmc , to perform pulse shaping . by carefully choosing the circular filter , the out - of - block leakage can be reduced even if the orthogonality is completely given up . we can flexibly adjust @xmath21 frequency samples and @xmath20 time samples for a gfdm block according to the application environment . the transmit signal for each gfdm block can be expressed as @xmath29 for @xmath30 , where @xmath22 is the transmit symbol on subcarrier @xmath23 at subsymbol @xmath24 and @xmath31 is the circular time and frequency shifted version of the prototype pulse shaping filter . in ( [ eq : gfdmsig ] ) , @xmath32 where @xmath33 denotes the @xmath34 modulo operation and @xmath25 is the prototype pulse shaping filter . similar to the traditional ofdm , the modulation process and demodulation process can be expressed by matrix operations . the idft and dft matrices in the traditional ofdm are substituted by some specific matrices corresponding to the modulation and demodulation for gfdm . but , the transceiver structure of gfdm is significantly different from the traditional ofdm . besides fbmc and gfdm , other modulations based on pulse shaping , such as pulse - shaped ofdm @xcite and qam - fbmc @xcite , have also been proposed for 5 g networks . generally , modulations based on pulse shaping try to restrict transmit signals within a narrow bandwidth and thus mitigate the oob leakage so that they can work in asynchronous scenarios with a narrow guard band . fbmc also uses oqam to achieve real - domain orthogonality , which saves the cost of the gi and interference cancellation . in addition , the circular shifted filters in gfdm avoid the long tail of the linear filters in the time domain , which makes gfdm fit for sporadic transmission . furthermore , gfdm is easily compatible to mimo technologies @xcite . subband filtering is another technique to reduce the oob leakage . universal filtered multicarrier ( ufmc ) @xcite and filtered ofdm ( f - ofdm ) @xcite are two typical modulations based on subband filtering , which will be introduced next . fig . [ fig : ufmc ] shows the transmitter and the receiver structures of ufmc @xcite . in ufmc , the subbands are with equal size , and each filter is a shifted version of the same prototype filter . ofdm is applied within a subband for this modulation as shown in the figure . since the bandwidth of the filter in ufmc is much wider than that of the modulations based on the pulse shaping , the length in time domain is much shorter . therefore , interference caused by the tail of the filter can be easily eliminated by adopting a zero - padding ( zp ) prefix with a reasonable length . assuming that @xmath35 subcarriers are divided into @xmath20 subbands , each with @xmath36 consecutive subcarriers , the transmit signal in ufmc can be expressed as @xmath37 where @xmath38 is the filter coefficient of subband @xmath23 , and @xmath39 is the ofdm modulated signal over subband @xmath23 that can be expressed as @xmath40 with @xmath41 denoting the length of the zp , @xmath21 denoting the number of symbol blocks and @xmath42 denoting the signal at subcarrier @xmath23 and symbol @xmath24 . in ( [ eq : ufmcsubsig ] ) , @xmath42 can be expressed as @xmath43 where @xmath44 denotes the @xmath45-th transmit symbol at the @xmath24-th symbol block . at the receiver , the signal at each symbol interval is with the length of @xmath46 and is zero - padded to have a length of @xmath47 so that a @xmath47-point fft can be performed . please note that only the even subcarriers are considered for signal detection after the @xmath47-point fft . f - ofdm has a similar transmitter structure as ufmc @xcite . the main difference is that f - ofdm employs a cp and usually allows residual inter - symbol interference ( isi ) @xcite . therefore , at the receiver , the mf is applied instead of the zp and decimation . besides , downsampling can be applied before the dft operation , which can reduce complexity significantly since the cp can mitigate most of interference caused by the tail of the filter ; the residual interference is with much lower power and can be treated as noise @xcite . thus , the filter in f - ofdm can be longer than that in ufmc and has better attenuation outside the band . with the aid of effective channel coding , the performance degradation caused by residual interference in f - ofdm can be negligible . another difference from ufmc is that the subcarrier spacing and the cp length do not have to be the same for different users in f - ofdm . the most widely used filter in f - ofdm is the soft - truncated sinc filter @xcite , which can be easily used in various applications with different parameters . therefore , f - ofdm is very flexible in the frequency multiplexing . besides ufmc and f - ofdm , other modulations based on subband filtering have also been proposed . for example , resource block f - ofdm ( rb - f - ofdm ) @xcite utilizes filters based on resource block instead of the whole band of users in f - ofdm . in general , modulations based on subband filtering can effectively reduce oob leakage and achieve better performance in comparison with the traditional ofdm . apart from pulse shaping and subband filtering , there are also some other techniques to suppress the oob leakage and meet the requirements of 5 g networks . in the following , we mainly introduce three other modulations , including guard interval discrete fourier transform spread ofdm ( gi dft - s - ofdm ) @xcite , spectrally - precoded ofdm ( sp - ofdm ) @xcite , and orthogonal time frequency and space ( otfs ) @xcite . in gi dft - s - ofdm @xcite , the known sequence is used as the gi instead of a cp . several types of the known sequences , such as the zero sequence @xcite and a well - designed unique word @xcite , can be used . by a fixed known sequence with constant amplitude in gi dft - s - ofdm , the peak - to - average power ratio ( papr ) of the modulated signal can be reduced . moreover , the known sequence can also be utilized to estimate the parameters , such as the carrier frequency offset ( cfo ) in the synchronization process . by utilizing a proper sequence as the gi , the discontinuity between the adjacent time blocks in the traditional ofdm / dft - s - ofdm can be avoided . as a result , the oob leakage is reduced . for gi dft - s - ofdm , the overall length of the gi and useful signal for different users is same . thus , the dft windows for different users at the receiver can still be aligned even if the lengths of the gis are different . therefore , the mutual interference due to asynchronization of users can be mitigated @xcite . [ fig : spofdm ] shows the diagram of sp - ofdm @xcite . from the figure , it consists idft and dft , spectral precoder , and iterative detector . generally , the data symbols mapped on subcarriers are precoded by a rank - deficient matrix in order to project the signal into a properly selected lower dimensional subspace so that the precoded signal can be high - order continuous , and results in much lower leakage compared with the traditional ofdm @xcite . even if precoded by a rank - deficient matrix can reduce the capacity of the channel , the oob leakage of the ofdm signals can be significantly suppressed at the cost of only few reduced dimensions . compared to the modulations based on filtering , sp - ofdm has the following three advantages : * the isi caused by the tail of the filters can be removed without filtering . therefore , the cp applied to combat the multipath of the wireless channels can be shorter , and se is improved . * when fragmented bands are used , sp - ofdm can easily notch specific well chosen frequencies without requiring multiple narrow subband filters @xcite . * furthermore , precoding and filtering can be combined to further improve the performance . the structure of otfs is similar to sp - ofdm , as can be seen in fig . [ fig : spofdm ] . the main difference is that the spectral precoder and the iterative detector are substituted by the two - dimensional ( 2d ) symplectic fourier transform and the corresponding inverse transform modules . otfs maps the symbols in the delay - doppler domain @xcite . through a 2d symplectic fourier transform , the corresponding data in the time - frequency domain can be calculated . then , the calculated data can be transmitted via a time - frequency - domain modulation method as in ofdm . since the 2d symplectic fourier transform is relatively independent of the time - frequency - domain modulation method , pulse shaping and subband filtering can also be applied together to further reduce the leakage in otfs . when a mobile is with a high speed , the channel experiences fast fading . channel parameters need to be estimated and tracked very often therefore , which significantly increases resource costs . moreover , most of the modulations are designed assuming that channels are constant within a symbol block . with a high mobility speeds , extra interference is introduced , which degrades the performance . however , in the delay - doppler domain , the high doppler channel can be expressed in a stable model , which saves the cost of tracking the time - varying fading and improves performance therefore . otfs can be also applied to estimate channel state information ( csi ) of different antennas in mimo systems @xcite . generally , the delay and doppler dispersions are still relatively small compared to the system scale . in this case , the channel can be expressed in a compact and stable form in the delay - doppler domain . as a result , the spread of the pilots caused by the channel are local , which enables to estimate the csi of different antennas in mimo systems by different pilots within a small area of the delay - doppler plane . in addition , a number of modulations based on other techniques have been also proposed , such as windowed ofdm ( w - ofdm ) @xcite , which utilizes windowing to deal with the discontinuity between adjacent ofdm symbols . we compare the power spectral density ( psd ) and bit - error rate ( ber ) of different modulations . suppressing the oob leakage is a key purpose for most of the modulation candidates for 5 g networks . the psds of the some modulations are shown in fig . [ fig : psd ] . from the figure , all modulations achieve much lower leakage compared to the traditional ofdm . among them , ufmc applies subband filtering and also has low leakage , and fbmc and f - ofdm have the lowest leakage . gfdm , gi dft - s - ofdm , and sp - ofdm , although do not reduce the leakage as much as fbmc and f - ofdm , can still achieve much better performance than the traditional ofdm . in order to reduce the oob leakage , many modulations utilize techniques , such as pulse shaping and subband filtering , which may introduce isi and ici . hence , the ber performance of different modulations is compared here . [ fig : ber ] shows the ber performance versus signal - to - noise ratio ( snr ) when the doppler spread @xmath48 and @xmath49 hz . from fig . [ fig : ber ] ( a ) , the traditional ofdm has the best performance when the doppler spread is zero ( @xmath48 ) since the isi caused by the multipath has been completely canceled by the cp . since the bandwidth of each subcarrier is small enough to make the corresponding channel approximately flat , the isi introduced by pulse shaping in fbmc is nearly pure imaginary . therefore , fbmc is approximately orthogonal in the real domain and achieves good ber performance . the performance of ufmc , gfdm , and sp - ofdm is similar to that of fbmc , which is degraded slightly due to noise enhancement and low - projection precoding . however , f - ofdm introduces extra isi that can not be completely canceled , and as a result , it has slightly worse performance , especially in the high snr region . gi dft - s - ofdm and otfs , which are different from the modulation schemes that directly map the symbols on subcarriers , apply spreading before mapping so that their performance does not approach that of ofdm . since the fast - fading channel is difficult to be estimated and tracked accurately , the performance of the most modulation schemes degrades significantly as we can see from fig . [ fig : ber ] ( b ) . while otfs can still achieve good performance due to its specific channel estimation method . moreover , its performance in the high - mobility scenario is even better than that in the zero doppler shift scenario because of doppler diversity . in this section , modulation techniques for 5 g networks will be be discussed . these techniques can be used with oma to effectively deal with the oob leakage in 5 g networks . however , there are still many open issues in the area . a potential application of f - ofdm is iot . in this scenario , the subbands are narrow and therefore , interference caused by a short cp can significantly degrade the performance and should be considered in the detection . to improve the detection performance , additional processing , such as filtering or successive interference cancellation ( sic ) , is needed . residual isi cancellation ( risic ) @xcite could be helpful . the existing designs for subband filtering , such as dolph - chebyshev filter in ufmc and soft - truncated filter in f - ofdm , are with fixed length . however , different users and different application scenarios will have different requirements on the leakage levels , filter lengths , etc . according to the heisenberg - gabor uncertainty principle @xcite , the time and frequency dispersions are dual variables that can not be reduced at the same time . therefore , how to balance the time and frequency dispersions and design an efficient prototype filter according to application scenarios is interesting . similar to the traditional ofdm , multi - carrier based new modulation candidates , such as fbmc and ufmc also have a large papr . in order to improve the efficiency of the power amplifier , the papr should be reduced . the traditional papr reduction methods @xcite applied in the traditional ofdm usually introduce distortions that degrade the performance . therefore , how to properly extend the papr reduction methods in the traditional ofdm to the new modulations is an interesting and meaningful issue . in order to support higher throughput and massive and heterogeneous connectivity for 5 g networks , we can adopt novel modulations discussed in section ii for oma , or directly use noma with effective interference mitigation and signal detection methods . the key features of noma can be summarized as follows : 1 . improved se : noma exhibits a high se , which is attributed to the fact that it allows each resource block ( e.g. , time / frequency / code ) to be exploited by multiple users . 2 . ultra high connectivity : with the capability to support multiple users within one resource block , noma can potentially support massive connectivity for billions of smart devices . this feature is quite essential for iot scenarios with users that only require very low data rates but with massive number of users . 3 . relaxed channel feedback : in noma , perfect uplink csi is not required at the base station ( bs ) . instead , only the received signal strength needs to be included in the channel feedback . low transmission latency : in the uplink of noma , there is no need to schedule requests from users to the bs , which is normally required in oma schemes . as a result , a grant - free uplink transmission can be established in noma , which reduces the transmission latency drastically . existing noma schemes can be classified into three categories : power - domain noma , code - domain noma , and noma multiplexing in multiple domains . we will introduce them subsequently with emphasis on power - domain noma . power - domain noma is considered as a promising ma scheme for 5 g networks @xcite . specifically , a downlink version of noma , named multiuser superposition transmission ( must ) , has been proposed for the 3gpp long - term evolution advanced ( 3gpp - lte - a ) networks @xcite . it has been shown that system capacity and user experiences can be improved by noma . more recently , a new work item ( wi ) outlining downlink multiuser superposition transmission for lte has been approved by 3gpp lte release 14 @xcite , which aims to identify the necessary techniques to enable lte to support the downlink intra - cell multiuser superposition transmission . here , we will expand upon the basic principles of various power - domain noma related techniques , including multiple antenna based noma , power allocation in noma , and cooperative noma . power - domain noma , as illustrated in fig . [ fig : noma ] for the two user case , deviates from conventional oma that uses tdma / fdma / cdma / ofdma allocating orthogonal resource blocks for different users to avoid the multiple access interference ( mai ) . instead power - domain noma can support multiple users within the same resource block by distinguishing them with different power levels . as a result , noma is able to support more connectivity and provide higher throughput with limited resources . the downlink transmission of noma for the two user case is shown in fig . [ fig : noma ] where the users are served at the same time / frequency / code resource block with a total power constraint . specifically , the bs sends a superimposed signal containing the two signals for the two users . this differs from conventional power allocation strategies , such as water filling , as noma allocates less power for the users with better downlink csi , to guarantee overall fairness and to utilize diversity in the time / frequency / code domains . sic is used for signal detection at the receiver . the user with more transmit power , that is , the one with smaller downlink channel gain , is first to be decoded while treating the other user s signal as noise . once the signal corresponding to the user with the larger transmit power is detected and decoded , its signal component will be subtracted from the received signal to facilitate the detection of subsequent users . it should be noted that the first detected user is with the largest inter - user interference and also the detection error in the first user will pass to the other user , which is why we have to allocate sufficient power to the first user to be detected . the extension of noma from two to multiple user cases is straightforward . for the uplink transmission of noma , the transmit power is limited by each individual user . different from the downlink , the transmit powers of the users using the same resource block are carefully controlled so that the received signal components at the bs corresponding to the users with the better csi , have more powers . at the receiver ( the bs ) , the user with the best csi is decoded first . after that , the corresponding component is removed from the received signal . the sic receiver works in a descending order of the csi , which is the opposite to the downlink case . [ fig : noma_oma ] compares noma and oma where two users are served by the same bs if noma is adopted . from the figure , the noma scheme achieves a lower outage probability . however , by adopting noma , a more complex transmitter and receiver are required to mitigate the interference . furthermore , power - domain noma usually works well when only two or a few users share the same resource block . as the number of users multiplexing in power domain increases , the mai becomes severe and the performance of noma degrades . multiple antenna techniques can provide an additional degree of freedom on the spatial domain , and bring further performance improvements to noma . recently , multiple antenna based noma has attracted lots of attention @xcite . different from single - input - single - output ( siso ) based noma , where the channels are normally represented by scalars , one of the research challenges in multiple antenna based noma comes from user ordering ; as the channels are generally in form of vectors or matrices . currently , the possible designs of multiple antenna based noma fall into two categories where one or multiple users are served by a single beamforming vector . by allocating different users with different beams in the same resource block , the quality of service ( qos ) of each user can be guaranteed in multiple antenna based noma systems forcing the beams to satisfy a predefined order . this type of multiple antenna based noma scheme has been first proposed by sun _ et al . _ in @xcite to investigate power optimization to maximize the ergodic capacity . this proposed multiple antenna based noma scheme has proved to be able to achieve significant performance improvement compared with conventional oma schemes . a cluster of users can share the same beam . the spatial channels of different users within the same cluster are considered to be highly correlated . therefore , beams for different clusters should be carefully designed to guarantee that the channels for different clusters are orthogonal to each other in order to suppress the inter - cluster interference . for multiple - input - single - output ( miso ) based noma , a two - stage multicast beamforming scheme has been proposed by choi in @xcite , where zf beamforming has been employed to mitigate interference from adjacent clusters first and then the optimal beamforming vectors have been designed to minimize the total transmit power within each cluster . for mimo based noma , a scheme to simultaneously apply open - loop random beamforming and intra - beam sic , has been proposed by higuchi and kishiyama in @xcite . however , here the system performance is considerably degraded as the random beamforming can bring uncertainties at the user side . more recently , a precoding and detection framework with fixed power allocation has been proposed by ding _ et al . _ @xcite to solve these problems caused by random beamforming , and demonstrated that mimo based noma can achieve better outage performance than mimo based oma even for users who experience strong co - channel interference . a comprehensive summary for the state - of - the - art work on multiple antenna based noma is given in table [ table : mimo noma ] , where `` bf '' , `` op '' , `` su '' and `` mu '' are used to represent beamforming , outage probability , two - user and multi - users cases , respectively . [ cols="<,<,<,<,<",options="header " , ] & no need for user clustering & specific channel coding + [ table : comparison ] several noma schemes have been discussed in this section . even if using different techniques , these schemes share the same spirit to utilize non - orthogonality to increase the system capacity and support more users by the limited resource blocks . beyond the existing work , more research is necessary to improve the performance of these noma schemes from the following aspects . the mpa - sic detection method is usually applied in scma and pdma , in which the user clustering mechanism affects the performance of the method significantly . when users are asynchronous , those with similar time delays should be divided into the same cluster for better performance . if the delays vary a lot among the users within the same cluster , interference among different users becomes large and may break the sparse structure . multi - branch technique @xcite can be applied to improve the performance by regarding each cluster as a branch . by calculating each branch in parallel and selecting the best result as the final one , the performance could be improved compared to the single clustering approach . the joint design of new modulation and noma schemes is an important direction to be explored in 5 g networks . some of the noma schemes , especially the lds based code - domain noma , are based on ofdm , where the output of the sparse spreading matrix is mapped into orthogonal subcarriers . in general , how to properly combine the modulation and noma scheme is under research . for example , for the combination of scma and f - ofdm , the short cp of f - ofdm could introduce isi and ici when the subband is narrow and degrade the detection performance of scma . if the risic algorithm is adopted to cancel the interference introduced by f - ofdm , the multiuser detection of scma should be included in the iteration of cp reconstruction , which poses a requirement of joint design approaches for the receivers . the design of modulation and ma schemes for high frequency bands ( above 40 ghz ) is beginning to receive increased iterest . the millimeter - wave ( mmwave ) and terahertz ( thz ) bands appear to be good candidates to decrease spectrum sacristy due to the availabilities in current circuit design @xcite . however , the propagation properties of mmwave and thz bands have shown to be quite poor , which brings new challenges on system designs . for example , noise is the major limitation of mmwave and thz bands , which makes the transmit power levels extremely important and ultimately impacts the classes of applications that can use them ( e.g. iot ) . moreover , high level impairments including carrier frequency offset ( cfo ) and phase noise also need to be considered in mmwave and thz bands as they are noise - limited . nevertheless , there is already a study on noma based mmwave communications @xcite , we may further see analyses of such systems based on practical scenarios in the future . in this article , we provide a comprehensive survey covering the major promising candidates for modulation and multiple access ( ma ) in fifth generation ( 5 g ) networks . from our discussion , we can see that new modulations for orthogonal ma can be adopted to reduce out - of - band leakage while meeting the diverse demands of 5 g networks . non - orthogonal ma is another promising approach that marks a deviation from the previous generations of wireless networks . by utilizing non - orthogonality , we have convincingly shown that 5 g networks will be able to provide enhanced throughput and massive connectivity with improved spectral efficiency . h. sampath , s. talwar , j. tellado , v. erceg , and a. paulraj , `` a fourth - generation mimo - ofdm broadband wireless system : design , performance , and field trial results , '' _ ieee commun . mag . _ , vol . 40 , no . 9 , pp . 143149 , sep . 2002 . v. vakilian , t. wild , f. schaich , s.brink , and j. f. frigon , `` universal - filtered multi - carrier technique for wireless systems beyond lte , '' in _ proc . ieee globecom workshops ( gc wkshps ) _ , atlanta , ga , usa , dec . 2013 , pp . 223228 . f. schaich , t. wild , and y. chen , `` waveform contenders for 5 g - suitability for short packet and low latency transmissions , '' in _ proc . ( vtc spring ) _ , seoul , korea , may 2014 , pp . j. abdoli , m. jia , and j. ma , `` filtered ofdm : a new waveform for future wireless systems , '' in _ proc . ieee 16th int . workshop signal process . adv . wireless commun . ( spawc ) _ , stockholm , sweden , jun . 2015 , pp . 6670 . x. zhang , m. jia , l. chen , j. ma , and j. qiu , `` filtered - ofdm - enabler for flexible waveform in the 5th generation cellular networks , '' in _ proc . ieee global commun . ( globecom ) _ , san diego , ca , usa , dec . 2015 , pp . 16 . m. bellanger , m. renfors , t. ihalainen , and c.a.f . da rocha , `` ofdm and fbmc transmission techniques : a compatible high performance proposal for broadband power line communications , '' in _ int . . power line commun . and its applications ( isplc ) _ , rio de janeiro , brazil , mar . 2010 , pp . 154159 . n. michailow , m. matth , i.s . gaspar , a. n. caldevilla , l. l. mendes , a. festag , and g. fettweis , `` generalized frequency division multiplexing for 5th generation cellular networks , '' _ ieee trans . _ , vol . 62 , no . 9 , pp . 30453061 , sep . 2014 . a. sahin , i. guvenc , and h. arslan , `` a survey on multicarrier communications : prototype filters , lattice structures , and implementation aspects , '' _ ieee commun . surveys tu . _ , vol . 16 , no . 3 , pp . 13121338 , third quarter , 2014 . z. zhao , m. schellmann , q. wang , x. gong , r. boehnke , and w. xu , `` pulse shaped ofdm for asynchronous uplink access , '' in _ asilomar conf . signals , systems and computers _ , monterey , usa , nov . 2015 , pp . 37 . c. kim , k. kim , y. h. yun , z. ho , b. lee , and j. y. seol , `` qam - fbmc : a new multi - carrier system for post - ofdm wireless communications , '' in _ proc . ieee global commun . conf . ( globecom ) _ , san diego , ca , usa , dec . 2015 , pp . 16 . g. berardinelli , f. m. l. tavares , t. b. sorensen , p. mogensen , and k. pajukoski , `` zero - tail dft - spread - ofdm signals , '' in _ proc . ieee globecom workshops ( gc wkshps ) _ , atlanta , ga , usa , dec . 2013 , pp . 229234 . a. sahin , r. yang , m. ghosh , and r. l. olesen , `` an improved unique word dft - spread ofdm scheme for 5 g systems , '' in _ proc . ieee globecom workshops ( gc wkshps ) _ , san diego , ca , usa , dec . 2015 , pp . p. achaichia , m. l. bot , and p. siohan , `` windowed ofdm versus ofdm / oqam : a transmission capacity comparison in the homeplug av context , '' in _ int . . power line commun . and its applications ( isplc ) _ , udine , italy , apr . 2011 , pp . 405410 x. li and l. j. cimini , `` effects of clipping and filtering on the performance of ofdm , '' _ ieee commun . lett . _ , vol . 2 , no . 5 , pp . 131133 , may 1998 . k. higuchi and y. kishiyama , `` non - orthogonal access with random beamforming and intra - beam sic for cellular mimo downlink , '' in _ proc . ( vtc fall ) _ , las vegas , nv , usa , sep . 2013 , pp . 15 . k. higuchi and a. benjebbour , `` non - orthogonal multiple access ( noma ) with successive interference cancellation for future radio access , '' _ ieice trans . _ , vol . 98 , no . 3 , pp . 403414 , 2015 . z. ding , r. schober , and h. v. poor , `` a general mimo framework for noma downlink and uplink transmission based on signal alignment , '' _ ieee trans . wireless commun . 15 , no . 6 , pp . 4438 - 4454 , jun . 2016 . z. qin , y. liu , z. ding , y. gao and m. elkashlan , `` physical layer security for 5 g non - orthogonal multiple access in large - scale networks , '' in _ proc . conf . on commun . ( icc ) _ , kuala lumpur , malaysia , may 2016 , pp . 1 - 6 . y. liu , z. qin , m. elkashlan , y. gao , and l. hanzo , `` enhancing the physical layer security of non - orthogonal multiple access in large - scale networks , '' _ ieee trans . wireless commun . _ , to appear in 2017 . y. saito , y. kishiyama , a. benjebbour , t. nakamura , a. li , and k. higuchi , `` non - orthogonal multiple access ( noma ) for cellular future radio access , '' in _ proc . ( vtc spring ) _ , dresden , germany , jun . 2013 , pp.15 . y. saito , a. benjebbour , y. kishiyama , and t. nakamura , `` system - level performance evaluation of downlink non - orthogonal multiple access ( noma ) , '' in _ proc . personal , indoor , and mobile radio commun . ( pimrc ) _ , london , uk , sep . 2013 , pp . 611 - 615 . z. ding , z. yang , p. fan , and h. v. poor , `` on the performance of non - orthogonal multiple access in 5 g systems with randomly deployed users , '' _ ieee signal process . _ , vol . 21 , no . 12 , pp . 15011505 , dec . 2014 . y. liu , z. ding , m. elkashlan , and h. v. poor , `` cooperative non - orthogonal multiple access with simultaneous wireless information and power transfer , '' _ ieee j. sel . areas commun . _ , vol . 34 , no . 4 , apr . 2016 . f. liu , p. mahonen , and m. petrova , `` proportional fairness - based user pairing and power allocation for non - orthogonal multiple access , '' in _ proc . personal , indoor , and mobile radio commun . ( pimrc ) _ , hong kong , p.r . china , aug . 2015 , pp . 11271131 . f. liu , p. mahonen , and m. petrova , `` proportional fairness - based power allocation and user set selection for downlink noma systems , '' in _ proc . on commun . ( icc ) _ , kuala lumpur , malaysia , may 2016 , pp . j. mei , l. yao , h. long , and k. zheng , `` joint user pairing and power allocation for downlink non - orthogonal multiple access systems , '' in _ proc . conf . on commun . ( icc ) _ , kuala lumpur , malaysia , may 2016 , pp . n. otao , y. kishiyama , and k. higuchi , `` performance of non - orthogonal access with sic in cellular downlink using proportional fair - based resource allocation , '' in _ proc . wireless commun . ( iswcs ) _ , paris , france , aug . 2012 , pp . 476480 . y. sun , d. w. k. ng , z. ding , and r. schober , `` optimal joint power and subcarrier allocation for mc - noma systems , '' in _ proc . ieee global commun . ( globecom ) _ , washington , dc , usa , 2016 , pp . 1 - 6 . m. al - imari , p. xiao , m. a. imran , and r. tafazolli , `` uplink non - orthogonal multiple access for 5 g wireless networks , '' in _ proc . int . wireless commun . ( iswcs ) _ , barcelona , spain , aug . 2014 , pp . 781785 . l. lei , d. yuan , c. k. ho , and s. sun , `` power and channel allocation for non - orthogonal multiple access in 5 g systems : tractability and computation , '' _ ieee trans . wireless commun . 8580 - 8594 , dec . 2016 . d. diamantoulakis , k. n. pappi , z. ding , and g. k. karagiannidis , `` wireless powered communications with non - orthogonal multiple access , '' _ ieee trans . wireless commun . 8422 - 8436 , dec . 2016 d. lee , h. seo , b. clerckx , e. hardouin , d. mazzarese , s. nagata and k. sayana , `` coordinated multipoint transmission and reception in lte - advanced : deployment scenarios and operational challenges , '' _ ieee commun . mag . 148 - 155 , feb . 2012 . y. liu , z. ding , m. elkashlan and j. yuan , `` non - orthogonal multiple access in large - scale underlay cognitive radio networks , '' _ ieee trans . veh . 10152 - 10157 , dec . 2016 y. liu , z. qin , m. elkashlan , y. gao , and n. arumugam , `` non - orthogonal multiple access in massive mimo aided heterogeneous networks , '' in _ proc . of global commun . ( globecom ) _ , washington d.c , usa , dec . 2016 . f. fang , h. zhang , j. cheng , and v. c. m. leung , `` energy - efficient resource allocation for downlink non - orthogonal multiple access network , '' _ ieee trans . commun . _ , vol . 64 , no . 9 , pp . 37223732 , sep . 2016 . s. m. r. islam , n. avazov , o. a. dobre , k. s. kwak , `` power - domain non - orthogonal multiple access ( noma ) in 5 g systems : potentials and challenges , '' in ieee commun . surveys tutorials , vol.pp , no.99 , pp.1 - 1 , nov . 2016 . m. al - imari , p. xiao , m. a. imran , and r. tafazolli , `` uplink non - orthogonal multiple access for 5 g wireless networks , '' in _ proc . wireless commun . ( iswcs ) _ , barcelona , spain , aug . 2014 , pp . 781785 . d. cai , p. fan , x. lei , y. liu , and d. chen , `` multi - dimensional scma codebook design based on constellation rotation and interleaving , '' in _ proc . technol . conf . ( vtc spring ) _ , nanjing , china , may 2016 , pp . 15 . s. chen , b. ren , q. gao , s. kang , s. sun , and k. niu , `` pattern division multiple access ( pdma ) - a novel non - orthogonal multiple access for 5 g radio networks , '' _ ieee trans . _ , vol . pp , no . 99 , pp . 11 , 2016 . d. fang , y. huang , z. ding , g. geraci , s. l. shieh , and h. claussen , `` lattice partition multiple access : a new method of downlink non - orthogonal multiuser transmissions , '' in _ proc . of global commun . ( globecom ) _ , washington d.c , usa , dec . j. zeng , b. li , x. su , l. rong , and r. xing , `` pattern division multiple access ( pdma ) for cellular future radio access , '' in _ proc . wireless commun . signal process . ( wcsp ) _ , nanjing , jiangsu , china , oct . 2015 , pp . 15 . b. ren , x. yue , w. tang , y. wang , s. kang , x. dai , and s. sun , `` advanced idd receiver for pdma uplink system , '' in _ proc . ieee / cic int . conf . commun . in china ( iccc ) _ , chengdu , china , jul . 2016 , pp . 1 - 6 . y. huang and k. r. narayanan , `` construction @xmath50 and @xmath51 lattices : construction , goodness , and decoding algorithms , '' _ arxiv preprint _ , [ online ] . available : http://arxiv.org/abs/1506.08269 , aug . 2016 . y. cai and r. c. de lamare , `` multistage mimo receivers based on multi - branch interference cancellation for mimo - cdma systems , '' in _ proc . wireless commun . ( iswcs ) _ , tuscany , italy , sep . 2009 , pp .
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fifth generation ( 5 g ) wireless networks face various challenges in order to support large - scale heterogeneous traffic and users , therefore new modulation and multiple access ( ma ) schemes are being developed to meet the changing demands . as this research space is ever increasing , it becomes more important to analyze the various approaches , therefore in this article we present a comprehensive overview of the most promising modulation and ma schemes for 5 g networks .
we first introduce the different types of modulation that indicate their potential for orthogonal multiple access ( oma ) schemes and compare their performance in terms of spectral efficiency , out - of - band leakage , and bit - error rate .
we then pay close attention to various types of non - orthogonal multiple access ( noma ) candidates , including power - domain noma , code - domain noma , and noma multiplexing in multiple domains . from this exploration
we can identify the opportunities and challenges that will have significant impact on the design of modulation and ma for 5 g networks .
5 g , modulation , non - orthogonal multiple access .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the u - cas is an ingenious device that hovers in mid - air while spinning . it is marketed as a kit in japan under the trade name u - cas @xcite , and in the u.s.a . and europe under the trade name levitron@xmath14 @xcite . the whole kit consists of three main parts : a magnetized top which weighs about @xmath15 gr , a thin ( lifting ) plastic plate and a magnetized square base plate ( base ) . to operate the top one should set it spinning on the plastic plate that covers the base . the plastic plate is then raised slowly with the top until a point is reached in which the top leaves the plate and spins in mid - air above the base for about 2 min . the hovering height of the top is approximately @xmath16 cm above the surface of the base whose dimensions are about 10 cm @xmath17 10 cm @xmath17 2 cm . the kit comes with extra brass and plastic fine tuning weights , as the apparatus is very sensitive to the weight of the top . it also comes with two wedges to balance the base horizontally . the physical principle underlying the operation of the u - cas relies on the so - called ` adiabatic approximation ' @xcite : as the top is launched , its magnetic moment points _ antiparallel _ to the magnetization of the base in order to supply the repulsive magnetic force which will act against the gravitational pull . as the top hovers , it experiences lateral oscillations which are slow ( @xmath18 hz ) compared to its precession ( @xmath19 hz ) . the latter itself , is small compared to the top s spin ( @xmath20 hz ) . since @xmath21 the top is considered ` fast ' and acts like a classical spin . furthermore , as @xmath22 this spin may be considered as experiencing a _ slowly _ rotating magnetic field . under these circumstances the spin precesses around the _ local _ direction of the magnetic field @xmath23 ( adiabatic approximation ) and , on the average , its magnetic moment @xmath24 points _ antiparallel _ to the local magnetic field lines . in view of this discussion , the magnetic interaction energy which is normally given by @xmath25 is now given approximately by @xmath26 . thus , the overall effective energy ` seen ' by the top is @xmath27 where @xmath10 is the mass of the top , @xmath11 is the free - fall acceleration and @xmath28 is the height of the top above the base . by virtue of the adiabatic approximation , two of the three rotational degrees of freedom are coupled to the transverse translational degrees of freedom , and as a result the rotation of the axis of the top is already incorporated in eq.([energy ] ) . thus , under the adiabatic approximation , the top may be considered as a _ point - like _ particle whose only degrees of freedom are translational . the important point of this discussion is the following : the energy expression written above _ possesses a minimum _ for certain values of @xmath29 . thus , when the mass is properly tuned , the apparatus acts as a trap , and stable hovering becomes possible . a detailed description of this device , extending beyond the adiabatic approximation , may be found elsewhere @xcite . for the purpose of this paper , the adiabatic approximation will suffice . in this paper we focus on the question : @xmath30 first , we define what do we mean by ` height ' : we assume that the top hovers above some horizontal plane . the height of the top is measured with respect to that plane . the ` rules of the game ' are , that we can put permanent magnet _ below _ the plane , but never _ above _ this plane . whatever is below this plane , will be called ` the base ' . the method we use to answer the question above will be by getting upper and lower _ bounds _ for this height , denoted by @xmath0 . we begin by pointing out the factors that limit the hovering height of the u - cas . first , we set aside the question of stability and assume that the top is guided by a vertical axis , allowing it to move only axially ( up and down ) . in this case , the easiest way to increase @xmath13the hovering height , is by using a more powerful magnet for the base . this technique however , can not be applied indefinitely since there is no way to increase the magnetization of a given substance without limit . we are therefore forced to limit the strength of @xmath31the magnetization density of the base , to some maximum value , say @xmath9 , and design the shape and magnetization density of the base so as to maximize @xmath13 . one more factor that limits the levitation height is the amount of magnetic substance , or _ , in our disposal . clearly , the more material we use , the larger is the levitation height that can be achieved . but still , even if the volume is infinite the levitation height is bounded , since @xmath32 is bounded . when stability is taken into account , things get more complicated . stability sets another limitation on the design of the base , in _ addition _ to the limitations that were discussed above : a brief look at eq.([energy ] ) tells us that for the top to hover _ stably _ over the plate , the effective energy @xmath33 should possess a _ minimum_. this means , in particular , that @xmath34 where the derivatives are evaluated at the equilibrium position of the top . since these are _ homogenous _ inequalities , it is clear that the region in space where a minimum _ may _ occur , does not depend on the _ strength _ of the magnetic field . as a consequence , the hovering height of the u - cas is _ not _ determined by the strength of the magnetic field but on its _ geometry , _ or alternatively , by the _ shape _ of the base . as an example , it has been shown @xcite that for a vertically magnetized base in the shape of a disk of radius @xmath35 , the range of heights @xmath13 for which stable hovering is possible , is very narrow and is around @xmath36 . this result agrees roughly with the parameters of the u - cas , for which @xmath37 cm and @xmath38 cm , when @xmath13 is measured from the center - of - mass of the base . the _ strength _ of the field then comes into play by tuning it in order to achieve equilibrium for a given mass of the top . equilibrium prevails when the total force on the top vanishes , i.e. when @xmath39 both in the guided case and in the stable case , we see that in order to increase the hovering height of the u - cas , we should design a better base . we do expect however , that the hovering height can not be increased indefinitely . in connection with the redesigning of the base , we have recently shown @xcite , both theoretically and experimentally , that the use of a vertically magnetized _ ring _ of radius @xmath35 as a base , increases the hovering height by more than three times to about @xmath40!. this increase in height did not came without cost , as the tolerance on the mass of the top became more stringent , being @xmath41 for the ring vs. @xmath42 in the case of the disk @xcite . all the calculations that we do are outlined in sec.([sec2 ] ) . we will now describe what we calculate and what is the motivation behind it . we start in sec.([sec2.1 ] ) , by deriving a close form expression for the first and second derivatives of the magnetic field , in terms of the magnetization density of the base . the first derivative is the magnetic force on the top , which is used throughout the paper , while the second derivative is exploited when we discuss stability . in sec.([sec2.2 ] ) we consider the problem of maximizing the hovering height in the case where the _ volume _ of the base may be _ infinite _ , and may be _ arbitrarily _ magnetized under the constraint that @xmath43 . we still _ do not _ require that the top be stable against lateral translations , and assume that it is guided along a vertical axis . we show that in this case @xmath44 where @xmath5 is a characteristic height for the problems that are discussed in this paper . it is defined as @xmath45 where @xmath7 is the magnetic permeability of the vacuum , @xmath8 is the magnetic moment of the top , @xmath10 is the mass of the top , and @xmath11 is the free - fall acceleration . in sec.([sec2.3 ] ) we consider the same problem as before but for a base which is _ uniformly _ magnetized along the vertical direction , namely @xmath46 , where @xmath47 is a unit vector in the vertical direction . in this case we find that @xmath48 in order to arrive to better bounds on @xmath0 , our next step is to limit the amount of material available for the base . thus , in sec.([sec2.4 ] ) we maximize the hovering height in the case where the base may be _ arbitrarily _ magnetized under the constraint that @xmath43 , and that its _ volume _ @xmath49 is given . here , the result is given in the form of a plot , showing the dependence @xmath50 on @xmath49 . in the limit @xmath51 , we recover the result of sec.([sec2.2 ] ) . our derivation is also _ constructive _ in the sense that it shows how to _ construct _ the optimal base for a given @xmath49 . in this paper however , we do not discuss this matter thoroughly due to space shortage . for completeness , we study in sec.([sec2.5 ] ) the same problem as in sec.([sec2.4 ] ) , but for a _ uniformly _ magnetized base . here , again we find the dependence of @xmath50 on @xmath49 and recover the result of sec.([sec2.3 ] ) . sec.([sec2.6 ] ) is the first part where stability is added into play : we begin by explaining how to _ test _ for stability of the top under the adiabatic approximation , and study the possible levitation heights of a _ uniformly _ magnetized base whose shape is _ cylindrical _ of volume @xmath49 . as the stability condition is given in the form of inequality relations ( see eq.([stab ] ) ) , it is not suffice to determine uniquely the levitation height . we therefore choose to study two particular stable points : an _ isotropic _ stable point and the _ highest _ possible stable point . the meaning of these points would become clear later . in this case we also give a plot , showing the dependence of @xmath50 on @xmath49 . this result should also be considered as a lower bound for @xmath0 , since the cylindrical base is a special case of all the base configurations that are possible . in section([sec2.7 ] ) we take one step further , and generalize the result of section([sec2.6 ] ) to the case of a base which may be _ arbitrarily _ magnetized , and look for the _ lowest _ possible stable point . in this case however , we assume that @xmath52 , to ease the solution of the problem . we also show that in this case , where @xmath53 and @xmath52 , the _ lowest _ possible stable point and the _ highest _ one coincide , so this result represent not a _ bound _ for @xmath0 , but is actually @xmath0 itself , for the case @xmath52 . in sec.([sec3 ] ) we discuss interesting aspects of our results . in particular we estimate the value of @xmath5 for modern permanent magnet materials and calculate the values of the bounds for @xmath0 . we comment on the implications of our results and discuss other related questions . a simplified model of the u - cas is shown in fig.([fig1 ] ) . it consists of a point - like particle of mass @xmath10 and magnetic moment @xmath8 ( pointing downward ) , hovering at a height @xmath13 above the @xmath54 plane . for the moment , we consider the top as if it was _ guided _ along the @xmath28-axis so that only its vertical motion is allowed . other degrees of freedom are considered ` frozen ' . the region @xmath55 may be partially or wholly filled with a magnetized substance ( `` base '' ) , whose magnetization density is denoted by @xmath31 , producing a magnetic field throughout space which we denote by @xmath56 . in what follows , we assume that the base ( and hence the magnetic field ) has a cylindrical symmetry around the @xmath28-axis . thus , along this axis the magnetic field possesses only a @xmath28-component . we further assume that this component is _ positive _ , namely that the magnetic field along the @xmath28-axis is pointing upward . using eqs.([energy]),([force ] ) we find that , when the top is in equilibrium , the total vertical component of the force on the top vanishes , @xmath57 where the term @xmath58 is the gravitational force pulling the top towards the base , and the term @xmath59 is the magnetic force which pushes the top upward . note that @xmath60 should be _ since the top is allowed to move only along the @xmath28-axis , we may write @xmath61 ( in what follows we deserve the notation @xmath62 to denote the @xmath28-component of the magnetic field along the @xmath28-axis , namely @xmath63 ) . now , the magnetic force on the top simplifies to @xmath64 and hence , when the top is in equilibrium at @xmath65 , @xmath66 reciprocity allows us to express @xmath67 in terms of @xmath31 as @xmath68 here , @xmath69 is the field at the point @xmath70 produced by a unit magnitude dipole pointing _ upward _ , located at a height @xmath28 along the @xmath28-axis . taking @xmath71 and @xmath72 as depicted in fig.([fig1 ] ) , we can write @xmath73 as @xmath74 where @xmath75 is a polar unit vector , also defined in fig.([fig1 ] ) , and @xmath7 is the magnetic permeability of the vacuum . it is worth to note that @xmath76 is nothing but the field produced at @xmath70 by a _ quadruple _ located at a height @xmath28 along the @xmath28-axis . this quadruple is made out of a pair of identical dipoles : one is located at @xmath28 and pointing downward and the other is at an infinitesimally higher position @xmath77 and pointing upward , with their magnetic moment being @xmath78 . when eq.([eq1.3 ] ) is substituted into eq.([eq1.2 ] ) we find that , at @xmath65 , the field derivative is given by @xmath79 where @xmath80 and where @xmath81 is a polar unit vector orthogonal to @xmath75 , as is shown in fig.([fig1 ] ) . we would also need the _ second _ derivative of the field , when we discuss stability . it is given by @xmath82 where @xmath83 note that in both eqs.([eq1.4 ] ) and ( [ eq1.41 ] ) , @xmath84 is defined with respect to a polar coordinate system whose origin is at @xmath65 and _ not _ at @xmath54 . also note , that in going from eq.([eq1.4 ] ) to eq.([eq1.41 ] ) , one should take into account the dependence of the unit vectors @xmath81 and @xmath85 on @xmath28 . we consider an infinitesimal volume element at some point @xmath86 within the base . it is essentially a magnetic dipole whose magnitude is @xmath9 , and whose direction we will now find by the requirement that the magnetic force on the top is maximized . we have already shown that the _ force _ on the top , contributed by one elemental dipole , is proportional to the magnetic field @xmath73 that would be produced at @xmath86 by a quadruple located at @xmath65 . to make this force maximal we assign to each point @xmath86 in @xmath55 , a magnetization density @xmath31 whose magnitude is @xmath9 and whose direction is _ antiparallel _ to the field line of @xmath73 at that point . the magnetization density so defined will _ maximize _ the force on the top , and is therefore the requested answer . mathematically , this procedure amounts to replacing @xmath31 in eq.([eq1.4 ] ) by @xmath87 , where @xmath88 is a unit vector _ parallel _ to @xmath89 , the latter being defined in eq.([n ] ) . the result is @xmath90 for which the @xmath71 integration is trivial and the @xmath72 integration may be brought to a simpler form by the transformation @xmath91 . this gives @xcite @xmath92 which together with eq.([eq1.1 ] ) shows that @xmath93 where @xmath94 is the characteristic length in our problem which will reappear in the next sections . the value of @xmath5 may be interpreted as the _ distance _ between two colinear dipoles , one of them is of strength @xmath8 while the other is of strength @xmath95 , for which the mutual force between them is @xmath96 . eq.([eq1.6 ] ) shows that even though the magnetization direction is allowed to vary everywhere inside the base , the levitation height @xmath13 is bounded . eq.([eq1.6 ] ) presents the maximum height that can be accomplished with a given substance provided that the top is not allowed to move laterally . clearly , it also serve as an _ upper _ bound for @xmath0 , as stability was not considered yet . moreover , note that we have calculated the maximum magnetic force , which is proportional to @xmath67 . the magnetic field @xmath97 on the other hand , becomes _ infinite _ at @xmath98 as can be seen by the following simple argument : since each elemental dipole within the plate contributes a field that goes as @xmath99 and since the volume of integration goes as @xmath100 , the integrand goes as @xmath101 , for which the integral diverges as @xmath102 . later we show that the divergence of @xmath97 implies that the top _ can not _ be stable when placed in such a point . in any case , it would be impossible to _ spin _ the top . uniformly magnetized plates are clearly easier to construct . in this section we find out what upper bound does this restriction sets on the highest levitation point . in another sense , the result of this section may also be considered as a _ lower _ bound on the height of levitation ( for a top which is guided ! ) , provided one considers all possible base configurations . we again consider an infinitesimal volume element within the base . it is now oriented along the @xmath28-direction and interacting with the top s magnetic dipole . if this element is exactly below the top ( i.e. it is located somewhere on the negative * * * * @xmath28-axis ) , it exerts a repelling @xmath28-directed force on the top , pushing it away . if the element is not exactly below the top then the nature of this force ( i.e. weather it is repulsive or attractive ) is determined by the angle formed between the direction of the two dipoles ( in this case both point in the @xmath28-direction ) , and the direction defined by the line joining these dipoles . we call this angle @xmath72 , and measure it with respect to the positive @xmath28-direction , as is shown in fig.[fig1 ] . there exists a critical angle @xmath103 ( see fig.([fig2 ] ) ) , for which the @xmath28-directed force vanishes such that for @xmath104 the force becomes attractive . it is therefore useless to put any material within the region @xmath104 and @xmath105 since this would _ reduce _ the magnetic force and decrease the levitation height . to find @xmath103 we eliminate the @xmath28-component of eq.([n ] ) , and find the value of @xmath72 for which it vanishes we pick the value of @xmath72 that lies between @xmath106 and @xmath107 . this gives @xmath108 we now evaluate the upward force exerted on the top , by summing the contributions of _ all _ the elements in the base located within @xmath109 . the ( uniform ) magnetization density within this region is taken to be @xmath46 . using eq.([eq1.4 ] ) , we find that @xmath110\text{.}%\end{aligned}\ ] ] the integration over @xmath71 is again trivial , and we are left with an integration over @xmath72 . the latter is brought to a simpler form by changing the @xmath72 variable into @xmath111 , hence @xmath112 this result , together with the equilibrium condition eq.([eq1.1 ] ) , suggests that in this case the maximal levitation height possible is @xmath113 \equiv4\pi\left ( \dfrac{3}{5}\right ) ^{5/2}l_{0}\cong3.5l_{0}\text{. } \label{eq1.5}%\ ] ] note , however , that in order to realize the levitation heights found in this section and in the previous one we would need an unlimited supply of magnetic material . in the following sections we find how good can we do when the _ volume _ of the base is constrained . under a given volume @xmath49 , we find the optimum _ shape _ and _ magnetization _ of the base , that will maximize the levitation height . we use the lagrange s multipliers method to treat this variational problem @xcite . first , we parametrize the _ shape _ of the plate : let @xmath114 be the upper integration radius , as is shown in fig.([fig3 ] ) . the lower integration radius is @xmath115 . utilizing the cylindrical symmetry of the problem , the volume may be written as @xmath116{c}% v=2\pi% % tcimacro{\dint \limits_{\theta_{1}}^{\pi}}% % beginexpansion { \displaystyle\int\limits_{\theta_{1}}^{\pi } } % endexpansion \sin\theta d\theta% % tcimacro{\dint \limits_{-h/\cos\theta}^{r_{e}(\theta)}}% % beginexpansion { \displaystyle\int\limits_{-h/\cos\theta}^{r_{e}(\theta ) } } % endexpansion r^{2}dr\\ = \dfrac{2\pi}{3}% % tcimacro{\dint \limits_{\theta_{1}}^{\pi}}% % beginexpansion { \displaystyle\int\limits_{\theta_{1}}^{\pi } } % endexpansion d\theta\sin\theta\left ( r_{e}^{3}(\theta)+\dfrac{h^{3}}{\cos^{3}\theta } \right ) , \end{array } \label{eq1.7}%\ ] ] where the angle @xmath117 is determined by the condition that @xmath118 and is the value of the angle to the upper - right corner of the base , as is shown in fig.([fig3 ] ) . the upward force eq.([eq1.4 ] ) , which in this case is given by setting @xmath119 is obtained from @xmath120 according to the lagrange s multipliers method @xcite , the target function that is to be maximized is @xmath121 where @xmath122 is the lagrange s multiplier and @xmath49 is the given volume of the base . the target function is a functional of @xmath114 . we require that a variation of @xmath123 with respect to @xmath114 would vanish , thus @xmath124 + \frac{1}{\lambda^{4}}\frac{\delta v}{\delta r_{e}(\theta)}=0 . \label{eq1.11}%\ ] ] using eqs.([eq1.8]),([eq1.9 ] ) we find that @xmath125 therefore , the required parametrization @xmath114 is obtained by equating the integrand in eq.([eq1.12 ] ) to zero , giving @xmath126 ^{1/4}. \label{eq1.13}%\ ] ] eq.([eq1.13 ] ) defines the _ shape _ of the optimal base . we see that it is universal in the sense that as @xmath49 is changed , the optimal new plate s shape is only a scaled version of the original one . recall however , that @xmath117 is _ different_. the value of @xmath117 , determined by eq.([eq1.8 ] ) , may be combined with eq.([eq1.13 ] ) to read @xmath127 ^{1/4}=f^{-1}\left ( \theta_{1}\right ) , \label{eq1.14}%\ ] ] which implicitly expresses @xmath117 in terms of @xmath128 . we denote this function by @xmath129 , and write @xmath130 in addition , we use eq.([eq1.13 ] ) to arrive to an explicit expression for the volume @xmath49 , given in eq.([eq1.7 ] ) : @xmath131 where@xmath132{c}% \left [ 3\sqrt{4\cos^{4}\theta+\sin^{4}\theta}\right ] ^{3/4}\\ + \dfrac{x^{3}}{\cos^{3}\theta}% \end{array } \right ) \text{.}%\ ] ] similarly , applying eq.([eq1.13 ] ) to eq.([eq1.9 ] ) yields @xmath133 with the definition@xmath134 \text{.}%\end{aligned}\ ] ] using eq.([eq1.16 ] ) with the equilibrium condition eq.([eq1.1 ] ) , we find that @xmath135 and with few more steps we arrive to @xmath116{c}% \dfrac{h}{l_{0}}=\dfrac{h}{\lambda}s\left ( \dfrac{h}{\lambda}\right ) , \\ \dfrac{v_{0}}{l_{0}^{3}}=g\left ( \dfrac{h}{\lambda}\right ) s^{3}\left ( \dfrac{h}{\lambda}\right ) . \end{array } \label{eq1.18}%\ ] ] we see that @xmath136 depends on @xmath137 through an intermediate variable @xmath128 . it can be solved numerically by `` running '' over a wide range of @xmath128 and evaluating @xmath136 and @xmath137 for each of its values . the result is given by the dash - dotted line in fig.([plots ] ) . note that at the limit @xmath51 we find that @xmath138 , in agreement with the result of section([sec2.2 ] ) . the asymptotic behavior of @xmath13 as @xmath139 is quite interesting also : this limit may be evaluated from the above equations but it is much simpler ( and more instructive ) to use the following argument : at the low volume limit , the base may be considered as a dipole centered at the origin . such an assumption is valid only if @xmath140 . in this case it is easy to see that the magnetic force , acting on the top , is just @xmath141 and hence @xmath142 using the definition of @xmath5 we may rewrite the last result as @xmath143 ^{1/4}.\ ] ] we thus conclude that @xmath144 ^{1/4}. \label{eq1.19}%\ ] ] note that since @xmath145 we see that our assumption is confirmed . the asymptotic line of eq.([eq1.19 ] ) is also plotted in fig.([plots ] ) by the dash - dot - dotted line . it is conspicuous that it is indeed an asymptotic . the solution for this case is essentially similar to the solution presented in the previous section . the only difference is in the form of the magnetization that is used . here we take @xmath46 instead of @xmath146 . the relation between @xmath136 and @xmath137 is again given by eqs.([eq1.18 ] ) with the following new definitions : @xmath116{c}% f^{-1}(\theta)\equiv\cos\theta\left [ 3\cos\theta\left ( 5\sin^{2}% \theta-2\right ) \right ] ^{1/4},\\ g(x)\equiv\dfrac{2\pi}{3}% % tcimacro{\dint \limits_{f(x)}^{\pi}}% % beginexpansion { \displaystyle\int\limits_{f(x)}^{\pi } } % endexpansion d\theta\sin\theta\left ( \begin{array } [ c]{c}% \left [ 3\cos\theta\left ( 5\sin^{2}\theta-2\right ) \right ] ^{3/4}\\ + \dfrac{x^{3}}{\cos^{3}\theta}% \end{array } \right ) , \\ s(x)\equiv6\pi% % tcimacro{\dint \limits_{f(x)}^{\pi}}% % beginexpansion { \displaystyle\int\limits_{f(x)}^{\pi } } % endexpansion d\theta\sin\theta\cos\theta\left ( 5\sin^{2}\theta-2\right ) \\ \times\left [ \dfrac{-1}{\left [ 3\cos\theta\left ( 5\sin^{2}\theta-2\right ) \right ] ^{1/4}}-\dfrac{\cos\theta}{x}\right ] . \end{array}\ ] ] the result of this calculation is given by the dotted line in fig.([plots ] ) . note that in this case @xmath147 as @xmath51 , which is again in agreement with the result of section([sec2.3 ] ) . the asymptotic behavior of @xmath13 as @xmath139 is identical to the result that was found in the previous section . in this case we take the base to be a uniformly magnetized cylinder with a magnetization @xmath46 . the radius of the cylinder is @xmath35 and its thickness is @xmath148 , with its upper base at the @xmath54 plane , as is shown in fig.([fig4 ] ) . the magnetic field outside the cylinder is essentially the field outside a solenoid with similar dimensions . thus , the @xmath28-component of the magnetic field along the @xmath28-axis is given by@xcite : @xmath149 where@xmath150 \text{.}%\ ] ] we also define the function @xmath151 , to be used later , as@xmath152 we now formulate the conditions for the _ stability _ of the spinning top against both axial and lateral translations : stability in the @xmath28-direction is determined by the sign of the ` spring - constant ' in that direction @xmath153 . under the adiabatic approximation , the latter is given by the _ curvature _ of the effective energy along that direction . hence , @xmath154 where @xmath155 is the radial distance , in cylindrical coordinate system , from the @xmath28-axis . similarly , stability in the lateral direction @xmath156 , is governed by the sign of @xmath157 , \nonumber\end{aligned}\ ] ] where in the last equality use has been made of the cylindrical symmetry of the magnetic field and the fact that @xmath62 and all of its cartesian derivatives are harmonic functions @xcite . for the spinning top to be stable against translations , both @xmath153 and @xmath158 should be _ positive_. comparing eq.([eq1.31 ] ) to eq.([eq1.30 ] ) , we see that when @xmath159 , then @xmath158 @xmath160 @xmath161 , and therefore one of the pair @xmath162,@xmath163 _ must _ be negative . thus , when the magnetic field diverges at a point , a top placed at that point can not be stable . this proves that the highest hovering height , found in section ( [ sec2.2 ] ) , is not under stable conditions . the restriction that both @xmath153 and @xmath158 should be positive defines a stable _ region _ along the @xmath28 axis . as an example it can be shown @xcite , that for a base in the shape of a thin disk of radius @xmath164 , the value of @xmath153 is positive whenever @xmath165 , whereas @xmath158 is positive for @xmath166 . the region of stability in this case is therefore @xmath167 . within that region there exists a point @xmath168 for which @xmath169 . in the case of the disk it is @xmath170 . we call this point the _ isotropically stable _ point because the restoring ( stabilizing ) force which acts on a top , which is tuned to hover at @xmath171 is isotropic , depending only on the deviation from the equilibrium position and not on its direction . for a general cylinder we now consider two distinct situations : the first is the one in which the stable point is _ isotropic _ , i.e. a hovering height @xmath13 for which @xmath172 . the second is the case where @xmath13 is at the verge of stability in the lateral direction . this is also the _ highest _ stable point , characterized by @xmath173 . using eqs.([eq1.30]),([eq1.31 ] ) we write each of the two distinct conditions as @xmath174 where @xmath175 for the first situation and @xmath176 for the second . substituting eq.([eq1.27 ] ) into eq.([eq1.32 ] ) defines a functional relationship between @xmath177 and @xmath178 denoted by the function @xmath179,@xmath180 differentiating eq.([eq1.27 ] ) with respect to @xmath28 , setting @xmath65 , and using the equilibrium condition eq.([eq1.1 ] ) , gives @xmath181 where@xmath182 combining it with an expression for the volume , gives @xmath116{c}% \dfrac{v_{0}}{l_{0}^{3}}=\dfrac{\pi r^{2}d}{l_{0}^{3}}=\pi\left ( \dfrac{d}% { r}\right ) \left ( \dfrac{r}{l_{0}}\right ) ^{3}=\pi\dfrac{d}{r}n^{3}\left ( d / r\right ) , \\ \dfrac{h}{l_{0}}=\dfrac{h}{r}\dfrac{r}{l_{0}}=g(d / r)n\left ( d / r\right ) , \end{array } \label{eq1.33}%\ ] ] which expresses the volume @xmath49 and the levitation height @xmath13 in terms of a common variable @xmath178 . running over @xmath178 , and evaluating the volume and height according to eqs.([eq1.33 ] ) , furnishes the required relation between @xmath136 and @xmath137 . this plot is shown in fig.([plots ] ) where the solid line corresponds to @xmath176 ( the isotropic case ) and the dotted line corresponds to @xmath175 ( the highest stable point ) . note that both of these plots are _ not _ monotonically increasing . they possess a maximum of @xmath13 at some optimal volume @xmath183 . for @xmath175 we find that @xmath184 and @xmath185 , whereas for @xmath176 these are @xmath186 and @xmath187 . this indicates that using too a much material _ worsen _ the largest height that can be achieved , which is reminiscent of our conclusion of sec.([sec2.3 ] ) . if the given volume is larger than @xmath183 however , we can always use only an amount of volume equal to @xmath183 and discard the rest of the material . hence , in principal at least , one _ can _ realize the largest possible height , which is why the plots of @xmath13 vs. @xmath49 had been artificially corrected by assigning the maximum value of @xmath13 for the values of @xmath49 that are larger than @xmath183 . in this last section we consider the case of an infinite base , which may be arbitrarily magnetized , such that the height of levitation is maximized , yet the top is stable against both axial and lateral translations . in order to solve this problem , one needs to maximize the magnetic force @xmath188 under the constraints that @xmath153 and @xmath158 , defined in eqs.([eq1.30 ] ) and ( [ eq1.31 ] ) respectively , are _ both _ positive . the last requirement however , results in a non - linear inequality , which can not be solved analytically . the method we take here is to use the constraint @xmath189 instead , which marks the lower end of the stability region along the @xmath28-axis . consider an infinitesimal magnetic dipole at the point @xmath86 within the base with magnetization @xmath9 . it is situated below the expected hovering position , as is shown in fig.([fig5 ] ) , and its direction makes an angle @xmath190 with the line joining the dipole to the equilibrium position of the top . substituting @xmath191 into eqs.([eq1.4]),([eq1.41 ] ) , gives @xmath192{c}% -6\sin\alpha\sin\theta\cos\theta\\ -3(3\cos^{2}\theta-1)\cos\alpha \end{array } \right\ } , \nonumber\end{aligned}\ ] ] and @xmath193{c}% -9\sin\theta\left ( 5\cos^{2}\theta-1\right ) \sin\alpha\\ -12\cos\theta(5\cos^{2}\theta-3)\cos\alpha \end{array } \right\ } .\nonumber\end{aligned}\ ] ] the target function to be extremized in this case , is given by @xmath194 = \left . \frac{\partial b_{z}}{\partial z}\right| _ { z = h}+\lambda\left . \dfrac{\partial^{2}b_{z}}{\partial z^{2}% } \right| _ { z = h } , \label{eq7}%\ ] ] and is a functional of @xmath190 , with @xmath195 being the lagrange s multiplier . since the variation of @xmath123 with respect to @xmath190 must vanish , we find , on substitution of eqs.([eq5 ] ) and ( [ eq6.1 ] ) into eq . ( [ eq7 ] ) , that @xmath190 is given by @xmath196 using eq.([eq7.1 ] ) inside eq.([eq6.1 ] ) , and requiring that @xmath197 , gives the equation for @xmath198 @xmath199{c}% 3\sin\theta\left ( 5\cos^{2}\theta-1\right ) \sin\alpha\\ + 4\cos\theta(5\cos^{2}\theta-3)\cos\alpha \end{array } \right\ } = 0\text { , } \label{eq9}%\ ] ] in which @xmath190 depends on @xmath200 according to @xmath201 note that in eq.([eq9 ] ) the variable @xmath71 has been changed to @xmath202 . this way the double integration is finite and the singularity in the integrand is eliminated . the numerical solution of eq.([eq9 ] ) gives @xmath203 using this value inside eq.([eq5 ] ) yields @xmath204 which , together with the equilibrium condition , gives @xmath205 note that , though the gradient of the field is finite , the field itself _ diverges _ , and hence @xmath206 . as @xmath207 in the case we studied here , we conclude that @xmath158 also vanishes . thus , the lowest stable hovering height and the highest one _ coincide _ , leaving no range of stability . a small relaxation in the conditions however , such as limiting the volume of the material to a finite , though very large value , results in a formation of a finite albeit small , range of stability . we showed that the maximum levitation height of the u - cas is bounded and is given in terms of a characteristic length @xmath5 , which depends on the properties of the substance that the base and the top are made of . note also , that @xmath208 where @xmath155 is the mass density of the top , and @xmath209 is its magnetization density . since @xmath210 , where @xmath211 is the residual induction , and since @xmath211 is related to the energy product @xcite via @xmath212 , we may also write @xmath5 as @xmath213 in this form , @xmath5 can be easily estimated from the knowledge of the energy product of the material . the best candidates for large hovering heights are the nd - fe - b magnets . an example of which is vacuumschmelze s vacodym 344 hr @xcite , whose remanence ( @xmath211 ) is @xmath214 kgauss and whose density is @xmath215 gr/@xmath216 . for this magnet , the magnetization is @xmath217 emu/@xmath216 . thus , with @xmath218 cm / sec@xmath219 , we find that @xmath220 meter . this gives , according to eq.([eq11 ] ) , a maximum stable hovering height of about @xmath221 meters!. typical values of the energy product , density and corresponding @xmath5 , for modern commercial magnets available today are listed in table i. [ c]|c|c||c|c|c||c| & & @xmath222[m ] + material & ( @xmath223)@xmath224@xcite & material & ( @xmath223)@xmath224@xcite & @xmath155[kg / m@xmath225 & + fe - nd - b & @xmath226[kj / m@xmath225 & fe - nd - b & @xmath226[kj / m@xmath225 & @xmath227 & @xmath228 + ferrite & @xmath229[kj / m@xmath225 & fe - nd - b & @xmath226[kj / m@xmath225 & @xmath227 & @xmath230 + strnat@xcite & @xmath231[kj / m@xmath225 & strnat@xcite & @xmath231[kj / m@xmath225 & @xmath232 & @xmath233 + according to eq.([eq1.61 ] ) the characteristic length is proportional to the _ energy product_. therefore , the larger the energy product is , the larger @xmath5 will be . a question is then asked : what is the highest conceivable energy product ? this question has been already discussed by strnat @xcite , according to which `` ... it seems reasonable to assume that the best room - temperature energy products will never exceed @xmath234 mgoe '' , which is about @xmath235 joules / m@xmath236 , and is also included in table i. these predictions , however , assume that a way might be found to give a fairly high @xmath237 to any magnetic material . table ii summarizes the upper and lower bounds for the maximum hovering height under a selected number of constraints . [ c]|lll| & & @xmath238{c}% upper\\ bound \end{tabular } $ ] + & & + & & + & & + & & + & & + it is important to note , that in the above derivation we assumed that the magnetization vector @xmath239 is _ independent _ of the field . though this is a very good approximation for modern permanent magnets like rare - earth magnets , its only an approximation even for these . furthermore , the maximum field at the top is limited by its mechanical and magnetic strength : stability considerations show @xcite that the spinning speed of the top while hovering must be greater than @xmath240 where @xmath241 and @xmath242 are the moments of inertia along the principal and secondary axes , respectively . also , the maximum field at the top is limited by its coercivity as the field is _ opposite _ to the magnetization . thus , in practice , eq.([eq11 ] ) should be considered as an _ upper bound _ for the maximum hovering height of the u - cas . yet another way to increase the hovering height of the top is to use current coil for the base , instead of a permanent magnet @xcite . the advantage of the coil over the magnetic plate is in the fact that one can raise the top to the levitation point by electrical means instead of raising the top mechanically , as in the permanent magnet case . here , the hovering height is not limited directly by the strength of the magnetic field , but rather by the amount of power that one can deliver into the coil to overcome electrical resistance . one might argue that the use of superconducting wires for the coil should lift this constraint , but the hovering height is bounded in this case as well , for if the field increases beyond a critical value , the superconductor goes into its normal phase . we have shown @xcite , that for a given height of levitation @xmath13 , the _ minimum _ power required for levitation is given by @xmath243 here , @xmath244 is the power in watts , @xmath71 is the resistivity in @xmath245 cm , @xmath11 is the free - fall acceleration in @xmath246sec@xmath247 , @xmath248 is the magnetization per unit mass of the top in emu / gr , and @xmath249 is a number of dimensions amp@xmath219emu@xmath219/erg@xmath219cm@xmath219 , and is determined by the _ shape _ and _ current distribution _ of the coil . we found that for a rectangular cross - section coil , the minimal value of @xmath251 . for the _ optimal _ coil however , we find that @xmath252 . the authors thank profs . m. milgrom and m. kugler for helpful discussions , and s. tozik and n. fernik for help in the calculations . one of the authors ( s. s. ) thanks riken , saitama in japan and in particular dr . y. kawamura for kindly hosting him during a short stay in japan that triggered this study , as it is then that he became acquainted with the u - cas . s. gov , s. shtrikman and h. thomas , `` on the dynamical stability of the hovering magnetic top '' , accepted for publication in _ physica d_. a copy may be found in _ http://xxx.lanl.gov/abs/physics/9803020 . _ p. flanders , s. gov , s. shtrikman and h. thomas , `` on the spinning motion of the hovering magnetic top '' , accepted for publication in _ physica d_. a copy may be found in _ http://xxx.lanl.gov / abs / physics/9803044_.
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the u - cas is a spinning magnetized top that is levitated in a static magnetic field .
the field is produced by a permanent magnet base , positioned below the hovering top . in this paper
we derive upper and lower bounds for @xmath0the maximum hovering height of this top .
we show that the bounds are of the form @xmath1 where @xmath2 is a dimensionless number ranging from about @xmath3 to @xmath4 , depending on the constraints on the shape of the base and on stability considerations , and @xmath5 is a characteristic length , given by @xmath6 here , @xmath7 is the permeability of the vacuum , @xmath8 is the magnetic moment of the top , @xmath9 is the maximum magnetization of the base , @xmath10 is the mass of the top , and @xmath11 is the free - fall acceleration . for modern permanent magnets
we find that @xmath12[meter ] , thus limiting @xmath13 to about few meters . 0.25 in _ index terms _ * * u - cas , levitron , magnetic trap , magnetic levitation , hovering magnetic top .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
consider lz transitions in a qubit coupled to a bath of @xmath0 quantum harmonic oscillators , as described by the hamiltonian @xmath1 with the qubit - oscillator coupling @xmath2(b_{j}+b_{j}^{\dag}).\ ] ] the energy difference between the diabatic qubit states changes linearly in time as @xmath3 ( with level - crossing speed @xmath4 and their intrinsic interaction amplitude is @xmath5 . the @xmath6 are pauli operators . the first two terms of ( [ e.ham1 ] ) define the standard landau - zener problem for an isolated two - level system . the @xmath0 harmonic oscillators in ( [ e.ham1 ] ) can have different frequencies @xmath7 , qubit - oscillator couplings @xmath8 , and interaction angles @xmath9 . the oscillators affect the qubit ( i ) by changing its energies via the diagonal coupling @xmath10 , and ( ii ) by inducing transitions between its levels via the transverse coupling @xmath11 . in order to calculate the probability that the qubit state flips due to the lz sweep , we will work in an interaction picture and split the hamiltonian ( [ e.ham1 ] ) into an interaction @xmath12 $ ] involving bit flips as described by @xmath13 , and the ( bit - flip-)free hamiltonian @xmath14 . a polaron transformation @xcite diagonalizes @xmath14 in terms of qubit - state dependent shifted oscillators , giving @xmath14 the form @xmath15 , with creation and annihilation operators for the shifted oscillators @xmath16 with @xmath17 ; the @xmath18 corresponds to the qubit state @xmath19 and the @xmath20 to @xmath21 . with bath oscillators shifting in this way , the reorganization energy @xcite gained by the system has the same value @xmath22 for both qubit states . eigenstates of the shifted oscillators are labelled as @xmath23 , where the @xmath0 components @xmath24 of the vector @xmath25 are single - oscillator excitation numbers . the free time - evolution operator @xmath26 can be written as @xmath27 , where the inner product of the vectors @xmath25 and @xmath28 shows up . we next define the interaction - picture hamiltonian as @xmath29 . in order to bring @xmath30 into a useful form , we write the oscillator operators @xmath31 in @xmath32 as @xmath33 . we choose to associate the `` @xmath18''-oscillators with the term @xmath34 of @xmath35 , and the `` @xmath20''-operators with the other term . we then write @xmath36 , where @xmath37 is the unit vector with @xmath38 component equal to @xmath39 , and likewise @xmath40 . in this form , the ` bra s ' of shifted oscillator states in @xmath32 correspond to the ` bra s ' of the qubit , while for the ` kets ' this must still be arranged by using the completeness relation @xmath41 . the interaction - picture hamiltonian then becomes @xmath42 involving the two infinite - dimensional matrices @xmath43 with @xmath44 we focus on the situation that at time @xmath45 the system starts in its ground state @xmath46 . we are now interested in the survival probability @xmath47 of the initial state @xmath19 of the qubit . @xmath47 equals the square of the norm of the projected final oscillator state @xmath48 , which can be written as @xmath49 , where @xmath50 $ ] is the time evolution operator in the interaction picture . in a time - ordered expansion of @xmath51 , only the even powers of @xmath52 will contribute to @xmath47 . as is well known , the @xmath53-order term in the expansion involves a @xmath54-fold time integral with variables @xmath55 in the interval ( @xmath56 ) . it is advantageous to make the variable transformation @xcite @xmath57 for @xmath58 , @xmath59 , and @xmath60 . we label the @xmath0-oscillator state after @xmath61 interactions as @xmath62 . for brevity , we define the frequencies @xmath63 and @xmath64 . the perturbation series for @xmath48 then becomes @xmath65 |{{\bf n}^{(2k)}_{+}}\rangle . \end{aligned}\ ] ] it would be convenient to symmetrize the @xmath66-integrals at this point , but in general the integrand is not symmetric under permutation of the @xmath66 . when transforming the @xmath66 to new variables @xmath67 and @xmath68 for @xmath69 , one finds that the @xmath70-integral yields the delta - function @xmath71 . now , since the initial state is @xmath72 , the sum over the @xmath73 in this delta function equals @xmath74 , which evidently is @xmath75 . likewise , the variables @xmath76 are all positive or zero . therefore , the delta - function can only `` click '' in the subspace @xmath77 and will do so only if the vector @xmath78 vanishes . the physical meaning of the latter statement is discussed below . performing the other @xmath79-integrals is cumbersome , so we quickly return to the integrals in eq . ( [ e.integralsxy ] ) , but this time armed with the knowledge that only the subspace @xmath80 contributes . since within this subspace the integrand _ is _ symmetric in the variables @xmath66 , it is correct to symmetrize the @xmath66-integrals in eq . ( [ e.integralsxy ] ) , i.e. we can replace them by @xmath81 . after performing these standard integrals , the @xmath76-integrals can be evaluated as well . for example , the @xmath82-integral @xmath83 vanishes unless @xmath84 vanishes , in which case it is indeed the `` subspace '' defined by @xmath85 that makes this integral equal to @xmath86 . from the time integrals we find the following _ selection rule _ : when starting in the ground state @xmath72 , the only @xmath53-order processes contributing to the survival probability @xmath87 are those with @xmath88 . hence the oscillators will end up in their initial state @xmath89 in case the qubit ends up in @xmath19 . this striking result agrees with the so - called no - go theorem @xcite , which we extended here to spin - boson problems . the time integrals do not forbid occupation of the states @xmath90 at intermediate times , nor do they restrict the intermediate oscillator states @xmath91 , but further restrictions may originate from vanishing matrix elements @xmath92 , see eq . ( [ wpm ] ) . only when the qubit ends up in @xmath21 can the lz driving dissipate energy into the bath . hence qubit and bath end up entangled . in line with the selection rule , we find that @xmath93 simplifies into @xmath94 , where the parameter @xmath95 is still to be determined by using eq . ( [ wpm ] ) . we finally obtain our central result : the exact landau - zener transition probability for a qubit arbitrarily coupled to an oscillator bath at @xmath96 is @xmath97 where the parameter @xmath98 is given by @xmath99 by introducing the spectral density @xmath100 and assuming that oscillators with equal frequencies @xmath101 have equal coupling angles @xmath102 , eq . ( [ w2final ] ) becomes @xmath103\cos[\theta(\omega)]j(\omega)/\omega \big)^2 \nonumber \\ & & + \frac{\hbar^{2}}{4\pi}\int_{0}^{\infty}\mbox{d}\omega\,\sin^{2}[\theta(\omega)]j(\omega).\end{aligned}\ ] ] no specific form of @xmath104 or @xmath102 is presupposed , but for the limit @xmath105 describing a continuum of oscillators , the two integrals in eq . ( [ w2finalcontinuum ] ) have to be finite . we first focus on purely transverse bath coupling and obtain @xmath107 , where @xmath108 is the integrated spectral density . the @xmath38 oscillator reduces the final survival probability by a factor @xmath109 $ ] , independent of the other oscillators or of the value of @xmath110 . for @xmath111 , the bath ends in @xmath89 when the qubit ends in @xmath19 , whereas the bath contains an odd number of bosons when the qubit ends in @xmath19 . qubit and bath therefore end up fully entangled , in the sense that the qubit coherence vanishes after tracing out the bath . nonzero final qubit coherence requires @xmath112 . notice also the interesting phenomenon of _ bath - assisted adiabatic following _ : for large bath coupling , i.e. @xmath113 , the qubit ends up in the initially unpopulated state , even in the absence of an intrinsic interaction @xmath5 . in solid - state environments , decoherence often occurs much faster than the bath - induced relaxation so that the latter is neglected . hence the `` standard model '' for lz transitions in dissipative environments @xcite is the single qubit diagonally coupled to an oscillator bath , which is obtained from our hamiltonian ( [ e.ham1 ] ) by setting all @xmath114 . by doing the same in eq . ( [ w2final ] ) , we find that for a qubit diagonally coupled to a bath at @xmath96 , the landau - zener transition probability @xmath115 $ ] , which is the well - known result for an _ isolated _ qubit ! thus , although the bath coupling does not commute with the qubit hamiltonian , there is no bath dependence in the transition probability , neither via qubit - bath couplings @xmath8 nor via the oscillator frequencies @xmath110 , no matter how fast the lz sweep is performed . we discuss this below . next assume that all oscillators couple to the qubit with the same angle @xmath116 . this may follow from microscopic considerations or may be engineered . for example , when all oscillators couple diagonally to the qubit in a basis described by pauli matrices @xmath117 , then an experiment where the qubit is driven via @xmath118 while @xmath119 is kept constant , is described by our hamiltonian ( [ e.ham1 ] ) with all angles @xmath116 . the reorganization energy then becomes @xmath120 , where @xmath121 occurs for diagonal coupling . from eq . ( [ w2final ] ) we deduce @xmath122 unlike for transverse coupling , the oscillators do not affect @xmath123 independently . relaxation @xmath124 , dephasing @xmath125 , and intrinsic interaction act together . in the continuum limit @xmath126 , eq . ( [ w2finaltheta ] ) holds for spectral densities @xmath104 that give rise to finite reorganization energies @xmath127 and finite integrated spectral densities @xmath128 . as a main application of our calculations , we propose to gauge the dissipative environment of a qubit via lz transitions . by gauging we mean in short : the measurement of @xmath127 and @xmath128 and the subsequent parameter fixing of appropriate model spectral densities . more in detail , we propose to perform lz sweeps and to determine transition probabilities for several fixed values of the tunable intrinsic interaction @xmath5 . notice that @xmath129 as a function of @xmath5 is a parabola in eq . ( [ w2finaltheta ] ) . figure [ fig : gauging ] as a function of intrinsic interaction @xmath5 , for several values of the coupling angle @xmath130 . parameters : @xmath131 and @xmath132 . ] depicts corresponding final transition probabilities for several coupling angles . diagonal ( @xmath133 ) and transverse ( @xmath106 ) coupling are limiting cases which have in common that @xmath129 and hence @xmath134 assume their minima for @xmath135 . neither limiting case is suitable for gauging @xmath127 . since for intermediate cases @xmath136 we find a minimal transition probability for a _ nonzero _ internal interaction @xmath137 , one can determine the reorganization energy @xmath127 by measuring @xmath138 . moreover , the integrated spectral density @xmath128 can then be identified as @xmath139 . as a consistency test , one can check whether @xmath140 holds . consistency can be further tested by changing the basis in which the lz sweep is performed : the values of @xmath127 and @xmath128 should come out independent of the angle @xmath130 . if not , then @xmath102 is not constant and eq . ( [ w2finalcontinuum ] ) applies . with the values of @xmath127 and @xmath128 thus determined , one can fix parameters in appropriate model spectral densities . for example , suppose @xmath104 has the known form @xmath141 with a power @xmath142 given by the physical nature of the environment @xcite , but with unkown strength @xmath143 and cutoff frequency @xmath144 . ( @xmath145 is the ohmic case and for @xmath146 , @xmath127 would diverge . ) then @xmath147 , in terms of the gamma function , and @xmath148 . hence lz gauging fixes @xmath149 and @xmath143 . other spectral densities may depend on more parameters . lz gauging fixes two of them . the efficient use of our gauging scenario requires the following : for @xmath134 to change considerably , one chooses @xmath150 such that @xmath151 and varies @xmath5 on the scale of @xmath152 . then @xmath127 can be measured accurately if @xmath153 is not much smaller than @xmath154 , or @xmath155 not much smaller than unity for an ohmic bath . so @xmath134 is robust under dephasing , even if dephasing is much faster than relaxation ( @xmath156 ) , unless the qubit - bath coupling is strong . this robustness has found its use in experiments on molecular nanomagnets @xcite . we presented in eqs . ( [ transitionsexact ] ) and ( [ w2final ] ) exact landau - zener transition probabilities for a qubit with arbitrary linear coupling to a zero - temperature bath . we found that qubit and bath end up entangled . our results apply to experiments where the initial and final qubit energies are off - resonant with relevant bath frequencies , for example in circuit qed @xcite , where qubit energies can be varied over a broad range and spectral densities are peaked @xcite . indeed , our predictions for transverse coupling generalize our detailed study of lz sweeps in circuit qed @xcite to more realistic situations where peaked spectral densities have nonzero widths . other applications include tunable atoms in optical cavities and in photonic crystals . for diagonal coupling we find that the transition probability does not depend on the bath at all . our exact result settles a long - standing discussion @xcite , at least for zero temperature . it corroborates and interpolates between what ao and rammer @xcite found for the fast - passage limit @xmath157 and for the opposite adiabatic limit @xmath157 , which both have been confirmed numerically @xcite . for fast lz sweeps , the bath clearly has no time to affect the transition , but the absence of any bath influence also for slower sweeps is a highly nontrivial property of this standard model . a qubit undergoing a lz sweep measures ` global ' properties of the bath , namely the frequency - integrated spectral density , and for strong coupling the reorganization energy . sample - dependent spikes in spectral densities are therefore averaged out , even in a single sweep , so that model parameters can be determined . we therefore propose to employ lz transitions for a valuable gauging of the dissipative environment of a tunable qubit .
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we calculate the _ exact _ landau - zener transitions probabilities for a qubit with arbitrary linear coupling to a bath at zero temperature . the final quantum state exhibits a peculiar entanglement between the qubit and the bath . in the special case of a diagonal coupling , the bath does not influence the transition probability , whatever the speed of the landau - zener sweep . it is proposed to use landau - zener transitions to determine both the reorganization energy and the integrated spectral density of the bath .
possible applications include circuit qed and molecular nanomagnets .
quite a number of quantum two - state systems are presently tested as candidate qubits , the units of quantum information .
good qubits are well isolated from their environment , but easy to manipulate .
this somewhat conflicting requirement has spurred renewed interest in the dynamics of qubits coupled to an environment or heat bath @xcite .
qubits can be seen as bath detectors .
for example , static qubits probe via their decay rates the bath spectral density at their transition frequencies . in solid - state environments
these rates can be strongly frequency- and sample - dependent .
we discuss an in this respect superior ` bath detection mode ' of the qubit .
one way of changing the state of a two - level system involves the forced crossing of its diabatic energies . for constant level - crossing speed
this is known as the landau - zener ( lz ) problem @xcite , which for a two - level system can be solved exactly @xcite .
this is no longer the case when taking its environment into account @xcite that may cause thermal excitation and quantum tunneling . in the low - temperature tunneling regime , analytical estimates for transition probabilities exist only for very fast and very slow sweeps , and the literature is not unanimous about the latter limit @xcite . in another line of research , incited by the paper @xcite
, it was recently proven that lz transition probabilities can be calculated exactly for some many - level systems as well , although only for some initial states @xcite . in this letter
, we extend the analysis to quantum dissipative systems and study lz transitions in spin - boson problems in a new way .
first we calculate zero - temperature lz transition probabilities exactly .
second , by making use of this exact dependence on the bath parameters , we propose to gauge the dissipative environment of a qubit by performing lz sweeps .
one advantage of this bath detection mode of the qubit is that effects of spiky bath spectral densities are averaged out in every sweep .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
both , the bohr - mottelson ( bm ) collective model @xcite and the interacting boson model ( ibm ) @xcite have thoroughly been used to study the same kind of nuclear structure problems . although very different in their formulation , both models present clear relationships . in an approximate way , the ibm can be interpreted as the second quantization of the bm shape variables @xcite . more detailed connections between both models were studied during the eighties by several authors @xcite and , more recently , by rowe and collaborators @xcite . both models have three particular cases that can be easily solved and for which a clear correspondence can be done . these three cases are : i ) the bm anharmonic vibrator and the dynamical symmetry @xmath7 ibm limit , ii ) the bm @xmath5-unstable deformed rotor and the dynamical @xmath8 ibm limit , and iii ) the bm axial rotor and the dynamical symmetry @xmath8 ibm limit including @xmath9 interactions @xcite ) . note that although it is traditionally accepted the correspondence of the dynamical symmetry @xmath10 ibm limit to a submodel of the bm , this fact has never been explicitly probed @xcite . each of these cases are assigned to a particular shape using the hill - wheeler variables @xmath11 @xcite : spherical , deformed with @xmath5-instability , and axially deformed , respectively . for transitional situations the correspondence between the two models is difficult , as rowe said `` what is simple in one model will be complicated when expressed in terms of the observables of the other '' . this situation suggests , for the case of transitional hamiltonians , to look for the connection between bm and ibm through numerical studies . among the transitional hamiltonians , a specially interesting case occurs when it describes a critical point in the transition from a given shape to another . in general , for such a situation , where the structure of the system can change abruptly by applying a small perturbation , both , the bm and the ibm , have to be solved numerically . however , recently iachello has proposed schematic bohr hamiltonians that intend to describe different critical points and that can be solved exactly in terms of the zeros of bessel functions . the first of these models is known as @xmath1 @xcite . @xmath1 is designed to describe the critical point at the transition from spherical to deformed @xmath5-unstable shapes . the potential to be used in the differential bohr equation is assumed to be @xmath12independent and , for the @xmath13 degree of freedom an infinite square well is taken . similar models were proposed later on by iachello , called @xmath14 and @xmath15 @xcite , to describe the critical points between spherical and axially deformed shapes and between axial and triaxial deformed shapes , respectively . all these models give rise to spectra and electromagnetic transition rates that are parameter free , up to a scale . in spite of their simplicity , some experimental examples were found @xcite , just after the appearance of these models . in this work , we concentrate on @xmath1 and related models . it will be published elsewhere the corresponding study for @xmath16models @xcite . the formulation of @xmath1 attracted immediately attention both experimentally and theoretically . soon after the introduction of the @xmath1 model , the nucleus @xmath17ba was proposed by casten and zamfir @xcite as a realization of it . other experimental examples proposed are : @xmath18ru @xcite , @xmath19pd @xcite and , @xmath20pd @xcite . concerning theoretical extensions of @xmath1 , first , arias @xcite proposed a generalization of the e2 operator to be used with the @xmath1 model , then caprio @xcite checked that a substitution of the original infinite well in the @xmath13 variable by a finite one , which makes the model not exactly solvable anymore , provides similar results . it showed that the @xmath1 description is `` robust in nature '' , _ i.e. _ the main features of the model remain almost unchanged under strong modification of the depth of the potential . arias and collaborators @xcite were the first authors who tried to analyze in a quantitative way the connection between the @xmath6 ibm critical point and the @xmath1 model . in particular , they established , looking to few observables that the ibm , at the critical point , gives results close to @xmath1 for a small ( @xmath21 ) number of bosons . however , the ibm results for large @xmath22 nicely reproduce the spectra and electromagnetic transition rates of a bohr hamiltonian with a @xmath23 potential ( in the following @xmath2 ) . once more , the model is not analytically solvable anymore . lvai and arias @xcite solved the bohr equation with a sextic potential with a centrifugal barrier @xcite , arriving to almost closed analytical formulae for the energies and wavefunctions . immediately after , bonatsos and collaborators explored the possibility of getting numerical solutions for the @xmath5-independent bohr hamiltonian with potentials of the type @xmath24 , with @xmath25 @xcite . these sequences of potentials allow to go from the vibrational limit , @xmath26 , to @xmath1 , @xmath27 . in particular , in ref . @xcite spectra and transition rates for the potentials @xmath28 and , @xmath29 , are given explicitly and compared with the original @xmath1 ( infinite square well potential ) case . as mentioned above , all these models are produced in the bm scheme and a natural question is to ask for the corresponding equivalence in the ibm . is the ibm able for producing the same spectra and transition rates ? if yes , does the ibm hamiltonian correspond to a critical point ? this work is intended to answer these questions for the @xmath1 and related models ( @xmath30 and , @xmath31 potentials ) and analyze the convergence as a function of the boson number . for that purpose , a large set of @xmath1 and related models results for excitation energies and transition rates are taken as reference for numerical fits of the general @xmath6 ibm transitional hamiltonian . this procedure will allow to establish the ibm hamiltonian which best fit the different @xmath0models and their relation with the critical points . the paper is organized as follows : in section [ sec - fit ] the fitting procedure is described and the obtained results are commented . section [ sec - crit ] is devoted to study the energy surfaces of the fitted ibm hamiltonians and to analyze these in relation to the critical point . in section [ sec - quasi ] the connection between the present results and the concept of quasidynamical symmetry is discussed . finally , in section [ sec - conclu ] the summary and conclusions of this work are presented . the most general , including up to two - body terms , ibm hamiltonian can be written in multipolar form as , @xmath32 where @xmath33 is the @xmath34 boson number operator , and @xmath35 the symbol @xmath36 stands for the scalar product , defined as @xmath37 where @xmath38 corresponds to the @xmath39 component of the operator @xmath40 . the operator @xmath41 ( where @xmath5 refers to @xmath42 and @xmath34 bosons ) is introduced to ensure the correct tensorial character under spatial rotations . the electromagnetic transitions can also be analyzed in the framework of the ibm . in particular , in this work we will focus on the @xmath43 transitions . the most general @xmath43 transition operator including up to one body terms can be written as , @xmath44,\ ] ] where @xmath45 is the boson effective charge and @xmath46 is a structure parameter . the @xmath0models are intended to be of use for @xmath5-unstable nuclei having @xmath47 as symmetry algebra . for the construction of an ibm @xmath5-unstable transitional hamiltonian it is sufficient to impose in eq . ( [ ham1 ] ) @xmath48 ( this implies that no casimir operator from the @xmath10 algebra is included ) as can be observed if the hamiltonian ( [ ham1 ] ) is rewritten in terms of casimir operators ( the definition for the casimir operators have been taken from @xcite ) : @xmath49 + \frac{18}{35 } \kappa_4 \,\hat c_2[u(5)]\\ & + & \big(\kappa_1-\frac{\kappa_3}{10}-\frac{\kappa_4}{14}\big)\,\hat c_2[o(3)]+ \big(\frac{\kappa_3}{2}-\frac{3}{14 } \kappa_4\big)\,\hat c_2[o(5 ) ] -\frac{\kappa_0}{4}\,\hat c_2[o(6 ) ] . \label{ham - cas}\end{aligned}\ ] ] if additionally , we want to construct an ibm transitional hamiltonian that preserves the @xmath47 symmetry , casimir operators for @xmath7 , @xmath8 and @xmath47 can be included but not the quadratic @xmath50 casimir operator . this condition , translated to the multipolar form language used in eq . ( [ ham1 ] ) , leads to the constraint @xmath51 ( see eq . ( [ ham - cas ] ) ) . in addition , the structure parameter , @xmath46 , in the @xmath52 operator is usually taken as zero in the standard ibm calculations for @xmath5-flat hamiltonians . in our calculations we will impose @xmath48 , _ i.e. _ the @xmath5 flatness . to make more simple the later analysis , we will restrict ourself to the case @xmath53 , leaving as free parameters @xmath54 , @xmath55 , @xmath56 ( plus @xmath57 that fixes the energy scale ) . in practice , we do not impose the constraint @xmath58 but , as it will be shown , the condition will be fulfilled in every fit . in this section we describe the procedure for getting the ibm hamiltonian which best fit the different @xmath0models . the @xmath59 test is used to perform the fitting . the @xmath59 function is defined in the standard way , @xmath60 where @xmath61 is the number of data , from a specific @xmath1-model , to be fitted , @xmath62 is the number of parameters used in the ibm fit , @xmath63 is an energy level ( or a @xmath64 value ) taken from a particular @xmath0model , @xmath65 is the corresponding calculated ibm value , and @xmath66 is an arbitrary error assigned to each @xmath63 . in order to perform the fit , we minimize the @xmath59 function for the energies , using @xmath57 , @xmath54 , @xmath55 and @xmath56 as free parameters and @xmath67 and @xmath68 fixed to zero . for doing this task we use minuit @xcite , which allows to minimize any multi - variable function . the labels for the energy levels follow the usual notation introduced for the @xmath1 model : @xmath69 enumerates the zeros of the @xmath13 part of the wave function , and @xmath70 is the label for the @xmath47 algebra , _ i.e. _ the @xmath47 seniority quantum number , which is a good quantum number along all the transition from @xmath7 to @xmath8 . the selected set of levels included in the fit for the different @xmath0models are : * for the @xmath71 band , all the states with angular momentum lower than @xmath72 and @xmath73 . an arbitrary @xmath74 is used for these states except for the @xmath75 state for which @xmath76 is used . this latter value allows to normalize all the ibm energies to @xmath77 . note that the energy of the state @xmath75 is fixed arbitrarily to @xmath78 ( remind that the spectrum is calculated up to a global scale factor ) . * for the @xmath79 band , all the states with angular momentum lower than @xmath80 and @xmath81 . an arbitrary @xmath82 is used for these states . * for the @xmath83 band , just the states with ( @xmath84 ) and ( @xmath85 ) are included . an arbitrary @xmath86 is used for these states . with this selection , the number of energy levels included in the fit , @xmath61 , is equal to @xmath87 . note that the state @xmath88 is not a real data to be reproduced because we are interested just in excitation energies and therefore the ground state is naturally fixed to zero in both , @xmath0models and ibm . in table [ tab - energ - fit ] the states included in the fit are explicitly given . .states included in the energy fit . [ cols="^,^,^,^",options="header " , ] one of the most attractive features of the @xmath0models treated in this work is that they are supposed to describe , at different approximation levels , the critical point in the transition from spherical to deformed @xmath5-unstable shapes . since they are connected to a given ibm hamiltonian , as shown in the preceding section , this should correspond to the critical point in the transition from @xmath7 to @xmath8 ibm limits , _ i.e. _ this hamiltonian should produce an energy surface with @xmath89 . is this the case for the fitted ibm hamiltonians obtained in the preceding section ? before starting with the discussion it is necessary to establish a measure on how close is a given ibm hamiltonian to the critical point . an energy surface can be associated to a given ibm hamiltonian by using the intrinsic state formalism @xcite which introduces the shape variables @xmath11 in the ibm . to define the intrinsic state one has to consider that the dynamical behavior of the system can be approximately described in terms of independent bosons moving in an average field @xcite . the ground state of the system is written as a condensate , @xmath90 , of bosons that occupy the lowest - energy phonon state , @xmath91 : @xmath92 where @xmath93 @xmath13 and @xmath5 are variational parameters related with the shape variables in the geometrical collective model @xcite . the expectation value of the hamiltonian ( [ ham1 ] ) in the intrinsic state ( [ gs ] ) provides the energy surface of the system , @xmath94 . this energy surface in terms of the parameters of the hamiltonian ( [ ham1 ] ) and the shape variables can be readily obtained @xcite , @xmath95.\end{aligned}\ ] ] the shape of the nucleus is defined through the equilibrium value of the deformation parameters , @xmath13 and @xmath5 , which are obtained minimizing the ground state energy , @xmath96 . a spherical nucleus has a minimum in the energy surface at @xmath97 , while a deformed one presents the minimum at a finite value of @xmath13 . the parameter @xmath5 represents the departure from axial symmetry , _ i.e. _ @xmath98 and @xmath99 stand for an axially deformed nucleus , prolate and oblate respectively , while any other value corresponds to a triaxial shape . an additional situation appears when the energy surface is independent on @xmath5 but presents a minimum in @xmath13 , being the nucleus @xmath5-unstable . it should be noted that for a general ibm hamiltonian including up to two body terms the shape is either axially symmetric or @xmath5-unstable . moreover , the hamiltonians considered in this work correspond always to the @xmath5-unstable situation . with the tools described above one can study phase transitions in the ibm @xcite . first , the parameters that define the hamiltonian are the control parameters and normally are chosen in such a way that only one of them is a variable , while the rest remain constant . the deformation parameters @xmath13 and @xmath5 become the order parameters , although in our case the only order parameter is @xmath13 . roughly speaking , a phase transition appears when there exists an abrupt change in the shape of the system when changing smoothly the control parameter . the phase transitions can be classified according to the ehrenfest classification @xcite . first order phase transitions appear when there exists a discontinuity in the first derivative of the energy with respect to the control parameter . this discontinuity appears when two degenerate minima exist in the energy surface for two values of the order parameter @xmath13 . second order phase transitions appear when the second derivative of the energy with respect to the control parameter displays a discontinuity . this happens when the energy surface presents a single minimum for @xmath97 and the surface satisfies the condition @xmath89 . in a more modern classification , second order phase transitions belongs to the high order or continuous phase transitions @xcite . to determine whether a given hamiltonian corresponds to a critical point or not , the flatness or the existence of two degenerate minima in the energy surface should be investigated . for the case of one parameter ibm hamiltonian , _ e.g. _ consistent q ( cqf ) hamiltonians @xcite , it is simple to find an analytical expression for the critical control parameter in the hamiltonian . however , for a general ibm hamiltonian it is necessary to rewrite the energy surface in a special way , as the one presented in ref . @xcite . there , the authors manage to write the energy surface of a general ibm hamiltonian in terms of two parameters . the authors make use of some concepts from the catastrophe theory @xcite to define the two essential parameters , ( @xmath100 ) . in terms of these they find expressions for the locus , in the essential parameter space , that gives a critical point at the origin in @xmath13 , called bifurcation set , and for the locus that gives rise to two degenerate minima , called maxwell set . for the hamiltonians considered in section [ sec - fit ] @xmath48 and @xmath53 , in these cases @xmath101 and @xmath102 can be written as , @xmath103 where @xmath104 note that in the large @xmath22 limit , @xmath57 is proportional to @xmath22 ( see figure [ fig - par - not4t4 ] ) and therefore ( [ r1 ] ) can be approached by , @xmath105 this expression agrees with the use of an energy surface derived through a holstein - primakoff expansion @xcite . in this language , a critical hamiltonian corresponds to @xmath106 . in figure [ fig - r1-not4t4 ] the values of @xmath102 as a function of @xmath22 for the ibm hamiltonians obtained from the fit are presented for the different @xmath0models studied . for the @xmath1 model , the fitted ibm hamiltonian produces @xmath106 for @xmath107 . in the case of the @xmath4 the value @xmath106 is obtained for @xmath108 , while for @xmath3 it is obtained for @xmath109 . for the @xmath2 model it is known that @xmath106 is reached for very large number of bosons @xcite . ( see text for definition ) as a function of @xmath22 for the fitted ibm hamiltonians.,width=377 ] it is worth to show ( see figure [ fig - ener - r1 ] ) that for all the fitted ibm hamiltonians , the resulting energy surfaces are quite flat in a large interval of @xmath22 values and , therefore , it is justified to say that the fitted ibm hamiltonians are very close to the critical area . as a consequence , the @xmath0models will be appropriated to describe phase transition regions close to the critical point . , for selected values of @xmath102 ( see text for definition).,width=377 ] the concept of quasidynamical symmetry ( qds ) was introduced in refs . @xcite and has been used in the study of phase transitions . this concept is very useful for working with hamiltonians that present as limits two dynamical symmetries ( depending on the value of a control parameter ) . in this situation , the system shows the tendency to hold onto a given symmetry until the control parameter reach a critical value , passing the system , at this moment , onto the other symmetry . the remarkable feature is that the system can present a set of states that behave as belonging to irreducible representations ( irreps ) of the corresponding symmetry group , although in fact , they do not belong to a given irrep but to a mixture of them . in mathematical terms , a qds can be defined through the _ embedded representations _ @xcite : `` if a subset of states of a system are in one - to - one correspondence with the states of an irrep of a group g and if all the properties of the subset of states associated with observables in the lie algebra of g ( including their relationships to one another but not necessarily their relationships to states outside of the subset ) are as they would be if the states actually belonged to an irrep of the group g , then the subset of states is said to span an embedded representation of g '' . therefore , in the case of a qds , there exist a set of states that behave as belonging to a unique irrep of g , although that is only apparent , because they correspond to a superposition of irreps , but all their observables ( up to certain degree of accuracy ) are identical to the ones of states within a given irrep . in summary , the states can be expressed as a coherent superposition of irreps that behave as a single one . note that to show that a qds exists , one has to fix a subset of states and the degree of accuracy for the comparison with the observables of the dynamical symmetry . in our comparison between the ibm and the @xmath0models we observe a phenomenon which resembles the qds , _ i.e. _ part of the ibm spectrum behaves as having @xmath0symmetry , although , indeed they do not have such a symmetry . we should emphasize that this is not a real qds for two reasons : i ) @xmath0cases are not dynamical symmetry limits of the ibm and ii ) the bm and the ibm have different hilbert spaces . indeed , it is not possible to define irreps in @xmath0models and therefore embedded representations . we will call this situation quasi - critical point symmetry ( qcps ) @xcite . in order to study in detail the qcps one has to fix the degree of accuracy to be demanded to the observables . in our study , for the energies an accuracy of @xmath110 for all the states belonging to a given @xmath69 is set while for the @xmath64 values an accuracy of @xmath111 for all the studied intra - band transitions in a given @xmath69 is selected . tables [ table - states - not4t4 ] and [ tab - be2-comp ] , which correspond to @xmath112 are analyzed below , * @xmath1 : only the states in the @xmath71 band present @xmath1 qcps . * @xmath4 : only the @xmath71 states present @xmath4 qcps . * @xmath3 : only the @xmath71 states present @xmath3 qcps . * @xmath2 : all the studied states , @xmath71 , @xmath79 and @xmath83 , present @xmath2 qcps . these results , regarding the energies , can be extended to larger values of @xmath22 too ( see figure [ fig - chi2-e5-not4t4 ] ) , _ i.e. _ the values of the energies remain stable when @xmath22 increases , while for the @xmath64 values the observed differences become larger , specially in the @xmath1 case . in this paper , we have studied the connection between the @xmath0models and the ibm on the basis of a numerical mapping between models . to establish the mapping we have performed a best fit of the general @xmath6 transitional ibm hamiltonian to a selected set of energy levels produced by several @xmath0models . later on , a check to the wavefunctions , obtained with the best fit parameters , has been done by calculating relevant @xmath64 transition rates . all calculations have been done as a function of the number of bosons . once the best fit ibm hamiltonians to the different @xmath0models are obtained , their energy surfaces are constructed and analyzed with the help of the catastrophe theory so as to know how close they are to a critical point . finally , the concept of quasi - critical point symmetry is introduced , as similar to the idea of quasidynamical symmetry . we have shown that it is possible , in all cases , to establish a one - to - one mapping between the @xmath0models and the ibm with a remarkable agreement for both the energies and the @xmath64 transition rates . in general , the goodness of the fit to the energies is independent on the number of bosons , but the corresponding @xmath64 transition rates are indeed sensitive to @xmath22 . this is so specially in the @xmath1 , for which the @xmath59 value reaches a minimum for @xmath22 small ( @xmath107 ) and from there on increases notably as a function of @xmath22 . globally , the best agreement is obtained for the @xmath2 hamiltonian and the worst for the @xmath1 case . for the case of very large number of bosons and hamiltonians with @xmath47 symmetry we have confirmed the results of @xcite , _ i.e. _ the only @xmath0model that can be reproduced exactly by the ibm is @xmath2 , corresponding such a hamiltonian with the critical point of the model ( @xmath106 ) . a consequence of this excellent agreement is that it is impossible , from a experimental point of view , to discriminate between a @xmath1-model and its corresponding ibm hamiltonian when only few low - lying states are considered ( usually the four lowest states in the ground state band , plus @xmath113 and @xmath114 in the @xmath79 band ) . we have also proved that all the @xmath0models correspond to ibm hamiltonians very close to the critical area , @xmath115 . therefore , one can say that the @xmath0 models are appropriate to describe transitional @xmath12unstable regions close to the critical point . we have found that the results presented in this paper are consistent with the existence of something similar to a quasidynamical symmetry , we call this phenomenon quasi - critical point symmetry . finally , it should be noted that the use of a more general @xmath6 hamiltonian , _ e.g. _ using @xmath68 as free parameter , do not change the main conclusions of this work . we are grateful to d.j . rowe for a careful reading of the manuscript and for his valuable comments . this work has been partially supported by the spanish ministerio de educacin y ciencia and by the european regional development fund ( feder ) under projects number fis2005 - 01105 , fpa2006 - 13807-c02 - 02 and fpa2007 - 63074 , and by the junta de analuca under projects fqm160 , fqm318 , p05-fqm437 and p07-fqm-02962 . 25 a. bohr , mat . vidensk . ( 14 ) ( 1952 ) . a. bohr , b.r . mottelson , mat . vidensk . selsk . 27 ( 16 ) ( 1953 ) . a. bohr , b.r . mottelson , _ nuclear structure _ , vol . ii , ( benjamin , elmsford , ny , 1969 ) . a. arima , f. iachello , ann . phys . * 99 * , 253 ( 1976 ) . a. arima , f. iachello , ann . phys . * 111 * , 201 ( 1978 ) ; o. scholten , f. iachello , a. arima , ann . phys . , 325 ( 1978 ) ; f. iachello and a. arima , _ the interacting boson model _ ( cambridge university press , cambridge , 1987 ) . d. janssen , r.v . jolos , f. dnau , nucl . a * 224 * , 93 ( 1974 ) . dieperink , o. scholten , f. iachello , phys . rev . lett . * 44 * , 1747 ( 1980 ) . a.e.l . dieperink and o. scholten , nucl . a * 346 * , 125 ( 1980 ) . ginocchio , m.w . kirson , phys . lett . , 1744 ( 1980 ) . ginocchio , m.w . kirson , nucl . a * 350 * , 31 ( 1980 ) . kirson , ann . * 143 * , 448 ( 1982 ) . ginocchio , nucl . a * 376 * , 438 ( 1982 ) . rowe and g. thiamanova , nucl . a * 760 * , 59 ( 2005 ) . p. van isacker , phys . lett . * 83 * , 4269 ( 1999 ) . d.l . hill and j.a . wheeler , phys . 89 * , 1102 , ( 1953 ) . f. iachello , phys . lett . * 85 * , 3580 , ( 2000 ) . f. iachello , phys . 87 * , 052502 , ( 2001 ) . f. iachello , phys . lett . * 91 * , 132502 , ( 2003 ) . casten and n.v . zamfir , phys . lett . , 3584 , ( 2000 ) . casten and n.v . zamfir , phys . lett . , 052503 ( 2001 ) . j.e . garca - ramos and j.m . arias , in preparation . a. frank , c.e . alonso , and j.m . arias , phys . c * 65 * , 014301 ( 2001 ) . zamfir , _ et al _ , phys . c * 65 * , 044325 ( 2002 ) . da - li zhang and yu - xin liu , phys . rev . c * 65 * , 057301 ( 2002 ) . arias , phys . c * 63 * , 034308 ( 2001 ) . caprio , phys . c * 65 * , 031304 ( 2002 ) . arias , c.e . alonso , a. vitturi , j.e . garca - ramos , j. dukelsky , and a. frank , phys . c * 68 * , 041302(r ) ( 2003 ) . garca - ramos , j. dukelsky , and j.m . arias , phys . c * 72 * , 037301 ( 2005 ) . g. lvai and j.m . arias , phys . c * 69 * , 014304 ( 2004 ) . ushveridze , _ quasi - exactly solvable models in quantum mechanics _ ( iop publishing , bristol , 1994 ) . d. bonatsos , d. lenis , n. minkov , p.p . raychev , and p.a . c * 69 * , 044316 ( 2004 ) . a. frank and p. van isacker , _ algebraic methods in molecular and nuclear structure physics _ ( john wiley & sons , ny , 1994 ) . f. james , minuit : function minimization and error analysis reference manual , version 94.1 , cern , 1994 . j. dukelsky , g.g . dussel , r.p.j . perazzo , s.l . reich , and h.m . sofia , nucl . phys . a * 425*,93 ( 1984 ) . p. van isacker and j.q . chen , phys . c * 24 * , 684 ( 1981 ) . h.e . standley , `` introduction to phase transitions and critical phenomena '' , oxford university press , oxford ( 1971 ) . warner and r.f . casten , phys . * 48 * , 1385 ( 1982 ) . e. lpez - moreno and o. castaos , , ( 1996 ) . r. gilmore . _ catastrophe theory for scientists and engineers _ ( wiley , new york , 1981 ) . arias , j. dukelsky , j.e . garca - ramos , and j. vidal , phys . c * 75 * , 014301 ( 2007 ) . rowe , phys . lett . * 93 * , 122502 ( 2004 ) . rowe , p.s . turner , and g. rosensteel , phys . * 93 * , 232502 ( 2004 ) . rowe , nucl . a * 745 * , 47 ( 2004 ) . turner and d.j . rowe , nucl . phys . a * 756 * , 333 ( 2004 ) . rowe , private comm .
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the connections between the @xmath0models ( the original @xmath1 using an infinite square well , @xmath2 , @xmath3 and @xmath4 ) , based on particular solutions of the geometrical bohr hamiltonian with @xmath5-unstable potentials , and the interacting boson model ( ibm ) are explored . for that purpose ,
the general ibm hamiltonian for the @xmath6 transition line is used and a numerical fit to the different @xmath0models energies is performed , later on the obtained wavefunctions are used to calculate b(e2 ) transition rates .
it is shown that within the ibm one can reproduce very well all these @xmath0models .
the agreement is the best for @xmath2 and reduces when passing through @xmath3 , @xmath4 and @xmath1 , where the worst agreement is obtained ( although still very good for a restricted set of lowest lying states ) .
the fitted ibm hamiltonians correspond to energy surfaces close to those expected for the critical point . a phenomenon similar to the quasidynamical symmetry
is observed .
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in classical statistics , it is often assumed that the outcome of an experiment is precise and the uncertainty of observations is solely due to randomness . under this assumption , numerical data are represented as collections of real numbers . in recent years , however , there has been increased interest in situations when exact outcomes of the experiment are very difficult or impossible to obtain , or to measure . the imprecise nature of the data thus collected is caused by various factors such as measurement errors , computational errors , loss or lack of information . under such circumstances and , in general , any other circumstances such as grouping and censoring , when observations can not be pinned down to single numbers , data are better represented by intervals . practical examples include interval - valued stock prices , oil prices , temperature data , medical records , mechanical measurements , among many others . in the statistical literature , random intervals are most often studied in the framework of random sets , for which the probability - based theory has developed since the publication of the seminal book matheron ( 1975 ) . studies on the corresponding statistical methods to analyze set - valued data , while still at the early stage , have shown promising advances . see stoyan ( 1998 ) for a comprehensive review . specifically , to analyze interval - valued data , the earliest attempt probably dates back to 1990 , when diamond published his paper on the least squares fitting of compact set - valued data and considered interval - valued input and output as a special case ( see diamond ( 1990 ) ) . due to the embedding theorems started by brunn and minkowski and later refined by radstrm ( see radstrm ( 1952 ) ) and hrmander ( see hrmander ( 1954 ) ) , @xmath0 , the space of all nonempty compact convex subsets of @xmath1 , is embedded into the banach space of support functions . diamond ( 1990 ) defined an @xmath2 metric in this banach space of support functions , and found the regression coefficients by minimizing the @xmath2 metric of the sum of residuals . this idea was further studied in gil et al . ( 2002 ) , where the @xmath2 metric was replaced by a generalized metric on the space of nonempty compact intervals , called `` w - distance '' , proposed earlier by krner ( 1998 ) . separately , billard and diday ( 2003 ) introduced the central tendency and dispersion measures and developed the symbolic interval data analysis based on those . ( see also carvalho et al . ( 2004 ) . ) however , none of the existing literature considered distributions of the random intervals and the corresponding statistical methods . it is well known that normality plays an important role in classical statistics . but the normal distribution for random sets remained undefined for a long time , until the 1980s when the concept of normality was first introduced for compact convex random sets in the euclidean space by lyashenko ( 1983 ) . this concept is especially useful in deriving limit theorems for random sets . see , puri et al . ( 1986 ) , norberg ( 1984 ) , among others . since a compact convex set in @xmath3 is a closed bounded interval , by the definition of lyashenko ( 1983 ) , a normal random interval is simply a gaussian displacement of a fixed closed bounded interval . from the point of view of statistics , this is not enough to fully capture the randomness of a general random interval . in this paper , we extend the definition of normality given by lyashenko ( 1983 ) and propose a normal hierarchical model for random intervals . with one more degree of freedom on `` shape '' , our model conveniently captures the entire randomness of random intervals via a few parameters . it is a natural extension from lyashenko ( 1983 ) yet a highly practical model accommodating a large class of random intervals . in particular , when the length of the random interval reduces to zero , it becomes the usual normal random variable . therefore , it can also be viewed as an extension of the classical normal distribution that accounts for the extra uncertainty added to the randomness . in addition , there are two interesting properties regarding our normal hierarchical model : 1 ) conditioning on the first hierarchy , it is exactly the normal random interval defined by lyashenko ( 1983 ) , which could be a very useful property in view of the limit theorems ; 2 ) with certain choices of the distributions , a linear combination of our normal hierarchical random intervals follows the same normal hierarchical distribution . an immediate consequence of the second property is the possibility of a factor model for multi - dimensional random intervals , as the `` factor '' will have the same distribution as the original intervals . for random sets models , it is important , in the stage of parameter estimation , to take into account the geometric characteristics of the observations . for example , tanaka et al . ( 2008 ) proposed an approximate maximum likelihood estimation for parameters in the neyman - scott point processes based on the point pattern of the observation window . for another model , heinrich ( 1993 ) discussed several distance functions ( called `` contrast functions '' ) between the parametric and the empirical contact distribution function that are used towards parameter estimation for boolean models . bearing this in mind , to estimate the parameters of our normal hierarchical model , we propose a minimum contrast estimator ( mce ) based on the hitting function ( capacity functional ) that characterizes the distribution of a random interval by the hit - and - miss events of test sets . see matheron ( 1975 ) . in particular , we construct a contrast function based on the integral of a discrepancy function between the empirical and the parametric distribution measure . theoretically , we show that under certain conditions our mce satisfies a strong consistency and asymptotic normality . the simulation study is consistent with our theorems . we apply our model to analyze a daily temperature range data and , in this context , we have derived interesting and promising results . the use of an integral measure of probability discrepancy here is not new . for example , the integral probability metrics ( ipms ) , widely used as tools for statistical inferences , have been defined as the supremum of the absolute differences between expectations with respect to two probability measures . see , e.g. , zolotarev ( 1983 ) , mller ( 1997 ) , and sriperumbudur et al . ( 2012 ) , for references . especially , the empirical estimation of ipms proposed by sriperumbudur et al . ( 2012 ) drastically reduces the computational burden , thereby emphasizing the practical use of the ipms . this idea is potentially applicable to our mce and we expect similar reduction in computational intensity as for ipms . the rest of the paper is organized as follows . section [ sec : model ] formally defines our normal hierarchical model and discusses its statistical properties . section [ sec : mce ] introduces a minimum contrast estimator for the model parameters , and presents its asymptotic properties . a simulation study is reported in section [ sec : simu ] , and a real data application is demonstrated in section [ sec : real ] . we give concluding remarks in section [ sec : conclu ] . proofs of the theorems are presented in section [ sec : proofs ] . useful lemmas and other proofs are deferred to the appendix . let @xmath4 be a probability space . denote by @xmath5 the collection of all non - empty compact subsets of @xmath6 . a random compact set is a borel measurable function @xmath7 , @xmath5 being equipped with the borel @xmath8-algebra induced by the hausdorff metric . if @xmath9 is convex for almost all @xmath10 , then @xmath11 is called a random compact convex set . ( see molchanov ( 2005 ) , p.21 , p.102 . ) denote by @xmath12 the collection of all compact convex subsets of @xmath6 . by theorem 1 of lyashenko ( 1983 ) , a compact convex random set @xmath11 in the euclidean space @xmath6 is gaussian if and only if @xmath11 can be represented as the minkowski sum of a fixed compact convex set @xmath13 and a @xmath14-dimensional normal random vector @xmath15 , i.e. @xmath16 as pointed out in lyashenko ( 1983 ) , gaussian random sets are especially useful in view of the limit theorems discussed earlier in lyashenko ( 1979 ) . that is , if the conditions in those theorems are satisfied and the limit exists , then it is gaussian in the sense of ( [ def_lsko ] ) . puri et al . ( 1986 ) extended these results to separable banach spaces . in the following , we will restrict ourselves to compact convex random sets in @xmath17 , that is , bounded closed random intervals . they will be called random intervals for ease of presentation . according to ( [ def_lsko ] ) , a random interval @xmath11 is gaussian if and only if a is representable in the form @xmath18 where @xmath19 is a fixed bounded closed interval and @xmath15 is a normal random variable . obviously , such a random interval is simply a gaussian displacement of a fixed interval , so it is not enough to fully capture the randomness of a general random interval . in order to model the randomness of both the location and the `` shape '' ( length ) , we propose the following normal hierarchical model for random intervals : @xmath20 where @xmath21 is another random variable and @xmath22 $ ] is a fixed interval in @xmath3 . here , the product @xmath23 is in the sense of scalar multiplication of a real number and a set . let @xmath24 denote the lebesgue measure of @xmath17 . then , @xmath25 that is , @xmath21 is the variable that models the length of @xmath11 . in particular , if @xmath26 , then a reduces to a normal random variable . obviously , @xmath15 and @xmath21 are `` location '' and `` shape '' variables . we assume that @xmath27 . then the normal hierarchical random interval is explicitly expressible as @xmath28.\ ] ] the parameter @xmath29 is indeed unnecessary , as the difference @xmath30 can be absorbed by @xmath21 . as a result , @xmath31\ ] ] compared to the naive " model @xmath32 $ ] , for which @xmath15 is precisely the center of the interval , ( [ mod - simple ] ) has an extra parameter @xmath33 . notice that the center of @xmath11 is @xmath34 , so @xmath33 controls the difference between @xmath15 and the center , and therefore is interpreted as modeling the uncertainty that the normal random variable @xmath15 is not necessarily the center . [ rmk:1 ] there are some existing works in the literature to model the randomness of intervals . for example , a random interval can be viewed as the crisp " version of the lr - fuzzy random variable , which is often used to model the randomness of imprecise intervals such as [ approximately 2 , approximately 5 ] . see krner ( 1997 ) for detailed descriptions . however , as far as the authors are aware , models with distribution assumptions for interval - valued data have not been studied yet . our normal hierarchical random interval is the first statistical approach that extends the concept of normality while modeling the full randomness of an interval . an interesting property of the normal hierarchical random interval is that its linear combination is still a normal hierarchical random interval . this is seen by simply observing that @xmath35 for arbitrary constants @xmath36 , where `` @xmath37 '' denotes the minkowski addition . this is very useful in developing a factor model for the analysis of multiple random intervals . especially , if we assume @xmath38 , then the `` factor '' @xmath39 has exactly the same distribution as the original random intervals . we will elaborate more on this issue in section [ sec : simu ] . without loss of generality , we can assume in the model ( [ def : a_1])-([def : a_2 ] ) that @xmath40 . we will make this assumption throughout the rest of the paper . according to the choquet theorem ( molchanov ( 2005 ) , p.10 ) , the distribution of a random closed set ( and random compact convex set as a special case ) a , is completely characterized by the hitting function @xmath41 defined as : @xmath42 writing @xmath43 $ ] with @xmath44 , the normal hierarchical random interval in ( [ def : a_1])-([def : a_2 ] ) has the following hitting function : for @xmath45 $ ] : @xmath46)\\ & = & p([a , b]\cap a\neq\emptyset)\\ & = & p([a , b]\cap a\neq\emptyset,\eta\geq 0)+p([a , b]\cap a\neq\emptyset,\eta < 0)\\ & = & p(a-\eta b_0\leq\epsilon\leq b-\eta a_0,\eta\geq 0)+p(a-\eta a_0\leq\epsilon\leq b-\eta b_0,\eta < 0).\end{aligned}\ ] ] the expectation of a compact convex random set @xmath11 is defined by the aumann integral ( see aumann ( 1965 ) , artstein and vitale ( 1975 ) ) as @xmath47 in particular , the aumann expectation of a random interval @xmath11 is given by @xmath48,\ ] ] where @xmath49 and @xmath50 are the interval ends . therefore , the aumann expectation of the normal hierarchical random interval @xmath11 is @xmath51i_{(\eta\geq 0)}+[b_0\eta , a_0\eta]i_{(\eta<0)}\right\}\\ & = & e\left[a_0\eta i_{(\eta\geq 0)}+b_0\eta i_{(\eta<0)},b_0\eta i_{(\eta\geq 0)}+a_0\eta i_{(\eta<0)}\right]\\ & = & \left[a_0e\eta_{+}+b_0e\eta_{-},b_0e\eta_{+}+a_0e\eta_{-}\right],\end{aligned}\ ] ] where @xmath52 notice that @xmath53 can be interpreted as the positive part of @xmath21 , but @xmath54 is not the negative part of @xmath21 , as @xmath55 when @xmath56 . the variance of a compact convex random set @xmath11 in @xmath6 is defined via its support function . in the special case when @xmath57 , it is shown by straightforward calculations that @xmath58 or equivalently , @xmath59 where @xmath60 and @xmath61 denote the center and radius of a random interval @xmath11 . see krner ( 1995 ) . again , as we pointed out in remark [ rmk:1 ] , a random interval can be viewed as a special case of the lr - fuzzy random variable . therefore , formulae ( [ var-1 ] ) and ( [ var-2 ] ) coincide with the variance of the lr - fuzzy random variable , when letting the left and right spread both equal to 0 , i.e. , @xmath62 . see krner ( 1997 ) . for the normal hierarchical random interval @xmath11 , @xmath63 ^ 2\\ & = & e\epsilon^2+a_0 ^ 2var(\eta_{+})+b_0 ^ 2var(\eta_{-})\\ & & + 2\left(a_0e\epsilon\eta_{+}+b_0e\epsilon\eta_{-}-a_0b_0e\eta_{+}e\eta_{-}\right),\end{aligned}\ ] ] and , analogously , @xmath64 the variance of @xmath11 is then found to be @xmath65\\ & & + ( a_0+b_0)e\epsilon\eta-2a_0b_0e\eta_{+}\eta_{-}.\end{aligned}\ ] ] assuming @xmath27 , we have @xmath66 with @xmath40 . this formula certainly includes the special case of the naive " model @xmath32 $ ] , by letting @xmath67 and @xmath68 . it is more general because it also accounts for the covariance between location " and length " in calculating the total variance of the random interval , while the naive " model simply has @xmath69 . we study minimum contrast estimation ( mce ) of the parameters of the normal hierarchical random interval ( @xmath70)-(@xmath71 ) , as well as its asymptotic properties . since @xmath57 , from now on we let @xmath5 be the space of all non - empty compact subsets in @xmath3 restrictively , and let @xmath72 be the borel @xmath8-algebra on @xmath5 induced by the hausdorff metric . let @xmath12 denote the space of all non - empty compact convex subsets , i.e. , bounded closed intervals , in @xmath3 . as mentioned in the previous section , a random interval @xmath73 is a borel measurable function from a probability space @xmath4 to @xmath74 such that @xmath75 almost surely . throughout this section , we assume observing a sample of i.i.d . random intervals @xmath76 . let @xmath77 denote a @xmath78 vector containing all the parameters in the model , which takes on a value from a parameter space @xmath79 . here @xmath80 is the number of parameters . let @xmath81 denote the true value of the parameter vector . denote by @xmath82)$ ] the hitting function of @xmath83 with parameter @xmath77 . in order to introduce the mce , we will need some extra notations . let @xmath84 be a basic set and @xmath85 be a @xmath8-field over it . let @xmath86 denote a family of probability measures on ( * x*,@xmath85 ) and @xmath87 be a mapping from @xmath86 to some topologial space @xmath41 . @xmath88 denotes the parameter value pertaining to @xmath89 , @xmath90 . the classical definition of mce given in pfanzagl ( 1969 ) is quoted below . @xmath91 $ ] a family of @xmath85-measurable functions @xmath92 is a family of contrast functions if @xmath93<\infty,\ ] ] @xmath94 , and @xmath95<e_p\left[f_t\right],\ ] ] @xmath96 . in other words , a contrast function is a measurable function of the random variable(s ) whose expected value reaches its minimum under the probability measure that generates the random variable(s ) . from the view of probability , with the true parameters , a contrast function tends to have a smaller value than with other parameters . adopting notation from pfanzagl ( 1969 ) , we let @xmath86 denote a family of probability measures on ( @xmath97 ) and @xmath87 be a mapping from @xmath86 to some topologial space @xmath41 . similarly , @xmath88 denotes the parameter value pertaining to @xmath89 , @xmath98 . in a similar fashion to the contrast function in heinrich ( 1993 ) for boolean models , we give our definition of contrast function for random intervals in the following . and then the mce is defined as the minimizer of the contrast function . [ def : cf ] a family of @xmath99-measurable functions @xmath100 : @xmath101 $ ] , @xmath102 , @xmath103 is a family of contrast functions for @xmath86 , if there exists a function @xmath104 : @xmath105 such that @xmath106 and @xmath107 [ def : mce ] a @xmath99-measurable function @xmath108 : @xmath109 , which depends on @xmath110 only , is called a minimum contrast estimator ( mce ) if @xmath111 we make the following assumptions to present the theoretical results in this section . [ aspt:1 ] @xmath112 is compact , and @xmath81 is an interior point of @xmath112 . [ aspt:2 ] the model is identifiable . [ aspt:3 ] @xmath113)$ ] is continuous with respect to @xmath77 . [ aspt:4 ] @xmath114)$ ] , @xmath115 , exist and are finite on a bounded region @xmath116 . [ aspt:5 ] @xmath117)$ ] , @xmath118)$ ] , and @xmath119)$ ] , @xmath120 , exist and are finite on @xmath121 for @xmath122 . assumptions 4 and 5 are essential to establish the asymptotic normality for the mce @xmath108 . they are rather mild and can be met by a large class of capacity functionals . for example , if @xmath121 is closed , then each @xmath123 with continuous up to third order partial derivatives satisfies both assumptions , as a continuous function on a compact region is always bounded . the following theorem gives sufficient conditions under which the minumum contrast estimator @xmath108 defined above is strongly consistent . [ thm : strong - consist ] let @xmath100 be a contrast function as in definition [ def : cf ] and let @xmath108 be the corresponding mce . under the hypothesis of assumption [ aspt:1 ] and in addition if @xmath100 is equicontinuous w.r.t . @xmath124 for all @xmath125 , then , @xmath126 let @xmath127\in\mathcal{k}_{\mathcal{c}}$ ] . define an empirical estimator @xmath128;x(n))$ ] for @xmath129)$ ] as : @xmath130;x(n))=\frac{\ # \left\{x_i : [ a , b]\cap x_i\neq\emptyset , i=1,\cdots , n\right\}}{n}.\ ] ] extending the contrast function defined in heinrich ( 1993 ) ( for parameters in the boolean model ) , we construct a family of functions : @xmath131)-\hat{t}([a , b];x(n))\right]^2w(a , b)\mathrm{d}a\mathrm{d}b,\ ] ] for @xmath103 , where @xmath132 , and @xmath133 is a weight function on @xmath127 $ ] satisfying @xmath134 , @xmath135\in\mathcal{k}_\mathcal{c}$ ] . we show in the next proposition that @xmath136 , @xmath137 defined in ( [ h_def ] ) is a family of contrast functions for @xmath77 . this , together with theorem [ thm : strong - consist ] , immediately yields the strong consistency of the associated mce . this result is summarized in corollary [ coro : consist ] . [ prop : cf ] suppose that assumption [ aspt:2 ] and assumption [ aspt:3 ] are satisfied . then @xmath138 , @xmath103 , as defined in ( [ h_def ] ) , is a family of contrast functions with limiting function @xmath139)-t_{\boldsymbol{\zeta}}([a , b])\right]^2w(a , b)\mathrm{d}a\mathrm{d}b.\ ] ] in addition , @xmath138 is equicontinuous w.r.t . @xmath124 . [ coro : consist ] suppose that assumption [ aspt:1 ] , assumption [ aspt:2 ] , and assumption [ aspt:3 ] are satisfied . let @xmath138 be defined as in ( [ h_def ] ) , and @xmath140 then @xmath141 as @xmath142 . next , we show the asymptotic normality for @xmath143 . as a preparation , we first prove the following proposition . the central limit theorem for @xmath143 is then presented afterwards . [ prop : parh ] assume the conditions of lemma 1 ( in the appendix ) . define @xmath144^{t},\nonumber\ ] ] as the @xmath78 gradient vector of @xmath145 w.r.t . then , @xmath146 \stackrel{\mathcal{d}}{\rightarrow}n\left(0,\xi\right),\nonumber\ ] ] where @xmath147 is the @xmath148 symmetric matrix with the @xmath149 component @xmath150\neq\emptyset , x_1\cap[c , d]\neq\emptyset\right ) -t_{{\boldsymbol}{\theta}_0}\left([a , b]\right)t_{{\boldsymbol}{\theta}_0}\left([c , d]\right)\right\}\nonumber\\ & & \frac{\partial t_{{\boldsymbol}{\theta}_0}}{\partial\theta_i}\left([a , b]\right ) \frac{\partial t_{{\boldsymbol}{\theta}_0}}{\partial\theta_j}\left([c , d]\right ) w(a , b)w(c , d)\mathrm{d}a\mathrm{d}b\mathrm{d}c\mathrm{d}d.\label{def : xi}\end{aligned}\ ] ] [ thm : clt ] let @xmath136 be defined in ( [ h_def ] ) and @xmath143 be defined in ( [ def : theta ] ) . assume the conditions of corollary [ coro : consist ] . if additionally assumption [ aspt:5 ] is satisfied , then @xmath151 where @xmath152)w(a , b)\mathrm{d}a\mathrm{d}b$ ] , and @xmath147 is defined in ( [ def : xi ] ) . we carry out a small simulation to investigate the performance of the mce introduced in definition [ def : mce ] . assume , in the normal hierarchical model ( [ def : a_1])-([def : a_2 ] ) , that @xmath153 and @xmath154 the bivariate normal distribution conveniently takes care of the variances and covariance of the location variable @xmath15 and the shape variable @xmath21 . the removal of the freedom of @xmath29 is for model identifiability purposes ; it is seen that the hitting function @xmath155 is defined via @xmath156 and @xmath157 only . for the simulation , we assign the following parameter values : @xmath158 under the bivariate normal distribution assumption , the hitting function of our normal hierarchical model is found to be @xmath159)\nonumber\\ & = & p(a-\eta b_0\leq\epsilon\leq b-\eta a_0,\eta\geq 0)+p(a-\eta a_0\leq\epsilon\leq b-\eta b_0,\eta < 0)\nonumber\\ & = & p\left(\epsilon\leq b-\eta a_0,\eta\geq 0\right)-p\left(\epsilon < a-\eta b_0,\eta\geq 0\right)\nonumber\\ & & + p\left(\epsilon\leq b-\eta b_0,\eta < 0\right)-p\left(\epsilon < a-\eta a_0,\eta<0\right)\nonumber\\ & = & p\left(\begin{bmatrix}1 & a_0\\ 0 & -1\end{bmatrix } \begin{bmatrix}\epsilon\\ \eta\end{bmatrix}\leq\begin{bmatrix}b\\0\end{bmatrix}\right ) -p\left(\begin{bmatrix}1 & b_0\\ 0 & -1\end{bmatrix } \begin{bmatrix}\epsilon\\ \eta\end{bmatrix}\leq\begin{bmatrix}a\\0\end{bmatrix}\right)\nonumber\\ & & + p\left(\begin{bmatrix}1 & b_0\\ 0 & 1\end{bmatrix } \begin{bmatrix}\epsilon\\ \eta\end{bmatrix}\leq\begin{bmatrix}b\\0\end{bmatrix}\right ) -p\left(\begin{bmatrix}1 & a_0\\ 0 & 1\end{bmatrix } \begin{bmatrix}\epsilon\\ \eta\end{bmatrix}\leq\begin{bmatrix}a\\0\end{bmatrix}\right)\nonumber\\ & = & \phi\left(\begin{bmatrix}b\\0\end{bmatrix } ; d_1\begin{bmatrix}0\\ \mu\end{bmatrix } , d_1\sigma d_1^{'}\right ) -\phi\left(\begin{bmatrix}a\\0\end{bmatrix } ; d_2\begin{bmatrix}0\\ \mu\end{bmatrix } , d_2\sigma d_2^{'}\right)\nonumber\\ & & + \phi\left(\begin{bmatrix}b\\0\end{bmatrix } ; d_3\begin{bmatrix}0\\ \mu\end{bmatrix } , d_3\sigma d_3^{'}\right ) -\phi\left(\begin{bmatrix}a\\0\end{bmatrix } ; d_4\begin{bmatrix}0\\ \mu\end{bmatrix } , d_4\sigma d_4^{'}\right),\label{eqn : hit - fct}\end{aligned}\ ] ] where @xmath160 is the bivariate normal cdf with mean @xmath161 and covariance @xmath162 , and @xmath163 after linear transformation of variables , the terms in formula ( [ eqn : hit - fct ] ) is calculated via the standard bivariate normal cdf . by absolute continuity , @xmath82)$ ] in this case is continuous and also infinitely continuously differentiable . therefore , all the assumptions are satisfied and the corresponding mce achieves the strong consistency and asymptotic normality . according to the assigned parameter values given in ( [ eqn : par - val ] ) , @xmath164 . therefore the hitting function is well approximated by @xmath159)\\ & \approx&p(a-\eta b_0\leq\epsilon\leq b-\eta a_0,\eta\geq 0)\\ & \approx&p(a-\eta b_0\leq\epsilon\leq b-\eta a_0)\\ & = & p\left ( \begin{bmatrix}1 & a_0\\ -1 & -a_0 - 1\end{bmatrix } \begin{bmatrix}\epsilon\\ \eta\end{bmatrix}\leq \begin{bmatrix}b\\-a\end{bmatrix}\right)\\ & = & \phi\left ( \begin{bmatrix}b\\-a\end{bmatrix } ; d\begin{bmatrix}0\\ \mu\end{bmatrix } , d\sigma d^{'}\right),\end{aligned}\ ] ] where @xmath165 we use this approximate hitting function to simplify computation in our simulation study . the model parameters can be estimated by the method of moments . in most cases it is reasonable to assume @xmath166 , and consequently , @xmath167 . so the moment estimates for @xmath168 and @xmath33 are approximately @xmath169 where @xmath170 and @xmath171 denote the sample means of @xmath50 and @xmath49 , respectively . denoting by @xmath60 the center of the random interval @xmath11 , we further notice that @xmath172 . by the same approximation we have @xmath173 . define a random variable @xmath174 then , the moment estimate for @xmath175 is approximately given by the sample variance - covariance matrix of @xmath176 and @xmath177 , i.e. @xmath178 our simulation experiment is designed as follows : we first simulate an i.i.d . random sample of size @xmath179 from model ( [ def : a_1])-([def : a_2 ] ) with the assigned parameter values , then find the initial parameter values by ( [ mm-1])-([mm-3 ] ) based on the simulated sample , and lastly the initial values are updated to the mce using the function _ fminsearch.m _ in matlab 2011a . the process is repeated 10 times independently for each @xmath179 , and we let @xmath180 , successively , to study the consistency and efficiency of the mce s . figure [ fig : sample_simu ] shows one random sample of 100 observations generated from the model . we show the average biases and standard errors of the estimates as functions of the sample size in figure [ fig : results_simu ] . here , the average bias and standard error of the estimates of @xmath175 are the @xmath2 norms of the average bias and standard error matrices , respectively . as expected from corollary [ coro : consist ] and theorem [ thm : clt ] , both the bias and the standard error reduce to 0 as sample size grows to infinity . the numerical results are summarized in table [ tab : mc_1 ] . finally , we point out that the choice of the region of integration @xmath181 is important . a larger @xmath181 usually leads to more accurate estimates , but could also result in more computational complexity . we do not investigate this issue in this paper . however , based on our simulation experience , an @xmath181 that covers most of the points @xmath182 such that @xmath127 $ ] hits some of the observed intervals , is a good choice as a rule of thumb . in our simulation , @xmath183 $ ] , by ignoring the small probability @xmath184 . therefore , we choose @xmath185 , and the estimates are satisfactory . + + .average biases and standard errors of the mce s of the model parameters in the simulation study . [ cols= " > , > , > , > , > , > , > , > , > " , ] in this section , we apply our normal hierarchical model and minimum contrast estimator to analyze the daily temperature range data . we consider two data sets containing ten years of daily minimum and maximum temperatures in january , in granite falls , minnesota ( latitude 44.81241 , longitude 95.51389 ) from 1901 to 1910 , and from 2001 to 2010 , respectively . each data set , therefore , is constituted of 310 observations of the form : [ minimum temperature , maximum temperature ] . we obtained these data from the national weather service , and all observations are in fahrenheit . the plot of the data is shown in figure [ fig : real ] . the obvious correlations of the data play no roles here . + + same as in the simulation , we assume a bivariate normal distribution for @xmath186 and @xmath187 $ ] has length 1 . the initial parameter values are computed according to ( [ mm-1])-([mm-3 ] ) , and the weight function @xmath188 . the minimum contrast estimates for the model parameters are : * data set 1 ( 1901 - 1910 ) : @xmath189 * data set 2 ( 2001 - 2010 ) : @xmath190 recall that the center and the length of the normal hierarchical random interval are @xmath191 and @xmath192(@xmath193 for the two considered data sets ) , respectively . therefore , they are assumed to follow normal distributions with means @xmath194 and @xmath168 , and variances @xmath195 and @xmath196 , respectively . to assess the goodness - of - fit , we compare the fitted normal distributions with the corresponding empirical distributions for both the center and the length of the two data sets . the results are shown in figure [ fig : pdf_plot ] . for the interval length of data 2 ( 2001 - 2010 ) , the fitted normal distribution is slightly more deviated from the empirical distribution , due to the skewness and heavy tail of the data . all the other three plots show very good fittings of our model to the data . + + + denote by @xmath197 and @xmath198 respectively the random intervals from which the two data sets are drawn . the model fitted mean and variance for @xmath197 and @xmath198 are found to be : @xmath199 , \widehat{\text{var}}(a_1)=221.2313;\\ & & \hat{\text{e}}(a_2)=\left[5.3335 , 25.8416\right ] , \widehat{\text{var}}(a_2)=247.3275.\end{aligned}\ ] ] both mean and variance of the recent data are larger than those of the data 100 years ago . the two model fitted means are also shown on the data plots blue as the intervals between the solid horizontal lines in figure [ fig : real ] . in addition , the correlation coefficient of @xmath186 is @xmath200 for data set 1 and @xmath201 for data set 2 , suggesting a negative correlation between the location and the length for the january temperature range data in general . that is , colder days tend to have larger temperature ranges , and , this relationship is stronger in the more recent data . + finally , we point out that some of the parameters can be easily estimated by simple traditional methods . for example , by averaging the two interval ends respectively , we get the moment estimates for the two means : @xmath202,\\ & & \hat{\text{e}}_{m}(a_2)=\left[3.8323 , 23.6903\right].\end{aligned}\ ] ] they are shown in figure [ fig : real ] as the intervals between the dashed horizontal lines , in comparison with our model fitted means . further , the sample correlations between the interval centers and lengths are computed as @xmath203 and @xmath204 for data sets 1 and 2 , respectively . these estimates can be viewed as a preliminary analysis . our model and the mce of the parameters refine it and provide a more systematic understanding of the data , by examining their geometric structure in the framework of random sets . in this paper we introduced a new model of random sets ( specifically for random intervals ) . in many practical situations data are not completely known , or are only known with some margins of error , and it is a very important issue to consider a model which extends normality for ordinary ( numerical ) data . our hierarchical normal model extends normality for point - valued random variables , and is quite flexible in the sense that it is well suited for both theoretical investigations and for simulations and real data analysis . to these goals we have defined a minimum contrast estimator for the model parameters , and we have proved its consistency and asymptotic normality . we carry out simulation experiments , and , finally we apply our model to a real data set ( daily temperature range data obtained from the national weather service ) . our approach is suitable for extensions to models in higher dimensions , e.g. , a factor model for multiple random intervals , or more general random sets , including possible extensions to spherical random sets . assume by contradiction that @xmath108 does not converge to @xmath205 almost surely . then , there exists an @xmath206 such that @xmath207 let @xmath208 and @xmath209 . by the compactness of @xmath210 , for every @xmath211 , there exists a convergent subsequence @xmath212 of @xmath213 such that @xmath214 as @xmath215 . since @xmath81 is the true underlying parameter vector that generates @xmath110 , from definition [ def : cf ] , @xmath216 converges to @xmath217 almost surely , and any subsequence converges too . so we have @xmath218 on the other hand , almost surely , @xmath219 equation ( [ equicon ] ) follows from the equicontinuity of @xmath100 . therefore , @xmath220 where @xmath221 and consequently @xmath222 . but from the assumptions , @xmath223 . this contradicts ( [ contra ] ) . hence the desired result follows . from taylor s theorem , we have @xmath224 ^ 2 \frac{\partial h}{\partial\theta_i}\left(x\left(n\right);{\boldsymbol}{\epsilon}_n\right)\nonumber\\ & = & \frac{\partial h}{\partial\theta_i}\left(x\left(n\right);{\boldsymbol}{\theta}_0\right)\nonumber\\ & & + \sum\limits_{j=1}^{p}\left(\theta^h_{n , j}-\theta_{0,j}\right)\left [ \frac{\partial^2h}{\partial\theta_j\partial\theta_i}\left(x(n);{\boldsymbol}{\theta}_0\right)+\frac{1}{2 } \sum\limits_{l=1}^{p}\left(\theta^h_{n , l}-\theta_{0,l}\right)\frac{\partial^3h } { \partial\theta_l\partial\theta_j\partial\theta_i}\left(x\left(n\right);{\boldsymbol}{\epsilon}_n\right ) \right],\nonumber\end{aligned}\ ] ] for @xmath115 , where @xmath225 lies between @xmath81 and @xmath143 . writing the above equations in matrix form , we get @xmath226 \left({\boldsymbol}{\theta}_n^h-{\boldsymbol}{\theta}_0\right)\nonumber\\ & & = 0\label{thm3:eqn1}.\end{aligned}\ ] ] observe , by taking derivatives under the integral sign , that @xmath227 , @xmath228)-\hat{t}([a , b];x(n))\right]^2w(a , b)\mathrm{d}a\mathrm{d}b,\nonumber\\ & = & \frac{\partial}{\partial\theta_j}2\iint\limits_{s}\left[t_{\boldsymbol{\theta}}([a , b])-\hat{t}([a , b];x(n))\right ] \frac{\partial t_{{\boldsymbol}{\theta}_0}}{\partial\theta_i}([a , b])w(a , b)\mathrm{d}a\mathrm{d}b,\nonumber\\ & = & 2\iint\limits_{s}\left[t_{\boldsymbol{\theta}}([a , b])-\hat{t}([a , b];x(n))\right ] \frac{\partial^2t_{{\boldsymbol}{\theta}_0}}{\partial\theta_j\partial\theta_i}([a , b])w(a , b)\mathrm{d}a\mathrm{d}b\nonumber\\ & & + 2\iint\limits_{s}\left(\frac{\partial t_{{\boldsymbol}{\theta}_0}}{\partial\theta_j } \frac{\partial t_{{\boldsymbol}{\theta}_0}}{\partial\theta_i}\right)([a , b])w(a , b)\mathrm{d}a\mathrm{d}b\nonumber\\ & : = & i+ii.\nonumber\end{aligned}\ ] ] the first term is @xmath229\right)-\frac{1}{n}\sum_{k=1}^{n}y_k\left(a , b\right)\right ) \frac{\partial^2 t_{{\boldsymbol}{\theta}_0}}{\partial\theta_j\partial\theta_i}([a , b])w(a , b)\mathrm{d}a\mathrm{d}b\nonumber\\ & = & \frac{2}{n}\sum_{k=1}^{n}\iint\limits_{s}\left[t_{{\boldsymbol}{\theta}_0}\left([a , b]\right)-y_k\left(a , b\right)\right ] \frac{\partial^2 t_{{\boldsymbol}{\theta}_0}}{\partial\theta_j\partial\theta_i}([a , b])w(a , b)\mathrm{d}a\mathrm{d}b\nonumber\\ & = & o_p(1),\nonumber\end{aligned}\ ] ] according to the strong law of large numbers for i.i.d . random variables . therefore , @xmath230)w(a , b)\mathrm{d}a\mathrm{d}b,\nonumber\ ] ] @xmath227 . in matrix form , @xmath231)w(a , b)\mathrm{d}a\mathrm{d}b.\ ] ] observe again that @xmath232 , @xmath233)-\hat{t}([a , b];x(n))\right ] \frac{\partial^3t_{{\boldsymbol}{\epsilon}_n}}{\partial\theta_j\partial\theta_k\partial\theta_l } ( [ a , b])w(a , b)\mathrm{d}a\mathrm{d}b\right|\nonumber\\ & & + 2\left|\iint\limits_{s}\left[\left(\frac{\partial t_{{\boldsymbol}{\epsilon}_n}}{\partial\theta_j } \frac{\partial^2t_{{\boldsymbol}{\epsilon}_n}}{\partial\theta_k\partial\theta_l}\right ) + \left(\frac{\partial^2t_{{\boldsymbol}{\epsilon}_n}}{\partial\theta_j\partial\theta_k}\frac{\partial t_{{\boldsymbol}{\epsilon}_n } } { \partial\theta_l}\right ) + \left(\frac{\partial^2t_{{\boldsymbol}{\epsilon}_n}}{\partial\theta_j\partial\theta_l}\frac{\partial t_{{\boldsymbol}{\epsilon}_n } } { \partial\theta_k}\right)\right]([a , b])w(a , b)\mathrm{d}a\mathrm{d}b\right|\nonumber\\ & \leq&4\iint\limits_{s}\left|\frac{\partial^3t_{{\boldsymbol}{\epsilon}_n}}{\partial\theta_j\partial\theta_k\partial\theta_l } ( [ a , b])w(a , b)\mathrm{d}a\mathrm{d}b\right|\nonumber\\ & & + 2\left|\iint\limits_{s}\left[\left(\frac{\partial t_{{\boldsymbol}{\epsilon}_n}}{\partial\theta_j } \frac{\partial^2t_{{\boldsymbol}{\epsilon}_n}}{\partial\theta_k\partial\theta_l}\right ) + \left(\frac{\partial^2t_{{\boldsymbol}{\epsilon}_n}}{\partial\theta_j\partial\theta_k}\frac{\partial t_{{\boldsymbol}{\epsilon}_n } } { \partial\theta_l}\right ) + \left(\frac{\partial^2t_{{\boldsymbol}{\epsilon}_n}}{\partial\theta_j\partial\theta_l}\frac{\partial t_{{\boldsymbol}{\epsilon}_n } } { \partial\theta_k}\right)\right]([a , b])w(a , b)\mathrm{d}a\mathrm{d}b\right|\nonumber\\ & : = & c_1({\boldsymbol}{\epsilon}_n)\leq c_2,\nonumber\end{aligned}\ ] ] @xmath234 , by the compactness of @xmath112 . this , together with the strong consistency of @xmath143 , gives @xmath235 @xmath236 . equivalently , in matrix form , @xmath237 by the multivariate slutsky s theorem , proposition [ prop : parh ] , together with equation ( [ thm3:eqn1 ] ) , ( [ thm3:eqn2 ] ) , and ( [ thm3:eqn3 ] ) , yields the desired result . + 3 and vitale , r.a . ( 1975 ) . a strong law of large numbers for random compact sets . _ annals of probability _ , 5 , 879882 . integrals and set - valued functions . _ journal of mathematical analysis and applications _ , 12 , 112 . and diday , e. ( 2003 ) . from the statistics of data to the statistics of knowledge : symbolic data analysis . _ journal of the american statistical association _ , 98 , 462 , review article . , neto , e.a.l . , and tenorio , c.p . ( 2004 ) . a new method to fit a linear regression model for interval - valued data . _ lecture notes in computer science _ , 3238 , 295306 . least squares fitting of compact set - valued data . _ journal of mathematical analysis and applications _ , 147 , 531544 . , lubiano , m.a . , montenegro , m. , and lopez , m.t . least squares fitting of an affine function and strength of association for interval - valued data . _ metrika _ , 56 , 97111 . asymptotic properties of minimum contrast estimators for parameters of boolean models . _ metrika _ , 40 , 6974 . sur la fonction dappui des ensembles convexes dans un espace localement convexe . _ arkiv for mat . _ , 3 , 181186 . foundations of a theory of random sets . _ in stochastic geometry harding , e.f . and kendall , d.g . john wiley @xmath238 sons , new york . a variance of compact convex random sets . working paper , institut fr stochastik , bernhard - von - cotta - str . 2 09599 freiberg . ( 1997 ) . on the variance of fuzzy random variables . _ fuzzy sets and systems _ , 92 , 8393 . and nther , w. ( 1998 ) . linear regression with random fuzzy variables : extended classical estimates , best linear estimates , least squares estimates . _ information sciences _ , 109 , 95118 . ( 1979 ) . on limit theorems for sums of independent compact random subsets in the euclidean space . translated from _ zapiski nauchnykh seminarov leningradskogo otdeleniya matematicheskogo instituta _ , 85 , 113128 . statistics of random compacts in euclidean space . translated from _ zapiski nauchnykh seminarov leningradskogo otdeleniya matematicheskogo instituta _ , 98 , 115139 . _ lments pour une thorie des milieux poreux_. masson , paris . _ random sets and integral geometry_. john wiley @xmath238 sons , new york . ( 2005 ) . _ theory of random sets_. springer - verlag , london . integral probability metrics and their generating classes of functions . _ advances in applied probability _ , 29 , 429 - 443 . convergence and existence of random set distributions . _ the annals of probability _ , 12 , 3 , 726732 . ( 1969 ) . on the measurability and consistency of minimum contrast estimates . _ metrika _ , 14 , 249272 . , ralescu , d.a . , and ralescu , s.s . gaussian random sets in banach space . _ theory of probablity and its applications _ , 31 , 526529 . ( 1952 ) . an embedding theorem for spaces of convex sets . _ , 3 , 165169 . ( 1993 ) . _ convex bodies : the brunn - minkowski theory_. cambridge university press , cambridge . on the empirical estimation of integral probability metrics . _ electronic journal of statistics _ , 6 , 1550 - 1599 . random sets : models and statistics . _ international statistical review _ , 66 , 1 , 1 - 27 . parameter estimation and model selection for neyman - scott point processes . _ biometrical journal _ , 50 , 43 - 57 . probability metrics . _ theory of probability and its applications _ , 28 , 278 - 302 . notice that @xmath128;x(n))$ ] is the sample mean of i.i.d . random variables @xmath239 defined as : @xmath240\neq\emptyset , \\ 0 , & \text{otherwise}. \end{cases}.\ ] ] therefore , an application of the strong law of large numbers in the classical case yields : @xmath241\neq\emptyset\right ) = t_{\boldsymbol{\theta}_0}\left([a , b]\right),\ \text{as}\ n\to\infty,\ ] ] @xmath242 , and assuming @xmath205 is the true parameter value . that is , @xmath243;x(n)\right)\stackrel{a.s.}{\rightarrow}t_{\boldsymbol{\theta}_0}\left([a , b]\right ) , \nonumber\ ] ] as @xmath244 . it follows immediately that @xmath245;x(n))-t_{\boldsymbol{\theta}_0}\left([a , b]\right)\right]^2w(a , b)\stackrel{a.s.}{\rightarrow}0 . \nonumber\ ] ] notice that @xmath242 , @xmath246;x(n))-t_{\boldsymbol{\theta}_0}\left([a , b]\right)\right]^2w(a , b)$ ] is uniformly bounded by @xmath247 . by the bounded convergence theorem , @xmath248;x(n))-t_{\boldsymbol{\theta}_0}\left([a , b]\right)\right]^2w(a , b)\mathrm{d}a\mathrm{d}b \stackrel{a.s.}{\rightarrow}\iint\limits_{s}0\cdot \mathrm{d}a\mathrm{d}b=0 , \nonumber\ ] ] given any @xmath249 with finite lebesgue measure . this verifies that @xmath250 similarly , we also get @xmath251)-t_{\boldsymbol{\zeta}}([a , b])\right]^2w(a , b)\mathrm{d}a\mathrm{d}b\right\}=1,\ ] ] @xmath252 . equations ( [ eqn : n1 ] ) and ( [ eqn : n2 ] ) together imply @xmath253)-t_{\boldsymbol{\zeta}}([a , b])\right]^2w(a , b)\mathrm{d}a\mathrm{d}b,\ \boldsymbol{\theta } , \boldsymbol{\zeta}\in\theta.\ ] ] by assumption [ aspt:2 ] , @xmath254)\neq t_{\boldsymbol{\zeta}}([a , b])$ ] , for @xmath255 , except on a lebesgue set of measure 0 . this together with ( [ eqn : n ] ) gives @xmath256 which proves that @xmath138 , @xmath257 is a family of contrast functions . to see the equicontinuity of @xmath138 , notice that @xmath258 , we have @xmath259)-\hat{t}([a , b];x(n))\right)^2w(a , b)\mathrm{d}a\mathrm{d}b\\ & & -\iint\limits_{s}\left(t_{\boldsymbol{\theta}_2}([a , b])-\hat{t}([a , b];x(n))\right)^2w(a , b)\mathrm{d}a\mathrm{d}b|\\ & = & |\iint\limits_{s}\left(t_{\boldsymbol{\theta}_1}([a , b])-t_{\boldsymbol{\theta}_2}([a , b])\right ) \left(t_{\boldsymbol{\theta}_1}([a , b])+t_{\boldsymbol{\theta}_2}([a , b ] ) -2\hat{t}([a , b];x(n))\right)w(a , b)\mathrm{d}a\mathrm{d}b|\\ & \leq&4c\iint\limits_{s}\left|t_{\boldsymbol{\theta}_1}([a , b])-t_{\boldsymbol{\theta}_2}([a , b])\right|\mathrm{d}a\mathrm{d}b,\end{aligned}\ ] ] since , by definition ( [ h_def ] ) , @xmath260 is uniformly bounded by @xmath261 , @xmath262 then the equicontinuity of @xmath138 follows from the continuity of @xmath254)$ ] . let @xmath138 be the contrast function defined in ( [ h_def ] ) . under the hypothesis of assumption [ aspt:4 ] , @xmath263 \stackrel{\mathcal{d}}{\rightarrow } n\left(0,\delta_i\right),\ \text{as}\ n\to\infty,\ ] ] for @xmath115 , where @xmath264\neq\emptyset , x_1\cap[c , d]\neq\emptyset\right ) -t_{{\boldsymbol}{\theta}_0}\left([a , b]\right)t_{{\boldsymbol}{\theta}_0}\left([c , d]\right)\right\}\nonumber\\ & & \times\frac{\partial t_{{\boldsymbol}{\theta}_0}}{\partial\theta_i}\left([a , b]\right ) \frac{\partial t_{{\boldsymbol}{\theta}_0}}{\partial\theta_i}\left([c , d]\right ) w(a , b)w(c , d)\mathrm{d}a\mathrm{d}b\mathrm{d}c\mathrm{d}d.\nonumber\end{aligned}\ ] ] we will write @xmath265\right)}{\partial\theta_i}= t_{{\boldsymbol}{\theta}_0}^i\left(a , b\right)$ ] to simplify notations . exchanging differentiation and integration by the bounded convergence theorem , we get @xmath266\right)-\hat{t}\left([a , b];x(n)\right)\right)^2w(a , b)\mathrm{d}a\mathrm{d}b\nonumber\\ & = & \iint\limits_{s}\frac{\partial}{\partial\theta_i } \left(t_{{\boldsymbol}{\theta}_0}\left([a , b]\right)-\hat{t}\left([a , b];x(n)\right)\right)^2w(a , b)\mathrm{d}a\mathrm{d}b\nonumber\\ & = & \iint\limits_{s}2\left(t_{{\boldsymbol}{\theta}_0}\left([a , b]\right)-\hat{t}\left([a , b];x(n)\right)\right ) t_{{\boldsymbol}{\theta}_0}^i\left(a , b\right)w(a , b)\mathrm{d}a\mathrm{d}b.\nonumber\end{aligned}\ ] ] define @xmath267 as in ( [ y_def ] ) . then , @xmath268\right)-\frac{1}{n}\sum_{k=1}^{n}y_k\left(a , b\right)\right ) t_{{\boldsymbol}{\theta}_0}^i\left(a , b\right)w(a , b)\mathrm{d}a\mathrm{d}b\nonumber\\ & = & \frac{2}{n}\iint\limits_{s}\sum_{k=1}^{n}\left(t_{{\boldsymbol}{\theta}_0}\left([a , b]\right)-y_k\left(a , b\right)\right ) t_{{\boldsymbol}{\theta}_0}^i\left(a , b\right)w(a , b)\mathrm{d}a\mathrm{d}b\nonumber\\ & = & \frac{1}{n}\sum_{k=1}^{n}2\iint\limits_{s}\left(t_{{\boldsymbol}{\theta}_0}\left([a , b]\right)-y_k\left(a , b\right)\right ) t_{{\boldsymbol}{\theta}_0}^i\left(a , b\right)w(a , b)\mathrm{d}a\mathrm{d}b\label{eqn : parh}\\ & : = & \frac{1}{n}\sum_{k=1}^{n}r_k.\nonumber\end{aligned}\ ] ] notice that @xmath269 s are i.i.d . random variables : @xmath270 . + let @xmath271 be a partition of @xmath181 , and @xmath272 be any point in @xmath273 , @xmath274 . let @xmath275 . denote by @xmath276 the area of @xmath273 . by the definition of the double integral , @xmath277\right)-y_k\left(a , b\right)\right ) t_{{\boldsymbol}{\theta}_0}^i\left(a , b\right)w(a , b)\mathrm{d}a\mathrm{d}b\nonumber\\ & = & \lim_{\lambda\rightarrow 0}\left\{\sum_{j=1}^{m}\left(t_{{\boldsymbol}{\theta}_0 } \left([a_j , b_j]\right)-y_k\left(a_j , b_j\right)\right ) t_{{\boldsymbol}{\theta}_0}^i\left(a_j , b_j\right)w(a_j , b_j)\delta\sigma_j\right\}.\nonumber\end{aligned}\ ] ] therefore , by the lebesgue dominated convergence theorem , @xmath278\right)-y_k\left(a_j , b_j\right)\right ) t_{{\boldsymbol}{\theta}_0}^i\left(a_j , b_j\right)w(a_j , b_j)\delta\sigma_j\right\}\\ & = & 2\lim_{\lambda\rightarrow 0}\left\{\sum_{j=1}^{m}\left[e\left(t_{{\boldsymbol}{\theta}_0 } \left([a_j , b_j]\right)-y_k\left(a_j , b_j\right)\right)\right ] t_{{\boldsymbol}{\theta}_0}^i\left(a_j , b_j\right)w(a_j , b_j)\delta\sigma_j\right\}\label{eqn_1}\\ & = & 2\lim_{\lambda\rightarrow 0}\left\{\sum_{j=1}^{m}0\right\}=0.\end{aligned}\ ] ] moreover , @xmath279\right)-y_k\left(a_j , b_j\right)\right ) t_{{\boldsymbol}{\theta}_0}^i\left(a_j , b_j\right)w(a_j , b_j)\delta\sigma_j\right\}\right\}^2\\ & = & 4e\lim_{\lambda_1\rightarrow 0}\lim_{\lambda_2\rightarrow 0 } \left\{\sum_{j_1=1}^{m_1}\left(t_{{\boldsymbol}{\theta}_0 } \left([a_{j_1},b_{j_1}]\right)-y_k\left(a_{j_1},b_{j_1}\right)\right ) t_{{\boldsymbol}{\theta}_0}^i\left(a_{j_1},b_{j_1}\right)w(a_{j_1},b_{j_1})\delta\sigma_{j_1}\right\}\\ & & \left\{\sum_{j_2=1}^{m_2}\left(t_{{\boldsymbol}{\theta}_0 } \left([a_{j_2},b_{j_2}]\right)-y_k\left(a_{j_2},b_{j_2}\right)\right ) t_{{\boldsymbol}{\theta}_0}^i\left(a_{j_2},b_{j_2}\right)w(a_{j_2},b_{j_2})\delta\sigma_{j_2}\right\}\\ & = & 4e\lim_{\lambda_1\rightarrow 0}\lim_{\lambda_2\rightarrow 0}\sum_{j_1=1}^{m_1}\sum_{j_2=1}^{m_2 } \left(t_{{\boldsymbol}{\theta}_0}\left([a_{j_1},b_{j_1}]\right)-y_k\left(a_{j_1},b_{j_1}\right)\right ) \left(t_{{\boldsymbol}{\theta}_0}\left([a_{j_2},b_{j_2}]\right)-y_k\left(a_{j_2},b_{j_2}\right)\right)\\ & & t_{{\boldsymbol}{\theta}_0}^i\left(a_{j_1},b_{j_1}\right)t_{{\boldsymbol}{\theta}_0}^i\left(a_{j_2},b_{j_2}\right ) w(a_{j_1},b_{j_1})w(a_{j_2},b_{j_2})\delta\sigma_{j_1}\delta\sigma_{j_2}\\ & = & 4\lim_{\lambda_1\rightarrow 0}\lim_{\lambda_2\rightarrow 0}\sum_{j_1=1}^{m_1}\sum_{j_2=1}^{m_2 } e\left(t_{{\boldsymbol}{\theta}_0}\left([a_{j_1},b_{j_1}]\right)-y_k\left(a_{j_1},b_{j_1}\right)\right ) \left(t_{{\boldsymbol}{\theta}_0}\left([a_{j_2},b_{j_2}]\right)-y_k\left(a_{j_2},b_{j_2}\right)\right)\\ & & t_{{\boldsymbol}{\theta}_0}^i\left(a_{j_1},b_{j_1}\right)t_{{\boldsymbol}{\theta}_0}^i\left(a_{j_2},b_{j_2}\right ) w(a_{j_1},b_{j_1})w(a_{j_2},b_{j_2})\delta\sigma_{j_1}\delta\sigma_{j_2}\label{eqn_2}\\ & = & 4\lim_{\lambda_1\rightarrow 0}\lim_{\lambda_2\rightarrow 0}\sum_{j_1=1}^{m_1}\sum_{j_2=1}^{m_2 } cov\left(y_k\left(a_{j_1},b_{j_1}\right),y_k\left(a_{j_2},b_{j_2}\right)\right)\\ & & t_{{\boldsymbol}{\theta}_0}^i\left(a_{j_1},b_{j_1}\right)t_{{\boldsymbol}{\theta}_0}^i\left(a_{j_2},b_{j_2}\right ) w(a_{j_1},b_{j_1})w(a_{j_2},b_{j_2})\delta\sigma_{j_1}\delta\sigma_{j_2}\\ & = & 4\iiiint\limits_{s\times s}cov\left(y_k\left(a , b\right),y_k\left(c , d\right)\right ) t_{{\boldsymbol}{\theta}_0}^i\left(a , b\right)t_{{\boldsymbol}{\theta}_0}^i\left(c , d\right ) w(a , b)w(c , d)\mathrm{d}a\mathrm{d}b\mathrm{d}c\mathrm{d}d\\ & = & 4\iiiint\limits_{s\times s}\left\{p\left(x_k\cap[a , b]\neq\emptyset , x_k\cap[c , d]\neq\emptyset\right ) -t_{{\boldsymbol}{\theta}_0}\left([a , b]\right)t_{{\boldsymbol}{\theta}_0}\left([c , d]\right)\right\}\\ & & t_{{\boldsymbol}{\theta}_0}^i\left(a , b\right)t_{{\boldsymbol}{\theta}_0}^i\left(c , d\right ) w(a , b)w(c , d)\mathrm{d}a\mathrm{d}b\mathrm{d}c\mathrm{d}d.\end{aligned}\ ] ] from the central limit theorem for i.i.d . random variables , the desired result follows . by the cramr - wold device , it suffices to prove @xmath280 for arbitrary real numbers @xmath281 . it is easily seen from ( [ eqn : parh ] ) in the proof of lemma 1 that @xmath282\right)-y_k\left(a , b\right)\right ) \frac{\partial t_{{\boldsymbol}{\theta}_0}}{\partial\theta_i}\left([a , b]\right)w(a , b)\mathrm{d}a\mathrm{d}b\right)\nonumber\\ & : = & \frac{1}{n}\sum\limits_{k=1}^{n}\left(2\sum\limits_{i=1}^p\lambda_iq_k^i\right).\nonumber\end{aligned}\ ] ] by lemma 1 , @xmath283 in view of the central limit theorem for i.i.d . random variables , ( [ prop1:target ] ) is reduced to proving @xmath284 by a similar argument as in lemma 1 , together with some algebraic calculations , we obtain @xmath285\right)-y_k\left(a , b\right)\right ) \frac{\partial t_{{\boldsymbol}{\theta}_0}}{\partial\theta_i}\left([a , b]\right)w(a , b)\mathrm{d}a\mathrm{d}b\right)\\ & & \left(\iint\limits_{s}\left(t_{{\boldsymbol}{\theta}_0}\left([a , b]\right)-y_k\left(a , b\right)\right ) \frac{\partial t_{{\boldsymbol}{\theta}_0}}{\partial\theta_j}\left([a , b]\right)w(a , b)\mathrm{d}a\mathrm{d}b\right)\\ & = & 4\sum\limits_{1\leq i , j\leq p}\lambda_i\lambda_j\iiiint\limits_{s\times s}\left\{p\left(x_1\cap[a , b]\neq\emptyset , x_1\cap[c , d]\neq\emptyset\right ) -t_{{\boldsymbol}{\theta}_0}\left([a , b]\right)t_{{\boldsymbol}{\theta}_0}\left([c , d]\right)\right\}\\ & & \frac{\partial t_{{\boldsymbol}{\theta}_0}}{\partial\theta_i}\left([a , b]\right ) \frac{\partial t_{{\boldsymbol}{\theta}_0}}{\partial\theta_j}\left([c , d]\right ) w(a , b)w(c , d)\mathrm{d}a\mathrm{d}b\mathrm{d}c\mathrm{d}d.\end{aligned}\ ] ] this validates ( [ prop1:target2 ] ) , and hence finishes the proof .
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many statistical data are imprecise due to factors such as measurement errors , computation errors , and lack of information . in such cases , data are better represented by intervals rather than by single numbers .
existing methods for analyzing interval - valued data include regressions in the metric space of intervals and symbolic data analysis , the latter being proposed in a more general setting .
however , there has been a lack of literature on the parametric modeling and distribution - based inferences for interval - valued data . in an attempt to fill this gap ,
we extend the concept of normality for random sets by lyashenko and propose a normal hierarchical model for random intervals . in addition
, we develop a minimum contrast estimator ( mce ) for the model parameters , which we show is both consistent and asymptotically normal .
simulation studies support our theoretical findings , and show very promising results .
finally , we successfully apply our model and mce to a real dataset .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
transiting planets provide a wealth of information about exoplanetary systems . short - term variations in the mid - eclipse times of the transits may reveal the presence of moons , trojans or other planets ( ( * ? ? ? * holman & murray 2005 ) , ( * ? ? ? * agol 2005 ) , ( * ? ? ? * ford & holman 2007 ) ) , whereas long - term variations could result from orbital precession ( ( * ? ? ? * miralda escud 2002 ) ) . this provides further constraints on theories of planetary system formation and evolution , as well on theories of planetary atmospheres and their composition . we observed three transits of tres-1 on ut 2007 september 12 , 15 , and 18 , corresponding to epochs @xmath0 , 153 , and 154 of the ephemeris given by ( * ? ? ? * winn ( 2007 ) ) : @xmath1 and one transit of wasp-2 on ut 2007 september 13 , corresponding to epoch @xmath2 of the ephemeris given by ( * ? ? ? * charbonneau ( 2007 ) ) : @xmath3 we used ag2 , a frame - transfer ccd mounted at the folded cassegrain focus of the william herschel telescope of the isaac newton group , la palma , spain . ag2 is an ing - designed autoguider head with e2v ccd having a field of view ( fov ) @xmath4 and a pixel scale of @xmath5 . we used a kitt peak r filter in order to minimize the effect of color - dependent atmospheric extinction on the differential photometry and the effect of limb - darkening on the transit light curve . we observed under nearly perfect conditions on the nights of ut september 12 , 13 , and 18 . on the night of ut september 15 , we observed under scattered clouds in the second half of the night . on all nights , autoguiding ensured the positions of all stars on the ccd varied by no more than 3 pixels over the course of each night . we strongly defocused the telescope in order to minimize the effect of the pixel - to - pixel sensitivity variations , and also to enhance the duty cycle . the full - width at half - maximum ( fwhm ) of stellar images was 7 pixels for wasp-2 and ranged from 8 to 13 pixels for tres-1 . for tres-1 for the three nights , we acquired 5 , 10 and 10 second exposures , while for wasp-2 we used 7 second exposures . we used standard iraf procedures for the overscan correction , bias subtraction and performing the differential photometry . we did not use flat - fields to minimise the impact of position - angle dependent scattered light entering the aperture . different aperture sizes were tried in order to find out the one that produced the minimum noise in the out - of - transit data . for tres-1 for the three nights , the optimum aperture radii were 31 , 31 , and 34 pixels , while for wasp-2 it was 30 pixels . the sky background was subtracted , using an estimate of its brightness determined within an annulus centered on each star . because of the strong defocusing and other field stars , we set an inner radius of the annulus equal to the aperture radius and used the width of 10 pixels . the comparison stars used were 2mass : j19041058 + 3638409 , j19040934 + 3639195 , and j19040792 + 3640116 for tres-1 , and 2mass : j20304846 + 0627355 , and j20305168 + 0628008 for wasp-2 . for each night , differential photometry was obtained by taking the ratio of the signal of the variable to the mean of the comparisons , both normalized by a smooth function of time . to estimate appropriate error bars for our data , we used a procedure similar to that by ( * ? ? ? * narita ( 2007 ) ) . we first fitted the light curves with the analytic formulae of ( * ? ? ? * mandel & agol ( 2002 ) ) to find the differences between the data and the best - fitting model . we rescaled the error bars to satisfy @xmath6 , where @xmath7 is the number of the measurements in each light curve . for the three nights of tres-1 we found that the true errors are higher by factors of 2.2 , 1.6 and 1.9 respectively than the errors including only the poisson noise , and for the one night of wasp-2 by a factor of 2.2 . differential photometry with these rescaled error bars and the best - fitting model ( solid line ) are shown in fig . [ wt_lc ] . in order to estimate the amount of the time - correlated red noise , we solved the equations given e.g. by ( * ? ? ? * narita ( 2007 ) ) : @xmath8 @xmath9 where @xmath10 is the standard deviation of each residual and @xmath11 is the standard deviation of the average of the successive @xmath12 points . we selected @xmath12 to correspond to about 20 - 50 minutes depending on the length of the interval with the correlated errors . @xmath13 is the white noise , which is uncorrelated and averages down as @xmath14 , @xmath15 is the red noise , which is correlated and remains constant for specified @xmath12 . we adjusted the error bars by multiplying @xmath16^{1/2}$ ] and used these rescaled uncertainties for the subsequent fitting procedure . .resulting barycentric transit times of tres-1 and wasp-2 systems with their uncertainties given by the 68% confidence limits . [ cols="<,^,^,^",options="header " , ] we used the markov chain monte carlo simulations with the metropolis - hastings algorithm ( ( * ? ? ? * ford 2005 ) ) to estimate the statistical uncertainties of the resulting parameters . we assumed a quadratic law of the limb darkening . for tres-1 , we fixed the system parameters according to ( * ? ? ? * winn ( 2007 ) ) and solved only for the transit times . for wasp-2 , we fixed the parameters derived by ( * ? ? ? * charbonneau ( 2007 ) ) and fitted the limb - darkening coefficients and the transit time . for each night , we created 10 independent chains , with the length typically 100,000 points in each chain . we checked the convergence of generated chains using the ( * ? ? ? * gelman & rubin ( 1992 ) ) r statistic . the resulting barycentric transit times can be found in table [ times ] . their uncertainties are given by the 68% confidence limits . the resulting residuals are plotted in fig . [ tres_oc ] for tres-1 , and in fig . [ wasp_oc ] for wasp-2 , using the ephemerides ( [ efe - tres ] ) , and ( [ efe - wasp ] ) , respectively . in summary , our data provide no conclusive evidence of other bodies in either system but the systematic trend of residuals for tres-1 , found by ( * ? ? ? * winn ( 2007 ) ) , is worthy of further study . for tres-1 , our 3 new transit times plus 1 by ( * ? * narita ( 2007 ) ) extend the interval covered by data for more than 150 cycles . we therefore used a weighted linear fit to the transit times as a function of epoch ( weights inversely proportional to the squares of rms errors of individual transit times ) to derive a more accurate ephemeris for the future transit predictions : @xmath17 + the research was supported by the grants 205/08/h005 and 205/06/0304 of the czech science foundation and also from the research program msm0021620860 of the ministry of education of the czech republic .
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searching for transit timing variations in the known transiting exoplanet systems can reveal the presence of other bodies in the system . here
we report such searches for two transiting exoplanet systems , tres-1 and wasp-2 .
their new transits were observed with the 4.2 m william herschel telescope located on la palma , spain . in a continuing programme ,
three consecutive transits were observed for tres-1 , and one for wasp-2 during september 2007 .
we used the markov chain monte carlo simulations to derive transit times and their uncertainties .
the resulting transit times are consistent with the most recent ephemerides and no conclusive proof of additional bodies in either system was found .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
bekenstein and hawking have showed that the entropy of black holes is proportional to the area of their event horizon @xcite . in units of @xmath0 and @xmath1 , the black hole entropy is given as @xmath2 where @xmath3 is the area of event horizon of the black hole . hawking have shown that the black hole can evaporate by emitting radiation , consequently it s event horizon area decreases . he had also shown that the event horizon of the black hole posses temperature , which is inversely proportional to it s mass or proportional to it s surface gravity . during the process of evaporation the entropy of the black hole will decrease . but due to the emitted radiation , the entropy of the surrounding universe will increase . hence the second law of thermodynamics was modified in such a way that , the entropy of the black hole plus the entropy of the exterior environment of the black hole will never decrease , this is called as the generalized second law(gsl ) , which can be represented as , @xmath4 where @xmath5 is the entropy of environment exterior to the black hole and @xmath6 is the entropy of the black hole . the thermodynamic properties of the event horizon , was shown to exist in a more basic level@xcite , by recasting the einstein s field equation for a spherically symmetric space time as in the form of the first law of thermodynamics . in references @xcite one can find investigations on the applicability of the first law of thermodynamics to cosmological event horizon . jacobson @xcite showed that , einstein s field equations are equivalent to the thermodynamical equation of state of the space time . in cosmology the counter part of black hole horizon is the cosmological event horizon . gibbons and hawking @xcite proposed that analogous to black hole horizon , the cosmological event horizon also do possess entropy , proportional to their area . they have proved it particularly for de sitter universe for which an event horizon is existing . for cosmological horizon , gsl implies that , the entropy of the horizon together with the matter enclosed by the event horizon of the universe will never decrease . that is the rate of change of entropy of the cosmological event horizon together with that of material contents within the horizon of the universe , must be greater than or equal to zero , @xmath7 where @xmath8 is the entropy of the cosmological event horizon and @xmath9 represents the entropy of the matter or radiation ( or both together ) of the universe . the validity of gsl for cosmological horizon was confirmed and extended to universe consisting of radiation by numerical analyses by davies @xcite and others @xcite . in reference @xcite , the authors analyzed the gsl with some variable models of f(t ) gravity . in reference @xcite gsl was analyzed with reference to brane scenario . ujjal debnath et . @xcite have analyzed the gsl for frw cosmology with power - law entropy correction . there are investigations connecting the entropy and hidden information . in the case of black hole horizon , the observer is outside the horizon , and the entropy of the black hole is considered as measure of the information hidden within the black hole . while regarding cosmological horizon , the observer is inside the horizon . this will cause problems in explaining the entropy of the cosmological horizon as the measure of hidden information as in the case of black hole . in the case of black hole the hidden region is finite , while in the case of cosmological horizon , there may be infinite region beyond the event horizon of the universe , which causing problems in explaining the cosmological horizon entropy as the hidden information . another important fact is regarding the existence of dominant energy condition for the non decreasing horizon area . in the case of black hole , hawking proved an area theorem , that the area of the black hole will never decrease if it is not radiating @xcite . davies @xcite proved an analogous theorem for cosmological event horizon that the area of the cosmological event horizon will never decrease , provided it satisfies the dominant energy condition , @xmath10 where @xmath11 is the density of the cosmic fluid and @xmath12 is its pressure . regarding the applicability of the generalized second law to the friedmann universe , analysis were done by considering the friedmann universe as a small deviation form the de sitter phase@xcite . in these works the authors calculated the horizon entropy through a numerical computation of the comoving distance to the event horizon . in the present work we obtained an analytical equation for the hubble parameter and proceeded to the calculation of the entropy of the cosmological event horizon in an analytical way . we also checked the validity of dominant energy condition by using the derived hubble parameter . our analysis is for a flat universe which consists of ( i ) radiation and positive cosmological constant and ( ii ) non - relativistic matter and positive cosmological constant . we have considered the flat universe because of the fact that , the inflationary cosmological models predicts flat universe and more over the flatness of the space is confirmed by observations , for example , the current value of the curvature parameter is @xmath13 @xcite . the paper is arranged as follows . in section two , we consider the flat friedmann universe with radiation and a positive cosmological constant . we are presenting the calculation of the entropy of radiation , event horizon and the total entropy of universe and the respective time evolutions . we have also checked the validity of the generalized second law in this section . in section three we present the analogous calculations for the flat friedmann universe with non - relativistic matter and a positive cosmological constant . in section four we present the particular behaviour of the radiation entropy in the friedmann universe with reference to the development of the event horizon . in section five we present the discussion followed by conclusions . for a flat friedmann universe with frw metric , the dynamics are governed by the friedmann equations(by choosing @xmath14 ) , @xmath15 and @xmath16 where @xmath17 is the radiation density , @xmath18 is the radiation pressure , @xmath19 is the constant cosmological constant , @xmath20 is the hubble parameter and the dot over the density represents derivative with respect to time . for radiation , the pressure is , @xmath21 . from equations ( [ eqn : friedmann1 ] ) and ( [ eqn : friedmann2 ] ) the scale factor of this universe can be obtained as , @xmath22 where @xmath23 , @xmath24 and @xmath25 is the present value of the hubble parameter . this equation shows that as @xmath26 the scale factor @xmath27 the radiation dominated phase of the friedmann universe and as @xmath28 the scale factor @xmath29 , with time @xmath30 in gyrs , the blue line ( lower line ) corresponds to the friedmann universe and red line ( upper line ) for de sitter universe . _ _ ] the de sitter phase . the behaviour of the scale factor with time is shown in figure [ fig : af1 ] , in comparison with the scale factor of the de sitter phase . from the plot it is evident that the scale factor of the friedmann universe tends to the de sitter phase at large times . so at smaller times the universe is in the radiation dominated phase and it is decelerating , consequently it does nt have event horizon . at larger times the universe enters the accelerated expansion phase , where it posses an event horizon . the co - moving distance to the event horizon , can be obtained by using the relation , @xmath31 thus the proper distance to the event horizon is @xmath32 for the existence of the event horizon , the integral has to converge . with the scale factor in equation ( [ eqn : afact1 ] ) , the integral in the equation for comoving distance to the event horizon can not be solved analytically . so as a first step we made a numerical computation of the comoving distance to the event horizon , as it is necessary to understand the time evolution of the comoving distance and the result is shown in figure [ fig : codist1 ] . [ fig : codist1 ] to the event horizon with time in gyrs for friedmann universe with radiation and cosmological constant , title="fig : " ] the plot shows that the comoving distance to the event horizon is decreasing with time . since the comoving horizon distance is decreasing , the comoving volume of the universe within the horizon also decreases . the radiation density behaves as @xmath33 , therefore the radiation content within the horizon is decreasing with time . which nevertheless implies that the radiation is crossing the horizon , hence the radiation entropy within the horizon is decreasing . this method and conclusion is in line with the result of t m davies et . one can also find investigations of the same spirit regarding the heat flow through the cosmic horizon in references @xcite . in fact this result is true for any model of the universe having an event horizon . the horizon entropy can be obtained as per the bekenstein equation ( [ eqn : hentro ] ) . for that the area of the event horizon can be taken as @xmath34 in the work of davies et , al . the entropy was calculated in a numerical way , but we are obtaining the entropy of the horizon using the hubble parameter obtained form the scale factor . we are substituting @xmath35 in terms of the hubble parameter . the scale factor in equation ( [ eqn : afact1 ] ) shows that , at large time the scale factor is approaching to that of de sitter phase . for de sitter phase , it can be shown that , @xmath36 since the friedmann universe considered here is approaching the de sitter phase at large times , it will not be unfair in taking , the comoving distance @xmath37 for the friedmann universe in consideration . for the scale factor in equation ( [ eqn : afact1 ] ) , the hubble parameter is , @xmath38 before going for a calculation of the entropy of the event horizon , we will check here the validity of the area theorem proposed by davies , with the obtained hubble parameter . from equation ( [ eqn : friedmann1 ] ) and ( [ eqn : h1 ] ) , the condition for non - decreasing horizon area , equation ( [ eqn : condition1 ] ) , leads to @xmath39 using equation ( [ eqn : h1 ] ) we have plotted @xmath40 versus time in figure [ fig : cond1 ] . we have used the parameter values , @xmath41 @xcite and a standard value @xmath42 through out for our calculations . the plot shows that the area of the event horizon of the friedmann universe with radiation and a positive cosmological constant will never decrease , hence the entropy of horizon will never decrease . ) ] on the other hand , the entropy of the radiation is decreasing with time as we have argued earlier . in oder to satisfy the gsl , the decrease in the entropy of the radiation is to be balanced by the increase in the horizon entropy . the horizon area is always increasing , implies that there exist some kind of trading of the entropy between the horizon and the radiation content of the universe . the entropy of the event horizon is @xmath43 as we have argued earlier , taking @xmath44 , the horizon entropy become , @xmath45 the entropy of the radiation can be obtained using the relation , @xmath46 where @xmath47 is the volume of the event horizon and @xmath48 is the temperature of the radiation . taking @xmath49 , substituting temperature form radiation energy density , @xmath50 , @xmath51 which after substituting @xmath20 parameter form equation ( [ eqn : h1 ] ) and @xmath17 in terms of @xmath20 , using the friedmann equations , become @xmath52 where @xmath53 the radiation constant . we have plotted the time variation of @xmath8 , @xmath54 and @xmath55 in figure 4 . the figure shows that , at sufficiently large times the radiation entropy is decreasing , while the horizon entropy is increasing . the increase in the horizon entropy is more than required to compensate for the decrease in the radiation entropy because of that , total entropy comprising the entropy of the radiation and horizon will increase . this is confirming the validity of the generalized second law for the cosmological horizon , that the entropy of the horizon plus the entropy of the fluid within the horizon will never decrease . this result is in confirmation with the earlier works of davies and others , but they have arrived at the conclusion through straight numerical work , on the other hand our work is more of an analytical way . the conditions for satisfying the generalized second law can be obtained by analysing the validity of the exact statement of the law as given equation ( [ eqn : gsl ] ) . the time rate of the horizon entropy is , @xmath56 where the dot over @xmath20 represents the derivative with time given as @xmath57 , leads to @xmath58 the time rate of radiation entropy can be given from ( [ eqn : sgamma1 ] ) as , @xmath59 the above two equations reveal that the time rate of horizon entropy is positive hence the horizon entropy is at the increase , while the time rate of radiation entropy is negative hence the radiation entropy is at the decrease . the generalized second law , can then be represented as , @xmath60 [ eqn : gslcond3 ] replacing @xmath20 and @xmath61 using the equation ( [ eqn : h1 ] ) , the above condition become , @xmath62 expressing @xmath17 , in terms of @xmath20 , using the friedmann equation , we have evaluated the time evolution of the left hand side of the above inequality condition and the result is shown in figure [ fig : gslcon1 ] . ] the figure shows that the condition for the gsl is always satisfied . from the gsl condition in equation([eqn : gslcon3 ] ) , we can obtain a condition regarding the temperature of the radiation within the horizon . with @xmath63 , the condition ( [ eqn : gslcon3 ] ) , become , @xmath64 taking @xmath65 , then an inequality condition constraints the present value of the temperature of the radiation can be obtained as , @xmath66 the above condition leads to a numerical value , @xmath67k . compared to the present temperature of the radiation @xmath68 , this is very much in favour of the validity of second law in the friedmann universe . this result is agreeing with the result obtained by davies et . al , that @xmath69 , where @xmath20 is now taken as the temperature of the horizon . by using the fundamental constants , the temeprature of the horizon , is @xmath70 where @xmath71 is the boltzmann constant , which implies a present value , @xmath72k . so the temperature of the horizon is less than that of the event horizon , which indicates that , the radiation can aproach the horizon . in this section we are analysing the friedmann universe with matter and a positive cosmological constant , regarding the horizon entropy and the generalized second law . the friedmann equation , in this case is , @xmath73 the scale factor can then be obtained as , @xmath74 as in the previous case , here also the solution will tends to the de sitter phase , @xmath75 as @xmath76 which means that the model posses an event horizon . the hubble parameter corresponds to the scale factor is , @xmath77 it can be seen that the dominant energy condition is being satisfied , as in the case of friedmann universe with radiation , such that @xmath78 at all time . the comoving distance to the horizon is evaluated using the scale factor in the equation ( [ eqn : a2 ] ) , and is deceasing with time as shown is shown in figure [ fig : dist1 ] . so the matter entropy within the horizon ] will decrease and hence the matter will crosses the event horizon . entropy of the event horizon is calculated as , @xmath79 entropy of matter can calculated using an analogous relation corresponds to equation ( [ eqn : radentro ] ) , and taking temperature of matter approximately as @xmath80 , @xmath81 which after substitution of @xmath20 parameter become , @xmath82 the behaviour of @xmath83 and @xmath84 with time is shown in figure [ fig : totent2 ] . and @xmath84 with time in gyrs for friedman universe with matter and cosmological constant . the continuous line representing entropy of horizon plus that of matter , dashed line representing the entropy of horizon and dash - dot line is for entropy of matter ] the figure shows that the total entropy of the universe is increasing and the increase in the entropy of the horizon is more than that required for compensating the decrease in the matter entropy . the general behaviour is the same as that of friedmann universe with radiation , that universe with matter also will satisfy the generalized second law . the condition for satisfying the generalized second law for this universe can be obtained by incorporating the time derivatives of the corresponding entropies into the second law , as @xmath85 using the hubble parameter equation ( [ eqn : h2 ] ) , the above condition become , @xmath86 substituting @xmath87 in terms of the hubble parameter using the friedmann equation , we have plotted the time evolution of the left hand side of the above equation and is shown in figure [ fig : secndlaw2 ] . as in the case of the friedmann universe with radiation , here also the plot shows that the inequality condition corresponds to the generalized second law is satisfied . as in the previous section , the generalized second law can leads to constraint on the temperature of matter . equation ( [ eqn : gslma ] ) can be recast , by taking @xmath88 , as @xmath89 from this it can be shown that , the present temperature of matter in the universe satisfies , @xmath90 for the standard value @xmath91 the above condition also gives , @xmath92 the temperature of the horizon is @xmath93 @xcite , and with proper parameters , @xmath94 so the present temperature of the matter is greater than the temperature of the horizon , which supports the conclusion that the matter can cross the event horizon . in this section we will restrict our analysis to firedmann universe with radiation and cosmological constant only . however one can easily see that the conclusions made are in general true for firedmann universe with matter also , but with different numerical values . our aim here is to show that the entropy of the contend of the universe does have a small increase before the development of event horizon . in the last two sections we have discussed the behaviour of horizon entropy and entropy of the material within the horizon . we have concentrated on checking the validity of the generalized second law . we have shown that the total entropy of the universe is always increasing , and the cosmological event horizon is satisfying the generalized second law . however it is to be noted from the figure [ fig : entrorad ] ( from figure [ fig : totent2 ] ) that the entropy of the radiation ( matter ) is increasing first , attaining a maximum , then after it is decreasing . and the plot on the right represents the time evolution of the @xmath95factor of the same friedmann universe with time in giga years.,title="fig : " ] and the plot on the right represents the time evolution of the @xmath95factor of the same friedmann universe with time in giga years.,title="fig : " ] for clarity regarding this we will show in figure [ fig : radentro ] , the time evolution of the radiation entropy for a friedmann universe having radiation and a positive cosmological constant . the figure shows that the radiation entropy first increases and then decreases to zero at very large times . in the previous section we have concluded that the decrease in the entropy of radiation ( or matter ) is due to the escape of the radiation ( or matter ) from within the horizon . the horizon will exist only when the universe is accelerating . from the bahaviour of the scale factor we have noted that , the universe will be in the radiation dominated ( or matter dominated ) phase as time @xmath96 in the radiation dominated or matter dominated phase the expansion of the universe is decelerating , hence no horizon . the horizon will develop only when the universe enters the @xmath19 dominated phase . a clear demarcation between the deceleration and acceleration phases during the evolution of the universe can be obtained by calculating the deceleration parameter @xmath97 , which is defined as @xmath98 we have calculated the @xmath95factor using the hubble parameter in equation ( [ eqn : h1 ] ) for the friedmann universe with radiation and @xmath19 and the time evolution of which is shown in figure [ fig : radentro ] along with the time evolution of the radiation entropy for an easy comparison . the universe enters the accelerating phase , corresponding to the time at which the @xmath95factor starts to have negative values and as per the figure that is around a time @xmath99 as per the above analysis the universe enters the accelerating phase at around @xmath100 , and at around this time the event horizon starts developing . at this transition time the horizon was tiny . even at this time the difference in the entropy of the event horizon and radiation was very high . entropy of the radiation given in the equation ( [ eqn : sgamma2 ] ) , gives a value for @xmath101 , @xmath102 . while the entropy of the event horizon , as in equation ( [ eqn : ceh2 ] ) , leads to value of @xmath103 , for the same time . these shows that , even at the formation of the event horizon , the entropy of it is eight orders of magnitude greater than the radiation entropy . so even at the tiny stage of the event horizon the entropy of radiation is not so significant . a comparison of the plots in figure [ fig : radentro ] shows that the entropy of radiation is increasing at first and is start decreasing at the same time when @xmath95factor become negative , as the universe entering the accelerating phase . in the decelerating phase , corresponds to positive values of @xmath95factor the radiation entropy is increasing as there is no horizon for the radiation to escape . since the radiation entropy is increasing during the initial stages , the generalized second law is still valid such that @xmath104 will become the gsl as there in no event horizon . when the universe enters the accelerated expanding phase , where it has event horizon , the radiation entropy is decreasing , because now the radiation is crossing the event horizon . but nevertheless , in the accelerating phase , the horizon entropy is increasing at a faster rate compensated to the decrease in the radiation entropy , which in turn leads to the increase in the total entropy of the universe , guaranteeing the validity of the generalized second law . the time rate of radiation entropy is given in equation ( [ eqn : radentrot ] ) . substituting for @xmath105 for friedmann universe with radiation , the equation can be reduced to , @xmath106 when the radiation entropy is maximum , the time rate is zero , then the above equation leads to the condition , @xmath107 where we have used the friedmann equation to substitute for the @xmath108 from which the corresponding time can be obtained as @xmath109 for the standard parameters , the value of the above time , corresponds to the decreasing of radiation entropy , is around @xmath110 which is in confirmation with the figure [ fig : radentro ] . in the case friedmann universe with matter and a positive cosmological constant also , it is evident form the figure [ fig : totent2 ] , that the entropy of matter too have an increase before the formation of the event horizon . so one can easily see that in the case matter also , the above conclusions are true in general . gibbons and hawking have conjectured that cosmological event horizon of the de sitter universe have entropy like black hole event horizon , and the total entropy of such a universe will never decrease , that is it satisfies the gsl . later davies and others have extended this conjecture to friedmann universe with radiation and dust such that the friedmann universe satisfies the gsl . however their work is mainly based on the numerical computation . in this paper we have presented an analytical analysis of the entropy of the event horizon and fluid within the horizon and also the constraints followed from the validity of the gsl . we have considered two types of friedmann universes . type one is the friedmann universe with radiation and a positive cosmological constant . the other type is the friedmann universe with non - relativistic matter and a positive cosmological constant . we have obtained the expansion scale factor and the hubble parameter for the friedmann universe with radiation ( and matter ) and cosmological constant . the time evolution of the scale factor is plotted and have found that at sufficiently small times the friedmann universe is radiation(or matter ) dominated and is in the decelerating phase . but at large times , the universe become dominated by the cosmological constant , hence in the accelerated expansion and will approach de sitter phase at very large times . during the accelerated expansion phase , the universe has got an event horizon . we have numerically evaluated the time evolution of the comoving distance to the event horizon and verified that the comoving distance is decreasing with time in both types of the universes . as a result the comoving volume of the event horizon decreases , subsequently the radiation ( matter ) can cross the event horizon . this implies that the entropy of the radiation ( or matter ) is decreasing consequent to the escaping of radiation ( or matter ) through the horizon . analogous to the area theorem in black hole , davies proposed a corresponding theorem for the cosmological event horizon which implies a dominant energy condition as given in equation ( [ eqn : condition1 ] ) . in the present case of the friedmann universe , the dominant energy condition implies that , @xmath111 the plot in figure [ fig : cond1 ] conclusively proves this . so once the event horizon is formed it s area will never decrease . so unlike in the case of the black holes , where the area of the event horizon decreases when it is radiating , the area of the cosmological event horizon increases when radiation ( matter ) crosses the horizon . we have obtained the analytical relations for the entropy of the event horizon and radiation ( matter ) for the friedman universe . the entropy of the event horizon is given in equation ( [ eqn : ceh2 ] ) , according to which the present value of the event horizon entropy will be around , @xmath112 we have plotted the time evolution of these entropies and found that the net entropy of the radiation ( or matter ) is decreasing but the entropy of the event horizon is increasing at faster rate as the universe expands . this implies that the total entropy of the friedmann universe , that is the sum of the entropy of the radiation ( or matter ) and event horizon , is increasing . this indicate the validity of the gsl for both types of the friedmann universes . the constraints imposed by the gsl is obtained . for the friedmann universe with radiation and cosmological constant , the gsl constraint the present temperature of the radiation as , @xmath67k in standard units . compared to the latest value of the radiation temperature form cobe , [email protected] k @xcite , the above constraint implies the friedmann universe in consideration is very well in the purview of the gsl . the temperature of the horizon , is @xmath114 in the standard units . for the the present case , this temperature is around @xmath115k . the comparison of the above temepratures shows that there is a radiation drain form within the horizon . at this point one should note the result of davies et . al.@xcite that , the temperature of the radiation is higher than that of the horizon . so there is natural flow direction towards the horizon . therefore in the present context , we can conclude that the temperature of the horizon of the friedmann universe with radiation and cosmological constant at present is less than @xmath116 k. for the universe with non - relativistic matter and cosmological constant , the gsl constraint the matter temperature as @xmath117k , which in turn implies that the horizon temperature of the friedmann universe with matter and cosmological constant is less than @xmath116k , so that there is matter flow towards the event horizon . the time evolution of the radiation ( matter ) entropy shows that , it increases first , attains a maximum and then decreases as shown in the figure [ fig : radentro ] . it is seen that the increase in the radiation ( matter ) entropy is during the deceleration phase of the universe , when the radiation ( matter ) is dominating the cosmological constant . it is to be noted that there is no event horizon when the universe is decelerating . so the corresponding increase in the entropy of the radiation is due to the non - existence of the of the event horizon . if the event horizon is absent , there is no crossing of the radiation over the horizon , and it retained within causal region of our universe , which facilitate the small increase in the radiation entropy . we made this point clear by comparing the time evolution of the radiation entropy and @xmath95factor , such that the entropy of the radiation is start decreasing when the @xmath95factor become negative , consequently the expansion is accelerating at which condition the universe posses an event horizon . we have computed the the time corresponding to the maximum of the radiation entropy at which the @xmath95factor is critically become negative , as @xmath118 , and is evident form figure [ fig : radentro ] . bekenstein j d , phys . d * 7 * , 2333 ( 1973 ) . hawking s w _ commun . phys . _ * 43 * 199 ( 1975 ) j m bardeen , b carter and s w hawking , commun . math . phys . * 31 * , 161 ( 1973 ) t padmanabhan , phys . rept . * 49 * , 406 ( 2005 ) t padmanabhan , phys . rept . * 73 * , 046901 ( 2010 ) t jacobson , phys . lett . * 75 * , 1260 ( 1995 ) r g cai , phys . b * 525 * , 331 ( 2002 ) r g cai and s p kim , jhep . textbf0502 , 050 ( 2005 ) gibbons g w and hawking s w , phys . d * 15 * , 2738 ( 1977 ) davies p c w , class . quantum grav . * 4 * , l225 ( 1987 ) t m davies , p c w davies and c h lineweaver , class . quantum grav . * 20 * , 2753 ( 2003 ) davies p c w , class . quantum grav . * 5 * , 1349 ( 1988 ) r amanullah et . j. * 716 * , 712 ( 2010 ) m d pollock , t p singh , class . quantum grav . * 6 * , 901 ( 19 ) r brustein , phys . lett . * 84 * , 2072 ( 2000 ) diego pavan , class . quantum grav . * 7 * , 487 ( 1990 ) r bousso , phys . d * 71 * , 064024 ( 2005 ) b wang , y gong and e abdalla , phys . d * 74 * , 083520 ( 2006 ) k karami and a abdomaleki , jcap * 04 * , 007 ( 2012 ) rahul ghosh , antonio pasqua and surajit chattopadhyay , eur . j. plus * 128 * , 12 ( 2013 ) nairwita mazunder and subenoy chakraborthy , eur . j. c * 70 * , 329 ( 2010 ) ujjal debnath , surajith chattopadhyay , ibrar hussain , mubasher jamil and ratbay myrzakulov , eur . j. c * 72 * , 1875 ( 2012 ) sergio del campo , ramon herrera and diego pavon , phys . b * 707 * , 8 ( 2012 ) a frolov and lev kofman jcap , * 0305 * , 009 ( 2003 ) hawking s w and ellis g f r , _ the large scale structure of space - time _ cambridge university press , ( 1973 ) e komatsu et . ( wmap collaborations ) , astrophys . . ser . * 192 * 18 ( 2011 ) c l bennet et . al . astrophys . * 464 * , l1 ( 1996 )
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we discuss the generalized second law ( gsl ) and the constraints imposed by it for two types of friedmann universes .
the first one is the friedmann universe with radiation and a positive cosmological constant , and the second one consists of non - relativistic matter and a positive cosmological constant .
the time evolution of the event horizon entropy and the entropy of the contents within the horizon are analyses in an analytical way by obtaining the hubble parameter .
it is shown that the gsl constraint the temperature of both the radiation and matter of the friedmann universe .
it is also shown that , even though the net entropy of the radiation ( or matter ) is decreasing at sufficiently large times as the universe expand , it exhibit an increase during the early times when universe is decelerating .
that is the entropy of the radiation within the comoving volume is decreasing only when the universe has got an event horizon .
keywords : friedmann universe , entropy , generalised second law .
pacs : 04.70.dy,97.60lf,98.80jk,02.60jh,04.20cv
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You are an expert at summarizing long articles. Proceed to summarize the following text:
proto - planetary nebulae ( ppne ) are the long - sought - after missing link between the end of the asymptotic giant branch ( agb ) phase and the beginning of planetary nebula phase of stellar evolution . after the _ infrared astronomical satellite _ ( _ iras _ ) mission , a number of objects were proposed as candidate ppne based on their infrared colors and other spectral properties . these are typically stars of g to b spectral type with significant infrared excesses due to the remnant circumstellar dust shell ejected in the agb phase . of particular interest among these candidates are a number of carbon - rich objects whose abundances show a strong enhancement of s - process elements , as expected from the dredge - up of material in thermal pulses during the agb evolution @xcite . for other candidates there is some possibility of confusion with massive supergiants , but those in this carbon - rich group are almost certainly bona - fide ppne . one of these objects is iras 16594@xmath04656 . it is a bright mid - infrared source which has typical colors of a ppn @xcite . optically it is associated with a southern emission - line star . it was found to be of spectral type b7 with v magnitude 14.6 , subject to about 7.5 magnitudes of visual extinction @xcite . it was not clear from the original _ iras _ spectral observations whether the object was oxygen - rich or carbon - rich , but subsequent _ infrared space observatory _ ( _ iso _ ) spectral observations showed that it has carbon - based dust features , including the 21 @xmath1 m feature @xcite . optical images obtained with the _ hubble space telescope _ showed that the star has a surrounding reflection nebula with a complex structure @xcite . the relative faintness of the reflection nebula compared to the star , together with the optical morphology , led @xcite to conclude that the nebula is intrinsically bipolar ( or multi - polar ) viewed at an intermediate angle to the bipolar axis . both optical and near - infrared spectral observations show emission lines , which are thought to be shock - excited rather than radiatively excited by the star @xcite . no radio emission has been detected from iras 16594@xmath04656 @xcite . the distance to this object is uncertain . the dust shell model of @xcite suggests a distance of 2.6 kpc if the total luminosity of the star is 10@xmath2 @xmath3 . a slightly smaller estimate of ( [email protected] ) kpc is given by @xcite for the same assumed total luminosity . most of the luminosity is being emitted in the infrared where extinction effects are smaller than in the optical , so these estimates are not strongly affected by the non - spherical morphology of the dust shell . the optical morphology of the nebula in iras 16594@xmath04656 appears complex , with what appear to be pairs of symmetric structures at multiple position angles , for which it was named the `` water lily nebula '' @xcite . the optical images also showed several concentric arcs centered on the star @xcite . however , near - infrared observations with the _ nicmos _ instrument on _ hst _ showed a somewhat simpler morphology , although it was not clear whether this was simply an effect of reduced dynamic range compared to the optical observations @xcite . an initial n - band observation of iras 16594@xmath04656 with the timmi2 camera on the eso 3.6 m telescope failed to resolve the dust shell @xcite , while more recent timmi2 observations did marginally resolve the structure @xcite . in this paper we present new , higher sensitivity and higher angular resolution mid - infrared images of iras 16594@xmath04656 which resolve the dust shell in thermal emission . we present the observations in section 2 . we then derive a dust color temperature map in section 3 , and compare the morphology as observed in the mid - infrared to that seen at other wavelengths in section 4 . we give a brief discussion of the results in the final section . the observations reported here were obtained with the t - recs instrument on gemini south under program gs-2004a - q-56 . the sky and telescope background were removed by chopping and nodding during the observations . images of iras 16594@xmath04656 were obtained in three filters . in the n - band window : 360 second ( total on - source exposure times ) images with the `` si-5 11.66um '' filter on 2004 march 11 and with the `` si-6 12.33um '' filter on 2004 may 8 ; and in q - band window : a 600 second image with the `` qa 18.30um '' filter on 2004 may 10 . these filters will be referred to as the `` si5 '' , `` si6 '' , and `` qa '' filters , respectively . all these filters have widths @xmath5 . information about these filters , including the filter profiles , can be found on the gemini observatory public www pages ( see _ http://www.gemini.edu/sciops/instruments/miri/t-recsfilters.html_ ) . on each night , standard star observations for flux calibration were carried out immediately after the observation of the science target . these stars were hd 123139 on march 11 , hd 169916 on may 8 , and hd 175775 on may 10 . all of these stars are included among the mid - infrared spectrophotometric standards of @xcite . comparisons of the point - spread functions ( psfs ) of these stars with observations of @xmath6 cen a / b or @xmath6 cma on various nights from december 2003 through may 2004 indicate that these stars are suitable as psf references as well as spectral standards . the n - band observations were made under good conditions , as judged from the level of sky cancellation obtained while chopping . the q - band observations were made under marginal conditions ; however iras 16594@xmath04656 is a very bright target at 18 @xmath1 m and it was detected with good signal - to - noise ratio despite the less than ideal conditions . the image quality was good as estimated from the standard stars . the full width at half maximum ( fwhm ) was 0@xmath737 for the si5 filter image , 0@xmath739 for the si6 image , and 0@xmath760 for the qa image . these correspond to strehl values of about 0.6 , fairly typical of better seeing conditions for the n - band filters but somewhat lower than usual for the qa filter . the raw images on and off of the target were subtracted and then summed to produce raw images of iras 16594@xmath04656 in the three filters . flux calibration of the images was done in two different ways . first , the standard star observations were used to find the conversion from in - band counts to jy using the assumed spectral energy distribution from @xcite integrated over the filter profiles , and then these scaling factors were applied to the images of iras 16594@xmath04656 . the pixel by pixel brightnesses so obtained were then converted to jy / square arc - second using the pixel size of t - recs . second , the estimated filter flux densities were generated from the _ iso _ spectrum of iras 16594@xmath04656 in the same way as was used to get the expected flux densities in jy for the standard stars . these values were also used to convert from counts to jy / square arc - second in the images . it was found that for the si5 filter these two methods agreed within less than 1% . for the other filters the agreement was poorer . using the _ iso _ spectrum for the si6 filter estimate gave a value 22% higher than that from the standard star . this discrepancy is too large to be due to an atmospheric extinction effect , judging from some estimates of the extinction coefficient in this filter made on other nights . it probably indicates that the sky conditions were not uniform in the directions to the standard star and to our target , as in other observations we have obtained with t - recs the inter - comparison with _ iso _ spectra gives 2% agreement for filters in the n - band window . the qa brightness calculated from the standard star came out 11% lower than that calculated from the _ iso _ spectrum . this is probably within the uncertainties caused by the variable sky conditions . there is also a color effect due to differences in spectral shape between iras 16594@xmath04656 and the standard stars , but since the filters are relatively narrow this is a small correction and it was neglected . in what follows , we have chosen to use the _ iso _ spectrum as the basis for creating surface brightness images , since this minimizes the effect of variable sky conditions . this assumes that the _ iso _ spectrum gives the correct absolutely calibrated total brightness and the t - recs observations give the correct relative brightness distribution , or equivalently that all the atmospheric effects are uniform over the small t - recs field of view . figure [ fig1 ] shows the three flux - calibrated images in jy / square arc - second . the region shown in the figure is @xmath8 and contains all the detected emission from iras 16594@xmath04656 . the total flux density for iras 16594@xmath04656 was calculated to be 44.0 , 56.9 , and 177 jy for the si5 , si6 , and qa filters , respectively . in all three cases , the circumstellar shell of iras 16594@xmath04656 appears as a bipolar nebula of dimension about 4@xmath75 @xmath9 2@xmath725 , with a bright central region orientated roughly north - south and two lobes extending east and west . there is a clear difference in size between the east and west lobes . the east lobe is about 20% smaller than the west lobe both in width and maximum detected radius from the star in these images . the optical depth at these wavelengths , especially in q - band , must be small ; thus this size difference must be caused by either a physical size difference between the two lobes or distinctly different projection angles for the two lobes . the central bright region of the mid - infrared images appears to be some type of thin `` equatorial '' torus , perpendicular to the axis of the two lobes . the northern end of the this structure is brighter than the southern end in all the images , but the ratio is much closer to 1:1 in the qa image . this shows that the dust in the torus is cooler in the southern region than in the northern region . the sharp edge of the torus in the north is particularly striking , as the images are very bright there but there is a sudden edge beyond which no emission is observed along the line of the torus . lucy deconvolution of the raw images was carried out using the standard star observations as psf templates . this was done using the stsdas.analysis.restore.lucy task in the stsci reduction package under iraf version 2.12a . a 61 by 61 pixel box was used to define the psf . pixels outside this psf box but within a 181 by 181 pixel box centered on the star were used to derive the background level , which was subtracted from the stellar profile . it was found that most of the improvement in the image resolution was obtained in the first few lucy iterations , after which there was no significant change in the derived structure , so the deconvolution was stopped after 20 iterations . the deconvolved images were then re - smoothed with a gaussian of fwhm 0@xmath71 , slightly larger than the original pixel size . the resulting images are sharpened by about a factor of 3 in psf width compared to the original images . figure [ fig2 ] shows these sharpened images . the two n - band images have almost identical structure . the sharpened q - band image has the same general morphology as the n - band images but both lobes are seen to be smaller by about 0.4 arc - seconds than in the n - band images . however , we believe that this size difference is not real . comparison of the raw images in the qa and n - band filters shows that the emission region is just slightly larger for the qa image than for the si5 and si6 filters . this indicates that the lucy deconvolution for the lobes introduced a small artifact into the image . as the strehl value for the standard star in the qa filter was lower than usual , it is possible that the seeing changed between the target observation and the standard star observation . there was some indication of variable q - band conditions during these observations . if the seeing did get worse for the standard star observation , that would explain the decrease in lobe size for the deconvolved image compared to the raw image , although the magnitude of the decrease is larger than one would expect based upon the fwhm value for the standard star qa filter image . the structure of the brightest regions is similar for all three filters . the bright part suggests a torus of some sort , and it appears to be thinner at the center than at the edges . this suggests that the torus is seen nearly edge on . if it were oriented at some intermediate angle to the plane of the sky , as was earlier asserted based upon the visible images , then one would expect the torus to appear as a small ellipse in these mid - infrared images . this is clearly not seen . however , if we are indeed viewing the torus edge - on then the visible and mid - infrared images suggest it to be quite asymmetric with position . in each panel of figure [ fig2 ] , the estimated star position is marked with a small black dot . this position was found by cross - comparing the t - recs si5 image with an _ hst nicmos _ image taken in the narrow - band filter centered on the h@xmath10 2.12 @xmath1 m line @xcite . the estimated stellar position is very close to the geometrical center of the bright `` bar '' in the dust emission . it is located in a region of relatively low brightness in the three filters , especially in the si5 and si6 filters . this probably means that in the optical images we are seeing the star through some type of hole in the toroid , where little dust is present . from the surface brightness images in two filters it is possible to construct a ratio map , and if one assumes that the surface brightness is due to optically - thin thermal emission from dust grains of a known type , then the surface brightness ratio can be converted into color temperatures between the two wavelengths . denoting the surface brightness in jy / square arc - second by @xmath11 , the color temperature , @xmath12 , is defined by solving @xmath13 for each brightness ratio value in the image . the optical depth values @xmath14 for the two filters are assumed to be proportional to the absorption cross - sections , so we can replace the optical depths with the actual q@xmath15 values for this calculation . the scattering component of the dust extinction is expected to be small at these long wavelengths . the ratio of @xmath14 values is a constant as long as the dust grain properties are uniform in the dust shell , so then the equation gives a one - to - one transformation between surface brightness ratio and @xmath12 . we have used the si5 and qa images to produce a brightness ratio map , after convolving the si5 image with a gaussian profile of fwhm of 0@xmath7195 to match the angular resolution of the qa image . regions of the two images which had `` low '' brightness values , taken to be less than 0.1% of the respective peak values , were masked out of the ratio image . the ratio image was then transformed to dust color temperature , assuming that the dust grains are 0.1 @xmath1 m amorphous carbon grains with the opacity function for ac type 2 grains from @xcite . these grain properties were the basis of the spectral model for iras 16594@xmath04656 presented by @xcite . for another assumed dust grain size or type the dust color temperature values would change , but the relative variations over the image should remain the same . while the @xmath12 values do not indicate the physical temperatures of the dust grains , since they are some type of average along the line of sight for each pixel , they do indicate the global dust temperature variations in the circumstellar shell as long as the dust grains are not drastically different than assumed for the calculation . the @xmath12 map is shown in the lower right panel of figure [ fig1 ] . the @xmath12 values are confined to a relatively narrow range . the bright part of the dust shell has a range of @xmath12 from about 140 k to 160 k. it was found that the @xmath12 map was much the same whether or not the si5 image was convolved to the resolution of the qa image . the @xmath12 map shows that the region of highest dust color temperature is in the north part of the central bright region , while the color temperature is much lower directly to the south on the other side of the stellar position . there is also a region of high dust color temperature in the southern wall of the west lobe . there is a cap at the end of the east lobe that is seen in the color temperature map , which is also apparent in the si5 image . this cap does not appear to be associated with a higher dust color temperature than elsewhere in lobe , which suggests that the dust optical depth is higher here than for other positions in the lobe . it is possible that the east lobe is smaller than the west lobe because its expansion is impeded by external material , and that the cap represents a boundary between the lobe and the external medium . figure [ fig3 ] shows the deconvolved si5 image with overlaid contour plots in the optical i - band ( 0.8 @xmath1 m ; su , hrivnak , & kwok 2001 ) and near - infrared h@xmath10 filter images ( 2.12 @xmath1 m ; hrivnak , kelly , & su 2004 ) . this allows direct comparison of the morphology of the nebula in these different wavelengths . the h@xmath10 image was matched to the si5 filter image since they were immediately seen to have very similar morphologies . this was used to obtain a reasonably accurate estimate of the central star position in the t - recs image . that position was then used to match the optical image to the mid - infrared image . the si5 image is plotted in absolute surface brightness units , jy / square arc - second . as shown in the upper right panel of figure [ fig3 ] , there is a close match of features in the lobes between the _ nicmos _ h@xmath10 filter image and the t - recs image . the bright lobe edges are regions of strong h@xmath10 emission . the cap in the east lobe is clearly visible in this panel , and it also corresponds to a region of strong h@xmath10 emission . it is more difficult to determine if any of the mid - infrared bright waist structure is also detected in the h@xmath10 image , because the star is saturated in the h@xmath10 image , but it does not look as if anything but the edges of the lobes are detected in the h@xmath10 image . since the h@xmath10 emission is mainly shock excited @xcite , this raises the possibility that the dust emission from these regions are partially shock - excited . another possibility that could explain why the dust emission region so closely matches the shock is that the dust grains may be much smaller downstream from the shock than they are before the shock , and that as a result the small grains are transiently heated to relatively high temperatures . comparison of the t - recs images with the _ hst _ optical images @xcite shows that the two lobes seen in the t - recs images correspond closely to the central brightest part of the reflection nebula . the total size of the optical reflection nebula is 12.3@xmath16 by 8.8@xmath16 , which is much larger than the size of the n - band or q - band images . the optical image overlay in the lower left panel of figure [ fig3 ] uses logarithmically spaced contours ranging from 0.0085% of the stellar peak brightness up to the peak brightness , with each contour at a level 2.7 times the previous contour . the optical reflection nebula is at most about 0.75% of the stellar peak brightness . the lowest contours show what has been suggested to be point - symmetric morphology , with three pairs of oppositely directed features at position angles of about 40@xmath17 , 57@xmath17 , and 87@xmath17 east of north as can be measured from the i - band image presented in @xcite . the optical lobes have been suggested to be caused by a rapidly precessing , columnated high - speed wind from the star @xcite . the t - recs image shows just the two lobes at position angle 75@xmath17 . while there seems to be some very faint mid - infrared emission detected in the si5 filter on a size scale roughly three times that of the main mid - infrared lobes , over - plotting this with the optical image does not show any correspondences with the faint extended optical structure . in particular , none of this emission is seen just outside the bright torus either to the north or to the south . kinematic observations of the individual optical lobes is needed to determine if they are distinct structures or not . if these point symmetric features are not distinct kinematic structures , then perhaps there are just two lobes but the optical appearance is due to structure in the walls of these lobes which make them look more complex . the optical emission is due to reflection by dust , and so the larger size of the optical images compared to the mid - infrared images shows that the dust shell is much larger than is obvious from the n - band or q - band images . the dust outside the central bright region delineated by the shocked h@xmath10 emission is clearly much colder than that inside the shocks . at the ends of mid - infrared torus , in particular , there must be a large discontinuity in temperature and optical depth so that there is a large change in mid - infrared surface brightness but a much smaller change in the optical brightness of the scattered light . these mid - infrared images suggest that the torus is perpendicular to the plane of the sky . this is consistent with recent observations of the bipolar lobes , which indicate that they are nearly in the plane of the sky . unpublished high resolution long - slit spectra in the near - infrared @xcite have been obtained with the phoenix spectrograph on gemini south . analysis of these spectra , which map the molecular hydrogen line at 2.12 @xmath1 m for cuts at different position angles through the star and the lobes , shows similar velocities for the two lobes , indicating that they are oriented very close to the plane of the sky . this was also concluded by @xcite based upon near - infrared polarimetry along with dust shell modeling of the spectral energy distribution of the object . since the evidence indicates that the nebular axis is oriented very close to the plane of the sky , the east lobe must actually be smaller than the west lobe . the t - recs images show that the central bright region of the circumstellar dust shell is quite different than that of other well - known bipolar nebulae iras 17150@xmath03224 , iras 17441@xmath02411 , roberts 22 , and hen 3 - 401 , in all of which the dust emission is strongly peaked at the stellar position . for iras 16594@xmath04656 the mid - infrared brightness is at a minimum near the stellar position and is much higher to the north and the south along the central waist . possibly the torus is of much lower optical depth in this object than in the others , or it is highly asymmetric with a low optical depth along our line of sight . the latter suggestion is consistent with the visibility of the central star . certainly for iras 17150@xmath03224 and iras 17441@xmath02411 the spectral energy distribution implies a relatively high optical depth along our line of sight to the star , and the star is not seen in visible light . the spectral type of the star in iras 16594@xmath04656 is much earlier than that in the latter two objects , so it may simply be more evolved and the torus may have had more time to disperse . unlike the hour - glass or open lobes observed in most bipolar planetary nebulae ( e.g. ngc 6302 ) , the bipolar lobes of iras 16594@xmath04656 as shown in figure [ fig2 ] are closed and resemble the lobes seen in the ppn iras 17106@xmath03046 and the young planetary nebula hen 2 - 320 . the morphology of the lobes clearly shows that the lobes are confined by the circumstellar medium , and the fast collimated outflow which creates the bipolar lobes has not yet broken through the stellar wind of the agb progenitor . the interaction between the fast and slow winds is clearly delineated by dust distribution in the lobes . we also note that there is `` bulge '' at the tip of the western lobe , which suggests that the high - velocity flow is on the verge of breaking out . in contrast to the cap at the tip of the eastern lobe ( which represents a pile - up at the wind interface ) , the western lobe may represent a slightly more advanced stage of the breakout , and therefore explains the difference in sizes between the two lobes . in a few hundred years , we expect that both lobes will open up into butterfly morphology . we are therefore witnessing a critical phase of morphological transformation of pns . we have successfully detected the bipolar lobes and a central bright waist in mid - infrared images of iras 16594@xmath04656 . the bright waist suggests that we are seeing a central torus nearly edge - on . while this is consistent with published polarization and unpublished kinematic results , it differs from earlier published interpretations of the visible image that concluded that the lobes are seen at an intermediate orientation . this emphasizes the need for multi - wavelength observations to confidently understand the structure of proto - planetary nebulae . this result may well be applicable to our understanding of other bipolar phenomenon such as ysos and agns . b.j.h acknowledges support by the national science foundation under grant no . this work was supported in part by grants to s.k from the natural sciences and engineering research council of canada . cohen , m. , et al . 1999 , , 117 , 1864 garca - hernndez , d. , _ et al . _ 2004 , in _ asymmetrical planetary nebulae iii _ , eds . m. meixner , j. h. kastner , b. balick , & n. soker , ( asp : san francisco ) , 367 garca - lario , p. , manchado , a. , ulla , a. , & manteiga , m. 1999 , , 513 , 941 hrivnak , b. j. , kelly , d. m. , & su , k. y. l. 2004 , in _ asymmetrical planetary nebulae iii _ , eds . m. meixner , j. h. kastner , b. balick , & n. soker , ( asp : san francisco ) , 175 hrivnak , b. j. , kwok , s. , & su , k. y. l. 1999 , , 542 , 849 hrivnak , b. j. , kwok , s. , & su , k. y. l. 2001 , , 121 , 2775 hrivnak , b. j. , volk , k. , & kwok , s. 2000 , , 535 , 275 hrivnak , b. j. , et al . , 2006 , paper in preparation kwok , s. 1993 , , 31 , 63 rouleau , f. , & martin , p. g. 1991 , apj , 377 , 526 santander - garca , m. , corradi , r. l. m. , balick , b. , & mampaiso , a. 2004 , , 426 , 185 su , k. y. l. , hrivnak , b. j. , & kwok , s. 2001 , , 122 , 1585 su , k. y. l. , hrivnak , b. j. , kwok , s. , & sahai , r. 2003 , , 126 , 848 ueta , t. , murakawa , k. , & meixner , m. 2005 , aj , 129 , 1625 van de steene , g. c. & pottasch , s. r. 1993 , , 274 , 895 van de steene , g. c. , van hoof , p. a. m. , & wood , p. 2000 , , 362 , 984 van de steene , g. c. & van hoof , p. 2003 , , 406 , 773 volk , k. , & kwok , s. 1989 , , 342 , 345 van winckel , h. 2003 , , 43 , 391
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high - resolution mid - infrared images have been obtained in n - band and q - band for the proto - planetary nebula iras 16594@xmath04656 .
a bright equatorial torus and a pair of bipolar lobes can clearly be seen in the infrared images .
the torus appears thinner at the center than at the edges , suggesting that it is viewed nearly edge - on .
the infrared lobes correspond to the brightest lobes of the reflection nebula seen in the hubble space telescope ( _ hst _ ) optical image , but with no sign of the point - symmetric structure seen in the visible image .
the lobe structure shows a close correspondence with a molecular hydrogen map obtained with _
hst _ , suggesting that the dust emission in the lobes traces the distribution of the shocked gas .
the shape of the bipolar lobes shows clearly that the fast outflow is still confined by the remnant circumstellar envelope of the progenitor asymptotic giant branch ( agb ) star .
however , the non - detection of the dust outside of the lobes suggests that the temperature of the dust in the agb envelope is too low for it to be detected at 20 @xmath1 m .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
let @xmath5 be the quantum enveloping algebra associated to an affine algebra @xmath6 without derivation . let @xmath7 be finite - dimensional @xmath5-modules . suppose @xmath8 is irreducible and @xmath7 have crystal bases @xmath9 . then it is known @xcite that there exists a unique map @xmath0 from @xmath10 to @xmath11 commuting with any crystal operators @xmath12 and @xmath13 . there also exists an integer - valued function @xmath14 on @xmath10 , called energy function , satisfying a certain recurrence relation under the action of @xmath12 ( see ) . combinatorial @xmath0-matrices or energy functions play an important role in the affine crystal theory . in the kyoto path model @xcite , that realizes the affine highest weight crystal in terms of a semi - infinite tensor product of perfect crystals , the energy function is an essential ingredient for the computation of the affine weight . in the box - ball system @xcite or its generalizations @xcite in the formulation of crystal bases , the time evolution of the system is defined by using the combinatorial @xmath0-matrix . energy functions are also crucial in the calculation of conserved quantities . in @xcite a new connection was revealed between the energy function and the kkr or kss bijection @xcite that gives a one - to - one correspondence between highest weight paths and rigged configurations . recently , for all nonexceptional affine types , all kr crystals , crystal bases of kirillov reshetikhin ( kr ) modules ( if they exist ) , were shown to exist and their combinatorial structures were clarified @xcite . hence , it is natural to consider the problem of obtaining a rule to calculate the combinatorial @xmath0-matrix and energy function . in this paper , for type @xmath2 we calculate the image of the combinatorial @xmath0-matrix for any classical highest weight element in the tensor product of kr crystals @xmath1 ( theorem [ th : main ] ) . ( note that the first upper index of the second component is 1 . ) we also obtain the value of the energy function for such elements . although we get the rule only for highest weight elements , there is an advantage from the computational point of view , since it is always easy to calculate the action of crystal operators @xmath15 for @xmath16 not only by hand but also by computer . to identify highest weight elements in the image @xmath4 the notion of @xmath3-diagrams , introduced in @xcite , is used effectively . the paper is organized as follows . in section 2 we briefly review crystals and @xmath3-diagrams . in section 3 we recall the kr crystal @xmath17 for type @xmath18 and @xmath19 , and the notion of combinatorial @xmath0-matrix and energy function . the condition for an element of @xmath1 or @xmath4 to be classically highest is also presented . the main theorem is given in section 4 . in section 5 we prove a special case of the theorem , and reduction to this case is discussed in section 6 according to whether @xmath20 is odd or even . mo was supported by grant jsps 20540016 . the work of rs is supported by the core research for evolutional science and technology of japan science and technology agency . let @xmath6 stand for a simple lie algebra or affine kac moody lie algebra with index set @xmath21 and @xmath22 the corresponding quantized enveloping algebra . axiomatically , a @xmath6-crystal is a nonempty set @xmath23 together with maps @xmath24 where @xmath25 is the weight lattice associated to @xmath6 . the maps @xmath12 and @xmath13 are called kashiwara operators and @xmath26 is the weight function . to each crystal one can associate a crystal graph with vertices in @xmath23 and an arrow colored @xmath27 from @xmath28 to @xmath29 if @xmath30 or equivalently @xmath31 . for @xmath32 and @xmath27 , let @xmath33 in this paper we only consider crystal bases coming from @xmath22-modules . for a complete definition of crystal bases see for example @xcite . let @xmath34 be crystals . then @xmath35 can be endowed with the structure of crystal . the actions of kashiwara operators and the value of the weight function are given by @xmath36 the multiple tensor product is defined inductively . in order to compute the action of @xmath15 on multiple tensor products , it is convenient to use the rule called signature rule " @xcite . let @xmath37 be an element of the tensor product of crystals @xmath38 . one wishes to find the indices @xmath39 such that @xmath40 to do it , we introduce ( @xmath41-)signature by @xmath42 we then reduce the signature by deleting the adjacent @xmath43 pair successively . eventually we obtain a reduced signature of the following form . @xmath44 then the action of @xmath12 ( resp . @xmath13 ) corresponds to changing the rightmost @xmath45 to @xmath46 ( resp . leftmost @xmath46 to @xmath45 ) . if there is no @xmath45 ( resp . @xmath46 ) in the signature , then the action of @xmath12 ( resp . @xmath13 ) should be set to @xmath47 . the value of @xmath48 ( resp . @xmath49 ) is given by the number of @xmath45 ( resp . @xmath46 ) in the reduced signature . consider , for instance , an element @xmath50 of the 3 fold tensor product @xmath51 . suppose @xmath52 . then the signature and reduced one read @xmath53 thus we have @xmath54 we denote by @xmath55 the highest weight crystal of highest weight @xmath56 , where @xmath56 is a dominant integral weight . let @xmath57 with @xmath27 be the fundamental weights associated to a simple lie algebra . in this paper , we consider the types of @xmath58 and @xmath59 . as usual , a dominant integral weight @xmath60 is identified with a partition or young diagram with columns of height @xmath61 for @xmath62 , except when @xmath63 is a spin weight , namely , @xmath64 for type @xmath65 and @xmath66 and @xmath64 for type @xmath59 . to represent elements of @xmath55 we use kashiwara nakashima ( kn ) tableaux , a generalization of semistandard young tableaux for type @xmath67 . for kn tableaux refer to @xcite . ( see also @xcite for a summary . ) contrary to the original one , we use the french notation where parts are drawn in increasing order from top to bottom . to calculate the actions of @xmath15 on a kn tableau it is convenient to use so - called the japanese reading word of a tableau . for a kn tableau @xmath68 move from right to left and on each column move from bottom to top . during this process we read letters , thereby obtaining a word @xmath69 . a letter can be identified with an element of @xmath70 , crystal of the vector representation . hence @xmath69 can be viewed as an element of @xmath71 with @xmath72 being the number of nodes in @xmath68 or length of @xmath69 . then the action of @xmath12 or @xmath13 is calculated by using the signature rule . we still need to remember the crystal graph of @xmath70 for type @xmath73 , but it is easy as described in @xcite . let @xmath74 be @xmath58 or @xmath59 . for a subset @xmath75 , we say that @xmath32 is @xmath76-highest if @xmath77 for all @xmath78 . we set @xmath79 . we describe @xmath76-highest elements in terms of a notion of @xmath3-diagram @xcite . a @xmath3-diagram @xmath25 of shape @xmath80 is a sequence of partitions @xmath81 such that @xmath82 and @xmath83 are horizontal strips . we depict this @xmath3-diagram by the skew tableau of shape @xmath80 in which the cells of @xmath83 are filled with the symbol @xmath46 and those of @xmath82 are filled with the symbol @xmath45 . write @xmath84 and @xmath85 for the outer and inner shapes of the @xmath3-diagram @xmath25 . for type @xmath86 we have a further requirement : the outer shape @xmath56 contains columns of height at most @xmath87 , but the inner shape @xmath88 is not allowed to be of height @xmath87 ( hence there are no empty columns of height @xmath87 ) . as we have discussed we identify a young diagram with a weight . @xcite [ p : branch ] let @xmath56 be an @xmath74 weight that does not contain spin weights . then there is an isomorphism of @xmath89-crystals @xmath90 that is , the multiplicity of @xmath91 in @xmath92 , is the number of @xmath3-diagrams of shape @xmath80 . there is a bijection @xmath93 from @xmath3-diagrams @xmath25 of shape @xmath80 to the set of @xmath76-highest elements @xmath28 of @xmath89-weight @xmath94 . for any columns of height @xmath87 containing @xmath46 , place a column @xmath95 otherwise , place @xmath96 in all positions in @xmath25 that contain a @xmath45 , and fill the remainder of all columns by strings of the form @xmath97 . we move through the columns of @xmath28 from top to bottom , left to right . each @xmath46 in @xmath25 ( starting with the leftmost moving to the right ignoring @xmath46 at height @xmath87 ) will alter @xmath28 as we move through the columns . suppose the @xmath46 is at height @xmath98 in @xmath25 . if one encounters a @xmath96 , replace @xmath96 by @xmath99 . if one encounters a @xmath100 , replace the string @xmath97 by @xmath101 . [ ex : pm - diag ] let us consider the following @xmath3-diagram . @xmath102 to obtain @xmath103 we first draw the tableau @xmath104 reading from left there are @xmath46 s at height 4,3,2,2,1 . each @xmath46 alter the above tableau as follows . the 1st @xmath46 changes the first column as @xmath105 ( reading from bottom ) , the 2nd and 3rd change the second column as @xmath106 , the 4th changes the third column as @xmath107 and the 5th changes the fourth column as @xmath108 . therefore , @xmath103 is given by @xmath109 for a word @xmath110 let @xmath111 . we use this convention also for @xmath112 . note that the order in @xmath113 is reversed from @xmath114 . next proposition shows how we get to the highest element from a @xmath3-diagram by applying @xmath12 s . [ prop : to highest ] let @xmath25 be a @xmath3-diagram whose outer shape has depth @xmath20 . suppose @xmath115 for @xmath65 , @xmath116 for @xmath86 , @xmath117 for @xmath59 . let @xmath118 be the number of columns of the outer shape with height @xmath41 . let @xmath119 ( resp . @xmath120 ) be the number of @xmath45 ( resp . @xmath46 ) at height @xmath41 . define a word @xmath114 by @xmath121 where @xmath122 where @xmath123 for @xmath65 , @xmath124 for the other cases , and @xmath125 for @xmath59 , @xmath126 for the other cases . then @xmath127 is the hightest weight element with highest weight given by its outer shape . moreover , at each step when we apply @xmath128 or @xmath129 , including @xmath130 , the action is maximal , namely , if we apply @xmath131 or @xmath132 , the outcome turns out @xmath133 . suppose @xmath134 for type @xmath86 . we first prove the claim when there is no @xmath46 in @xmath25 . set @xmath135 . then the japanese reading word of the tableau corresponding to @xmath25 is given by @xmath136 the 1-signature is just given by @xmath137 , where @xmath138 , and there is no need to reduce . hence one can apply @xmath139 . calculating similarly for @xmath140 one always has a simple @xmath41-signature of the form @xmath141 , and we arrive at the highest weight element as desired . next we consider the general case . we prove by induction on @xmath72 , the number of @xmath46 . if @xmath142 , the claim is proven . suppose @xmath143 and let @xmath98 be the height of the lowest @xmath46 in @xmath25 . let @xmath144 be the same @xmath3-diagram as @xmath25 except that there are one less @xmath46 s at height @xmath98 . compare the japanese reading word of the corresponding tableaux of @xmath25 and @xmath144 . the difference is : @xmath145 apart from this difference in two words , there are subwords of the form @xmath146 or letters @xmath96 on the left and subwords of the form @xmath147 or @xmath148 for some @xmath149 on the right . let us calculate the 1-signatures of both words . they are @xmath150 for @xmath25 and @xmath151 for @xmath144 . ( there are no @xmath43 pairs . ) after applying @xmath152 on both @xmath25 and @xmath144 , the 2-signatures also turn out of the form @xmath153 for @xmath25 and @xmath154 for @xmath144 . the difference is that there is @xmath155 or @xmath99 in @xmath25 but @xmath156 or @xmath157 in @xmath144 . similar situations continue until we apply @xmath158 , and after applying @xmath159 , the two results coincide . hence we should have the desired result . the proof in the case of @xmath160 for type @xmath86 is almost the same . the only difference is that we first treat the case when there is no @xmath46 in @xmath25 except at height @xmath87 , since there is no empty column of height @xmath87 . hence we omit the proof . for a @xmath3-diagram given in example [ ex : pm - diag ] set @xmath161 . then , according to the previous proposition @xmath127 is a highest weight element . later in this paper we will need to apply @xmath162 to a @xmath3-diagram @xmath25 . since @xmath163 is no longer @xmath76-highest , we have to use a pair of @xmath3-diagrams @xmath164 to consider @xmath165-highest elements . namely , @xmath25 represents a @xmath76-highest element and @xmath166 represents a @xmath165-highest element in the @xmath89-component whose highest weight vector correponds to @xmath25 . under this bijection we identify a @xmath165-highest element @xmath28 with a pair of @xmath3-diagram @xmath164 . to describe the action of @xmath162 on @xmath164 perform the following algorithm : 1 . successively run through all @xmath46 in @xmath166 from left to right and , if possible , pair it with the leftmost yet unpaired @xmath46 in @xmath25 weakly to the left of it . 2 . successively run through all @xmath45 in @xmath166 from left to right and , if possible , pair it with the rightmost yet unpaired @xmath45 in @xmath25 weakly to the left . 3 . successively run through all yet unpaired @xmath46 in @xmath166 from left to right and , if possible , pair it with the leftmost yet unpaired @xmath45 in @xmath166 . * lemma 5.1 ) [ prop : e1 action ] if there is an unpaired @xmath46 in @xmath166 , @xmath162 moves the rightmost unpaired @xmath46 in @xmath166 to @xmath25 . else , if there is an unpaired @xmath45 in @xmath25 , @xmath162 moves the leftmost unpaired @xmath45 in @xmath25 to @xmath166 . else @xmath162 annihilates @xmath164 . let @xmath6 be an affine lie algebra of type @xmath167 , @xmath168 , or @xmath169 with the underlying finite - dimensional simple lie algebra @xmath170 of type @xmath171 , or @xmath86 , respectively . we label the vertices of the corresponding dynkin diagram according to @xcite , so the index set of @xmath6 ( resp . @xmath170 ) is @xmath172 ( resp . @xmath173 ) . in this section we review kr crystals @xmath17 of type @xmath6 given in @xcite for @xmath174 and @xmath175 for @xmath167 , @xmath176 for @xmath168 and @xmath177 for @xmath169 . as an @xmath74-crystal , @xmath17 is given by @xmath178 here @xmath55 is the @xmath74-crystal of highest weight @xmath56 and the sum runs over all dominant weights @xmath56 that can be obtained from @xmath179 by the removal of vertical dominoes , where @xmath57 are the @xmath41-th fundamental weights of @xmath74 . in order to define the actions of @xmath180 and @xmath181 we first consider an automorphism @xmath182 on the kr crystal @xmath17 . the dynkin diagrams of type @xmath183 , and @xmath169 all have an automorphism interchanging nodes @xmath133 and @xmath184 . @xmath182 corresponds to this dynkin diagram automorphism . by construction @xmath182 commutes with @xmath185 and @xmath186 for @xmath187 . hence it suffices to define @xmath182 on @xmath76-highest elements . because of the bijection @xmath188 from @xmath3-diagrams to @xmath76-highest elements as described in section [ subsec : pm diag ] , it suffices to define the corresponding map @xmath189 on @xmath3-diagrams . let @xmath25 be a @xmath3-diagram of shape @xmath80 . let @xmath190 be the number of columns of height @xmath41 in @xmath94 for all @xmath191 with @xmath192 . if @xmath193 , then in @xmath25 , above each column of @xmath94 of height @xmath41 , there must be a @xmath46 or a @xmath45 . interchange the number of such @xmath46 and @xmath45 symbols . if @xmath194 , then in @xmath25 , above each column of @xmath94 of height @xmath41 , either there is no sign or a @xmath195 pair . suppose there are @xmath196 @xmath195 pairs above the columns of height @xmath41 . change this to @xmath197 @xmath195 pairs . the result is @xmath198 , which has the same inner shape @xmath94 as @xmath25 but a possibly different outer shape . let @xmath199 and @xmath200 be such that @xmath201 is a @xmath76-highest element . then , @xmath202 is given by @xmath203 where @xmath204 . the affine crystal operators @xmath205 and @xmath206 are then defined as @xmath207 if @xmath208 , the structure of the kr crystal turns out simple . an crystal element of @xmath209 can be identified with one - row kn tableau of length @xmath210 with letters from @xmath211 ( @xmath212 ) ( and @xmath133 for @xmath168 ) . denoting the number of letters @xmath211 or @xmath133 by @xmath213 or @xmath214 , we have the so - called coordinate representation of @xmath209 @xcite . @xmath215 the action of @xmath15 for @xmath216 can be calculated as we explained in the last paragraph of section [ subsec : crystals ] . the action of @xmath217 is given by @xmath218 we list the values of @xmath219 below . @xmath220 where @xmath221 . let us now consider a tensor product of kr crystals @xmath222 . it is known @xcite that there exists a unique bijection @xmath0 , called combinatorial @xmath0-matrix , commuting with kashiwara operators @xmath15 for any @xmath223 . since @xmath0 preserves the weight , @xmath224 should be sent to @xmath225 by @xmath0 , where @xmath226 ( resp . @xmath227 ) is the @xmath228-highest elements of @xmath229 ( resp . @xmath230 ) in @xmath17 ( resp . @xmath231 ) . for the other elements the image is uniquely determined , since @xmath222 is known to be connected @xcite . next we explain the energy function @xmath14 . let @xmath232 correspond to @xmath233 by @xmath0 . suppose @xmath234 . applying @xmath12 on both sides of @xmath235 , we are led to consider the following four cases : @xmath236 then the function @xmath14 is uniquely determined , up to adding a constant , by @xmath237 although it is not obvious that such a function exists , it is shown to exist @xcite . next we investigate conditions for an element of @xmath1 or @xmath4 to be @xmath228-highest . recall the following fundamental fact : @xmath238 in particular , if @xmath239 is @xmath228-highest , then @xmath28 has to be @xmath228-highest . [ prop : ht cond 1 ] let @xmath240 be a dominant integral weight that appears in as highest weight . by abuse of notation let @xmath241 also stand for the highest kn tableau of weight @xmath241 . let @xmath242 be an element of @xmath209 represented by coordinates . then , an element @xmath243 of @xmath1 is @xmath228-highest , if and only if @xmath244 for some @xmath241 as above and the following conditions for @xmath242 are satisfied . * @xmath245 if @xmath246 , * @xmath247 if @xmath246 or @xmath248 , * @xmath249 if @xmath250 and @xmath251 , * @xmath252 if @xmath253 and @xmath251 . in the case of @xmath160 where @xmath254 , @xmath255 appearing in ( iii ) should be understood as @xmath133 . apply for @xmath216 and use the formula for @xmath256 in . in what follows , for a @xmath3-diagram @xmath25 we use the following notation . let @xmath257 be one of @xmath258 ( @xmath259 stands for emptiness ) . we denote by @xmath260 the number of columns of the outer shape of @xmath25 of height @xmath41 that contain @xmath257 . ( 30,10 ) ( 0,0)(1,0)30 ( 5,6)(1,0)20 ( 0,0)(5,0)4 ( 5,6)(5.3,6.5)(6.7,6.8 ) ( 8.3,6.8)(9.7,6.5)(10,6 ) ( 5,6)(0,1)2 ( 25,6)(0,-1)2 ( 25.3,3.9)(0.3,-0.1)16 ( 4.7,8.1)(-0.3,0.1)16 ( 7.1,6.8)@xmath261 ( 12.1,6.8)@xmath262 ( 17.1,6.8)@xmath263 ( 22.1,6.8)@xmath264 ( 5,3.7)(0,1)2.3 ( 5,2.3)(0,-1)2.3 ( 4.8,2.7)@xmath41 ( 10,5.2)@xmath46 ( 14.2,5.2)@xmath46 ( 15,5.2)@xmath45 ( 19.2,5.2)@xmath45 ( 20,5.2)@xmath45 ( 24.2,5.2)@xmath45 ( 20,4.3)@xmath46 ( 24.2,4.3)@xmath46 ( 0,0)(5,0)3 ( 10.95,5.42)(0.3,0)11 ( 20.95,4.52)(0.3,0)11 [ prop : ht cond 2 ] an element @xmath239 of @xmath4 is @xmath228-highest , if and only if @xmath265 and @xmath29 is a @xmath76-highest element whose corresponding @xmath3-diagram @xmath25 satisfies @xmath266 apply for @xmath216 and use proposition [ prop : e1 action ] to calculate @xmath267 of the @xmath3-diagram @xmath25 . we consider the combinatorial @xmath0-matrix @xmath268 let @xmath269 be @xmath228-highest and @xmath270 . then , @xmath271 is also @xmath228-highest , and from propositions [ prop : ht cond 1 ] and [ prop : ht cond 2 ] @xmath244 for some dominant integral weight @xmath241 , @xmath272 and there exists a @xmath3-diagram @xmath25 such that @xmath273 . thus we have @xmath274 for an element of @xmath209 we use both the coordinate representation and the japanese reading word of the corresponding one - row tableau . let @xmath260 ( @xmath275 ) be data corresponding to @xmath25 as in the previous section . then our main result is : [ th : main ] with the notations above we have the following formulas . @xmath276 here @xmath277 , and @xmath278 . we also note @xmath279 ( mod 2 ) . moreover , the value of the energy function is given by @xmath280 if we normalize @xmath14 in such a way as @xmath281 . here @xmath282 is the first part of the partition corresponding to the weight of @xmath283 . solving the formulas for @xmath260 with respect to @xmath241 and @xmath242 , we obtain the coordinates of the image @xmath283 of the inverse of @xmath0 for an element @xmath284 of @xmath4 are given by @xmath285 here we should understand @xmath286 . in what follows in this section we prove theorem [ th : main ] by assuming technical propositions in later sections . we give a proof only for type @xmath169 , since the difference from the other cases is very small as we have seen in proposition [ prop : to highest ] . suppose we need to apply @xmath113 with such a word @xmath114 as @xmath287 for type @xmath169 ( see e.g. ) . then for type @xmath168 we replace it with @xmath288 and for type @xmath167 @xmath289 consider first the case when @xmath20 is odd . suppose @xmath290 and @xmath25 are related as in the statement of the theorem . we are to show @xmath291 by proposition [ reduction_odd3 ] showing is reduced to the case where @xmath242 is of the form @xmath292 . applying this proposition again to this case , it is then reduced to the case where @xmath293 , since there is no @xmath100 or @xmath96 in @xmath242 of the previous case . hence proposition [ prop_special ] completes the proof of . ( notice that when @xmath294 for any odd @xmath41 and the other @xmath260 are all zero . ) using these propositions we can calculate @xmath14 as @xmath295 since @xmath296 . the case when @xmath20 is even can be proven similarly by using propositions [ reduction_even3 ] and [ prop_special ] . let @xmath297 be a dominant integral weight whose corresponding young diagram is depicted as follows . ( 25,11.5)(-1,0 ) ( -2.3,5)@xmath298 ( 0,0)(1,0)25 ( 0,0)(0,1)10 ( 0,10)(1,0)5 ( 5,10)(0,-1)2 ( 5,8)(1,0)5 ( 10,8)(0,-1)1 ( 10.3,7)(0.5,-0.1)20 ( 20,4)(0,1)1 ( 25,4)(-1,0)5 ( 25,0)(0,1)4 ( 2.3,4.8)@xmath299 ( 2.5,6)(0,1)4 ( 2.5,4)(0,-1)4 ( 7.3,3.8)@xmath300 ( 7.5,5)(0,1)3 ( 7.5,3)(0,-1)3 ( 22.3,1.8)@xmath301 ( 22.5,3)(0,1)1 ( 22.5,1)(0,-1)1 ( 0,10)(1,10.6)(1.8,10.7 ) ( 5,10)(4,10.6)(3.2,10.7 ) ( 2.1,10.6)@xmath302 ( 5,8)(6,8.6)(6.8,8.7 ) ( 10,8)(9,8.6)(8.2,8.7 ) ( 7.1,8.6)@xmath303 ( 20,4)(21,4.6)(21.8,4.7 ) ( 25,4)(24,4.6)(23.2,4.7 ) ( 22.0,4.6)@xmath304 here one can assume @xmath305 and @xmath306 . we also assume that @xmath307 then the claim of this section is the following special version of the main theorem : [ prop_special ] let @xmath241 be as above . then we have @xmath308 the condition ( [ s : eq : sumc_i = k ] ) for @xmath309 is necessary . for example , in type @xmath310 , the image of the combinatorial @xmath0-matrix and the value of the energy function for @xmath311 are @xmath312 and @xmath313 . we divide the proof of this proposition into three parts . let us define two words that will be used in the proof . @xmath314 where underbraces are introduced to show a unit of repetitions , and @xmath315 are defined as follows . set @xmath316 , then @xmath317 where @xmath318 and for @xmath319 , @xmath320 @xmath321 is defined by @xmath322 and @xmath323 where for @xmath324 , @xmath325 and @xmath326 is @xmath327 finally , @xmath328 in the process of proof , we use a dominant integral weight @xmath329 given by @xmath330 . we assume that @xmath304 is even . the proof for odd @xmath304 is similar . during the proof , we often identify a kn tableau with its japanese reading word . the goal of this subsection is the following lemma : [ lem_special1 ] @xmath331 this is a direct consequence of the following three sublemmas . @xmath332 [ lem : special1 ] @xmath333 to begin with we apply @xmath180 on @xmath334 for maximal times . define a word @xmath335 by @xmath336 then the @xmath76-highest element of @xmath334 is @xmath337 by the map @xmath338 we get the corresponding @xmath3-diagram and @xmath189 acts on it as follows : ( 31.2,8 ) ( 0,0)(18,0)2 ( 0,0)(1,0)13.2 ( 0,0)(0,1)8 ( 0,8)(1,0)3.3 ( 3.3,8)(0,-1)2 ( 3.3,6)(1,0)3.3 ( 6.6,6)(0,-1)2 ( 6.6,4)(1,0)3.3 ( 13.2,2)(0,-1)2 ( 9.9,4)(0,-1)2 ( 9.9,2)(1,0)3.3 ( 27.9,4)(1,0)3.3 ( 31.2,4)(0,-1)2 ( 0.15,7.2)@xmath339 ( 3.45,5.2)@xmath339 ( 6.75,3.2)@xmath339 ( 14,3)@xmath340 ( 18.15,7.2)@xmath341 ( 21.35,5.2)@xmath341 ( 24.65,3.2)@xmath341 ( 28.1,3.2)@xmath341 ( 28.1,2.2)@xmath339 note that there are @xmath304 @xmath46 s at height @xmath342 of the right @xmath3-diagram . assume that @xmath304 satisfies @xmath343 . then @xmath344 is @xmath345 and @xmath346 is @xmath347 from this expression , we get @xmath348 . applying @xmath349 we get @xmath350 to convert the action of @xmath351 into that of @xmath352 , we need to define the words @xmath353 as follows . @xmath354 where the subwords @xmath355 are @xmath356 define @xmath357 and @xmath358 where the subwords @xmath359 are @xmath360 then , starting from @xmath361 , one calculates @xmath362 here @xmath363 means @xmath364 . with @xmath338 , this corresponds to the following @xmath3-diagram and @xmath189 acts on it as follows : ( 25,8 ) ( 0,0)(15,0)2 ( 0,0)(1,0)9.9 ( 0,0)(0,1)7 ( 0,7)(1,0)3.3 ( 3.3,7)(0,-1)2 ( 3.3,5)(1,0)3.3 ( 6.6,5)(0,-1)2 ( 6.6,3)(1,0)3.3 ( 9.9,3)(0,-1)3 ( 0.1,6.3)@xmath339 ( 3.4,4.3)@xmath339 ( 6.7,2.3)@xmath339 ( 10.9,3)@xmath340 ( 15.1,6.3)@xmath341 ( 18.4,4.3)@xmath341 ( 21.7,2.3)@xmath341 thus , starting from @xmath365 , we calculate @xmath366 where the final formula gives @xmath367 . from @xmath368 and @xmath369 , we see that the 0-signature of @xmath370 is @xmath371 . therefore we get @xmath372 which gives the desired expression . [ lem : special2 ] starting from @xmath373 , we have @xmath374 where the final expression is equal to @xmath375 . since @xmath376 , we have finished the proof of lemma [ lem_special1 ] . the goal of this subsection is the following lemma : [ lem_special2 ] @xmath377 to begin with , we have @xmath378 here , we need to divide the calculation into two cases . + _ case 1 : _ if @xmath379 , we have @xmath380 _ case 2 : _ if @xmath381 , we have @xmath382 to begin with we remark that in this case we have @xmath383 assume that @xmath210 satisfies @xmath384 . then @xmath385 define a word @xmath386 by @xmath387 . then we have @xmath388 by the map @xmath338 , @xmath389 corresponds to the following @xmath3-diagram , @xmath189 acts on it as follows : ( 35,10 ) ( 0,1)(20,0)2 ( 0,0)(1,0)15 ( 0,0)(0,1)8 ( 0,8)(1,0)3.3 ( 3.3,8)(0,-1)2 ( 3.3,6)(1,0)3.3 ( 11.6,4)(1,0)3.4 ( 15,4)(0,-1)4 ( 7.7,-1)@xmath390 ( 13.2,-1)@xmath210 ( 6.6,-1)(0,0.4)18(0,1)0.2 ( 11.6,-1)(0,0.4)13(0,1)0.2 ( 15,-1)(0,0.4)5(0,1)0.2 ( 6.6,7)(0,-1)2 ( 6.6,5)(1,0)5 ( 26.6,7)(1,0)5 ( 31.6,7)(0,-1)2 ( 0.15,8.3)@xmath339 ( 3.45,6.3)@xmath339 ( 11.85,4.3)@xmath339 ( 15.8,4)@xmath391 ( 20.1,8.3)@xmath341 ( 23.4,6.3)@xmath341 ( 26.95,5.3)@xmath392 ( 26.95,6.3)@xmath393 ( 31.8,4.3)@xmath341 there are @xmath394 @xmath46 s at height @xmath342 in the right @xmath3-diagram . assume that @xmath394 satisfies @xmath395 . we also assume that this @xmath41 satisfies @xmath396 for the sake of simplicity . then @xmath397 is @xmath398 and @xmath399 is @xmath400 from this expression , we have @xmath401 . applying @xmath402 we get @xmath403 note that the length of the string @xmath404 is @xmath301 whereas that of the string @xmath405 is @xmath406 . in order to convert the action of @xmath351 into that of @xmath352 , we define the word @xmath407 as follows . @xmath408 , @xmath409 where subwords are defined by @xmath410 @xmath411 , @xmath412 where subwords are defined by @xmath413 and @xmath414 . computation of @xmath415 proceeds as follows : @xmath416 by @xmath338 , this corresponds to the following @xmath3-diagram , and @xmath189 acts on it as follows : ( 35,11 ) ( 0,1)(20,0)2 ( 0,0)(1,0)15 ( 0,0)(0,1)9 ( 0,9)(1,0)3.3 ( 3.3,9)(0,-1)2 ( 3.3,7)(1,0)3.3 ( 6.6,7)(0,-1)2 ( 6.6,5)(1,0)5 ( 15,3)(0,-1)3 ( 7.7,-1)@xmath390 ( 13.1,-1)@xmath210 ( 6.6,-1)(0,0.4)15(0,1)0.2 ( 11.6,-1)(0,0.4)15(0,1)0.2 ( 15,-1)(0,0.4)5(0,1)0.2 ( 11.6,6)(0,-1)2 ( 11.6,4)(1,0)3.4 ( 35,6)(0,-1)2 ( 31.6,6)(1,0)3.4 ( 0.2,9.3)@xmath339 ( 3.5,7.3)@xmath339 ( 7.0,5.3)@xmath392 ( 15.8,4)@xmath391 ( 20.1,9.3)@xmath341 ( 23.5,7.3)@xmath341 ( 27.0,5.3)@xmath393 ( 31.9,5.3)@xmath341 ( 31.9,4.3)@xmath339 let us assume that @xmath417 . then the right @xmath3-diagram corresponds to the expression ( [ eq : w_3 ] ) with @xmath418 and @xmath419 in ( [ eq : w_3 ] ) being replaced with @xmath420 and @xmath421 . application of @xmath422 is similar to that of @xmath423 on ( [ eq : w_3 ] ) and we obtain @xmath424 as @xmath425 the remaining computation of @xmath426 is almost the same as the computation of @xmath427 given in the final part of the proof of lemma [ lem : special1 ] . the only difference in @xmath428 is caused by the fact that letters @xmath184 and @xmath429 appear @xmath210 times in @xmath424 . as for @xmath430 , the beginning two steps @xmath431 gives @xmath432 by comparing this with @xmath433 , we see that the rest of the computation of @xmath430 is almost the same as that given in lemma [ lem : special2 ] . this completes the proof for case 1 . note that in this case we have @xmath434 . action of @xmath352 is obtained by formally setting @xmath435 in case 1 . therefore we have @xmath436 where @xmath437 is given in ( [ eq : special3 ] ) . when we further apply @xmath430 on this formula , we realize that there are extra exponents originating from @xmath438 in the first tensor component of the right hand side . these extra contributions coincide with the exponents @xmath439 in @xmath440 . we have completed the proof of lemma [ lem_special2 ] . now we can prove proposition [ prop_special ] . we prove the first relation by descending induction on @xmath301 . if @xmath441 ( this is the maximal possible value ) , we have @xmath442 . in this case we see @xmath443 by weight consideration . the induction proceeds by using lemmas [ lem_special1 ] and [ lem_special2 ] . as for the energy function , we have to look carefully the action of @xmath180 in lemmas [ lem_special1 ] and [ lem_special2 ] . if @xmath180 acts on the second component of the tensor product , we write @xmath0 , and @xmath444 on the first component . we summarize actions of @xmath180 to get @xmath375 and @xmath445 in two lemmas as follows ( proceeds from left to right ) : @xmath446 the diagram is drawn in the case of @xmath447 . including the other inequality case , we see that we have exactly the same number of @xmath448 and @xmath449 cases ( see ) . therefore we have @xmath450 . using the same induction as above we obtain @xmath451 . this completes the proof of proposition [ prop_special ] . let @xmath453 be @xmath228-highest . recall that we defined @xmath454 by @xmath240 . ( readers are warned that it is not the multiplicity of @xmath41 in the corresponding partition @xmath241 but its conjugate @xmath455 . ) note that @xmath456 unless @xmath457 and @xmath41 is odd . we also know that the coordinates other than @xmath458 are all @xmath133 by proposition [ prop : ht cond 1 ] . let us define a word @xmath459 by @xmath460 where the exponents are defined as follows . for @xmath461 , @xmath462 define @xmath463 . then @xmath464 . for @xmath465 , @xmath466 set @xmath467 and define other @xmath468 by @xmath469 for @xmath470 . note that @xmath471 . set @xmath472 , @xmath473 and define other @xmath474 by @xmath475 for @xmath476 . since @xmath482 is the @xmath3-diagram of outer shape @xmath241 such that all the columns have @xmath46 as symbol , we see @xmath483 . thus one has @xmath484 . we have @xmath485 . to convert the result into that for @xmath486 we define a word @xmath487 as follows : @xmath488 where @xmath489 , @xmath490 , @xmath491 . then we have @xmath492 applying @xmath493 further , we obtain @xmath494 finally , applying @xmath495 we obtain the desired relation . let us consider the operation of @xmath502 in @xmath503 . since @xmath504 , @xmath505 acts on the second component at most @xmath506 times and the rest goes to the first . the 2-signature of the second component of ( [ tochu6_2 ] ) is @xmath507 . from the highest condition for @xmath508 we have @xmath509 , thus @xmath505 acts on @xmath510 only . we can continue similarly and obtain the desired result . finally , we consider the action of @xmath511 . the 1-signature of eq.@xmath501 is @xmath512 . by applying @xmath513 , we get @xmath514 the 2-signature of the above element is @xmath515 . from the highest condition for @xmath508 we have @xmath516 , thus @xmath505 does not act on @xmath517 . therefore @xmath518 acts on the first component and obtain @xmath519 we can continue the computation and arrive at proposition [ reduction_odd ] . in this subsection , let @xmath25 and @xmath144 be the @xmath3-diagrams . as before , corresponding to @xmath25 and @xmath144 , we use the parametrization @xmath260 and @xmath521 ( @xmath522 ) respectively . note that by definition @xmath523 . define a word @xmath524 by @xmath525 here the exponents for @xmath526 are @xmath527 the exponents for @xmath528 are @xmath529 we use proposition [ prop : e1 action ] . schematically , the pair of @xmath3-diagrams corresponding to @xmath198 looks as follows : @xmath541{0,0,0,0.2 } \put(2,9){\rule{20pt}{20pt } } \put(6,8){\rule{20pt}{20pt } } \put(13,5){\rule{20pt}{20pt } } \put(17,4){\rule{20pt}{20pt } } \put(21,3){\rule{20pt}{20pt } } \put(28,0){\rule{20pt}{20pt } } \color{black } \thicklines \put(0,0){\line(1,0){36 } } \put(0,0){\line(0,1){11 } } \put(0,11){\line(1,0){8 } } \put(8,11){\line(0,-1){2 } } \put(8,9){\line(1,0){2 } } \multiput(10,9)(0.3,-0.1){10}{\circle*{0.1 } } \put(13,8){\line(1,0){2 } } \put(15,8){\line(0,-1){2 } } \put(15,6){\line(1,0){8 } } \put(23,6){\line(0,-1){2 } } \put(23,4){\line(1,0){2 } } \multiput(25,4)(0.3,-0.1){10}{\circle*{0.1 } } \put(28,3){\line(1,0){2 } } \put(30,3){\line(0,-1){2 } } \put(30,1){\line(1,0){6 } } \put(36,1){\line(0,-1){1 } } \thinlines \put(2,11){\line(0,-1){1 } } \put(2,10){\line(1,0){4 } } \put(6,10){\line(0,-1){1 } } \put(6,9){\line(1,0){2 } } \put(13,6){\line(1,0){2 } } \put(17,6){\line(0,-1){1 } } \put(17,5){\line(1,0){4 } } \put(21,5){\line(0,-1){1 } } \put(21,4){\line(1,0){2 } } \put(28,1){\line(1,0){2 } } \put(32,1){\line(0,-1){1 } } \put(0.2,10.3){$+$ } \put(1.2,10.3){$+$ } \put(2.2,10.3){$+$ } \put(3.2,10.3){$+$ } \put(4.2,10.3){$-$ } \put(5.2,10.3){$-$ } \put(6.2,10.3){$-$ } \put(7.1,10.3){$-$ } \put(2.2,9.3){$+$ } \put(3.2,9.3){$+$ } \put(4.2,9.3){$+$ } \put(5.2,9.3){$+$ } \put(6.2,9.3){$+$ } \put(7.1,9.3){$+$ } \put(6.2,8.3){$+$ } \put(7.1,8.3){$+$ } \put(8.2,8.3){$+$ } \put(9.2,8.3){$+$ } \put(13.2,7.3){$-$ } \put(14.1,7.3){$-$ } \put(13.2,6.3){$+$ } \put(14.1,6.3){$+$ } \put(13.2,5.3){$+$ } \put(14.1,5.3){$+$ } \put(15.2,5.3){$+$ } \put(16.2,5.3){$+$ } \put(17.2,5.3){$+$ } \put(18.2,5.3){$+$ } \put(19.2,5.3){$-$ } \put(20.2,5.3){$-$ } \put(21.2,5.3){$-$ } \put(22.2,5.3){$-$ } \put(17.2,4.3){$+$ } \put(18.2,4.3){$+$ } \put(19.2,4.3){$+$ } \put(20.2,4.3){$+$ } \put(21.2,4.3){$+$ } \put(22.2,4.3){$+$ } \put(21.2,3.3){$+$ } \put(22.2,3.3){$+$ } \put(23.2,3.3){$+$ } \put(24.2,3.3){$+$ } \put(28.2,2.3){$-$ } \put(29.2,2.3){$-$ } \put(28.2,1.3){$+$ } \put(29.2,1.3){$+$ } \put(28.2,0.3){$+$ } \put(29.2,0.3){$+$ } \put(30.2,0.3){$+$ } \put(31.2,0.3){$+$ } \put(32.2,0.3){$+$ } \put(33.2,0.3){$+$ } \put(34.2,0.3){$-$ } \put(35.2,0.3){$-$ } \put(0.5,11.6){$p_r^\cdot$ } \put(2.5,11.6){$p_r^-$ } \put(4.5,11.6){$p_r^+$ } \put(6.0,11.6){$p_{r-2}^\cdot$ } \put(8.5,9.6){$p_{r}^\mp$ } \put(13.5,8.6){$p_i^\cdot$ } \put(15.2,6.6){$p_{i+2}^\mp$ } \put(17.5,6.6){$p_i^-$ } \put(19.5,6.6){$p_i^+$ } \put(21.1,6.6){$p_{i-2}^\cdot$ } \put(23.5,4.6){$p_i^\mp$ } \put(28.6,3.6){$p_1^\cdot$ } \put(30.5,1.6){$p_3^\mp$ } \put(32.6,1.6){$p_1 ^ -$ } \put(34.6,1.6){$p_1^+$ } \end{picture } \nonumber\end{aligned}\ ] ] here the thick lines represent outer shape of @xmath198 and the thin lines represent the inner @xmath3-diagram . ( since we are to consider the @xmath162 action , we need such a pair of @xmath3-diagrams . ) the numbers @xmath260 represent the numbers of columns which have the same pattern of @xmath46 and @xmath45 indicated below @xmath260 . according to proposition [ prop : e1 action ] , we make pairs of two @xmath46 symbols which we indicate by gray squares in the diagram . then we see that we can apply @xmath162 up to @xmath542 times , which gives the value for @xmath543 . the pair of @xmath3-diagrams corresponding to @xmath544 looks as follows : @xmath545 note that the numbers of columns of height 1 have changed from @xmath546 , @xmath547 , @xmath548 to @xmath548 , @xmath546 , @xmath547 . in order to compute @xmath549 , we usually make @xmath544 into @xmath550-highest by applying suitable @xmath113 , apply @xmath189 and then apply @xmath551 ( see ) . however , since @xmath189 commutes with the action of @xmath12 @xmath552 , we can apply @xmath189 on the pair of @xmath3-diagrams directly . namely , @xmath189 changes the outer @xmath3-diagram only . the pair of @xmath3-diagrams corresponding to @xmath553 looks as follows : @xmath554 note that the outer shape has also been changed at @xmath548 . we use proposition [ prop : to highest ] . the quantities @xmath118 , @xmath119 and @xmath120 there should be used for the corresponding numbers of the inner @xmath3-diagram of ( [ pair+-diagram ] ) . since we are considering the inner @xmath3-diagram , we have to understand the word @xmath114 there as follows : @xmath556 and the formula for @xmath557 and @xmath558 are the same in terms of @xmath118 , @xmath119 and @xmath120 . then , @xmath559 and differences @xmath560 are @xmath561 and differences @xmath562 are @xmath563 we see that the word @xmath114 computed here coincides with @xmath526 . again , we use proposition [ prop : to highest ] . in this case , the quantities @xmath118 , @xmath119 and @xmath120 there mean those for the outer @xmath3-diagram of ( [ pair+-diagram ] ) . let us compute the word @xmath565 there in the case of our ( [ pair+-diagram ] ) . to begin with , @xmath566 is @xmath567 and differences @xmath560 are @xmath568 and differences @xmath562 are @xmath569 we see that the word @xmath114 computed here coincides with @xmath528 except for @xmath570 which does not appear in @xmath528 . let @xmath571 be the @xmath228-highest weight element whose outer shape coincides with ( [ pair+-diagram ] ) . then the above lemma shows that @xmath572 . since there are exactly @xmath573 columns of height 1 in @xmath571 , we see that the content of columns of height 1 in the tableau @xmath574 are all 2 and that the other columns are the same as @xmath571 . from the shape of ( [ pair+-diagram ] ) we see that @xmath574 coincides with @xmath144 given in proposition [ reduction_odd2 ] . to summarize , we have @xmath575 , hence we complete the proof of proposition [ reduction_odd2 ] . \(i ) we have @xmath586 where we have used @xmath587 in the final line . thus @xmath588 . + ( ii ) to begin with let us show @xmath589 . we compute @xmath590 thus @xmath591 , which shows the coincidence of the first letters of @xmath592 and @xmath593 . as for the other @xmath594 and @xmath595 , note that @xmath596 when @xmath41 is odd , we see @xmath597 . when @xmath41 is even , we have @xmath598 and thus we have @xmath597 , i.e. , @xmath599 for all @xmath41 . we have @xmath600 , i.e. , @xmath601 . similarly , we have @xmath602 . as for other @xmath603 and @xmath604 , we have @xmath605 thus we have @xmath606 for all @xmath41 and obtain @xmath589 . similarly we can show @xmath607 . we compute @xmath608 thus @xmath609 , i.e. , the coincidence of the first letters of @xmath592 and @xmath593 . next , we have @xmath610 . on the other hand , we have @xmath467 and @xmath611 , thus @xmath612 , i.e. , @xmath613 . similarly , we can recursively show @xmath614 for all @xmath41 , @xmath615 , @xmath616 for all @xmath41 . so we have @xmath607 , and therefore we get the final result @xmath617 . + ( iii ) the 0-signature of @xmath508 is @xmath618 for some @xmath619 and that of @xmath579 is @xmath620 for some @xmath621 . here we divide into two cases . let us first assume @xmath622 . then the actions of @xmath180 on two tensor products look as follows ( proceed from left to right ) : @xmath623 thus we have @xmath624 ( rr ) pairs and @xmath210 ( ll ) pairs . therefore we have @xmath625 which gives the desired relation . next assume @xmath626 . then we have @xmath627 ( rr ) pairs and @xmath628 ( ll ) pairs and again we obtain @xmath625 . let @xmath453 be @xmath228-highest and @xmath240 . @xmath456 unless @xmath457 and @xmath41 is even . we also know that the coordinates other than @xmath629 are @xmath133 by proposition [ prop : ht cond 1 ] . let us set @xmath630 throughout this subsection . let us define a word @xmath631 by @xmath632 where the exponents are defined as follows . for @xmath633 , @xmath634 define @xmath635 . then @xmath464 . for @xmath636 , @xmath637 set @xmath638 and define other @xmath468 by @xmath639 for @xmath640 . note that @xmath471 . set @xmath472 and define other @xmath474 by @xmath641 for @xmath642 . in this subsection , let @xmath25 and @xmath144 be the @xmath3-diagrams . as before , corresponding to @xmath25 and @xmath144 , we use the parametrization @xmath260 and @xmath521 ( @xmath522 ) respectively . define a word @xmath649 by @xmath650 here the exponents for @xmath651 are @xmath652 the exponents for @xmath653 are @xmath654 s. v. kerov , a. n. kirillov , n. yu . reshetikhin , _ combinatorics , the bethe ansatz and representations of the symmetric group _ , zap.nauchn . ( lomi ) * 155 * ( 1986 ) 5064 ( english translation : j. sov . math . * 41 * ( 1988 ) 916924 ) .
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we calculate the image of the combinatorial @xmath0-matrix for any classical highest weight element in the tensor product of kirillov reshetikhin crystals @xmath1 of type @xmath2 .
the notion of @xmath3-diagrams is effectively used for the identification of classical highest weight elements in @xmath4 .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the problem of community detection in networks has received wide attention @xcite . it has proved to be a problem of remarkable subtlety , computationally challenging and with deep connections to other areas of research including machine learning , signal processing , and spin - glass theory . a large number of algorithmic approaches to the problem have been considered , but interest in recent years has focused particularly on statistical inference methods @xcite , partly because they give excellent results , but also because they are mathematically principled and , at least in some cases , provably optimal @xcite . in this paper we study two of the most fundamental community inference methods , based on the so - called stochastic block model or its degree - corrected variant @xcite . we show that it is possible to map both methods onto the well - known minimum - cut graph partitioning problem , which allows us to adapt any of the large number of available methods for graph partitioning to solve the community detection problem . as an example , we apply the laplacian spectral partitioning method of fiedler @xcite to derive a community detection method competitive with the best currently available algorithms in terms of both speed and quality of results . the first method we consider is based on the stochastic block model , sometimes also called the planted partition model , a well studied model of community structure in networks @xcite . this model supposes a network of @xmath0 vertices divided into some number of groups or communities , with different probabilities for connections within and between groups . we will here focus on the simplest case of just two groups ( of any size , not necessarily equal ) . in the commonest version of the model edges are placed independently at random between vertex pairs with probability @xmath1 for pairs in the same group and @xmath2 for pairs in different groups . in this paper we use the slightly different poisson version of the model described in @xcite , in which we place between each pair of vertices a poisson - distributed number of edges with mean @xmath3 for pairs in the same group and @xmath4 for pairs in different groups . in essentially all real - world networks the fraction of possible edges that are actually present in the network is extremely small ( usually modeled as vanishing in the large-@xmath0 limit ) , in which case the two versions of the model become indistinguishable , but the poisson version is preferred because its analysis is more straightforward . at its heart , the statistical inference of community structure is a matter of answering the following question : if we assume an observed network is generated according to our model , what then must the parameters of that model have been ? in other words , what were the values of @xmath3 and @xmath4 used to generate the network and , more importantly , which vertices fell in which groups ? even though the model is probably not a good representation of the process by which most real - world networks are generated , the answer to this question often gives a surprisingly good estimate of the true community structure . to answer the question , we make use of a maximum likelihood method . let us label the two groups or communities in our model group 1 and group 2 , and denote by @xmath5 the group to which vertex @xmath6 belongs . the edges in the network will be represented by an adjacency matrix having elements @xmath7 then the likelihood of generating a particular network or graph @xmath8 , given the complete set of group memberships , which we ll denote by the shorthand @xmath9 , and the poisson parameters , which we ll denote by @xmath10 , is @xmath11 where @xmath12 denotes the expected number of edges between vertices @xmath6 and @xmath13either @xmath3 or @xmath4 , depending on whether the vertices are in the same or different groups . we are assuming there are no self - edges in the network edges that connect vertices to themselves so @xmath14 for all @xmath6 . given the likelihood , one can maximize it to find the most likely values of the group labels and parameters , which can be done in a number of different ways . in ref . @xcite , for example , the likelihood was maximized first with respect to the parameters @xmath3 and @xmath4 by differentiation . applying this method to eq . gives most likely values of @xmath15 where @xmath16 and @xmath17 are the observed numbers of edges within and between groups respectively for a given candidate division of the network , and @xmath18 and @xmath19 are the numbers of vertices in each group . substituting these values back into eq . gives the profile likelihood , which depends on the group labels only . in fact , one typically quotes not the profile likelihood itself but its logarithm , which is easier to work with . neglecting an unimportant additive constant , the log of the profile likelihood for the present model is @xmath20 the communities can now be identified by maximizing this quantity over all possible assignments of the vertices to the groups . this is still a hard task , however . there are an exponentially large number of possible assignments , so an exhaustive search through all of them is unfeasible for all but the smallest of networks . one can apply standard heuristics like simulated annealing to the problem , but in this paper we take a different approach . in the calculation above , the likelihood is maximized over @xmath10 first , for fixed group assignments , then over the group assignments . but we can also take the reverse approach , maximizing first over the group assignments , for given @xmath10 , and then over @xmath10 at the end . this approach is attractive for two reasons . first , as we will show , the problem of maximizing with respect to the group assignment when @xmath10 is given is equivalent to the standard problem of minimum - cut graph partitioning , a problem for which many excellent heuristics are already available . second , after maximizing with respect to the group assignments the remaining problem of maximizing with respect to @xmath10 is a one - parameter optimization that can be solved trivially . the net result is that the problem of maximum - likelihood community detection is reduced to one of performing a well - understood task graph partitioning plus one undemanding extra step . the resulting algorithm is fast and , as we will see , gives good results . so consider the problem of maximizing the likelihood , eq . , with respect to the group labels @xmath5 , for given values of the parameters @xmath3 and @xmath4 . we will actually maximize the logarithm @xmath21 of the likelihood , @xmath22 , \label{eq : logl1}\ ] ] which gives the same result but is usually easier . to proceed we write @xmath12 and @xmath23 as @xmath24 where @xmath25 is the kronecker delta . substituting these into eq . and dropping overall additive and multiplicative constants , which have no effect on the position of the maximum , the log - likelihood can be rearranged to read @xmath26 where @xmath27 which is positive whenever @xmath28 , meaning we have traditional community structure in our network . ( it is possible to repeat the calculations for the case @xmath29 and derive methods for detecting such structure as well , although we will not do that here . ) the quantity @xmath30 is the cut size of the network partition represented by our two communities , i.e. , the number of edges connecting vertices in different communities , which we previously denoted @xmath17 , and @xmath31 where as previously @xmath18 and @xmath19 are the numbers of vertices in communities 1 and 2 . thus we can also write the log - likelihood in the form @xmath32 the maximization of this log - likelihood corresponds to the minimization of the cut size , with an additional penalty term @xmath33 that favors groups of equal size . this is similar , though not identical , to the so - called ratio cut problem , in which one minimizes the ratio @xmath34 , which also favors groups of equal size , although the nature of the penalty for unbalanced groups is different . the catch with maximizing eq . is that we do nt know the value of @xmath35 , which depends on the unknown quantities @xmath3 and @xmath4 via eq . , but we can get around this problem by the following trick . we first perform a limited maximization of in which the sizes @xmath18 and @xmath19 of the groups are held fixed at some values that we choose . this means that the term @xmath36 is a constant and hence drops out of the problem and we are left maximizing only @xmath37 , or equivalently minimizing the cut - size @xmath17 . this problem is now precisely the standard minimum - cut problem of graph partitioning the minimization of the cut size for divisions of a graph into groups of given sizes . there are @xmath38 possible choices of the sizes of the two groups , ranging from putting all vertices in group 1 to all vertices in group 2 , and everything in between . if we solve the minimum - cut problem for each of these @xmath38 choices we get a set of @xmath38 solutions and we know that one of these must be the solution to our overall maximum likelihood problem . it remains only to work out which one . but choosing between them is easy , since we know that the true maximum also maximizes the profile likelihood , eq . . so we can simply calculate the profile likelihood for each solution in turn and find the one that gives the largest result . in effect , our approach narrows the exponentially large pool of candidate divisions of the network to a one - parameter family of just @xmath38 solutions ( parametrized by group size ) , from which it is straightforward to pick the overall winner by exhaustive search . moreover , the individual candidate solutions are all themselves solutions of the standard minimum - cut partitioning problem , a problem that has been well studied for many years and about which a great deal is known @xcite . although partitioning problems are , in general , hard to solve exactly , there exist many heuristics that give good answers in practical situations . the approach developed here allows us to apply any of these heuristics directly to the maximum - likelihood community detection problem . as an example of this approach , we demonstrate a fast and simple spectral algorithm based on the laplacian spectral bisection method for graph partitioning introduced by fiedler @xcite a description of this method can be found , for example , in @xcite , where it is shown that a good approximation to the minimum - cut division of a network into two parts of specified sizes can be found by calculating the fiedler vector , which is the eigenvector of the graph laplacian matrix @xmath39 corresponding to the second smallest eigenvalue . ( the graph laplacian is the @xmath40 symmetric matrix @xmath41 , where @xmath42 is the adjacency matrix and @xmath43 is the @xmath40 diagonal matrix with @xmath44 equal to the degree of vertex @xmath6 . ) having calculated the fiedler vector one divides the network into groups of the required sizes @xmath18 and @xmath19 by inspecting the vector elements and assigning the @xmath18 vertices with the largest ( most positive ) elements to group 1 and the rest to group 2 . although the method gives only an approximation to the global minimum - cut division , practical experience ( and some rigorous results ) show that it gives good answers under commonly occurring conditions @xcite . a nice feature of this approach is that , in a single calculation , it gives us the entire one - parameter family of minimum - cut divisions of the network . we need calculate the fiedler vector only once , sort its elements in decreasing order , then cut them into two groups in each of the @xmath38 possible ways and calculate the profile likelihood for the resulting divisions of the network . the one with the highest score is ( an approximation to ) the maximum - likelihood community division of the network . these developments are for the standard stochastic block model . as shown in ref . @xcite , however , the standard block model gives poor results when applied to most real - world networks because the model fails to take into account the broad degree distribution such networks possess . this problem can be fixed by a relatively simple modification of the model in which the expected number @xmath12 of edges between vertices @xmath6 and @xmath13 is replaced by @xmath45 where @xmath46 is the degree of vertex @xmath6 and @xmath12 again depends only on which groups the vertices @xmath6 and @xmath13 belong to . all the developments for the standard block model above generalize in straightforward fashion to this `` degree - corrected '' model . the log - likelihood and log - profile likelihood become @xmath47 where @xmath48 and @xmath49 are the sums of the degrees of the vertices in the two groups . in other words , the expressions are identical to those for the uncorrected model except for the replacement of the group sizes @xmath50 by @xmath51 . the maximization of @xmath21 is thus once again reduced to a generalized minimum - cut partitioning problem , with a penalty term proportional to @xmath52 , which again favors balanced groups . although we do nt know the value of @xmath35 , we can reduce the problem to a variant of the minimum - cut problem by the equivalent of our previous approach , holding @xmath48 and @xmath49 constant . and again we can derive a spectral algorithm for this problem based on the graph laplacian . by a derivation analogous to that for the standard spectral method we can show that a good approximation to the problem of minimum - cut partitioning with fixed @xmath51 ( as opposed to fixed @xmath50 ) is given not by the second eigenvector of @xmath39 but by the second eigenvector of the generalized eigensystem @xmath53 , where , as previously , @xmath43 is the diagonal matrix of vertex degrees . once again we calculate the vector and split the vertices into two groups according to the sizes of their corresponding vector elements and once again this gives us a one - parameter family of @xmath38 candidate solutions from which we can choose an overall winner by finding the one with the highest profile likelihood , eq . . vertices , generated using the standard ( uncorrected ) stochastic block model with equal group sizes of 5000 vertices each and a range of strengths of the community structure . defining @xmath54 , @xmath55 , the curves are ( top to bottom ) @xmath56 , 75 , 70 , 65 , and 60 , and @xmath57 . the dashed vertical line indicates the true size of the planted communities . the curves have been displaced from one another vertically for clarity . the vertical axis units are arbitrary because additive and multiplicative constants have been neglected in the definition of the log - likelihood . ( b ) profile likelihoods for the same parameter values but unequal groups of size @xmath58 and @xmath59 . ( c ) the average fraction of vertices classified correctly for networks of @xmath60 vertices each and two equally sized groups . each point is an average over 100 networks . statistical errors are smaller than the points in all cases . the vertical dashed line indicates the position of the `` detectability threshold '' at which community structure becomes formally undetectable @xcite . ] we have tested this method on a variety of networks , and in practice it appears to work well . figure [ fig : synthetic ] shows results from tests on a large group of synthetic ( i.e. , computer - generated ) networks . these networks were themselves generated using the standard stochastic block model ( which is commonly used as a benchmark for community detection @xcite ) . the two left panels in the figure show the value of the profile likelihood for the families of @xmath38 candidate solutions generated by the spectral calculation for networks with two equally sized groups ( top ) and with unequal groups ( bottom ) . in each case there is a clear peak in the profile likelihood at the correct group sizes , suggesting that the algorithm has correctly identified the group membership of most vertices . the third panel in fig . [ fig : synthetic ] tests this conclusion by calculating the fraction of correctly identified vertices as a function of the strength of the community structure for equally sized groups ( which is the most difficult case ) . as the figure shows , the algorithm correctly identifies most vertices over a large portion of the parameter space . the vertical dashed line represents the `` detectability threshold '' identified by previous authors @xcite , below which it is believed that every method of community detection must fail . our algorithm fails below this point also , but appears to work well essentially all the way down to the transition , and there are reasons to believe this result to be exact , at least for networks that are not too sparse @xcite . + + figure [ fig : results ] shows the results of applications of the algorithm to two well - studied real - world networks , zachary s `` karate club '' network @xcite and adamic and glance s network of political blogs @xcite . both are known to have pronounced community structure and the divisions found by our spectral algorithm mirror closely the accepted communities in both cases . in addition to being effective , the algorithm is also fast . the computation of the eigenvector can be done using , for instance , the lanczos method , an iterative method which takes time @xmath61 per iteration , where @xmath62 is the number of edges in the network . the number of iterations required is typically small , although the exact number is not known in general . the search for the division that maximizes the profile likelihood can also be achieved in @xmath61 time . of the @xmath38 different divisions of the network that must be considered , each one differs from the previous one by the movement of just a single vertex from one group to the other . the movement of vertex @xmath6 between groups causes the quantities appearing in eq . to change according to @xmath63 where @xmath64 equals the number of edges between @xmath6 and vertices in group 1 minus the number between @xmath6 and vertices in group 2 . these quantities and the resulting change in the profile likelihood can be calculated in time proportional to the degree of the vertex and hence all @xmath0 vertices can be moved in time proportional to the sum of all degrees in the network , which is equal to @xmath65 . thus , to leading order , the total running time of the algorithm goes as @xmath62 times the number of lanczos iterations , the latter typically being small , and in practice the method is about as fast as the best competing algorithms . in this paper we have shown that the widely - studied maximum likelihood method for community detection in networks can be reduced to a search through a small family of candidate solutions , each of which is itself the solution to a minimum - cut graph partitioning problem , which is a well studied problem about which much is known . this mapping allows us to use trusted partitioning heuristics to solve the community detection problem . as an example we have adapted the method of laplacian spectral partitioning to derive a spectral likelihood maximization algorithm and tested its performance on both synthetic and real - world networks . in terms of both accuracy and speed we find the algorithm to be competitive with the best current methods . a number of extensions of our approach would be possible , including extensions with more general forms for the parameters @xmath10 , such as different values of @xmath3 and @xmath4 for different groups , or extensions to more than two groups , but we leave these for future work . the author would like to thank charlie doering , tammy kolda , and raj rao nadakuditi for useful conversations and lada adamic for providing the data for the network of political blogs . this work was funded in part by the national science foundation under grant dms1107796 and by the air force office of scientific research ( afosr ) and the defense advanced research projects agency ( darpa ) under grant fa95501210432 .
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many methods have been proposed for community detection in networks . some of the most promising are methods based on statistical inference , which rest on solid mathematical foundations and return excellent results in practice . in this paper
we show that two of the most widely used inference methods can be mapped directly onto versions of the standard minimum - cut graph partitioning problem , which allows us to apply any of the many well - understood partitioning algorithms to the solution of community detection problems .
we illustrate the approach by adapting the laplacian spectral partitioning method to perform community inference , testing the resulting algorithm on a range of examples , including computer - generated and real - world networks . both the quality of the results and the running time rival the best previous methods .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the set of dyson schwinger equations form a poincar covariant framework within which to study hadrons @xcite . in rainbow - ladder truncation , they have been successfully applied to calculate a range of properties of the light pseudoscalar and vector mesons , see ref . @xcite and references therein . the dse for the renormalized quark propagator @xmath1 in euclidean space is @xcite @xmath2 where @xmath3 and @xmath4 are the renormalized dressed gluon propagator and quark - gluon vertex , respectively . the most general solution of eq . ( [ gendse ] ) has the form , renormalized at spacelike @xmath5 according to and with @xmath6 the current quark mass . mesons are described by solutions of the homogeneous bethe salpeter equation ( bse ) @xmath7 at discrete values of @xmath8 , where @xmath9 is the meson mass . in this equation , @xmath10 and @xmath11 are the outgoing and incoming quark momenta respectively , and similarly for @xmath12 . the kernel @xmath13 is the renormalized , amputated @xmath14 scattering kernel that is irreducible with respect to a pair of @xmath14 lines . together with the canonical normalization condition for @xmath14 bound states , eq . ( [ hombse ] ) completely determines the bound state bethe salpeter amplitude ( bsa ) @xmath15 . different types of mesons , such as pseudoscalar or vector mesons , are characterized by different dirac structures . to solve the bse , we use the ladder truncation , @xmath16 in conjunction with the rainbow truncation for the quark dse , eq . ( [ gendse ] ) : and , with @xmath17 . here , @xmath18 is the free gluon propagator in landau gauge , and @xmath19 an effective quark - quark interaction , which reduces to the one - loop running coupling of perturbative qcd ( pqcd ) in the perturbative region . this truncation preserves both the vector ward takahashi identity ( wti ) for the @xmath20 vertex and the axial - vector wti , independent of the details of the effective interaction . the latter ensures the existence of massless pseudoscalar mesons associated with dynamical chiral symmetry breaking ( d@xmath21sb ) in the chiral limit @xcite . in combination with an impulse approximation , the former ensures electromagnetic current conservation @xcite . a momentum - dependent quark mass function @xmath22 is central to qcd . in the perturbative region this mass function gives the one - loop perturbative running quark mass @xmath23\right)^{\gamma_m } } \ , , \label{eq : currentm}\end{aligned}\ ] ] with the anomalous mass dimension @xmath24 . dynamical chiral symmetry breaking means that this mass function is nonzero even though the current - quark masses are zero . in the chiral limit the mass function is @xcite @xmath25 \right)^{1-\gamma_m}}\ , , \label{eq : chiralm}\end{aligned}\ ] ] with @xmath26 the renormalization - point - independent vacuum quark condensate @xcite . it is a longstanding prediction of dse studies in qcd that the dressed quark propagator receives strong momentum - dependent corrections at infrared momenta , see e.g. refs . @xcite and references therein . provided that the ( effective ) quark - quark interaction reduces to the perturbative running coupling in the ultraviolet region , it is also straightforward to reproduce the asymptotic behavior of eqs . ( [ eq : currentm ] ) and ( [ eq : chiralm ] ) @xcite . both these phenomena are illustrated in the left panel of fig . [ fig : quark ] . in this figure one can also see that the dynamical mass function of the @xmath27 and @xmath28 quarks becomes very similar to that of quarks in the chiral limit in the infrared region . this is a direct consequence of d@xmath21sb , and leads to a mass function of the order of several hundred mev for the light quarks in the infrared , providing one with a constituent mass for quarks inside hadrons , even though the corresponding current quark masses are only a few mev . the dynamical quark mass function @xmath29 for different quark flavors ( left , adapted from ref . @xcite ) and a comparison of the chiral - limit quark mass function with the lattice data @xcite ( right , adapted from ref . @xcite).,title="fig : " ] the dynamical quark mass function @xmath29 for different quark flavors ( left , adapted from ref . @xcite ) and a comparison of the chiral - limit quark mass function with the lattice data @xcite ( right , adapted from ref . @xcite).,title="fig : " ] these predictions were recently confirmed in lattice simulations of qcd @xcite . quantitative agreement between the lattice simulations and the dse results for the quark propagator functions can be obtained within the rainbow truncation via a suitable choice for the effective quark - quark interaction @xcite . pointwise agreement for a range of quark masses requires this interaction to be flavor - dependent @xcite , suggesting that dressing the quark - gluon vertex @xmath30 is important . indeed , both lattice simulations @xcite and dse studies @xcite of this vertex indicate that @xmath30 deviates significantly from a bare vertex in the nonperturbative region . a ( flavor - dependent ) nonperturbative vertex dressing could make a significant difference for the solution of the quark dse @xcite , as can be seen from the right panel of fig . [ fig : quark ] . the consequences of a dressed vertex for the meson bses are currently being explored @xcite and indications are that in the pseudoscalar and vector channels , the effects are small @xcite . the meson spectrum contains three pseudoscalars with quantum numbers @xmath31 and masses below @xmath32gev : @xmath33 ; @xmath34 ; and @xmath35 . in a constituent - quark model , these mesons are viewed as the first three members of a @xmath14 @xmath36 trajectory , where @xmath37 is the principal quantum number with the ground state @xmath38 ( the @xmath33 ) , and the others are its first two radial excitations , @xmath39 and @xmath40 . the pseudoscalar trajectory is particularly interesting , because its lowest mass member is qcd s goldstone mode . therefore an explanation should describe both chiral symmetry and its dynamical breaking as well as a correlation of the ground and excited states via an approximately linear radial regge trajectory . the latter is easily realized in poincar invariant quantum mechanics @xcite but the former is not . the chiral properties of qcd are manifest in the axial - vector wti , which reads @xmath41 with @xmath42 and @xmath43 the renormalized dressed axial - vector and pseudoscalar vertices , each satisfying an inhomogeneous extension of eq . ( [ hombse ] ) . equation ( [ eq : avwti ] ) is an exact statement in qcd implying that the kernels of the quark dse , eq . ( [ gendse ] ) , and of the bses have to be intimately related . a weak coupling expansion of the dses yields perturbation theory and satisfies this constraint , but is not useful in the study of intrinsically nonperturbative phenomena . however , a systematic and symmetry preserving nonperturbative truncation scheme exists @xcite , allowing for both elucidation and illustration of the consequences of the axial - vector wti . pseudoscalar mesons appear as pole contributions to the axial - vector and pseudoscalar vertices at @xmath44 . the residues of these poles are @xmath45 \ , , \\ i \rho_{\pi_n}\!(\zeta ) & = & z_4(\zeta)\,\ , \int\!\!\frac{d^4q}{(2\pi)^4 } { \rm tr}\left [ \gamma_5 \ ; s(q_+ ) \ , \gamma_{\pi_n}(q_+,q_-;p ) \ , s(q_-)\ , \right ] \,,\end{aligned}\ ] ] both of which are gauge invariant and cutoff independent . it follows from eq . ( [ eq : avwti ] ) that these residues satisfy the following exact identity in qcd @xcite @xmath46 valid for every @xmath47 meson @xcite , irrespective of the magnitude of the current quark mass @xcite . for the @xmath38 , d@xmath21sb yields : @xmath48 and @xmath49 . hence , the gell - mann oakes renner relation emerges as a corollary of eq . ( [ eq : gmorgen ] ) and the ground state pion is qcd s goldstone mode @xcite . for the @xmath50 pseudoscalar mesons one has @xmath51 by assumption , and hence @xmath52 in the chiral limit . furthermore , the ultraviolet behavior of the quark - antiquark scattering kernel in qcd guarantees that @xmath53 is finite in the chiral limit . hence , it is a necessary consequence of chiral symmetry and the axial - vector wti that @xmath54 vanishes in the chiral limit for all excited pseudoscalar mesons ( i.e. for all @xmath55 ) @xcite . we now illustrate the exact results reviewed above in a model that both preserves qcd s ultraviolet properties and exhibits d@xmath21sb , namely the rainbow - ladder truncation of the set of dses . for the infrared behavior of the effective quark - quark interaction , @xmath19 , we employ an ansatz @xcite that is sufficiently enhanced in the infrared to produce a realistic value for the vacuum quark condensate of about @xmath56 . the model parameters , along with the quark masses @xcite , are fitted to give a good description of the chiral condensate , @xmath57 and @xmath58 . the obtained quark propagator functions agree qualitatively with lattice simulations . l|ll|ll|ll|l|ll|ll @xmath59 & @xmath60 & @xmath61 & @xmath62 & @xmath63 & @xmath64 & @xmath61 & @xmath65 & @xmath66 & @xmath61 & @xmath67 & @xmath63 + @xmath68 & 0.14 & 0.14 & 0.131 & 0.131 & 1.10 & 1.3 & -0.002 & 0.74 & 0.77 & 0.206 & 0.22 + @xmath69 & 0.70 & & 0.182 & & 1.41 & 1.47 & -0.033 & 1.08 & 1.02 & 0.257 & 0.23 + @xmath70 & 2.98 & 2.98 & 0.33 & 0.34 & 3.45 & 3.65 & -0.15 & 3.13 & 3.10 & 0.33 & 0.41 + the main results for the pseudoscalar and vector meson masses and leptonic decay constants are summarized in table [ tab : excpion ] and illustrated in fig . [ fig : mesons ] . regarding the table one should note that the rainbow - ladder truncation gives `` ideal '' flavor mixing and the study uses @xmath71 . thus the first row simultaneously describes four degenerate mesons for each column , namely , a \{@xmath72 , @xmath73,@xmath74 } isotriplet and a @xmath75 isosinglet . the second row describes @xmath76 mesons , and the third row @xmath77 mesons . for vector mesons ideal mixing is a very good approximation , though this is not the case for the pseudoscalar ground states . however , the experimental degeneracy of the @xmath34 and @xmath78 suggests @xcite that ideal mixing is almost realized for the excited pseudoscalar mesons . this supports the interpretation of the @xmath78 and @xmath79 as the radial excitations of the @xmath80 and @xmath81 , with quark content almost entirely @xmath75 and @xmath76 respectively @xcite . the masses @xmath82 of the pseudoscalar ground and first radially excited states ( left ) and the corresponding weak decay constants ( right ) as functions of the current - quark mass @xmath83 . for comparison , we also include the mass of the ground - state vector mesons.,title="fig : " ] the masses @xmath82 of the pseudoscalar ground and first radially excited states ( left ) and the corresponding weak decay constants ( right ) as functions of the current - quark mass @xmath83 . for comparison , we also include the mass of the ground - state vector mesons.,title="fig : " ] figure [ fig : mesons ] shows bound - state masses and leptonic decay constants for the ground and excited states of pseudoscalar mesons as well as ground - state vector - meson masses as functions of the current - quark mass . it illustrates that @xmath84 vanishes in the chiral limit , while @xmath85 and @xmath86 do not ; on the other hand , @xmath65 vanishes in the chiral limit , while @xmath62 does not . satisfaction of the axial - vector wti can be checked by virtue of eq . ( [ eq : gmorgen ] ) for any pseudoscalar meson in table [ tab : excpion ] . for the ground and excited states the inaccuracies are below 1% and 5% , the latter are larger because of : ( 1 ) the need to project out the ground state in order to calculate the excited state , ( 2 ) the smallness of @xmath65 close to the chiral limit , and ( 3 ) the larger bound - state mass , leading to a larger region of analytic continuation for the quark dse solution in the complex @xmath87 plane . meson - meson - photon interactions can be described in generalized impulse approximation by @xmath88 \ , , \label{eq : generictri}\ ] ] where @xmath89 , @xmath90 , @xmath91 , and momentum conservation dictates @xmath92 . in eq . ( [ eq : generictri ] ) , @xmath93 is the dressed quark propagator with flavor index @xmath94 , and @xmath95 stands for a generic vertex function with incoming quark flavor @xmath96 and momentum @xmath97 , and outgoing quark flavor @xmath94 and momentum @xmath98 . depending on the specific process under consideration , this vertex function could be a meson bsa or a quark - photon vertex . the quark - photon vertex , , with @xmath99 the photon momentum and @xmath100 the quark momenta , is the solution of the inhomogeneous bse @xmath101 solutions of the homogeneous version of eq . ( [ verbse ] ) define vector meson bound states at timelike photon momenta . it follows that @xmath102 has poles at these locations @xcite ( see also the vector - meson masses in table [ tab : excpion ] ) . there are two diagrams that contribute to meson electromagnetic form factors : one with the photon coupled to the quark and one with the photon coupled to the antiquark respectively . with photon momentum @xmath99 , and incoming and outgoing meson momenta @xmath103 , we can define a form factor for each of these diagrams @xcite @xmath104 we work in the isospin symmetric limit , and thus . the @xmath105 and @xmath106 form factors are given by and , respectively . left : the pion form factor @xmath107 , compared to the experimental results from refs . right : the transition form factor @xmath108 together with experimental data @xcite.,title="fig : " ] left : the pion form factor @xmath107 , compared to the experimental results from refs . right : the transition form factor @xmath108 together with experimental data @xcite.,title="fig : " ] our result for @xmath109 are shown in the left panel of fig . [ fig : emff ] . in the timelike region , and in the spacelike region up to about @xmath110 , @xmath111 can be described very well by a monopole with our calculated @xmath112-mass , @xmath113 ( note that our calculated @xmath112-mass is slightly below the experimental value ) . above this value , our curve starts to deviate more and more from this naive vector meson dominance ( vmd ) monopole . our result is in excellent agreement with the most recent jlab data @xcite ; it would be very interesting to compare with future jlab data in the 3 to 5 gev@xmath114 range , where we expect to see a significant deviation from the naive monopole behavior . the pqcd prediction is significantly smaller than our results at 4 gev@xmath114 ; we anticipate that true perturbative behavior sets in somewhere between 10 and 20 gev@xmath114 @xcite . recent lattice simulation also indicate that in the region between 0 and 2 gev@xmath114 the pion form factor can be represented by a vmd - like monopole @xcite . this vmd - like behavior of the pion form factor in the spacelike region appears to be valid almost independent of the pion mass , both in our calculations and in the lattice simulations , though a monopole fit results in a vmd - mass which is slightly less than the actual vector meson mass . current lattice simulations are not accurate enough , nor do they extend to large enough values of @xmath115 , to detect a significant deviation from vmd - like behavior above 2 gev@xmath114 . l|lll|ll|ll|ll|ll & @xmath116 & @xmath117 & @xmath118 & @xmath119 & @xmath120 & @xmath121 & @xmath120 & @xmath122 & @xmath120 & @xmath123 & @xmath120 + calc . & 0.44 & 0.38 & -0.085 & 53 & 0.69 & 479 & 2.07 & 90 & 0.99 & 130 & 1.19 + expt . & 0.44 & 0.34(5 ) & -0.052(26 ) & 68 & 0.74 & 757 & 2.38 & 50 & 0.84 & 116 & 1.27 + also our results @xcite for @xmath124 agree quite well with the available experimental data , as do both the neutral and the charged kaon charge radius @xcite , see table [ tab : radiiradec ] . it should be noted here that the @xmath106 form factor is obtained by taking the difference of two numbers , @xmath125 and @xmath126 , which are both close to one for @xmath115 near zero . it is therefore much more sensitive to details of the model : both kaon loops and a flavor dependence of the ( effective ) quark - quark interaction will have a significantly bigger effect on the @xmath106 form factor than on the @xmath105 form factor . this could explain why it does not agree as well with experiment as our other results . in this respect one should note that the absolute deviation between our calculated charge radius and the experimental charge radius is similar for the charged and neutral kaon . we can describe the radiative decay of the vector mesons using the same loop integral , eq . ( [ eq : generictri ] ) , this time with one vector meson bsa , one pseudoscalar bsa , and one @xmath127-vertex @xcite . the generic structure of a vector - pseudoscalar - photon vertex is @xmath128 where @xmath129 is the vector momentum and @xmath99 the photon momentum . the on - shell value gives us the coupling constant , and can be used to calculate the partial decay width of the vector mesons . for virtual photons , we can define a transition form factor @xmath130 , normalized to 1 at @xmath131 , which can be used in estimating meson - exchange contributions to hadronic processes @xcite . in the isospin limit , both the @xmath132 and @xmath133 vertices are given by @xmath134 the @xmath135 vertex is a factor of 3 larger , due to the difference in isospin factors ; however , the form factor @xmath130 is the same for @xmath136 and @xmath135 . in contrast to the elastic form factors , this transition form factor falls off significantly faster than a vmd - like monopole , as can be seen in the right panel of fig . [ fig : emff ] . only in the timelike region , near the vector meson pole , do we see a true vmd - like behavior . as eq . ( [ eq : rpgver ] ) shows , it is @xmath137 that is the natural outcome of our calculations . therefore , it is this combination that we give in table [ tab : radiiradec ] , together with the corresponding partial decay widths @xcite . as anticipated , the partial decay width @xmath138 is indeed ( approximately ) nine times larger than the @xmath138 partial decay width . note that part of the difference between the experimental and calculated decay width comes from the phase space factor because our calculated vector meson masses deviate up to 5% from the physical masses . for the @xmath139 decays ( and corresponding form factors ) @xcite the situation is more complicated owing to the interference of the diagrams with the photon coupled to the @xmath69-quark and to the @xmath27- or @xmath28-quark . in the su(3 ) flavor limit , the charged @xmath140 vertex becomes equal to the @xmath136 vertex , whereas the neutral @xmath141 vertex is twice as large in magnitude @xmath142 in table [ tab : radiiradec ] we see indeed that the partial decay width of the neutral @xmath143 is larger than that of the charged @xmath143 , though not by a factor of four as it would in the su(3 ) flavor limit . furthermore , we see that the deviation between our calculation and experiment is largest for the charged @xmath143 decay . again , this can be understood since this decay is sensitive to the difference between the impulse diagrams , and therefore more sensitive to details of the model and its omissions . we would like to thank craig roberts and peter tandy for useful discussions . this work was supported by the austrian research foundation _ fwf , erwin - schrdinger - stipendium _ no.j2233-n08 , the department of energy , office of nuclear physics , contract no . w-31 - 109-eng-38 , and benefited from the anl computing resource center s facilities ; part of the computations were performed on the national science foundation terascale computing system at the pittsburgh supercomputing center . 99 c. d. roberts and a. g. williams , _ prog . nucl . phys . _ * 33 * , 477 ( 1994 ) ; c. d. roberts and s. m. schmidt , _ prog . part . nucl . phys . _ * 45s1 * , 1 ( 2000 ) ; r. alkofer and l. von smekal , _ phys . _ * 353 * , 281 ( 2001 ) . p. maris and c. d. roberts , _ int . j. mod . _ e * 12 * , 297 ( 2003 ) . maris and c. d. roberts , _ phys . _ c * 56 * , 3369 ( 1997 ) . p. maris , c. d. roberts and p. c. tandy , _ phys . _ b * 420 * , 267 ( 1998 ) . c. d. roberts , _ nucl . _ a * 605 * , 475 ( 1996 ) . p. maris and p. c. tandy , _ phys . _ c * 62 * , 055204 ( 2000 ) ; p. maris and p. c. tandy , _ phys . c * 61 * , 045202 ( 2000 ) . h. d. politzer , _ nucl . b * 117 * , 397 ( 1976 ) . k. higashijima , _ phys . _ d * 29 * , 1228 ( 1984 ) . i. fomin , v. p. gusynin , v. a. miransky and y. a. sitenko , _ riv . nuovo cim . _ * 6n5 * , 1 ( 1983 ) . j. i. skullerud and a. g. williams , _ phys . rev . _ d * 63 * , 054508 ( 2001 ) ; j. skullerud , d. b. leinweber and a. g. williams , _ phys . _ d * 64 * , 074508 ( 2001 ) . o. bowman , u. m. heller and a. g. williams , _ phys . _ d * 66 * , 014505 ( 2002 ) . j. b. zhang , p. o. bowman , d. b. leinweber , a. g. williams and f. d. r. bonnet _ phys . _ d * 70 * , 034505 ( 2004 ) . p. maris , a. raya , c. d. roberts and s. m. schmidt , _ eur . j. _ a * 18 * , 231 ( 2003 ) . m. s. bhagwat , m. a. pichowsky , c. d. roberts and p. c. tandy , _ phys . _ c * 68 * , 015203 ( 2003 ) . j. skullerud and a. kizilersu , _ jhep _ * 0209 * , 013 ( 2002 ) ; j. i. skullerud , p. o. bowman , a. kizilersu , d. b. leinweber and a. g. williams , _ jhep _ * 0304 * , 047 ( 2003 ) . m. s. bhagwat , a. holl , a. krassnigg , c. d. roberts and p. c. tandy , _ phys . _ c * 70 * , 035205 ( 2004 ) . m. s. bhagwat and p. c. tandy , arxiv : hep - ph/0407163 . c. s. fischer , f. j. llanes - estrada and r. alkofer , arxiv : hep - ph/0407294 ; f. j. llanes - estrada , c. s. fischer and r. alkofer , arxiv : hep - ph/0407332 . c. s. fischer and r. alkofer , _ phys . _ d * 67 * , 094020 ( 2003 ) . a. bender , c. d. roberts and l. von smekal , _ phys . _ b * 380 * , 7 ( 1996 ) . a. bender , w. detmold , c. d. roberts and a. w. thomas , _ phys . _ c * 65 * , 065203 ( 2002 ) . a. krassnigg , w. schweiger and w. h. klink , _ phys . _ c * 67 * , 064003 ( 2003 ) . a. krassnigg and c. d. roberts , arxiv : nucl - th/0308039 . a. holl , a. krassnigg and c. d. roberts , _ phys . c * 70 * , 042203 ( 2004 ) . m. a. ivanov , y. l. kalinovsky and c. d. roberts , _ phys . _ d * 60 * , 034018 ( 1999 ) . p. maris and p. c. tandy , _ phys . _ c * 60 * , 055214 ( 1999 ) . a. hoell , a. krassnigg , c. d. roberts and s. v. wright , arxiv : nucl - th/0411065 . s. eidelman _ et al . _ [ particle data group collaboration ] , _ phys . lett . _ b * 592 * , 1 ( 2004 ) . k. w. edwards _ et al . _ [ cleo collaboration ] , _ phys . lett . _ * 86 * , 30 ( 2001 ) . j. volmer _ et al . _ [ the jefferson lab f(pi ) collaboration ] , _ phys . lett . _ * 86 * , 1713 ( 2001 ) . s. r. amendolia _ et al . _ [ na7 collaboration ] , _ nucl . _ b * 277 * , 168 ( 1986 ) . maris and c. d. roberts , _ phys . _ c * 58 * , 3659 ( 1998 ) . j. van der heide , j. h. koch and e. laermann , _ phys . _ d * 69 * , 094511 ( 2004 ) . f. d. r. bonnet , r. g. edwards , g. t. fleming , r. lewis and d. g. richards arxiv : hep - lat/0411028 . s. r. amendolia _ et al . _ , _ phys . _ b * 178 * , 435 ( 1986 ) . w. r. molzon _ et al . _ , _ phys . lett . _ * 41 * , 1213 ( 1978 ) [ erratum - ibid . * 41 * , 1523 ( 1978 errat,41,1835.1978 ) ] . p. maris and p. c. tandy , _ phys . _ c * 65 * , 045211 ( 2002 ) . j. w. van orden , n. devine and f. gross , _ phys . lett . _ * 75 * , 4369 ( 1995 ) . p. c. tandy , _ prog . _ * 39 * , 117 ( 1997 ) . r. i. dzhelyadin _ et al . _ , _ phys . b * 102 * , 296 ( 1981 ) [ _ jetp lett . _ * 33 * , 228 ( 1981 ) ] . p. maris , arxiv : nucl - th/0209048 .
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two - body bound states such as mesons are described by solutions of the bethe salpeter equation .
we discuss recent results for the pseudoscalar and vector meson masses and leptonic decay constants , ranging from pions up to @xmath0 bound states .
our results are in good agreement with data .
essential in these calculation is a momentum - dependent quark mass function , which evolves from a constituent - quark mass in the infrared region to a current - quark mass in the perturbative region .
in addition to the mass spectrum , we review the electromagnetic form factors of the light mesons . electromagnetic current conservation is manifest and the influence of intermediate vector mesons is incorporated self - consistently .
the results for the pion form factor are in excellent agreement with experiment .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the spin - orbit ( so ) interaction in mesoscopic and nano - scale semiconductor structures has been at the center stage of research in condensed matter theory and device engineering in recent times . the principal reason is its potential application in spintronics , where the possibility of manipulating and controlling the spin of the electron rather than its charge , plays the all important role @xcite . the spin - orbit fields in a solid are called the rsoi or the dsoi depending on whether the electric field originates from a structural inversion asymmetry or the bulk inversion asymmetry respectively @xcite . quantum rings formed at the interface of two semiconducting materials are ideal candidates where the interplay of the two kinds of soi might be observed . a quantum ring in a heterojunction is realized when a two dimensional gas of electrons is trapped in a quantum well due to the _ band offset _ at the interface of two different semiconducting materials . this _ band offset _ creates an electric field which may be described by a potential gradient normal to the interface @xcite . the potential at the interface is thus asymmetric , leading to the presence of a rsoi . on the other hand , at such interfaces , the bulk inversion symmetry is naturally broken . in addition to this , the topology of such a ring gives rise to remarkable properties typical of low dimensional systems , for example , the persistent current . the phenomenon of persistent current in a conducting mesoscopic ring threaded by an aharonov - bohm ( ab ) flux @xmath0 has been established over many decades . the existence of discrete energy levels and large phase coherence length allow a non - decaying current upon the application of an external magnetic flux @xmath0 . et al . _ @xcite first studied the behavior of persistent current in a metallic ring , and then many attempts have been made to explore the basic mechanisms of persistent current in mesoscopic ring and cylindrical systems @xcite . later , several experiments @xcite have also been performed to verify the existence of non - decaying current in these systems . although the studies involving the mesoscopic rings have already generated a wealth of literature there is still need to look deeper into the problem both from the point of view of fundamental physics and to resolve a few issues that have not yet been answered in an uncontroversial manner . in the present communication we undertake an in - depth analysis of the band structure of a mesoscopic ring in the presence of both the rsoi and the dsoi within a tight - binding ( tb ) formalism . we wish to look for a possible method of determining the strength of the dsoi in a simply connected mesoscopic ring . in recent past there have been a few experiments to measure the strength of the dsoi by optically monitoring the spin precession of the electrons @xcite , through a measurement of the electrical conductance of narrow quantum wires defined in a 2deg @xcite or , by photogalvanic methods @xcite . several studies already exist which deal with electron spin states in mesoscopic rings in presence of both rsoi and dsoi and the interplay between them @xcite , the persistent current in presence of so interaction @xcite and the spin - hall conductance @xcite using the continuum as well as the tight - binding hamiltonians . however , an analytical proposal to measure the strength of the dsoi , particularly within a tight - binding formalism , is still lacking , to the best of our knowledge . in the present work we consider a mesoscopic ring threaded by an ab flux , and work within a tight - binding approximation . both the rsoi and the dsoi are included in the hamiltonian . while looking for a way to estimate the strength of the dsoi , we work out a method to calculate the energy dispersion of the mesoscopic ring either with rsoi or with the dsoi , thus gaining a clean insight into the band structure . this method leads finally to the fact that by making the strengths of the rsoi and the dsoi equal , one comes across an absolute minimum in the conductance of the ring . as the rsoi is tuned , and hence estimated by controlling an external gate voltage , the estimation of the dsoi becomes obvious . the conductance is calculated via the drude weight @xcite numerically , and is found to support the analytical understanding . in the second part , we show that the presence of the so interaction leads to an enhancement of persistent current in one - dimensional rings . in addition to these , we present the detailed energy band structures and the oscillations of persistent current as the rsoi is varied to get a complete picture of the phenomena at the microscopic level . let us start by referring to fig . [ ring ] , where a mesoscopic ring is subject to an ab flux @xmath0 ( measured in unit of the elementary flux quantum @xmath1 ) . within a tb framework the hamiltonian for such an @xmath2-site ring is @xcite ( and references therein ) , @xmath3 here , @xmath4 and , @xmath5 \label{equ3}\ ] ] where , @xmath6 @xmath7 , @xmath8 , @xmath9 , @xmath2 is the site index along the azimuthal direction @xmath10 of the ring . the other factors in eqs . [ equ22 ] and [ eq44 ] are as follows . 0.2 cm = @xmath11 = @xmath12 = @xmath13 . + + here @xmath14 is the site energy of each atomic site of the ring . @xmath15 is the nearest - neighbor hopping integral and @xmath16 is the phase factor due to the ab flux @xmath0 threaded by the ring . @xmath17 and @xmath18 are the isotropic nearest - neighbor transfer integrals which measure the strengths of rashba and dresselhaus so couplings , respectively , and @xmath19 , where @xmath20 . and are the pauli spin matrices . @xmath21 ( @xmath22 ) is the creation ( annihilation ) operator of an electron at the site @xmath23 with spin @xmath24 ( @xmath25 ) . throughout the presentation we choose the units where @xmath26 and measure the so coupling strength in unit of @xmath15 . an elementary analysis of the effect of the so interaction on the spectral and transport properties of the hamiltonian may now be in order . to this end , we consider a mesoscopic ring with rsoi and dsoi , but without any ab flux threading the ring . the so part of the hamiltonian is then re - written as , @xmath27 \label{equ99}\ ] ] which , on expansion becomes , @xmath28 where , @xmath29 a straightforward algebra helps us to recast the above hamiltonian in eq . [ equ99 ] in the form , @xmath30 where , @xmath31 and , @xmath32 with @xmath33 and , + @xmath34 . 0.1 cm we now analyze eq . [ hso ] for two different cases . 0.1 cm @xmath35 * * in this limit eq . [ hso ] leads to the full hamiltonian of a ring with @xmath2 sites . with @xmath36 , and with rsoi alone , this reads , @xmath37 to get the energy dispersion relations analytically , we define a unitary operator @xmath38 which transforms the old operators to a set of new operators . with the operators defined in the new basis , and using the discrete fourier transform , @xmath39 where , @xmath40 is the lattice spacing ( and set equal to unity in the subsequent discussion ) , one obtains explicit expressions of eigenvalues as , @xmath41 and , @xmath42 where , @xmath43 . in presence of a magnetic flux @xmath44 is replaced by @xmath45 . a similar analysis can be done to extract the eigenvalues for the dispersion relations of a ring with dsoi only , that is , with rsoi set equal to zero . however , in presence of both the rsoi and dsoi one has to resort to numerical methods for calculating the eigenvalues and eigenstates . 0.1 cm @xmath35 * * the expression of @xmath46 ( eq . [ factor ] ) plays a key - role in providing a method of estimation of the strength of the dsoi . in the expression of @xmath46 , the presence of the term @xmath47 under the square root ` generates ' an effective site - dependent hopping integral in the hamiltonian . this site dependence of the hopping integral enters through the @xmath48 term , and is responsible for scattering as the electron circulates along the arm of the ring . the scattering reduces the conductivity ( the drude weight ) . from the expression of the factor @xmath49 , we see that , in the absence of @xmath18 , @xmath50 , and the the effective hopping integral becomes site - independent . the transport is ballistic in this case . on the other hand , when @xmath51 the _ fluctuation _ part under the square - root is maximum . in this case @xmath52 . the site dependent scattering is strongest in this case , and should produce a minimum in the conductivity . this is precisely what we observe in the numerical calculations . 0.1 cm @xmath35 * * we now present numerical results in support of this argument , and in addition to this , on the effect of so interaction on the persistent current in such rings . while the results in the former case completely agrees with the predictions from the analytical calculations , the latter results exhibit remarkable variations including a marked enhancement in the amplitude of the persistent current in the presence of rsoi alone . we obtain numerical results for the conductance of a ring with different values of @xmath17 and @xmath18 by calculating the drude weight @xmath53 in accordance with the idea originally put forward by kohn @xcite . the drude weight for the ring is given by the relation , @xmath54 where , @xmath2 gives total number of atomic sites in the ring . kohn has shown that for an insulating system @xmath53 decays exponentially to zero , while it becomes finite for a conducting system . we have investigated the variation of the drude weight @xmath53 as a function of the strength of the rsoi for an ordered ring at half - filling . the results are depicted in fig . [ drude ] . a remarkable feature of the results presented is that , the conductivity ( drude weight ) exhibits an absolute minimum whenever the strength of the dsoi becomes equal to the strength of the rsoi . this has been verified by choosing different values of @xmath18 for a @xmath55-site ordered ring and with electron number @xmath56 . in each case , the conductance minimum appears exactly at @xmath57 confirming , as argued earlier , that a clear estimate of the strength of the dsoi may be obtained by noting the conductance minimum of a mesoscopic ring formed at the heterojuncion of two semiconduncting materials . as the strength of the rsoi is controlled by applying suitable gate voltage , the measurement of the dsoi becomes obvious . before we end this section we would like to point our that the conclusion regarding the minimum in the drude weight remains independent of the numerical values chosen for @xmath17 , and will occur whenever @xmath51 . however , the question is about the detectability of this minimum " . in this regard , we would like to emphasize that observing the minimum numerically depends strongly on the size of the ring ( that is , the number of atoms taken in the ring ) . one can bring down the minimum in the drude weight to any desirable value of @xmath18 by appropriately increasing the size of the ring . for example , though in the paper we have presented results for a @xmath55-site ring with the minima beginning at @xmath58 , we have checked that the minimum can be brought down to say , @xmath59 by choosing a ring with @xmath60 sites . at absolute zero temperature ( @xmath61k ) , the persistent current in the ring described with fixed number of electrons @xmath62 is determined by , @xmath63 where , @xmath64 is the ground state energy . we compute this quantity to understand unambiguously the role of the rsoi interaction alone on persistent current . before presenting the results for @xmath65 , to make the present communication a self contained study , we first take a look at the energy spectrum of both an ordered and a disordered ring with and without the so interactions , as the flux through the ring is varied . in fig . [ energy ] the flux dependent spectra are shown for a @xmath66-site ordered ring and a randomly disordered one ( with diagonal disorder ) in the left and the right columns respectively . clearly , disorder destroys the band crossings observed in the ordered case . the presence of the rsoi and the dsoi also lifts the degeneracy and opens up gaps towards the edges of the spectrum . 0.1 cm @xmath35 * * in fig . [ current ] we examine the effect of the rsoi on the persistent current of an ordered ring with @xmath67 sites . the dsoi is set equal to zero . we have examined both the non - half - filled and half - filled band cases , but present results for the latter only to save space . with increasing strength of the rsoi the persistent current exhibits a trend of an increase in its amplitude . local phase reversals take place together with the appearance of kinks in the current - flux diagrams which are however , not unexpected even without the rsoi , and are results of the band crossings observed in the spectra of such rings . the amplitude of the persistent current at a specific value of the magnetic flux is of course not predictable in any simple manner , and is found to be highly sensitive to the number of electrons @xmath62 ( i.e. , the filling factor ) . issues related to the dependence of the persistent current on the filling factor have been elaborately discussed by splettstoesser _ et al . _ @xcite . the persistent current in an ordered ring also exhibits interesting oscillations in its amplitude as the rsoi is varied keeping the magnetic flux fixed at a particular value . the oscillations persist irrespective of the band - filling factor @xmath62 , with or without the presence of the dsoi . the current exhibits oscillations with growing amplitude as the strength of the rsoi is increased . 0.1 cm @xmath35 * * we now present the results for a disordered ring of @xmath67 sites in fig . [ currentdisorder1 ] . disorder is introduced via a random distribution ( width @xmath68 ) of the values of the on - site potentials ( diagonal disorder ) , and results averaged over sixty disorder configurations have been presented . the dsoi remains zero . without any spin - orbit interaction , disorder completely suppresses the persistent current ( an effect of the localization of the electronic states in the ring ) , as it is observed in fig . [ currentdisorder1 ] ( black curve ) . with the introduction of the rsoi , the current starts increasing , and for @xmath69 ( blue curve ) , increases significantly , attaining a magnitude comparable to that in a perfectly ordered ring . it is to be noted that the strength of the rsoi is strongly dependent on gate voltage . an enhancement of the persistent current in the presence of disorder can be achieved even with much lower values of the rsoi parameter compared to what have already been presented in the figures . to achieve this one needs to increase the size of the mesoscopic ring . we have checked this with a @xmath70-site ring where even with @xmath71 the current increases by an order of magnitude compared to the case when @xmath72 . however , we present the results using a somewhat larger values of @xmath17 for a better viewing of the results . similar observations are made by setting @xmath72 and varying @xmath18 . disorder introduces quantum interference which leads to localization of the electronic states . rsoi , on the other hand , introduces spin flip scattering in the system , which can destroy quantum interference effect , leading to a possible delocalization of the electronic states . this leads to an enhancement of the persistent current in the presence of disorder . the competition between the strength of disorder and the rsoi is also apparent in fig . [ typicalcurrentdisorder ] . for small values of the rsoi , the disorder dominates . as the strength of the rsoi is increased , the spin flip scattering starts dominating over the quantum interference effect , and finally the oscillations become quite similar to that in a ballistic ring . as the so interaction is a natural interaction for a quantum ring grafted at a heterojunction , we are thus tempted to propose that the spin - orbit interaction is responsible for an enhanced persistent current in such mesoscopic disordered rings . before we end this section , we would like to mention that the presence of dsoi alone leads to exactly similar results as expected , since the rashba and the dresselhaus hamiltonians are related by a unitary transformation . this does not change the physics . we have also computed the persistent current in the presence of both the interactions . the amplitude of the current does not increase significantly compared to the case where only one interaction is present . however , the precise magnitude of the current is sensitive to the strength of the magnetic flux threading the ring . the observation remains valid even when the strengths of the rashba and dresselhaus spin - orbit interactions are the same . in conclusion , we have investigated the spectrum and the magnetic response of a tight - binding mesoscopic ring with rashba and dresselhaus spin - orbit interactions both analytically and numerically . two principal results have been obtained and discussed . first , after an exact analysis of the spin dependent hamiltonian we argue that a minimum in the conductance of the ring system when the dsoi equal the strength of the rsoi provides a method of estimating the strength of the former . second , we present numerical results of the band structure of the ring system when it is threaded by an ab flux @xmath0 . the effect of the spin - orbit terms on the energy bands are shown . this is followed by an exhaustive numerical calculation of the persistent current in ordered and disordered rings in the presence of the spin - orbit interactions . the result for the disordered ring exhibits a large persistent current , of the same order of magnitude as that in a perfectly ordered ring , in the presence of rashba spin - orbit interaction . i. zutic , j. fabian , and s. das sarma , rev . mod . phys . * 76 * , 323 ( 2004 ) . s. datta and b. das , appl . . lett . * 56 * , 665 ( 1990 ) . , edited by t. ando , y. arakawa , k. furuya , s. komiyama , and h. nakashima ( springer , berlin , 1998 ) . g .- h . ding and b. dong , phys . b * 76 * , 125301 ( 2007 ) . s. bellucci and p. onorato , phys . rev . b * 78 * , 235312 ( 2008 ) . a. aharony , o. e .- wohlman , y. tokura , and s. katsumoto , phys . b * 78 * , 125328 ( 2008 ) . a. aharony , o. e .- wohlman , y. tokura , and s. katsumoto , physica e * 42 * , 629 ( 2010 ) . r. citro and f. romeo , phys . b * 75 * , 073306 ( 2007 ) . j. s. sheng and k. chang , phys . rev . b * 74 * , 235315 ( 2006 ) . j. splettstoesser , m. governale , and u. zlicke , phys . b * 68 * , 165341 ( 2003 ) . f. e. meijer , a. f. morpurgo , and t. m. klapwijk , phys . b * 66 * , 033107 ( 2002 ) . l. meier , g. salis , i. shorubalko , e. gini , s. schn , and k. ensslin , nature physics * 3 * , 650 ( 2007 ) . j. premper , m. trautmann , j. henk , and p. bruno , phys . b * 76 * , 073310 ( 2007 ) . m. bttiker , y. imry , and r. landauer , phys . lett . * 96a * , 365 ( 1983 ) . h. f. cheung , y. gefen , e. k. riedel , and w. h. shih , phys . rev . b * 37 * , 6050 ( 1988 ) . b. l. altshuler , y. gefen , and y. imry , phys . lett . * 66 * , 88 ( 1991 ) . a. schmid , phys . lett . * 66 * , 80 ( 1991 ) . v. ambegaokar and u. eckern , phys . * 65 * , 381 ( 1990 ) . p. a. orellana and m. pacheco , phys . b * 71 * , 235330 ( 2005 ) . l. k. castelano , g .- q . hai , b. partoens , and f. m. peeters , phys . b * 78 * , 195315 ( 2008 ) . m. abraham and r. berkovits , phys . . lett . * 70 * , 1509 ( 1993 ) . g. bouzerar , d. poilblanc , and g. montambaux , phys . rev . b * 49 * , 8258 ( 1994 ) . l. p. levy , g. dolan , j. dunsmuir , and h. bouchiat , phys . * 64 * , 2074 ( 1990 ) . v. chandrasekhar , r. a. webb , m. j. brady , m. b. ketchen , w. j. gallagher , and a. kleinsasser , phys . * 67 * , 3578 ( 1991 ) . e. m. q. jariwala , p. mohanty , m. b. ketchen , and r. a. webb , phys . 86 * , 1594 ( 2001 ) . r. deblock , r. bel , b. reulet , h. bouchiat , and d. mailly , phys . . lett . * 89 * , 206803 ( 2002 ) . w. rabaud , l. saminadayar , d. mailly , k. hasselbach , a. benoit , and b. etienne , phys . lett . * 86 * , 3124 ( 2001 ) . d. mailly , c. chapelier , and a. benoit , phys . * 70 * , 2020 ( 1993 ) . h. bluhm , n. c. koshnick , j. a. bert , m. e. huber , and k. a. moler , phys . lett . * 102 * , 136802 ( 2009 ) . m. studer , m. p. walser , s. baer , h. rusterholz , s. schn , d. schuh , w. wegscheider , k. ensslin , and g. salis , phys . b * 82 * , 235320 ( 2010 ) . m. scheid , m. kohda , y. kunihashi , k. richter , and j. nitta , phys . * 101 * , 266401 ( 2008 ) . a. r. dedigama , d. jayathilaka , s. h. gunawardana , s. q. murphy , m. edirisoorya , n. goel , t. d. mishima , and m. b. santos , springer proceedings in physics * 119 * , 35 ( 2008 ) . c. yin , b. shen , q. zhang , f. xu , n. tang , l. cen , x. wang , y. chen , and j. yu , appl . * 97 * , 181904 ( 2010 ) . c. p. moca and d. c. marinescu , j. phys . : condens . matter * 18 * , 127 ( 2006 ) . w. kohn , phys . * 133 * , a171 ( 1964 ) .
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electronic transport in a one - dimensional mesoscopic ring threaded by a magnetic flux is studied in presence of rashba and dresselhaus spin - orbit interactions .
a completely analytical technique within a tight - binding formalism unveils the spin - split bands in presence of the spin - orbit interactions and leads to a method of determining the strength of the dresselhaus interaction .
in addition to this , the persistent currents for ordered and disordered rings have been investigated numerically .
it is observed that , the presence of the spin - orbit interaction , in general , leads to an enhanced amplitude of the persistent current .
numerical results corroborate the respective analytical findings .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the accreted stellar halo ( ash ) of a galaxy represents a record of the accretion history of the galaxy itself . its assembly is determined by a large number of free parameters , including the structural properties of each accreted satellite ( virial mass , concentration , stellar content , morphology ) , the orbital properties of each accretion event ( energy and angular momentum at infall ) , the structural properties of the host itself during accretion . this implies a significant degree of stochasticity , as shown by the observed halo - to - halo scatter ( e.g. , van dokkum et al . 2014 ) and by the dichotomy between the ` broken ' and sharply declining density profile of the stellar halo of the milky way ( e.g. , deason et al . 2013 ) and the more extended halo of andromeda , whose density profile is well described by a single power - law ( e.g. , gilbert et al . 2012 , ibata et al . in amorisco ( 2015 ) i have isolated the main ingredients that shape the contribution of each accreted satellite to the ash . i adopted a simplified approach and assumed that the contributing dwarfs are dark matter dominated ( as expected for the case of an @xmath1 host ) and ignored the gravitational influence of the host s disk . combined with the halo mass - concentration relation ( e.g. , gao et al . 2008 , ludlow et al . 2014 ) , this reduces the structural properties of each minor merger to two dimensionless parameters . two additional parameters characterise the orbital properties of the satellite at accretion ( e.g. , benson 2005 , jiang et al . the locus of this parameter space that is relevant to a @xmath0cdm cosmology is explored with a library of minor merger n - body simulations , in which stars are assigned to the most bound 5@xmath2 of the satellite s particles , using a standard particle tagging technique ( e.g. , bullock & johnston 2005 , cooper et al . this study shows that dynamical friction is a major player in shaping stellar deposition , allowing only the most massive ( and/or concentrated ) accreted satellites to deposit their stars in the innermost regions of the host . orbital radialisation by dynamical friction causes the stellar populations deposited by such most massive accretion events ( virial satellite - to - host mass ratio at accretion @xmath3 ) to bear little memory of the details of the orbital properties of the progenitor at infall . galaxies sharing a final virial mass of @xmath4 . panels b ) and c ) : an example for the procedure of assembly of the accreted stellar halo using the combination of a given accretion history and of a library of minor merger simulations ; each contribution to the halo is color - coded by accretion redshift . panel c ) : the spherically averaged density profiles of the 150 accreted stellar haloes corresponding to the accretion histories of panel a ) , color - coding refers to the local mass - weighed virial satellite - to - host mass . panel e ) : the correlation between the total accreted stellar mass in the halo and the number of main accretion events ( see text for details ) . ] i use the library just described to assemble 500 ashs , for galaxies that share a virial mass of @xmath5 . i use monte carlo generated accretion histories ( fakhouri et al . 2010 ) , 150 of which are displayed in _ panel a ) _ of fig . 1 , color - coded by the total accreted stellar mass . satellite stellar masses are assigned based on a redshift - independent abundance matching relation ( garrison - kimmel et al . 2014 , 0.3 dex scatter ) . _ panels b ) _ and _ c ) _ exemplify the assembly of an individual ash : each accreted and disrupted satellite ( color - coded by its accretion redshift ) is associated to the spherically averaged density profile of the stars it deposits in the halo . these are retrieved from the library using the relevant time - interval between accretion and @xmath6 , and are re - scaled to physical quantities according to the dimensional properties of the merger at hand . _ panel d ) _ displays the spherically averaged density profiles of 150 ashs built in this manner . at each radius , the halo - to - halo scatter approaches 2 dex , and increases at @xmath7 , together with an increasing amount of not fully phase - mixed substructure from recent accretion events . although with a significant scatter , on average , ashs share a logarithmic density slope @xmath8 within 20 kpc , and become steeper with radius , as shown by the dashed guiding lines . the details of this steepening are highly variable : some profiles have marked and sharp breaks , others ` roll ' gently towards steeper and steeper slopes , other remain comparatively shallower . the radii where such transitions take place are equally variable . the color - coding in _ panel d ) _ indicates the local mass - weighted satellite - to - host virial mass ratio . on average , the innermost regions of the ash are contributed by satellites that have larger virial mass ratios at accretion . this gradient has been observed in cosmological hydrodynamical simulations ( rodriguez - gomez et al . 2016 ) and i conclude is a direct consequence of dynamical friction ( e.g. , amorisco 2015 ) . color - coding in _ panel d ) _ reveals that the local mean virial mass ratio also correlates positively with the local density . _ panel e ) _ confirms this link by showing a scatter plot of the total accreted stellar mass of the ash against the ` number of main accretion events ' @xmath9 ( i.e. the ratio between the total accreted stellar mass of the ash and the mean stellar mass of the contributing satellites , mass - weighted by stellar mass itself ) . the most massive ashs result from the accretion of just one / two particularly massive satellites , which dominate the ex - situ mass . although this technique represents a highly simplified approach , it allows for an efficient exploration of the significant stochasticity of ashs . physical ingredients that are neglected here ( the host s stellar disk , the morphologies of the accreted dwarfs , any post - accretion star formation etc . ) will result in even increased variability . detailed analysis of a large sample of ashs concentrating on the systematic correlations that connect density profile and accretion history is the subject of a forthcoming work ( amorisco 2016 ) .
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i use a library of controlled minor merger n - body simulations , a particle tagging technique and monte carlo generated @xmath0cdm accretion histories to study the highly stochastic process of stellar deposition onto the accreted stellar halos ( ashs ) of @xmath1 galaxies .
i explore the main physical mechanisms that drive the connection between the accretion history and the density profile of the ash .
i find that : i ) through dynamical friction , more massive satellites are more effective at delivering their stars deeper into the host ; ii ) as a consequence , ashs feature a negative gradient between radius and the local mass - weighed virial satellite - to - host mass ratio ; iii ) in @xmath1 galaxies , most ashs feature a density profile that steepens towards sharper logarithmic slopes at increasing radii , though with significant halo - to - halo scatter ; iv ) the ashs with the largest total ex - situ mass are such because of the chance accretion of a small number of massive satellites ( rather than of a large number of low - mass ones ) .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
binary star systems are of astrophysical importance for various reasons . first , they compose an important portion of stars in the galaxy @xcite and thus theories about stellar formation and evolution should account for the binary nature of stars . second , binary stars allow us to directly measure the masses of their component stars . the determined masses in turn allow other stellar parameters , such as radius and density , to be indirectly estimated . these physical parameters help us to understand the processes by which binary stars form @xcite . in particular , the separation and mass of a binary system tell us about the amount of angular momentum in the system . because it is a conserved quantity , binaries with measured angular momentum give us important clues about the conditions under which the stars were formed . despite the importance , broad ranges of separations , distances , and component masses make it hard to detect and measure all binaries . nearby systems with wide separations may be directly resolved using high - resolution imaging , while systems with small separations can be detected as eclipsing or spectroscopic binaries . however , binaries with intermediate separations are difficult to be detected by the conventional methods . in addition , it is difficult to detect binaries if they are located at large distances or either of the binary components is faint . as a result , samples are restricted to binaries in the solar neighborhood and are not complete down to low - mass stars . for a complete view of stellar multiplicity across a broad range of physical parameters , therefore , it is necessary to use a variety of observational techniques . gravitational microlensing can provide a complementary method that can detect and measure binaries that are difficult to be detected by other methods . microlensing occurs when an astronomical object is closely aligned with a background star . the gravity of the intervening object ( lens ) causes deflection of the light from the background star ( source ) , resulting in the brightening of the source star . if the lens is a single star , the light curve of the source star brightness is characterized by smooth rising and fall . however , if the lens is a binary , the light curve can be dramatically different , particularly for caustic - crossing events , which exhibit strong spikes in the light curve . among caustic - crossing binary - lens events , those with long time scales are of special importance because it is often possible to determine the physical parameters of lenses ( see more details in section 2 ) . the binary separations for which caustic crossings are likely to occur are in the range of order au , for which binaries are difficult to be detected by other methods . in addition , due to the nature of the lensing phenomenon that occurs regardless of the lens brightness , microlensing can provide an important channel to study binaries composed of low - mass stars . furthermore , most microlensing binaries are located at distances of order kpc and thus microlensing can expand the current binary sample throughout the galaxy . in this paper , we report the detections and measurements of 2 binaries discovered from observations of long time - scale caustic - crossing binary microlensing events moa-2011-blg-090 and ogle-2011-blg-0417 . in 2 , we describe the basic physics of binary lensing and the method to determine the physical parameters of binary lenses . in 3 , we describe the choice of sample , observations of the events , and data reduction . in 4 , we describe the procedure of modeling the observed light curves . in 5 , we present the results from the analysis . we discuss about the findings and conclude in 6 . for a general lensing event , where a single star causes the brightening of a background source star , the magnification of the source star flux depends only on the projected separation between the source and the lens as @xmath1 where the separation @xmath2 is normalized in units of the angular einstein radius of the lens , @xmath3 . for a uniform change of the lens - source separation , the light curve of a single - lens event is characterized by a smooth and symmetric shape . the normalized lens - source separation is related to the lensing parameters by @xmath4^{1/2 } , \label{eq2}\ ] ] where @xmath5 represents the time scale for the lens to cross the einstein radius ( einstein time scale ) , @xmath6 is the time of the closest lens - source approach , and @xmath7 is the lens - source separation at that moment . among these lensing parameters @xmath6 , @xmath5 , and @xmath7 , the only quantity related to the physical parameters of the lens is the einstein time scale . however , it results from the combination of the lens mass , distance , and transverse speed of the relative lens - source motion and thus the information about the lens from the time scale is highly degenerate . when gravitational lensing is caused by a binary , the gravitational field is asymmetric and the resulting light curves can be dramatically different from that of a single lensing event @xcite . the most prominent feature of binary lensing that differentiates it from single lensing is a caustic . a set of caustics form a boundary of an envelope of rays as a curve of concentrated light . the gradient of magnification around the caustic is very large . as a result , the light curve of an event produced by the crossing of a source star over the caustic formed by a binary lens is characterized by sharp spikes occurring at the time of caustic crossings . caustic - crossing binary - lens events are useful because it is often possible to measure an additional lensing parameter appearing in the expression of the einstein radius . this is possible because the caustic - crossing part of the light curve appears to be different for events associated with source stars of different sizes @xcite . by measuring the deviation caused by this finite - source effect , it is possible to measure the source radius in units of the einstein radius , @xmath8 ( normalized source radius ) . then , combined with the information about the source angular size , @xmath9 , the einstein radius is determined as @xmath10 . the einstein radius is related to the mass , @xmath11 , and distance to the lens , @xmath12 , by @xmath13 where @xmath14 , @xmath15 is the distance to the source , and @xmath16 represents the relative lens - source proper motion . unlike the einstein time scale , the einstein radius does not depend on the transverse speed of the lens - source motion and thus the physical parameters are less degenerate compared to the einstein time scale . among caustic - crossing events , those with long time scales are of special interest because it is possible to completely resolve the degeneracy of the lens parameters and thus uniquely determine the mass and distance to the lens . this is possible because an additional lensing parameter of the lens parallax can be measured for these events . the lens parallax is defined as the ratio of earth s orbit , i.e. 1 au , to the physical einstein radius , @xmath17 , projected on the observer plane , i.e. @xmath18 with simultaneous measurements of the einstein radius and the lens parallax , the mass and distance to the lens are uniquely determined as @xmath19 and @xmath20 respectively @xcite . the lens parallax is measured by analyzing deviations in lensing light curves caused by the deviation of the relative lens - source motion from a rectilinear one due to the change of the observer s position induced by the orbital motion of the earth around the sun @xcite . this deviation becomes important for long time - scale events , which endure for a significant fraction of the orbital motion of the earth . therefore , the probability of measuring the lens parallax is high for long time - scale events . we searched for long time - scale caustic - crossing binary events among lensing events discovered in the 2011 microlensing observation season . we selected events to be analyzed based on the following criteria . 1 . the overall light curve was well covered with good photometry . at least one of caustic crossings was well resolved for the einstein radius measurement . 3 . the time scale of an event should be long enough for the lens parallax measurement . from this search , we found 2 events of moa-2011-blg-090 and ogle-2011-blg-0417 . besides these events , there exist several other long time - scale caustic - crossing events , including moa-2011-blg-034 , ogle-2011-blg-0307/moa-2011-blg-241 , and moa-2011-blg-358/ogle-2011-blg-1132 . we did not include moa-2011-blg-034 and moa-2011-blg-358/ogle-2011-blg-1132 in our analysis list because the coverage and photometry of the events are not good enough to determine the physical lens parameters by measuring subtle second - order effects in the lensing light curve . for ogle-2011-blg-0307/moa-2011-blg-241 , the signal of the parallax effect was not strong enough to securely measure the physical parameters of the lens . ll moa-2011-blg-090 & moa moa - ii 1.8 m , new zealand + & ogle warsaw 1.3 m , chile + & @xmath21fun ctio / smarts2 1.3 m , chile + & @xmath21fun pest 0.3 m , australia + & mindstep danish 1.54 m , chile + & robonet fts 2.0 m , australia + ogle-2011-blg-0417 & ogle warsaw 1.3 m , chile + & @xmath21fun ctio / smarts2 1.3 m , chile + & @xmath21fun auckland 0.4 m , new zealand + & @xmath21fun fco 0.36 m , new zealand + & @xmath21fun kumeu 0.36 m , new zealand + & @xmath21fun opd 0.6 m , brazil + & planet canopus 1.0 m , australia + & planet saao 1.0 m , south africa + & mindstep danish 1.54 m , chile + & robonet ftn 2.0 m , hawaii + & robonet lt 2.0 m , spain the events moa-2011-blg-090 and ogle-2011-blg-0417 were observed by the microlensing experiments that are being conducted toward galactic bulge fields by 6 different groups including moa , ogle , @xmath21fun , planet , robonet , and mindstep . among them , the moa and ogle collaborations are conducting survey observations for which the primary goal is to detect a maximum number of lensing events by monitoring a large area of sky . the @xmath21fun , planet , robonet , and mindstep groups are conducting follow - up observations of events detected from survey observations . the events were observed by using 12 telescopes located in 3 different continents in the southern hemisphere . in table [ table : one ] , we list the telescopes used for the observation . reduction of the data was conducted by using photometry codes developed by the individual groups . the ogle and moa data were reduced by photometry codes developed by @xcite and @xcite , respectively , which are based on the difference image analysis method @xcite . the @xmath21fun data were processed using a pipeline based on the dophot software @xcite . for planet and mindstep data , a pipeline based on the pysis software @xcite is used . for robonet data , the dandia pipeline @xcite is used . to standardize error bars of data estimated from different observatories , we re - scaled them so that @xmath22 per degree of freedom becomes unity for the data set of each observatory , where @xmath22 is computed based on the best - fit model . for the data sets used for modeling , we eliminate data points with very large photometric uncertainties and those lying beyond @xmath23 from the best - fit model . we present the light curves of events in figure [ fig : one ] and [ fig : two ] . to be noted is that the overall light curves including caustic crossings of both events are well covered by survey observations . this demonstrates that the observational cadence of survey experiments is now high enough to characterize lensing events based on their own data . the parts of light curves with 2455880 < hjd < 2455960 were not covered because the galactic bulge field could not be observed . although not included in the selection criteria , both events showed a common bump to those involved with caustic crossings : at hjd@xmath242455655 for moa-2011-blg-090 and at hjd@xmath242455800 for ogle-2011-blg-0417 . these bumps were produced during the approach of the source trajectory close to a cusp of a caustic . @xcite pointed out that such triple - peak features help to better measure the lens parallax . in modeling the light curve of each event , we search for a solution of lensing parameters that best characterizes the observed light curve . describing the basic feature of a binary - lens light curve requires 6 parameters including the 3 single - lensing parameters @xmath6 , @xmath7 , and @xmath5 . the 3 additional binary - lensing parameters include the mass ratio between the lens components , @xmath25 , the projected separation in units of the einstein radius , @xmath26 , and the angle between the source trajectory and the binary axis , @xmath27 ( trajectory angle ) . in addition to the basic binary lensing parameters , it is required to include additional parameters to precisely describe detailed features caused by various second - order effects . the first such effect is related to the finite size of the source star . this finite - source effect becomes important when the source is located at a position where the gradient of magnification is very high and thus different parts of the source surface experience different amounts of magnification . for binary - lens events , this happens when the source approaches or crosses the caustic around which the gradient of magnification is very high . to describe the light curve variation caused by the finite - source effect , it is necessary to include an additional parameter of the normalized source radius , @xmath8 . for long time - scale events , such as those analyzed in this work , it is required to additionally consider the parallax effect . consideration of the parallax effect in modeling requires to include 2 additional parameters @xmath28 , and @xmath29 , which represent the two components of the lens parallax vector @xmath30 projected on the sky along the north and east equatorial coordinates , respectively . the direction of the lens parallax vector corresponds to the relative lens - source relative motion in the frame of the earth at a specific time of the event . the size of the vector corresponds to the ratio of the earth s orbit to the einstein radius projected on the observer s plane , i.e. @xmath31 $ ] , where @xmath17 is the physical size of the einstein radius . another effect to be considered for long time - scale events is the orbital motion of the lens @xcite . the lens orbital motion affects lensing light curves in two different ways . first , it causes to change the binary separation and thus the magnification pattern . second , it also causes the binary axis to rotate with respect to the source trajectory . in order to fully account for the lens orbital motion , 4 additional parameters are needed . the first two of these parameters are @xmath32 and @xmath33 , which represent the change rates of the projected binary separation and the trajectory angle , respectively . the other two orbital parameters are @xmath34 and @xmath35 , where @xmath34 represents the line - of - sight separation between the binary components in units of @xmath3 and @xmath35 represents its rate of change . for a full description of the orbital lensing parameters , see the summary in the appendix of @xcite . the deviation in a lensing light curve affected by the orbital effect is smooth and long lasting and thus is similar to the deviation induced by the parallax effect . this implies that if the orbital effect is not considered , the measured lens parallax and the resulting lens parameters might be erroneous . therefore , considering the orbital effect is important not only for constraining the orbital motion of the lens but also for precisely determining the physical parameters of the lens . with all these parameters , we test three different models . in the first model , we fit the light curve with standard binary lensing parameters considering the finite - source effect ( standard model ) . in the second model , we additionally consider the parallax effect ( parallax model ) . finally , we take the orbital effect into consideration as well ( orbital model ) . when the source trajectory is a straight line , the two light curves resulting from source trajectories with positive ( @xmath36 ) and negative ( @xmath37 ) impact parameters are identical due to the symmetry of the magnification pattern with respect to the binary axis . when either the parallax or the orbital effect is considered , on the other hand , the source trajectory deviates from a straight line and thus the light curves with @xmath36 and @xmath37 are different from each other . we , therefore , consider both the @xmath36 and @xmath37 cases for each of the models considering the parallax and orbital effects . in modeling , the best - fit solution is obtained by minimizing @xmath22 in the parameter space . we conduct this in 3 stages . in the first stage , grid searches are conducted over the space of a subset of parameters and the remaining parameters are searched by using a downhill approach @xcite . we then identify local minima in the grid - parameter space by inspecting the @xmath22 distribution . in the second stage , we investigate the individual local minima by allowing the grid parameters to vary and find the exact location of each local minimum . in the final stage , we choose the best - fit solution by comparing @xmath22 values of the individual local minima . this multiple stage procedure is needed for thorough investigation of possible degeneracy of solutions . we choose of @xmath26 , @xmath25 , and @xmath27 as the grid parameters because they are related to the light curve features in a complex way such that a small change in the values of the parameters can lead to dramatic changes in the resulting light curve . on the other hand , the other parameters are more directly related to the light curve features and thus they are searched for by using a downhill approach . for the @xmath22 minimization in the downhill approach , we use the markov chain monte carlo ( mcmc ) method . once a solution of the parameters is found , we estimate the uncertainties of the individual parameters based on the chain of solutions obtained from mcmc runs . ccc @xmath38 & 0.52 & 0.71 + @xmath39 & 0.45 & 0.61 + @xmath40 & 0.37 & 0.51 + source type & fv & kiii + @xmath41 ( k ) & 6650 & 4660 + @xmath42 ( @xmath43 ) & 2 & 2 + @xmath44 ( @xmath45 ) & 4.5 & 2.5 to compute lensing magnifications affected by the finite - source effect , we use the ray - shooting method . @xcite . in this method , rays are uniformly shot from the image plane , bent according to the lens equation , and land on the source plane . then , a finite magnification is computed by comparing the number densities of rays on the image and source planes . precise computation of finite magnifications by using this numerical technique requires a large number of rays and thus demands heavy computation . to minimize computation , we limit finite - magnification computation by using the ray - shooting method only when the lens is very close to caustics . in the adjacent region , we use an analytic hexadecapole approximation @xcite . in the region with large enough distances from caustics , we use a point - source magnifications . in the finite magnification computation , we consider the variation of the magnification caused by the limb - darkening of the source star s surface . we model the surface brightness profile of a source star as @xmath46,\ ] ] where @xmath47 is the linear limb - darkening coefficients , @xmath48 is the source star flux , and @xmath49 is the angle between the normal to the source star s surface and the line of sight toward the star . the limb - darkening coefficients are set based the source type that is determined on the basis of the color and magnitude of the source . in table [ table : two ] , we present the used limb - darkening coefficients , the corresponding source types , and the measured de - reddened color along with the assumed values of the effective temperature , @xmath41 , the surface turbulence velocity , @xmath42 , and the surface gravity , @xmath44 . for both events , we assume a solar metallicity . in table [ table : three ] , we present the solutions of parameters for the tested models . the best - fit model light curves are drawn on the top of the observed light curves in figures [ fig : one ] and [ fig : two ] . in figure [ fig : three ] , we present the geometry of the lens systems where the source trajectory with respect to the caustic and the locations of the lens components are marked . we note that the source trajectories are curved due to the combination of the parallax and orbital effects . we also note that the positions of the lens components and the corresponding caustics change in time due to the orbital motion and thus we present caustics at 2 different moments that are marked in figure [ fig : three ] . these moments correspond to those of characteristic features on the light curve such as the peak involved during a cusp approach or a caustic crossing . to better show the differences in the fit between different models , we present the residuals of data from the best - fits of the individual models . for a close - up view of the caustic - crossing parts of the light curves , we also present enlargement of the light curve . for both events , the parallax and orbital effects are detected with significant statistical confidence levels . it is found that inclusion of the second - order effects of the parallax and orbital motions improves the fits with @xmath50 and @xmath51 for moa-2011-blg-090 and ogle-2011-blg-0417 , respectively . to be noted is that the orbital effect is considerable for ogle-2011-blg-0417 and thus the difference between the values of the lens parallax measured with ( @xmath52 ) and without ( @xmath53 ) considering the orbital effect is substantial . since the lens parallax is directly related to the physical parameters of the lens , this implies that considering the orbital motion of the binary lens is important for the accurate measurement of the lens parallax and thus the physical parameters . lrrrrrr @xmath54 & 5207/5164 & 4718/5162 & 4636/5158 & 4415/2627 & 2391/2625 & 1735/2621 + @xmath6 ( hjd ) & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] + @xmath7 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] + @xmath5 ( days ) & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] + @xmath56 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] + @xmath25 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] + @xmath27 ( rad ) & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] + @xmath57 ( @xmath58 ) & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] + @xmath28 & & [email protected] & [email protected] & & [email protected] & [email protected] + @xmath29 & & [email protected] & [email protected] & & [email protected] & [email protected] + @xmath59 ( @xmath60 ) & & & [email protected] & & & [email protected] + @xmath33 ( @xmath60 ) & & & [email protected] & & & [email protected] + @xmath61 & & & [email protected] & & & [email protected] + @xmath62 ( @xmath60 ) & & & [email protected] & & & [email protected] lrr @xmath63 ( @xmath64 ) & [email protected] & [email protected] + @xmath65 ( @xmath64 ) & [email protected] & [email protected] + @xmath66 ( @xmath64 ) & [email protected] & [email protected] + @xmath3 ( mas ) & [email protected] & [email protected] + @xmath21 ( mas yr@xmath67 ) & [email protected] & [email protected] + @xmath12 ( kpc ) & [email protected] & [email protected] + @xmath68 ( au ) & [email protected] & [email protected] + @xmath69 ( yr ) & [email protected] & [email protected] + @xmath70 & [email protected] & [email protected] + @xmath71 ( @xmath72 ) & [email protected] & [email protected] the finite - source effect is also clearly detected and the normalized source radii are precisely measured for both events . to obtain the einstein radius from the measured normalized source radius , @xmath8 , additional information about the source star is needed . we obtain this information by first locating the source star on the color - magnitude diagram of stars in the field and then calibrating the source brightness and color by using the centroid of the giant clump as a reference under the assumption that the source and clump giants experience the same amount of extinction and reddening @xcite . the measured @xmath73 colors are then translated into @xmath74 color by using the @xmath75 relations of @xcite and the angular source radius is obtained by using the @xmath74 color and the angular radius given by @xcite . in figure [ fig : four ] , we present the color - magnitudes of field stars constructed based on ogle data and the locations of the source star . we find that the source star is a f - type main - sequence star with a de - reddened color of @xmath76 for moa-2011-blg-090 and a k - type giant with @xmath77 for ogle-2011-blg-0417 . here we assume that the de - reddened color and absolute magnitude of the giant clump centroid are @xmath78 and @xmath79 @xcite , respectively . the mean distances to clump stars of @xmath247200 pc for moa-2011-blg-090 and @xmath247900 pc for ogle-2011-blg-0417 are estimated based on the galactic model of @xcite . the measured einstein radii of the individual events are presented in table [ table : four ] . also presented are the relative lens - source proper motions as determined by @xmath80 . with the measured lens parallax and the einstein radius , the mass and distance to the lens of each event are determined by using the relations ( [ eq5 ] ) and ( [ eq6 ] ) . the measured masses of the binary components are 0.43 @xmath0 and 0.39 @xmath0 for moa-2011-blg-090 and 0.57 @xmath0 and 0.17 @xmath0 for ogle-2011-blg-0417 . it is to be noted that both lens components of moa-2011-blg-090 and one component of ogle-2011-blg-0417 are m dwarfs which are difficult to be detected by other conventional methods due to their faintness . it is found that the lenses are located at distances @xmath81 kpc and @xmath82 kpc for moa-2011-blg-090 and ogle-2011-blg-0417 , respectively . since full keplerian motion of the binary lens is considered in our modeling , we also determine the orbital parameters of the semi - major axis @xmath68 , period @xmath69 , eccentricity @xmath70 , and inclination @xmath71 . we find that the binary lens components of moa-2011-blg-090 are orbiting each other with a semi - major axis of @xmath83 au and an orbital period of @xmath84 yrs . for ogle-2011-blg-0417 , the semi - major axis and the orbital period of the binary lens are @xmath85 au and @xmath86 yrs , respectively . in figure [ fig : five ] , we present the distributions of the physical and orbital parameters constructed based on the mcmc chains . in table [ table : four ] , we summarize the measured parameters of the binary lenses for both events . we note that the uncertainties of the parameters are based on the standard deviations of the mcmc distributions . we note that the blended light of ogle-2011-blg-0417 comes very likely from the lens itself , implying that the lens can be directly observed . based on the measured mass of 0.57 @xmath0 , the primary of the binary lens corresponds to a late k - type main sequence star with an absolute magnitude and a de - reddened color of @xmath87 and @xmath88 , respectively . considering the distance of 0.89 kpc and assuming an extinction of @xmath89 and the color index of @xmath90 , the apparent brightness and color of the lens correspond to @xmath91 and @xmath92 , respectively . these values match very well with the location of the blend marked on the right panel of figure [ fig : four ] , implying that the blend is very likely the lens . the visibility of the lens signifies this event because it is possible to check the microlensing orbital solution by spectroscopic radial - velocity observation . we reported detections and measurements of 2 binaries discovered from observations of microlensing events moa-2011-blg-090 and ogle-2011-blg-0417 . the light curves of the events have common characteristics of long durations with caustic - crossing features , which enabled to determine the physical parameters of the lenses . it was found that both lens components of moa-2011-blg-090 and one component of ogle-2011-blg-0417 are m dwarfs . therefore , the discovered microlensing binaries demonstrate the usefulness of gravitational lensing in detecting and characterizing binaries composed of low - mass stars . by considering full keplerian binary motion , we also determined the orbital parameters of the binaries . for ogle-2011-blg-0417 , the blended light comes very likely from the lens itself and thus it would be possible to check the orbital solution from follow - up radial - velocity observation . studies of m dwarfs are important not only because they are the most abundant stars in the milky way but also they form a link between solar - type stars and brown dwarfs ; two mass regimes that might exhibit very different multiplicity characteristics . precise knowledge of multiplicity characteristics and how they change in this transitional mass region provide constraints on low - mass star and brown dwarf formation @xcite . despite the importance of m - dwarf binaries , only a few measurements of the binary fraction and distribution of low - mass stars have been made , e.g. , @xcite , and the samples are restricted to only binaries in the solar neighborhood . as a result , there are still large uncertainties about their basic physical properties as well as their formation environment . considering the rapid improvement of lensing surveys both in equipment and strategy , it is expected that the number of microlensing binaries with measured physical parameters will increase rapidly . this will contribute to the complete view of stellar multiplicity across a wide range of binary parameters . work by ch was supported by creative research initiative program ( 2009 - 0081561 ) of national research foundation of korea . the ogle project has received funding from the european research council under the european community s seventh framework programme ( fp7/2007 - 2013 ) / erc grant agreement no . the moa experiment was supported by grants jsps22403003 and jsps23340064 . ts was supported by the grant jsps23340044 . y. muraki acknowledges support from jsps grants jsps23540339 and jsps19340058 . the mindstep monitoring campaign is powered by artemis ( automated terrestrial exoplanet microlensing search ; dominik et al . 2008 , an 329 , 248 ) . mh acknowledges support by the german research foundation ( dfg ) . dr ( boursier fria ) and j. surdej acknowledge support from the communaut franaise de belgique actions de recherche concertes acadmie universitaire wallonie - europe . the robonet team is supported by the qatar foundation through qnrf grant nprp-09 - 476 - 1 - 78 . cs received funding from the european union seventh framework programme ( fpt/2007 - 2013 ) under grant agreement 268421 . this work is based in part on data collected by mindstep with the danish 1.54 m telescope at the eso la silla observatory . the danish 1.54 m telescope is operated based on a grant from the danish natural science foundation ( fnu ) . a. gould and b.s . gaudi acknowledge support from nsf ast-1103471 . gaudi , a. gould , and r.w . pogge acknowledge support from nasa grant nng04gl51 g . work by j.c . yee is supported by a national science foundation graduate research fellowship under grant no . 2009068160 . s. dong s research was performed under contract with the california institute of technology ( caltech ) funded by nasa through the sagan fellowship program . research by tch was carried out under the krcf young scientist research fellowship program . tch and cul acknowledge the support of korea astronomy and space science institute ( kasi ) grant 2012 - 1 - 410 - 02 .
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despite astrophysical importance of binary star systems , detections are limited to those located in small ranges of separations , distances , and masses and thus it is necessary to use a variety of observational techniques for a complete view of stellar multiplicity across a broad range of physical parameters . in this paper , we report the detections and measurements of 2 binaries discovered from observations of microlensing events moa-2011-blg-090 and ogle-2011-blg-0417 . determinations of the binary masses are possible by simultaneously measuring the einstein radius and the lens parallax .
the measured masses of the binary components are 0.43 @xmath0 and 0.39 @xmath0 for moa-2011-blg-090 and 0.57 @xmath0 and 0.17 @xmath0 for ogle-2011-blg-0417 and thus both lens components of moa-2011-blg-090 and one component of ogle-2011-blg-0417 are m dwarfs , demonstrating the usefulness of microlensing in detecting binaries composed of low - mass components . from modeling of the light curves considering full keplerian motion of the lens
, we also measure the orbital parameters of the binaries .
the blended light of ogle-2011-blg-0417 comes very likely from the lens itself , making it possible to check the microlensing orbital solution by follow - up radial - velocity observation . for both events ,
the caustic - crossing parts of the light curves , which are critical for determining the physical lens parameters , were resolved by high - cadence survey observations and thus it is expected that the number of microlensing binaries with measured physical parameters will increase in the future .
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inclined air showers are conventionally defined as those arriving at ground with zenith angles @xmath1 above @xmath2 . at large zenith angles the electromagnetic ( em ) component in air showers , mainly produced by the decay of @xmath3s , is largely absorbed in the greatly enhanced atmospheric depth the shower needs to cross before reaching ground , so that in a first approximation only the more penetrating particles such as muons survive to ground . muons are accompanied by an electromagnetic component produced mainly by muon decay in flight and muon interactions such as bremsstrahlung , pair production and nuclear interactions . a full characterisation of this so - called electromagnetic halo " is given in this work . the study of inclined showers is of great interest because their detection immediately enhances the exposure of existing air shower detectors by up to about @xmath4 with respect to that achieved with vertical showers @xmath5 , extending the field of view to sky directions otherwise inaccessible . inclined showers have been detected in the past in arrays of ground detectors such as the haverah park @xcite and agasa @xcite experiments . modern detectors such as the pierre auger observatory @xcite or the telescope array @xcite can also be used to detect showers with large zenith angles . in particular , the surface detector array ( sd ) of the pierre auger observatory is well suited to detect very inclined showers at energies above about @xmath6 ev , with high efficiency and unprecedented statistical accuracy . the energy spectrum of ultra - high energy cosmic rays ( uhecr ) with inclined showers was recently measured @xcite . the surface detector is an extended array of deep water cherenkov detectors that act like volume detectors adequate for recording particles arriving at ground at all zenith angles . as in all ground arrays , the distribution of the detector signals produced by shower particles is used to estimate shower observables such as the primary energy and to perform composition studies . as shown in previous works @xcite , the specific characteristics of inclined showers entail that their analysis requires a different approach from the standard one for vertical showers . the study of the particle densities of the electromagnetic and muonic components at ground level becomes essential in the reconstruction @xcite and analysis of events at large zenith angles . inclined showers have also been studied for years for other reasons . in the 70 s , it was suggested that showers induced by neutrinos could be identified in the background of inclined proton or nuclei induced showers by searching for deeply penetrating inclined showers @xcite which should exhibit a significant electromagnetic component at ground . in this respect the study and characterisation of the electromagnetic halo in inclined showers is also of great importance . in this paper we have used monte carlo generators simulating the air shower development to obtain the particle densities induced by the electromagnetic and muonic components of inclined air showers . in contrast to other works @xcite we have performed a comprehensive characterisation of the ratio of the electromagnetic to muonic densities as a function of distance to the shower core for different zenith angles . we have studied the dependences of this ratio on the primary energy , mass composition and hadronic interaction model assumed in the simulations . we have also studied the effect of the geomagnetic field that deviates charged particles along their paths to the detector . as a result of this study we give parameterisations of the average ratio of the em and muonic components as a function of the distance to the shower core , shower zenith angle ( @xmath1 ) and azimuth angle ( @xmath7 ) of the position at which the particle arrives at ground with respect to the incoming shower direction , as well as on the distance to the shower core in the shower plane . the effect of the intrinsic shower - to - shower fluctuations on the electromagnetic and muonic contributions to the density has also been studied . our results are relevant for the reconstruction of inclined showers in ultra - high energy cosmic ray detectors such as the pierre auger observatory and the telescope array . we have generated a library of inclined air showers induced by proton and iron primaries using the monte carlo shower propagation code aires 2.6.0 @xcite and the hadronic interaction models qgsjet01 @xcite and sibyll 2.1 @xcite . showers were generated with energies of 1 , 10 and , with zenith angle ranging from 60@xmath8 to 88@xmath8 in steps of 2@xmath8 and random azimuth angle ( unless otherwise indicated ) . we have simulated 100 showers for each combination of energy and zenith angle . showers were simulated with and without geomagnetic field assuming the location of the surface detector of the pierre auger observatory @xcite ( ground level placed at a depth of @xmath9 ) and using a curved atmosphere based on the linsley s atmospheric model @xcite . the number of particles that are produced in an air shower at eev energies and above and the computing time needed to follow all the secondaries is typically excessively large . to avoid this problem , current air shower simulation codes use a statistical sampling algorithm ( thinning algorithm ) which allows to propagate only a small representative fraction of the total number of particles @xcite . statistical weights are assigned to the sampled particles in order to compensate for the rejected ones . therefore , the output of the air shower simulation is a detailed data file with weighted entries containing the particle information at ground level . in this work , showers were simulated with a relative thinning level of @xmath10 and a weight factor for the electromagnetic ( heavy ) particles of 12 ( 0.14 ) . since the thinning process itself introduces artificial fluctuations , for the study of the intrinsic shower - to - shower fluctuations we have also simulated sets showers with a thinning level of @xmath11 . all the particles with kinetic energies above the following thresholds were tracked : 80 kev for electrons , positrons and gammas , 10 mev for muons , 60 mev for mesons and 120 mev for nucleons and nuclei . since the ground arrays differ in detection thresholds and detector response , we have performed particle energy cuts at ground level corresponding to the particular case of water cherenkov detector thresholds : 264 kev for electrons and positrons , 1.286 mev for gammas and 54.6 mev for muons . note that the kinetic energy thresholds used in the aires simulations for electromagnetic particles are all below the cherenkov thresholds , implying that we are not artificially eliminating electrons , positrons or photons that could contribute to the electromagnetic densities . concerning muons , one can see by extrapolating the energy spectrum of muons shown in fig . [ muonspectrum ] that the number of muons below 10 mev in inclined showers not accounted for in our aires simulations but still being able to produce electrons above the cherenkov threshold in water represents a small fraction of the total number of muons . to calculate the particle densities induced by the electromagnetic and muonic components of the shower , we perform a simple resampling procedure ( also called _ unthinning _ procedure ) . the particle density is calculated as the sum of the statistical weights assigned by the thinning algorithm of the particles falling in a sampling region divided by its area . the sampling area must be large enough so that a significant fraction of particles falls inside it , but at the same time it should be small enough so that particle properties are representative of their expected properties in the particular region of interest @xcite . in this work , we have chosen two different sampling regions in the plane perpendicular to the incoming shower direction ( shower plane ) . on the one hand following the previous criterion , , we have considered particles falling in concentric rings with width of 0.08 in @xmath12 in the shower plane to obtain the particle densities as a function of the distance to the shower axis ( lateral density ) . on the other hand , we have considered square cells of @xmath13 m@xmath14 in the shower plane when calculating the two dimensional distributions of the particle densities . the conventional separation between vertical and inclined showers is based on the zenith angle @xmath1 of the arrival direction of the cosmic ray particle that induces the shower . this separation stems from the different atmospheric grammage that the showers have to cross before reaching the ground , which increases approximately as @xmath15 . this fact implies that the showers arrive at ground at different stages in their evolution depending on @xmath1 . a hadronic cosmic ray typically initiates an air shower at the top of the atmosphere in the first few 100 g @xmath16 . the em component of the shower rises as the shower develops and reaches a maximum at a depth of @xmath17 for a proton shower , close to the total vertical depth of atmosphere for the altitude of the pierre auger observatory . after @xmath18 , the em component is rapidly absorbed in the atmosphere mainly due to low - energy ionization processes and the photoelectric effect . meanwhile , non - decaying muons propagate almost unattenuated to the ground , except for ionization energy losses and deflections in the geomagnetic field . consequently , a shower arriving with @xmath19 reaches the ground shortly after shower maximum and the electromagnetic component dominates at ground . however , for showers arriving with @xmath20 the atmospheric slant depth increases from @xmath21 to up to more than @xmath22 for completely horizontal showers . this results in the dominance of the muonic component at ground @xcite accompanied by a small electromagnetic component consisting mainly of the remnant of the electromagnetic shower due to cascading processes from @xmath3 decay of hadronic origin , and the electromagnetic halo due to muon decay in flight and hard muon interactions . low energy muons ( below a few gev ) typically decay along their paths to the ground generating small electromagnetic subshowers contributing to the particle densities at ground . this component mimics the muon spatial distribution and is proportional to the muon density @xcite . hard muon interactions ( pair production , bremsstrahlung and hadronic interactions ) become more and more relevant as muon energy increases . to illustrate when these processes are expected to contribute larger to the em halo , in the left panel of fig . [ muonspectrum ] we show the energy spectrum of muons at distances of 30 m and 1000 m to the shower axis for 10 eev proton - induced showers at different zenith angles . as shown here , these processes are expected to contribute to the em halo especially in highly inclined showers in which most of the muons are typically very energetic ( hundreds of gev ) , because the lower energetic muons typically decay before traveling the enlarged distances from their production height to the ground . hard muon interactions are also expected to be more frequent close to the shower core where a larger content of energetic muons is expected since energetic muons deviate less from the shower axis . a vertical shower typically exhibits a symmetric pattern of particle densities around the shower core in the shower plane . in the case of inclined showers there are several effects that produce an asymmetry on the pattern of particle densities . the most important sources of which are described below . muons in inclined showers travel along sufficiently long paths in the atmosphere to be affected by the earth s magnetic field . positive and negative muons are deviated in opposite directions with respect to the rectilinear trajectories they would follow in the absence of geomagnetic field . as a consequence the field distorts the patterns of the muonic densities in the shower plane producing elliptical or even 2-lobed patterns in very inclined showers at ground , an effect extensively studied in @xcite . the degree of distortion and the shape of the pattern depend on the strength and orientation of the magnetic field with respect to the shower axis , namely on the component of the field perpendicular to the shower axis @xmath23 , as well as on the distance traveled by the muons . this introduces a dependence of the asymmetry on the shower zenith ( @xmath1 ) and azimuth ( @xmath24 ) angles . as it was shown in @xcite the effect on the muonic distributions is only significant for @xmath25 . in fig . [ compbperp ] we show @xmath23 as a function of @xmath24 for the particular location of the southern pierre auger observatory site . for a given zenith angle , the distortion is expected to be maximal close to @xmath26 and @xmath27 and minimal around @xmath28 . @xmath29 points eastwards and is oriented counterclockwise . as an example of the effect of the geomagnetic field on the particle densities , in fig . [ muonmapwithandnob ] we show 2-dimensional maps of the muonic and electromagnetic particle densities in the shower plane in a eev proton induced shower arriving at @xmath30 and @xmath31 , with ( right ) and without ( left ) the geomagnetic field induced . the distortion of the cylindrical symmetry of the muonic component is apparent , and since the electromagnetic halo preserves the spatial distribution of muons in inclined showers , the pattern of the electromagnetic particle density exhibits a shape similar to the muonic one , although slightly blurred due to multiple coulomb scatterings suffered by the electrons before reaching ground . besides the asymmetries induced by the geomagnetic field , there is also an azimuthal asymmetry in the muonic and electromagnetic particle densities due to the combination of geometrical and shower evolution effects @xcite . as illustrated in fig . [ asymschema ] shower particles do not travel parallel to the shower axis in general and therefore they cross different amounts of atmospheric depth before reaching ground . the crossed depth depends on the azimuthal angle ( @xmath7 ) of the position on the ground at which the particle arrives with respect to the incoming shower direction . in particular , particles arrive at the ground in the _ early _ region of the shower ( the portion of the shower front that hits the ground first corresponding to @xmath32 ) more vertically than those in the _ late _ region ( corresponding to @xmath33 ) . this is essentially the basis for the geometrical effect , which depends strongly on the specific characteristics of the detectors sampling the shower front . the asymmetry induced by the geometrical effect is typically small in showers with high zenith angles . the more inclined a shower , the more energetic the muons arriving at ground are and hence the smaller the difference in the arrival angle distributions between the early and late regions of the shower . the asymmetry induced by the shower evolution can be understood as follows . particles at the same distance to the shower axis in the shower plane , but arriving with different azimuthal angles @xmath7 travel along different paths , and belong to different stages in the evolution of the shower . the importance of this effect depends on the evolution of the lateral particle distribution and on the attenuation of the total number of particles with the atmospheric depth . in fig . [ asymschema ] the detector in the early region of the shower is hit by a younger stage in the evolution of the shower than the detector in the late region . more quantitatively , for instance in a 10 eev proton shower at @xmath34 , there is a difference of @xmath35 g @xmath16 ( for notation see fig . [ asymschema ] ) between the atmosphere crossed by particles produced at the same height , traveling in straight line and hitting the ground in the late and early regions at a distance @xmath36 m to the core in the shower plane . this difference increases with the distance from the shower core . the asymmetry induced by the evolution of the shower affects more the electromagnetic component produced by @xmath37 decay than the muonic component or its associated electromagnetic halo . the reason is that this component is exponentially suppressed after the shower maximum , and small variations of the depth crossed by the shower induce large differences in the number of electromagnetic particles on the ground . however , the muonic component is less attenuated and therefore the asymmetry induced by this effect is smaller . as a consequence , shower evolution is expected to induce a much smaller asymmetry in the density in showers with zenith angles @xmath38 , because the electromagnetic component from @xmath37 decay is practically suppressed , and the electromagnetic halo is simply following the pattern of the muonic component where the asymmetry is small . this is shown in fig . [ asymdensit ] where we plot the relative difference between the electromagnetic and muonic particle densities in the early ( @xmath39 ) and late ( @xmath40 ) regions in the shower plane as obtained in our simulations . for the moment the geomagnetic field effect is neglected in this study . in the left panel of fig . [ asymdensit ] it can be seen that for showers at @xmath34 the difference in the electromagnetic density between the early and the late regions is very large even at small distances to the core ( for example a factor 1.2 at @xmath41 100 m ) , because the electromagnetic component from @xmath37 decay is still significant in the early region , and it is significantly absorbed before reaching ground in the late region . this difference increases steeply with distance @xmath42 . however , at @xmath43 the difference between the densities in the early and late regions is smaller , because the electromagnetic component from @xmath37 decay is absorbed at all @xmath7 and only the electromagnetic halo remains , except for at very far distances from the core where , besides inheriting the already quite large asymmetry due to the muonic component , there is still some remnant of the electromagnetic shower present in the early region but not in the late one . this explains why the early - late asymmetry follows essentially the behaviour of the corresponding muonic density ( right panel of fig . [ asymdensit ] ) . in the muonic case , the relative difference between the early and late densities is always small regardless of zenith angle . the electromagnetic and muonic particle densities have a characteristic behaviour with distance to the shower axis , shower zenith angle and azimuthal position with respect to the incoming shower direction . also the different contributions to the electromagnetic particle densities differ from each other as will be shown below . firstly , we have studied the em and muon number densities as a function of the distance to the shower axis ( lateral distributions ) in 10 eev proton - induced showers at different zenith angles , averaging over the azimuthal angle @xmath7 on the ground . we have also parameterised the lateral distributions of the particle densities as a function of shower zenith angle for practical applications , using 10 eev proton showers simulated with the hadronic model qgsjet01 as reference . these parameterisations are presented in appendix a. the results from these parameterisations are shown as solid lines in the top panels of fig . [ emmusignal ] compared to the simulations . the simulations are reproduced by the fits without significant deviations as shown in the bottom panels of fig . [ emmusignal ] ( @xmath44 for muons and @xmath45 for the em component ) . as shown in the left panel of fig . [ emmusignal ] , the muonic density @xmath46 decreases with @xmath1 , because muons need to travel larger distances before reaching ground arriving at larger distances to the core and being spread over a larger area . this is mainly due to the transverse momentum inherited from its parent hadron . muons of the lowest energies typically decay , decreasing further the density with theta , especially at large distances . both effects also explain the behaviour of the muon energy distribution with @xmath1 and distance to the shower axis shown in fig . [ muonspectrum ] . for example , in the right panel of this figure one can see that the mean muon energy increases with @xmath1 for a fixed @xmath42 , because only the more energetic muons survive and besides , these deviate more from the shower axis . the behaviour of the electromagnetic lateral distribution in inclined showers ( right panel of fig . [ emmusignal ] ) , can be qualitatively understood as a combination of the different behaviour of the two contributions to the em component namely , that produced by @xmath3 decay ( from hadronic origin ) and the electromagnetic halo ( from muonic origin ) . separating the two main contributions to the total em density is not possible in aires simulations because the information on the mother particle ( a @xmath3 or a muon ) producing the em subshower is lost . however we have devised a procedure to obtain in an approximate way the various contributions to the total em density . the total em density @xmath47 can be obtained as : @xmath48 where @xmath49 is the em density due to decay of @xmath3s produced in hadron and meson interactions , @xmath50 is the em density due to muon decay and @xmath51 is the em density due to muon bremsstrahlung , pair production and nuclear interactions . using aires we have performed a special subset of shower simulations in which we artificially set the muon energy threshold above which muons are explicitly followed in the simulations to a very high energy ( 10 tev ) , and at the same time we set the muon lifetime to infinity so that muons never decay . the last condition needs to be forced otherwise muons are artificially decayed in aires . the em density obtained in this way has no contribution from muons ( i.e. @xmath52 ) and can only be generated in @xmath3 decays ( produced in hadron and meson interactions ) . in fig . [ emremnant ] we show the lateral distribution of the em density as obtained in these simulations for different zenith angles . it can be seen that as the zenith angle increases from @xmath2 the em density due to @xmath3 decay is increasingly absorbed , until the zenith angle reaches @xmath53 and it practically disappears , except for a small contribution still reaching ground very near the core at @xmath54 m and far from it at @xmath55 km reaching only the early region of the shower . for @xmath38 the em component from @xmath37 decay is negligible and the em halo dominates at essentially all distances to the core . this is reflected in the fact that the em lateral distribution follows the behaviour of the muonic one . this is more apparent in fig . [ emmuratio ] where the ratio of the electromagnetic and muonic densities is shown . note that the results shown in this figure are obtained with standard aires simulations . we have also studied the behaviour of the ratio of the electromagnetic to the muonic densities : @xmath56 in fig . [ emmuratio ] , we show @xmath57 averaged over azimuth angle @xmath7 as a function of the distance to the core @xmath42 for different shower zenith angles @xmath1 and as a function of the @xmath1 for different fixed distances as obtained in the simulations , and the ratio predicted by the parameterisations proposed in appendix a. near the core , the ratio decreases with zenith angle from @xmath58 to @xmath59 due to the increasing absorption of the em component from @xmath3 decay , and then increases again with @xmath1 as can be seen in fig . [ emmuratio ] , mainly due to muon hard interactions processes that are expected to dominate near the core in very inclined showers . far from the core the lateral distribution of the ratio tends to flatten due to the dominant contribution of the em halo produced by muon decay in flight . the larger the zenith angle , the closer to shower core the ratio levels off . the slight increase of the ratio for @xmath60 and far from the core ( @xmath61 km ) is attributed to the combination of two effects , one is that the number of low energy muons decreases more rapidly at large distances because they decay before reaching the ground , and only energetic muons survive , and on the other hand the presence of the contribution to the em density due to @xmath3 decay , particularly in the early region of the shower . in addition to the dependence on zenith angle and distance to shower core , we have also studied the azimuthal asymmetry of @xmath62 . for the moment the effect of the geomagnetic field is neglected . in the left panel of fig . [ ratiomap ] , we show the 2-dimensional map of the ratio @xmath57 in the shower plane for 10 eev proton showers at @xmath34 . the shower incoming direction is from east to west in the figure . as expected , there is a clear azimuthal asymmetry at a fixed distance to the core . the contribution of the electromagnetic component is larger in the early region . however , as shown in the right panel of fig . [ ratiomap ] , at zenith angles @xmath63 the azimuthal asymmetry is less significant because only muons accompanied by the electromagnetic halo arrive at ground . since these two components approximately have the same asymmetry ( section 3.1.2 ) , the final asymmetry is practically canceled out when making their ratio . to study further the azimuthal dependence of the asymmetry we divide the shower plane in bins of width @xmath64 centered around @xmath7 , and we calculate the lateral distributions of the ratio in each bin for a fixed zenith angle : @xmath65 , and we compare these distributions to the distribution @xmath66 obtained averaging over @xmath7 . for this purpose we define the asymmetry parameter @xmath67 as @xmath68 in fig . [ exampleasyemmuratio ] , we show the lateral distribution of @xmath57 in different @xmath7 bins compared to the mean value ( left panel ) and their corresponding asymmetry parameter @xmath67 ( right panel ) for showers at @xmath34 . @xmath69 increases with distance to the core and it is larger in the early region than in the late region as expected . in addition to the dependence on position in the shower plane ( @xmath70 ) , the asymmetry parameter @xmath67 also depends on the shower zenith angle . in fig . [ amplitudeasy ] we show @xmath67 as a function of @xmath7 for a fixed distance @xmath71 m and different zenith angles . the amplitude of the asymmetry decreases as the zenith angle increases for the reasons explained above . this plot illustrates the importance of accounting for the asymmetry in the ratio when dealing with inclined showers with @xmath72 . we have parameterised the asymmetry @xmath67 as a function of the distance to the core , zenith angle and the azimuthal angle @xmath7 accounting for all the dependences above ( see appendix b ) . as an example , we show the results of the fit as solid lines in fig . [ exampleasyemmuratio ] . the simulations are reproduced by the fit without significant deviations as shown in the bottom panel of fig . [ exampleasyemmuratio ] . in the parameterisation , we have assumed that the azimuthal asymmetry @xmath73 is negligible when @xmath74 . as discussed before ( section 3.1.1 ) , the muon distributions are distorted by the presence of the geomagnetic field for zenith angles greater than @xmath75 . at these angles , the dominant contribution to the electromagnetic component at ground is due to the electromagnetic halo , which inherits the muon spatial distribution and is proportional to the muonic density , and we expect the ratio of the electromagnetic to muonic densities to maintain the symmetry in the azimuthal angle @xmath7 . for this reason , we have only studied the effect of the geomagnetic field on the ratio averaging over the azimuthal angle @xmath7 in the shower plane , for different shower zenith @xmath1 and azimuth @xmath24 angles . more precisely , for each @xmath1 we study the extreme cases where the effect is minimal , @xmath76 , and maximal , @xmath77 ( or @xmath78 ) , where @xmath24 is defined as in fig . [ compbperp ] . in the left panels of fig . [ beffratio ] we show the lateral behaviour of @xmath57 in the presence of the geomagnetic field for 10 eev proton showers . we also plot the relative difference between @xmath62 with and without geomagnetic field effect , namely : @xmath79 as can be seen in the right panels of fig . [ beffratio ] , the effect of the geomagnetic field on @xmath57 is more important near the core at all zenith angles . the reason for this is that only the highest energy muons are not significantly deflected and stay close to the core , and these are more likely to suffer hard interactions and induce an electromagnetic shower . as a consequence @xmath47 increases and at the same time @xmath80 decreases because lower energy muons are being deflected away from the core . these two effects produce an overall increase in @xmath57 . this increase is small for @xmath81 and @xmath82 m , with @xmath83 . when @xmath84 the effect starts to be important in the case of maximal deviation ( @xmath85 ) where @xmath86 for @xmath87 m , whereas for the case of minimal expected deviation ( @xmath31 ) the relative difference remains smaller at all distances . at larger angles @xmath88 , the geomagnetic field has a strong influence on @xmath57 , even when @xmath89 and the effect is expected to be minimal . it can also be seen that the larger the zenith angle , the farther from the shower core the influence of the geomagnetic field is still important for the reasons explained before . in conclusion , the effect of the geomagnetic field on the ratio of the electromagnetic to muonic number densities can be considered relevant for showers at @xmath90 . it should be noted that the rate of events at such high zenith angles detected at ground level by an array of detectors is small due to the reduced solid angle and the small size of the projection of the array onto the direction of the shower , so that this effect can be ignored for the purposes of data analysis in a first approximation . in this section we study the dependence of @xmath57 on primary energy , mass of the primary cosmic ray initiating the shower , and hadronic interaction model used to perform the simulations . these dependences are studied neglecting the effect of the geomagnetic field and averaging over azimuthal angle @xmath7 in the shower plane . the more energetic a shower , its maximum occurs deeper in the atmosphere , and therefore , the shower components are in a younger stage of evolution . the lateral distribution of the electromagnetic and muonic densities exhibits a characteristic behaviour with energy and depth of shower maximum @xcite . the lateral distribution of the charged electromagnetic component due to the cascading processes ( mainly due to @xmath3 decay ) is approximately given by , @xmath91 where the total number of particles @xmath92 depends on primary energy @xmath93 as @xmath94 . at shower maximum , @xmath95 @xcite . the lateral behaviour of the electromagnetic component @xmath96 depends on shower age ( @xmath97 ) . on the other hand , the lateral distribution of muons can be expressed as : @xmath98 where the total number of muons @xmath99 ( with @xmath100 ) increases slower with the energy than the electromagnetic component @xcite . also the lateral behaviour @xmath101 is approximately independent of the shower age @xcite . finally , as discussed before , the electromagnetic component due to muon decay in flight is proportional to the muonic density . hence , the lateral distribution of this component is expected to have the same energy dependence as the muonic one . combining all these facts , we expect the ratio @xmath102 to have a different behaviour depending on whether the electromagnetic component due to @xmath3 decay or the em halo contributes more to the total electromagnetic density . we study the energy dependence of @xmath57 performing the relative difference between the ratio at a given energy with respect to that obtained for 10 eev proton showers , @xmath103 ( see previous section ) : @xmath104 in fig . [ edep ] we show @xmath105 for @xmath106 1 eev and 100 eev showers . at @xmath107 , the energy dependence of the ratio is small @xmath108 , because the electromagnetic density being dominated by the contribution from the em halo , is roughly proportional to the muonic density , regardless of the shower energy . however for @xmath58 there is a dependence of @xmath109 on the shower energy , especially close to the shower core where the em component due to cascading processes contributes more to the em density ( see also fig . [ emremnant ] ) . in this angular range , we expect @xmath57 to behave as @xmath110 and hence to depend on shower energy . as can be seen in fig . [ edep ] , @xmath109 is larger ( smaller ) for 100 eev ( 1 eev ) showers than at 10 eev because of the larger ( smaller ) em component coming from the cascading processes in the shower which penetrates more ( less ) in the atmosphere . the dependence of @xmath105 on @xmath1 is studied in more detail in fig . [ edepvsth ] , where we plot @xmath105 in different bins of @xmath42 , as a function of @xmath1 for 1 eev and 100 eev proton showers . here , we confirm what was said before , for zenith angles above @xmath111 the em component is only due to the em halo and the ratio @xmath57 remains constant at the same level with energy , while if @xmath112 there is a dependence on energy that increases as the distance to the core decreases . at present , the chemical composition of the cosmic rays at the highest energies ( @xmath113 eev ) remains uncertain . some authors claim that cosmic rays at these energies are mainly protons @xcite , and others discuss the possibility of heavier elements such as iron nuclei @xcite . the most recent results from extensive air shower experiments do not allow to rise any firm conclusion @xcite . for this reason we have studied the dependence of the ratio @xmath114 on the mass of the primary particle initiating the shower accounting for protons and iron nuclei in our simulations . compared to a proton , an iron nucleus typically interacts higher in the atmosphere , producing a shower with a smaller depth of maximum . also applying a simple superposition model an iron nuclei produces a shower with @xmath115 more muons than a proton . as a result , an iron - initiated shower is expected to have a smaller electromagnetic density from cascading processes and a larger muonic density than a proton shower , and therefore a smaller @xmath57 as long as the em component due to cascading processes contributes more than that due to the em halo ( @xmath116 and close to the shower core ) . to demonstrate this , we have calculated the relative difference between @xmath57 in iron showers at 10 eev with respect to that obtained in 10 eev proton shower simulations @xmath117 in fig . [ massdepvsth ] we show @xmath118 as a function of zenith angle in different bins of @xmath42 . for reasons very similar to those that explain the energy dependence of @xmath57 studied before , we can explain the mass dependence . for angles larger than @xmath59 the dependence on the mass is negligible , because only the em halo is present and it is proportional to the muonic density . although the latter increases due to the larger mass of the primary , the former increases accordingly and the ratio stays roughly constant at the same level with mass . however when @xmath119 we can see a clear dependence on mass , with iron showers having smaller values of @xmath57 due to the fact that the em component is dominated by cascading processes and the iron - induced shower penetrates less in the atmosphere . at the highest energies , there is a lack of empirical knowledge about the hadronic interactions which greatly influence shower development @xcite . laboratory experiments have studied particle collisions only at centre - of - mass energies equivalent to fixed target energies of @xmath120 ev , so assumptions must still be made to perform the interactions needed in the shower simulations at the energies of interest ( @xmath121 ev ) . this fact leads to discrepancies between the different hadronic models on predictions of inelastic cross - sections and inelasticity ( multiplicity and energy of the secondaries ) ( see for instance @xcite for more details ) . these quantities determine to a large extent the longitudinal development of the air shower and , as a consequence , the number densities of the em and muonic components at ground . in this work , we compare two high energy interaction models currently used in cosmic ray physics : qgsjet01 @xcite and sibyll 2.1 @xcite . for proton primaries at 10 eev , the qgsjet model predicts showers that on average develop higher in the atmosphere and have @xmath122 more muons than showers simulated with sibyll . as a result , qgsjet predicts more muons at ground and a smaller electromagnetic component due to cascading processes . following a similar line of reasoning as in the case of the energy and mass dependence of the ratio of the em to muonic densities , @xmath57 is expected to be larger for showers simulated with sibyll at zenith angles @xmath123 , and roughly independent on the model for @xmath1 larger than @xmath111 . this behaviour can be seen in fig . [ haddepvsth ] , where we plot the relative difference @xmath124 between @xmath125 for 10 eev proton showers simulated with sibyll 2.1 with respect to the one obtained in showers simulated with qgsjet01 : @xmath126 @xmath124 is shown as a function of zenith angle in different bins of @xmath42 , to make the increasingly larger differences between sibyll and qgsjet for small distances to the shower axis more apparent , as expected from the dominance of the em component due to cascading processes near the core . summarizing this section , we have studied the effect of energy , mass composition and hadronic interaction model on the ratio of electromagnetic to muonic densities in absence of geomagnetic field effect , using the ratio obtained for 10 eev proton showers simulated with qgsjet01 as reference . combining all these dependences , we find that the extreme differences with respect to the reference ratio correspond to : * maximum : 100 eev proton showers simulated with sibyll 2.1 . * minimum : 1 eev iron showers simulated with qgsjet01 . in fig . [ largestdepvsth ] we show the relative differences between both cases and the reference ratio ( @xmath127 and @xmath128 ) as a function of @xmath1 in different bins in @xmath42 . the figure illustrates the extreme cases in the dependency of @xmath57 on energy , mass and model one should expect . in this section , we attempt to estimate the effect of physical fluctuations on the muon and electromagnetic number densities using simulations . the particle densities in individual showers are affected by different sources of fluctuation , namely : * physical ( intrinsic ) `` shower - to - shower '' fluctuations due to fluctuations in the atmospheric depth of the first interactions in the shower , fluctuations in secondary particle production , etc . , in general fluctuations in the cascading processes during the development of the shower . * `` artificial '' fluctuations due to the thinning procedure in the shower simulation . the ideal way of computing the `` shower - to - shower '' fluctuations of the electromagnetic and muonic components is to perform full ( non - thinned ) shower simulations . this is unfortunately not feasible due to the large computing time and huge disk space needed to store the information on particles at ground at the highest energies ( even for a single shower ) . one has to rely on thinned simulations in which artificial fluctuations are introduced . to overcome this problem , we have devised a simple approach based on a method given in @xcite to estimate how physical fluctuations affect the electromagnetic and muonic particle distributions when obtained from tracked particles with weights @xmath129 . here , we neglect the azimuthal asymmetry of the densities and obtain the densities in concentric rings in @xmath42 around the shower core of area @xmath130 . the particle density corresponding to @xmath131 particles with individual weights @xmath129 falling in an area @xmath130 is calculated as : @xmath132 so that the fluctuations of the particle densities stem from the fluctuations in the total particle number after accounting for the weights : @xmath133 . defining the average weight of the @xmath131 particles falling in a ring of area @xmath130 as @xmath134 , the unweighted particle number can be approximated as @xmath135 and the standard deviation of @xmath136 can be expressed as : @xmath137 assuming that the mean particle weight does not fluctuate from shower to shower as in the ideal case of an unthinned shower , and taking it to be equal to the average weight over all the particles followed explicitly in the shower simulation , we obtain : @xmath138 eq . [ sigman_shtosh ] represents an approximate way of obtaining the intrinsic physical shower - to - shower fluctuations of the muonic and electromagnetic particle densities from monte carlo simulations , which gives a good account of the physical fluctuations for thinning levels smaller than @xmath139 @xcite . in fact , in the case of an unthinned shower , in which only the intrinsic fluctuations should be present , @xmath140 , @xmath141 and eq . 8 gives @xmath142 as expected . in fig . [ signalfluct ] we show the relative physical fluctuations ( @xmath143 ) of the electromagnetic ( top ) and muon ( bottom ) densities as a function of the distance to the shower core for 10 eev proton showers at different zenith angles . these have been computed obtaining first @xmath144 and @xmath145 from the simulations and then applying eq . [ sigman_shtosh ] . the fluctuations in the muonic component are roughly independent of @xmath42 in the wide range of distances to the core plotted in the figure , and remain at the level of @xmath146 . on the other hand , the fluctuations in the electromagnetic component seem to depend on the contribution of the em halo . at @xmath34 the decrease of the relative fluctuations as the distance to the core increases could be interpreted as the em halo is becoming the dominant contribution to the em density and mimicking the behaviour of the fluctuations in the muonic component . similarly , for @xmath63 the fluctuations mimic the behaviour of those of the muonic component , they are roughly independent of @xmath42 and remain at the level of @xmath146 . we have studied the characteristics of the particle densities of the electromagnetic and muon components of inclined showers at the ground level using monte carlo simulations . we have shown that the electromagnetic component is composed of several sub - components originated in cascading and muonic processes . these different contributions to the electromagnetic component differ from each other on their behaviour with distance to the core , zenith angle and angular position ( @xmath7 ) . we have studied the ratio of the electromagnetic to muon densities ( @xmath57 ) as a function of several parameters . firstly , we have characterised the dependence of this ratio on the distance to the core and shower zenith angle ( see fig . [ emmuratio ] ) . near the core up to @xmath147 its behaviour is explained by the increasing absorption of the contribution to the em component due to @xmath3 decay and beyond @xmath148 by the dominance of the contribution to the em component due hard muon processes . far from the core ( @xmath113 km ) , the ratio is compatible with an almost constant value because the electromagnetic component due to muon decay in flight dominates the electromagnetic density at ground . then , we have studied the dependence of @xmath57 on the azimuthal position @xmath7 . for showers with @xmath72 we have found an azimuthal asymmetry , which is mostly due to the longitudinal development effect . moreover , we have studied the effect of the earth magnetic field in this ratio finding that is important for showers at @xmath90 . considering all these dependences above and neglecting the geomagnetic field effect , we propose a parameterisation of the @xmath65 that predicts the simulated data to a good precision level ( see fig . [ exampleasyemmuratio ] ) . the fits are valid for 10 eev proton showers simulated with the hadronic model qgsjet01 , and in the ranges @xmath149 $ ] and @xmath150 $ ] m. we have characterised the dependence of this ratio with the primary energy , the primary mass composition and the hadronic interaction model used in the simulations . the general result is that at zenith angles @xmath151 the ratio remains constant because only the electromagnetic halo contributes to the electromagnetic component . finally , we have estimated the effect of physical fluctuations on the electromagnetic and muon densities . the muon fluctuations remain roughly constant at the level of @xmath152 . while , the em fluctuations depend on the dominance of the electromagnetic halo on the em component , and tend to be constant @xmath146 for higher zenith angles . in table [ tabla1 ] we summarise some results for two different distances to the core and different shower zenith angles . note that in the angular region @xmath72 the presence of the remnant of the electromagnetic shower due to cascading processes from @xmath3 decay of hadronic origin entails important dependences of the ratio on the energy , mass composition and hadronic model . therefore , this region could be the most suitable one for studies about composition and hadronic interaction models by means of the electromagnetic component in the shower at ground level . however in the region @xmath153 the ratio is not sensitive to composition and hadronic model , and this angular region might be in principle more suitable to study the uhecr energy spectrum with inclined events detected by ground detectors . .summary of the dependences of the ratio of em to muonic component on the geomagnetic field , energy , primary mass and hadronic model , using protons at @xmath154 ev simulated with the qgsjet01 model and with no geomagnetic field as reference ( see eqs . [ deltab ] , [ deltae ] , [ deltam ] and [ deltah ] for the definitions of the different @xmath155 ) . the dependences are given at different distances from the shower axis 100 m and 1 km , and different zenith angles @xmath156 , @xmath148 and @xmath157 . we also give the relative physical fluctuations of the em and muonic components as obtained with eq . [ sigman_shtosh ] . the results are given in percentage @xmath158.[tabla1 ] [ cols="^,^,^,^,^,^,^,^ " , ] the authors thank gonzalo rodrguez - fernndez and our colleagues of the pierre auger collaboration for support and discussions on this topic . the helmholtz association , germany ( hhng-128 grant ) , ministerio de ciencia e innovacin ( fpa 2007 - 65114 ) , the spanish consolider - ingenio 2010 programme cpan ( csd2007 - 00042 ) , xunta de galicia ( pgidit 06 pxib 206184 pr ) and consellera de educacin ( grupos de referencia competitivos consolider xunta de galicia 2006/51 ) , and feder funds , spain , are acknowledged for providing funding . ins valio gratefully acknowledges the financial support from `` fundacin pedro barri de la maza '' ( spain ) . the authors also thank cesga ( centro de supercomputacin de galicia ) for computing resources . we give here the parameterisations performed for the lateral distributions of the electromagnetic and muonic densities at ground level projected onto the shower plane . the densities are given as a function of distance to the core @xmath42 in the shower plane and shower zenith angle @xmath1 , averaging over the azimuthal angle @xmath7 . in appendix b we also give a parameterisation of the azimuthal asymmetry . the fits are valid for 10 eev proton showers simulated with the hadronic model qgsjet01 , and in the ranges @xmath149 $ ] and @xmath150 $ ] m. we have used two different functional forms to account for the different behaviour of the lateral distributions at large distances from the core . near the core , we use a form based on the vernov functional form @xcite and for larger distances a function based on the nishimura - kamata - greisen functional form @xcite : @xmath159 with @xmath160 m. all the parameters , except @xmath161 m and @xmath162 , depend on @xmath1 . for distances @xmath163 , the parameters can be fitted with : @xmath164 @xmath165 @xmath166 for distances @xmath167 , the parameters can be fitted with : @xmath168 @xmath169 @xmath170 we have fitted the lateral distribution of the electromagnetic component to the same functional forms used for the muonic component ( see eq . [ musignals_fit ] ) and in the same radial ranges . in this case , the parameters show a different dependence on the zenith angle below and above @xmath107 , reflecting the dominance of the contribution of the em halo to the total em density . for distances @xmath163 , @xmath171 m and the remaining parameters can be fitted with : @xmath172 @xmath173 @xmath174 @xmath175 for distances @xmath167 , @xmath176 m and the remaining parameters can be fitted with : @xmath177 @xmath178 we have also parameterised the asymmetry parameter @xmath179 . the following fits are valid for 10 eev proton showers simulated with qgsjet01 , in the ranges @xmath180 $ ] , @xmath181 $ ] and @xmath150 $ ] m. note that the asymmetry is only important for @xmath182 . the parameters @xmath185 and @xmath186 can be parameterised using a cauchy - type function modified : @xmath187 \hspace{1.cm } \beta = \beta_1 \left [ \frac{1}{\pi } \left ( \frac{\beta_2}{\zeta^2 + \beta_2 ^ 2 } \right ) - \beta_3\right]\ ] ] with @xmath181 $ ] and : @xmath188 @xmath189 @xmath190 @xmath191 @xmath192 @xmath193 t. yamamoto [ pierre auger collaboration ] , proc . 30@xmath194 int . cosmic ray conference , merida , 4 ( 2007 ) , p. 335 ; j. bellido [ pierre auger collaboration ] , proc . 31@xmath194 int . cosmic ray conference , lodz ( 2009 ) , astro - ph/0906.2319 .
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inclined air showers those arriving at ground with zenith angle with respect to the vertical @xmath0 are characterised by the dominance of the muonic component at ground which is accompanied by an electromagnetic halo " produced mainly by muon decay and muon interactions . by means of monte carlo simulations
we give a full characterisation of the particle densities at ground in ultra - high energy inclined showers as a function of primary energy and mass composition , as well as for different hadronic models assumed in the simulations .
we also investigate the effect of intrinsic shower - to - shower fluctuations in the particle densities .
cosmic rays , extensive air showers , ground detector , simulation , muon component , electromagnetic component 96.50.s , 96.50.sd , 13.85.tp
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You are an expert at summarizing long articles. Proceed to summarize the following text:
we continue our development on algorithms to count the points of various representation varieties of a quiver with relations . in @xcite , we applied serval counting characters to the harder - narasimhan identity in the hall algebra of a quiver and obtained several interesting formulas . all characters that we considered are originated from reineke s counting character @xmath3 from the hall algebra to certain quantum power series ring . unfortunately @xmath3 fails to be an algebra morphism for non - hereditary algebras , though harder - narasimhan identity exists quite generally . however , applying the same map @xmath3 to the hn - identity can still generate effective counting formulas . we will follow the similar line as the first paper . the only change is that we replace algebraic manipulations in the hall algebras by corresponding geometric constructions . we first state the main results of this notes . let @xmath4 to be the finite field @xmath0 with @xmath5 elements and @xmath6 be any basic algebra presented by @xmath7 . fix a slope function @xmath8 , and we denote by @xmath9 the variety of @xmath10-dimensional @xmath8-semistable representations of @xmath6 , and by @xmath11 its git quotient . @xmath12 where the sum runs over all decomposition @xmath13 of @xmath10 into non - zero dimension vectors such that @xmath14 for @xmath15 . we will define the key varieties @xmath16 in section 1 . in particular , if all @xmath17 varieties can be effectively counted , then so are @xmath9 . the map @xmath3 have so - called @xmath18 and @xmath19 analogs . they are defined in @xcite as @xmath20 and @xmath21 . here , @xmath18 and @xmath19 are related to the comultiplication and the antipode @xcite in the hall algebra . in our geometric setting , lemma 0.1 and @xmath17 varieties have @xmath18 and @xmath19 analogs as well . recall that a variety @xmath22 is called _ polynomial - count _ ( or has a counting polynomial ) if there exists a ( necessarily unique ) polynomial @xmath23 $ ] such that for every finite extension @xmath24 , we have @xmath25 . we are especially interested in when all @xmath17 varieties are polynomial - count . if this is the case , it is clear that each @xmath11 is polynomial - count when it is a geometric quotient . in this notes , we will mainly focus on a class of algebras called one - point extensions from a quiver . let @xmath1 be any finite quiver and @xmath26 a representation of @xmath1 . the one - point extension of @xmath1 by @xmath26 is the triangular algebra @xmath27:={\left(\begin{smallmatrix}kq\ 0\\\,e\:\ k\end{smallmatrix}\right)}$ ] . we also interested in the tensor product algebra @xmath28 , where @xmath29 is the dynkin quiver of type @xmath29 . for @xmath30 $ ] or @xmath31 , we have explicit counting formulas for @xmath17 varieties and their @xmath18 and @xmath19 analogs . [ t : intro ] 1 . for @xmath30 $ ] , all git quotients @xmath11 can be explicitly counted in terms of quiver grassmannians of @xmath26 . if @xmath26 is add - polynomial - count , then all @xmath11 are polynomial - count . 2 . for @xmath32 , all @xmath11 have counting polynomials , which can be explicitly computed . 3 . if @xmath26 is add - polynomial - count , @xmath33)$ ] is polynomial - count for certain choice of @xmath10 and @xmath8 . this notes are organized as follows . in section [ s : prelim ] , we provide necessary background on the representation theory of quivers with relations and points counting . in section [ s : hn ] , we introduce the @xmath17 variety and the notion of f - polynomial - count . after recalling the harder - narasimhan identity in the hall algebra , we conclude our key lemma ( lemma [ l : tao ] ) . in section [ s : ext ] , we first review the trivial extension of algebras in general , then specialize to the case of one - point extensions from a quiver . we describe the relations of these algebras from the projective presentation of @xmath26 . in section [ s : frep ] , we show in lemma [ l : frep ] that their @xmath17 varieties can be counted in terms of the usual representation varieties . in section [ s : rep ] , we show in lemma [ l : rep ] that these usual representation varieties can be counted in terms of the grassmannians of @xmath34 . motivated by this , we introduce add - polynomial - count property for a representation . we conclude by our fist main result theorem [ t : ext ] ( theorem [ t : intro].(1 ) ) . many examples will follow in section [ s : example ] . in section [ s : hs ] , we apply our algorithm to count homological strata on the geometric quotients . the method is outlined in theorem [ t : hs ] . in section [ s : a2 ] , we work with the algebra @xmath35 . our second main result theorem [ t : a2q ] ( theorem [ t : intro].(2 ) ) gives an analogous counting formula , which is independent of grassmannians of representations . in section [ s : delta ] , we consider the @xmath18-analog of counting . we introduce the @xmath18-analog of the @xmath17 varieties . lemma [ l : frep2 ] is the @xmath18-analog of lemma [ l : frep ] . we conclude by our third main results theorem [ t : a2ext ] ( theorem [ t : intro].(3 ) ) . finally in section [ s : s ] , we consider the @xmath19-analog of counting . our final main results is theorem [ t : final ] , which removes the assumption of being a geometric quotient in our previous results . most of our constructions can be easily generalized to the motivic setting . since the main application of this theory will be in the quantum algebra , we will not pursue that generality . the geometry of these moduli spaces will be studied in another series of notes @xcite . let @xmath1 be a finite quiver with the set of vertices @xmath36 and the set of arrows @xmath37 . if @xmath38 is an arrow , then @xmath39 and @xmath40 denote its tail and its head respectively . fix a _ dimension vector _ @xmath10 , the space of all @xmath10-dimensional representations over a field @xmath4 is @xmath41 the group @xmath42 acts on @xmath43 by the natural base change . two representations @xmath44 are isomorphic if they lie in the same @xmath45-orbit . let @xmath46 be the _ path algebra _ of @xmath1 over @xmath4 , then @xmath47 is naturally a ( right ) @xmath46-module . fix a set @xmath48 of homogeneous elements in @xmath46 with respect to the bigrading : @xmath49 here , we abuse the vertex @xmath50 for the trivial path @xmath51 . if @xmath52 for all @xmath53 , then we say @xmath54 is a _ representation of @xmath1 with relations @xmath48_. the _ path algebra of @xmath1 with relations @xmath48 _ is the quotient algebra @xmath55 . a representation of @xmath1 with relations @xmath48 naturally becomes an @xmath6-module . the assignment @xmath56 defines a polynomial map @xmath57 which is represented by an @xmath58 matrix with entries in @xmath59 $ ] . let @xmath60 $ ] be the ideal generated by the entries of all @xmath61 for which @xmath53 . the representation space @xmath62 is the scheme @xmath63/\tilde{r})$ ] . as a variety , @xmath62 consists of all @xmath10-dimensional representations of @xmath6 . a _ weight _ @xmath64 is an integral linear functional on @xmath65 . a _ slope function _ @xmath8 is a quotient of two weights @xmath66 with @xmath67 for any dimension vector @xmath10 . a representation @xmath54 is called _ @xmath8-semi - stable ( resp . @xmath8-stable ) _ if @xmath68 ( resp . @xmath69 ) for every non - trivial subrepresentation @xmath70 . we denote by @xmath9 the variety of @xmath10-dimensional @xmath8-semistable representations of @xmath6 . by the standard git construction @xcite , there is a _ categorical quotient _ @xmath71 and its restriction to the stable representations @xmath72 is a _ geometric quotient_. a slope function @xmath8 is called _ coprime _ to @xmath10 if @xmath73 for any @xmath74 . so if @xmath8 is coprime to @xmath10 , then there is no strictly semistable ( semistable but not stable ) representation of dimension @xmath10 . in this case , @xmath11 must be a geometric quotient . note that the semi - stable objects with a fixed slope @xmath75 form an exact subcategory @xmath76 . for any dimension vector @xmath10 , we can always modify @xmath8 to get a new slope function @xmath77 such that @xmath78 and @xmath79 . if @xmath80 , then we can take @xmath81 , where @xmath82 . _ * proposition 3.3 ) _ 1 . _ harder - narasimhan filtration : _ every representation @xmath54 has a unique filtration @xmath83 such that @xmath84 is @xmath8-semi - stable and @xmath85 . _ jordan - holder filtration : _ every @xmath8-semi - stable representation @xmath54 has a filtration @xmath83 such that @xmath84 is @xmath8-stable with the same slope . the set @xmath86 is uniquely determined . + let @xmath22 be a variety over @xmath87 and @xmath88 . we denote by @xmath89 the @xmath90-th @xmath91-adic cohomology group with compact support of @xmath92 . the key method for counting rational points on @xmath22 is given by the grothendieck - lefschetz trace formula : @xmath93 where @xmath94 is the frobenius morphism @xmath95 . @xmath22 is called _ @xmath91-pure _ if the eigenvalues of @xmath94 on @xmath89 have absolute value @xmath96 . it is known that if @xmath22 is smooth and proper over @xmath97 then @xmath22 is @xmath91-pure . in fact , the condition of being proper can be weakened as follows . * proposition a.2 ) assume that @xmath22 is smooth quasi - projective and that there is an action @xmath98 such that for every @xmath99 the limit @xmath100 exists . assume in addition that @xmath101 is projective , then z is @xmath91-pure . _ * proposition 6.1 ) _ [ l : euler ] if @xmath22 is counted by a rational function @xmath102 , then @xmath102 must lie in @xmath103 $ ] . its specialization at @xmath104 gives the @xmath91-adic euler characteristic of @xmath92 . the _ poincar polynomial _ @xmath105 $ ] of @xmath22 is @xmath106 ( * ? ? ? * lemma a.1 ) [ l : polycount ] assume that @xmath22 is @xmath91-pure and polynomial - count . then @xmath107 . in particular , @xmath108 $ ] . let @xmath6 be any basic algebra presented by @xmath7 for @xmath4 the finite field @xmath0 . for any decomposition of dimension vector @xmath109 , we define @xmath110 , where @xmath111 is the usual flag variety parameterizing flags of subspaces of dimension @xmath112 in @xmath113 . to simplify the notation , we denote @xmath114 . [ d : frep ] we define the _ frep _ variety : @xmath115 let @xmath116 be the projection , the _ flag variety _ @xmath117 of @xmath54 is the fibre @xmath118 , and its subvarieties @xmath119 is @xmath120 when the flag is only 2-step , we may use the usual grassmannian notation . for example , @xmath121 and @xmath122 , where @xmath123 . for any three @xmath6-modules @xmath124 and @xmath125 with dimension vector @xmath126 and @xmath127 , the _ hall number _ @xmath128 is by definition @xmath129 . for any module @xmath54 , we denote @xmath130 let @xmath131 be the space of all formal ( infinite ) linear combinations of isomorphism classes @xmath132 $ ] in @xmath6-@xmath133 . * proposition 1.1 ) the completed _ hall algebra _ @xmath131 is the associative algebra with multiplication @xmath134[v]:=\sum_{[w]}f_{uv}^w[w],\ ] ] and unit @xmath135 $ ] . we fix a slope function @xmath8 . for a dimension vector @xmath10 , let @xmath136 $ ] and @xmath137 $ ] . our convention is that they contain zero representation @xmath138 $ ] . we consider a simple counting map @xmath139 defined by @xmath132\mapsto a_m^{-1}$ ] . since @xmath140 , we have that @xmath141 . we denote the function @xmath142 by @xmath143 . in general , this function may not be rational in @xmath5 . the existence of the harder - narasimhan filtration yields the following identity in the hall algebra @xmath131 . * proposition 4.8 ) [ l : hnid ] @xmath144 where the sum running over all decomposition @xmath13 of @xmath10 into non - zero dimension vectors such that @xmath145 . in particular , solving recursively for @xmath146 , we get @xmath147 where the sum runs over all decomposition @xmath13 of @xmath10 into non - zero dimension vectors such that @xmath14 for @xmath15 . the key observation is that @xmath148 so the problem boils down to counting these @xmath17 varieties . in this paper , we will only focus on a class of algebras , for which these varieties can be effectively counted . [ d : fpc ] we say an algebra @xmath6 is _ polynomial - count _ if each @xmath149 is polynomial - count . it is called _ f - polynomial - count _ if each @xmath16 is polynomial - count . we do not know a single example where @xmath6 is polynomial - count but not f - polynomial - count . we conjecture that if each @xmath150 is polynomial - count , then @xmath6 is f - polynomial - count . [ l : tao ] @xmath151 in particular , if @xmath6 is f - polynomial - count , then each @xmath11 is polynomial - count when it is a geometric quotient . the assumption of being a geometric quotient in the lemma can be dropped . given two finite - dimensional @xmath4-algebras @xmath152 and a @xmath6-@xmath153-bimodule @xmath26 , we get the _ trivial extension algebra _ @xmath154={\left(\begin{smallmatrix}b & 0\\ _ ae_b & a\end{smallmatrix}\right)}$ ] . in the meanwhile , we can form the category @xmath155 of representations of the bimodule @xmath156 as follows : the objects are triples @xmath157 . a morphism from @xmath158 to @xmath159 is a pair @xmath160 making the following diagram commutes @xmath161^{\varphi } \ar[d]_{f_a\otimes{\operatorname{id } } } & m_b\ar[d]_{f_b}\\ n\otimes_a e \ar[r]^{\varphi } & n_b } \ ] ] ( * ? ? ? * a.2.7 ) the two categories @xmath162)$ ] and @xmath155 are equivalent . the equivalence is given by @xmath163 , where @xmath164 and @xmath165 . in particular , if @xmath166)$ ] corresponds to @xmath158 , then the dimension vector of @xmath54 is given by @xmath167 . one particular case we interested in is when @xmath26 is simply a right @xmath153-module . think @xmath26 as a @xmath4-@xmath153-bimodule , the triangular algebra @xmath168 is called ( trivial ) _ one - point extension _ of @xmath153 by @xmath26 . there is an obvious dual notion of one - point coextension @xmath169:={\left(\begin{smallmatrix}k & 0\\ e^ * & b\end{smallmatrix}\right)}$ ] . [ l : variety ] 1 . @xmath170)$ ] is the subvariety of @xmath171 defined as @xmath172 2 . @xmath173)$ ] is the subvariety of @xmath174 defined as @xmath175 suppose that @xmath176 is presented by @xmath177 with @xmath178 and @xmath179 , where @xmath180 is the indecomposable projective representation corresponding to the vertex @xmath50 . then the algebra @xmath30 $ ] can be presented by a new quiver @xmath181 , which is obtained from @xmath1 by adjoining a new vertex @xmath182 " and for each @xmath180 in @xmath183 a new arrow from @xmath182 " to the vertex @xmath50 . the relations are clearly given by the matrix @xmath184 . in reality , the presentation is always chosen to be minimal . by abuse of notation , we also use @xmath185 $ ] to denote the new quiver @xmath181 with those new relations . the one - point coextension @xmath186 $ ] can be similarly described using injective presentation of @xmath26 . by convention , the newly adjoined vertex is denoted by + " . it is clear that @xmath26 is the first syzygy of @xmath187 , so @xmath188 . moreover , a simple representation of @xmath185 $ ] is either @xmath187 or a simple representation of @xmath1 . so we conclude that @xmath27 $ ] has global dimension @xmath189 . it is easy to compute the matrices @xmath190 . let @xmath191 , then @xmath192 so the euler matrix @xmath193 of @xmath6 is @xmath194 , where @xmath195 and @xmath196 is the euler matrix of @xmath1 . throughout this notes , @xmath197 is the multiplicative euler form of the quiver @xmath1 , that is , @xmath198 . similarly , we define @xmath199 for @xmath200 . to simplify our notation , we always use letter with tilde to indicate that @xmath201)$ ] can be represented by @xmath202 , where @xmath203 and @xmath204 . a dimension vector with tilde , say @xmath205 , consists of two components @xmath206 , or @xmath207 for coextension . [ l : frep ] @xmath208)\to { \operatorname{fl}}_{{\tilde{\beta},\tilde{\gamma}}}$ ] is a fibre bundle with fibre @xmath209)\times { \operatorname{rep}}_{\tilde{\beta}}(q[e])\times \prod_{a\in q_1}{\operatorname{hom}}(k^{\beta(ta)},k^{\gamma(ha)})\ ] ] so @xmath210):=\frac{|{\operatorname{frep}}_{{\tilde{\beta},\tilde{\gamma}}}(q[e])|}{|{\operatorname{gl}}_{\tilde{\alpha}}| } = { \langle\beta,\gamma\rangle}^{-1}\bigl [ \begin{smallmatrix } \alpha_- \\ \gamma_-\end{smallmatrix}\bigr]|{\operatorname{gl}}_{\beta_-}|r_{\tilde{\beta}}(q[e])r_{(\alpha_-,\gamma)}(q[e]).\ ] ] we will sketch the fibre bundle construction by a picture . after fixing an elements in @xmath211 , we need to fill in the missing part for a @xmath212-dimensional representation of @xmath185 $ ] . the missing part consists of a @xmath213-dimensional representation @xmath19 , a @xmath205-dimensional representation @xmath214 , and a bunch of linear maps from @xmath215 to @xmath216 , as indicated below . @xmath217 we can stuff the space in the order below independently . the linear maps from @xmath218 together with all representations @xmath19 can be identified with @xmath219)$ ] ; all representations @xmath214 can be identified with @xmath220)$ ] , and the rest of the linear maps are @xmath221 . for the last formula , we only need to notice that @xmath222}|{\operatorname{gl}}_{\alpha}|}{|{\operatorname{gl}}_{\beta}||{\operatorname{gl}}_{\gamma}|{\langle\beta,\gamma\rangle}_0}.\ ] ] the above 2-step case can be recursively generalized to the @xmath223-step case . we only state the analog for the last formula . @xmath224)=\prod_{i=1}^s { \bigl[\begin{smallmatrix}\dot{\alpha}_{i,- } \\ \alpha_{i,- } \end{smallmatrix}\bigr]}|{\operatorname{gl}}_{\dot{\alpha}_{i-1,-}}|r_{(\dot{\alpha}_{i,-},\alpha_i)}(q[e]).\ ] ] we also state this formula for the dual case . @xmath225)=\prod_{i=2}^s { \bigl[\begin{smallmatrix}\dot{\alpha}_{i,+ } \\ \alpha_{i,+}\end{smallmatrix}\bigr]}|{\operatorname{gl}}_{\dot{\alpha}_{i-1,+}}|r_{(\alpha_i,\dot{\alpha}_{i,+})}(q^\circ[e]).\ ] ] now the problem boils down to count those affine representation varieties @xmath226)$ ] . for any dimension vector @xmath227 , we denote by @xmath228 the variety parameterizing all @xmath227-dimensional quotient representations of @xmath26 , and define @xmath229 [ l : rep ] @xmath230 is a fibre bundle with fibre @xmath231 so @xmath232):=\sum_{\alpha=\gamma+\beta}\frac{|{\operatorname{gr}}^{\beta}(ne)|}{{\langle\gamma,\beta\rangle}|{\operatorname{gl}}_n|}r_{\gamma}(q).\ ] ] dually , we have a formula for the one - point coextension : @xmath233)=\sum_{\alpha=\gamma+\beta}\frac{|{\operatorname{gr}}_{\gamma}(ne)|}{{\langle\gamma,\beta\rangle}|{\operatorname{gl}}_n|}r_{\beta}(q).\ ] ] the fibre bundle construction is not hard to verify . since we only need the last formula , we give a hall algebra proof for that . we denote by @xmath234 and @xmath235 the set of all monomorphisms and epimorphisms from @xmath54 to @xmath236 respectively . fix a representation @xmath125 , then the following identities clearly holds in the hall algebra @xmath237}[u]\big)\big(\sum_{[v]}|{\operatorname{epi}}_q(m , v)|[v]\big ) = \sum_{[w]}|{\operatorname{hom}}_q(m , w)|[w].\ ] ] let @xmath238 be the completed quantum polynomial algebra @xmath239 $ ] , where the multiplication rule is @xmath240 . then the map @xmath241 sending @xmath132\to a_m^{-1}x^{{\overline{m}}}$ ] is an algebra morphism @xcite . here , we use the slanted @xmath242 to distinguish the one with target @xmath243 . apply @xmath3 to both sides , we get @xmath244}[u ] \oldint \sum_{[v]}|{\operatorname{epi}}_q(m , v)|[v ] = \oldint\sum_{[w]}|{\operatorname{hom}}_q(m , w)|[w ] \\ \leftrightarrow & \sum_\gamma r_\gamma(q)x^\gamma\sum_{[v]}a_v^{-1}|{\operatorname{epi}}_q(m , v)|x^{{\overline{v } } } = \sum_{[w]}a_w^{-1}|{\operatorname{hom}}_q(m , w)|x^{{\overline{w } } } \\ \leftrightarrow & \sum_\gamma r_\gamma(q)x^\gamma\sum_\beta |{\operatorname{gr}}^\beta(m)| x^\beta= \sum_{[w]}\frac{|{\mathcal{o}}_w|}{|{\operatorname{gl}}_\alpha|}|{\operatorname{hom}}_q(m , w)|x^\alpha \quad ( \alpha={\overline{w } } ) \\ \leftrightarrow & \sum_{\beta+\gamma=\alpha}{\langle\gamma,\beta\rangle}^{-1}r_{\gamma}(q)|{\operatorname{gr}}^{\beta}(m)|=\sum_{w\in{\operatorname{rep}}_\alpha(q)}\frac{|{\mathcal{o}}_w|}{|{\operatorname{gl}}_\alpha|}|{\operatorname{hom}}_q(m , w)|.\end{aligned}\ ] ] now we set @xmath245 , then the formula follows from lemma [ l : variety ] . the dual formula can be obtained by applying @xmath3 to the identity : @xmath237}|{\operatorname{mon}}_q(u , m)|[u]\big)\cdot\big(\sum_{[v]}[v]\big ) = \sum_{[v]}|{\operatorname{hom}}(w , m)|[w].\ ] ] [ r : gen ] let @xmath246 be the generating functions @xmath247 , and @xmath248 if we set @xmath249 then lemma [ l : rep ] can be rewritten as @xmath250)=r(q)f^\infty(e ) , \text{\quad and\quad } r(q^\circ[e])=f_\infty(e)r(q).\ ] ] a representation @xmath176 is called _ polynomial - count _ , if all its grassmannians @xmath251 are polynomial - count . it is called _ add - polynomial - count _ , if each @xmath34 is polynomial - count . @xmath34 is polynomial - count if and only if @xmath252)$ ] is polynomial - count for any @xmath10 . if @xmath26 is add - polynomial - count , then @xmath27 $ ] is f - polynomial - count by . it follows from lemma [ l : tao ] , [ l : frep ] , and [ l : rep ] that [ t : ext ] @xmath253)$ ] can be explicitly counted in terms of @xmath254 s . in particular , if @xmath26 is add - polynomial - count , then each @xmath255)$ ] is polynomial - count when it is a geometric quotient . we will see in the last section that the assumption of being a geometric quotient can be dropped . polynomial - count representations of quivers will be studied in detail in @xcite . this class includes rigid representations because of ( * ? ? ? * corollary 6.4 ) , but there are many more . is there a representation , which is polynomial - count but not add - polynomial - count ? for any representation @xmath26 , the @xmath26-homological stratification of @xmath43 is the decomposition of @xmath43 into ( finite ) disjoint union of locally closed subvarieties @xmath256 , where @xmath257 by lemma [ l : variety ] , we know that for @xmath258 , @xmath259)|=\sum_h |{\operatorname{rep}}_\alpha(q;e , h)|q^{nh}.\ ] ] the coefficient matrix of above linear system is a non - degenerated vandermonde - type matrix , so we can solve all @xmath256 . in particular , @xmath256 is polynomial - count if and only if @xmath26 is add - polynomial - count . we will see that the above is still true for @xmath260 let @xmath261 be any slope function for @xmath1 . the ( negative ) extension @xmath262 to @xmath185 $ ] with respect to an dimension vector @xmath10 is @xmath263 , where @xmath264 and @xmath265 for some sufficiently small positive @xmath266 . similarly , we define the ( positive ) extension of @xmath267 to @xmath268 $ ] with respect to @xmath10 as @xmath269 , where @xmath270 , and @xmath271 . the following lemma was proved in ( * ? ? ? * theorem 5.2 ) for @xmath26 projective , but the argument goes through for any @xmath26 . we have the following identity in @xmath238 : @xmath272)x^\gamma\big ) = \sum_\alpha \big(\sum_{m\in{\operatorname{rep}}_\alpha^\mu(q)}\frac{|{\mathcal{o}}_m||{\operatorname{hom}}_q(ne , m)|}{|{\operatorname{gl}}_{\alpha}||{\operatorname{gl}}_n|}\big)x^\alpha.\ ] ] [ t : hs ] @xmath273 can be explicitly computed from @xmath274 . when @xmath26 is add - polynomial - count and @xmath275 is a geometric quotient , each homological strata on @xmath275 is polynomial - count . according to theorem [ t : ext ] , all @xmath276)$ ] s can be computed from @xmath274 , and so does the right hand side of . notice that @xmath277 we can invert the same vandermonde - type matrix as before to solve @xmath273 . consider the @xmath223-arrow kronecker quiver @xmath278 and its extension by an @xmath279-dimensional representation @xmath26 . then we can view the algebra @xmath280 $ ] as an algebra coextended from @xmath281 by a @xmath282-dimensional representation @xmath283 . it follows from remark [ r : gen ] that [ p : dual ] @xmath284 and @xmath285 are related by @xmath286)=f_\infty(e^\circ)r(k_m).\ ] ] in particular , if @xmath26 is add - polynomial - count , then so is @xmath283 . let @xmath287 $ ] be the algebra coextended from @xmath281 by a representation @xmath26 of dimension @xmath288 . for any dimension vector @xmath289 of @xmath281 , there is a unique choice of weight @xmath64 up to scalar such that @xmath290 . for the rest of this section , we always take @xmath261 for different @xmath10 . the first two isomorphisms below can be easily established by lemma [ l : variety ] . @xmath291 and @xmath292 . + assume that @xmath26 is not too special so that @xmath293 is empty . @xmath294-[\epsilon_2 - 1])|{\operatorname{gr}}_{(1,1)}(e)|,\\ & |{\operatorname{mod}}_{(2,2,1)}^{\mu_-}(a)|=|{\operatorname{gr}}_{(2,2)}(e)|+([2m-1]-[\epsilon_2 - 1])|{\operatorname{gr}}_{(2,1)}(e)|,\\ & \quad\quad\cdots\cdots\quad \ ] ] where @xmath295 $ ] is the quantum number . consider the quiver @xmath296 with relation @xmath297 . the corresponding algebra @xmath6 is one - point - extended from the dynkin quiver @xmath2 by the simple @xmath298 . so @xmath299 can be computed by lemma [ l : tao ] . note that @xmath300 is just the usual grassmannian variety @xmath301 . it follows that the quiver @xmath302 with relations @xmath303 is polynomial - count . this algebra is extended from the kronecker quiver @xmath304 by a decomposable non - rigid representation of dimension @xmath305 . [ ex:2n ] fix @xmath306 , we consider the quiver @xmath307 @xmath308 with relation @xmath309 , where @xmath310 and @xmath311 . it is extended form @xmath278 by @xmath312 presented by @xmath313 it is also coextended from @xmath304 by the exceptional @xmath314 presented by @xmath315 where @xmath316 . although @xmath312 is not rigid , it follows from proposition [ p : dual ] that it is add - polynomial - count . @xmath317 can be recursively computed by the cluster theory and lemma [ l : gralg ] . a closed formula was given in ( * ? ? ? * theorem 4.3 ) . @xmath318}{\bigl[\begin{smallmatrix}\gamma_2 + 1 \\ \gamma_1\end{smallmatrix}\bigr ] } & \text { otherwise , } \end{cases}\end{aligned}\ ] ] where @xmath319}$ ] is the quantum binomial coefficient . now we recall ( * ? ? ? * proposition 2.8 ) . we also draw some easy consequences , which are useful for counting the grassmannian of representations . [ l : gralg ] assume that @xmath320 and @xmath321 . @xmath322}\frac{|{\operatorname{ext}}_q(u , v)_w|}{|{\operatorname{ext}}_q(u , v)|}|{\operatorname{gr}}_{\gamma}(w)|.\ ] ] now suppose that @xmath323 . then @xmath324 hence , if both @xmath325 and @xmath326 are ( add)-polynomial - count , then so is @xmath327 . moreover , if @xmath328 and @xmath125 is the only non - trivial middle term of the extensions , then @xmath329 we add one arrow to example [ ex:2n ] : @xmath330 then it is extended from @xmath278 by @xmath331 , or coextended from @xmath304 by @xmath332 . let @xmath333 $ ] . since @xmath334 , we can use lemma [ l : gralg ] or compute directly @xmath335 . so we are able to find all @xmath336 . for example , @xmath337+[3]-1,\\ & |{\operatorname{mod}}_{(2,2,1)}^{\mu_-}(a_n)|=q^4 + 2q^3 + 4q^2 + 2q + 1 . \ ] ] however , all @xmath338 are different , they are hirzebruch surfaces @xmath339 @xcite . consider quiver @xmath340 with relation @xmath341 . it is coextended from @xmath342 by a rigid module presented by @xmath343 . similar calculation as before gives @xmath344[3],\\ & |{\operatorname{mod}}_{(2,1,1)}^{\mu_\pm}(a)|=|{\operatorname{mod}}_{(1,1,2)}^{\mu_\pm}(a)|=[3],\\ & |{\operatorname{mod}}_{(1,2,1)}^{\mu_\pm}(a)|=[3][5],\\ & |{\operatorname{mod}}_{(2,2,1)}^{\mu_-}(a)|=|{\operatorname{mod}}_{(1,2,2)}^{\mu_-}(a)|=[3][5](1,0,1),\\ & |{\operatorname{mod}}_{(1,2,2)}^{\mu_-}(a)|=|{\operatorname{mod}}_{(2,2,1)}^{\mu_+}(a)|=[3](1,1,3,3,3,1,1).\end{aligned}\ ] ] the first one is @xcite a divisor @xmath345 on @xmath346 of bidegree @xmath347 , or equivalently the complete flag variety @xmath348 of @xmath349 . now consider the deformation @xmath350 of @xmath26 , where @xmath351 . since @xmath352 with @xmath26 the only non - trivial middle term , we can compute @xmath353 using lemma [ l : gralg ] . @xmath354x^{(1,0)}+[2]^2x^{(1,1)}+[2]x^{(2,1)}+x^{(0,2)}+[5]x^{(1,2)}+{\bigl[\begin{smallmatrix}5\\2\end{smallmatrix}\bigr]}x^{(2,2)}+\cdots.\ ] ] @xmath355[5],\\ & |{\operatorname{mod}}_{(2,2,1)}^{\mu_-}(a)|=|{\operatorname{mod}}_{(1,2,2)}^{\mu_+}(a)| = [ 3][5](1,0,1),\\ & |{\operatorname{mod}}_{(1,2,2)}^{\mu_-}(a)|=|{\operatorname{mod}}_{(2,2,1)}^{\mu_+}(a)|=[3](1,1,4,4,3,1,1).\end{aligned}\ ] ] note that the first one is irreducible and singular by lemma [ l : polycount ] . consider quiver @xmath340 with relation @xmath309 , where @xmath356 . it is coextended from @xmath342 by @xmath26 presented by the following base diagram . the black dots are a basis in @xmath357 ; while the white dots are a basis in @xmath358 . the letter on an arrow means the identity map on the arrow of the same letter . @xmath359 it is known @xcite that for a general representation @xmath360 of dimension @xmath361 , @xmath362 is an elliptic curve . so @xmath360 is _ not _ polynomial - count . however , for this special @xmath26 , @xmath363 is three @xmath364 s intersecting at a point . with a little effort one can show that @xmath26 is actually polynomial - count . let us consider a category , which is universal in the sense that it contains all one - point extensions of @xmath1 as its full subcategories . it is clearly the module category of @xmath366 , where @xmath2 is the dynkin quiver @xmath367 . the quiver of @xmath35 is composed of two copies of @xmath1 corresponding to two idempotents of @xmath368 , and _ morphism arrows _ connecting the same vertices in two different copies . the relations are obviously the commuting diagram relations . by abuse of notation , we use @xmath365 to denote such a quiver with relations . so the dimension vector of @xmath365 is composed of two dimension vectors of @xmath1 , say @xmath369 . by convention , @xmath10 correspond to the quiver sending morphism arrows . let @xmath326 be an @xmath10-dimensional @xmath4-vector space . we denote by @xmath370 the incidence variety @xmath371 and by @xmath372 the incidence variety @xmath373 [ l : a2q ] @xmath374 is a fibre bundle with fibre @xmath375 so @xmath376 is equal to @xmath377 where @xmath378 } { { \langlec - c_e , c_e - c_d\rangle}{\langled - c_d , c_d\rangle}{\langlee - d - c_e+c_d , d+c_e - c_d\rangle}{\langle\gamma_d - c - e+c_e , c+e - c_e\rangle}}$ ] . we sketch the fibre bundle construction by a picture . after fixing an elements in @xmath379 , we need to fill in the missing part for a @xmath380-dimensional representation of @xmath365 . similar to lemma [ l : frep ] , the missing part consist of a @xmath381-dimensional representation @xmath19 , a @xmath382-dimensional representation @xmath214 , and a bunch of linear maps from @xmath215 to @xmath216 , as indicated below . @xmath383 the first step is to choose a configuration of image spaces of the vertical and diagonal morphism arrows . let @xmath384 be the rank vector of the morphism arrows of @xmath214 and @xmath19 respectively , and @xmath385 be the rank vector of the diagonal morphism arrows . let @xmath386 be the rank vector of the diagonal morphism arrows restricted on the kernel of the morphism arrows of @xmath214 . the second step is to stuff in the following order the lower part of @xmath19 , the upper part of @xmath214 , the rest part of @xmath19 and @xmath214 , and other diagonal arrows . they corresponds to @xmath387 , , , and respectively . we leave the details to the readers . to compute @xmath388 , we only need to count the incidence varieties . we use the transitive action of @xmath45 and count the stabilizers . the following formulas are immediate . @xmath389 put the fibre bundle structure and these equations together , and we obtain what we desire . this result can be generalized to the @xmath390-step frep varieties . so we conclude that the algebra @xmath365 is f - polynomial - count . for the @xmath391-step case , it suffices to set @xmath392 . @xmath393 this formula has a dual version : @xmath394 alternatively , this corollary can be proved by a hall algebra method similar to lemma [ l : rep ] . consider the following identity in the algebra @xmath395 . @xmath396\otimes\sum_{[u]}[u]\big)\big(\sum_{[m],[v]}|{\operatorname{epi}}_q(m , v)|[m]\otimes [ v]\big ) = \sum_{[m],[w]}|{\operatorname{hom}}_q(m , w)|[m]\otimes [ w].\ ] ] applying the character @xmath397 to the both sides , we see the result immediately . it follows from lemma [ l : tao ] and [ l : a2q ] that [ t : a2q ] if @xmath398 is a geometric quotient , then it has a counting polynomial , which can be explicitly computed . we will see in the last section that the assumption of being a geometric quotient is unnecessary . this result is known ( * ? ? ? * theorem 4.3 ) for some special choices of @xmath10 and @xmath8 . consider the 3-arrow kronecker quiver @xmath342 with dimension vectors @xmath399 and @xmath400 . let @xmath54 be a general representation of dimension @xmath10 , then @xmath54 has no subrepresentation of dimension @xmath401 . so the projection @xmath402 is an isomorphism . we can use the algorithm in ( * ? ? ? * corollary 4.4 ) to find that @xmath403[2]^2(1,4,2,8,5,8,2,4,1),\end{aligned}\ ] ] where @xmath404 is the slope function constructed in ( * ? ? ? * section 1 ) . recall that @xmath405 for some sufficiently small @xmath288 . now we change @xmath406 to @xmath407 , then @xmath408 there is no difficulty to generalize the above results to @xmath409 . in fact , we will do it more generally in @xcite . if @xmath26 is add - polynomial - counting , then @xmath27\otimes ka_2 $ ] is f - polynomial counting . let us come back to general @xmath7 . consider the map @xmath410=\frac{|{\operatorname{gr}}_\gamma(w)|}{a_w}$ ] as in ( * ? ? ? * section 2 ) . if we apply this map to @xmath146 , we get @xmath411 we known from ( * ? ? ? * lemma 1.2 ) that when @xmath11 is a geometric quotient , this number is equal to @xmath412 for some slope @xmath413 . to compute @xmath414 , we apply @xmath415 to as before . we define @xmath416 then @xmath417 let @xmath418 where @xmath419 is the projection . then @xmath420 is stratified by the locally closed subvarieties @xmath421 when @xmath422 , for any decompositions @xmath127 and @xmath423 , @xmath424 is the same as the incidence variety @xmath425 the proof of the following lemma is similar to that of lemma [ l : rep ] and [ l : a2q ] , so we leave it for the readers . [ l : frep2 ] @xmath426)\to { \operatorname{gr}}_{{\tilde{a}\cap\tilde{\gamma}}}^{\tilde{c}}(\tilde{\alpha})$ ] is a fibre bundle with fibre @xmath427)\times { \operatorname{rep}}_{(b , a_+)}(q^\circ[e])\times { \operatorname{rep}}_{(\gamma - c,\gamma_+)}(q^\circ[e])\times { \operatorname{rep}}_{\tilde{c}}(q^\circ[e])\\ & \times \prod_{a\in q_1}{\operatorname{hom}}(k^{(\beta - b)(ta)},k^{(\gamma+b)(ha)})\times { \operatorname{hom}}(k^{b(ta)},k^{c(ha ) } ) \times { \operatorname{hom}}(k^{(\gamma - c)(ta)},k^{c(ha)}).\end{aligned}\ ] ] so @xmath428)=\sum_{\tilde{b}+\tilde{c}=\tilde{a}}t_{(\beta,\gamma , b , c)}\cdot r_{(\beta - b,\alpha_+)}r_{(b , a_+)}r_{(\gamma - c,\gamma_+)}r_{\tilde{c}},\ ] ] where @xmath429}{\bigl[\begin{smallmatrix}a_+\\b_+\end{smallmatrix}\bigr]}{\bigl[\begin{smallmatrix}\gamma_+\\(\gamma - c)_+\end{smallmatrix}\bigr]}|{\operatorname{gl}}_{(\gamma+b)_+}||{\operatorname{gl}}_{c_+}|}{{\langle\beta - b,\gamma+b\rangle}{\langleb , c\rangle}{\langle\gamma - c , c\rangle}},$ ] and @xmath430)$ ] . readers can easily write out the formula for the dual case @xmath185 $ ] . this lemma can be recursively generalized to the @xmath390-step case : @xmath431 . [ t : a2ext ] if @xmath26 is add - polynomial - count and @xmath255)$ ] is a geometric quotient , then @xmath432)}|{\operatorname{gr}}_\gamma(m)|$ ] is polynomial - count for any @xmath433 . as in ( * section ) , we can also consider the @xmath434-step analog of @xmath435 : @xmath436={a_w}^{-1}|{\operatorname{fl}}_{\gamma_t\cdots\gamma_1}(w)|.\ ] ] everything can be generalized to this case without any essential difficulty . finally we consider the map @xmath21 from the hall algebra @xmath131 to the formal power series algebra @xmath437 $ ] in @xmath438 variables as in ( * ? ? ? * section 8) : @xmath439=a_w^{-1}\sum_{i=0 } ( -1)^{i+1 } f_i(w)x^\alpha,\ ] ] where @xmath440 is the number of @xmath90-step filtrations of @xmath125 . we recall from ( * ? ? ? * lemma 8.3 ) that the number @xmath441 has a neat formula in terms of the multiplicities of simple summands of @xmath125 . fix a slope function @xmath8 and a slope @xmath442 . let @xmath76 be the abelian subcategory of all semistable representations with slope @xmath75 , and @xmath443 $ ] . let us denote by @xmath444 the number of @xmath10-dimensional absolutely stable representations and @xmath445 . the absolute ( resp . relative ) poincar series of @xmath446 at @xmath75 is @xmath447 ( resp . @xmath448 ) . our convention is that the relative ones have constant term @xmath391 , but @xmath449 for the absolute ones . it was proved in ( * ? ? ? * theorem 4.1 ) that @xmath450 where @xmath451 is the plethystic exponential in the @xmath452-ring @xmath437 $ ] ( * ? ? ? * section 2 ) . moreover , it is known ( * ? ? ? * theorem 8.3 ) that @xmath453 actually , for both the argument would work for any algebra not necessarily hereditary . to compute @xmath454 , we apply @xmath21 to each individual @xmath455 for @xmath456 using . it follows from lemma [ l : frep2 ] and its @xmath434-step generalization that the series @xmath457 has all coefficients polynomials in @xmath5 , and so are @xmath458 . 99 i. assem , d. simson , a. skowroski , _ elements of the representation theory of associative algebras , _ london mathematical society student texts 65 , cambridge university press , 2006 . w. crawley - boevey , m. van den bergh , _ absolutely indecomposable representations and kac - moody lie algebras , _ invent . ( 2004 ) , no . 3 , 537559 . h. derksen , j. weyman , _ the combinatorics of quiver representation , _ ann . fourier ( grenoble ) 61 ( 2011 ) , no . 3 , 10611131 . j. engel , m. reineke , _ smooth models of quiver moduli , _ math . z. 262 ( 2009 ) , no . 4 , 817848 . j. fei , _ counting using hall algebras i. quivers , _ j. algebra 372 ( 2012 ) , 542559 . _ counting using hall algebras iv . tensor products of quivers , _ in preparation . _ moduli and tilting ii . extensions from quivers , _ preprint . _ polynomial - count representations of quivers , _ in preparation king , _ moduli of representations of finite - dimensional algebras , _ quart . oxford ser . ( 2 ) 45 ( 1994 ) , no . 180 , 515530 . s. mozgovoy , m. reineke , _ on the number of stable quiver representations over finite fields , _ j. pure appl . algebra 213 ( 2009 ) , no . 4 , 430439 . m. reineke , _ the harder - narasimhan system in quantum groups and cohomology of quiver moduli , _ invent . ( 2003 ) , no . 2 , 349368 . , _ counting rational points of quiver moduli , _ int . math . res . not . 2006 , art.id 70456 , 19 pp . , _ moduli of representations of quivers , _ trends in representation theory of algebras and related topics , 589637 , ems ser . soc . , zrich , 2008 . o. schiffmann , _ lectures on hall algebras , _ preprint , arxiv:0611617 . c. sznt _ on the cardinalities of kronecker quiver grassmannians , _ math . z. 269 ( 2011 ) , no . 3 - 4 , 833846 . j. xiao , _ drinfeld double and ringel - green theory of hall algebras , _ j. algebra 190 , no . 1 , 100144 ( 1997 ) .
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we count the @xmath0-rational points of git quotients of quiver representations with relations .
we focus on two types of algebras
one is one - point extended from a quiver @xmath1 , and the other is the dynkin @xmath2 tensored with @xmath1 . for both
, we obtain explicit formulas .
we study when they are polynomial - count .
we follow the similar line as in the first paper but algebraic manipulations in hall algebra will be replaced by corresponding geometric constructions .
[ multiblock footnote omitted ]
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it has been considered that the origin of heavy neutron - rich elements like uranium is mainly due to the @xmath0-process nucleosynthesis that occurs during the supernova explosions and/or neutron star mergers @xcite . the main issue concerning the @xmath0-process research is to reproduce the three peaks ( @xmath3 80 , 130 , and 195 ) in the abundance pattern for the @xmath0-elements in the solar system . among models of the @xmath0-process , it has been believed that supernovae are the most plausible astrophysical site @xcite . the explosion is triggered by the gravitational collapse of massive stars of @xmath4(e.g . * ) . since a proto - neutron star is formed after the explosion , neutron - rich elements seem to be easily ejected by the supernova shock . unfortunately all realistic numerical simulations concerning the collapse - driven supernovae have failed to explode the outer layer outside the fe - core @xcite . therefore , plausible site / mechanism of the @xmath0-process has not yet been clarified . on the other hand , explosive nucleosynthesis that produces most elements up to fe - group nuclei has been calculated under the assumption that the explosion is triggered from outside the fe - core whose location is defined as the _ mass cut _ @xcite ; the calculated abundances from c to ge are consistent with the supernova observations and the chemical evolution of galaxies @xcite . however , this situation can not be applied to the @xmath0-process due to the large electron fraction @xmath5 and the low entropy distribution above the @xmath6 . a specific model of neutrino - wind with the very high entropy per baryon ( @xmath7 and @xmath8 ) has been suggested to reproduce up to the third peak in the abundance pattern of the @xmath0-process @xcite . it should be noted that this model includes artificial parameters such as mass loss rates and initial conditions of hydrodynamical calculations @xcite . on the other hand , detailed @xmath0-process calculations that include the fission have not been fully performed . if the @xmath0-process occurs along the paths of the neutron - drip line , the fission process should become important @xcite . the explosive nucleosynthesis under the jet - like explosion is investigated with use of the two - dimensional hydrodynamical code @xcite , where strong @xmath9-rich freezeout region has emerged . recently , two - dimensional magnetohydrodynamical ( mhd ) calculations have been performed under the various initial parameters concerning the rotation and magnetic field @xcite . the zeus-2d code developed by @xcite has been modified to include a tabulated equation of state @xcite , electron captures , and neutrino transport @xcite . various combinations have been investigated for the initial ratio of the rotational energy ( @xmath10 ) and/or magnetic energy ( @xmath11 ) to the gravitational energy ( @xmath12 ) . in case of the cylindrical profiles of the rotation and magnetic field , it is found that the shape of the shock wave becomes prolate compared to the case without magnetic fields though detailed studies of neutrino transport must be developed . furthermore , the confined magnetic fields behind the shock front push the shock wave strongly . the aspect ratio of the polar to the equatorial radius in the stalled shock front becomes 1.4 for the initial values of @xmath13 and @xmath14 . it is noted that whether the initial models of stellar evolution include magnetorotational effects @xcite or not @xcite , no significant differences in the hydrodynamical features are found . in the present paper , we will carry out the calculations of the mhd explosion of the he - core of 3.3 @xmath1 whose mass in the main sequence stage is about 13 @xmath1 star . thereafter , we investigate the r - process nucleosynthesis using the results of mhd calculations . we find new @xmath0-process site thanks to a large nuclear reaction network that includes fission ; we obtain the region that produces the @xmath0-process elements under the low @xmath15 . in 2 the full nuclear reaction network necessary to the @xmath0-process calculation is constructed . we describe initial models in 3 and give explosion models based on the mhd simulations . the results of the @xmath0-process nucleosynthesis calculations are presented in 4 . we summarize and discuss the results in 5 . we have developed the nuclear reaction network that had been constructed for the rapid - proton capture process @xcite . the network is extended toward the neutron - rich side to the neutron - drip line . the full network consists of about 4000 nuclear species up to @xmath16 = 100 . we include two body reactions , i.e. , ( n,@xmath17 ) , ( p,@xmath17 ) , ( @xmath9,@xmath17 ) , ( p , n ) , ( @xmath9,p ) , ( @xmath9,n ) , and their inverses . this network contains specific reactions such as three body reactions , heavy ion reactions and weak interactions . as shown in table [ tab : ntwk ] we construct two kinds of the network a and b that consist of different nuclear data set . for nuclear masses , the experimental data @xcite is used if available ; otherwise , the theoretical data by mass formula frdm @xcite is adopted in the range @xmath18 , and/or etfsi @xcite in @xmath19 . most reaction rates are taken from the compilation ( reaclib ) of @xcite that includes experimental and theoretical data for the reaction rates and partition functions with use of frdm ( network a ) or etfsi ( network b ) . reaction rates for @xmath20 that are not available in reaclib are taken from @xcite . the rates on decay channels , @xmath9- , @xmath21-decay , and @xmath2-delayed neutron emission , are taken from jaeri @xcite , that includes experimental and theoretical decay rates of nuclei near the stability line . on the @xmath2-decay rates not available in jaeri , theoretical rates by @xcite for network a or those by @xcite are adopted for network b. the same fission data is adopted for both network a and b. ( a ) _ spontaneous fission _ : experimental half lives and branching ratios of spontaneous fission are taken from @xcite and @xcite . while theoretical formula of half life ( * ? ? ? * eq . ( 23 ) ) with empirical fission barrier @xcite is adopted for nuclei whose half lives are not known experimentally , for all nuclei of both @xmath22 155 and @xmath23 240 , the life times of the decay are set to be @xmath24 s @xcite . @xmath2-delayed fission _ : branching ratios of @xmath2-delayed fission are taken from @xcite . ( c ) _ fission yields _ : empirical formula ( * ? ? ? ( 5 ) ) is adopted about decay products . since many charged particles participate in the nucleosynthesis during the explosion , we have included the screening effects for all relevant reactions @xcite . we also use theoretical weak interaction rates that are the function of the density and temperature @xcite . cr|cr|cr|cr|cr & & & & & & & & & + h & 1 3 & sc & 39 67 & nb & 83 125 & pm & 143 187 & tl & 203 263 + & 3 & & 67 & & 129 & & 187 & & 255 + he & 3 6 & ti & 40 70 & mo & 86 126 & sm & 144 188 & pb & 204 264 + & 6 & & 72 & & 132 & & 188 & & 259 + li & 6 8 & v & 43 73 & tc & 90 129 & eu & 151 189 & bi & 209 265 + & 8 & & 76 & & 133 & & 193 & & 263 + be&7 12 & cr & 44 74 & ru & 96 130 & gd & 152 190 & po & 210 266 + & 12 & & 78 & & 136 & & 196 & & 267 + b & 8 14 & mn & 46 77 & rh & 101 141 & tb & 155 198 & at & 211 269 + & 14 & & 81 & & 137 & & 197 & & 269 + c & 11 18 & fe & 47 78 & pd & 102 142 & dy & 156 212 & rn & 215 270 + & 18 & & 84 & & 138 & & 202 & & 270 + n & 12 21 & co & 50 81 & ag & 105 149 & ho & 161 215 & fr & 218 271 + & 21 & & 85 & & 147 & & 203 & & 271 + o & 14 22 & ni & 51 82 & cd & 106 150 & er & 162 216 & ra & 221 272 + & 22 & & 86 & & 148 & & 208 & & 272 + f & 17 26 & cu & 56 91 & in & 111 155 & tm & 167 221 & ac & 224 273 + & 26 & & 89 & & 149 & & 215 & & 273 + ne & 7 30 & zn & 57 94 & sn & 112 156 & yb & 168 222 & th & 227 274 + & 34 & & 92 & & 154 & & 218 & & 274 + na&20 34 & ga & 60 95 & sb & 119 162 & lu & 173 224 & pa & 230 278 + & 37 & & 97 & & 161 & & 225 & & 277 + mg&20 36 & ge & 61 102 & te & 120 164 & hf & 174 226 & u & 232 280 + & 38 & & 100 & & 164 & & 228 & & 280 + al&22 41 & as & 64 103 & i & 123 171 & ta & 179 235 & np & 235 284 + & 41 & & 101 & & 165 & & 229 & & 284 + si&24 44 & se & 65 106 & xe & 124 180 & w & 180 236 & pu & 238 287 + & 46 & & 104 & & 168 & & 232 & & 288 + p & 27 45 & br & 68 117 & cs & 129 181 & re & 183 239 & am & 241 290 + & 49 & & 117 & & 181 & & 235 & & 292 + s & 28 48 & kr & 69 118 & ba & 130 182 & os & 184 240 & cm & 244 294 + & 50 & & 118 & & 182 & & 236 & & 296 + cl&31 51 & rb & 74 119 & la & 135 183 & ir & 189 241 & bk & 247 298 + & 51 & & 119 & & 183 & & 239 & & 300 + ar&32 56 & sr & 77 120 & ce & 136 184 & pt & 190 242 & cf & 250 302 + & 54 & & 120 & & 184 & & 243 & & 304 + k & 35 55 & y & 79 121 & pr & 141 185 & au & 195 257 & es & 253 306 + & 57 & & 121 & & 185 & & 247 & & 308 + ca&36 62 & zr & 81 122 & nd & 142 186 & hg & 196 258 & fm & 256 310 + & 60 & & 124 & & 186 & & 251 & & 312 + the presupernova model has been calculated from the evolution of he - core of 3.3 @xmath25 that corresponds to 13 @xmath25 in the main sequence stage @xcite . the mass of the fe - core is 1.18 @xmath1 that is the smallest fe - core in massive stars obtained from the stellar evolutionally calculation with the limitation of the spherical symmetry . the edge of the fe - core that has steep density gradient is at @xmath26 cm from the center . the mass of the si - rich layer is @xmath27 and the layer extends to @xmath28 cm above the fe - core . since the central density exceeds @xmath29 ( @xmath30 ) and temperature @xmath31 in units of @xmath32 k , the fe - core just begins to collapse . initial models for the collapse calculations ( precollapse models ) are constructed by using the density and temperature distributions of the original fe+si core . we adopt cylindrical properties of the angular velocity @xmath33 and the toroidal component of the magnetic field @xmath34 as follows @xcite : @xmath35 where @xmath36 and @xmath16 are the distances from the rotational axis and the equatorial plane with @xmath37 and @xmath38 being model parameters . both @xmath39 and @xmath40 are the initial values at @xmath41 and @xmath42 . initial parameters of four precollapse models are given in table [ tab : ini ] . the spherically symmetric case is denoted by model 1 . in model 2 , the profiles of rotation and magnetic field in the fe - core are taken to be nearly uniform . we present model 4 as the case having a differentially rapid rotating core and strong magnetic fields . an intermediate example , model 3 between model 2 and model 4 is prepared for reference . since the value of @xmath43 is higher compared to that used in @xcite by a few percents , we regard the present case of @xmath44 % as rather rapid rotating stars with the moderate magnetic field . in all computations , spherical coordinates @xmath45 are adopted . the computational region is set to be @xmath46 km and @xmath47 , where the included mass in the precollapse models amounts to @xmath48 . the first quadrant of the meridian section is covered with @xmath49 mesh points . to get information of mass elements , five thousand tracer particles are placed within the region of @xmath50 between @xmath51 ( @xmath52 km ) and @xmath53 ( @xmath54 km ) . .initial parameters of precollapse models . [ tab : ini ] [ cols="<,^,^,^,^,^,^",options="header " , ] we perform the calculations of the collapse , bounce , and the propagation of the shock wave with use of zeus-2d in which the realistic equation of state @xcite has been implemented by @xcite . we do not include the neutrino transport , since our aim is to clarify the differences in the nucleosynthesis between spherical and mhd jet explosion . it is noted that the contribution of the nuclear energy generation is usually negligible compared to the shock energy . in table [ tab : res ] , our results of mhd calculations are summarized . @xmath55 is the explosion energy when the shock reaches the edge of the fe - core @xcite . in model 3 , the explosion is failed due to the specific combination of rotation and magnetic field between the values of model 2 and model 4 ; although @xmath55 still exists , the radial distance from the center in the generated shock front at the bounce shrinks gradually after a few oscillations of the front . therefore , it does not always depend on @xmath43 and/or @xmath56 whether the explosion succeeds or not . in figs . [ fig : part1 ] and [ fig : part2 ] trajectories of tracer particles are shown for some specified values of @xmath15 . while the jet - like explosion occurs along the equator in model 2 ( fig . [ fig : part1 ] ) , collimated jet is emerged from the rotational axis in model 4 ( fig . [ fig : part2 ] ) . figure [ fig : rts ] shows the density , temperature , and entropy per baryon in @xmath57 of the tracer particles in fig . [ fig : part2 ] . we find after the jet - like explosion of model 4 that it remains 1.24 @xmath1 proto - neutron star inside the radius 300 km accompanying successive accretion onto the star with @xmath58 at @xmath59 s. ) of @xmath60 ms during the simulation ( model 2 ) . the edge of the fe - core in the precollapse model is shown by the thick - dotted line . the values of @xmath15 correspond to those in the last stage of nse for each tracer particle . [ fig : part1],title="fig:",width=604 ] + but for the final stage ( @xmath61 ) of @xmath62 ms ( model 4 ) . [ fig : part2],title="fig:",width=1133 ] + . each curve corresponds to that in fig . [ fig : part2 ] ( model 4 ) . [ fig : rts],title="fig:",width=340 ] . each curve corresponds to that in fig . [ fig : part2 ] ( model 4 ) . [ fig : rts],title="fig:",width=340 ] + . each curve corresponds to that in fig . [ fig : part2 ] ( model 4 ) . [ fig : rts],title="fig:",width=340 ] + during the explosion , the temperature exceeds @xmath64 k around the layers of the si+fe core , where the region of the nuclear statistical equilibrium ( nse ) is realized as shown in fig . [ fig : rts ] . therefore , we follow the change in @xmath15 of the ejected tracer particle due to the weak interactions of electron / positron captures , and @xmath65-decays until the last stage of nse . we set this stage to be @xmath66 ; afterward the temperature decreases in time as shown in fig . [ fig : rts ] . the change in @xmath15 is calculated from the relation @xmath67y_i , \ ] ] where for the abundance @xmath68 , @xmath69 consists of the @xmath70 and positron capture rates , and @xmath71 consists of the @xmath72 and electron capture rates , respectively . time evolutions of @xmath15 relevant to the @xmath0-process are shown in fig . [ yetm ] ( left panels ) . the trajectories of tracer particles with @xmath73 are depicted in fig . [ fig : part1 ] for model 2 and those with @xmath74 in fig . [ fig : part2 ] for model 4 , respectively ; the values of @xmath15 indicate those of the last stage of the nse calculation . in model 4 , the polar region is ejected having rather high value of @xmath8 . the lowest value of @xmath75 is discovered from around the region inclined at 20 30 degrees from the rotational axis . thereafter , using the compositions obtained from the last nse stage and the profiles of the density and temperature during the explosion , we perform the @xmath0-process nucleosynthesis with the nuclear reaction network described in 2 . we remark that after the last stage of the nse , the changes in @xmath15 for fig . [ yetm ] are obtained from the calculations by the full network . after the time @xmath76 , both the temperature and density are extrapolated to @xmath77 s ( @xmath78 ) in proportion to @xmath79 with @xmath80 for the temperature and @xmath81 for the density , respectively @xcite . we consider that the qualitative results of the @xmath0-process nucleosynthesis do not depend much on the value of @xmath82 . figures [ yetm ] ( right panels ) show the ejected mass in @xmath1 against @xmath15 in the range @xmath83 . for the spherical explosion , materials with @xmath84 is ejected . the ejection for @xmath85 occurs from inside the fe - core in the range of @xmath86 km . on the other hand , ejection occurs in the direction of the equator with @xmath87 for model 2 . as shown in fig . [ fig : part2 ] , materials with @xmath88 are emerged for the jet - like explosion along the rotational axis ( model 4 ) . for both models 2 and 4 , the ejection for @xmath89 comes from the si - rich layer . we recognize that as against the spherical explosion , jet - like explosion of model 4 decreases @xmath15 significantly . + + + we calculated the @xmath0-process nucleosynthesis during the mhd explosion using networks a and b. the progress in the @xmath0-process is shown in fig . [ rpro ] at three epochs in model 4 . since @xmath15 decreases to 0.158 at the last stage of nse , the @xmath0-process paths reach to the neutron drip line . figures [ modelfrdm ] and [ modelet ] show the comparison of the solar @xmath0-process abundances with obtained abundances . generally , compared to the spherical explosion the jet - like explosion results in the increase of the nuclei for @xmath90 . the reproduction of the peaks in the distributions of the @xmath0-elements depends on the decrease in @xmath15 during the early phase of the explosion ( @xmath91 ) . in model 1 , the produced @xmath0-elements are not enough to explain even the second peak in the @xmath0-process pattern . in model 2 , although the second peak is reproduced well , the amount of the produced @xmath0-elements are too small to explain the third peak . for model 4 we succeed in making the global abundance pattern of the @xmath0-elements from the first to the third peak . moreover , we find that the fission cycling leads to the normal @xmath0-process nucleosynthesis based on ( n,@xmath17 ) @xmath92 ( @xmath17,n ) equilibrium accompanied with @xmath2-decays for the low @xmath15 region ( @xmath93 ) . on the other hand , distribution of final products are found to be much sensitive to the mass formula as seen in the differences between figs . [ modelfrdm ] and [ modelet ] . this is because the global @xmath0-process pattern and the profiles of the peaks in the abundance pattern depend on the @xmath2-decay rates that have been calculated using the mass formulae explained in 2 . -process nucleosynthesis during the jet - like explosion of model 4 ; ( @xmath94 , @xmath15 ) = ( 0.489 s , 0.161 ) : upper , ( 1.650 s , 0.312 ) : middle , and ( 2.760 s , 0.372 ) : lower . the tracer particle has @xmath95 in the last stage of nse : @xmath96 at @xmath97 s from the beginning of the collapse.,title="fig:",width=566 ] -process nucleosynthesis during the jet - like explosion of model 4 ; ( @xmath94 , @xmath15 ) = ( 0.489 s , 0.161 ) : upper , ( 1.650 s , 0.312 ) : middle , and ( 2.760 s , 0.372 ) : lower . the tracer particle has @xmath95 in the last stage of nse : @xmath96 at @xmath97 s from the beginning of the collapse.,title="fig:",width=566 ] -process nucleosynthesis during the jet - like explosion of model 4 ; ( @xmath94 , @xmath15 ) = ( 0.489 s , 0.161 ) : upper , ( 1.650 s , 0.312 ) : middle , and ( 2.760 s , 0.372 ) : lower . the tracer particle has @xmath95 in the last stage of nse : @xmath96 at @xmath97 s from the beginning of the collapse.,title="fig:",width=566 ] + + + + + + + + using zeus-2d , we find the thermodynamical conditions for the @xmath0-process to occur . the appropriate condition of the density , temperature and @xmath15 can be satisfied under the initial rotation law and the magnetic field strength of the precollapse model ; the jet between the polar and the equatorial radius produces the @xmath0-process elements which emerge from the si - rich layers above the fe - core . contrary to the calculation under an artificial explosion energy and a @xmath6 located outside the fe - core in the spherical model @xcite , we have pursued the calculation from the collapse to the bounce , and shock wave propagation that attains to the edge of the fe - core . while the spherical explosion reproduces at most to the beginning of the second peak in the abundance pattern of the solar @xmath0-elements , jet - like explosion due to the effects of mhd in the direction of the rotational axis succeeds in reproducing the third peak . however , our results depend on the initial parameters for both the strength of the rotation and magnetic field . for the jet - like explosion from the equator , the reproduction of the @xmath0-elements is limited to the second peak . we stress that the @xmath0-process nucleosynthesis can have many variations if the mhd effects play an important role in the supernova explosion . the shape and the position of peaks depend crucially on the mass formula and @xmath2-decay rates . however , it is considered that theoretical data on @xmath2-decay rates has large uncertainty . therefore , model 4 can reproduce well the solar @xmath0-process abundance pattern within the uncertainties of @xmath2-decay rates . the @xmath2-decay rates calculated by the gross theory are tend to become small compared to those obtained by shell model computations . as a consequence , network a makes the three peaks more clearly than network b. small abundance peak around @xmath98 is not built enough in the present calculations for both networks . it is hoped that the information of the @xmath0-process acquired through theoretical and observational studies may give severe constraints to the nuclear data . in the present calculation , the fission process does not play an important role , since the ejected mass of the very low @xmath15 region is small . however , production of abundances for @xmath99 may need the region of @xmath100 as suggested in figs . [ modelfrdm ] and [ modelet ] . furthermore , fission should be included in the @xmath0-process calculation for the situation of very low @xmath15 such as a neutron star merger with @xmath101 @xcite . as shown in table [ tab : res ] ejected mass of the @xmath0-elements amounts to @xmath102 in case of model 4 that is one tenth of the total ejected mass . since the ejected mass of oxygen is @xmath103 for the spherical explosion of the @xmath104 helium core @xcite , the ratio @xmath105 is large by a factor of 100 compared to the corresponding solar ratio of @xmath106 . it has been pointed out for the explosion of massive stars the underproduction of the p - nuclides with respect to oxygen , when normalized to the solar values @xcite . however it is found that as far as the case in the 3.3 @xmath1 helium core , produced p - nuclides are free from the problem of the underproduction @xcite . considering uncertainties neglected in the present simulations and the differences in the nucleosynthesis between the spherical and jet - like explosion , the problem of the overproduction in the @xmath0-elements should be worth while to examine in detail . in the present investigation , as the first step we ignore the effects of neutrino transport . it is known that neutrinos take off the explosion energy significantly . therefore , delayed explosion by neutrino heating is the most promising scope of the supernova explosion . however , there exists difficulties related to the two dimensional treatment of neutrino transport @xcite : even one dimensional simulations that include the detailed neutrino transport process do not succeed in the explosion @xcite . our purpose has been to study the effects of the mhd jet on the @xmath0-process and elucidate the differences in the produced @xmath0-elements from the spherical explosion . for the next step , some investigations that include the neutrino transport will be pursued . audi , g. , & wapstra , a. h. 1995 , , 595 , 409 freiburghaus , c. , rosswog , s. , thielmann , f .- k . 1999 , apj , 525 , l121 fuller , g. m. , fowler , w. a. , & newman , m. 1980 , apjs 42 447 fuller , g. m. , fowler , w. a. , & newman , m. 1982 , apjs 48 279 goriely , s. ( 2002 ) , in proc . of nuclei in the cosmos vii ( japan , july 2002 ) in press , most : http://www-astro.ulb.ac.be/nucdata hashimoto , m. , nomoto , k. , @xmath107 shigeyama , t. 1989 , a@xmath107a , 210 , l5 . hashimoto , m. 1995 , prog . theor . phys . heger , a. , woosley , s. e. , langer , n. , & spruit , h. c. in iau symp . 215 , stellar rotaion , in press ( astro - ph/0301374 ) horiguchi , t. , tachibana , t. , & katakura , t. 1996 , chart of the nuclides ( ibaraki : nucl . data center ) k@xmath108ppeler , f. , beer , h. and wisshak , k. 1989 , rep . 52 945 janka , h .- th . 192 , in press ( astro - ph/0401461 ) kinsey , r. r. et al . , the nudat / pcnudat program for nuclear data , the 9 th international symposium of capture - gamma - ray spectroscopy and related topics , budapest , hungary , october ( 1996 ) kodama , t. , @xmath107 takahashi , k. 1975 nucl . koike , o. , hashimoto , m. , kuromizu , r. , fujimoto , s. , 2004 , , 603 , 242 kotake , k. , sawai , h. , yamada , s. , & sato , k. 2004 , , 608 , 391 kotake , k. , yamada , s. , sato , k. , sumiyoshi , k. , ono , h. , & suzuki , h. 2004 , phys . d. , 609 , 124004 kotake , k. , yamada , s. , & sato , k. 2005 , , 618 , 474 mamdouh , a. , pearson , j.m . , rayet , m. , & tondeur , f. 1999 , . 644 , 389 mamdouh , a. , pearson , j.m . , rayet , m. , & tondeur , f. 2001 , . 679 , 337 m@xmath109ller , p. , myers , w. d. , & swiatecki , w. j. 1995 , at . data nucl . data tables , 59 , 185 m@xmath109ller , p. , nix , j. r. , kratz , k .- 1997 , atomic data and nuclear data tables , vol . nagataki , s. , hashimoto , m. , sato , k. , & yamada , s. 1997 , , 486 , 1026 panov , i. v. , freiburghaus , c. , thielmann , f .- k . 2001 , nucl . a688 587c . qian y .- z . 2003 , prog . 50 , 153 qian , y - z . 2005 , first argonne / msu / jina / int ria wokshop rauscher , t. , & thielemann , f .- k . 2000 , at . data nucl . data tables 75 1 rauscher , t. , & thielemann , f .- k . 2001 , at . data nucl . data tables , 79 47 rayet , m. , arnould , m. , hashimoto , m. , prantzos , n. , & nomoto , k. 1995 , a&a 298 , 517 staudt , a. , klapdor - kleingrothaus , h. v. 1992 , nucl . a549 254 rosswog , s.k . , freiburghaus , c. , & thielemann , f .- k . 2001 , nucl . a688 , 344 shen , h. , toki , h. , oyamatsu , k. , & sumiyoshi , k. 1998 , nucl . phys . a. , 637 , 435 stone , j. m. , & norman , m. l. 1992 , , 80 , 791 sumiyoshi , k. et al . 2000 , pub . japan , 52 601 sumiyoshi , k. et al . 2001 , , 561 , 880 tachibana , t. , & yamada , m. proc . conf on exotic nuclei and atomic masses , arles , 1995 , p.763 takiwaki , t. , kotake , k. , nagataki , s. , & sato , k. 2004 , , 616 , 1086 thielemann , f .- k . , nomoto , k. , & hashimoto , m. , 1996 , , 460 , 408 thielemann , f .- k . proceedings of 1crc 2001 : 1copernicus gesellschaft 2001 timmes , f. x. , woosley , s. e. , & weaver , t. a. 1995 , , 98 , 617 tsujimoto , t. , iwamoto , k. , hashimoto , m. , nomoto , k. , & f - k thielemann + 1993 in origin and evolution of the elements , world scientific , 50 woosley , s. e. , & weaver , t. a. 1995 , , 101 , 181 yamada , s. , & sawai , h. , 2004 , , 608 , 907 yamada , s. , kotake , k. , & yamasaki , t. 2004 , new . j. phys . , 6 , 79
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we investigate @xmath0-process nucleosynthesis during the magnetohydrodynamical ( mhd ) explosion of supernova in a massive star of 13 @xmath1 .
contrary to the case of the spherical explosion , jet - like explosion due to the combined effects of the rotation and magnetic field lowers the electron fraction significantly inside the layers above the iron core .
we find that the ejected material of low electron fraction responsible for the @xmath0-process comes out from the silicon rich layer of the presupernova model .
this leads to the production up to the third peak in the solar @xmath0-process elements .
we examine whether the fission affects the @xmath0-process paths by using the full nuclear reaction network with both the spontaneous and @xmath2-delayed fission included .
moreover , we pay particular attention how the mass formula affects the @xmath0-process peaks with use of two mass formulae .
it is found that both formulae can reproduce the global abundance pattern up to the third peak though detailed distributions are rather different .
we point out that there are variations in the @xmath0-process nucleosynthesis if the mhd effects play an important role in the supernova explosion .
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let @xmath0 denote the heights of @xmath2 simple random walks on @xmath3 , conditioned on satisfying @xmath4 , \left| z^{(i+1)}_n - z^{(i)}_n \right| = 1.\ ] ] more precisely , the random walk is a markov chain on the state space of @xmath5-step walks in @xmath6 @xmath7 , |z^{(i+1 ) } - z^{(i)}| = 1 \}\ ] ] where the next step from @xmath8 is selected uniformly among the neighbours of @xmath9 in the usual lattice @xmath10 that belong to @xmath11 . in other words , we consider @xmath2 simple random walks on the lattice @xmath3 coupled under a shape condition . as in the case of a simple random walk , the rescaled trajectory of a walker , say @xmath12 , will converge in law to a brownian motion . however , it is interesting to note that the constraint between each coordinate only slow down the walk by decreasing its variance . since it is classical to illustrate for our students the simple random walk as the motion of a drunk man , we can illustrate the previous mathematical fact by considering the random walk as the motion of a chain of prisoners . it should convince even non mathematicians that the motion of the walk is slowed by the constraint . however it seems very hard to guess the variance from this comparison ! more precisely , denote @xmath13 where @xmath14 is the integer part of the real number @xmath15 [ t : var ] the rescaled random walk @xmath16 converges in law , as @xmath17 goes to infinity , to a brownian motion with variance @xmath18 convergence to the brownian motion is the usual invariance principle : the noteworthy statement here is that it is possible to give an explicit expression for the limit diffusivity of the process , and that its expression is particularly simple . our object of interest , the motion of @xmath19 is a non - markovian process that falls into the class of random walks with internal structure . related questions of limit diffusivity for random walks conditioned to respect some geometric shape have been studied in the literature , under the name of `` spider random walks '' or `` molecular spiders '' , see @xcite . the computation of the limit diffusion coefficient is also a central aim there , although the model and methods are different . our initial motivation was however more remote . actually we first addressed this question starting from combinatorial problems related to @xmath20-vertex model in relation with the razumov stroganov conjecture ( see @xcite for instance ) . the problem can also be related to random graph - homomorphisms ( see @xcite ) or the square ice model ( see @xcite where @xmath12 evolves on a torus . ) roughly speaking we can say that , in the literature we read , the authors consider questions related to the uniform distribution on a sequence of finite graphs @xmath21 and wonder about various asymptotics when @xmath22 later in the article the evolution of @xmath23 will be described as the simple random walk on a graph @xmath24 hence on the one hand our problem is a very simplified version of the problems stated above , on the other hand we were surprised to have such a simple formula for @xmath25 which is true for all @xmath26 and not only for @xmath22 we thought in the beginning that the proof of this fact should be simple but it turns out that , although elementary , the tools used to obtain the result are more sophisticated than expected . it is the aim of this note to show these tools . to prove the theorem , we will look for a decomposition @xmath27 where @xmath28 is a martingale , and where @xmath29 is a bounded function . we will then show that the following limit exists : @xmath30.\ ] ] and that it is indeed the desired diffusivity . the path to this conclusion is akin to classical results for central limit theorems for markov chains ( e.g. @xcite ) . we will use another equivalent , albeit more geometric , point of view on this decomposition . we split the chain in two parts : on the one hand , the motion of one of the walkers , and on the other hand , the relative positions of the walkers ( which we call the `` shape '' of the chain at a given time ) . the latter part is a markov chain over the state space @xmath31 and our quantity of interest is ( almost ) an observable of this chain . computing the martingale decomposition that we wish for amounts to decomposing a discrete vector field over this new state space into a divergence - free part ( corresponding to the martingale part ) and a gradient part ( corresponding to the function @xmath32 ) . owing to a particular geometric property of this vector field , for which we coin the term `` stationarity '' , it is indeed possible to perform this calculation explicitly . let us denote @xmath33 , y_n^{(i ) } = z_n^{(i+1)}- z_n^{(i)},\ ] ] and @xmath34 here @xmath35 describes the shape of @xmath0 , i.e. the position of each @xmath36 relatively to the previous one , and belongs to @xmath37 , whereas @xmath38 can be seen as the height of the first walker . obviously , the evolution of the chain of walkers may be described by the variables @xmath39 . for a convenient analysis , we will represent our process as the simple random walk on a ( multi)-graph @xmath40 , which we define below . set @xmath41 : the multi - graph @xmath40 is given as a triplet @xmath42 where @xmath43 are two edge sets , called respectively the set of positive `` and ' ' negative edges . a couple @xmath44 , @xmath45 , belongs to @xmath46 if the vector @xmath47 has nonzero entries of alternating signs , with the first one negative . moreover , @xmath48 also contains a loop from each @xmath49 to itself , noted @xmath50 . likewise , @xmath51 contains those couples @xmath44 , @xmath45 , such that @xmath47 has nonzero entries of alternating sign , with the first one positive , and self - loops noted @xmath52 for each @xmath49 . set @xmath53 . finally , we consider the following function on @xmath54 [ def : a ] let @xmath55 be the function that takes the value @xmath56 on @xmath48 and @xmath57 on @xmath51 . note that we have , for any @xmath49 , @xmath58 . let @xmath59 be the simple random walk on @xmath40 . the processes @xmath60 and @xmath61 have the same distribution . it is sufficient to prove that the two markov chains ( taking values in @xmath62 ) @xmath60 and @xmath61 have the same transition matrix . we claim that @xmath63 recall that @xmath64 , \left| z^{(i + 1 ) } - z^{(i ) } \right| = 1 \right\}$ ] . denote @xmath65 , \left| z_1^{(i ) } - z_2^{(i ) } \right| = 1 \right\}.\ ] ] since @xmath66 and @xmath67 are both simple random walks , the corresponding transition matrices are given respectively by @xmath68 denote now @xmath69 so that @xmath70 . to prove the proposition , it is sufficient to prove that @xmath71 where @xmath72 ( and @xmath73 is the sign function ) . first note that , if @xmath74 , and if @xmath75 , z_1^{(i ) } = z_2^{(i ) } - \epsilon$ ] , then @xmath76 . moreover , we have in this case @xmath77 , and @xmath78 as claimed . on the other hand , if @xmath77 and @xmath79 , then @xmath75 , z_1^{(i ) } = z_2^{(i ) } + \epsilon$ ] . for the chain , it means that , if @xmath80 then @xmath81 is @xmath82 with equal probability @xmath83 independently of @xmath84 . assume now that @xmath85 . we will prove that @xmath86 , where @xmath87 , i.e. @xmath88 , with nonzero entries of alternating signs , and a first one of the sign of @xmath89 . indeed , there is a first index @xmath90 , \ ; \mbox{such that } \quad \delta(z_{1})^{(i ) } \neq \delta(z_2)^{(i ) } \}.\ ] ] there are two possible cases @xmath91 @xmath92 in the two cases we have @xmath93 furthermore , we have @xmath94 then if @xmath95 let us define @xmath96 , \ ; \mbox{such that } \quad \delta(z_2)^{(i ) } \neq \delta(z_1)^{(i ) } \},\ ] ] where we set by convention @xmath97 if the condition defining the infimum is never satisfied . using the same arguments , we get @xmath98 by induction one can define @xmath99 , \ ; \mbox{such that } \quad \delta(z_1)^{(i)}\neq \delta(z_2)^{(i ) } \},\ ] ] until @xmath100 we have @xmath101 by definition , we get @xmath102 where @xmath103 . on the other hand , if @xmath104 , where @xmath105 , then one can recover explicitly @xmath106 from the definition of @xmath107 . moreover , the condition @xmath108 , \left| z_2^{(i ) } - z_1^{(i ) } \right|=1 $ ] is implied by the previous arguments ( following the definitions of the @xmath109 ) . we may denote , @xmath110 when @xmath111 ( @xmath112 ) and @xmath113 when @xmath114 ( @xmath115 ) . for a general @xmath116 the previous enumeration of its neighbors is surprisingly complicated but we can provide some simple examples . for instance if @xmath117 has only @xmath118 neighbors : @xmath119 , a & \rightarrow ( 1,\cdots , -1 , \cdots,1 ) , \end{aligned}\ ] ] where the @xmath57 is in the @xmath120-th position . note now that the graph @xmath40 can also be described inductively : there are only six following possibilities for @xmath121 , described below : in the figure , we have used the concatenation notation : given a string @xmath122 , the string @xmath123 , resp . @xmath124 , is obtained by adding a @xmath56 , resp . @xmath57 in front of @xmath122 . looking only at the @xmath125 cases such that @xmath126 , we can deduce the construction of @xmath127 from @xmath128 : @xmath129 figure 1 shows the first two graphs @xmath130 and @xmath131 . note that each edge of @xmath132 gives @xmath125 edges for @xmath133 , one on each facet @xmath134 , and @xmath135 and one crossing from the facet @xmath136 to the facet @xmath135 . [ f : constr ] and @xmath131 . ] we obtain the cardinality @xmath137 of @xmath138 ( as a multigraph ) by induction : @xmath139 we will also make use of the number @xmath140 of edges of the form @xmath141 which can be computed by induction : @xmath142 let us now describe vector fields on this graph . in the previous section a function @xmath143 has been defined on edges of @xmath144 we will consider here @xmath143 as a vector field on @xmath145 let us first recall some classical definitions . a vector field on @xmath146 is a function @xmath147 such that @xmath148 and such that , for any @xmath49 , @xmath149 . we say that the vector field @xmath150 on @xmath40 is a gradient vector field if there exists a function @xmath32 on the vertices of @xmath40 such that for each edge @xmath151 , @xmath152 . the gradient vector field associated with @xmath32 is denoted by @xmath153 . the divergence of a vector field @xmath150 at point @xmath154 is defined by @xmath155 we say that a vector field @xmath150 is divergence - free if its divergence vanishes at all points . we can endow the set of vector fields with a scalar product @xmath156 please note that the sum runs over all edges , including loops @xmath157 . denote also , for any subsets @xmath158 of @xmath159 , @xmath160 the flux of @xmath150 going from @xmath161 to @xmath162 . note that a divergence free field @xmath163 verifies @xmath164 in analogy with the case of vector fields in euclidean spaces , we can decompose any vector field on @xmath40 into the sum of a gradient vector field and a divergence - free field . the following proposition is well - known . [ prop_dec_usual ] let @xmath150 be a vector field on @xmath40 . there exist a unique gradient vector field @xmath153 and a unique divergence - free field @xmath163 such that @xmath165 moreover @xmath166 the last identity simply means that gradient fields and divergence - free fields are orthogonal complements of each other in the vector space of vector fields over @xmath40 . in our case we are interested in stationary vector fields . [ def : stat ] a subgraph @xmath167 of the complete graph on @xmath168 is _ stationary _ , if the following holds . for @xmath169 and @xmath170 such that @xmath171 if @xmath172 is an edge of @xmath167 , then @xmath173 is an edge of @xmath167 . a vector field @xmath150 defined on a subgraph of the complete graph on @xmath168 with a stationary domain is _ stationary _ if for all @xmath174 edges of @xmath167 , @xmath175 only depends on @xmath176 ( where @xmath168 is embedded in @xmath177 in an obvious way ) . note that , thanks to the construction of @xmath48 , it is stationary . moreover if @xmath178 and @xmath179 , then @xmath180 and thus the vector field @xmath181 taking values @xmath56 , resp . @xmath57 , on @xmath182 , resp . @xmath51 , is stationary . so we may expect the gradient vector field @xmath183 in the hodge decomposition of @xmath143 to be stationary . unfortunately if @xmath150 is a stationary vector field on @xmath40 and its decomposition is @xmath184 as per proposition [ prop_dec_usual ] , then @xmath153 is not always stationary . nevertheless it turns out that the gradient vector field @xmath183 in the hodge decomposition of @xmath143 is actually stationary as it will be shown in the next section . let us recall the definition [ def : a ] the vector field @xmath181 on @xmath40 is such that @xmath185 in this section our aim is to compute a function @xmath186 such that @xmath187 one can first remark that @xmath188 @xmath189 in the previous equation we used the notation @xmath190 for cardinality of sets . we will introduce various notations related to other cardinalities @xmath191 then we need also to define for @xmath192 @xmath193 the number of the vertices @xmath194 such that @xmath195 with @xmath196 digits @xmath197 such that @xmath198 similarly @xmath199 is the number of the vertices @xmath194 such that @xmath200 and @xmath196 digits @xmath197 such that @xmath201 then we consider @xmath202 and @xmath203 similarly @xmath204 and @xmath205 let us consider the function @xmath206 on @xmath207 such that @xmath208 the last equation is trivial since @xmath209 is the number of the @xmath210s equal to @xmath211 and @xmath212 the number of @xmath210s equal to @xmath213 obviously we also get @xmath214 for any vertex @xmath122 in @xmath215 let us remark that for any function @xmath186 on @xmath159 @xmath216 since @xmath217 yields the sum of the digits of any vertex , we first observe that if @xmath218 and the number of digits @xmath197 such that @xmath219 is even then @xmath220 if the number of digits @xmath197 such that @xmath219 is odd and @xmath221 then @xmath222 one can then deduce that @xmath223 it turns out that if we consider the function @xmath224 on @xmath207 such that @xmath225 then @xmath226 we will prove by induction on @xmath227 to do that we split @xmath228 into the sum of two functions @xmath229 to proceed the induction argument we remark that for any vertex @xmath122 in @xmath159 @xmath230 we will then compute @xmath231 the easiest computation is @xmath232 since @xmath233 then @xmath234 to evaluate @xmath235 we use that @xmath212 is the number of digits equal to @xmath236 in @xmath237 if @xmath238 and if the number of @xmath239 is even then @xmath240 if this number is odd then @xmath241 therefore @xmath242 hence @xmath243 one also get in the same way @xmath244 the induction is a bit more involved for @xmath245 because of @xmath246 then @xmath247 hence @xmath248 similarly we get @xmath249 we can now evaluate the functions @xmath250 [ lem : phi_barphi ] @xmath251 @xmath252 @xmath253 we will only sketch the proof performed by an easy induction on @xmath26 for @xmath254 computations are similar for @xmath255 let us assume that , hold for @xmath256 we have to compute @xmath257 and @xmath258 and check that they fulfill , for @xmath259 because of @xmath260 one can check that @xmath261 using for @xmath256 we get for @xmath262 and @xmath263 the computations for @xmath258 are left to the reader . by summing and we get , and we deduce that if we take @xmath264 @xmath265 is divergence free . please note that obviously the additive constant in is arbitrary but it yields the following convenient expression of @xmath266 in terms of the digits of @xmath267 [ lem : form - grad a ] @xmath268 where @xmath269 moreover @xmath183 is a stationary gradient vector field . since we already know that @xmath270 it is enough to show by induction on @xmath26 that @xmath271 obviously the formula is true for @xmath272 because of and , @xmath273 hence is proved for @xmath274 owing to @xmath270 and , it is obvious that @xmath275 are stationary and consequently @xmath183 is a stationary gradient vector field . we denote by @xmath276 the decomposition of @xmath143 as per proposition [ prop_dec_usual ] . back to the original problem , we recall that @xmath277 let us denote @xmath278 . let @xmath279 , we have @xmath280 = ( \nabla \cdot b)(y_n ) = 0,\ ] ] and @xmath281 is a martingale . we now sketch out how to apply the central limit theorem for markov chains . let @xmath282 @xmath283 is a markov chain on @xmath54 then our quantity of interest @xmath284 is an additive observable of the process @xmath283 , as @xmath285 the central limit theorem for markov chains ( see e.g. @xcite ) shows that @xmath286 converges as @xmath287 to a brownian motion , with variance given by @xmath288 \\ & = \lim_{n \rightarrow + \infty } \frac{1}{n } \mathbb{e } \left[m_n^2\right]+ \lim_{n \rightarrow + \infty } \frac{1}{n } \mathbb{e } \left [ f(y_n)^2\right ] + \lim_{n \rightarrow + \infty } \frac{2}{n } \mathbb{e } \left[\left(f(y_n ) m_n \right)\right].\end{aligned}\ ] ] let us first compute @xmath289.$ ] we can remark that @xmath290 is a @xmath291 martingale , hence @xmath292=\lim_{n \rightarrow + \infty } \frac{1}{n } \mathbb{e } \left [ \sum_{i = 0}^{n-1 } \big(b(y_i , y_{i+1 } ) \big)^2\right].$ ] if @xmath293 denotes the invariant measure for the random walk @xmath294 , by ergodicity , we get @xmath295=\mathbb{e}_\mu \left[\left(b(y_0 , y_1)\right)^2\right].\ ] ] under @xmath296 the distribution of @xmath297 is uniform on @xmath298 because @xmath299 is uniformly chosen among all neighbours of @xmath300 hence @xmath301 using that @xmath32 does not depend on @xmath302 we obtain @xmath303 = 0,\ ] ] and by cauchy schwarz inequality and @xmath304=0.\ ] ] so we get @xmath305,\ ] ] and by ergodicity , @xmath306.\ ] ] since , under @xmath307 the distribution of @xmath297 is uniform on @xmath298 @xmath308 by orthogonality of @xmath163 and @xmath153 , @xmath309 thus , it remains to compute @xmath310 ( since @xmath311 , by definition ) . at this point we use the fact that @xmath153 is a stationary field , in the sense of definition [ def : stat ] . then if we denote by @xmath312 ( where @xmath57 is in @xmath120-th position ) and , for @xmath313 , we have by @xmath314 now we compute @xmath310 as a function of @xmath315 , the value @xmath153 on the edge @xmath316 . by @xmath317 if @xmath318 and @xmath319 , then using the relation between @xmath320 and @xmath321 @xmath322 by stationarity of @xmath323 @xmath324 then , because of the definition of @xmath325 and @xmath326 @xmath327 then by stationarity of @xmath153 @xmath328 in the proof , the way we guessed is a bit mysterious . assuming that @xmath183 is a stationary gradient vector field , the family @xmath329}$ ] can be computed considering the system of equations given by @xmath330 , j_{m_i , n_i } = 0,\ ] ] where @xmath331 } \in v_k , a_i = 1\right\}$ ] and @xmath332 } \in v_k , a_i = -1\right\}$ ] . this leads to the following system : @xmath333 f = \left[\begin{matrix}3^{k-1 } \\ 3^{k-2 } \\ \vdots \\ 3 \\ 1\end{matrix } \right].\ ] ] the unique solution is given by : @xmath334 even if the guess is correct , we did not find another way as the techniques used in the section [ sec : hodge - a ] to show that @xmath183 is a stationary gradient vector field . in this final section we briefly outline some related problems . * it is possible to make sense of the process when @xmath5 is infinite . several questions arise : what happens to the process of one marked walker ? is there a scaling limit under equilibrium for the `` shape''process @xmath335 ? + another natural step would be to let @xmath5 grow with @xmath336 in a suitable way , so as to get a scaling limit for the two - parameter process @xmath337 . * one may also ask about different quantities , such as the diameter of the set of walkers under the invariant measure for the entire walk . * one may also consider random walkers conditioned on satisfying different shape constraints , and on graphs more general than @xmath3 . as a starting example , what happens if we work on a torus , i.e. if we force also @xmath338 ? the shape chain changes in this case and it is no longer irreducible over @xmath339 ( one may check that the number of @xmath57 symbols is fixed , and that this enumerates the recurrence classes ) . it is interesting to point out that this setup is the one chosen by e. lieb for the computation of the six - vertex constant in @xcite . we warmly thank charles bordenave for fruitful conversations and useful comments .
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we consider a random walk @xmath0 with the constraint that each coordinate of the walk is at distance one from the following one . in this paper , we show that this random walk is slowed down by a variance factor @xmath1 with respect to the case of the classical simple random walk without constraint .
_ keywords _ : random walk , graph , central limit theorem _ ams classification ( 2000 ) _ : 05c81 , 60f05 .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the pioneer 10/11 missions were the first spacecraft to explore the outer solar system @xcite . their objectives were to conduct , during the 1972 - 73 jovian opportunities , exploratory investigation beyond the orbit of mars of the interplanetary medium , the nature of the asteroid belt , and the environmental and atmospheric characteristics of jupiter and saturn ( for pioneer 11 ) . pioneer 10 was launched on 2 march 1972 on top of an atlas / centaur / te364 - 4 launch vehicle . the launch marked the first use of the atlas - centaur as a three - stage launch vehicle . the third stage was required to rocket pioneer 10 to the speed of 14.39 km / s needed for the flight to jupiter . its sister craft , pioneer 11 , was launched on 5 april 1973 , like pioneer 10 , also on top of an atlas / centaur / te364 - 4 launch vehicle . after safe passage through the asteroid belt on 19 april 1974 , pioneer 11 s thrusters were fired to add another 63.7 m / s to the spacecraft s velocity . this adjusted the aiming point at jupiter to 43,000 km above the cloudtops . the close approach also allowed the spacecraft to be accelerated by jupiter to a velocity of 48.06 km / s so that it would be carried across the solar system some 2.4 billion km to saturn . after the jupiter and saturn ( for pioneer 11 ) encounters ( see figure [ fig : pioneer_inner_path ] ) , the craft followed escape hyperbolic orbits near the plane of the ecliptic on opposite sides of the solar system , continuing their extended missions @xcite . ( see figure [ fig : pioneer_path ] . ) pioneer 10 explored the outer regions of the solar system , studying energetic particles from the sun ( solar wind ) , and cosmic rays entering our portion of the milky way . pioneer 10 continued to make valuable scientific investigations until its science mission ended on march 31 , 1997 . since that time , pioneer 10 s weak signal has been tracked by the nasa s deep space network ( dsn ) as part of an advanced concept study of communication technology in support of nasa s future interstellar probe mission . pioneer 10 eventually became the first man - made object to leave the solar system . the power source on pioneer 10 finally degraded to the point where the signal to earth dropped below the threshold for detection in its latest contact attempt on 7 february , 2003 . the previous three contacts had very faint signals with no telemetry received . the last telemetry data point was obtained from pioneer 10 on 27 april 2002 when the craft was 80 au from the sun . following its encounter with saturn , pioneer 11 explored the outer regions of our solar system , studying the solar wind and cosmic rays . pioneer 11 sent its last coherent doppler data on 1 october 1990 while at @xmath7 au from the sun . in september 1995 , pioneer 11 was at a distance of 6.5 billion km from earth . at that distance , it takes over 6 hours for the radio signal to reach earth . however , by september 1995 , pioneer 11 could no longer make any scientific observations . on 30 september 1995 , routine daily mission operations were stopped . intermittent contact continued until november 1995 , at which time the last communication with pioneer 11 took place . there has been no communication with pioneer 11 since . the earth s motion has carried our planet out of the view of the spacecraft antenna . the spacecraft can not be maneuvered to point back at the earth . it is not known whether the spacecraft is still transmitting a signal . no further tracks of pioneer 11 are scheduled . -10pt the pioneers were excellent craft for the purposes of precision celestial mechanics experiments @xcite . this was due to a combination of many factors , including their attitude control ( spin - stabilized , with a minimum number of attitude correction maneuvers using thrusters ) , power design ( the plutonium-238 powered heat - source rtgs radioisotope thermoelectric generators being on extended booms aided the stability of craft and also reduced thermal effects on the craft ) , and precise doppler tracking ( with the accuracy of post - fit doppler residuals at the level of mhz ) . the result was the most precise navigation in deep space to date @xcite . ( see figure [ fig : pio - craft ] for a design drawing of the spacecraft . ) because of the excellent health and navigational capabilities of pioneer 10 , nasa supported a proposal to initiate search for unmodeled accelerations in 1979 ( when the spacecraft was at a distance of some 20 au from the sun ) . the main motivation was the search for planet x. eventually both pioneers were used in the search for trans - neptunian objects ; their doppler tracking capabilities also yielded the first ever limits on low frequency gravitational radiation @xcite . the pioneer 10/11 spacecraft had exceptional `` built - in '' acceleration sensitivity naturally allowing them to reach the level of @xmath8 m / s@xmath2 . as indicated by their radiometric data received from heliocentric distances of 20 - 70 au , the accuracies of their orbit reconstructions were limited by a small , anomalous , doppler frequency drift @xcite . by 1980 , when pioneer 10 already had passed a distance of @xmath9 20 au from the sun and the acceleration contribution from solar radiation pressure on the craft ( away from the sun ) had decreased to less than @xmath10 m / s@xmath2 , the radiometric data started to show presence of the anomalous acceleration ( towards the sun ) . this blue - shifted drift is uniformly changing with a rate of @xmath11 hz / s , which can be interpreted as a sunward constant acceleration of @xmath12 m / s@xmath2 @xcite . the detailed study of this anomaly @xcite led to a better understanding of its properties ( summarized in section [ sec : anomaly ] ) ; however , the nature of this anomalous signal is still unknown . in this paper we are going to discuss recently recovered set of pioneer data and objectives for upcoming new investigation of the anomalous residuals detected in the radiometric doppler data of the pioneers 10 and 11 . we continue the paper in section [ sec : anomaly ] by discussing the pioneer anomaly and its known properties . we briefly review the original efforts to understand the signal as well as some recent proposals to explain the pioneer anomaly . section [ sec : pio - doppler ] presents the current status of the effort to retrieve the entire set of the pioneer radiometric doppler data . section [ sec : mdr ] reports on the newly recovered pioneer telemetry data in the form of master data records ( mdrs ) . because of their potentially high value in the investigation of the pioneer anomaly , we discuss the purpose , means of delivery , storage , processing and possible use of the mdrs . in section [ sec : objectives ] we present objectives for the upcoming study of the pioneer anomaly and particularly the use of both the extended set of doppler data and the mdrs to study the on - board systematics , including the development of an accurate thermal - electric - dynamical model . in section [ sec : conclude ] we conclude by presenting the next steps for the analysis of the pioneer anomaly . the analysis of the pioneer 10 and 11 data ( with some support from the ulysses and galileo spacecraft ) @xcite demonstrated the presence of an anomalous , doppler frequency blue shift drift , uniformly changing with a rate of @xmath13 hz / s @xcite . to understand the phenomenology of the effect , consider @xmath14 , the frequency of the retransmitted signal observed by a dsn antenna , and @xmath15 , the predicted frequency of that signal modeling conventional forces influencing the spacecraft s motion including both gravitational and non - gravitational in their origin . the observed , two - way ( round trip ) anomalous effect can be expressed to first order in @xmath16 as @xmath17 = -2\dot{f}_p\,t.$ ] after accounting for the gravitational and other large forces included in standard orbit determination programs this translates to @xmath18 _ { \tt dsn } = -f_{0}\frac{2a_p~t}{c}. \label{eq : delta_nu_syst}\end{aligned}\ ] ] here @xmath19 is the dsn reference frequency @xcite ( see discussion of the dsn sign convention in ref . [ 38 ] of @xcite ) . after accounting for all _ known _ ( not modeled ) sources of systematic error ( discussed in @xcite ) , the conclusion was reached that there exists an anomalous sunward constant acceleration signal of @xmath20 the nature of this anomaly remains unexplained . this signal has become known as the pioneer anomaly . note that there exist a dualism in interpreting the radiometric doppler data . the anomaly can be due to a force acting on the craft that produces acceleration eq . ( [ eq : ap ] ) , or , alternatively , this signal can also be interpreted as a time deceleration uniformly changing with rate @xmath3 s / s@xmath2 ( see discussion of this possibility in @xcite ) . by now several studies of the pioneer doppler navigational data have demonstrated that the anomaly is unambiguously present for both pioneer 10 and 11 . these studies were performed with four independent ( and different ! ) navigational computer programs @xcite , namely : * the jpl s orbit determination program ( odp ) developed in 1980 - 2005 , * the aerospace corporation s chasmp code extended for deep space navigation @xcite , * code written at the goddard space flight center @xcite that was used to analyze pioneer 10 data for the period 1987 - 1994 obtained from the national space science data center ( http://nssdc.gsfc.nasa.gov/ ) , and finally * code developed at the institute of theoretical astrophysics , university of oslo , norway , that was recently successfully used to analyze the set of the pioneer 10 data above @xcite . for the most detailed analysis of the pioneer anomaly to date , @xcite used the following pioneer 10/11 doppler data : * pioneer 10 : the data set had 20,055 data points obtained between 3 january 1987 and 22 july 1998 and covering heliocentric distances @xmath21 au . * pioneer 11 : the data set had 19,616 data points obtained between 5 january 1987 to 1 october 1990 and covering heliocentric distances @xmath22 au . the recent analyses of the pioneer 10 and 11 radiometric data @xcite have established the following basic properties of the pioneer anomaly : * _ distance : _ it is unclear how far out the anomaly goes , but the pioneer 10 data supports its presence at distances up to @xmath970 au from the sun . the pioneer 11 data shows the presence of the anomaly as close in as @xmath920 au . * _ direction : _ from typical angular uncertainty of doppler navigation at s - band and spacecraft high gain antennae pointing accuracy set within 3 db gain bandwidth , @xmath23 behaves as a line - of - sight constant acceleration of the craft generally pointing in the innermost region of the solar system . * _ constancy : _ both temporal and spatial variations of the anomaly s magnitude are of order 10% for each craft , while formal errors are significantly smaller . there are other pieces of information obtained from spot analyses @xcite ; they indicate that : * the anomalous acceleration was present in the pioneer 11 data at shorter distances , as close in as @xmath24 au ( see figures 6 and 7 in @xcite ) . * the pioneer 11 data also indicated that the anomaly may be much smaller at distances @xmath25 au . it appears to be amplified ( or turned on ) at a distance of @xmath24 au from the sun . this is approximately when the craft flew by saturn and entered an hyperbolic , escape trajectory . this information was used as guidance in investigating the applicability of proposals to explain the pioneer anomaly using both conventional and `` new '' physical mechanisms . in the next section we briefly review these proposals . initial announcement of the anomalous acceleration ( e.g. @xcite ) triggered many proposals that invoked various conventional physics mechanisms , all aimed at explaining the origin of the anomaly . finding a systematic origin of the proper magnitude and behavior was the main focus of these proposals , which is yet to be found . although the most obvious explanation would be that there is a systematic origin to the effect , perhaps generated by the spacecraft themselves from anisotropic heat rejection or propulsive gas leaks , the analysis performed did not find evidence for either of them ; that is , no unambiguous , on - board systematic has been discovered . this initial search was summarized in @xcite , where possible contributions of various mechanisms to the final solution for @xmath23 were given . the entire error budget was subdivided in three main types of effects , namely effects due to sources external to the spacecraft ; the contribution of on - board systematics , and computational systematic errors . in this section , we present a summary of this earlier work . the first group of effects includes those external to the spacecraft , such as the solar radiation pressure , effects of the solar wind , and the effect of the solar corona on the propagation of radio wave signals . @xcite also discuss the influences of the kuiper belt s gravity , galactic gravity , and electromagnetic lorentz forces . errors in the accepted values of the earth s orientation parameters , precession , and nutation are included in the discussion . the analysis evaluated the contributions of the mechanical instabilities and the location errors of the dsn antenna structures , the phase stabilities of the dsn antennae and clocks , and effects due to the troposphere and ionosphere . even though some of these mechanisms are near the limit for contributing to the final error budget , it was found that none of them could explain the behavior of detected signal . moreover , some were three orders of magnitude or more too small . in totality , they were insignificant . the second group of effects includes those that originated on - board and are tied to well - known sources ; this group , as expected , had the largest impact on the final error . among these effects , the radio beam reaction force produced the largest bias to the result , @xmath26 m / s@xmath2 . as the force exerted by the radio beam necessarily points away from the earth ( and thus from the sun ) , the correction of the measured data increases the amount of the observed anomalous attractive force and makes the pioneer effect larger . large uncertainties also came from a conjectured differential emissivity of the radioisotope thermoelectric generators ( rtgs ) , radiative cooling of the spacecraft , and propulsive gas leaks from thrusters of the attitude control system : @xmath27 m / s@xmath2 , @xmath28 m / s@xmath2 , and @xmath29 m / s@xmath2 , respectively . the effect due to expelled helium produced within the rtgs was also considered , as well as the small difference in anomaly determinations between the two pioneers . the effect of rejected thermal radiation was the second largest bias / uncertainty that has been the most critical systematic bias to quantify . if heat generated by the on - board power sources was asymmetrically reflected by the body of the craft , a fore / aft acceleration could be produced causing the measured anomaly . the pioneer spacecraft were powered by snap-19 rtgs ( space nuclear ancillary power ) mounted on long extended booms ( designed to protect the on - board electronics from heat and radiation impact ) @xcite . in principle , there was more than enough heat available on the craft to cause the anomaly . however , the spacecraft s spin - stabilized attitude control , special design of the rtgs and the length of the rtg booms that resulted in a relatively small spacecraft surface available for the preferential heat rejection significantly minimized the amount of heat for the mechanism to work . the analysis of the 11.5 years of pioneer doppler data @xcite can only support an effect as large as @xmath30 m / s@xmath2 . in summary , although this group represents conceptually the most likely sources for the anomaly , these mechanisms did not gain enough experimental support . at most one can obtain @xmath912% of the discovered effect by employing all of these mechanisms . the possibility exists that a combination of the factors above could amount to the measured effect @xcite , but again , the shortness of the data interval used in the analysis , actual spacecraft design and performance data at hand , and also the complexities of modeling thermal radiative processes on the craft made it difficult to unambiguously support this claim @xcite . the third group of effects was composed of contributions from computational errors . the effects in this group dealt with the numerical stability of least - squares estimations , accuracy of consistency / model tests , mismodeling of maneuvers , and the solar corona model used to describe the propagation of radio waves . it has also been demonstrated that the influence of annual / diurnal terms seen in the data on the accuracy of the estimates was all small @xcite . the total uncorrelated error associated with computational systematics is estimated to be less than @xmath31 m / s@xmath2 . these three groups of effects exhausted all available conventional explanations for the anomaly . the inability to explain the pioneer anomaly with conventional physics has led to a significant number of theoretical proposals that use more unusual mechanisms ( more details are in @xcite ) . the pioneer anomaly is an effect at the limit of what is detectable with radiometric tracking of a deep space probe , but it is huge in physical terms : the anomaly exceeds the corrections to newtonian motion predicted by general relativity by five orders of magnitude ( at 50 au ) . hence , if the effect is not a result of conventional systematics it would have a considerable impact on our models of fundamental forces , regardless of the anomaly being due to a deceleration of the spacecraft or a blue shift of the radio signal . attempts to verify the anomaly using other spacecraft proved disappointing . this is because the voyager , galileo , ulysses , and cassini spacecraft navigation data all have their own individual difficulties for use in an independent test of the anomaly @xcite . in addition , many of the deep space missions that are currently being considered either may not provide the needed navigational accuracy and trajectory stability sensitive to accelerations of under @xmath32 m / s@xmath2 or else they have significant on - board systematics that mask the anomaly . a requirement to have an escape hyperbolic trajectory makes a number of other missions @xcite less able to directly test @xmath23 . although these missions all have excellent scientific goals and technologies , nevertheless , their orbits lend them a less advantageous position to conduct a precise test of the detected anomaly . a number of alternative ground - based verifications of the anomaly have also been considered ; for example , using very long baseline interferometry ( vlbi ) astrometric observations . however , the trajectories of spacecraft like the pioneers , with small proper motions in the sky , make it presently impossible to use vlbi in accurately isolating an anomalous sunward acceleration of the size of @xmath23 . efforts to explain the anomaly were originally focused on conventional physics mechanisms generated on - board , such as gas leaks from the propulsion system or a recoil force due to the on - board thermal power inventory . so far , these mechanisms have been found to be either not strong enough to explain the magnitude of the anomaly or else to exhibit significant temporal or spatial variations contradicting the known properties of the anomaly presented above @xcite . a number of other conventional physics possibilities have also been addressed . in particular , it has been proposed that kuiper belt objects or dust could explain the anomaly by a gravitational acceleration , an additional drag force ( resistance ) and a frequency shift of the radio signals proportional to the distance . of course , one of the most natural mechanisms to generate a putative physical force is the gravitational attraction due to a known mass distribution in the outer solar system ; for instance , due to kuiper belt objects or dust . however , possible density distributions for the kuiper belt were studied in @xcite and found to be incompatible with the discovered properties of the anomaly . even worse , these distributions can not circumvent the constraint from the undisturbed orbits of mars and jupiter @xcite . the density of dust is not large enough to produce a gravitational acceleration on the order of @xmath23 @xcite and also it varies greatly within the kuiper belt , precluding any constant acceleration . hence , a gravitational attraction by the kuiper belt can , to a large extent , be ruled out . also , the data from the inner parts of the solar system taken by the pioneer 10/11 dust detectors favors a spherical distribution of dust over a disk in this inner region . ulysses and galileo measurements in the inner solar system find very few dust grains in the @xmath33 kg range @xcite . ir observations rule out more than 0.3 earth mass from kuiper belt dust in the trans - neptunian region . the resistance caused by the interplanetary dust is too small to provide support for the anomaly @xcite . any dust - induced frequency shift of the carrier signal is also ruled out . finally we note that , motivated by the numerical coincidence @xmath34 , where @xmath35 the speed of light and @xmath36 is the hubble constant at the present time , there have been many attempts to explain the anomaly in terms of the expansion of the universe . @xcite showed that such a mechanism would produce an opposite sign for the effect . a study of the effect of cosmic acceleration on the radio signals rather than on the spacecraft themselves was also undertaken . this mechanism might be able to overcome the apparent conflict that @xmath23 presents to modern solar system planetary ephemerides @xcite . the apparent difficulty to explain the anomaly within standard physics became a motivation to look for `` new physics . '' so far these attempts have not produced a clearly viable mechanism for the anomaly . in particular , the physics of mond ( modified newtonian dynamics ) represents an interesting possibility with its phenomenological long - range modification of gravity invoked to explain the rotation curves of galaxies @xcite . however , the numerical value for the mondian acceleration @xmath37 is almost an order of magnitude smaller then @xmath23 and is not likely to be observed on the scales of the solar system . there is also an attempt to explain the anomaly in the framework of a nonsymmetric gravitational theory @xcite . it has been argued that a skew - symmetric field with a suitable potential could account for @xmath23 @xcite as well as galaxy and cluster rotation curves . a modification of the gravitational field equations for a metric gravity field , by introducing a general linear relation between the einstein tensor and the energy - momentum tensor has also been claimed to account for @xmath23 @xcite . both these approaches are currently being further investigated . various distributions of dark matter in the solar system have been proposed to explain the anomaly , e.g. , dark matter distributed in the form of a disk in the outer solar system with of a density of @xmath38 kg / m@xmath39 , yielding the wanted effect . dark matter in the form of `` mirror matter '' @xcite is one example . however , it would have to be a special smooth distribution of dark matter that is not gravitationally modulated as normal matter so obviously is . @xcite have shown that a generic scalar field can not explain @xmath23 ; on the other hand they proposed that a non - uniformly coupled scalar might produce the wanted effect . although braneworld models with large extra dimensions offer a richer phenomenology than standard scalar - tensor theories , it is difficult to find a convincing explanation for the pioneer anomaly @xcite . other ideas include yukawa - like or higher order corrections to the newtonian potential and a theory of conformal gravity with dynamical mass generation , including the higgs scalar ( see discussion in @xcite ) . we must conclude that there are many interesting ideas proposed to explain the physics of the pioneer anomaly . however , most of them need more work before they can be considered to be viable . to summarize , the origin of the pioneer anomaly remains unclear . @xcite have advocated for an analysis of the entire set of existing pioneer doppler data , obtained from launch to the last useful data received from the vehicles . this data could yield critical new information about the anomaly , especially during the earlier mission phases @xcite . this led to the initiation of an effort to retrieve the early pioneer data that until recently existed in several places on various obsolete format and media ( e.g. , 9-track and even 7-track magnetic tapes ) . recovery of radiometric data for a mission operating for more then 30 years is an effort that was never attempted before . indeed , 30 years is a long time , presenting many unique challenges , including changes in the data formats , navigational software , as well as supporting hardware . even the dsn configuration had changes new stations were built , and some stations moved , upgraded and reassigned . the main asset of the entire mission support its people changed the most . by 2005 all the dsn data formats , navigational software used to support pioneers , all the hardware used to read , write and maintain the data have become obsolete and are no longer operationally supported by existing nasa protocols . despite the anticipated complexities , the transfer of the available pioneer doppler data to modern media formats had been initiated at jpl in early june 2005 and , as of december 2005 , it is nearing completion . in this section we discuss the current status of this effort to recover the pioneer 10/11 radiometric doppler data . doppler data is the measure of the cumulative number of cycles of a spacecraft s carrier frequency received during a user - specified count interval . the exact precision to which these measurements can be carried out is a function of the received signal strength and station electronics , but it is a small fraction of a cycle . raw doppler data is generated at the tracking station and delivered via a dsn interface to customers . in order to acquire doppler data , the user must provide a reference trajectory and information concerning the spacecraft s rf system to jpl s deep space mission system ( dsms ) , to allow for the generation of pointing and frequency predictions . the user specified count interval can vary from 0.1 sec to 10 minutes , with count times of 10 to 60 seconds being typical @xcite . the average rate of change of the cycle count over the count interval expresses a measurement of the average velocity of the spacecraft in the line between the antenna and the spacecraft . the accuracy of doppler data is quoted in terms of how accurate this velocity measurement is over a 60 second count . the accuracy of data improves as the square root of the count interval . -12pt 1 . _ non - coherent doppler data _ ( also known as one - way or f1 data ) is data received from a spacecraft where the downlink carrier frequency is not based on an uplink signal . the ability of the tracking station to measure the phase of the received signal is the same for non - coherent versus coherent data types , however the uncertainty in the value of the reference frequency used to generate the carrier is generally the dominant error source . pioneers provided a significant amount of f1 data that is unfortunately not useful for precision orbit determination . coherent doppler data _ is received from a spacecraft when the reference frequency of the received carrier signal was based on a transmitted uplink signal from the earth . this is commonly known as two - way ( or f2 ) data , when the receiving and transmitting ground stations are the same , and three - way ( or f3 ) data , when the transmitting and receiving stations are different . since the frequency of the original source signal is known , this error source does not affect data accuracy . the accuracy of this data is a function primarily of the carrier frequency , but is affected by transmission media effects . the f2 and f3 data are the primary source for the pioneer 10 and 11 orbit determination and were the primary focus of the data recovery efforts . the pioneers used s - band ( @xmath92.2 ghz ) radio signals to communicate with the nasa deep space network ( dsn ) . the s - band data is available from 26 m , 70 m , and some 34 m antennas of the dsn complex ( see baseline dsn configuration in the figure [ fig : pio - radio - science ] ) . the 1-@xmath40 accuracy of s - band data is approximately 1 mm / s for a 60 second count interval after being calibrated for transmission media effects . the dominant systematic error that can affect s - band tracking data is ionospheric transmission delays . when the spacecraft is located angularly close to the sun , with sun - earth - spacecraft angles of less than 10 degrees , degradation of the data accuracy will occur . s - band data is generally unusable for sun - earth - spacecraft angles of less than 5 degrees . ( see more details on the modern capabilities of the dsn , especially its radio science performance for extracting precision doppler signal demonstrated in @xcite . ) -10pt the pioneer radiometric data was received by the dsn in `` closed - loop '' mode , i.e. , it was tracked with phase lock loop hardware . ( `` open loop '' data is recorded to tape but not tracked by phase lock loop hardware . ) there are basically two types of data : doppler ( frequency ) and range ( time of flight ) , recorded at the tracking sites of the dsn as a function of ut ground received time @xcite . during their missions , the raw radiometric tracking data from pioneers were received originally in the form of intermediate data record ( idr ) tapes , which were then processed into special binary files called atdfs ( archival tracking data files , see description in @xcite ) , containing doppler data from the standard dsn tracking receivers ( see figure [ fig : pio - data - format - flow ] ) . note that the `` closed - loop '' data constitutes the atdfs that were used in @xcite . after a standard processing at the rmdc ( radio metric data conditioning group ) of jpl s navigation and mission design section , atdfs are transformed into odfs ( orbit data files , see description in @xcite ) . a program called stripper is used to produce the odfs that are , at this point , the main product that is distributed to the end users for their orbit determination needs ( see more discussion on the conversion process in @xcite ) . at jpl , after yet additional processing , these odfs are used to produce sequentially formatted input / output files in navio format that is used by navigators while working with the jpl orbit determination program . ( note that the navio input / output file format is used only at jpl ; other orbit determination programs convert odfs to their particular formats . ) atdfs are files of radiometric data produced by the network operations control center ( nocc ) navigation subsystem ( nav ) ( see figure [ fig : pio - radio - science ] ) . they are derived from intermediate data records by nav and contain all radiometric measurements received from the dsn station including signal levels ( agc = automatic gain control in dbm ) , antenna pointing angles , frequency ( often referred to simply as `` doppler '' ) , range , and residuals . doppler data is often used to infer spacecraft radial motion relative to the tracking antenna . data values in atdfs are reported at rates no higher than 10 per second . during their missions , the pioneers received frequencies at s - band were recorded at a nominal sample time of one per either 10 , 60 , 300 , 600 or 1980 seconds . in addition , for several years from 1980 to 1994 there were tracking passes scheduled for the jpl - led gravity wave experiments that delivered data at a smaller count time of 1 per second . each atdf consists of all tracking data types used to navigate a particular spacecraft ( pioneers had only doppler data type ) and typically include doppler , range and angular types ( in s- , x- , and l - frequency bands ) , differenced range versus integrated doppler , programmed frequency data , pseudo - residuals , and validation data . ( unfortunately there was no range capability implemented on the pioneers @xcite . early in the mission , jpl successfully simulated range data using modulation of sub - carrier signal and some range data is available , especially for planetary flybys . later on , this technique of simulating range did was limited by the bandwidth of the sub - carrier signal and did not produce any range data . ) also , atdfs contain data for a single spacecraft , for one or more ground receiving stations , and for one or more tracking passes or days . the atdf is described in section trk-2 - 25 of @xcite . each atdf data record contains 117 parameters , stored in records of 288 bytes . each atdf physical record is 2016 32-bit words in length ( 8064 bytes ) and consists of 28 72-word ( 288-byte ) logical records . each atdf contains an integer number of these blocks . the tracking data table format and content used until early 1997 are described by @xcite . the atdf records are arranged in a sequence that consists of one file identification record , one transponder logical record , tracking data logical records in time order , and software / hardware end - of - file markers . bit lengths of data fields are variable and cross word boundaries . ( further details of the atdf content and format are given in @xcite . ) each file contains 15 columns per data point specifying a time - tag , s - band , and x - band receiver parameters ( the latter not applicable for the pioneers . ) after standard processing at jpl s rmdc with the use of the stripper program ( see @xcite for more details ) , atdfs are transformed into odfs for use in determining spacecraft trajectories , gravity fields affecting them , and radio propagation conditions . odfs contain radiometric data that has been converted from the atdf format @xcite . each odf physical record is 2016 32-bit words in length and consists of 224 9-word logical records per data block . the odf records are arranged in a sequence that consists of one file label record , one file identifier logical record , orbit data logical records in time order , ramp data logical records in time order , clock offset data logical records in time order , data summary logical records in time order and software / hardware end - of - file markers . bit lengths of data fields are variable and cross word boundaries . an odf usually contains most types of records , but may not have them all . the first record in each of the 7 primary groups is a header record ; depending on the group , there may be from zero to many data records following each header . the first task was to identify the data storage facilities with potential inventory of the pioneer doppler data . in general , we expected to find data in four different locations , namely : 1 . archive of radiometric data at rmdc at jpl : our hope was that radiometric data of pioneers 10 and 11 ( in all formats , namely idr , atdf , and odf ) may still exist at the various data storage archives at jpl and especially at the rmdc archive . furthermore , we expected to find data in all possible media formats , including 7-track and 9-track magnetic tapes and also digital records on the rdc system . the next logical place was the national space science data center ( nssdc ) at the nasa goddard space flight center , http://nssdc.gsfc.nasa.gov/planetary/pioneer10-11.html . we knew that nssdc had copies of atdfs on magnetic tapes submitted there by john d. anderson of jpl , he being the principal investigator on the celestial mechanics experiments , as part of his agreement with nasa . the data coverage was expected to be from 1978 to 1994 . in addition to that , we knew that george null of jpl also submitted pioneer doppler data to nssdc prior to 1978 . it was also know that , in addition to raw doppler data , he sent some orbital solutions and auxiliary information ( maneuvers , spin rates , initial conditions , etc . ) that also would be of interest to us . we also expected that nasa ames research center ( arc ) , as the center with the project management responsibilities for all the pioneer missions , would have some information useful for our investigation . we later realized that no raw orbital information was provided to arc ; instead , the trajectory solutions derived by jpl were used for all project needs . 4 . several facilities throughout the country were used to provide overflow storage for magnetic tapes in the 1970s and 1980s . two of these federal data storage facilities are located in california : in long beach and in palo alto . we have not yet tried to approach these data archival centers , but plan to do this in the near future . all the sources of radiometric data listed above were used to recover the pioneer doppler data . this data was present in different media formats ( magnetic tapes , floptical disks , digital formats on various computer platforms with different hard drive formatting standards ) and were written in different data formats ( i.e. idr , atdf and odf ) . the goal was to recover as much of this data as possible . the next section summarizes the results of our retrieval efforts . to recover pioneer radiometric data we first went to the nssdc with a request to provide us with copies of all available navigational data ( idr , atdf , and odf ) for the period 1978 - 1994 that was previously archived at nssdc ( done by j.d . anderson and e.l . lau of jpl ) . the nssdc staff loaded the jpl - supplied idr and atdf tapes on their vms and unix machines . however , we realized that all the atdf data files that we were receiving from nssdc were corrupted . the problem was introduced by the process that was used to transfer the data across multiple computer platforms : an additional byte was inserted at record boundaries . in other words , every 8065th byte is a record marker and is not part of the atdf stream . this problem was first encountered by c. markwardt during his study of the pioneer anomaly @xcite . fortunately , we were able to recognize the problem early on and were able to fix most of the corrupted files following the procedure suggested by @xcite . when jpl retrieved back the data using ftp , every 8065th byte was removed from the each file record . the resulting data is good to use . another data segment retrieved from the nssdc corresponded to the very beginning of the pioneer missions , namely the period @xmath91972 - 1976 ( files were archived by g. null of jpl ) . unfortunately , it appears that not only are these data files corrupted , but they were also written in a format called `` type-66 '' ( or t66 ) which has not been used at jpl for more than 25 years , which adds unanticipated delay . this is why we are still in the process of trying to recover some segments in the hope of adding this data to the new data set . ( note that we were not able to recover any odf data files from nssdc due to unknown / unidentified data formats . ) -2pt -5pt a significant volume of data ( both idr and atdf ) was found at rmdc on 9-track magnetic tapes ( 1978 - 1988 ) that were read on a still available minivax computer . this segment of the data is in a good condition and is ready to use . the retrieval of idr data was reasonably fast as each tape contained only one file . however , the atdf 9-track tape had several atdfs per tape ( usually 10 - 15 files ) . this increased the time needed to read a single tape on the minivax computer . with regular cleaning of the head of the magnetic tape reader and also cleaning the tapes themselves it usually took almost 90 minutes to read a single tape on this hardware , which was built in the 1980s . with an inventory of more than 200 tapes this was a major effort during the summer of 2005 . there were a few additional and unexpected finds of pioneer data at jpl . several historical atdfs for both pioneer 10 and 11 were found and recovered from the rmdc s archival optical disks . also at rmdc , we found spr ( system performance record ) data ( format trk-2 - 15a ) . the raw spr data format was used to replace the idr format , and it was utilized after 1985 when jpl / nav upgraded the system from univac to dec / vms hardware . we were fortunate to recover several additional historical atdfs for both pioneers from the rmdc s system archive tapes ( sat ) ( written on 9-track and 4 mm tapes ) . these files were also added to the overall file inventory . the resulting atdf and odf retrieval methods are shown in figure [ fig : recovery - flow - chart ] . in addition to the data already recovered , we found and recovered some early odf data ( 1973 - 1974 ) that was still available in the personal data archives of our jpl colleagues who worked with pioneer data for other purposes ( e.g. , development of the solar system ephemerides ) . in fact , we were able to recover several odf files written in the navio input / output form . these files , obtained from several jpl users , were re - converted back to the odf data format . finally , we have added the data that was already analyzed by @xcite that corresponds to the period of 1987 - 1998 ( for pioneer 10 ) and 1987 - 1990 ( for pioneer 11 ) . lastly , we added the most recent data for the period 1998 - 2002 . the resulting 1972 - 2002 data has some redundancy , but mostly it is a very complete data set assembled for the first time . after necessary certification at jpl this same set will be available for distribution . the currently available expanded data set includes the following data segments for each spacecraft : * pioneer 10 : the entire available data set covers mission events from mid-1973 ( including jupiter encounter data ) to the last time a pioneer 10 contact returned telemetry data , 27 april 2002 . this interval spanned heliocentric distances from @xmath41 au to 87 au . the total @xmath930-year pioneer 10 data set might have @xmath960,000 data points and is about 20 gb in size . we also have most of the information on maneuvers , spin rate , and initial conditions . * pioneer 11 : the entire available data set covers the mission from mid-1974 to late 1994 ( including both jupiter and saturn encounters ) . this interval spanned heliocentric distances from @xmath94 au to 33 au . the total @xmath920-year pioneer 11 data set might have @xmath950,000 data points and is about 15 gb in size . we also have most of the information on maneuvers , spin rate , and initial conditions . to summarize , there exists @xmath930 years of pioneer 10 and @xmath920 years of pioneer 11 data , most of which had never been well studied for our purposes . we are still trying to increase the percentage of the f2 and f3 doppler data in the overal retreived doppler dataset , as f1 data is useless for our analysis . for this we are working together with rmdc in an attempt to improve the output of the atdf - to - odf file conversion effort , which we expect to complete by january 2006 . by the same time we also hope to close the t66 issue and prepare all input information critical for the upcoming re - processing of the doppler data for the entire pioneer 10 and 11 trajectories . to re - process the complete pioneer trajectories , one would first have to ( re)edit and ( re)process the entire data span ( from 1972 to 2002 ) , using the same editing strategy , initial conditions , and parameter estimation and noise propagation algorithms . one could also process the high rate doppler data ( i.e. , 1 record per second ) that previously was used very little . the high rate data can be used to better determine a spacecraft s spin rate and also to improve the maneuver data file information . also , the spin rate change was found to be highly correlated with a small but significant spacecraft - generated force , probably from gas leaks @xcite . therefore , one can also use the high - rate data to estimate and/or calibrate valve gas leaks and all the maneuvers . note that the use of the spacecraft telemetry data may be critical in addressing this issue ( see discussion of this possibility in section [ sec : mass - expulsion ] ) . since the previous analysis @xcite , physical models for the earth s interior and the planetary ephemeris have greatly improved . this is due to progress in gps- , slr- , llr- and vlbi - enabling technologies , doppler spacecraft tracking , and new radio science data processing algorithms . one would have to write and/or update existing orbit determination programs using these latest earth models ( adopted by the iers ) and also using the latest planetary ephemeris . this will improve the solutions for the dsn ground station locations by two orders of magnitude ( 1 cm ) over that of the previous analysis . additionally , this will allow a better characterization of not only the constant part of any anomalous acceleration , but also of the annual and diurnal terms detected in the pioneer 10 and 11 doppler residuals @xcite . as pointed out above , we were able to recover almost the entire set of doppler data obtained from both pioneers 10 and 11 . we were also able to obtain the entire archive of the spacecraft telemetry data in a form called master data records ( mdr ) for both craft . because of their potentially high value in the investigation of the pioneer anomaly , in this section we discuss the purpose , means of delivery , storage , processing and possible use of the mdrs . all data received from the pioneer 10 and 11 spacecraft by the nasa deep space network ( dsn ) was initially stored in the form of mdrs . these records contained information about the dsn station and reception characteristics , in addition to the actual , `` raw '' data records themselves , which contained both scientific measurements and engineering telemetry , as received from the spacecraft . ( as an example , figure [ fig : sensors ] shows locations of several thermal sensors in the spacecraft instrument compartment . ) -10pt -10pt normally , mdrs are seen to be of little use once the relevant information is extracted from them . scientific measurements are extracted , packaged in the appropriate form , and sent to the corresponding experimenters for further processing and evaluation . engineering telemetry is used by the spacecraft operations team to control and guide the spacecraft . mdrs are not necessarily considered worth preserving once the scientific data has been extracted , and the engineering telemetry is no longer needed for spacecraft operations . indeed , the mdr retention schedule prescribed that the tapes be destroyed after 7 years . fortunately , most of the mdrs for the pioneer 10 and 11 projects have been preserved nevertheless . long before we ( lrk and vtt ) heard about the pioneer anomaly investigation , we recognized the potential value of the mdrs for educational purposes ; we envisioned an educational web site where a visitor could observe the pioneers as they traveled through the solar system , while viewing telemetry information that showed the health status and operational characteristics of the two spacecraft . for this reason , lrk endeavored to preserve the complete set of mdrs for the two projects and copied the data to modern media . meanwhile , vtt began developing pc - based tools for processing the mdrs and extracting information . far beyond the original expectations , this effort is now seen to be of value for the investigation of the pioneer anomaly , as the mdrs , specifically the telemetry data contained therein , may help in the construction of an accurate model of the spacecraft during their decades long journey , including a precise thermal profile , the time history of propulsion system activation and usage , and many other potential sources of on - board disturbances . in the early days of pioneer operations , the original mdrs for the pioneer 10 and 11 missions were copied to magnetic tape at the dsn receiving stations , and then sent to the appropriate nasa office for further processing . after more than 20 years of storage , the medium started to show signs of deterioration and all of the recoverable raw data was copied to 128 megabyte magneto - optical ( `` floptical '' ) disks . these disks were readable by a microvax using a digital corporation rwz21 magneto optical scsi disk drive . digital corporation is now part of compaq , the available vax 4000 - 300 computer will soon cease to exist , and the scsi rwz21 drives will not be usable to read these `` floptical '' disks . at present , a near complete set of mdr `` floptical '' disks still exists , along with equipment capable of reading this media . during the course of 2004 and 2005 , kellogg has copied the files from `` floptical '' disks to a modern computer , and also created dvd - roms for distribution . the contents of the available mdr disks can be summarized as follows : pioneer 10 : 155 disks x 128 mb ( 16.33 gb ) + pioneer 11 : 217 disks x 128 mb ( 23.01 gb ) all mdr files are named using a common naming convention . each file contains the mdrs for one spacecraft for an entire day . the file name begins with the letter ` m ' , followed by a two - digit spacecraft identifier ( 23 for pioneer 10 , 24 for pioneer 11 ) , and a 5-digit date ( 2 digits for the year , 3-digit doy . ) the filename extension is .mdr . thus , m2495003.mdr , for instance , would be the name of the pioneer 11 file for january 3 , 1995 . when kellogg copied the files to modern computers , he established a separate directory for each `` floptical '' disk copied . he also used a consistent naming convention : the directory name began with the two - digit spacecraft identifier , followed by the letter ` p ' , a two - digit year value , and a two - digit sequence number that simply identified the number of the disk for the given year . so for instance , 23p9602 would be disk 2 for pioneer 10 , for the year 1996 . the total amount of data stored in these mdr files is approximately 40 gb . according to the original log sheets that record the transcription from tape to `` floptical '' media , only a few days worth of data is missing , some due to magnetic tape damage . one notable exception is the `` jupiter encounter period '' of pioneer 10 . according to the transcription log sheets , doy 332 - 341 from 1973 were not available at the time the `` floptical '' disks were made . kellogg s investigation revealed that the tapes in question may have been misplaced long before the transcription has taken place . unfortunately , it is unlikely that those tapes will ever be located , and even if they re found , chances are that they will no longer be readable due to media deterioration . other significant periods of missing data are listed in table [ tb : missing_mdr ] . it is not known why these records are not present , except that we know that very few days are missing due to unreadable media ( i.e. , the cause is missing , not damaged , tapes . ) -12pt .pioneer 10/11 missing mdrs ( periods of missing data shorter than 1 - 2 days not shown . ) [ cols="<,<,<",options="header " , ] [ tb : power ] -10pt to utilize this data for our investigation , one would have to reconstruct the direction of heat flow : absorption and re - emission by , and reflection off the craft surfaces . we see this as the main challenge of the upcoming analysis . how to convincingly use a reading at several points on the craft to reconstruct the heat dynamics ? ( similar questions exist for gas leaks and all other parameters . ) if we do nt see their signature in the doppler signal , we should be able to put an upper limit on the potential contribution from these sources to the pioneer anomaly . similarly , the temperature data in the mdrs must put an upper limit on the changes of the optical surfaces of the rtgs ( i.e. their emissivity and reflectivity ) . because of the complexities involved , clearly more resources are needed ; we invite researchers to join us for this part of the upcoming study . another suggestion related to the rtgs is based on the idea that during the early parts of the missions , there might have been a differential change of the radiant emissivity of the sun - pointing sides of the rtgs as compared to the sides facing deep space . note that , especially closer to the sun , the inner sides were subjected to the solar wind . on the other hand , the outer sides were sweeping through the solar system dust cloud . these two processes could result in different levels of degradation of the optical surfaces of the rtgs . in turn this degradation could result in asymmetric patterns of heat radiation away from the rtgs in the fore / aft directions along the spin axis . therefore , it can be argued that such an anisotropy may have caused the anomaly . using the actual design information on the rtgs and a sophisticated model of the fin structure , @xcite estimate an upper limit of 6.12w to be the uncertainty in the contribution from the differential emissivity of the rtgs to the anomalous acceleration of the craft . this value , in turn , results in an acceleration uncertainty of @xmath42 the authors also point out the significance of radioactive decay for this mechanism and especially its expected contribution to the decrease of the magnitude of @xmath23 , which has not been observed . the effect of the radioactive decay on the thermal output of the rtgs can be directly observed from telemetry . all rtgs had a temperature sensor located at the root of one of the radiating fins ( `` fin root temperature '' ) . preliminary analysis suggests that these readings are in agreement with what would be expected from the radioactive decay of the plutonium fuel ( see figure [ fig : root - fin ] ) . one can try to detect changes in the measurements of fin root temperature during each of the three planetary encounters and see if the rtgs had developed an anisotropic radiating pattern . ( note that the same test can be conducted for the electrical energy dissipation . ) thus , if the temperatures at the fins change during the fly - bys , this may indirectly indicate an alteration of the optical properties of the rtgs during the planetary encounters . this should allow us to ask the question : `` if the system survived flawlessly than why would it change long after the encounter ? '' one can also look at the different segments of the thermal history in an attempt to identify different heat decay rates . however , the planetary flybys might be the strongest test for the system s design and operational longevity . if the mechanisms above were treated as separate constituents to the anomalous accelerations , one can ask what if a combination of them is responsible for the effect . it has been recently suggested that most , if not all , of the unmodeled acceleration of pioneer 10 and 11 is due to an essentially constant supply of heat coming from the central compartment , directed out the front of the craft through the closed louvers @xcite . this is a more subtle version of an earlier proposal @xcite calling on the total electrical power as a mechanism . that proposal was argued against because of the observed lack of decay of the acceleration with time @xcite . for this mechanism to work , one essentially needs to find a constant supply of heat radiated off the back of the spacecraft that would produce a thermal recoil force with the well established properties ( discussed in section [ sec : anomaly ] ) . however , the assumption of constancy came against the actual design of the pioneers and experimental data on the performance of the vehicles @xcite . in fact , the lack of constancy of heat dissipated during the longest doppler segment analyses ( i.e. 11.5 years of the pioneer 10 data ) invalidated the hypothesis . however , this claim can and will be further investigated with the newly recovered data , both doppler and telemetry ( see tables [ tab : telemetry],[tb : power ] ) . the newly acquired data ( both doppler and telemetry ) can significantly contribute to addressing this possibility . for one , we also have a much larger segment of doppler data that will be used to analyze these heat dissipation processes on the vehicles . in addition , we now have the actual design , fabrication , testing , pre- and in - flight calibration data that characterize the pioneer craft performance for the duration of their missions . this data can tell precisely at what time the louvers were open and closed , when a certain instrument was powered on and off , what was the performance of the battery , shunt current and all electric parts of the spacecraft . this is truly a unique data set that will be used to reconstruct the physics of the anomalous acceleration , if it is due to on - board systematics . in fact we have all the detailed information on properties of the spacecraft and the data needed to reconstruct the behavior of its major components , including electrical power and thermal subsystems . this information may be used in developing a thermodynamical model of the spacecraft that would help us to establish the true thermal and electrical power dissipation history of the vehicles and also correlate major events on the pioneers ( such as powering `` on '' or `` off '' certain instruments or performing a maneuver ) with the available doppler data ; this possibility is addressed in section [ sec : thermal - model ] . another possible on - board systematic error may be coming from the expulsion of the he being created in the rtgs from the @xmath43-decay of @xmath44pu ( see discussion in @xcite ) . the pioneer rtgs were designed so that the he pressure is not totally contained within the pioneer heat source over the life of the rtgs . instead , the pioneer heat source contains a pressure relief device that allows the generated he to vent out of the heat source and into the thermoelectric converter . the thermoelectric converter housing - to - power output receptacle interface is sealed with a viton o - ring . the o - ring allows the helium gas within the converter to be released by permeation into the space environment throughout the mission life of the pioneer rtgs . for this mechanism to work , what one needs is all he produced to preferentially leave the rtgs in one direction at a rate of 0.77g / yr . assuming a single elastic reflection , @xcite ruled out helium permeating through the o - rings as the cause of @xmath23 . they also give the estimate for acceleration contribution in the direction away from the sun due to this systematic expulsion @xmath45 it is highly unlikely that this mechanism is responsible for the pioneer anomaly seen on both spacecraft ; the use of mdrs can further support this conclusion . the possible use of a similar strategy to the above to study the effect of propulsive mass expulsion due to gas leakage has to be assessed . it is known that , although this effect is largely unpredictable , many spacecraft have experienced gas leaks producing accelerations on the order of @xmath46m / s@xmath2 . gas leaks generally behave differently after each maneuver . the leakage often decreases with time and becomes negligible . ( see more discussion of these effects in @xcite . ) -6pt -5pt -6pt -5pt -6pt -5pt gas leaks can originate from the pioneers propulsion system , which is used for mid - course trajectory maneuvers , for spinning up or down the spacecraft , and for orientation of the spinning spacecraft . ( consult @xcite for design of the pioneer propulsion and attitude control systems . ) by studying time histories of the spin rates of the pioneers , @xcite estimated the effect of the gas leakage mechanism on the anomalous acceleration . this mechanism was found to produce an acceleration uncertainty of @xmath47 furthermore , they concluded that the gas leak mechanism is very unlikely to be the explanation for the anomalous acceleration . this is because it is difficult to understand why this effect , being a stochastic variable obeying a poisson distribution , would affect pioneers 10 and 11 by the same amount @xcite . the new analysis can refine the conclusions above . we have three readings from the propellant tank on board that can help establish the propellant loss history of the two spacecraft , namely propellant tank temperature ( figure [ fig : thrusters - c-130 ] ) , propellant supply pressure ( figure [ fig : thrusters - c-210 ] ) , and propellant supply temperature ( figure [ fig : thrusters - c-327 ] ) . we also have several temperature readings from the thruster clusters and individual thrusters . not only may this new information be used to precisely establish the time of the maneuvers , it can also provide some insight as to why the two identically designed systems behaved so differently , especially when changes to their spin rates are concerned ( see figures 11 and 12 of @xcite . ) the corresponding analysis had been initiated and results will be reported . in this section we will argue that , based on the information provided by the mdrs , one can develop a high accuracy thermal / electrical / dynamical model of the pioneer spacecraft . such a model can be used to further improve our understanding of the anomalous acceleration and especially to study contribution from the on - board thermal environment to the anomaly . in this section we will comment on our preliminary evaluation of what is needed to develop such an accurate model . it is clear that a thermal model for the pioneers spacecraft would have to account for all heat radiation within the spacecraft and out to deep space . specifically , this model would have start from the rtgs and include description of the heat radiated , absorbed , reradiated , and conducted by each and every instrument , all cables , support structures and etc . the objective is to develop a broadband ( ir and optical ) thermal map of the spacecraft s exterior , accurate to few degrees kelvin . as input , this model will use electrical telemetry readings from the on - board sensors that were installed on the spacecraft , while temperature readings ( see figure [ fig : sensors ] ) can be used for its calibration . we will then attempt to verify this model by using the radiometric doppler data from the attitude maneuvers performed at the earlier mission phases . it is worth noting that no other mission in the past used telemetry data to improve its navigational capabilities ; what we are going to implement with the pioneers is a novel navigational technique . pioneer 10/11 may be the first missions to benefit from this technique , designed to reduce the impact of on - board sources to non - gravitational accelerations on the accuracy of trajectory reconstruction . our proposal uses the fact that the pioneers already have very good `` built - in '' navigational capabilities ( i.e. guidance , navigation and control ) ; our goal is to reach the absolute limit allowed by these capabilities . as a quantitative objective , we hope to improve acceleration contribution due to on - board systematic sources to the level below @xmath48 m / s@xmath2 . to reach the stated level of accuracy , one would have to adjust navigational data by using the developed thermal / electrical / dynamical model of the spacecraft . to achieve this we will utilize the following available information : 1 . dynamical properties of the spacecraft , i.e. geometry , dimensions , and weights of all the components of the spacecraft , including cabling , fuel and thermal blankets . in short , we will need to have precise mass and power budget data for the entire craft . ideally , we would prefer to use not only design numbers , but rather actual values established during the final pre - launch tests at the kennedy space center , florida ( ksc ) . this information is still to be identified and retrieved . 2 . thermal , electric and optical ( infrared ) properties of the spacecraft materials used to fabricate all major components of the spacecraft , including thermal blankets and cabling . again , we would prefer to have these values from the tests at the ksc . note that we will have access to the initial values of these parameters ; their degradation during the 30 years in space is unknown . there is very limited information available from long - term exposure tests . furthermore , for obvious reasons ( tests conducted in a different environment , e.g. , in low earth orbit ( loe ) , for different durations ) , even these results will not be directly applicable to our model . 3 . we will benefit from the design information of the spacecraft , especially information concerning their thermal , power , propulsion , and communication subsystems . we will use the knowledge on the location of each and every telemetry sensor placed on the spacecraft and use all the recently retrieved pioneer telemetry data . this information will be used to build and calibrate the thermal model of the spacecraft . in particular , the following telemetry information will be critical for the upcoming investigation : 1 . rtg fin root temperature , as an indicator of rtg thermal output 2 . rtg current and voltage , providing a measure for total electrical power 3 . all thermal data from sensors on the spacecraft and instrument compartment 4 . temperatures and pulse counts from thruster cluster assemblies 5 . battery current and temperature 6 . shunt current 7 . radio beam gain and transmitted power , twt amplifier temperature 8 . propellant tank and supply lines temperature 4 . we would have to analyze the design and thermal properties of the snap-19 rtgs used in the pioneers . much information is already available , but additional data on their thermal behavior will be very helpful in building a 3-dimensional thermal radiation pattern . once the thermal model is completed , it must be calibrated for in - flight conditions . for this , we may need to analyze all the attitude control ( i.e. conscan ) maneuvers in the earlier mission phases @xcite . the earlier phases had significant variations in the sun - earth line to spin axis angle . these variations will help us to calibrate the effect of thermal recoil forces on the radiometric data received from the craft and , thus , to precisely evaluate the magnitude of the thermal recoil force . the closer the craft to the earth is , the better the ultimate accuracy of the model will be . this is due to the fact that we know the properties of the solar radiation and thermal environment in the inner solar system to a much better accuracy than in the deep space . in fact , it will be hard to do the required calibration after the jupiter flyby , and it will be impossible at distances greater than 10 au . clearly we will benefit from the larger size of the earth - probe - sun angle , which is getting much smaller after 10 au . after 20 au , both the sun and the earth will be in the same field of view , making it difficult to decouple any thermally induced force from other physical effects . however , given the significant uncertainties on the degradation of the thermodynamical and optical properties of the materials used to build the pioneers , the outcome of this model is difficult to predict . nevertheless , we expect that , given the simplicity of the pioneers design , one should be able to build a model with an uncertainty of no more than 15% . we have initiated this work with anticipated involvement of several navigational and thermal engineers at jpl ; results of this work will be reported . the plan outlined here is ambitious . before we embark on this complex exercise , we shall also attempt to build a simpler ( but enthusiastically not `` simplistic '' ) thermal model of the pioneers based on available telemetry data . such a model would necessarily utilize simplifying assumptions on the spacecraft s geometry , and ignore contributions from small components ( e.g. , structural elements , struts , cables as heat sources ) . nevertheless , it is possible that even such a model may lead to a much better understanding of the contribution of spacecraft systematics to the anomalous acceleration , and may also guide us to better direct our efforts as we carry out a more elaborate effort . on march 2 , 2002 nasa s dsn made the last contact with pioneer 10 and confirmed that the spacecraft was still operational thirty years after its launch on march 3 , 1972 ( ut ) . the uplink signal was transmitted on march 1 from the dsn s goldstone , california facility and a downlink response was received twenty - two hours later by the 70-meter antenna at madrid , spain . at this time the spacecraft was 11.9 billion kilometers from earth at about 79.9 au from the sun and heading outward into interstellar space in the general direction of aldebaran at a distance of about 68 light years from the earth , and a travel time of two million years . although science operations were officially terminated on march 31 , 1997 , pioneer s onboard rtgs still provided just enough electrical power for operations of the radio transmitter and receiver systems and for the geiger tube telescope experiment of the university of iowa . after 1997 , nasa arc continued to support occasional tracking of pioneer 10 for development and testing of communications technology that might be used for a future interstellar probe mission . pioneer 10 s signal was also used as a standard test source for seti radio telescope studies ( i.e. project phoenix , see details at http://www.seti.org/ ) . by 2005 , the existence of the pioneer anomaly is no longer in doubt . further , after much understandable hesitancy , a steadily growing part of the community has concluded that the anomaly should be subject to further investigation and interpretation . the results of the investigation of the pioneer anomaly would be win - win ; improved navigational protocols for deep space at the least , exciting new physics at the best . a strong international collaboration ( e.g. , http://www.issi.unibe.ch/teams/pioneer/ ) that extends in broader areas of fundamental physics , technology and mission design is an additional outcome of the discussed program of the study of the pioneer anomaly @xcite . concluding , we would like to emphasize one immediate impact of the study of the pioneer anomaly , this time on the pioneer 10 mission itself . recently it became clear that there exists one last opportunity to contact pioneer 10 in the deep space . specifically , in february - march 2006 , the mutual motions of the earth and the spacecraft will again put the earth in the field of view of pioneer 10 s antenna , thus making it possible to establish a radio contact with the craft . at this time , pioneer 10 will be at a heliocentric distance of @xmath990.1 au , moving at a speed of nearly 12.1 km / s with round trip light time from the earth of almost 25 hours ( thus , the same dsn antenna , dss 14 at goldstone , is planned for the operations . ) this would be the last time when dsn will initiate contact with pioneer 10 , as the on - board power system on the craft is at its limits . this intriguing possibility to re - acquire a coherent doppler signal from pioneer 10 is currently being investigated at jpl and results will be reported . we would like to express our gratitude to our many colleagues who have either collaborated with us on this manuscript or given us their wisdom . firstly we must acknowledge the many people who have helped us with suggestions , comments , and constructive criticisms . we specifically thank john d. anderson , sami asmar , and timothy p. mcelrath , who provided us with very valuable comments while this manuscript was in preparation . invaluable information on the history , spacecraft design and mission operations of the pioneers 10 and 11 , as well as the structure of their telemetry data , came from lawrence lasher and david lozier of the nasa ames research center . we also thank belinda arroyo , peter j. breckheimer , john e. ekelund , jordan ellis , scott e. fullner , gene l. goltz , olga king , george d. lewis , robert a. jacobson , margaret medina , neil mottinger , teresa thomas of jpl for their help in obtaining , understanding and conditiononing of the pioneer doppler data . we thank moustafa t. chahine , william folkner , ulf e. israelsson , tomas j. martin - mur , thomas a. prince , and michael m. watkins of jpl for encouragement and stimulating discussions regarding the pioneer doppler data retrieval effort . john f. cooper and sharlene rhodes of nssdc provided pivotal help in the retrieval and understanding of the historical pioneer doppler data archived with nssdc . louis k. scheffer of cadence design systems contributed with useful observations on the thermal modeling and analysis . we especially thank the planetary society for support and , in particular , louis d. freidman , charlene m. anderson , and bruce betts for their interest , stimulating conversations and encouragement . this work was partially performed at the international space science institute ( issi ) , bern , switzerland , when two of us ( sgt and vtt ) visited issi as part of an international team program . in this respect we would like to thank roger m. bonnet , vittorio manno , brigitte fasler and saliba f. saliba of issi for their hospitality and support . the work of sgt was supported in part by the office of the jpl chief scientist under the research and technology development program and carried out at the jet propulsion laboratory , california institute of technology , under a contract with the national aeronautics and space administration . asmar , s.w . , armstrong , j.w . , iess , l. , and tortora , p. , `` spacecraft doppler tracking : noise budget and accuracy achievable in precision radio science observations , '' _ radio science * 40 * _ , rs2001 ( 2005 ) . anderson , j.d . , `` quarterly report to nasa / ames research center , _ celestial mechanics experiment , pioneer 10/11 , _ '' jpl interoffice memorandum , 8 july 1992 ; _ celestial mechanics experiment , pioneer 10/11 , _ 22 july 1992 . also see the later quarterly report for the period 1 oct . 1992 to 31 dec . 1992 , dated 17 dec . 1992 , letter of agreement arc / pp017 . the first and last , specifically , contain figure 7 of ref . @xcite . anderson , j.d . , laing , p.a . , lau , e.l . , liu , a.s . , nieto , m.m . , and turyshev , s.g . , `` indication , from pioneer 10/11 , galileo , and ulysses data , of an apparent anomalous , weak , long - range acceleration , '' _ phys . lett . _ * 81 * 2858 ( 1998 ) , gr - qc/9808081 . anderson , j.d . , laing , p.a . , lau , e.l . , liu , a.s . , nieto , m.m . , and turyshev , s.g . , `` anderson et al . reply ( to the comment by katz on pioneer 10/11 ) , '' _ phys . lett . _ * 83 * , 1893 ( 1999 ) , gr - qc/9906112 . anderson , j.d . , laing , p.a . , lau , e.l . , liu , a.s . , nieto , m.m . , and turyshev , s.g . , `` anderson et al . reply ( to the comment by murphy on pioneer 10/11 ) , '' _ phys . rev . lett . _ * 83 * , 1891 ( 1999 ) , gr - qc/9906113 . anderson , j.d . , laing , p.a . , lau , e.l . , liu , a.s . , nieto , m.m . , and turyshev , s.g . , `` study of the anomalous acceleration of pioneer 10 and 11 , '' _ phys . d. _ * 65 * , 082004/1 - 50 ( 2002a ) , gr - qc/0104064 . deep space network , system requirements detailed interface design , jpl document 820 - 13 , rev . a , jet propulsion laboratory , pasadena , ca ( november 1 , 1991 ) , see _ dsn tracking system interfaces , archival tracking data file interface , trk-2 - 25 . _ an example of an atdf for ulysses mission ( part of sfoc - nav-2 - 25 ) at http://www.igpp.ucla.edu/pds3/uly_5101/data/atdf/2040040a.lbl deep space network , system requirements detailed interface design , jpl document 820 - 13 , rev . a , jet propulsion laboratory , pasadena , ca ( november 1 , 1991 ) , see chapter on _ dsn tracking system interfaces , orbit data file interface , trk-2 - 18 . _ an example of an odf for near mission ( part of sfoc - nav - trk-2 - 18 ) may be found at + http://pdssbn.astro.umd.edu/data3/near/nreros_2001/erosfb/odf/8357357c.lbl defrere , d. , rathke , a. , `` pioneer anomaly : what can we learn from lisa ? '' in proceedings of `` lasers , clocks , and drag - free , '' zarm , bremen , germany , 30 may 1 june 2005 , ed . c. lmmerzahl , h. dittus , s.g . turyshev , to be published , ( 2006 ) gr - qc/0509021 dittus , h. , turyshev , s.g . , lmmerzahl , c. , theil , s. , foerstner , r. , johann , u. , ertmer , w. , rasel , e. , dachwald , b. , seboldt , w. , hehl , f.w . , kiefer , c. , blome , h .- j . , kunz , j. , giulini , d. , bingham , r. , kent , b. , sumner , t.j . , bertolami , o. , pramos , j. , rosales , j.l . , christophe , b. , foulon , b. , touboul , p. , bouyer , p. , reynaud , s. , brillet , a. , bondu , f. , samain , e. , de matos , c.j . , erd , c. , grenouilleau , j.c . , izzo , d. , rathke , a. , anderson , j.d . , asmar , s.w . , lau , e.e . , nieto , m.m . , and mashhoon , b. , `` a mission to explore the pioneer anomaly , '' in proceedings of 2005 eslab symposium `` trends in space science and cosmic vision 2020 , '' esa / estec , noordwijk , the netherlands , 19 april 2005 , esa publication sp-588 , 3 - 10 ( 2005 ) , gr - qc/0506139 . fimmel , r. o. , swindell , w. , and burgess , e. , _ pioneer odyssey : first into the outer solar system _ , nasa publication sp-349/396 ( nasa , washington , d.c . , 1977 ) , see electronic version at http://history.nasa.gov/sp-349/sp349.htm turyshev , s.g . , laing , p.a . , lau , e.l . , liu , a.s . , and nieto , m.m . : 1999 , `` the apparent anomalous , weak , long - range acceleration of pioneer 10 and 11 . '' in : _ gravitational waves and experimental gravity _ , proceedings of the xviiith workshop of the rencontres de moriond , les arcs , savoi , france ( january 23 - 30 , 1999 ) , ed . by j. dumarchez and j. tran thanh van ( world publishers , hanoi - vietnam , 2000 ) . 481 - 486 ( 1999 ) , gr - qc/9903024 . turyshev , s.g . , nieto , m.m . , and anderson , j.d , `` lessons learned from the pioneers 10/11 for a mission to test the pioneer anomaly . '' `` 35th cospar scientific assembly , '' july 18 - 24 , 2004 , paris . france . to be published in _ advances in space res . _ , ( 2004 ) , gr - qc/0409117 . turyshev , s.g . , nieto , m.m . , and anderson , j.d , `` a route to understanding of the pioneer anomaly . '' `` the xxii texas symposium on relativistic astrophysics , '' stanford university , dec . 13 - 17 , 2004 , edited by p. chen , e. bloom , g. madejski , and v. petrosian . slac - r-752 , stanford e - conf # c041213 , # 0310 , ( 2005a ) , eprint : http://www.slac.stanford.edu/econf/c041213/ , gr - qc/0503021 . turyshev , s.g . , nieto , m.m . , and anderson , j.d . , `` study of the pioneer anomaly : a problem set . '' _ amer . j. phys . * 73 * _ , 1033 - 1044 ( 2005b ) , physics/0502123 . turyshev , s.g . , nieto , m.m . , and anderson , j.d . , `` the pioneer anomaly and its implications . '' xxist iap colloquium on `` mass profiles and shapes of cosmological structures '' at the institute dastrophysique de paris , paris , france , 9 july 2005 , to be published by edp sciences , paris , france , ( 2006 ) , gr - qc/0510081 .
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the pioneer 10/11 spacecraft yielded the most precise navigation in deep space to date .
however , their radiometric tracking data has consistently indicated the presence of a small , anomalous , doppler frequency drift .
the drift is a blue shift , uniformly changing with a rate of @xmath0 hz / s and can be interpreted as a constant sunward acceleration of each particular spacecraft of @xmath1 m / s@xmath2 ( or , alternatively , a time acceleration of @xmath3 s / s@xmath2 ) .
this signal has become known as the pioneer anomaly ; the nature of this anomaly remains unexplained .
we discuss the current state of the efforts to retrieve the entire data sets of the pioneer 10 and 11 radiometric doppler data .
we also report on the availability of recently recovered telemetry files that may be used to reconstruct the engineering history of both spacecraft using original project documentation and newly developed software tools .
we discuss possible ways to further investigate the discovered effect using these telemetry files in conjunction with the analysis of the much extended pioneer doppler data . in preparation for this new upcoming investigation , we summarize the current knowledge of the pioneer anomaly and review some of the mechanisms proposed for its explanation .
we emphasize the main objectives of this new study , namely analysis of the early data that could yield the true direction of the anomaly and thus , its origin , analysis of planetary encounters , that should tell more about the onset of the anomaly ( e.g. pioneer 11 s saturn flyby ) , analysis of the entire dataset , that should lead to a better determination of the temporal behavior of the anomaly , comparative analysis of individual anomalous accelerations for the two pioneers with the data taken from similar heliocentric distances , the detailed study of on - board systematics , and development of a thermal - electric - dynamical model using on - board telemetry .
the outlined strategy may allow for a higher accuracy solution for the anomalous acceleration of the pioneer spacecraft and , possibly , will lead to an unambiguous determination of the origin of the pioneer anomaly .
slava g. turyshev,@xmath4 viktor t. toth,@xmath5 larry r. kellogg,@xmath6 eunice .
l. lau,@xmath4 and kyong j. lee@xmath4 _ @xmath4jet propulsion laboratory , california institute of technology , + 4800 oak grove drive , pasadena , ca 91109 , usa + @xmath5vttoth.com , 3 - 575 old st patrick st . , ottawa on k1n 9h5 , canada + @xmath6nasa ames research center , moffett field , ca 94035 , usa _
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You are an expert at summarizing long articles. Proceed to summarize the following text:
codes , proposed by arikan @xcite , are proved to achieve the symmetric capacities of the binary - input discrete memoryless channels ( b - dmcs ) . this capacity - achieving code family is based on a technique called channel polarization . by performing the channel splitting and channel combining operations on independent copies of a given b - dmc , a set of synthesized binary - input channels can be obtained . let @xmath0 denote the symmetric capacity of a b - dmc @xmath1 . it is proved in @xcite that : with @xmath2 uses of @xmath1 , @xmath3 , when @xmath4 is large enough , it is possible to construct @xmath4 synthesized channels such that @xmath5 of them are completely unreliable and @xmath6 of them are noiseless . by transmitting free bits ( called information bits ) over the noiseless channels and transmitting a sequence of fixed bits ( called frozen bits ) over the others , polar codes can achieve the symmetric capacity under a successive cancellation ( sc ) decoder with both encoding and decoding complexity @xmath7 . in @xcite , it is proved that the block error probability of polar code under sc decoding satisfies @xmath8 for any @xmath9 when code length @xmath4 is large enough and code rate @xmath10 . furthermore , it was shown by korada et al . @xcite that the error exponent @xmath11 can be arbitrarily close to 1 for large @xmath4 with a general construction using larger kernel matrices than the @xmath12 matrix proposed by arikan . to construct polar codes , the channel reliabilities can be calculated efficiently using bhattacharyya parameters for binary - input erasure channels ( becs ) @xcite . but for channels other than becs , density evolution is required @xcite . more practical methods for calculating the channel reliabilities are discussed in @xcite and @xcite , and these techniques are extended to input channels @xcite . the channel polarization phenomenon is believed to be universal in many other applications , such as parallel communications @xcite @xcite , coded modulation systems @xcite , multiple access communications @xcite @xcite , source coding @xcite @xcite , information secrecy @xcite @xcite and other settings . although polar codes have astonishing asymptotic performance , the finite - length performance of polar code under sc decoding is not satisfying . with the factor graph representation of polar codes , a belief propagation ( bp ) decoder is introduced by arian in @xcite . and in @xcite , hussami et . al . show that bp decoder significantly can outperform sc decoder , and point out that , for channels other than bec , the schedule of message passing in bp plays an important role . and they also show that the performance of bp decoder can be further improved by utilization of overcomplete factor graph representations over bec . unfortunately , due to the sensitivity of bp decoder to message - passing schedule , this is not realized on other channels . in @xcite a linear programming ( lp ) decoder is introduced without any schedule , and also , by using the overcomplete representations can improve the performance of lp decoder . but lp decoder can not work on channels other than bec . maximum likelihood ( ml ) decoders are implemented via viterbi and bcjr algorithms on the codeword trellis of polar codes @xcite , but because of their high complexity , they can only work on very short code blocks . successive cancellation ( sc ) decoding of polar codes essentially shares the same idea with the recursive decoding of rm codes @xcite . like the recursive decoders can be improved by using a list @xcite or a stack @xcite , sc can also be enhanced in the same way . as an improved version of sc , successive cancellation list ( scl ) decoding algorithm is introduced to approach the performance of maximum likelihood ( ml ) decoder with an acceptable complexity @xcite , @xcite . and later , an other improved decoding algorithm based on sc named successive cancellation stack ( scs ) decoding algorithm is proposed whose computational complexity will decrease with the increasing of signal - to - noise ratio ( snr ) and can be very close to that of the sc decoding in the high snr regime @xcite . compared with scl , scs will have a much lower computational complexity . but it comes at the price of much larger space complexity and it will fail to work when the stack is too small . combining the ideas of scl and scs , a new decoding algorithm named successive cancellation hybrid ( sch ) is proposed in this paper , and it can achieve a better trade - off between computational complexity and space complexity . in this paper , all the three improved sc decoding algorithms , scl , scs and sch , are described under a unified manner of a path searching procedure on the code tree . further , to reduce the complexity , a pruning technique is proposed to avoid unnecessary path searching operations . the remainder of the paper is organized as follows . section [ section_preliminaries ] reviews the basics of polar coding and describes the sc decoding algorithm as a path searching procedure on a code tree using _ a posteriori _ probabilities ( apps ) as metrics . then the three improved successive cancellation ( isc ) decoding algorithms and the pruning technique are introduced in section [ section_isc ] . section [ section_simulations ] provides the performance and complexity analysis based on the simulation results of polar codes under isc decoders with different parameters . finally , section [ section_conclusion ] concludes the paper . in this paper , we use blackboard bold letters , such as @xmath13 and @xmath14 , to denote sets , and use @xmath15 to denote the number of elements in @xmath13 . we write the cartesian product of @xmath13 and @xmath14 as @xmath16 , and write the @xmath17-th cartesian power of @xmath13 as @xmath18 . we use calligraphic characters , such as @xmath19 to denote a event . and let @xmath20 denote the event that @xmath19 is not happened . we use notation @xmath21 to denote a @xmath4-dimension vector @xmath22 and @xmath23 to denote a subvector @xmath24 of @xmath21 , @xmath25 . particularly when @xmath26 , @xmath23 is a vector with no elements in it and the empty vector is denoted by @xmath27 . we write @xmath28 to denote the subvector of @xmath21 with odd indices ( @xmath29 ; @xmath30 is odd ) . similarly , we write @xmath31 to denote the subvector of @xmath21 with even indices ( @xmath29 ; @xmath30 is even ) . for example , for @xmath32 , @xmath33 , @xmath34 and @xmath35 . further , given a index set @xmath36 , @xmath37 denote the subvector of @xmath21 which consists of @xmath38s with @xmath39 . only square matrices are involved in this paper , and they are denoted by bold letters . the subscript of a matrix indicates its size , e.g. @xmath40 represents a @xmath41 matrix @xmath42 . we write the kronecker product of two matrices @xmath42 and @xmath43 as @xmath44 , and write the @xmath17-th kronecker power of @xmath42 as @xmath45 . let @xmath46 denote a b - dmc with input alphabet @xmath13 and output alphabet @xmath47 . since the input is binary , @xmath48 . the channel transition probabilities are @xmath49 , @xmath50 , @xmath51 . for code length @xmath52 , @xmath53 , and information length @xmath54 , i.e. code rate @xmath55 , the polar coding over @xmath1 proposed by arikan can be described as follows : after channel combining and splitting operations on @xmath4 independent uses of @xmath1 , we get @xmath4 successive uses of synthesized binary input channels @xmath56 , @xmath57 , with transition probabilities @xmath58 where @xmath59 and the source block @xmath60 are supposed to be uniformly distributed in @xmath61 . let @xmath62 denote the probability of maximum - likelihood ( ml ) decision error of one transmission on @xmath63 , @xmath64 where @xmath65 , @xmath66 and the indicator function @xmath67 and @xmath68 is the module-@xmath69 addition . the reliabilities of polarized channels @xmath70 are usually measured by ( [ equ_pe ] ) , and can be evaluated using bhattacharyya parameters @xcite for binary erasure channels ( becs ) or density evolution @xcite for other channels . to transmit a binary message block of @xmath54 bits , the @xmath54 most reliable polarized channels @xmath70 with indices @xmath39 are picked out for carrying these information bits ; and transmit a fixed bit sequence called frozen bits over the others . the index set @xmath71 is called information set and @xmath72 . and the complement set of @xmath36 is called frozen set and is denoted by @xmath73 . alternatively , the polar coding can be described as follow : a binary source block @xmath60 which consists of @xmath54 information bits and @xmath74 frozen bits is mapped to a code block @xmath75 via @xmath76 . the matrix @xmath77 , where @xmath78 $ ] and @xmath79 is the bit - reversal permutation matrix . the binary channel @xmath80 are then sent into channels which are obtained by @xmath4 independent uses of @xmath1 . as mentioned in @xcite , polar codes can be decoded by successive cancellation ( sc ) decoding algorithm . let @xmath81 denote the estimate of the source block @xmath60 . after receiving @xmath82 , the bits @xmath83 are determined successively with index @xmath84 from @xmath85 to @xmath4 in the following way : @xmath86 where @xmath87 the block error rate ( bler ) of this sc decoding is upper bounded by @xmath88 this successive decoding can be represented as a path searching process on a code tree . for a polar code with code length @xmath4 , the corresponding code tree @xmath89 is a full binary tree . more specifically , @xmath89 can be represented as a 2-tuple @xmath90 where @xmath91 and @xmath92 denote the set of nodes and the set of edges respectively , @xmath93 , @xmath94 . the depth of a node @xmath95 is the length of the path from the root to the node . the set of all nodes at a given depth @xmath96 is denoted by @xmath97 , @xmath98 . the root node has a depth of zero . all the edges @xmath99 are partitioned into @xmath4 levels @xmath100 , @xmath101 , such that the edges in @xmath100 incident with the nodes at depth @xmath102 and the nodes at depth @xmath103 . except the nodes at the @xmath4-th depth @xmath104 , each @xmath105 has two descendants which belong to @xmath106 , and the two corresponding edges are label as @xmath107 and @xmath85 respectively . the nodes @xmath108 are called leaf nodes . fig . [ fig_tree ] gives a simple example of code tree with @xmath109 . a @xmath84-length decoding path @xmath110 consists of @xmath84 edges , with @xmath111 , @xmath112 . a vector @xmath113 is used to depict the above decoding path , where @xmath38 is corresponding to the binary label of edge @xmath114 . the reliability of a decoding path @xmath113 can be measured using _ a posteriori _ probability @xmath115 the apps can be regarded as normalized versions of the channel transition probabilities defined in ( [ equ_polarized_channels ] ) . the two kinds of probabilities are related by a multiplicative factor @xmath116 . by eliminating the factor , the apps take values in a more stable range , and all the decoding paths with the same lengths have the sum probability equals to one , i.e. @xmath117 this property will help in understanding the path searching procedure in the code tree and is more suitable for hardware implementation . similar to the recursive expressions of ( [ equ_polarized_channels ] ) given in @xcite , the apps can also be calculated recursively . for any @xmath118 , @xmath2 , @xmath119 , @xmath120 @xmath121 sc decoding can be seen as a greedy search algorithm on the code tree . in each level , only the one of two edges with larger probability is selected for further processing . . the bold branches show a decoding path of sc with @xmath122 . ] the red bold edges in fig . [ fig_tree ] shows the sc decoding path . the number written next to each of the nodes provides the app metric of the decoding path from the root to that node . the nodes which are extended during the sc decoding procedure are represented by the numbered circles , and the corresponding numbers indicate the processing order . the black circles represent the nodes which are visited ( whose app metric is calculated ) but failed in competition for further exploring . and the gray ones are those which are not visited during the searching process . in the example , four times of calculations of equation ( [ equ_app ] ) are required , one for each level . however , the decoding path is not guaranteed to be the most probable one . as shown in the example , the one labeled @xmath123 has the largest probability of all the @xmath4-length paths , but it failed in the competition at the first level . for further practical considerations , we use the logarithmic apps as the path metrics : @xmath124 for @xmath125 , the path metric can be recursively calculated as @xmath126 and @xmath127 where function @xmath128 is the jacobian logarithm and @xmath129 , @xmath130 . then , the decision function of sc in ( [ equ_sc_h ] ) is rewritten as @xmath131 using the space - efficient structure @xcite to implement a sc decoder , the time and space complexity are @xmath132 and @xmath133 respectively . the performance of sc is limited by the bit - by - bit decoding strategy . since whenever a bit is wrongly determined , there is no chance to correct it in the future decoding procedure . theoretically , the performance of the maximum _ a posteriori _ probability ( map ) decoding ( or equivalently ml decoding , since the inputs are assumed to be uniformly distributed ) can be achieved by traversing all the @xmath4-length decoding paths in the code tree @xmath89 . but this brute - force traverse takes exponential complexity and is difficult to be implemented . two improved decoding algorithms called successive cancellation list ( scl ) decoding and successive cancellation stack ( scs ) are proposed in @xcite @xcite and @xcite . both of these two algorithms allow more than one edge to be explored in each level of the code tree . during the scl(scs ) decoding , a bunch of candidate paths will be obtained and stored in a list(stack ) . since for every single candidate path , the metric calculations and bit determinations are still performed bit - by - bit successively , scl and scs can be regarded as two improved versions of conventional sc decoding . in this section , we will restate scl and scs under a unified framework with the help of app metrics and the code tree representations . then to overcome the own shortages of scl and scs , a new hybrid decoding algorithm named successive cancellation hybrid ( sch ) decoding is proposed . furthermore , to reducing the computational complexities , we propose a pruning technique to eliminate the unnecessary calculations during the path searching procedure on the code tree . . ] as an enhanced version of sc , the successive cancellation list ( scl ) decoder @xcite @xcite searches level - by - level on the code tree , which is just the same with sc . however , unlike sc where only one path is reserved after processing at each level , scl allows at most @xmath134 candidate paths to be further explored at the next level . scl can be regarded as a breadth - first searching on the code tree @xmath135 with a searching width @xmath134 . at each level , scl doubles the number of candidates by appending a bit @xmath107 or a bit @xmath85 to each of the candidate paths , and then selects at most @xmath134 ones with largest metrics and stores them in a list for further processing at the next level . finally , when reaching the leaf nodes , the binary labels @xmath21 corresponding to the edges in path @xmath136 which has the largest metric in the list , are assigned to the estimated source vector @xmath81 . let @xmath137 denotes the set of candidate paths corresponding to the level-@xmath84 of code tree in a scl decoder . the @xmath137s are stored and updated in a list structure . the scl decoding algorithm with searching width @xmath134 , denoted by scl(@xmath134 ) , can be described as follows : ( a.1 ) initialization . a null path is included in the initial list and its metric is set to zero , i.e. @xmath138 , @xmath139 . ( a.2 ) expansion . at the @xmath84-th level of the code tree , the number of candidate paths in the list are doubled by concatenating new bits @xmath140 taking values of @xmath107 and @xmath85 respectively , that is , @xmath141 for each @xmath142 , the corresponding path metric(s ) are updated according to ( [ equ_metric_info_or_froz ] ) , ( [ equ_metric_odd ] ) and ( [ equ_metric_even ] ) . ( a.3 ) competition . if the number of paths in the list after ( a.2 ) is no more than @xmath134 , just skip this step ; otherwise , reserve the @xmath134 paths with the largest metrics and delete the others . ( a.4 ) determination . repeat ( a.2 ) and ( a.3 ) until level-@xmath4 is reached . then , the decoder outputs the estimated source vector @xmath143 , where @xmath21 is the binary labels of the path with the largest metric in the list . fig . [ fig_scl ] gives a simple example of the tree searching under scl decoding with @xmath144 . compare with sc in fig . [ fig_tree ] , scl find the most probable path @xmath123 . but the times of metric computations is increased from four to seven . scl maintains @xmath134 decoding paths simultaneously , each path consumes a @xmath133 space , the space complexity of scl then is @xmath145 . during the decoding process at each level , each of the @xmath134 candidates is copied once and extended to two new paths , these copy operations require @xmath145 computations . moreover , since the code tree has @xmath4 levels , a direct implementation of scl decoder will take @xmath146 computations . in @xcite , a so called `` lazy copy '' technique based on the memory sharing structure among the candidate paths is introduced to reduce this copy complexity . therefore , the scl decoder can be implemented with computational complexity @xmath147 . note that , the path metric ( [ equ_metric_info_or_froz ] ) of a certain decoding path with binary label vector @xmath113 will not be smaller than that of any of its descendants , i.e. for any @xmath148 and @xmath149 , @xmath150 hence , if the metric of a @xmath4-length decoding path is larger than that of another path with length @xmath151 , it must also be larger than the metric of any of the @xmath4-length descendant path of the latter . so rather than waiting after processing at each level , we can keep on searching along the single candidate path until its metric is no longer the largest . once a @xmath4-length path is found with the largest metric among all the candidate paths , its binary label vector is simply output as the final estimation , the unnecessary computations for extending other paths are then saved . the scs decoder @xcite uses a ordered stack @xmath152 to store the candidate paths and tries to find the optimal estimation by searching along the best candidate in the stack . whenever the top path in the stack which has the largest path metric reaches length @xmath4 , the decoding process stops and outputs this path . unlike the candidate paths in the list of scl which always have the same length , the candidates in the stack of scs have difference lengths let @xmath153 denote the maximal the stack @xmath152 in scs decoder . a little different from the original scs in @xcite , an additional parameter @xmath134 is introduced to limit the number of extending paths with certain length in decoding process . a counting vector @xmath154 is used to record the number of the popping paths with specific length , i.e. @xmath155 means the number of popped paths with length-@xmath84 during the decoding process . the scs decoding algorithm with the searching width @xmath134 and the maximal stack depth @xmath153 , denoted by scs @xmath156 , is summarized as follows : ( b.1 ) initialization : push the null path into stack and set the corresponding metric @xmath139 . initialize the counting vector @xmath157 with all - zeros , and the instantaneous stack depth @xmath158 . ( b.2 ) popping : pop the path @xmath159 from the top of stack , and if the path is not null , set @xmath160 . ( b.3 ) expansion : if @xmath140 is a frozen bit , i.e. @xmath161 , simply extend the path to @xmath162 ; otherwise , if @xmath140 is an information bit , extend current path to @xmath163 and @xmath164 . then calculate path metric(s ) by ( [ equ_metric_info_or_froz ] ) , ( [ equ_metric_odd ] ) and ( [ equ_metric_even ] ) . ( b.4 ) pushing : for information bit @xmath165 , if @xmath166 , delete the path from the bottom of the stack . then push the two extended paths into the stack . otherwise , for frozen bit @xmath140 , push the path @xmath167 into stack directly . ( b.5 ) competition : if @xmath168 , delete all the paths with length less than or equal to @xmath169 from the stack @xmath152 . ( b.6 ) sorting : resort paths in the stack from top to bottom in descending metrics . ( b.7 ) determination : if the top path in the stack reaches to the leaf node of the code tree , pop it from the stack . the decoding algorithm stops and outputs @xmath143 as the decision sequence . otherwise go back and execute step ( b.2 ) . [ fig_scs ] gives a simple example of the tree searching under scs . compare with scl in fig . [ fig_scl ] , scs can also find the most probable path @xmath123 with two fewer metric computations . similar to sc and scl , the space efficient structure and `` lazy copy '' technique are applied in the implementation of scs decoders . the time and space complexity of scs are @xmath147 and @xmath170 respectively . however , under the same searching width @xmath134 , the actual computations of scs@xmath171 will be much fewer than that of scl@xmath171 when workding in the moderate or high snr regime . compared with scl , scs decoding can save a lot of unnecessary computations especially when working in the high signal - to - noise ( snr ) regime @xcite . however , the stack used in scs consumes a much larger space than scl . theoretically , to prevent performance deterioration , the stack depth @xmath153 needs to be as large as @xmath172 , thus the space complexity will becomes @xmath173 . fortunately , as shown in @xcite , a much smaller stack - depth @xmath153 is enough for moderate and high snr regimes . but the most appropriate value of @xmath153 is relied on the specific snr and is hard to determine . in this paper , a new hybrid decoding algorithm called successive cancellation hybrid ( sch ) is proposed . sch , as the name suggests , is a hybrid of scl and scs . sch has two working modes called _ on - going _ and _ waiting_. at first , sch decoder works on the on - going mode , it searches along the best candidate path using a ordered stack just the same as that scs does . but when the stack is about to be full , sch stops searching forward and switches to the waiting mode . under the waiting mode , sch turns to extend the shortest path in the stack until all the candidate paths in the stack have the same length . the processing under waiting mode is somewhat similar to scl and it decreases the number of paths in stack . then , sch switches back to the on - going mode again . fig . [ fig_sch ] gives a graphic illustration . this decoding procedure goes on until an @xmath4-length path appears at the top of the stack . the sch algorithm with the searching width @xmath134 , the maximal stack depth @xmath153 , denoted by sch @xmath174 , is summarized as follows : ( c.1 ) initialization : push the null path into stack @xmath152 and set the corresponding metric @xmath139 . initialize the counting vector @xmath175 with all - zeros , and the instantaneous stack depth @xmath158 . the working mode flag @xmath176 is set to @xmath107 , where @xmath107 denote the on - going mode and @xmath85 denote the waiting mode . ( c.2 ) popping : when @xmath177 , pop the path @xmath159 from the top of stack ; else when @xmath178 , pop the path @xmath159 with the shortest path length in the stack . then , if the popped path is not null , i.e. @xmath179 , set @xmath160 . ( c.3 ) expansion : if @xmath140 is a frozen bit , i.e. @xmath180 , simply extend the path to @xmath181 ; otherwise , if @xmath140 is an information bit , i.e. @xmath39 , extend current path to @xmath163 and @xmath164 . then calculate path metric(s ) by ( [ equ_metric_info_or_froz ] ) , ( [ equ_metric_odd ] ) and ( [ equ_metric_even ] ) . ( c.4 ) pushing : for information ( frozen ) bit @xmath140 , push the new two paths ( one path ) into the stack . ( c.5 ) competition : if @xmath168 , delete all the paths with length less than or equal to @xmath169 from the stack @xmath152 . ( c.6 ) mode switching : when @xmath177 and @xmath182 , switch @xmath178 ; when @xmath178 and all the candidate pathes in the stack have equal lengths , @xmath178 ; ( c.7 ) sorting : resort paths in the stack from top to bottom in descending metrics . ( c.8 ) determination : if the top path in the stack reaches to the leaf node of the code tree , pop it from the stack . the decoding algorithm stops and outputs @xmath143 as the decision sequence . otherwise go back and execute step ( c.2 ) . the time and space complexity of sch are @xmath147 and @xmath170 respectively . the actual computations of sch decoding is less than that of scl but is usually more than that of scs . for sch decoding , since no path is dropped when the stack is about to be full , the performance will not affected by @xmath153 . however , when decoder stays in the waiting mode , unnecessary computations will be taken . and the smaller the maximum stack depth is @xmath153 , the more likely the decoder will switch to the waiting mode . so , the computational complexity grows with the decreasing of @xmath153 . to have enough space for waiting mode , the minimum value of @xmath153 is @xmath183 . particularly , when @xmath184 , sch(@xmath134,@xmath153 ) is equivalent to scl(@xmath134 ) ; and when @xmath185 , sch(@xmath134,@xmath153 ) is equivalent to scs(@xmath134,@xmath153 ) . candidates divide the probability space into @xmath134 partitions . ] during the path searching on the code tree , the candidate paths with too small metrics and their descendants will hardly have the chance to be reserved in the future process . in this subsection , we propose a pruning technique to reduce the computational complexity of the improved successive cancellation decoding algorithms . an additional vector @xmath186 is used to record the pruning reference for each level , where @xmath187 is the largest metric of all the traversed @xmath84-length decoding paths on the code tree . more specifically , for scl decoding , @xmath188 and equivalently , for scs and sch , @xmath187 is set to the metric of the first @xmath84-length path popped off the stack . we introduce a new parameter called probability ratio threshold @xmath189 . during the processing at level-@xmath84 on the code tree , a @xmath84-length path with metric smaller than @xmath190 is dropped directly . recall that the path metrics are defined as the logarithmic apps ( [ equ_metric_info_or_froz ] ) . therefore , the pruned paths are those whose apps @xmath191 intuitively , the correct path will possibly be dropped in this pruning operation . in the following part of this subsection , an upperbound of the additional performance deterioration brought by @xmath189 is derived and a conservative configuration of @xmath189 is given . hereafter , scl , scs and sch are collectively referred to as improved successive cancellation ( isc ) decoding algorithms . the block error event of polar code with information set @xmath36 under isc decoding is defined as @xmath192 by introducing pruning operations , the error events can be classified into two kinds . the first kind is the correct path is not lost until the final decision phase , i.e. the correct path is contained in the final list(or stack ) but does not have the largest metric . the second kind is the correct path is lost before the decision step . so , the block error rate ( bler ) of isc can be decomposed as @xmath193 where @xmath194 means the correct path loss . the event @xmath195 can be further decomposed as @xmath196 where @xmath197 is the event that the correct path is not lost until the processing at the @xmath84-th level . there are three kinds of event which will lead to path loss at the @xmath84-th level . the first is brought by the searching width limitation , i.e. the correct path is excluded from the @xmath134 best paths in @xmath84-th decoding step , and is denoted by @xmath198 . the second is brought by the maximum probability ratio limitation , i.e. the metric of the correct path is much smaller than that of the best one , and is denoted by @xmath199 . the third is brought by the maximum stack depth limitation , which only exist in the scs decoding that the correct path is abandoned when the path length equals @xmath84 and the metric is much smaller than the maximum one at that moment , and this event is denoted by @xmath200 . then @xmath201 for scl , sch decoding or scs with a large enough stack depth @xmath153 , @xmath202 . the additional bler performance deterioration brought by pruning is @xmath203 during the processing on the code tree @xmath89 , we will have at most @xmath134 paths at level-@xmath84 with apps @xmath204 which is calculated by ( [ equ_app ] ) , and @xmath205 without loss of generality , we assume that @xmath206 . by the assumption that the one of these paths is the correct path , the @xmath134 probability divided the whole probability space into @xmath134 parts as shown in fig . [ fig_prob_space ] . the event of correct path loss in the pruning processing at the @xmath84-th level has a probability @xmath207 for each of these eliminated paths , the corresponding probability @xmath208 where @xmath209 . so we have @xmath210 the additional error probability brought by @xmath189 is upper bounder by @xmath211 given a tolerable performance deterioration @xmath212 , the value of @xmath189 can be determined as @xmath213 in most cases , since the upperbound in ( [ equ_ptau_upperbound ] ) is very loose , the accrual performance deterioration is usually far less than @xmath212 . the configuration of @xmath189 in ( [ equ_tau_value ] ) is very conservative . in this section , the performance and complexity of the improved successive cancellation ( isc ) decoding algorithms will be discussed . to simplify the complexity evaluation of polar decoding , we measure the average computational complexity in terms of the number of metric recursive operations , which are defined in ( [ equ_metric_odd ] ) or ( [ equ_metric_even ] ) . for example , the computational complexity of sc decoder is @xmath214 . [ fig_diff_len_performance ] gives the simulation results with code length @xmath4 set as @xmath215 and @xmath216 , and the code rate @xmath217 . and [ fig_diff_rate_performance ] shows the bler performances with code rate @xmath218 set as @xmath219 and @xmath220 , and the code length @xmath4 is fixed to @xmath215 . the lowerbounds of bler performance under maximum - likelihood ( ml ) decoding are obtained by performing scl(@xmath221 ) decoding and counting the number of times the decoded codeword is more likely than the transmitted one . the probability ratio threshold @xmath189 for pruning operation is set by ( [ equ_tau_value ] ) with @xmath222 . as shown in the figures , under proper configurations , all the three decoding algorithms can achieve the performance very close to that of ml decoding . the average computational complexities under different decoding algorithms with code length @xmath223 and code rate @xmath217 are shown in fig . [ fig_diff_dec_complexity ] . we can see that the complexity of sch is not monotonically decreasing with the increasing of snr . this is because the switching between the two working modes is relied on the certain code construction and searching procedures . however , sch always has a much lower computational complexity than that of scl . although it needs more computations than scs , sch occupies less memory space without any deterioration in performance . in fact , under some specific configurations , sch can be equivalent to the other two decoding algorithms : when @xmath184 , sch(@xmath134 , @xmath183 ) is equivalent to scl(@xmath134 ) ; and when @xmath153 is very large , sch(@xmath134 , @xmath153 ) is equivalent to scs(@xmath134 , @xmath153 ) ; therefore , sch can achieve a better trade - off between computational complexity and space complexity . furthermore , by applying the pruning technique introduced in section [ subsec_prune ] , the computational complexity can be significantly reduced and very close to that of sc in the moderate and high snr regime . compared with sc , isc decoding algorithms introduce three more parameters : the searching width @xmath134 , the maximum stack depth @xmath153 and probability ratio threshold for pruning @xmath189 . in the following part of this section , we will analysis the impacts on performance and complexity of this three parameters one - by - one . ] ] fig . [ fig_diff_l_performance ] gives the performance comparisons under scl decoding with different @xmath134 . the code length and code rate are set as @xmath223 and @xmath224 respectively . the searching width @xmath134 varies from @xmath85 ( equivalent to sc ) to @xmath225 . note that , scl(@xmath134 ) is equivalent to sch(@xmath134 , @xmath183 ) and scs(@xmath134 , @xmath153 ) with a large enough @xmath153 . the affects brought by different @xmath134 in scl decoding are the same with that in scs and sch . the larger the searching width is , the less probable to lose the correct path , i.e. @xmath226 in ( [ equ_event_ci ] ) is a decreasing function of @xmath134 . but according to the results depicted in fig . [ fig_diff_l_complexity ] , the computational complexity is approximately proportional to @xmath134 . as shown in fig . [ fig_diff_l_performance ] , @xmath227 is good enough for @xmath223 and @xmath224 . ] for polar codes under scs decoding , a too small value of the maximum stack depth @xmath153 will lead to significant deterioration on performance . as shown in fig . [ fig_diff_d_performance ] , @xmath153 need to be larger than @xmath215 for scs decoding . but for sch , the different configurations of @xmath153 no longer affect the bler performance but the computational complexity . as shown in fig . [ fig_diff_d_complexity ] , the computational complexity of sch is decreasing with the increasing of @xmath153 . although it needs more computations than scs , sch occupies less memory space without any deterioration in performance . compared with scl , sch has much lower computational complexity and only require a little more memory space . in fact , under some specific configurations , sch can be equivalent to the other two decoding algorithms : when @xmath184 , sch(@xmath134 , @xmath183 ) is equivalent to scl(@xmath134 ) ; and when @xmath153 is very large , sch(@xmath134 , @xmath153 ) is equivalent to scs(@xmath134 , @xmath153 ) ; hence , sch can achieve a better trade - off between computational complexity and space complexity . ] ] ] fig . [ fig_diff_tau_performance ] and fig . [ fig_diff_tau_complexity ] give simulations of polar codes with code length @xmath223 code rate @xmath224 over binary - input additive gaussian noise channels ( bawgncs ) . the codes are decoded by sch decoders with @xmath227 , @xmath228 and @xmath189 varies from @xmath85 to @xmath229 . as shown in the figures , the computational complexity will be reduced when the increasing of @xmath189 , while the bler performance will be deteriorated with a too small @xmath189 . larger values of @xmath189 such as @xmath230 will introduce little deterioration in performance , but will lead to larger complexities . however , when the codes work in a moderate signal - to - noise ratio ( snr ) regime such as @xmath231db where the bler is less than @xmath232 , the computational complexity differences of sc and sch decoding under different @xmath189 in the simulated regime tends to negligible as shown in fig . [ fig_diff_tau_complexity ] . the successive cancellation ( sc ) decoding algorithm of polar codes and its improved versions , successive cancellation list ( scl ) and successive cancellation stack ( scs ) are restated as path searching procedures on the code tree of polar codes . combining the ideas of scl and scs , a new decoding algorithm named successive cancellation hybrid ( sch ) is proposed , which can achieve a better trade - off between computational complexity and space complexity . to avoid unnecessary path searching , a pruning technique which is suitable for all improved successive cancellation ( isc ) decoders is proposed . performance and complexity analysis based on simulations show that , with the help of the pruning technique , all the isc decoders can have a performance very close to that of maximum - likelihood ( ml ) decoding , and the computational complexities can be very close to that of sc in the moderate and high signal - to - noise ratio ( snr ) regime . e. arikan , `` channel polarization : a method for constructing capacity - achieving codes for symmetric binary - input memoryless channels , '' _ ieee trans . inf . theory _ 55 , no . 7 , pp . 3051 - 3073 , jul .
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as improved versions of successive cancellation ( sc ) decoding algorithm , successive cancellation list ( scl ) decoding and successive cancellation stack ( scs ) decoding are used to improve the finite - length performance of polar codes .
unified descriptions of sc , scl and scs decoding algorithms are given as path searching procedures on the code tree of polar codes .
combining the ideas of scl and scs , a new decoding algorithm named successive cancellation hybrid ( sch ) is proposed , which can achieve a better trade - off between computational complexity and space complexity .
further , to reduce the complexity , a pruning technique is proposed to avoid unnecessary path searching operations .
performance and complexity analysis based on simulations show that , with proper configurations , all the three improved successive cancellation ( isc ) decoding algorithms can have a performance very close to that of maximum - likelihood ( ml ) decoding with acceptable complexity .
moreover , with the help of the proposed pruning technique , the complexities of isc decoders can be very close to that of sc decoder in the moderate and high signal - to - noise ratio ( snr ) regime .
polar codes , successive cancellation decoding , code tree , tree pruning .
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fourier - transform infrared ( ftir ) spectroscopy is a widely utilized method to investigate the optical response of gasses , liquids and solids @xcite . in general , steady - state properties are measured , however , numerous approaches have been developed over the years to explore time - dependent phenomena by fourier - transform interferometry @xcite , many of them optimized for a certain time regime . standard rapid - scan techniques are limited by the mirror velocity to a fraction of a second ( typically 10 ms ) , depending on the spectral resolution @xmath0 required : @xmath1 . since this is often not sufficient , step - scanning is nowadays implemented in several high - end commercial fourier - transform instruments because there is no inherent time limit . it covers the largest dynamical range with a time resolution of the order of nanoseconds determined by the current detector and electronics technology @xcite and still achieving a high spectral resolution @xmath2 . the only restriction is the repeatability of the process : depending on the resolution , spectral range and desired signal - to - noise ratio , it has to be executed hundreds of times . in contrast to the continuously moving interferometer mirror in the rapid - scan configuration , the advantage of the step - scan method is the continuous recording of the signal at a fixed mirror position . this is repeated after the mirror has moved to the next position , eventually composing the complete interferogram . the step - scan technique is mainly applied in biophysics and polymer chemistry where , for instance , the photolysis processes of chemical reactions @xcite , bacteria systems @xcite and the time - dependent reorientation of liquid crystals under the influence of a short electric field @xcite are studied . but also temperature- and light - induced phase transitions can be investigated @xcite . furthermore , it is used to examine the characteristics of lasers and their mode spectra @xcite and for photo - reflection measurements of semiconducting materials as well as quantum wells @xcite . here we want to present and discuss our experimental setups employed to investigate the dynamics of phase transitions . tracing the vibrational spectra of molecules , we probe the configurational changes after the transition has been triggered either by a short laser pulse ( sec . [ sec : opticalswitching ] ) or by an electrical pulse ( sec . [ sec : electricalswitching ] ) . in the left panel of fig . [ fig:1-stepscan ] the data acquisition of a step - scan measurement is schematically depicted . at each mirror position @xmath3 along the traveling distance of the mirror the temporal varying reflection signal @xmath4 is recorded . the complete interferogram is sampled for various times and retardations by successive stepwise moving of the mirror . a subsequent fourier transformation for each measured time point @xmath5 in the intensity `` matrix '' @xmath6 derives the time - resolved spectrum @xmath7 . the process can be repeated several times to improve the signal - to - noise ratio whereas at each @xmath8 position the time - dependent signal is averaged ( typically 10 to 50 times ) . additionally , the stability of the mirror influences the signal - to - noise ratio significantly @xcite , therefore , one must take care that the spectrometer is located in a vibration - free and silent environment . for this reason we place our ftir - spectrometer on a heavy optical bench mounted on air attenuators to decouple the system from the environment . furthermore , the vacuum pumps were placed in a separated room to reduce vibrations and acoustic noise . this way we reached a mirror stability better than 3 nm . the detector signal @xmath9 is recorded as a function of time . after all mirror positions were passed through , the time - dependent signal is recieved by the execution of a fourier transformation for each measured time point @xmath5 . ( right panel ) illustration of the signal sequence within a step - scan experiment . ( a ) external trigger signal t1 given by the external perturbation source , for example a pulse generator or laser . ( b ) the second signal t2 is generated by the spectrometer and sent to the interferometer motor to move the mirror to the next position . ( c ) after a certain predefined stabilization time the third signal t3 waits for the next external trigger signal t1 and rises afterwards immediately . it stays high as long as all data points are captured . ( d ) signal t4 corresponds to each recorded time point within the signal t3.,title="fig:",scaledwidth=47.0% ] the detector signal @xmath9 is recorded as a function of time . after all mirror positions were passed through , the time - dependent signal is recieved by the execution of a fourier transformation for each measured time point @xmath5 . ( right panel ) illustration of the signal sequence within a step - scan experiment . ( a ) external trigger signal t1 given by the external perturbation source , for example a pulse generator or laser . ( b ) the second signal t2 is generated by the spectrometer and sent to the interferometer motor to move the mirror to the next position . ( c ) after a certain predefined stabilization time the third signal t3 waits for the next external trigger signal t1 and rises afterwards immediately . it stays high as long as all data points are captured . ( d ) signal t4 corresponds to each recorded time point within the signal t3.,title="fig:",scaledwidth=47.0% ] [ fig:2-trigger ] since the data acquisition at each mirror position @xmath3 has to start always at the same time , the data recording must be synchronized with the external stimulation source ( laser or pulse generator ) . thus , the trigger signal sequence is very important and crucial in a step - scan measurement . the temporal sequence of the ttl control signals is visualized in the right panel of fig . [ fig:2-trigger ] . an external or an internal trigger signal t1 , which is correlated with the beginning of the reaction , controls the data acquisition . yet , before the recording starts , the spectrometer sends a signal t2 to the interferometer , so that the mirror is moved to the next position . there , it is stabilized for a few milliseconds . after the stabilization procedure ( @xmath10 ms ) the next arising trigger signal t1 is used as a starting signal for the record window t3 . the signal t3 stays high for the total recording time @xmath11 ( @xmath12 total number of time slices ) which is defined at the beginning of each measurement . as soon as the signal t3 is on , at each ttl - signal t4 the detector signal is captured . depending on the number of averaging spectra this procedure is repeated several times starting again with the t3 signal . subsequently , the mirror moves to the next position . the time resolution @xmath13 depends mainly on the response time of the detector . standard pc - mct - detectors , operating in the photo - current mode , have a minimum response time of @xmath14s . their disadvantage is that the measured current becomes nonlinear above a certain incident threshold intensity . a photovoltaic ( pv ) detector has a two order of magnitudes shorter response time due to the small detector area . furthermore , the measured signal is always proportional to the incident light intensity . for the measurement a pc - mct of the model d316 ( bruker optics , ettlingen ) with a time resolution of about 1 @xmath15s and a pv - mct kmpv11 - 11 - 1-j1 from kolmar technologies with a theoretical rise time of 25 ns are utilized . the time resolution also depends on the amplifier and the a / d - converter . there are two amplifiers , the build - in amplifier of the spectrometer and the ka100-a1 from kolmar technologies with a bandwidth of 250 mhz . as an a / d - converter the internal converter of the spectrometer with its time resolution of 6 @xmath15s , with a dynamical range of 24 bit and a maximal input voltage of @xmath16 v , can be used , or a transient recorder m3i.4142 from spectrum systementwicklung microelectronic gmbh , grosshansdorf with a bandwidth of 400 mhz at 16 bit and @xmath17 v. the standard interferogram of a fourier - transform spectrometer consists of an ac- and dc - component : while only the ac - signal contains the important spectral information the dc - signal is usually removed by an electronic high - pass filter . the phase correction can only be performed directly for a rapid - scan measurement . in a step - scan experiment the spectrum can include positive as well as negative features . thus , a phase correction with the raw time - resolved ac - signal does not work . two options exist to resolve this complication : first , the simultaneously recording of the dc - signal and the ac - signal ; the dc - component provides the right phase correction for the ac - component . the second possibility is to use the phase of a previous rapid - scan measurement @xcite . electrically induced phenomena occur mainly in semiconductor leading to non - linear conductivity or negative differential resistance @xcite . high electric fields can also trigger phase transitions , for instance in charge - density wave systems @xcite or correlated insulators @xcite . in organic conductors electrical switching is subject of research for quite some time @xcite . as an example of electric switching , we here have investigated the polarization - dependent vibrational modes of liquid crystals , which allow us to trace the orientation of the molecule upon applying a voltage by time - resolved infrared measurements . in these molecules several vibrational modes exist which can be assigned to different parts of the molecules . by static and time - resolved polarization - dependent measurements we gain on the one hand information about the orientation of the individual molecular building blocks in static positions and on the other hand about the temporal evolution of the electric switching process . hence , we learn about the rotation of the different molecular constituents with the electric field . liquid crystals are widely used for electronic displays due to the possibility to manipulate the orientation of the complex organic molecules by applying an electric field and this way to control the transmitted light . we have chosen the felix 017/100 mixture from the clariant as a ferroelectric liquid crystal that constitutes a chiral smectic c phase ( smc@xmath18 ) at ambient conditions creating a helical structure with @xmath19 , as depicted in fig . [ fig:3-lc](a ) . the spontaneous electric polarization @xmath20 is perpendicular to the director vector @xmath21 and the layer normal vector @xmath22 . while in general the different orientations average and the total polarization is zero , a uniaxial rubbed surface aligns the molecules parallel to each other , leading to a net polarization , sketched in fig . [ fig:3-lc](b ) . the whole procedure is well - known as `` surface stabilized ferroelectric liquid crystal '' ( ssflc ) @xcite . two energetically equal configurations exist with @xmath23 arranged in opposite directions , but perpendicular to the surface with the same tilting angle @xmath24 . an applied electric field can now induce the transition between the two polarization states by changing the orientation of the molecule . ): going from layer to layer the director @xmath21 successively helically precedes around the stacking normal vector @xmath22 in an angle @xmath25 . the spontaneous polarization @xmath26 is perpendicular to the director and the normal vector . the total polarization @xmath27 is zero . ( b ) oriented smectic c phase : all directors @xmath21 are aligned towards the same direction . by symmetry the polarization @xmath28 can adopt two possible values . ( c ) side- and top view of the liquid crystal cell consisting of two caf@xmath29 plates ( light gray , transparent ) . a thin film of ito was sputtered onto the plates ( bright brown ) . polyimid ( green ) was deposited by the spin - coating process on top of the ito layer and serves as an orientation layer for the liquid crystals . uv - glue in connection with small plastic spheres ( shaded area ) with a thickness of @xmath30 m keeps the two windows at a fixed distance . ( d ) transmission spectrum of caf@xmath29(black ) measured at room temperature in comparison to caf@xmath29 coated with ito ( blue ) and caf@xmath29 in combination with ito and polyimid ( red ) . the dashed line indicates a spectral range where water absorption and ice on the detector window corrupt the data.,scaledwidth=80.0% ] while standard liquid - crystal cells consist of glass windows transparent in the visible , we have constructed a cell with two caf@xmath29 windows suitable for the infrared spectral range from 1000 to 50000 @xcite . in fig . [ fig:3-lc](d ) the transmission spectrum of a caf@xmath29 window is plotted in the mid - infrared frequency range . compared to alternative window materials , such as kbr , it is not hydroscopic , and with @xmath31 has a much lower refractive index than ge ( @xmath32 ) , for instance , reducing the reflection losses considerably . the caf@xmath29 windows ( @xmath33 mm@xmath34 , thickness of @xmath35 mm , purchased from korth kristalle gmbh , kiel ) are polished optically to get the optimal surface quality for the optical experiments . both caf@xmath29 windows are sputtered with a thin film of indium tin oxide ( ito ) commonly used as an electrode material for displays because it is lucent over a broad energy range and at the same time conductive . however , optical measurements plotted in fig . [ fig:3-lc](d ) reveal that the transmission decreases rapidly towards low frequencies @xcite . in the range of interesting for the c = c double bond vibrations the transmission is between 5 and 20% , which is still sufficient for our experiment . in order to preorient the liquid - crystal molecules a very thin film of polyimide was deposited on top of the ito layer by using the spin - coating technique which was rubbed afterwards unidirectional [ fig . [ fig:3-lc](c ) ] . as soon as the molecules were placed on the surface , they align themselves along the rubbing direction @xcite . as demonstrated in fig . [ fig:3-lc](d ) the very thin layer of polyimide does not affect the overall light transmission of the system in the relevant frequency range in agreement with previous results @xcite . both plates have been clued together with a uv glue mixed with small macroscopic plastic spheres with a diameter of @xmath30 m which serve as spacer to keep the windows on a constant distance , as sketched in fig . [ fig:3-lc](c ) . after the glue was cured , the cell was heated up above @xmath36c so that the liquid - crystal mixture transformed in the isotropic phase with a reduced viscosity . the liquid crystals were sucked in the chamber due to the capillary action . afterwards , the temperature of the melt was slowly lowered so that a single domain was formed . due to the large dimension of the cell the distance of the two plates was not constant and reduces to the center of the cell causing newton s rings . it changes again , when placed in a vacumm spectrometer . the contact wires were made out of copper and fixed on the exposed ito side areas with indium ; they are connected to a philips - pm 5768b pulse generator to switch the liquid crystals . the generator sends during the measurement simultaneously to the switching voltage pulse a second ttl signal to the spectrometer to start the data acquiring process . the cell is placed in a bruker vertex 66v / s fourier - transform infrared spectrometer together with a suitable infrared polarizer . the absorption becomes maximal if the infrared light is polarized parallel to the electronic transition dipole moment @xmath37 of the molecular vibrational mode . as soon as the external electric field @xmath38 is switched on , a force acts on the liquid crystal . in general , the switching process takes a few microseconds @xmath39 until most of the molecules are reoriented . the new orientation of the molecules and its director @xmath21 change the direction of the transition dipole moment @xmath37 leading to an increase or decrease of the absorption signal . if the electric field is switched on , the directors shown in fig . [ fig:3-lc](b ) spin collectively in one direction and therefore , the azimuthal angle changes . the switching velocity and -time can be derived from the equation of motion @xcite : @xmath40 with @xmath41 the damping due the viscosity of the liquid , @xmath42 is the angle between the electric field and the polarization of the molecule , @xmath43 is the coordinate along the direction perpendicular to the window , and @xmath44 the elasticity constant . due to the large moment of inertia @xmath45 kg / m the equation can be simplified to @xmath46 and solved for small angles @xmath42 @xmath47 with @xmath48 m as the spacing between the two cell windows . we can extract two time constants , @xmath39 for rise with the field and @xmath49 for fall after the electric field @xmath38 is turned off : @xmath50 the switching time for the here examined material felix 017/100 can be estimated using @xmath51 ns/@xmath52 , @xmath53 nc/@xmath52 , @xmath54 n , @xmath55 kv / cm , and @xmath48 m to yield @xmath56s and @xmath57 ms . the mid - infrared transmission spectrum of the liquid crystal cell filled with felix is displayed fig . [ fig:4-transmission ] . the dips in the spectrum between 1000 and 1700 as well as between 2800 and 3000 , enlarged in the insets ( b ) and(c ) , be assigned to the vibrational modes of the liquid - crystal molecules . the features @xmath58 are stretching modes of ch@xmath29- and ch@xmath59 end group . the modes @xmath60 are connected with the vibrations of the c = c double bonds belonging to the center part of the molecules . the features at 1439 and 1395 can be referred to the wiggling and torsion of the ch@xmath59 group . the low - lying resonances at 1269 and 1243 belong to an asymmetric stretch oscillation of the c - o - c bond . the position of the detected features perfectly coincides with the resonance frequencies determined by huang and shih @xcite . [ fig:5-angles ] shows a change of the infrared spectra when an electric field of @xmath61 v is applied . all measurements were performed at @xmath62c . due to the rearrangement of the liquid - crystal molecules a strong modification of the intensity could be observed . we can can deduce two different - oriented transition dipole moments : the first belongs to the vibrational mode of the rigid body of the molecule and the other points out from the molecule axis related to the end groups . the modes containing the ch@xmath29 and ch@xmath59 bonds exhibit a minimum at @xmath63 and a maximum at @xmath64 , correspondingly . the minimum of the modes belonging to the molecule body are slightly shifted and can be found at @xmath64 . the green curve , for instance , belongs to a vibrational mode with an aligned dipole moment parallel to the central molecular frame . at @xmath63 it reaches a maximum for a voltage of 5 v , this means that the dipole moment rotates away from the electric field vector of the infrared light . hence , the transmitted intensity increases at this specific wavelength . in contrast , the intensity of the dipole moments related to the end groups of the molecule is reduced because the transition dipole moments are aligned parallel to the incident radiation . for various polarization states of the incident light measured at room temperature . ( b ) polarization - dependent measurement of the change of the transmitted intensity after creating 5 v on the two windows for seven different vibrational modes . only the strongest modes are displayed . the maximal change of the transmission appears at @xmath63 , followed by a minimum at @xmath64 . the sinusoidal modification evidences that the liquid crystals react on the applied electric field . , scaledwidth=100.0% ] in order to investigate the dynamics of the liquid - crystal switching process the polarizer was fixed at an angle of @xmath65 , and voltages between 4 and 6 v applied between the two plates . the time - dependent infrared signal is presented in a contour plot in fig . [ fig:6-contour ] . the rectangular voltage pulse had a width of 2 ms with a repetition rate of 30 hz and the signal was recorded for a period of 4 ms . to improve the signal - to - noise ratio , five spectra were averaged . the time resolution was set to 25 @xmath15s and the spectral resolution was 2 . we can conclude that the liquid crystal molecules rotate and reorient like a stiff body under the influence of an external electric field . kv / cm with a pulse width of 2 ms . in a spectral region from 1200 to 1800 several features can be recognized corresponding to the orientation of their transition dipole moment . ( b ) time - evolution of the spectra demonstrate for four selected times after the voltage pulse of 6 v was applied . the spectrum ( blue ) , recorded 50 @xmath15s before the pulse is applied , exhibits almost no change and is flat . several features appear at time 200 @xmath15s that can be ascribed to certain vibrational modes . the signal saturates after 1150 @xmath15s and shows no further variation of the intensity.,scaledwidth=100.0% ] the extreme values of @xmath66 are displayed in blue or in red in the contour plot . the transition dipole moments @xmath37 are aligned parallel to @xmath67 lead to a reduced and , hence to a negative signal whereas in contrast for a positive @xmath37 is turned away from @xmath67 . this becomes more obvious in fig . [ fig:6-contour](b ) where spectra at different times are plotted . as a check , data were taken @xmath68s before the pulse ; no variation of the vibrational modes and no shift is observed . after applying the voltage pulse , however , the spectrum dramatically alters and reveals the feature exactly as in fig . [ fig:5-angles](a ) which marks the final state . also the absolute values of the change are the same . in accord with fig . [ fig:4-transmission](a ) and [ fig:5-angles](a ) no variation can be identified below 1000 because the ito layer absorbs most of the incident light . the switching process is completed after about 1 ms and the signal stays constant over a period of 800 @xmath15s . is displayed for the resonance frequency 1606 for three different voltages . the rise of the signal as well as the drop can be fitted well by a single exponential function @xmath69 . the signal increases with increasing voltage . at the same time the switching speed increases . in contrast to that the relaxation process retards with higher voltage . ( b ) voltage - dependence of the rise time @xmath39 for five different resonance frequencies . with increasing voltage the switching time decreases continuously . ( c ) whereas the relaxation time @xmath49 deceases , but no direct dependence on the voltage can be discovered . for 6 v @xmath49 reveals a larger variance.,scaledwidth=100.0% ] to determine the rise and fall time , the temporal evolution of @xmath70 of five vibrational modes were used . the change of the intensity of the @xmath71 mode is plotted as a function of time in fig . [ fig:7-time](a ) for three different voltages . with a rise time of less than 10 ns , the voltage pulse basically has a rectangular shape ; however , the infrared signal reacts slowly and requires several hundreds of microsecond until it saturates . due to the viscosity of the liquid crystals the molecules react with a certain delay to the electric field . the switching time @xmath39 diminishes with rising voltage whereas the amplitude increases slightly . however , the signal overshoots at the beginning for 6 v and relapses back to the value of the 5 v pulse . the rise time can be determined by fitting the experimental data with a single exponential function @xmath72 . the extracted values of @xmath39 are summarized in fig . [ fig:7-time](b ) and ( c ) as a function of the applied voltage . at the end of the pulse the signal relaxes slowly back to its initial values within 2 ms and is at least by a factor of four larger than the rise time @xmath39 . the time for reorientation @xmath49 can be determined from the temporal evolution of the recovery process by a single exponential function . at a voltage of 4 v almost all vibrational modes reveal a switching time @xmath39 between 210 @xmath15s and 230 @xmath15s ; with increasing voltage this range enlarges to maximum 70 @xmath15s for 6 v whereas the values are between 190 @xmath15s and 120 @xmath15s . the various time constants imply that constituents of the molecules react differently on the electric field . the vibrational modes related to the molecule body ( @xmath71-mode ) rotate slower than the ones ( @xmath73-mode ) which are connected to the end groups . the theoretically calculated value for @xmath74s ( for 5 v ) derived from eq . ( [ eq : tau ] ) agrees very well with the experimentally determined @xmath39 which is of about 200 @xmath15s . the reason for the small discrepancy may be due to uncertainties in the distance of the spacer plates or in the material specific parameters @xmath41 and @xmath26 ; they all depend strongly on temperature . from our limited data set , we can not verify the predicted @xmath75 characteristic of the rise time ; our data indicate a constant time . probably , @xmath39 exhibits the expected field dependence and the divergent behavior for smaller voltages . as it was expected from the theoretical calculations of @xmath49 the relaxation times in fig . [ fig:7-time](c ) are larger than the switching time . a time constant of 40 ms was predicted but not overserved . furthermore , the relaxation rate is a function of the applied voltage but according to eq . ( [ eq : tau ] ) it should be independent of any external parameter . one possible explanation therefor is the stronger tilting of the molecules with increasing voltage as well as the total amount of realigned molecules increases as well . thereby , the layer of oriented molecules increases which leads to a kind of self - stabilization effect resulting in the extended decay . for this reason , the relaxation rate @xmath49 is not a function of the intrinsic elastic constant @xmath44 , but also of the external eclectic field . over the last decades photo - induced phase transitions were investigated in several material classes , for instance , polymers @xcite , organic charge - transfer salts @xcite , transition - metal oxides like vanadium oxides @xcite , cuprates @xcite , and manganites @xcite . in the following we study the one - dimensional organic mixed - stack crystals tetrathiavulvalene - chloranil ( ttf - ca ) that undergoes a ionic - neutral phase transition at @xmath76 k that can also be induced by light . initiated by the groundbreaking experiments by koshihara _ @xcite , ultrafast pump - probe experiments have been performed to examine the photo - induced phase transition pipt in the femto- and picoseconds time range @xcite . the understanding of these phenomena was boosted by theories of nasu and yonemitsu @xcite . in the case of ttf - ca open questions concern the relaxation of metastable domains , the related time scale , and the modification of the infrared spectrum due to photo - excitation ; some of these have recently be addressed by time - resolved ftir - spectroscopy @xcite . cl@xmath77o@xmath29 ) and ( b ) tetrathiafulvalene ( ttf , c@xmath78s@xmath77h@xmath77 ) . ( c ) monoclinic unit cell of ttf - ca at room temperature . the ttf and ca molecules are ordered along the crystallographic @xmath79-axis . ( d ) at @xmath80 k the space group of the unit cell is p2@xmath81/n and the ca and ttf molecules are stacked equally spaced along the @xmath79-axis . a further stack is located at @xmath82 , respectively , at which the ttf - ca pairs are tilted opposite to the @xmath79-axis . in the ionic phase the ttf and ca molecules dimerize along the stack . by the charge transfer of about @xmath83 electric dipoles are formed along the stacking direction . , scaledwidth=100.0% ] at ambient conditions the planar molecules of ttf and ca are equidistantly arranged in alternating stacks along the @xmath79-direction ( fig . [ fig:10-ttf - ca ] ) . since the charge transfer @xmath84 as determined from optical experiments is rather small @xcite , this state is referred to as the neutral phase ; upon cooling the charge transfer increases slightly . at @xmath76 k a phase transition occurs where the molecules dimerize along the stack with intermolecular distances 3.504 and 3.685 @xcite , and the ionicity @xmath85 increases from 0.3 @xmath86 to about 0.6 @xmath86 ; fig . [ fig:10-ttf - ca](d ) displays a sketch of the arrangement . accordingly the dielectric constant diverges at the transition with a pronounced frequency dependent response @xcite . the application of 11 kbar pressure shifts the transition to room temperature @xcite . ( green ) of ttf - ca along the stacking direction . above the transition it behaves as a classic band insulator with an activation energy of @xmath87 ev.,scaledwidth=80.0% ] the resistivity of ttf - ca along the stacking direction increases upon cooling , following an activated behavior as displayed in the arrhenius plot of fig . [ fig:11-ttf - ca - rho ] . at @xmath88 , @xmath89 drops by one order of magnitude before it increase again at lower temperatures . the strong enhancement of the conductivity is attributed to the rising number of neutral - ionic domain walls in the neutral phase which also explains the dielectric behavior and non - linear transport @xcite . alternatively it was suggested , that the multiphonon peierls coupling leads upon approaching the phase transition to a progressive shift of spectral weight and of the coupling strength toward the phonons at lower frequencies , ending in a soft - mode behavior only for the lowest - frequency phonon near the transition temperature @xcite . in the proximity of the phase transition , the lowest - frequency phonon becomes overdamped due to anharmonicity induced by its coupling to electrons . in fig . [ fig:13-ttf - ca - refmodes ] the mid - infrared reflectivity and the optical conductivity of ttf - ca are presented for temperatures above and below the neutral ionic transition . in this energy range the intramolecular modes of the ttf and ca molecules can be identified and assigned @xcite ; some of them are sketched in panels ( c ) and ( d ) . along the @xmath79-direction the symmetric a@xmath90 as well as the infrared - active b@xmath91 modes of ca and ttf can be observed whereas the first one is only infrared active due to the emv - coupling . above the phase transition in the neutral phase the ttf and ca molecules are not dimerized and thus the a@xmath92 modes are only weakly infrared - active and the optical conductivity is low . k ( blue ) and 85 k ( red ) taken for the polarization along the stack . the maximum is mainly caused by emv - coupled modes which gain intensity in the dimerized ionic phase . ( c ) moleuclar vibrations of chloranil ( ca ) and ( b ) tetrathiofulvalene ( ttf ) : the upper rows depict the gerade a@xmath90-modes and the lower rows the ungerade b@xmath93-modes.,scaledwidth=100.0% ] in the ionic phase the point inversion symmetry is lost and the molecules become dimerized : the intensity of a@xmath90 modes is enhanced enormously . time - resolved measurements thus allow us to explore the dynamics at the neutral - ionic phase transition by looking at the change in dimerization of the ttf and ca molecules , the increase of the ionicity . we can investigate the building of metastable domains until they eventually annihilate . for photo - induced experiments a nd : yag laser ( b.m industries / thales , yag-502dns - dps920 ) was operated in the second harmonics @xmath94 nm ( 2.35 ev ) of the fundamental wavelength ( 1064 nm ) . the pulse length is 8 ns . the repetition rate can be selected internally and externally between 1 hz and 20 hz . the laser intensity is adjusted continuously by a brewster plate in addition to neutral density filters between 0.1 and 0.5 optical density . the laser intensity was checked by a power and energy meter in front of the sample . the long term stability of the laser power is better than 5% . a telescope arrangement reduces the diameter of the laser beam from 1 cm by a factor of 2 , illustrated in fig . [ fig:8-laser ] . the laser beam is directed from the optical bench to the infrared microscope via several mirrors ( see fig . [ fig:7-setup ] ) . there , it is deflected on the sample by a @xmath65 aluminum coated mirror mounted below the schmidt - cassegrain objective of the bruker hyperion infrared microscope . a lens ( @xmath95 mm ) focuses the beam on the sample , by varying the focal length . the light is circular polarized . the laser system and optical setup are spatially decoupled from the spectrometer and mounted on an optical table to suppress possible external vibrations . the laser pulse sequence is controlled externally to ensure the temporal synchronization between the laser , the pulse generators and the ftir - spectrometer , as depicted in fig . [ fig:8-laser ] . the charging of the flash lamps is triggered by an external signal as well as the pockels cell generating the laser pulse . therefore , a hp pulse generator ( pm 5786 b ) sends the trigger signal x1 to the delay generator ( eg&g princeton applied research model 4144 ) . one of the delay generator output signals x2 is forwarded to a second pulse generator ( hp 214b ) which releases a further delayed pulse x3 with a minimum length of 150 @xmath15s and minimum height of 5 v. it initializes the charging of the flash lamps of the laser system . after signal x3 drops to zero the lamps are charged with a delay of 1.5 ms . it also activates the discharge of the lamp with a delay of 15 @xmath15s . about 30 @xmath15s after the end of the charging and discharge pulse x3 a further voltage signal x4 from the first hp pm 5768 b pulse generator ( length 6 @xmath15s , 6 v ) is sent to the pockels cell generating the laser pulse . in the case of the photoconductivity measurements the delayed trigger signal x5 from the delay generator activates the pulse generator ( avtech electrosystems ltd . , ottawa ) to apply a voltage pulse x6 to the sample . the synchronizing pulse x7 of the avtech device goes to the tektronix oscilloscope to start the acquisition of the photocurrent . the third signal x8 from the delay generator is used to initialize the time - resolved infrared measurement of the ftir - spectrometer . s. the pulse x6 is the voltage pulse which is applied to the sample for the photoconductivity measurements , for instance . the oscilloscope records the variation of the sample current or resistance and is synchronized with the other instrument via the signal x7 . , title="fig:",scaledwidth=57.0% ] s. the pulse x6 is the voltage pulse which is applied to the sample for the photoconductivity measurements , for instance . the oscilloscope records the variation of the sample current or resistance and is synchronized with the other instrument via the signal x7 . , title="fig:",scaledwidth=36.0% ] for the pipt measurements we excite only one of the first transitions of ttf@xmath96 by the second harmonic of the nd : yag laser ; all intermolecular transitions of ca@xmath97 lie above present photon energy . also excitations from lower lying bands into the valence band are possible . _ compared the dependence of the conversion efficiency on the photon energy ; by excitation of intramolecular transitions no threshold intensity occurs to create neutral domains @xcite . and ca@xmath98 pairs in the ionic phase in ttf - ca . a ttf@xmath98 molecules is excited with a laser pulse with the photon energy @xmath99 . ( b ) vertical excitation of the homos of ttf@xmath98 according to the frank - condon - principle in the lumo of ttf@xmath98 . the excitation is strongly localized to the molecule . ( c ) creation of excitons , for instance frenkel - excitons , which are delocalized across the whole molecules . ( d ) via different relaxation processes and channels charge transfer excitons are created which triggers to the transition of the neutral phase . ( e ) the dimerization is suppressed and the charge between the molecules is redistributed . a neutral domain is created in the ionic host matrix which is separated by neutral - ionic domain walls . ( f ) afterwards the neutral domains extend along the one - dimensional chain . by electron - phonon - coupling also neighboring ionic chains are converted into neutral regions and a three - dimensional domain is established . ] in fig . [ fig:14-pipt ] the optical generation of neutral domains is schematically illustrated . the light pulse will create intramolecular frenckel - excitons , which are electron - hole pairs strongly localized on the excited molecules . they decay via various channels into several charge - transfer excitons @xcite and by that lead to a phase transition . in the case of direct excitation only one exciton is created , which is not enough to establish a macroscopic , metastable domain extended over several d@xmath100a@xmath100 pairs . with a sufficient number of photons a multiplicative , non - linear effect can be established , formating metastable domains . by the creation of the one - dimensional , neutral , non - dimerized region , neutral - ionic domain walls are formed between the neutral and ionic parts . the excitation energy of these domain walls is about 0.1 ev @xcite and corresponds to the activation energy of 120 mev and 65 mev in the ionic phase @xcite . this is in good agreement with predictions of an activation energy between 25 and 56 mev @xcite . afterwards , the electronic system couples to the crystal lattice and exciting phonons @xcite via electron - phonon coupling and hence , creates shock - waves . they can convert neighboring chains into neutral domains . ( solid lines ) after photo excitation is depicted for various times . the behavior resembles the static reflectivity difference @xmath101 presented in fig . [ fig:13-ttf - ca - refmodes](a ) . ( b ) and ( c ) normalized @xmath102 for various laser intensities recorded at 1390 for @xmath103 k and for different temperatures for 0.71 mj / cm@xmath34 . the time profile can be successfully fitted by a stretched explonential function @xmath105 ( dashed linies).,scaledwidth=100.0% ] in fig . [ fig:14-ttf - ca - transient](a ) the time - dependent behavior of the reflectivity change @xmath106 at @xmath103 k is plotted for a laser pulse intensity of 0.71 mj / cm@xmath34 . it directly compares to the static reflectivity change @xmath107 plotted in fig . [ fig:13-ttf - ca - refmodes](a ) . by photo excitation @xmath108 becomes negative within a short time which is below the experimental time resolution of 6 @xmath15s . the direct comparison of the @xmath108 shape and the static reflectivity change @xmath109 reveal that the ionic phase was not only dissolved , but also a transition into a neutral state was induced . within several hundreds of microseconds the signal @xmath108 relaxes back to zero which means that the ionic phase is reestablished . moreover , no change of the spectral shape with the elapsed time and laser pulse intensity could be detected . by comparing the shape of @xmath108 with the corresponding difference of the reflectivity @xmath109 between 78 k and 85 k , we conclude that the vanishing of this features indicates the dissolving of the dimerization state between the ttf and ca molecules . moreover , we suggest that metastable , non - dimerized neutral domains in the ionic matrix are created . to trace the temporal evolution of the pipt in dependence of the pump intensity and the sample temperature , we have chosen the very intense @xmath110 ( a@xmath111 ) mode of ttf residing at 1390 since we have asserted that the temporal evolution is the same for the whole spectra . the normalized @xmath102 is represented in fig . [ fig:14-pipt](b ) for different pulse intensities . at the beginning the signal decays very fast and at the end it flattens out . at the vicinity of @xmath88 the first component decays faster with decreasing laser intensity . a fit by a simple single or double exponential function @xcite does not yield satisfactory results . however , we obtain an excellet agreement when using a stretched - exponential function , which is also called kohlrausch - william - watt function @xmath112 as depicted in fig . [ fig:14-ttf - ca - transient](b ) and ( c ) . the fitting parameters @xmath113 and @xmath114 are a function of the laser intensity and decreases from 0.35 to 0.42 and from @xmath115 to @xmath116 s with decreasing laser intensities . in fig . [ fig:14-ttf - ca - transient](c ) @xmath102 is displayed for various temperatures @xmath117 , 73 , and 78 k. far below @xmath88 , the temporal dynamics of the reflectivity drops very fast within the first 20 @xmath15s and approaches asymptotically a constant value which is in contrast to the temporal profile at the vicinity ( @xmath118=78 k ) of @xmath88 which constantly diminishes . similar to the dependence of the fitting parameters on the laser intensity the effective recombination time @xmath114 as well as the stretching exponent decrease from @xmath116 s to @xmath119 s and from 0.42 to 0.23 , respectively , with decreasing sample temperature . the observed time profile can be explained by a random - walk annihilation process of the generated neutral - ionic domain walls . our comprehensive time - resolved infrared study and the random - walk model @xcite allow us to conclude that close to the phase transition , large domains are formed due to the valence instability . we find that the merger and interaction of the induced domains play an important role for the formation of the macroscopic domains and deduce from the model with decreasing laser intensity , the average domain size decreases . at lower temperatures the ionic phase is more robust ; the average domain size is much smaller and changes less with laser intensity . the random walk of the neutral - ionic domain walls is the dominant factor for the relaxation of the metastable domains in the temperature range considered . we have presented two examples where time - resolved infrared investigations using step - scan fourier - transform spectroscopy provide insight in the molecular dynamics at the phase transition and molecular orientation . in the case of liquid crystals , the application of an electric field switches the orientation of the molecular dipoles changing the transmission of the cell for polarized infrared light . watching the time evolution of the pronounced molecular vibrational modes allows us to extract the time constant of approximately 2 ms and its dependence on the applied voltage . the neutral - ionic phase transition of ttf - ca can be photo - excited with a short laser pulse , creating domain walls that are mobile and eventually annihilate . we follow the time - dependence of the vibrational modes activated by the changed dimerization and ionicity . the relaxation back to the initial state strongly depends on the temperature and laser intensity , it extends from a 3 @xmath15s to almost 1 ms . for both examples we give details of the experimental setups and limitations . we demonstrate the applicability of step - scan fourier - transform spectroscopy in a large dynamical range . we thank f. schrg and f. giesselmann for providing the liquid crystals and g. untereiner for continuous help ; e. kurz and n. frhauf supported us when building the cell and use of their clean room . funding by the deutsche forschungsgemeinschaft ( dfg ) is acknowledged . thanks the carl - zeiss - stiftung for financial support . m. mitrano , g. cotugno , clark , s.r . , r. singla , s. kaiser , j. sthler , r. beyer , m. dressel , l. baldassarre , d. nicoletti , a. perucchi , t. hasegawa , h. okamoto , d. jaksch , a. cavalleri , phys . lett . * 112 * , 117801 ( 2014 )
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the time - dependent optical properties of molecular systems are investigated by step - scan fourier - transform spectroscopy in order to explore the dynamics at phase transitions and molecular orientation in the milli- and microsecond range .
the electrical switching of liquid crystals traced by vibrational spectroscopy reveals a rotation of the molecules with a relaxation time of 2 ms .
the photo - induced neutral - ionic transition in ttf - ca takes place by a suppression of the dimerization in the ionic phase and creation of neutral domains .
the time - dependent infrared spectra depend on temperature and laser pulse intensity ; the relaxation of the spectra follows a stretched - exponential decay with relaxation times in the microsecond range strongly dependent on temperature and laser intensity .
we present all details of the experimental setups and thoroughly discuss the technical challenges .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
stellar mass and metallicity are two of the most fundamental physical properties of galaxies . the metal enrichment of the inter stellar medium ( ism ) of a galaxy is a consequence of supernova explosions and stellar winds and is therefore related to the star formation history ( sfh ) of the galaxy . also , the amount of mass in stars is a function of galaxy sfh . therefore , understanding the evolution of the two properties and the relation between them is fundamental to understand the formation and evolution of galaxies . observations have shown that for local and low redshift galaxies a tight relation exists between the galaxy mass and its metallicity ( see e.g. * ? ? ? * ; * ? ? ? the evolution of the mass - metallicity ( mz ) relation has been studied using emission lines from hii regions in galaxies out to @xmath2 and reveal for a galaxy of a given stellar mass a trend of a decreasing metallicity with increasing redshift @xcite . whether the numerous galaxies at the faint end of the luminosity function follow extrapolations of the mz relation can not easily be addressed using emission selected samples , and most studies have focused on composite spectra or a few individually selected massive galaxies @xcite . gravitational lensing which magnifies the flux of background sources is a powerful tool to probe the faint end of the galaxy luminosity function . studies of low - mass gravitationally lensed galaxies hint at a weaker evolution of the mz relation and an increasing scatter out to @xmath3 @xcite . [ cols="<,<,^,^ , < , < , < , < " , ] + + references for measured stellar masses : @xcite for 000926 ; @xcite for 050820a ; @xcite for 070802 ; this work for 090323 . _ { e , c}$ ] ) vs velocity width for qso - dlas ( small dots ) and grb host dlas ( large blue squares ) . best fits ( using equation [ c2 ] ) are shown as dotted lines ( qso - dlas ) , full lines ( grb - dlas ) . , title="fig:",width=328 ] -3 mm a prescription for computation of the stellar mass of qso - dla galaxies from only metallicity and redshift was given in @xcite . @xcite , improved this prescription by adding the effect of metallicity gradients and also performed a test comparing the computed stellar mass to the measured stellar mass from the sed fits . this test was carried out using the complete set of qso - dla galaxies for which the test is currently possible . they concluded that the prescription is confirmed for galaxies of stellar masses down to @xmath4 , while for lower stellar masses there are no available data . here we use the prescription from @xcite ( their equation ( 3 ) including the metallicity gradient term @xmath5 ) to compute the predicted stellar masses of all the host galaxies in our sample ( listed in table [ tab5 ] ) . for 3 of those , stellar masses have been determined directly via sed fitting ( also provided in table [ tab5 ] ) . for the host of grb 090323 , we use the photometric data given in @xcite and determine the stellar mass following the procedure described in @xcite and the initial mass function given in @xcite . in order to obtain the full distribution function of the allowed mass , a monte carlo simulation re - sampling the photometric errors we measure the stellar mass for this host to be @xmath6 . the large error bar is due to only having upper limits on the rest frame optical photometry . we find the measured stellar masses for these four hosts to be in complete agreement with our computed values provided we use the prescription including the metallicity gradient . if we instead use equation ( 1 ) from @xcite , which assumes a constant offset between absorption metallicity and emission metallicity , then the agreement is much poorer with the computed masses in the mean being 1 dex higher than the measured stellar masses . i.e. our data support the hypothesis that the stellar masses of grb - dla galaxies follow the same prescription as do qso - dla galaxies , and that they have metallicity gradients with a slope similar to that of qso - dlas . in a related study of a sample of 18 low redshift grb hosts with measured emission line metallicities and stellar masses from sed fits , @xcite showed that those host galaxies follow the same m - z relation as sdss galaxies , but only after correcting for the high sfr which is a result of the sfr weighted selection we discussed in section [ histos ] . it therefore appears that the available samples of emission selected galaxies , grb selected galaxies , and dla selected galaxies follow the same m - z relations ( when corrected for their specific selection function ) and likely are drawn from the same underlying galaxy sample . in section [ histos ] we described how the metallicity offset seen in figure [ vz - dist ] could be understood as a result of selection functions , but from figure [ zb - corrected ] it is seen that the offset could just as well be caused by the effect of metallicity gradients and different impact parameter distributions . our sample covers a range of stellar masses from @xmath7 to @xmath8 @xmath9 , with a median of @xmath10 ( table [ tab5 ] ) . this median mass is identical to that reported by @xcite for dla galaxies which , held together with the better fits using metallicity gradients described above , supports that at least part of the metallicity offset is a result of different impact parameter distributions . in that case the shift between the two samples seen in figure [ zb - corrected ] is most easily interpreted as the effect of different paths through the gravitational potentials . the interpretation of the observed distribution of data points in figure [ z - corrected ] is therefore complex . effects of redshift evolution , impact parameter distributions , metallicity gradients , and differently weighted selection functions all work to move the data - points , which causes at least part of the scatter of the relation . we here repeat from the conclusions of @xcite that in order to move forward towards an understanding of those objects we need to identify and understand the sources of the scatter . one of the sources ( redshift evolution ) has already been identified . @xcite recently found that half of the scatter in their sample was removed when the effect of metallicity gradients were included . here we have proposed that the effect of gravitational well depth could be an additional cause of scatter . most long duration grb host galaxies display strong intrinsic dla absorption systems similar in nature to the intervening dla systems seen in qso spectra . the grb host systems are , however , different in two ways : they originate inside the host galaxies rather than behind them and they are found at much smaller impact parameters . in addition they are also reported generally to have higher hi column densities and often to have higher metallicities than intervening dlas at the same redshift . it is important to establish if those differences simply are a result of two different selection functions applied to the same underlying sample of high redshift galaxies , or if the two types of galaxies are truly two different populations . we have here analysed the mass / metallicity / redshift relations of a complete literature sample of grb host galaxies and a sample of intervening dla galaxies in order to address this question . we have found that + 1 ) the two samples are fully consistent with being drawn from the same underlying population with a single mz relation , and a single redshift evolution of this relation with a break around @xmath11 . grb hosts are in better agreement with this redshift evolution compared to linear evolutions with constant slopes . + 2 ) there is evidence that the grb host galaxies have higher metallicities , but this is most likely a secondary correlation . the primary correlation is with either impact parameter , with stellar mass , or , presumably , with a combination of the two . the smaller impact parameters combined with a metallicity gradient will produce a metallicity offset , sfr selection bias is predicted to select galaxies of somewhat larger stellar mass than dla galaxies which will likewise cause an offset in metallicity . + 3 ) there is weak evidence that the @xmath12-metallicity relation for the grb hosts is offset towards larger @xmath12 values , as one would predict since their sightlines pass through a deeper part of the dark matter halo potential well than a random sightline to an intervening dla in a halo of the same mass . it has been shown previously that qso - dlas and lyman break galaxies ( lbg ) are consistent with being drawn from the same underlying population by two very different selection functions , where qso - dlas are drawn from the very low - mass end of the lbg population @xcite . with the results presented here we have now added long duration grb hosts to this list , which means that we have made another important step towards a global description of galaxies and galaxy evolution in the early universe . since the sample used in this pilot study is limited , it will be quite feasible to improve the accuracy of all results reported here simply by increasing the sample size . grb host are ideally suited to shed light on the structure of high redshift galaxies . they combine the data from emission selected galaxies directly with those of absorption selected galaxies . i.e. we obtain in the ideal case both absorption metallicity , emission metallicity , stellar mass from sed fits , impact parameter and @xmath12 for a single galaxy . we would like to thank jason x. prochaska for providing the hi column density for the host of grb 090313 prior to publication . we thank an anonymous referee for a careful reading of our manuscript and for many insightful and valuable comments which significantly improved the presentation of our results . we thank karl glazebrook and damien le borgne for helping with the stellar mass calculation of the host of grb 090323 . ma thanks max - planck - institut fr astrophysik for hosting her during the initial part of this work . jpuf acknowledges support from the erc - stg grant eggs-278202 . this work was funded by an eso dgdf grant to pm and wf . the dark cosmology centre is funded by the danish national research foundation . asplund m. , grevesse n. , sauval a.j . , scott p. , 2009 , ara&a , 47 , 481 baldry i.k . , glazebrok k. , 2003 , apj , 593 , 258 belli s. , jones t. , ellis r.s . , richard j. , 2013 , apj , 772 , 141 berger e. , penprase b.e . , cenko s.b . , kulkarni s.r . , , steidel c.c . , reddy n.a . , 2006 , apj , 642 , 979 bird s. , haehnelt m. , neeleman m. , genel s. , vogelsberger m. , hernquist l. , 2014 , arxiv1407.7858b bloom j.s . , kulkarni s.r . , djorgovski s.g . , 2002 , apj , 123 , 1111 castro s. , galama t.j . , harrison f.a . , holtzman j.a . , bloom j.s . , djorgovski s.g . , kulkarni s.r . , 2003 , apj , 586 , 128 chen h. et al . , 2009 , apj , 691 , 152 christensen l. , mller p. , fynbo j.p.u . , zafar t. , 2014 , arxiv1404.6529 christensen et al . , 2012 , mnras , 427 , 1973 christensen l. , hjorth j. , gorosabel j.,2004 , a&a , 425 , 913 chornock r. , berger e. , fox d.b . , fong w. , laskar t. , roth k.c . , 2014 , arxiv1405.7400 cucchiara a. , fumagalli m. , rafelski m. , kocevski d. , prochaska j.x . , cooke r.j . , becker g.d . , 2014 , arxiv1408.3578 cullen f. , cirasuolo m. , mclure r.j . , dunlop j.s . , bowler r.a.a , 2014 , mnras , 440 , 2300 de cia a. et al . , 2011 , mnras , 412 , 2229 delia v. et al . , 2014 , a&a , 564 , 38 delia v. , campana s. , covino s. , davanzo p. , piranomonte s. , tagliaferri g. , 2011 , mnras , 418 , 680 delia v. et al . , 2010 , a&a , 523 , 36 dekel a. , woo j. , 2003 , mnras , 344 , 1131 de ugarte postigo a. et el . , 2010 , a&a , 513 , 42 eladdttir a. et al . , 2009 , apj , 697 , 1725 ellison s. l. , kanekar n. , prochaska j.x . , momjian e. , worseck g. , 2012 , mnras , 424 , 293 erb d.k . , shapley a.e . , pettini m. , steide , c.c . , reddy n.a . , adelberger k.l . , 2006 , apj , 644 , 813 fox a.j . , ledoux c. , vreeswijk p.m. , smette a. , jaunsen a.o . , 2008 , a&a , 491 , 189 fruchter a.s . et al . , 2006 , natur , 441 , 463 fynbo , j.p.u . et al . , 2013 , mnras , 436 , 361 fynbo j.p.u . et al . , 2009 , apjss , 185 , 526 fynbo j.p.u . , prochaska j.x . , sommer - larsen j. , dessauges - zavadsky m. , mller p. , 2008 , apj , 683,321 fynbo j.p.u . et al . , 2006 , a&a , 451 , 47 fynbo j.p.u . , mller p. , warren s.j . , 1999 , mnras , 305 , 849 glazebrook k. et al . , 2004 , natur , 430 , 181 graham j.f . , fruchter a.s . , 2013 , apj , 774 , 119 henry a. et al . , 2013 , apj , 776 , 27h hjorth j. et al . , 2012 , apj , 756 , 187 krogager j .- k . et al . , 2013 , mnras , 433 , 3091 krogager j .- k . , fynbo j.p.u . , mller p. , ledoux c. , noterdaeme p. , christensen l. , milvang - jensen b. , sparre m. , 2012 , mnras , 424 , 1 krhler et al . , 2013 , a&a , 557 , 18 krhler et al . , 2011 , a&a , 534 , 108 ledoux c. , vreeswijk p.m. , smette a. , fox a.j . , petitjean p. , ellison s.l . , fynbo j.p.u . , savaglio s. , 2009 , a&a , 506 , 661 ledoux c. , petitjean p. , fynbo j.p.u . , mller p. , srianand r. , 2006 , a&a , 457 , 71 ledoux c. , bergeron j. , petitjean p. , 2002 , a&a , 385 , 802 le floch e. et al . , 2003 , a&a , 400 , 499 maiolino r. , et al . , 2008 , a&a , 488 , 463 mannucci f. , salvaterra r. , campisi m.a . , 2011 , mnras , 414 , 1263 michaleowskiowski m.j . et al . , 2012 , apj , 755 , 85 mcbreen s. et al . , 2010 , a&a , 516 , 71 mller p. , fynbo j.p.u . , ledoux c. , nilsson k.k . , 2013 , mnras , 430 , 2680 mller p. , warren s.j . , fall s.m . , fynbo j.p.u . , jakobsen p. , 2002 , apj , 574 , 51 mller p. , warren s.j . , 1998 , mnras , 299 , 661 neeleman m. , wolfe a.m. , prochaska j.x . , rafelski m. , 2013 , apj , 769 , 54 perley d.a . et al . , 2013 , apj , 778 , 128 proux c. , meiring j.d . , kulkarni v.p . , khare p. , lauroesch j.t . , vladilo g. , york d.g . , 2008 , mnras , 386 , 2209 proux c. , meiring j.d . , kulkarni v.p . , ferlet r. , khare p. , lauroesch j.t . , vladilo g. , york , d.g . , 2006 , mnras , 372 , 369 pettini m. , ellison s.l . , steidel c.c . , shapley a.e . , bowen d.v . , 2000 , apj , 532 , 65 pettini m. , smith l.j . , king d.l . , hunstead r.w . , 1997 , apj , 486 , 665 piranomonte s. et al . , 2008 , a&a , 492,775 pontzen a. et al . 2008 , mnras , 390 , 1349 price p.a . et al . , 2007 , apj , 663 , 57 prochaska j.x . , chen h. , wolfe a.m. , dessauges - zavadsky m. , bloom j.s . , 2008 , apj , 672 , 59 prochaska j.x . , chen h. , dessauges - zavadsky m. , bloom j.s . , 2007a , apj , 666 , 267 prochaska j.x . et al . , 2007b , apjs , 168 , 231 prochaska j.x . , gawiser e. , wolfe a.m. , castro s. , djorgovski s.g . , 2003 , apj , 595 , 9 prochaska j.x . , wolfe a.m. , 1997 , apj , 487 , 73 rafelski m. , wolfe a.m. , prochaska j.x . , neeleman m. , mendez a.j . , 2012 , apj , 755 , 89 richard j. , jones t. , ellis r. , stark d.p . , livermore r. , swinbank m. , 2011 , mnras , 413 , 643 savaglio s. et al . , 2012 , mnras , 420 , 627 savaglio s. , glazebrook k. , le borgne d. , 2009 , apj , 691 , 182 savaglio s. , 2006 , njp , 8 , 195 savaglio s. et al . , 2005 , apj , 635 , 260 savaglio s. , fall s.m . , fiore f. , 2003 , apj , 585 , 638 sparre et al . , 2014 , apj , 785 , 150 swinbank a.m. , sobral d. , smail ian , geach j.e . , best p.n . , mccarthy i.g . , crain r.a . , theuns t. , 2012 , mnras , 426 , 935 tescari e. , viel m. , tornatore l. , borgani s. , 2009 , mnras , 397 , 411 thne c.c . et al . , 2013 , mnras , 428 , 3590 tremonti et al . 2004 , apj , 613 , 898 troncoso p. , et al . , 2014 , a&a , 563 , 58 van zee l. , salzer j.j . , haynes m.p . , odonoghue a.a . , balonek t.j . , 1998 , aj , 116 , 2805 vreeswijk p.m. et al . , 2004 , a&a , 419 , 927 watson d. et al . , 2006 , apj , 652 , 1011 wuyts e. , rigby j.r . , sharon k. , gladders m.d . , 2012 , apj , 755 , 73 zwaan m.a . , van der hulst j.m . , briggs f.h . , verheijen m.a.w . , ryan - weber e.v . , 2005 , mnras , 364 , 1467
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we analyze a sample of 16 absorption systems intrinsic to long duration grb host galaxies at @xmath0 for which the metallicities are known .
we compare the relation between the metallicity and cold gas velocity width for this sample to that of the qso - dlas , and find complete agreement .
we then compare the redshift evolution of the mass - metallicity relation of our sample to that of qso - dlas and find that also grb hosts favour a late onset of this evolution , around a redshift of @xmath1 .
we compute predicted stellar masses for the grb host galaxies using the prescription determined from qso - dla samples and compare the measured stellar masses for the four hosts where stellar masses have been determined from sed fits .
we find excellent agreement and conclude that , on basis of all available data and tests , long duration grb - dla hosts and intervening qso - dlas are consistent with being drawn from the same underlying population .
grb host galaxies and qso - dlas are found to have different impact parameter distributions and we briefly discuss how this may affect statistical samples .
the impact parameter distribution has two effects .
first any metallicity gradient will shift the measured metallicity away from the metallicity in the centre of the galaxy , second the path of the sightline through different parts of the potential well of the dark matter halo will cause different velocity fields to be sampled .
we report evidence suggesting that this second effect may have been detected .
[ firstpage ] galaxies : high - redshift galaxies : ism galaxies : star formation galaxies :
evolution galaxies : formation quasars : absorption lines
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You are an expert at summarizing long articles. Proceed to summarize the following text:
this paper constitutes a survey of a collection of results from @xcite as well as the announcement of new results on the @xmath5-fractal billiard table @xmath6 ( see @xcite ) and a self - similar sierpinski carpet billiard table @xmath7 ( see @xcite ) . in [ sec : rationalbilliards ] and [ sec : fractalgeometry ] , we survey the necessary background material for understanding the remainder of the article . more specifically , in [ sec : rationalbilliards ] , we introduce the notion of a rational polygonal billiard , a translation surface determined from a rational polygonal billiard and discuss the consequence of a dynamical equivalence between the billiard flow and the geodesic flow . this dynamical equivalence allows us to express an orbit of a rational billiard table as a geodesic on an associated translation surface , and vice - versa , with the added benefit of being able to determine the reflection in certain types of vertices of a rational billiard table . furthermore , in [ sec : fractalgeometry ] , we provide additional background material from the subject of fractal geometry necessary for understanding the construction of the koch snowflake @xmath12 , @xmath5-fractal @xmath13,-fractal @xmath13 was previously studied in a different context in @xcite . ] and a sierpinski carpet @xmath15 , as well as particular orbits and _ nontrivial paths_. we then combine the background material presented in [ sec : rationalbilliards ] and [ sec : fractalgeometry ] to analyze the prefractal billiard tables @xmath8 , @xmath9 and @xmath10 , for @xmath11 . we begin by providing a general language for prefractal billiards and subsequently focus on determining sufficient conditions for what we are calling a _ sequence of compatible periodic orbits_. while [ subsec : theprefractalkochsnowflakebilliard][subsec : aprefractalselfsimilarsierpinskicarpetbilliard ] contain specific results and specialized definitions , there is an over - arching theme that is more fully developed in [ sec : fractalbilliards ] . in addition to providing a general language within which to analyze a fractal billiard , we discuss in [ subsec : thekochsnowflakefractalbilliard][subsec : aself - similarsierpinskicarpetbilliard ] how one can determine well - defined orbits of @xmath4 , @xmath6 and @xmath7 , as well as nontrivial paths of @xmath4 and @xmath6 that connect two elusive points of each respective billiard . relying on the main result of @xcite , the second author and joe p. chen have shown that it is possible to determine a periodic orbit of a self - similar sierpinski carpet billiard @xmath7 ; additional results and proofs are forthcoming in @xcite , but a synopsis is provided in [ subsec : aprefractalselfsimilarsierpinskicarpetbilliard ] and [ subsec : aself - similarsierpinskicarpetbilliard ] . many of the results in [ sec : prefractalrationalbilliards ] and [ sec : fractalbilliards ] are being announced for the first time . specifically , [ subsec : thetfractalprefractalbilliard ] and [ subsec : thetfractalbilliard ] contain new results on the prefractal @xmath5-fractal billiard @xmath9 and the @xmath5-fractal billiard @xmath6 ( see @xcite ) ; [ subsec : aprefractalselfsimilarsierpinskicarpetbilliard ] and [ subsec : aself - similarsierpinskicarpetbilliard ] contain new results for a prefractal sierpinski carpet billiard @xmath10 and self - similar sierpinski carpet billiard @xmath7 , where @xmath16 is the single underlying scaling ratio ( see @xcite ) . as these sections constitute announcements of new results on the respective prefractal and fractal billiards , we will provide in future papers @xcite detailed statements and proofs of the results given therein . given the nature of the subject of _ fractal billiards _ , we will close with a discussion of open problems and possible directions for future work , some of which are to appear in @xcite and @xcite . in this section , we will survey the dynamical properties of a billiard ball as it traverses a region in the plane bounded by a closed and connected polygon . in the latter part of this article , we will remove the stipulation that the boundary be a polygon and focus on billiard tables having boundaries that are fractal or containing subsets that are fractal ( while still being simple , closed and connected curves in the plane ) . under ideal conditions , we know that a point mass making a perfectly elastic collision with a @xmath17 surface ( or curve ) will reflect at an angle which is equal to the angle of incidence , this being referred to as the _ law of reflection_. consider a compact region @xmath1 in the plane with simple , closed and connected boundary @xmath0 . then , @xmath1 is called a _ planar billiard _ when @xmath0 is smooth enough to allow the law of reflection to hold , off of a set of measure zero ( where the measure is taken to be the arc length measure on @xmath0 ) . though the law of reflection implicitly states that the angles of incidence and reflection be determined with respect to the normal to the line tangent at the basepoint , we adhere to the equivalent convention in the field of mathematical billiards that the vector describing the position and velocity of the billiard ball ( which amounts to the position and angle , since we are assuming unit speed ) be reflected in the tangent to the point of incidence .. ] that is , employing such a law in order to determine the path on which the billiard ball departs after impact essentially amounts to identifying certain vectors . such an equivalence relation is denoted by @xmath18 and , in the context of a polygonal billiard , is discussed below in more detail . for the remainder of the article , unless otherwise indicated , when @xmath0 is a simple , closed , connected and piecewise smooth curve so as to allow the law of reflection to hold ( off finitely many points ) , we assume @xmath0 is a closed and connected polygon . in such a case , we will refer to @xmath1 as a _ polygonal billiard_. one may express the law of reflection in terms of equivalence classes of vectors by identifying two particular vectors that form an equivalence class of vectors in the unit tangent bundle corresponding to the billiard table @xmath1 ; see figure [ fig : billiardmap ] . ( see @xcite for a detailed discussion of this equivalence relation on the unit tangent bundle @xmath19 . ) denote by @xmath20 the unit circle , which we let represent all the possible directions ( or angles ) in which a billiard ball may initially move . to clearly understand how one forms equivalence classes from elements of @xmath19 , we let @xmath21 and say that @xmath22 if and only if @xmath23 and one of the following is true : 1 . @xmath23 is not a vertex of the boundary @xmath0 and @xmath24 ; 2 . @xmath23 is not a vertex of the boundary @xmath0 , but @xmath23 is a point on a segment @xmath25 of the polygon @xmath0 and @xmath26 , where @xmath27 denotes reflection in the segment @xmath25 ; 3 . if @xmath23 is a vertex of @xmath0 , then we identify @xmath28 with @xmath29 for every @xmath30 in the group generated by reflections in the two adjacent sides having @xmath31 ( or @xmath32 ) as a common vertex . for now , we shall denote by @xmath33 $ ] the equivalence class of @xmath28 , relative to the equivalence relation @xmath18 . the collection of vertices of @xmath1 forms a set of zero measure ( when we take our measure to be the arc - length measure on @xmath0 ) , since there are finitely many vertices . the phase space for the billiard dynamics is given by the quotient space @xmath34 . in practice , one restricts his or her attention to the space @xmath35 . the billiard flow on @xmath35 is determined from the continuous flow on @xmath36 as follows . let @xmath37 be an initial basepoint , @xmath38 be an initial direction and @xmath39 be a flow line corresponding to these initial conditions in the phase space @xmath34 . the values @xmath40 for which @xmath41 constitute the _ return times _ ( i.e. , times at which @xmath39 returns to the section , or intersects it in a non - tangential way ) . then , the discrete map @xmath42 constitutes the _ section map_. in terms of the configuration space , @xmath42 constitutes the point and angle of incidence in the boundary @xmath0 . since @xmath1 is the billiard table and we are interested in determining the collision points , it is only fitting that such a map be called the _ billiard map_. more succinctly , we denote @xmath43 by @xmath44 and , in general , such a map is called the _ poincar map _ and the section is called the _ poincar section_. furthermore , the obvious benefit of having a visual representation of @xmath45 in the configuration space is exactly why one restricts his or her attention to the section @xmath35 . specifically , all one really cares about in the end , from the perspective of studying a planar billiard , are the collision points , which are clearly determined by the billiard map . in order to understand how one determines the next collision point and direction of travel , we must further discuss the _ billiard map _ @xmath46 . as previously discussed , @xmath47 , where the equivalence relation @xmath18 is the one introduced above . more precisely , if @xmath38 is an inward pointing vector at a basepoint @xmath37 , then @xmath48 is the representative element of the equivalence class @xmath49 $ ] . the billiard map then acts on @xmath35 by mapping @xmath50 $ ] to @xmath51 $ ] , where @xmath52 and @xmath53 are collinear in the direction determined by @xmath54 and where @xmath55 is the reflection of angle @xmath54 through the tangent at @xmath53 . in general , we have @xmath56 = [ ( x^k,\theta^k)]$ ] , for every @xmath57 . in the sequel , we will simply refer to an element @xmath58\in ( \omega(d)\times s^1)/\sim$ ] by @xmath59 , since the vector corresponding to @xmath54 is inward pointing at the basepoint @xmath52 . so as not to introduce unnecessary notation , when we discuss the billiard map @xmath60 corresponding to the @xmath61th prefractal billiard @xmath62 approximating a fractal billiard @xmath3 , we will simply write @xmath60 as @xmath63 . when discussing the discrete billiard flow on @xmath64 , the @xmath65th point in an orbit @xmath66 will instead be denoted by @xmath67 , in order to keep track of the space such a point belongs to ( namely , with our present convention , @xmath64 ) . specifically , @xmath68 refers to the number of iterates of the billiard map @xmath63 necessary to produce the pair @xmath67 . an initial condition of an orbit of @xmath62 will always be referred to as @xmath69 . in what follows , we are presupposing an orbit can be formed by iterating the billiard map forward in time and backwards in time , whenever @xmath70 is defined . an orbit making finitely many collisions in the boundary is called a _ closed orbit_. if , in addition , there exists @xmath71 such that @xmath72 , then the resulting orbit is called _ periodic _ ; the smallest positive integer @xmath73 such that @xmath74 is called the _ period _ of the periodic orbit . in the event that a basepoint @xmath75 of @xmath76 is a corner of @xmath1 ( that is , a vertex of the polygonal boundary @xmath0 ) and reflection can not be determined in a well - defined manner , then the resulting orbit is said to be _ singular_. in addition , if there exists a positive integer @xmath65 such that the basepoint @xmath77 of @xmath78 is a corner of @xmath1 ( here , @xmath79 denotes the @xmath65th inverse iterate of @xmath46 ) , then the resulting orbit is closed and the path traced out by the billiard ball connecting @xmath75 and @xmath77 is called a _ saddle connection_. finally , we note that a periodic orbit with period @xmath73 is a closed orbit for which reflection is well defined at each basepoint @xmath80 of @xmath81 , @xmath82 and @xmath72 . we say that an orbit @xmath83 is _ dense _ in a rational billiard table @xmath1 if the path traversed ( forward and backward in time ) by the billiard ball in @xmath1 is dense in @xmath1 . that is , the closure of the set of points comprising the path traversed by the billiard ball is exactly @xmath1 . likewise , the points of incidence ( i.e. , the footprint ) of a dense orbit will be dense in the boundary @xmath0 , as explained in remark [ rmk : denseorbits ] . [ rmk : denseorbits ] consider a rational polygonal billiard @xmath1 . the associated translation surface @xmath84 can be constructed as described in [ subsec : translationstructuresandtranslationsurfaces ] . as we will show in [ subsec : unfoldingabilliardorbit ] , the geodesic flow on a translation surface is dynamically equivalent to the billiard flow . a dense orbit will have an initial direction preventing the path from being parallel to any side of @xmath1 ( except , possibly , for finitely many initial directions , and hence , for a measure - zero set ) . the corresponding path on the associated translation surface for an explanation of what constitutes a translation surface . ] must also be dense in the surface . since the path on the surface is arbitrarily close to every side appropriately identified with another side of a copy of @xmath1 and not parallel to any side , the path will be transversal with respect to each side . thus , the collection of basepoints of a dense orbit must be dense in @xmath0 . [ def : footprintofanorbit ] let @xmath85 be an orbit of a billiard @xmath1 with an initial condition @xmath86 . then the trace of an orbit on the boundary @xmath0 , @xmath87 is called the _ footprint _ of the orbit @xmath85 and is denoted by @xmath88 . when we are only interested in a prefractal billiard @xmath89 , we denote the footprint of an orbit by @xmath90 . for the remainder of the article , when discussing polygonal billiards , we will focus our attention on what are called _ rational polygonal billiards _ , or , more succinctly , _ rational billiards_. if @xmath0 is a nontrivial connected polygon such that for each interior angle @xmath91 of @xmath0 there are relatively prime integers @xmath92 and @xmath93 such that @xmath94 , then we call @xmath0 a _ rational polygon _ and @xmath1 a _ rational billiard_. [ def : ratbilliard ] in this subsection , we will discuss what constitutes a translation surface and how to construct a translation surface from a rational billiard . then , in [ subsec : unfoldingabilliardorbit ] , we will see how to relate the continuous billiard flow on @xmath34 with the geodesic flow on the associated translation surface . [ def : translationstructure ] let @xmath95 be a compact , connected , orientable surface . a _ translation structure _ on @xmath95 is an atlas @xmath96 , consisting of charts of the form @xmath97 , where @xmath98 is a domain ( i.e. , a connected open set ) in @xmath95 and @xmath99 is a homeomorphism from @xmath98 to a domain in @xmath100 , such that the following conditions hold : 1 . the collection @xmath101 covers the whole surface @xmath95 except for finitely many points @xmath102 , called _ singular points _ ; 2 . all coordinate changing functions are translations in @xmath100 ; 3 . the atlas @xmath96 is maximal with respect to properties @xmath103 and @xmath104 ; 4 . for each singular point @xmath105 , there is a positive integer @xmath106 , a punctured neighborhood @xmath107 of @xmath105 not containing other singular points , and a map @xmath108 from this neighborhood to a punctured neighborhood @xmath109 of a point in @xmath100 that is a shift in the local coordinates from @xmath96 , and is such that each point in @xmath109 has exactly @xmath106 preimages under @xmath108 . we say that a connected , compact surface equipped with a translation structure is a _ translation surface_. note that in the literature on billiards and dynamical systems , the terminology and definitions pertaining to this topic are not completely uniform ; see , for example , @xcite . ( we note that in @xcite and @xcite , ` translation surfaces ' are referred to as ` flat surfaces ' . ) we have adopted the above definition for clarity and the reader s convenience . we now discuss how to construct a translation surface from a rational billiard . consider a rational polygonal billiard @xmath1 with @xmath65 sides and interior angles @xmath110 at each vertex @xmath105 , for @xmath111 , where the positive integers @xmath112 and @xmath113 are relatively prime . the linear parts of the planar symmetries generated by reflection in the sides of the polygonal billiard @xmath1 generate a dihedral group @xmath114 , where @xmath115 ( the least common multiple of the @xmath113 s ) . next , we consider @xmath116 ( equipped with the product topology ) . we want to glue ` sides ' of @xmath117 together and construct a natural atlas on the resulting surface @xmath95 so that @xmath95 becomes a translation surface . as a result of the identification , the points of @xmath95 that correspond to the vertices of @xmath1 constitute ( removable or nonremovable ) conic singularities of the surface . heuristically , @xmath116 can be represented as @xmath118 , in which case it is easy to see what sides are made equivalent under the action of @xmath18 . that is , @xmath18 identifies opposite and parallel sides in a manner which preserves the orientation . see example [ exa : equilateraltriangleflatsurface ] and figure [ fig : sixequisurface ] for an example of a translation surface constructed from the equilateral triangle billiard @xmath119 . can be acted on by a particular group of symmetries to produce a translation surface that is topologically equivalent to the flat torus . in this figure , we see that opposite and parallel sides are identified in such a way that the orientation is preserved . this allows us to examine the geodesic flow on the surface . we will see in [ subsec : unfoldingabilliardorbit ] that the geodesic flow on the translation surface is dynamically equivalent to the continuous billiard flow . ] [ exa : equilateraltriangleflatsurface ] consider the equilateral triangle @xmath120 . the corresponding billiard is denoted by @xmath119 . the interior angles are @xmath121 . hence , the group acting on @xmath119 to produce the translation surface is the dihedral group @xmath122 . the resulting translation surface is topologically equivalent to the flat torus . we will make use of this fact in the sequel . consider a rational polygonal billiard @xmath1 and an orbit @xmath83 . reflecting the billiard @xmath1 and the orbit in the side of the billiard table containing the basepoint @xmath123 of the orbit ( or an element of the footprint of the orbit ) partially unfolds the orbit @xmath83 ; see figure [ fig : partiallyunfoldinganorbitofthesquarebilliard ] for the case of the square billiard @xmath124 . continuing this process until the orbit is a straight line produces as many copies of the billiard table as there are elements of the footprint ; see figure [ fig : unfoldinganorbitofthesquarebilliard ] . that is , if the period of an orbit @xmath83 is some positive integer @xmath125 , then the number of copies of the billiard table in the unfolding is also @xmath125 . we refer to such a straight line as the _ unfolding of the billiard orbit_. . the ` r ' is shown so as to provide the reader with a frame of reference . ] . ] given that a rational billiard @xmath1 can be acted on by a dihedral group @xmath114 to produce a translation surface in a way that is similar to unfolding the billiard table , we can quickly see how the billiard flow is dynamically equivalent to the geodesic flow ; see figure [ fig : equivalenceofgeodesicflow ] and the corresponding caption . and correctly identifying sides so as to recover the flat torus , we see that the unfolded orbit corresponds to a closed geodesic of the translation surface . ] one may modify the notion of `` reflecting '' so as to determine orbits of billiard tables tiled by a rational polygon @xmath0 . as an example , we consider the unit - square billiard table . an appropriately scaled copy of the unit - square billiard table can be tiled by the unit - square billiard table by making successive reflections in the sides of the unit square . one may then unfold an orbit of the unit - square billiard table into a larger square billiard table . when the unfolded orbit of the original unit - square billiard intersects the boundary of the appropriately scaled ( and larger ) square , then one continues unfolding the billiard orbit in the direction determined by the law of reflection ( that is , assuming the unfolded orbit is long enough to reach a side of the larger square ) . we will refer to such an unfolding as a _ reflected - unfolding_. we may continue this process in order to form an orbit of a larger scaled square billiard table . suppose that an orbit @xmath83 has period @xmath125 . the footprint of the orbit is then @xmath126 . if @xmath127 is a positive integer ( i.e. , @xmath128 ) , then the footprint @xmath129 of an orbit constitutes the footprint of an orbit that traverses the same path @xmath127-many times . for sufficiently large @xmath128 , an orbit that traverses the same path as an orbit @xmath83 @xmath127-many times can be reflected - unfolded in an appropriately scaled square billiard table to form an orbit of the larger billiard table ; see figure [ fig : areflectedunfoldinginthesquare ] . such a tool is useful in understanding the relationship between the billiard flow on a rational polygonal billiard @xmath1 and a billiard table tiled by @xmath0 , and will be particularly useful in understanding the nature of particular orbits of a self - similar sierpinski carpet billiard in [ subsec : aself - similarsierpinskicarpetbilliard ] . as one may expect , if @xmath0 is a rational polygon that tiles a billiard table @xmath130 , then an orbit of @xmath130 may be folded up to form an orbit of @xmath1 . this is done by making successive reflections in @xmath0 , the result being an orbit of @xmath1 ; see figure [ fig : foldeduporbit ] for the case of a square billiard table . . in the first image , we see an orbit of unit - square billiard table . partitioning the unit square into nine equally sized squares , we see that we can fold up the orbit by making successive reflections in the sides of the squares comprising the partition . using sufficiently many reflections results in an orbit of one of the squares of the partition . ] we are primarily interested in fractals with boundaries either partially or completely comprised of self - similar sets and fractals that are self - similar . so as to make the material discussed in [ sec : prefractalrationalbilliards][sec : fractalbilliards ] more accessible , we provide a few basic definitions from the subject of fractal geometry . [ def : contraction ] let @xmath131 be a metric space and @xmath132 . * ( contraction).if there exists @xmath133 such that @xmath134 for every @xmath135 , then @xmath136 is called a _ contraction _ ( or _ contraction mapping _ ) . * ( similarity contraction).if there exists @xmath137 such that @xmath138 for every @xmath135 , then @xmath136 is called a _ similarity contraction_. this unique value @xmath139 is called the _ scaling ratio _ of @xmath136 . [ def : iteratedfunctionsystemandselfsimilarset ] let @xmath131 be a complete metric space . * ( iterated function system and attractor).let @xmath140 be a family of contractions defined on @xmath141 . then @xmath140 is called an _ iterated function system _ ( ifs ) . + an iterated function system is so named because the map @xmath142 , given by @xmath143 and defined on the space @xmath144 of nonempty compact subsets of @xmath141 , can be composed with itself . indeed , for each @xmath145 , we have @xmath146 furthermore , there exists a unique nonempty compact set @xmath147 ( i.e. , @xmath148 ) , called the _ attractor _ of the ifs , such that @xmath149 * ( self - similar system and self - similar set).in the special case where each @xmath150 is a contraction similarity , for @xmath151 , then the ifs @xmath140 is said to be a _ self - similar system _ and its attractor @xmath2 is called a _ self - similar set _ ( or a _ self - similar subset _ of @xmath141 ) . if @xmath141 is complete , then so is @xmath144 ( equipped with the hausdorff metric ) and hence , since it can be shown that @xmath152 is a contraction , it follows from the contraction mapping theorem that @xmath153 has a unique fixed point ( thereby justifying the definition of the attractor @xmath2 above ) and that for any @xmath154 , @xmath155 , as @xmath156 ( where , as in equation ( [ eqn : phimiterate ] ) , @xmath157 is the @xmath73th iterate of @xmath153 ) . ( see @xcite . ) we state the next property in the special case which will be of interest to us , namely , that of an ifs in a euclidean space . [ thm : falconerstheorem ] consider an iterated function system given by contractions @xmath140 , each defined on a compact set @xmath158 , such that @xmath159 for each @xmath160 , and with attractor @xmath2 . then @xmath161 and in fact , @xmath162 for every set @xmath154 such that @xmath163 for all @xmath160 . here , the transformation @xmath142 is given as in part _ _ ( _ _ i _ ) _ of definition [ def : iteratedfunctionsystemandselfsimilarset ] . suppose @xmath2 is a fractal set . then , the @xmath61th prefractal approximation of @xmath2 is denoted by @xmath164 . in the case of a self - similar fractal @xmath2 , the @xmath61th _ prefractal approximation _ of @xmath2 is usually defined by @xmath165 , where @xmath166 . not every fractal is self - similar or embedded in euclidean space . however , such sets represent an important collection of examples of fractal sets . in the next subsection , we will discuss the fractal subsets ( self - similar or not ) of @xmath167 or of @xmath100 of direct interest to us in this paper . a cantor set is a set with very rich and counter - intuitive properties ; topologically , it is a compact and totally disconnected ( i.e. , perfect ) space . in order to illustrate some of the properties that make a cantor set so interesting , we refer to the canonical example of a cantor set : the ternary cantor set . we focus on three methods for constructing the ternary cantor set : 1 ) by tremas , 2 ) as the unique fixed point attractor of an iterated function system , and 3 ) in terms of an alphabet . before we discuss the ternary cantor set , we mention that this set was first discovered by henry j. s. smith in 1875 . later , in 1881 , vito volterra independently rediscovered the ternary cantor set . smith s and volterra s records being obscured over the years for one reason or another , it was the german mathematician georg cantor whom , in 1883 , history credits with the discovery of a bounded , totally disconnected , perfect and uncountable set with measure zero , that is now commonly referred to as `` the cantor set '' . we now proceed to construct the ternary cantor set , hereafter denoted by @xmath168 , by the method known as _ construction by tremas _ , which is latin for ` cuts ' . begin with the unit interval @xmath169 and remove the middle open third @xmath170 from @xmath169 , leaving the two closed intervals @xmath171 $ ] and @xmath172 $ ] . next , remove the middle open ninth from each closed subinterval . what remains are the closed intervals @xmath173 $ ] , @xmath174 $ ] , @xmath175 $ ] , @xmath176 $ ] . continuing this process ad infinitum , we construct the ternary cantor set ; see figure [ fig : ternarycantorset ] . one may also construct @xmath177 by utilizing an appropriately defined iterated function system . consider the following contraction maps defined on the real line @xmath178 : @xmath179 then , @xmath180 , where @xmath153 is given as in part ( i ) of definition [ def : iteratedfunctionsystemandselfsimilarset ] . moreover , since @xmath181 is a family of similarity contractions and @xmath182 , we have that @xmath168 is a self - similar set . a third and equivalent construction of the ternary cantor set can be given in terms of the symbols @xmath183 , @xmath184 , and @xmath185 . recall that the elements of @xmath167 can be expressed in terms of a base-@xmath186 number system . we focus our attention on elements of the unit interval @xmath169 . so - called ternary numbers is a _ ternary number _ if @xmath187 , @xmath188 , @xmath189 . ] in @xmath169 have two equivalent expansions : one that is finite and one that is infinite . for example , @xmath190 can be written in base-@xmath186 as @xmath191 or , equivalently , as @xmath192 ( where the overbar indicates that the digit @xmath193 is repeated infinitely often ) . we next discuss a similar _ addressing system _ that has the benefit of preventing ternary numbers from having a finite representation . the characters @xmath183 , @xmath184 and @xmath185 are to remind the reader of choosing _ left _ , _ center _ and _ right_. we identify an element of the unit interval @xmath169 by an infinite address that indicates _ where _ in @xmath169 the element is located . motivated by the construction of @xmath168 by tremas , one can identify any point of @xmath169 by an infinite address consisting of the characters @xmath183 , @xmath184 and @xmath185 . while elements of @xmath168 can be represented by infinite addresses consisting of @xmath184 s , we make the stipulation that no element of @xmath168 will be represented by an infinite address containing @xmath184 s . to be approximated by a sequence @xmath194 of elements of @xmath195 , where @xmath195 is the complement of the ternary cantor set in @xmath169 . ] moreover , this method of representing elements of @xmath169 ( or @xmath177 ) provides every element with an infinite representation and never a finite representation . the values @xmath196 , @xmath190 and @xmath197 have the ternary representations @xmath198 , @xmath199 and @xmath200 , respectively . has a representation given by @xmath201 . although , we will not consider this as a representation for @xmath190 on account of @xmath202 . ] while @xmath190 has a finite ternary expansion given by @xmath203 , it does not have a finite ternary representation . it should be noted that elements like @xmath196 and @xmath197 will play an important role in our analysis of the koch snowflake fractal billiard . the occurrence of infinitely many @xmath184 s or infinitely many @xmath183 s and @xmath185 s is critical to developing some of the theory regarding the koch snowflake fractal billiard . so that some of the results concerning the koch snowflake fractal billiard can be more succinctly expressed , we introduce a notation used for describing a value s _ type of ternary representation_. [ not : typeofternaryrepresentation ] the _ type of ternary representation _ can be defined as follows . if @xmath204 , then the first coordinate of @xmath205 $ ] describes the characters that occur infinitely often and the second coordinate of @xmath206 $ ] describes the characters that occur finitely often . if we want to discuss many different types of ternary representations , then we use ` or ' . that is , the notation @xmath206\vee [ \cdot,\cdot]\vee ... \vee[\cdot,\cdot]$ ] is to be read as _ @xmath206 $ ] or @xmath206 $ ] or ... or @xmath206$]_. if the collection of characters occurring finitely often is empty , then we denote the corresponding type of ternary representation by @xmath207}$ ] . the value @xmath197 has a ternary representation of @xmath200 . hence , @xmath197 has a type of ternary representation given by @xmath208}$ ] . moreover , the value @xmath209 has a ternary representation given by @xmath210 , which means that @xmath209 has a type of ternary representation given by @xmath211}$ ] . we note that `` the '' type of representation of a point @xmath204 is not unique , in general . for instance , the value @xmath190 has a ternary representation of type @xmath212}$ ] or @xmath213}$ ] . a thorough understanding of the ternary cantor set is not only important for understanding many of the results on the koch snowflake prefractal and fractal billiard . in general , cantor sets will be ever - present and instrumental in our analysis of other fractal billiard tables . in each example of a fractal billiard , we will clearly indicate where and how a particular cantor set is important in analyzing a particular fractal billiard table . the koch curve @xmath214 is constructed as shown in figure [ fig : constructionofkochcurvefrom2dcompactset ] and is the unique fixed point attractor of the following iterated function system on the euclidean plane ( here , @xmath215 ) : @xmath216 since each contraction map in the iterated function system is a similarity transformation ( i.e. , @xmath217 is a self - similar system ) and @xmath218 , we have that @xmath214 is a self - similar set ; see part ( ii ) of definition [ def : iteratedfunctionsystemandselfsimilarset ] . there are additional properties of the koch curve that are reminiscent of the cantor set ; this is more than just a coincidence and is discussed in more detail below . if we allow the iterated function system to act on the triangle @xmath219 , as shown in figure [ fig : constructionofkochcurvefrom2dcompactset ] , sequential iterates of the iterated function system very quickly produce a prefractal that is visually indiscernible from the true limiting set . but there is a more common construction that allows us to visualize the curve @xmath214 more readily , this being depicted in figure [ fig : kochcurveconstruction ] . the technical caveat which we are brushing under the carpet is that each polygonal approximation shown in figure [ fig : kochcurveconstruction ] does not contain the koch curve @xmath214 , while each approximation in the sequence shown in figure [ fig : constructionofkochcurvefrom2dcompactset ] does contain @xmath214 . and theorem [ thm : falconerstheorem ] that for a set @xmath2 to be the unique fixed point attractor of an ifs , each @xmath164 must be such that @xmath220 , so that @xmath221 . ] . here , the self - similar set @xmath214 is viewed as a limit of the prefractal approximations @xmath222 , where @xmath223 is the initial triangle and the map @xmath153 is defined in terms of the ifs given by equation ( [ eqn : ifsforthekochcurve ] ) , as in definition [ def : iteratedfunctionsystemandselfsimilarset ] . ( see theorem [ thm : falconerstheorem ] and the text preceding it . ) ] when learning about fractal sets . beginning with the unit interval @xmath169 , one removes the middle third and replaces it with the two other sides of an equilateral triangle , as shown . one then repeats this process infinitely often for every remaining interval ; the resulting limiting set is @xmath214 . such a sequence @xmath224 of approximations converges to @xmath214 , because it is a subsequence of the convergent sequence of prefractal approximations @xmath222 shown in figure [ fig : constructionofkochcurvefrom2dcompactset ] . ( here , we are using the notion of convergence in the sense of the hausdorff metric . ) ] for each integer @xmath225 , we denote by @xmath226 the @xmath61th ( inner ) polygonal approximation of the koch curve @xmath214 . intuitively , one expects the koch curve to have finite length , since it is the limit of a sequence of polygonal approximations . on the contrary , the koch curve @xmath214 has infinite length , which can be seen by the following calculation given in terms of the @xmath61th prefractal @xmath226 , where @xmath226 is one of the polygonal approximations indicated in figure [ fig : kochcurveconstruction ] : @xmath227 then , @xmath228 . the koch snowflake @xmath12 is a fractal comprised of three abutting copies of the self - similar koch curve ; see figure [ fig:3kochcurveslabeled ] . for each integer @xmath225 , we denote by @xmath229 the @xmath61th ( inner ) polygonal approximation of the koch snowflake @xmath12 . is the union of three abutting copies of @xmath214 . ] as a closed ( simple ) curve , the koch snowflake @xmath12 bounds a region of the plane ; furthermore , the area of this region can be calculated as follows : @xmath230 then , as @xmath61 increases , the right - hand side of ( [ eqn : areaofkochsnowflake ] ) tends to a finite value . the area bounded by the koch snowflake is thus given by @xmath231 , assuming the sides of @xmath232 have length one . as we noted at the end of [ subsec : cantorsets ] , cantor sets are ever present in the context of self - similarity . in the case of the koch snowflake , @xmath233 is the union of @xmath234 self - similar ternary cantor sets , each spanning a distance of @xmath235 . such a fact will be important in determining certain sequences of what we will call _ compatible orbits _ ( see definitions [ def : compatibleinitialconditions][def : sequenceofcompatibleorbits ] ) and certain families of well - defined orbits of @xmath4 . the @xmath5-fractal @xmath13 , discussed in @xcite in a different context , is not a self - similar set . however , @xmath13 contains , as a proper subset , a set that is constructed in a way that is reminiscent of an iterated function system acting on a compact set so as to produce a self - similar set . and theorem [ thm : falconerstheorem ] that each prefractal approximation @xmath164 must contain the unique fixed point attractor @xmath2 . ] as shown in figure [ fig : t - fractal ] , one constructs the @xmath5-fractal by appending scaled copies of the initial @xmath5 shape @xmath236 to each successive approximation . specifically , @xmath237 is constructed from @xmath238 by appropriately appending @xmath239 copies of @xmath240 to @xmath241 .. ] -fractal @xmath13 . ] the overall height of @xmath13 can be calculated and the total area bounded by @xmath242 can be shown to be finite , as shown in the following calculations ( we assume here that the base of @xmath236 is two units in length ) : @xmath243 then , @xmath244 , which is the height of @xmath13 . furthermore , the area bounded by @xmath238 is calculated as follows . there are eight squares , each with side - length one , comprising @xmath236 ; see figure [ fig : t0tiledbyunitsquare ] . hence , the area of @xmath236 is eight square - units . therefore , @xmath245 then , @xmath246 , which is the total area bounded by @xmath13 . can be tiled by the unit square @xmath247 . ] there is a natural fractal subset of @xmath13 , but , for each @xmath225 , no point of @xmath238 is in this fractal subset , which is unlike what we have seen in the case of the koch snowflake fractal @xmath12 . in fact , the fractal subset in question is given by @xmath248 . we note that this fractal subset is not self - similar and each point a priori fails to yield a well - defined tangent necessary for calculating the angle of reflection of a billiard ball traversing the billiard table . a sierpinski carpet can be constructed by systematically removing particular open subsquares from the unit square @xmath249 . depending on how one chooses the sizes of the open subsquares to be removed , one can either construct a self - similar sierpinski carpet or a non - self - similar sierpinski carpet , these being defined below . this method of construction is called _ construction by tremas _ and is described in the caption of figure [ fig : polygonalappxof1 - 3sierpinskicarpet ] , using the standard `` @xmath250-sierpinski carpet '' as an example . such a construction process should be very familiar , since `` removing middle thirds '' is exactly what we did to construct the ternary cantor set in [ subsec : cantorsets ] . -sierpinski carpet is a self - similar carpet constructed in one of two ways : 1 ) by tremas and 2 ) an iterated function system ( in fact , a self - similar system ) . we describe here the construction of the @xmath250-sierpinski carpet by tremas , the latter being further discussed in the main text . beginning with the unit square , one then removes the middle open square with side - length @xmath190 . from each remaining subsquare of side - length @xmath190 , one then removes the middle open square of side - length @xmath251 . one continues this procedure of removing subsquares of remaining squares until there is no area left . as one would expect , each step of the construction process can be emulated by applying the correct iterated function system , which is given in equation ( [ eqn : theifsfor1 - 3sierpinskicarpet ] ) . ] as referred to in the caption of figure [ fig : polygonalappxof1 - 3sierpinskicarpet ] , one may also construct the @xmath250-sierpinski carpet by applying an appropriately defined iterated function system to the unit square @xmath247 . consider the following iterated function system , which is a self - similar system . @xmath252 then , denoting the @xmath250-sierpinski carpet by @xmath253 , we have that @xmath254 . since each contraction in the iterated function system is a similarity contraction and @xmath255 , it follows that @xmath253 is a self - similar set . we discuss here the relevant results and material from @xcite . for our purposes , the first level approximation of a sierpinski carpet @xmath256 will always be the unit square @xmath247 and denoted by @xmath257 . since every ( self - similar and non - self - similar ) sierpinski carpet has the same zeroth level approximation and zero is never a scaling ratio , such notation will never cause any confusion . what follows is a general description on how to construct a sierpinski carpet by removing appropriately sized middle open squares . consider the unit square @xmath258 . let @xmath259 for some @xmath260 . partition @xmath257 into @xmath261 squares of side - length @xmath262 . next , remove the middle open subsquare . let @xmath263 for some @xmath264 . each subsquare may then be partitioned into @xmath265 many squares with side - length @xmath266 . we then remove each middle open subsquare of side - length @xmath266 ; see figure [ fig : polygonalappxof1 - 3sierpinskicarpet ] . continuing this process , let @xmath267 where @xmath268 and let @xmath269 for some @xmath270 . then we partition a subsquare of side - length @xmath271 into @xmath272 many squares . we then remove the middle open square from each subsquare in the partition . continuing in this manner ad infinitum , one constructs a sierpinski carpet denoted by @xmath273 , where @xmath274 . [ def : aselfsimilarsierpinskicarpet ] if @xmath275 , with @xmath276 and @xmath277 , is a periodic sequence of rational values , then the sierpinski carpet @xmath256 is called a _ self - similar sierpinski carpet_. we have described the construction of a self - similar sierpinski carpet @xmath15 in terms of the removal of particular open squares . as the name would suggest , there exists a suitably defined iterated function system @xmath140 such that @xmath278 . viewing @xmath279 as the unique fixed point attractor of an appropriately defined iterated function system will be useful in stating some of the results in the subsequent sections . more precisely , @xmath15 is viewed as the self - similar set associated with a self - similar system , as in part ( ii ) of definition [ def : iteratedfunctionsystemandselfsimilarset ] . while we do not discuss any results concerning non - self - similar sierpinski carpet billiards in this paper , we provide the definition for completeness . if @xmath275 , with @xmath276 and @xmath277 , is an aperiodic sequence of rational values , then the sierpinski carpet @xmath256 is called a _ non - self - similar sierpinski carpet_. [ def : acellofsai ] let @xmath280 , @xmath281 . consider a partition of the unit square @xmath258 into @xmath282 many squares of side - length @xmath262 . a subsquare of the partition is called a _ cell of @xmath257 _ and is denoted by @xmath283 . furthermore , let @xmath279 be a sierpinski carpet . consider a partition of the prefractal approximation @xmath284 into subsquares with side - length @xmath285 . a subsquare of the partition of @xmath284 is called a _ cell of @xmath284 _ and is denoted by @xmath286 and has side - length @xmath285 . in accordance with the convention adopted in @xcite , the boundary of an open square removed in the construction of @xmath256 is called a _ peripheral square _ of @xmath256 . furthermore , by convention , the unit square @xmath258 is not a peripheral square . [ def : nontriviallinesegmentofsa ] a _ nontrivial line segment of @xmath256 _ is a ( straight - line ) segment of the plane contained in @xmath256 and which has nonzero length . unless otherwise indicated , in what follows , we assume that @xmath256 is a self - similar sierpinski carpet with a single scaling ratio @xmath16 ; that is , @xmath287 , where @xmath288 for some fixed @xmath289 . in addition , when @xmath290 , @xmath15 is denoted by @xmath291 . we next state the following theorem , due to durand - cartagena and tyson in @xcite and which will be very useful to us in this context ( see [ subsec : aprefractalselfsimilarsierpinskicarpetbilliard ] and [ subsec : aself - similarsierpinskicarpetbilliard ] ) . [ thm : asetbset ] let @xmath291 be a self - similar sierpinski carpet . then the set of slopes @xmath292 of nontrivial line segments of @xmath291 is the union of the following two sets _ _ : _ _ @xmath293 moreover , if @xmath294 , then each nontrivial line segment in @xmath291 with slope @xmath295 touches vertices of peripheral squares , while if @xmath296 , then each nontrivial line segment in @xmath291 with slope @xmath295 is disjoint from all peripheral squares . [ not : aabaabbb ] let @xmath297 be odd positive integers such that @xmath298 and let @xmath292 and @xmath299 be the set of slopes of nontrivial line segments of @xmath291 and @xmath300 , respectively . we denote by @xmath301 ( resp . , @xmath302 ) the subset @xmath303 ( resp . , @xmath304 ) given in equation ( [ eqn : aset ] ) of theorem [ thm : asetbset ] . similarly , we denote by @xmath305 ( resp . , @xmath306 ) the subset @xmath307 ( resp . , @xmath299 ) given in equation ( [ eqn : bset ] ) of theorem [ thm : asetbset ] . ( resp . , @xmath306 ) , @xmath16 should of course be replaced by @xmath308 in equation ( [ eqn : aset ] ) ( resp . , equation ( [ eqn : bset ] ) ) . ] if @xmath291 and @xmath300 are self - similar sierpinski carpets with @xmath309 , then it is clear that @xmath310 . moreover , in this case , we also have that @xmath311 and @xmath312 . we note that if @xmath295 is the slope of a nontrivial line segment in @xmath291 , then so is @xmath313 , @xmath314 and @xmath315 by symmetry of the carpet . however , we restrict our attention in this paper to the slopes described in the above result of @xcite . in the previous sections , we surveyed basic facts and results from mathematical billiards and fractal geometry , with most of our attention being focused on the subject of rational billiards and sets exhibiting self - similarity . we also discussed the importance of examining the dynamically equivalent geodesic flow on an associated translation surface . in this section , we will examine examples from particular classes of prefractal ( rational ) billiards . we are interested in tables that can be tiled by a single polygon which can also tile the ( euclidean ) plane . the main examples we will discuss are the koch snowflake prefractal billiard table , the @xmath5-fractal prefractal billiard table and a self - similar sierpinski carpet prefractal billiard table . each example of a prefractal billiard table constitutes a rational billiard table , but is an element of a sequence of rational billiard tables approximating a fractal billiard table with radically different qualities when compared to the others . that is , the koch snowflake has an everywhere nondifferentiable boundary ; the @xmath5-fractal billiard table is certainly a fractal billiard table , since its boundary @xmath13 contains a fractal set , but the portion of the boundary that is nondifferentiable has lebesgue measure zero ; a sierpinski carpet billiard table can possibly have no area , yet yield billiard orbits of finite length . we restrict our attention to billiard tables with fractal boundary @xmath2 , where @xmath2 can be approximated by a suitably chosen sequence of rational polygons @xmath316 . more specifically , we are interested in a fractal billiard table @xmath317 with the property that , for every @xmath225 , @xmath89 can be tiled by a single polygon @xmath318 , where @xmath319 . here , @xmath320 is a suitably chosen scaling ratio and @xmath321 is a polygon that tiles both the ( euclidean ) plane as well as the rational billiard @xmath322 . is exactly @xmath321 . in general , however , @xmath321 does not always equal @xmath322 , but certainly tiles @xmath322 . an example of this situation is @xmath323 ; see [ subsec : thetfractalprefractalbilliard ] . such a billiard is tiled by the unit square , which is the associated polygon @xmath321 . ] the focus in this subsection is on developing a general framework for discussing billiards on prefractal approximations . if @xmath62 and @xmath324 are two prefractal billiard tables approximating a given fractal billiard table @xmath3 , then we want to have a systematic way of determining how and if two orbits @xmath325 and @xmath326 of @xmath62 and @xmath324 are related . we will primarily measure angles relative to a fixed coordinate system , with the origin being fixed at a corner of a prefractal approximation @xmath327 . however , we will sometimes measure an angle relative to a side of @xmath164 on which a billiard ball lies . in such situations , we will write the angle as @xmath328 in order to indicate that the inward pointing direction is @xmath329 , measured relative to the side on which the vector is based . to motivate our general discussion , consider the orbit @xmath330 of @xmath331 , where @xmath332 ; see the first image in figure [ fig : notapwfagnanoorbit ] ( and recall our earlier discussion in [ subsec : cantorsets ] ) . the same orbit , viewed as a continuous curve embedded in @xmath333 , does not constitute an orbit of @xmath333 ; see the second image in figure [ fig : notapwfagnanoorbit ] . consider the orbit @xmath334 shown in the third image in figure [ fig : notapwfagnanoorbit ] . such an orbit does intersect the boundary of @xmath333 and appears to be related to @xmath335 , but in what way we have not yet explicitly said . initially , we notice that , as a continuous curve embedded in @xmath333 , the orbit @xmath335 is a subset of @xmath334 . being eager to establish a proper notion of `` related '' , we may be inclined to declare that two orbits are related if one is a subset of the other , when viewed as continuous curves in the plane . unfortunately , we quickly see that such a definition is highly restrictive . a more general observation is that @xmath336 and @xmath337 are collinear in the direction of @xmath338 , without any portion of @xmath339 intersecting the segment @xmath340 . we then say that @xmath341 and @xmath342 are _ compatible initial conditions_. we state the formal definition as follows . of @xmath331 . in the second image , we see that the orbit @xmath335 , when embedded in @xmath333 , is not an orbit of @xmath333 . in the third image , the given orbit of @xmath333 intersects sides of @xmath333 and appears to be related to @xmath335 in some way . ] [ def : compatibleinitialconditions ] without loss of generality , suppose that @xmath61 and @xmath73 are nonnegative integers such that @xmath343 . let @xmath344 and @xmath345 be two initial conditions of the orbits @xmath325 and @xmath346 , respectively , where we are assuming that @xmath347 and @xmath348 are both inward pointing . if @xmath349 and if @xmath350 and @xmath351 lie on a segment determined from @xmath347 ( or @xmath348 ) that intersects @xmath89 only at @xmath350 , then we say that @xmath69 and @xmath352 are _ compatible initial conditions_. when two initial conditions @xmath69 and @xmath352 are compatible , then we simply write each as @xmath353 and @xmath354 . if two orbits @xmath346 and @xmath325 have compatible initial conditions , then we say such orbits are _ compatible_. depending on the nature of @xmath317 , not every orbit must pass through the region of @xmath62 corresponding to the interior of @xmath322 , let alone pass through the interior of @xmath355 , for any @xmath356 . because of this , it may be the case that an initial condition @xmath353 is not compatible with @xmath354 , for any @xmath356 . as such , in definitions [ def : sequenceofcompatibleinitialconditions ] and [ def : sequenceofcompatibleorbits ] , we consider sequences beginning at @xmath357 , for some @xmath358 . [ def : sequenceofcompatibleinitialconditions ] let @xmath359 be a sequence of initial conditions , for some integer @xmath358 . we say that this sequence is a _ sequence of compatible initial conditions _ if for every @xmath360 and for every @xmath361 , we have that @xmath69 and @xmath352 are compatible initial conditions . in such a case , we then write the sequence as @xmath362 . [ def : sequenceofcompatibleorbits ] consider a sequence of compatible initial conditions @xmath363 . then the corresponding sequence of orbits @xmath364 is called _ a sequence of compatible orbits_. if @xmath365 is an orbit of @xmath355 , then @xmath365 is a member of a sequence of compatible orbits @xmath364 for some @xmath358 . it is clear from the definition of a sequence of compatible orbits that such a sequence is uniquely determined by the first orbit @xmath366 . since the initial condition of an orbit determines the orbit , we can say without any ambiguity that a sequence of compatible orbits is determined by an initial condition @xmath367 . let @xmath368 be a property ( resp . , @xmath369 a list of properties ) . if every orbit in a sequence of compatible orbits has the property @xmath368 ( resp . , a list of properties @xmath369 ) , then we call such a sequence _ a sequence of compatible @xmath368 _ _ ( _ _ resp . , @xmath369 _ _ ) _ _ orbits_. the following theorem can be deduced from theorem 3 of gutkin s paper @xcite ; see @xcite . [ thm : topologicaldichotomyforfn ] consider a prefractal rational billiard @xmath62 . if @xmath370 is tiled by a rational polygon @xmath318 such that @xmath318 tiles the euclidean plane , then , for a fixed direction @xmath347 , every orbit @xmath371 of @xmath62 is closed or every orbit @xmath371 is dense in @xmath62 , . ] regardless of the initial basepoint @xmath350 . when @xmath62 is tiled by @xmath318 , where @xmath318 is a rational polygon tiling the plane , then @xmath62 is more generally referred to as an _ almost integrable billiard _ , this being the language used in @xcite . the following is a generalization to this broader setting of corollary 16 from @xcite . it is established in the same manner . [ thm : generaltopologicaldichotomyforsequencesofcompatibleorbits ] let @xmath317 be a fractal billiard table approximated by a suitable sequence of rational polygonal billiard tables @xmath372 . if there exists a polygon @xmath321 that tiles the plane and such that for every @xmath225 there exists @xmath320 with @xmath373 tiling @xmath62 , then any sequence of compatible orbits is either entirely comprised of closed orbits or entirely comprised of orbits that are dense in their respective billiard tables . the billiard @xmath8 can be tiled by equilateral triangles . specifically , if @xmath120 is the equilateral triangle with sides having unit length , then @xmath8 is tiled by @xmath374 , for every @xmath225 . moreover , as is well known , @xmath375 tiles the plane . therefore , theorems [ thm : topologicaldichotomyforfn ] and [ thm : generaltopologicaldichotomyforsequencesofcompatibleorbits ] hold for the prefractal billiard @xmath8 . our goal for this subsection and [ subsubsec : thecorrespondingprefractaltranslationsurface ] is to survey some of the main results of @xcite . we will focus on pertinent examples that will motivate a richer discussion in [ subsec : thekochsnowflakefractalbilliard ] . initially , we focus on properties of orbits with an initial direction of @xmath338 and @xmath376 . and @xmath377 , since @xmath377 is the rotation of @xmath376 through the angle @xmath338 , the angle @xmath338 being an angle that determines an axis of symmetry of @xmath229 , for @xmath225 . ] if @xmath330 is an orbit of @xmath331 , so long as @xmath378 is not a corner of @xmath232 , the orbit will be periodic , as expected . however , depending on the nature of the ternary representation of @xmath378 , the compatible orbit @xmath334 may be singular in @xmath333 . of [ subsec : cantorsets ] . ] let @xmath379 . if @xmath337 has a ternary representation of type @xmath380}\vee { [ r , lc]}$ ] , then there exists @xmath358 such that the compatible orbit @xmath381 will be singular in @xmath382 . moreover , for every @xmath383 , @xmath384 will also be singular in @xmath8 . if @xmath378 has a ternary representation of the form @xmath385}\vee{[lc , r]}\vee{[cr , l]}\vee{[lcr,\emptyset]}\vee{[lr , c]}$ ] , then the sequence of compatible orbits given by @xmath386 is a sequence of compatible periodic orbits . the length and period of an orbit @xmath387 is dictated by the ternary representation of @xmath378 . _ _ ( _ _ see @xcite for the corresponding specific formulas . _ ) _ see 4.4 of @xcite for a precise statement and proof of this result , as well as for additional properties of orbits with an initial direction of @xmath338 . [ exa : asequenceofcompatiblehookorbits ] let @xmath388 have a ternary representation given by @xmath389 . such a representation indicates that , in each prefractal approximation @xmath229 , @xmath337 is an element of an open , connected neighborhood contained in @xmath229 . the point @xmath337 corresponds to the value @xmath390 . if we consider an orbit of @xmath331 with an initial direction of @xmath376 , the ternary representation of the basepoints at which the billiard ball path forms right angles with the sides of @xmath331 is of the type @xmath385}$ ] . this is a degenerate periodic hybrid orbit , meaning that it doubles back on itself , and the next orbit in the sequence of compatible periodic hybrid orbits has the initial condition @xmath391 . since the ternary representation of the basepoint of @xmath392 is @xmath393 and @xmath394 , it follows that the basepoint of @xmath395 is a point which , for every prefractal approximation @xmath229 , is an element of an open , connected neighborhood contained in @xmath229 . then the basepoint of @xmath396 ( where @xmath397 denotes the second iterate of the billiard map @xmath398 ) has a ternary representation of type @xmath385}$ ] . this same pattern is repeated for every subsequent orbit in the sequence of compatible orbits . it follows that the resulting sequence of compatible orbits forms a sequence of orbits that is converging to a set which is well defined . that is , such a set will be some path in the fractal billiard table @xmath4 with finite length which is effectively determined by the law of reflection in each prefractal approximation of @xmath4 . such orbits are introduced in @xcite and referred to as _ hook orbits _ , because they appear to be `` hooking '' into the koch snowflake ; see figure [ fig : hybridpi6lv5 ] . the hook orbits of example [ exa : asequenceofcompatiblehookorbits ] are special cases of a general class of orbits called hybrid orbits , which were introduced , as well as studied , in @xcite . [ def : hybridorbit ] let @xmath325 be an orbit of @xmath8 . if all but at most two basepoints @xmath399 have ternary representations ( determined with respect to the side @xmath400 on which each point resides ) of type @xmath385}\vee{[cl , r]}\vee{[cr , l]}\vee{[lcr,\emptyset]}\vee{[lr,\emptyset]}$ ] , then we call @xmath371 a _ hybrid orbit _ of @xmath8 . a hybrid orbit is so named for the fact that it _ may _ have qualities reminiscent of an orbit @xmath326 that is identical to the compatible orbit @xmath325 and an orbit @xmath401 that is visually different from the compatible orbit @xmath402 ; see figure [ fig : hybrid012 ] and its caption . if @xmath325 is a hybrid orbit with property @xmath403 , then we say that it is a @xmath403 hybrid orbit . [ prop : denseorbitisadensehybridorbit ] if @xmath325 is a dense orbit of @xmath8 , then @xmath325 is a dense hybrid orbit . applying the results in theorem [ thm : generaltopologicaldichotomyforsequencesofcompatibleorbits ] and proposition [ prop : denseorbitisadensehybridorbit ] , we state the following result . [ thm : atopologicaldichotomy ] let @xmath364 be a sequence of compatible orbits . then we have that @xmath364 is either entirely comprised of closed orbits or is entirely comprised of dense hybrid orbits . , just before definition [ def : footprintofanorbit ] . ] [ thm : hybridorbitofks0implieshybridorbitofksn ] if @xmath404 is a periodic hybrid orbit of @xmath331 with no basepoints corresponding to ternary points _ _ ( _ _ i.e . , points having ternary representations of the types @xmath380}\vee{[r , lc]}$ ] _ _ ) _ _ , then for every @xmath225 , the compatible orbit @xmath325 is a periodic hybrid orbit of @xmath8 . in order to fully understand the following result , we define what it means for a vector to be _ rational with respect to a basis _ @xmath405 of @xmath100 . if @xmath406 , for some @xmath407 , then we say that @xmath408 is _ rational with respect to the basis _ otherwise , we say that @xmath408 is _ irrational with respect to _ @xmath405 . let @xmath388 and consider a vector @xmath409 that is rational with respect to the basis @xmath410 . then , we have the following _ _ : _ _ 1 . if @xmath16 and @xmath308 are both positive integers with @xmath308 being odd , @xmath411 , for some @xmath412 with @xmath413 , @xmath414 being odd and @xmath415 , then the sequence of compatible closed orbits @xmath416 is a sequence of compatible periodic hybrid orbits . if @xmath417 , @xmath308 is a positive odd integer , @xmath418 , for some @xmath412 with @xmath413 , @xmath419 being odd and @xmath415 , then the sequence of compatible closed orbits @xmath416 is a sequence of compatible periodic hybrid orbits . [ thm : bodd ] we want to emphasize that the angle @xmath38 in part ( 1 ) and part ( 2 ) of theorem [ thm : bodd ] is not necessarily @xmath338 , @xmath377 or @xmath376 , but can assume countably infinitely many values . [ exa : asequenceofcompatibleperiodichybridorbits ] in figure [ fig : hybrid012 ] , three periodic hybrid orbits are displayed . these three orbits constitute the first three terms in a sequence of compatible periodic hybrid orbits.(2 ) , the angle @xmath420 determined by the initial segment of the orbit and the initial basepoint @xmath421 both guarantee that the sequence of compatible orbits @xmath416 is a sequence of compatible periodic hybrid orbits . ] if we choose @xmath422 and @xmath423 to be an angle such that @xmath337 connects with the midpoint of the lower one - third interval on the side of @xmath331 , we can see that @xmath404 is a periodic hybrid orbit . more importantly , there are elements of the footprint @xmath424 with ternary representations of type @xmath211}$ ] . this observation is key for constructing what we call nontrivial paths of @xmath4 , a topic which is discussed in more detail in [ subsec : thekochsnowflakefractalbilliard ] . . in order to understand exactly what is discussed in the paragraph immediately following definition [ def : hybridorbit ] , compare and contrast the hybrid orbits shown here with the hybrid orbits shown in figures [ fig : notapwfagnanoorbit ] , [ fig : hybridpi6lv5 ] and [ fig : compatiblecantororbit7 - 12 ] . certain segments of the hybrid orbits shown here remain intact and become subsets of subsequent compatible periodic hybrid orbits , yet the orbits are visually different from one another . ] given a nonnegative integer @xmath425 , we say that a sequence of compatible orbits @xmath364 is a _ constant sequence of compatible orbits _ if the path traversed by @xmath326 is identical to the path traversed by @xmath325 , for every @xmath383 . furthermore , we say that a sequence of compatible orbits @xmath416 is _ eventually constant _ if there exists a nonnegative integer @xmath425 such that @xmath364 is constant , in the above sense . [ thm : sufficientconditionforcantororbit ] let @xmath404 be an orbit of @xmath331 such that every @xmath426 has a ternary representation of type @xmath211}$ ] . then @xmath416 is a sequence of compatible periodic hybrid orbits . moreover , there exists @xmath358 such that @xmath364 is a constant sequence of compatible periodic hybrid orbits . [ exa : aconstantsequenceofcompatibleperiodichybridorbits ] consider @xmath427 in the base of the equilateral triangle . such a value has a ternary representation of type @xmath211}$ ] . consider the initial condition @xmath428 . then the sequence of compatible orbits @xmath429 is a constant sequence . this follows from the fact that the ternary representation of @xmath336 is @xmath389 . moreover , the representation of every basepoint of @xmath384 is @xmath198 . in figure [ fig : compatiblecantororbit7 - 12 ] , we show the first three orbits in this ( eventually ) constant sequence of compatible periodic hybrid orbits . as of now , the only examples of constant sequences of compatible periodic hybrid orbits are those for which the initial direction is @xmath338 and @xmath376 ( and , equivalently , @xmath377 ) . when the initial angle of an orbit of a constant sequence of compatible periodic hybrid orbits is @xmath376 ( or , equivalently , @xmath377 ) , then the orbit will be degenerate . for example , the orbit @xmath430 traverses a path that is a vertical line . this orbit has period @xmath431 . while @xmath432 is an important example of a constant sequence of compatible periodic hybrid orbits , it is arguably less interesting than the constant sequence of compatible periodic hybrid orbits @xmath433 . lies on the middle third of the unit interval . the basepoint @xmath336 of the compatible initial condition @xmath342 has a ternary representation of type @xmath434}$ ] . ] in [ subsec : translationstructuresandtranslationsurfaces ] we saw how to construct a translation surface from a rational billiard table . in the case of the equilateral triangle billiard table @xmath436 , there are @xmath437 copies of @xmath119 used in the construction of the associated translation surface @xmath438 ; see example [ exa : equilateraltriangleflatsurface ] and the associated figure [ fig : sixequisurface ] . in the case of the prefractal billiard table @xmath8 , only six copies of @xmath8 are needed in the construction of the associated translation surface @xmath439 , for every @xmath225 ; see figure [ fig : thethreecorrespondingflatsurfaces ] . ( we refer to @xcite for further discussion of the topics in the present subsection . ) , @xmath440 and @xmath441 associated with the koch snowflake prefractal approximations @xmath339 , @xmath442 and @xmath443 , respectively.,title="fig : " ] , @xmath440 and @xmath441 associated with the koch snowflake prefractal approximations @xmath339 , @xmath442 and @xmath443 , respectively.,title="fig : " ] , @xmath440 and @xmath441 associated with the koch snowflake prefractal approximations @xmath339 , @xmath442 and @xmath443 , respectively.,title="fig : " ] the vertices of @xmath8 correspond to conic singularities of the translation surface . however , only certain singularities are removable . the vertices with angles measuring @xmath338 ( measured from the interior ) , constitute removable singularities of the translation surface . that is , the geodesic flow can be appropriately defined at these points . the vertices with angles measuring @xmath444 constitute nonremovable singularities . hence , it is possible to define reflection at certain vertices of the prefractal billiard @xmath8 , but impossible to define at others . moreover , defining reflection at acute corners of @xmath8 in this way is independent of @xmath61 . that is , for a given vertex @xmath445 of @xmath8 with an acute angle @xmath338 , the general rule for reflection in @xmath445 states that an incoming trajectory reflect through the angle bisector of @xmath445 . a billiard ball entering @xmath445 along the same path in @xmath446 as in @xmath8 will then reflect in @xmath445 in @xmath446 in exactly the same way as it did when considering @xmath445 as a vertex of @xmath8 . such insight is clearly helpful in further understanding the behavior of a billiard ball on the koch snowflake fractal billiard @xmath4 , but we must be careful not to extrapolate more than is possible from this observation . knowing that we can determine an orbit of a prefractal billiard @xmath8 by unfolding the orbit of @xmath331 in @xmath8 , we are inclined to allow orbits of @xmath331 that make collisions with corners . however , a priori , we can not conclude that such orbits do not unfold to form saddle connections in @xmath8 connecting two nonremovable singularities . in the event an orbit @xmath346 of @xmath447 intersects the boundary @xmath448 solely in acute corners , then such an orbit is an element of a sequence of compatible orbits @xmath364 with @xmath449 , for every @xmath450 . we refer to [ subsec : thetfractal ] for a discussion of the @xmath5-fractal @xmath13 and of its prefractal approximations @xmath451 , for @xmath452 ; see , in particular , figure [ fig : t - fractal ] . recall that the base of @xmath236 has a length of two units . the prefractal billiard @xmath323 can be tiled by the unit square @xmath247 ; see figure [ fig : t0tiledbyunitsquare ] . in general , for every @xmath225 , @xmath9 can be tiled by the square @xmath453 . as such , and since @xmath247 obviously tiles the plane , we can apply theorems [ thm : topologicaldichotomyforfn ] and [ thm : generaltopologicaldichotomyforsequencesofcompatibleorbits ] . much like the case of the prefractal koch snowflake billiard @xmath8 , we are interested in forming sequences of compatible orbits of prefractal billiards exhibiting particular properties . the results in this subsection appear here for the first time and will be further discussed in @xcite . it is true that if a periodic orbit has an initial condition @xmath454 , then there may exist a compatible orbit @xmath455 that forms a saddle connection if @xmath378 has a finite binary expansion . this is not to suggest that @xmath455 _ must _ form a saddle connection . however , if every basepoint @xmath456 of a periodic orbit @xmath457 of the unit square has an infinite binary expansion ( with no equivalent finite binary expansion ) , then viewing @xmath458 in @xmath323 as the reflected - unfolding of @xmath457 , the corresponding sequence of compatible orbits @xmath416 will be a sequence of compatible periodic orbits . we state this formally in the following theorem . let @xmath454 be an initial condition of an orbit @xmath459 of @xmath124 . suppose every element of the footprint @xmath460 has an infinite binary expansion _ _ ( _ _ and no equivalent finite binary expansion _ _ ) _ _ and @xmath454 is then the initial condition of an orbit of @xmath323 that constitutes the reflected - unfolding of @xmath457 in @xmath323 . then the sequence of compatible orbits @xmath416 _ _ ( _ _ where @xmath461 _ _ ) _ _ of the prefractal billiards @xmath9 is a sequence of compatible periodic orbits . [ exa : t - fractalsequenceofcompatibleorbits ] let @xmath462 and @xmath463 . then , @xmath464 is a nonconstant sequence of compatible periodic orbits ; see figure [ fig : t - fractalsequenceofcompatibleorbits ] . , @xmath465 and @xmath466 , respectively . ] the following two theorems are ultimately concerning the prefractal billiard @xmath9 . determining which intercepts and slopes yield line segments in the plane that avoid lattice points of the form @xmath467 is equivalent to specifying an initial condition of an orbit of a square billiard table that avoids corners of the billiard table . then , using the fact that an appropriately scaled square billiard table tiles @xmath9 , we can reflect - unfold such an orbit in @xmath9 in order to determine an orbit of @xmath9 . [ thm : whattheslopecannotbeintfrac ] let @xmath468 with @xmath469 , @xmath470 and @xmath186 relatively prime , @xmath471 and @xmath472 . further , let @xmath473 . if for every @xmath474 , @xmath475 , we have that @xmath476 then the line @xmath477 does not contain any point of the form @xmath467 , @xmath478 , with @xmath479 . note that the condition ( [ eqn : whatmcannotbe ] ) above is automatically satisfied if the slope @xmath73 is irrational . [ thm : whattheslopecanbeintfrac ] let @xmath468 , with @xmath480 , @xmath470 and @xmath186 relatively prime , @xmath471 and @xmath472 . if @xmath481 with @xmath482 , @xmath483 , then , for every @xmath474 with @xmath475 , the point @xmath484 does not lie on the line @xmath477 . finally , theorems [ thm : whattheslopecannotbeintfrac ] and [ thm : whattheslopecanbeintfrac ] combined with the fact that an initial condition of an orbit of @xmath485 , @xmath358 , determines a sequence of compatible orbits @xmath364 , allows us to determine a countably infinite family of sequences of compatible periodic orbits . for every @xmath225 , the interior angles of @xmath451 are @xmath377 and @xmath487 . to form the associated translation surface @xmath488 , we appropriately identify four copies of @xmath9 ; see figure [ fig : t - fractalflatsurfaces ] for a depiction of the first three translation surfaces . , @xmath489 and @xmath490 associated with the @xmath5-fractal prefractal approximations @xmath491 , @xmath492 and @xmath493 , ] then , every point of @xmath488 associated with a vertex of @xmath9 measuring @xmath377 constitutes a removable singularity of @xmath488 . similarly , every point of @xmath488 associated with a vertex of @xmath9 of interior angle measuring @xmath487 constitutes a nonremovable singularity of @xmath488 . therefore , not every vertex of @xmath9 will present a problem for the billiard flow . consider an orbit of @xmath124 , where the orbit has basepoints corresponding to vertices of @xmath247 , the unit square . since such vertices correspond to removable singularities in the corresponding translation surface ( this being the flat torus , see [ subsec : translationstructuresandtranslationsurfaces ] ) , we see that the same orbit reflected - unfolded in the billiard @xmath9 ( if one first scales the billiard @xmath124 and the orbit contained therein by @xmath494 , see [ subsec : unfoldingabilliardorbit ] ) can potentially intersect vertices of @xmath451 that are associated with nonremovable singularities in the corresponding translation surface . let @xmath14 be a self - similar sierpinski carpet , as defined in definition [ def : aselfsimilarsierpinskicarpet ] , and let us denote its natural prefractal approximations by @xmath495 for @xmath496 ( as in [ subsec : asierpinskicarpet ] ) . the corresponding billiard is then denoted by @xmath7 . in this subsection , we examine the behavior of the billiard flow on the rational polygonal billiard given by the prefractal approximations @xmath497 . in the event a billiard ball collides with a corner of a peripheral square , we must terminate the flow and such a trajectory is then called _ singular_. in addition to being singular , such a trajectory will form a saddle connection ( see the beginning of [ sec : rationalbilliards ] for a discussion of closed billiard orbits that form saddle connections ) . as we have discussed , an examination of the corresponding translation surface may prove useful in determining whether or not a billiard ball can reflect in a vertex . [ def : obstacletobilliardflow ] let @xmath1 be a polygonal billiard . then @xmath1 can be modified by placing in its interior a piecewise smooth segment that inhibits the billiard flow and causes a billiard ball to reflect . such a segment is called an _ obstacle _ of @xmath1 . clearly , each prefractal billiard @xmath497 can be interpreted as a _ square billiard with obstacles_. [ nota : alphafortheta ] due to the fact that theorem [ thm : asetbset ] refers to the slope of a nontrivial line segment and we make heavy use of this theorem , we will denote the initial condition @xmath69 of an orbit of @xmath10 by @xmath498 , where @xmath499 . consider the boundary of a cell @xmath500 of @xmath501 as a barrier . is a cell of the @xmath65th prefractal approximation @xmath502 , as given in definition [ def : acellofsai ] with all numbers @xmath503 equal to @xmath16 . ] then an orbit with an initial condition contained in the cell is called an _ orbit of the cell @xmath500 of @xmath501_. so as to be clear , the boundary of the cell does not form an obstacle to the billiard flow , as defined in definition [ def : obstacletobilliardflow ] . rather , we are treating the cell @xmath500 as a billiard table in its own right , embedded in the larger prefractal approximation @xmath501 . recall from [ subsec : asierpinskicarpet ] that a self - similar sierpinski carpet @xmath291 is the unique fixed point attractor of a suitably chosen iterated function system @xmath504 consisting of similarity contractions . in light of this , an orbit of a cell @xmath500 of @xmath501 is the image of an orbit @xmath505 of the unit - square billiard @xmath506 under the action of a composition of contraction mappings @xmath507 , with @xmath508 and @xmath509 , determined from the iterated function system @xmath510 of which @xmath291 is the unique fixed point attractor . [ lem : segmentbeginningatmidpointisnontrivialifalphafromb ] consider a self - similar sierpinski carpet @xmath291 . let @xmath57 and @xmath511 be a prefractal approximation of @xmath291 . if @xmath512 , that @xmath305 is the set of slopes given by equation ( [ eqn : bset ] ) . ] then the line segment beginning at a midpoint of a cell @xmath500 of @xmath511 is a nontrivial line segment _ _ ( _ _ in the sense of definition [ def : nontriviallinesegmentofsa]_)_. moreover , such a segment avoids the boundary of the peripheral squares of @xmath291 with side - length @xmath513 , @xmath514 . the statement in lemma [ lem : segmentbeginningatmidpointisnontrivialifalphafromb ] asserts that a segment beginning at a midpoint of a cell with slope @xmath512 will be a nontrivial line segment in @xmath291 . in addition to this , any line segment contained in @xmath100 that contains a nontrivial line segment of @xmath291 must necessarily avoid the peripheral squares in a tiling of @xmath100 by @xmath291 . otherwise , there exists @xmath515 such that scaling the line segment in @xmath100 and the tiling of @xmath100 by @xmath516 results in a segment contained in the nontrivial line segment which intersects peripheral squares of @xmath291 . this is a contradiction of the fact that the segment beginning at @xmath517 with slope @xmath512 is a nontrivial line segment of @xmath291 . we then deduce the following result . [ thm : constantsequenceofcompatibleorbitsinsa ] consider a self - similar sierpinski carpet @xmath291 . let @xmath57 and @xmath511 be a prefractal approximation of @xmath291 . furthermore , let @xmath512 and @xmath518 with @xmath519 a positive , odd integer . if @xmath520 is an orbit of @xmath501 , then the initial condition @xmath521 determines a sequence of compatible periodic orbits @xmath522 of the prefractal approximations @xmath10 . as one may suspect , there exists @xmath523 such that a sequence of compatible orbits @xmath524 is a constant sequence of compatible orbits . moreover , @xmath525 , for every @xmath383 . this is not any different from the case of a constant sequence of compatible orbits of prefractal billiards @xmath8 , as discussed in theorem [ thm : sufficientconditionforcantororbit ] and example [ exa : aconstantsequenceofcompatibleperiodichybridorbits ] . however , in the context of a self - similar sierpinski carpet billiard table , every sequence of compatible orbits we will examine will be a sequence for which there exists @xmath358 such that @xmath524 is a constant sequence of compatible orbits . in much the same way the billiard @xmath497 can be interpreted as a square billiard with obstacles , the corresponding translation surface can be interpreted as a `` torus with obstacles '' ; see figure [ fig : squaretoruswithobstacles ] . as a flat torus with obstacles . ] in light of the fact that @xmath527 can be interpreted as a torus with obstacles and the presence of a dynamical equivalence between the billiard flow and the geodesic flow on the corresponding translation surface ( see [ subsec : unfoldingabilliardorbit ] ) , we see that reflection in the vertices with angles measuring @xmath377 ( relative to the interior ) can be defined . more specifically , the geodesic flow can be defined at points corresponding to vertices with angles measuring @xmath377 , because these points constitute removable singularities of the geodesic flow . this fact is crucial in determining orbits of @xmath7 for which the slope @xmath295 is an element of @xmath301 and not @xmath305 ( see notation [ not : aabaabbb ] ) , and the orbit avoids all peripheral squares of @xmath7 . while one may say that this contradicts part of theorem [ thm : asetbset ] ( and he / she would be right ) , in @xcite a more precise formulation of theorem [ thm : asetbset ] is given that clarifies which slopes are permissible and which ones are not . that is , if @xmath528 , it may be possible for an orbit @xmath529 to begin at the origin and avoid the peripheral squares of each billiard @xmath530 , for every @xmath531 . we do not give here an explicit reformulation of theorem [ thm : asetbset ] , but example [ exa : orbitexamplecontradictingtysonstheoremslope2 - 3origins7 ] in [ subsec : aself - similarsierpinskicarpetbilliard ] exhibits a situation showing that the latter half of theorem [ thm : asetbset ] is not stated precisely enough . the theme that will tie together all of the examples in [ sec : prefractalrationalbilliards ] is that suitable limits of sequences of compatible orbits may constitute billiard orbits of each respective fractal billiard table . we have shown that in the case of @xmath4 , @xmath6 and @xmath7 , we can determine a sequence of compatible periodic orbits . we will see that in each case of a fractal billiard , under certain conditions , a sequence of compatible periodic orbits ( or a proper subset of points from each footprint @xmath532 ) will converge to a set which can be thought of as a true orbit of a fractal billiard table ( or such a sequence will yield a subsequence of basepoints converging to what we are calling an _ elusive point _ in @xcite ) . we restrict our attention to the family of fractal billiard tables @xmath3 where @xmath2 is a fractal approximated by a suitable sequence of rational polygons @xmath316 , with each @xmath164 tiled by @xmath534 for suitably chosen @xmath535 $ ] and @xmath321 a polygon that tiles the plane . specifically , we are interested in developing a general framework for dealing with a fractal billiard table @xmath317 which is similar to that of @xmath4 , @xmath6 and @xmath533 . before we begin our discussion of the fractal billiard tables @xmath4 , @xmath6 and @xmath7 , we define certain terms . the following definitions were initially motivated by the work in @xcite , but later generalized for this paper in order to account for a larger class of fractal billiard tables . ( from now on , we assume that @xmath317 is a fractal billiard table with prefractal billiard approximations @xmath536 as described just above . ) [ def : acorner ] let @xmath537 . if there exists @xmath225 such that @xmath538 and @xmath408 is a vertex of @xmath164 , then @xmath408 is called a _ corner _ of @xmath2 . [ def : acantorpoint ] let @xmath537 be such that @xmath408 is not a corner of @xmath2 . if there exists @xmath358 such that for every @xmath383 , @xmath538 and every connected neighborhood of @xmath408 contained in @xmath164 becomes totally disconnected when intersected with @xmath2 , then @xmath408 is called a _ cantor point _ of @xmath2 . in the koch snowflake @xmath12 , every cantor point is a smooth point of infinitely many prefractals @xmath229 approximating @xmath12 . that is , if @xmath408 is a cantor point in @xmath12 , then there exists @xmath539 such that for every @xmath383 , there exists a well - defined tangent at @xmath540 . below , @xmath408 is viewed as a point of the smooth subarc of @xmath164 to which it belongs . ] we deduce from this that the law of reflection holds at @xmath540 , for every @xmath383 . moreover , since the billiard ball reflects at @xmath540 at the same angle for every @xmath383 , we deduce that the tangent at @xmath408 is the same for each @xmath229 , @xmath383 . this observation then prompts us to generalize the definition of a cantor point in order to account ( for example ) for points of the @xmath5-fractal which are not cantor points , but are points for which a well - defined tangent can be found in infinitely many prefractal approximations . [ def : smoothfractalpoint ] let @xmath537 and @xmath358 be such that @xmath538 for every @xmath383 . if there exists a well - defined tangent at @xmath538 for every @xmath383 , then @xmath408 is called a _ smooth fractal point_. to be clear , a cantor point of @xmath2 is an example of a smooth fractal point of @xmath2 . the special nature of a cantor point warrants a formal definition . in the @xmath5-fractal billiard , there are certainly corners and elusive points . there are also smooth connected segments contained in the boundary of @xmath6 . points contained in such segments that do not correspond to corners are then called smooth fractal points . [ def : anelusivepoint ] let @xmath537 . if @xmath541 , then @xmath408 is called an _ elusive point _ of @xmath2 . consider a piecewise linear path in @xmath3 , such that every linear segment of the path is joined at the endpoint of another segment with the coincidental endpoints intersecting the boundary @xmath2 at a smooth fractal point of @xmath2 ( in the sense of definition [ def : smoothfractalpoint ] ) . in the following definition , we define a particular type of piecewise linear curve in a fractal billiard @xmath3 . suppose that there exists a piecewise linear curve in @xmath3 as described immediately above . if at each point @xmath408 for which the piecewise linear path intersects the boundary @xmath2 , the angle formed by the first segment is equal to the angle formed by the second segment , relative to the side of @xmath164 on which @xmath408 lies , that a smooth fractal point @xmath408 of @xmath2 is necessarily a point of infinitely many prefractal approximations @xmath542 . hence , there is a least nonnegative integer @xmath61 such that @xmath538 . ] then the piecewise linear path is called a _ nontrivial path _ of @xmath3 . in @xcite , a nontrivial path was called a _ nontrivial polygonal path_. the change in name is purely based on aesthetics . [ def : acantororbit ] suppose @xmath455 is an orbit of @xmath543 , for some @xmath358 , such that every point of the footprint @xmath544 corresponds to a smooth fractal point of @xmath2 . this then readily implies that @xmath325 is the same as @xmath455 for every @xmath383 . is a constant sequence of compatible orbits , where a _ sequence of compatible orbits _ was defined in definition [ def : sequenceofcompatibleorbits ] . ] then @xmath325 is called a _ cantor orbit _ of @xmath3 and is denoted by @xmath83 . if @xmath3 is a fractal billiard table , then it may or may not be possible to construct cantor orbits or nontrivial paths of @xmath317 . we will next discuss three examples of fractal billiard tables with different dynamical properties that lend themselves well ( or not ) to determining well - defined billiard orbits . we note that applying definitions [ def : acorner ] , [ def : acantorpoint ] and [ def : anelusivepoint ] to @xmath4 and the sequence of rational polygon prefractal approximations @xmath8 which we have discussed in [ subsec : thekochcurveandkochsnowflake ] yields exactly the sets of points we are considering as corners , cantor points and elusive points of @xmath4 , respectively . moreover , applying definitions [ def : acorner ] , [ def : smoothfractalpoint ] and [ def : anelusivepoint ] to @xmath6 and the prefractal approximations @xmath9 which we discussed in [ subsec : thetfractal ] yields exactly the sets of points that we are considering as corners , smooth fractal points and elusive points of @xmath6 . finally , applying definitions [ def : acorner ] and [ def : smoothfractalpoint ] to @xmath533 and the prefractal approximations @xmath10 which we discussed in [ subsec : asierpinskicarpet ] yields exactly the set of points we are considering as corners and smooth fractal points of @xmath533 . as we have noted before at the end of [ subsec : thekochcurveandkochsnowflake ] , for each @xmath225 , @xmath545 can be realized as the union of @xmath234 self - similar ternary cantor sets , each spanning a distance of @xmath235 . within each cantor set , we find cantor points and corners of the koch snowflake . we begin our discussion of orbits of @xmath4 by examining the limiting behavior of a particular sequence of compatible orbits with the initial condition @xmath546 , where @xmath547 is a cantor point of @xmath12 ( i.e. , @xmath547 is a point of @xmath548 with a well - defined tangent in @xmath229 for every @xmath383 ) . for the sake of simplicity , we let @xmath549 and @xmath550 be on the base of the equilateral triangle @xmath232 ( recall that we are assuming that the left corner of @xmath232 is at the origin and the length of each side is one unit ) . then , @xmath330 is an orbit that remains fixed as one constructs @xmath333 from @xmath331 . more correctly , @xmath551 is a sequence of compatible orbits with @xmath552 for every @xmath225 ( that is , with the same footprint in each prefractal approximation ) . in general , if @xmath546 is an initial condition of an orbit of @xmath382 and @xmath547 is a cantor point , then the sequence of compatible orbits is such that for every @xmath383 , the footprints @xmath553 and @xmath554 are the same . if @xmath555 is a cantor point , then there exists a well - defined orbit of @xmath4 with an initial condition @xmath556 , where the angle @xmath557 is determined with respect to the side on which @xmath31 lies in a prefractal approximation @xmath8 . there are many more properties of @xmath558 which we could discuss here . these properties largely rely on the nature of the ternary representation of @xmath547 , and are elaborated upon in @xcite . we now proceed to illustrate how we can connect two elusive points of @xmath4 . such a result has already been presented in greater detail in @xcite , so we will be brief . in [ subsec : thetfractalbilliard ] , we will show that an identical construction holds for the billiard table @xmath6 . recall from example [ exa : asequenceofcompatibleperiodichybridorbits ] that we were able to construct a sequence of compatible periodic hybrid orbits . from such a sequence we can derive a sequence of basepoints that is converging to an elusive point of @xmath4 . the latter sequence of basepoints constitutes the vertices of a nontrivial path ; see figure [ fig : nontrivialpath ] . one may consider a direction @xmath559 that is the reflection of @xmath423 through the normal at @xmath378 . then , the resulting sequence of compatible periodic hybrid orbits @xmath560 yields a sequence of basepoints converging to another elusive point . again , such a sequence of basepoints constitutes the vertices of a nontrivial path of @xmath4 ; see figure [ fig : twonontrivialpaths ] . together , these two nontrivial paths constitute a single nontrivial path connecting two elusive points of @xmath4 . beginning at @xmath561 . ] . ( as is explained in the text , these two paths can be concatenated to obtain a single nontrivial path connecting the two elusive points . ) in the first figure , we only show the relevant portions of the koch snowflake . in the second figure , we magnify the regions containing the nontrivial paths so as to highlight the fact that such paths are converging to elusive points . actually , there is an obvious geometric similarity one can take advantage of in order to produce more segments of the nontrivial path . ] in conjunction with theorem [ thm : bodd ] , we can determine countably infinitely many initial conditions @xmath69 , each of which determines a sequence of compatible periodic hybrid orbits yielding a sequence of basepoints converging to an elusive point of @xmath4 . the results in this subsection appear here for the first time and will be further discussed in @xcite . we begin our discussion of the billiard @xmath6 by recalling ( and referring the reader back to ) example [ exa : t - fractalsequenceofcompatibleorbits ] from [ subsec : thetfractalprefractalbilliard ] . the sequence of compatible periodic orbits provided by example [ exa : t - fractalsequenceofcompatibleorbits ] gives rise to a nontrivial path that connects @xmath562 with an elusive point of @xmath6 . furthermore , considering the sequence of compatible periodic orbits @xmath563 , we determine another nontrivial path that connects @xmath562 with another elusive point of @xmath6 ; see figure [ fig : t - fractalnontrivialpaths ] . this behavior is analogous to the one which we observed for the koch snowflake billiard in [ subsec : thekochsnowflakefractalbilliard ] . . ] as was the case with @xmath4 , we can analogously build upon theorems [ thm : whattheslopecannotbeintfrac ] and [ thm : whattheslopecanbeintfrac ] in order to determine a sequence of basepoints converging to an elusive point . that is , theorems [ thm : whattheslopecannotbeintfrac ] and [ thm : whattheslopecanbeintfrac ] guide our search for a sequence of compatible periodic orbits which yields a sequence of basepoints converging to an elusive point of @xmath6 . let @xmath364 be a sequence of compatible orbits . then , there are countably infinitely many directions and countably infinitely many points from which to choose so that @xmath364 is a sequence of compatible periodic orbits yielding a sequence of basepoints @xmath564 that converges to an elusive point of @xmath6 . the collection of basepoints @xmath564 constitute the vertices of a nontrivial path of @xmath6 . moreover , once such a nontrivial path is constructed , letting @xmath565 , an additional nontrivial path can be determined from a sequence of compatible periodic orbits @xmath566 in exactly the same fashion . in @xcite , nontrivial line segments of sierpinski carpets are constructed . building on the main results of @xcite , the second author and joe p. chen have been able to construct a family of cantor periodic orbits of a self - similar sierpinski carpet , in the sense of @xcite recalled in definition [ def : acantororbit ] . such orbits constitute cantor orbits of the self - similar sierpinski carpet . as of yet , we have not attempted to construct a nontrivial path of a sierpinski carpet . in light of theorem [ thm : constantsequenceofcompatibleorbitsinsa ] , we say that the trivial limit of a constant sequence of compatible periodic orbits constitutes a periodic orbit of a self - similar sierpinski carpet billiard @xmath7 . in the event an orbit has an initial direction @xmath567 , we may still be able to determine a constant sequence of compatible periodic orbits . the trivial limit of such a sequence then constitutes a periodic orbit of @xmath7 . using the fact that reflection can be defined in the vertices with interior angles measuring @xmath377 , we can state the following result . ( recall from [ subsec : aprefractalselfsimilarsierpinskicarpetbilliard ] that @xmath568 is the @xmath61th prefractal approximation of @xmath291 . ) [ thm : alphainaformsorbitofsa ] recall from notation [ nota : alphafortheta ] that if @xmath329 is the initial direction of a billiard orbit , then @xmath569 . let @xmath570 , @xmath571 and let @xmath572 be an orbit of @xmath506 . if @xmath573 , as an orbit of @xmath574 , avoids the middle peripheral square , then the initial condition @xmath575 will determine an orbit of @xmath576 . specifically , the path traversed by the orbit @xmath573 of @xmath574 is exactly the path traversed by the orbit of @xmath7 determined by @xmath575 . [ exa : orbitexamplecontradictingtysonstheoremslope2 - 3origins7 ] let @xmath570 , @xmath577 . consider an orbit of @xmath578 with an initial condition @xmath575 ; see figure [ fig:2 - 3orbitstartingfrom1 - 2 ] . we see that the orbit avoids the peripheral square of @xmath579 . by theorem [ thm : alphainaformsorbitofsa ] , the initial condition @xmath575 determines an orbit of @xmath580 . the path traversed by the orbit of @xmath580 is exactly the path traversed by the orbit @xmath573 . and with an initial direction constituting a slope of @xmath581 , where @xmath582 is defined as in notation [ not : aabaabbb ] . while it would appear that this orbit intersects corners of peripheral squares , it in fact remains away from all peripheral squares . the same is true for finer approximations . ] it is clear from the preceding sections that much work remains to be developed in order to determine a well - defined phase space @xmath583 for the yet to be defined fractal billiard flow . we have discussed several examples of what clearly constitute periodic orbits of @xmath4 and @xmath7 . furthermore , for both @xmath4 and @xmath6 , we were able to connect two elusive points of each billiard table via suitably chosen nontrivial paths . these nontrivial paths were determined from suitably chosen sequences of compatible periodic orbits . [ ques : determineadditionalnontrivialpaths ] let @xmath2 be either @xmath12 or @xmath13 . suppose that two suitably chosen nontrivial paths converge to two distinct elusive points of @xmath3 . for each of the two elusive points , is it possible to determine another nontrivial path converging to a different elusive point ? if we can answer question [ ques : determineadditionalnontrivialpaths ] in the affirmative ( or answer it in the affirmative under specific conditions ) , will this help us gain insight into how to determine a well - defined phase space for the billiard flow on @xmath3 ? an alternate approach , discussed in the concluding remarks of @xcite , entails determining a well - defined fractal translation surface . following this line of thought to its logical end , for certain fractal billiard tables ( e.g. , @xmath6 ) , is it possible to determine which directions produce recurrent orbits ? more generally , can one prove that , in almost every direction , the billiard flow is ergodic in @xmath317 ? regarding a self - similar sierpinski carpet billiard @xmath533 , we have determined a countable set of points from which a periodic billiard orbit can begin . can we show that the set of points from which a periodic orbit can begin is in fact uncountable and , furthermore , a set of full _ _ ( _ _ lebesgue _ ) _ measure in the base of the unit square @xmath257 ? it is possible to construct a nontrivial line segment of @xmath291 beginning from @xmath584 with slope @xmath585 , that , when translated to @xmath586 , no longer lies entirely in @xmath14 . however , if we consider the sequence of compatible periodic orbits given by @xmath587 , is it possible to determine a well - defined limit ? the work of @xcite may prove useful in further exploring the behavior of a sequence of compatible periodic orbits . building on the work of @xcite , the author of @xcite has examined the behavior of nonperiodic orbits in what is an example of what is called a _ wind - tree billiard _ , and what is also strongly suggestive of a sierpinski carpet . such work may provide insight into examining the behavior of a sequence of compatible dense orbits . [ ques : determiningawelldefinedbilliardflow ] in analogy with the prefractal billiard and associated translation surface , can a thorough understanding of the geodesic flow on the limiting _ _ ( _ _ and still to be mathematically defined _ _ ) _ _ _ _ _ _ fractal translation surface__ _ _ @xmath588 aid us in determining a well - defined billiard flow on @xmath3 ? approaching the problem of determining a well - defined billiard flow on a fractal billiard table from many different points of view may prove useful . the theories of translation surfaces and rational billiards are intimately tied together and more deeply understood by knowing the structure of what is called the _ veech group _ ( this being the group studied in , for example , @xcite ) . in short , the veech group of a translation surface @xmath84 determined from a rational polygon @xmath0 is the stabilizer of @xmath84 . let @xmath317 be a fractal billiard table , with @xmath2 being approximated by a suitably chosen sequence of rational polygons @xmath316 . is it then possible to construct a veech group for @xmath317 _ _ ( _ _ or rather , of @xmath588 _ _ ) _ _ , presumably in terms of the veech groups for the prefractal approximations @xmath62 _ _ ( _ _ or rather , of the associated translation surfaces @xmath589 _ _ ) _ _ ? will the knowledge of such a group aid us in determining a well - defined billiard flow on @xmath317 ? barnsley , m. f. : _ superfractals _ : _ patterns of nature _ , cambridge univ . press , new york , 2006 . chen , j. p. , niemeyer , r. g. : periodic billiard orbits of self - similar sierpinski carpets , 29 pages , e - print , arxiv:1303.4032v1 , 2013 . ( to appear in the j. of math . anal . and appl . ) lapidus , m. l. , niemeyer , r. g. : towards the koch snowflake fractal billiard computer experiments and mathematical conjectures , in : _ gems in experimental mathematics _ ( t. amdeberhan , l. a. medina and v. h. moll , eds . ) , contemporary mathematics , amer . math . soc . , providence , ri , * 517 * ( 2010 ) , pp . 231263 . [ e - print : arxiv : math.ds.0912.3948v1 , 2009 . ] lapidus , m. l. , niemeyer , r. g. : sequences of compatible periodic hybrid orbits of prefractal koch snowflake billiards , _ discrete and continuous dynamical systems ser . a _ , in press , 2012 . [ e - print : ihes / m/12/16 , 2012 ; arxiv:1204.3133v1 [ math.ds ] , 2012 . ] smillie , j. : dynamics of billiard flow in rational polygons , in : _ dynamical systems _ , encyclopedia of math . sciences , vol . 100 , math . physics 1 ( ya . g. sinai , ed . ) , springer - verlag , new york , 2000 , pp .
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if @xmath0 is a rational polygon , then the associated rational billiard table is given by @xmath1 .
such a billiard table is well understood .
if @xmath2 is a closed fractal curve approximated by a sequence of rational polygons , then the corresponding fractal billiard table is denoted by @xmath3 . in this paper
, we survey many of the results from [ * lapnie1 - 3 * ] for the koch snowflake fractal billiard @xmath4 and announce new results on two other fractal billiard tables , namely , the @xmath5-fractal billiard table @xmath6 ( see @xcite ) and a self - similar sierpinski carpet billiard table @xmath7 ( see @xcite ) . we build a general framework within which to analyze what we call a sequence of compatible orbits .
properties of particular sequences of compatible orbits are discussed for each prefractal billiard @xmath8 , @xmath9 and @xmath10 , for @xmath11 . in each case
, we are able to determine a particular limiting behavior for an appropriately formulated sequence of compatible orbits .
such a limit either constitutes what we call a nontrivial path of a fractal billiard table @xmath3 or else a periodic orbit of @xmath3 with finite period . in our examples ,
@xmath2 will be either @xmath12 , @xmath13 or @xmath14 .
several of the results and examples discussed in this paper are presented for the first time .
we then close with a brief discussion of open problems and directions for further research in the emerging field of fractal billiards .
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traditionally the ( semi ) phenomenological high precision two - nucleon ( nn)potentials av18 @xcite , cd bonn @xcite and nijm i , ii @xcite go together with the nonrelativistic operator for the kinetic energy @xmath2 in the nn c.m.system . nevertheless , as is well known , this nonrelativistic schrdinger equation @xmath3 can be related to an underlying relativistic nn schrdinger equation @xmath4 by a simple algebraic step@xcite,@xcite . applying @xmath5 to ( [ 2 ] ) from the left one obtains @xmath6 which can be identically rewritten into ( [ 1 ] ) if one defines @xmath7 with @xmath8 . ( we use @xmath9 in order to distinguish the momentum operator from the number @xmath10 ) . therefore adjusting @xmath11 in ( [ 1 ] ) to the nn phase shift and mixing parameters from a phase shift analysis and relating the c.m . momentum @xmath12 to the lorentz invariant lab energy @xmath13 via @xmath14 ( a relation identically valid for relativistic and nonrelativistic kinematics ) one has in fact solved a relativistic equation . we also see that @xmath15 equals @xmath16 .the question remains , what is @xmath17 given @xmath18 ? the formal solution of that quadratic equation ( [ 4 ] ) is @xmath19 why is @xmath17 of interest ? if one turns to the 3n system and would like to investigate relativistic effects @xcite,@xcite the knowledge of @xmath17 is very useful . it defines together with the relativistic kinetic energy the interacting nn mass operator , which is a key ingredient for building the interactive 3n mass operator @xcite . therefore we focus in this paper on the determination of @xmath20 related to the high precision nn potentials via ( [ 4 ] ) or ( [ 6 ] ) . in @xcite,@xcite a potential @xmath20 has been determined directly fitting ( [ 2 ] ) to nn phase shifts . thereby the urbana @xmath21 potential has been readjusted achieving a fair fit ( though not of the quality of the high precision potentials ) . in @xcite a momentum scale transformation @xmath22 has been introduced which provides an analytical relation between @xmath11 and @xmath20 and guarantees that the s - matrix related to ( [ 1 ] ) at c.m.momentum k equals the s - matrix related to ( [ 2 ] ) at c.m . momentum q. in other words the relativistic and nonrelativistic s - matrices agree at the same energy . this , however , is misleading since the equality of the two s - matrices should hold at the same c.m.momenta @xcite . a better , though still approximate approach to relate @xmath11 and @xmath20 has been given in @xcite . on the other hand there is the possibility to add an interaction to the square of the free nn mass operator @xmath23 . then @xmath24 has exactly the same form as ( [ 1 ] ) with @xmath25 provided we identify @xmath26 with the mass operator of the interacting nn system @xcite , @xcite . this is of course also obvious from the relations ( [ 1 ] ) - ( [ 4 ] ) . the construction of the relativistic 3n hamiltonian requires , however , the 3n mass operator @xmath27 rather than its square . therefore our aim here is to solve ( [ 4 ] ) and ( [ 6 ] ) exactly for @xmath20 . this is outlaid in section ii . the validity of the resulting @xmath20 is verified by demonstrating that it provides exactly the same phase shift parameters using ( [ 2 ] ) as the underlying @xmath11 using ( [ 1 ] ) . next we regard the 3n mass operator where nn c.m . forces enter in the form @xcite @xmath28 with @xmath29 the total nn momentum . the @xmath29-dependence arises since in a 3-body system the nn subsystems are not at rest . in section iii we propose a simple manner to determine @xmath30 in a numerical precise way . this opens now the door to use @xmath31 which are equivalent to the underlying high precision potentials in a relativistic context in 3n bound and scattering problems . we summarize in section iv . a technical derivation is given in the appendix . the determination of @xmath20 using eq . ( [ 6 ] ) can be achieved by a spectral decomposition . one can proceed in close analogy to the representation derived in @xcite for @xmath30 given in eq . ( [ 8 ] ) . we regard a specific partial wave state ( or coupled ones ) with given orbital angular momentum(a ) , total spin and total angular momentum . for the sake of a simpler notation we will not show these quantum numbers explicitely . using the completeness relation of bound and scattering states for the potential @xmath18 one obtains @xmath33 where @xmath34 is the nonrelativistic deuteron wave function of ( [ 1 ] ) and @xmath35 the mass of the deuteron . here we introduced a different definition of the binding energy , namely an implicit one : @xmath36 in lowest order it agrees with the usual one @xmath37 . this new definition of the binding energy has in addition the feature that it can naturally be written as @xmath38 in agreement with the form of the energy eigenvalue of ( [ 1 ] ) at the bound state pole @xmath39 of the s - matrix . the expression ( [ 9 ] ) can be identically rewritten into the form @xmath40\cr & - & 2 \omega(k~ ' ) ' \re [ t ( k , k ' ; { { k'}^2 \over m } ) ] \ } \cr & + & { m^2 \over { k ^2 - { k ' } ^2 } } \times \cr & \ { & { \cal p } \int_0^\infty d k '' ( k~'')^2 { 2 \omega(k~ '' ) \over { { { k '' } ^2 - { k } ^2 } } } \cr & & t ( k , k '' ; { { k''}^2 \over m } ) t^ { * } ( k ' , k '' ; { { k''}^2 \over m } ) \cr & - & { \cal p } \int_0^\infty d k '' ( k~'')^2 { 2 \omega(k~ '' ) \over { { k '' } ^2 - { k ' } ^2 } } \cr & & t ( k , k '' ; { { k''}^2 \over m } ) t^ { * } ( k ' , k '' ; { { k''}^2 \over m } ) \}. \label{11 } \end{aligned}\ ] ] where @xmath41 is the standard nn t - matrix related to @xmath42 via the nonrelativistic lippmann schwinger equation . the derivation of that form is defered to the appendix . a numerical implementation has not yet been performed , but we expect no problem . a second more simple way is to directly solve the quadratic operator equation ( [ 4 ] ) . in momentum space it reads @xmath43 or @xmath44 we verified numerically that for all the realistic high precision potentials av18 , cd bonn , and nijm i , ii the following very simple iterative scheme works @xmath45 @xmath46 with @xmath47 . for certain partial waves the iteration to converge requires an additional step , namely @xmath48 where the constants @xmath49 and @xmath50 are typically 1 . we display in table [ tab:1 ] an example documenting the convergence . in figs [ fig.1]-[fig.3 ] we show for an example the original nonrelativistic potential @xmath51 , the resulting relativistic potential @xmath52 and the difference @xmath53 . we see in that example that @xmath54 . av18 in the state @xmath55 . [ fig.1 ] ] related to av18 in the state @xmath55.[fig.2 ] ] in the state @xmath55 . [ fig.3 ] ] to further characterize the difference between @xmath56 and @xmath57 one can regard the asymptotic behavior @xmath58 against @xmath59 as examples , for av18 we find @xmath60 , @xmath61 , whereas cd bonn delivers @xmath62 , @xmath63 . the value of @xmath64 for cd bonn can easily be understood @xcite : @xmath65 with the two @xmath66 s resulting from the meson propagator and the choice of the strong form factor , the @xmath67 arising from transforming the blankenbeclar - sugar equation to the nonrelativistic lippmann schwinger equation and the @xmath68 from the two dirac spinors . in both cases , av18 and cd bonn , @xmath69 is larger than @xmath64 by one unit , which is suggested by eq.([12 ] ) . .convergence of @xmath70 to the iteration in eq . ( [ 14 ] ) . + we choose the coupled partial waves ( @xmath71-@xmath72 ) of the argonne v18 potential@xcite . the momenta @xmath10 and @xmath73 are 1.0 @xmath74 and the potential unit is [ @xmath75 . [ tab:1 ] [ cols="^,^,^,^,^",options="header " , ] if one defines @xmath76 then using ( [ 4 ] ) eq . ( [ 8 ] ) can be written @xmath77 or @xmath78 between free states it yields @xmath79 this has the same structure as ( [ 13 ] ) replacing @xmath80 by @xmath81 . the iteration procedure described in section ii works equally well . we compare in fig [ fig.4 ] @xmath82 and @xmath83 . we see a weakening of @xmath84 against @xmath20 . this is a fact known from previous calculations @xcite and from the approximate ( but very useful ) expression @xcite @xmath85 this is demonstrated in fig [ fig.4 ] , where also an even simpler approximation @xmath86 is shown . that latter approximate version , however , is somewhat worse and is not recommended . against @xmath87 for av18 in the state @xmath55 for fixed @xmath88 @xmath74 . the solid curve shows the relativistic potential @xmath89 ( @xmath90 ) . the other curves show @xmath91 for @xmath92@xmath74 . the long dashed curve is the exact one , the short dashed and dotted curves show the approximations for @xmath30 given in eqs.([approx1 ] ) and ( [ approx2 ] ) , respectively . [ fig.4 ] ] relativistic calculations in the instant form of dynamics proposed in @xcite requires an interacting 3n mass operator . as has been shown in @xcite and used in @xcite,@xcite,@xcite , @xcite , @xcite , and @xcite the nn potential in a moving frame @xmath30 enters in the form given in eq.([8 ] ) , where the nn force in the nn c.m.system @xmath20 enters into the relativistic nn schrdinger equation ( [ 2 ] ) . we showed that the quadratic operator relation ( [ 4 ] ) for @xmath89 can be solved directly in an iterative manner and this very precisely . this has been documented by evaluating nn phase shift and mixing parameters using the standard nonrelatvistic schrdinger equation ( [ 1 ] ) and the relativistic nn schrdinger equation ( [ 2 ] ) . this opens the way to get @xmath20 s related to any nn potential @xmath42 adjusted in a nonrelativistic frame work like the high precision nn potentials . by the same iterative procedure also @xmath30 can be gained . it turned out that @xmath17 is smaller in magnitude than @xmath18 and @xmath30 is smaller than @xmath89 . applications to the 3n bound and scattering states are planned . the numerical calculations has been performed on the ibm regatta p690 + of the nic in jlich , germany . derivation of eq.([11 ] ) we start from ( [ 9 ] ) and use the well known decomposition @xmath93 to arrive at @xmath94 the integral requires some care and we keep the limiting processes for the two scattering states separately by putting @xmath95 this allows us to perform one limit firstly with the result @xmath96 thus we get for some well behaved function @xmath97 @xmath98 the principal value prescription is denoted as `` @xmath99 '' . in our case @xmath100 and therefore @xmath101 this is part of the unitary relation @xmath102 ( note we used the symmetry of the off- the- energy- shell t - matrix ) . consequently @xmath103 \label{a10}\end{aligned}\ ] ] and @xmath104 + \omega ( k ' ) \im [ t ( k~ ' , k ; { { k'}^2 \over m } ) ] \right ) \label{a11}\end{aligned}\ ] ] combined with eq . ( [ a2 ] ) certain terms cancel and one arrives at eq.([11 ] ) .
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the potentials @xmath0 in the nonrelativistic ( relativistic ) nucleon - nucleon ( nn ) schrdinger equation are related by a quadratic equation .
that equation is numerically solved , thus providing phase equivalent @xmath1 - potentials related for instance to the high precision nn potentials , which are adjusted to nn phase shift and mixing parameters in a nonrelativistic schrdinger equation .
the relativistic nn potentials embedded in a three - nucleon ( 3n ) system for total nn momenta different from zero are also constructed in a numerically precise manner .
they enter into the relativistic interacting 3n mass operator , which is needed for relativistic 3n calculations for bound and scattering states .
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the free energy ( or potential of mean force ) is the thermodynamic force driving structural processes such as conformational changes of macromolecules in aqueous solution , ligand binding at the active site of an enzyme , protein - protein association , etc . the free energy gives information about both the rate at which these processes occur and the mechanism by which they occur . this makes free energy calculations a central issue in biophysics . molecular dynamics ( md ) simulations provide a tool for performing such calculations on a computer in a way which is potentially both precise and inexpensive ( e.g. @xcite ) . since a free energy is in essence the logarithm of a probability density function ( see ( [ eq : free ] ) below for a precise definition ) it can in principle be calculated by histogram methods based on the binning of an md trajectory . this direct approach , however , turns out to be unpractical in general because the time scale required for the trajectory to explore all the relevant regions of configuration space is prohibitively long . probably the best known and most widely used technique to get around this difficulty is the weighted histogram analysis method ( wham ) @xcite . following @xcite , wham adds artificial biasing potentials to maintain the md system in certain umbrella sampling windows . wham then recombines in an optimal way the histograms from all the biased simulations to compute the free energy . wham is much more efficient than the direct sampling approach , and generalizations such as @xcite alleviate somewhat the problem of where to put the umbrella windows ( usually , this requires some _ a priori _ knowledge of the free energy landscape ) . in practice , however , wham remains computationally demanding and it only works to compute the free energy in 2 or 3 variables . an interesting alternative to wham is metadynamics @xcite . in essence metadynamics is a way to use an md trajectory to place inverted umbrella sampling windows on - the - fly and use these windows both to bias the md simulation and as histogram bins to sample the free energy directly ( thereby bypassing the need of further histogram analysis in each window ) . both wham and metadynamics compute the free energy directly by histogram methods , but an alternative approach is possible . unlike the free energy which is a global quantity , its negative gradient ( known as the mean force ) can be expressed in terms of a local expectation and thereby computed at a given point in the free energy landscape . this is the essence of the blue moon sampling strategy @xcite and it offers the possibility to calculate first the mean force at a given set of locations , then use this information to reconstruct the free energy globally . in one dimension , this approach is known as thermodynamic integration and it goes back to kirkwood @xcite . in higher dimensions , however , this way to compute free energies has been impeded by two issues . the first is where to place the points at which to compute the mean force , and the second is how to reconstruct the free energy from these data in this paper , we propose a method , termed single - sweep method , which addresses both of these issues in two complementary but independent steps . in a first step , we use the temperature - accelerated molecular dynamics ( tamd ) proposed in @xcite ( see also @xcite ) to quickly sweep through the important regions of the free energy landscape and identify points in these regions where to compute the mean force . in the second step we then reconstruct the free energy globally from the mean force by representing the free energy using radial - basis functions , and adjusting the parameters in this representation via minimization of an objective function . the single - sweep method is easy to use and implement , does not require _ a priori _ knowledge of the free energy landscape , and can be applied to map free energies in several variables ( up to four , as demonstrated here , and probably more ) . the single - sweep method is also very efficient , especially since the mean force calculations can be performed using independent calculations on distributed processors ( i.e. using grid computing facilities @xcite ) . the reminder of this paper is organized as follows . in sec . [ sec : theory ] , we describe the two steps of the single - sweep method in detail , starting with the second one for convenience . in sec . [ sec : app1 ] we illustrate the method on a simple two - dimensional example . this example is then used for comparison with metadynamics in sec . [ sec : metadynamics ] . in sec . [ sec : ad ] we use the single - sweep method to compute the free energy of alanine dipeptide ( ad ) in solution in two and in four of its dihedral angles . finally , concluding remarks are made in sec . [ sec : conclu ] and the details of the md calculation on ad are given in appendix [ sec : md ] we shall consider a molecular system with @xmath0 degrees of freedom whose position in configuration space @xmath1 will be denoted by @xmath2 . we also introduce a set of @xmath3 collective variables @xmath4 which are functions of @xmath2 such as torsion angles , interatomic distances , etc . if @xmath5 denotes the potential energy of the system and @xmath6 its temperature , the free energy @xmath7 in the variables @xmath8 is defined as @xmath9 so that @xmath10 is , up to a proportionality constant , the probability density function ( pdf ) of the variables @xmath8 . as mentioned in the introduction , the negative gradient of the free energy , @xmath11 , is known as the mean force , and it can be computed locally at point @xmath12 via calculation of an expectation ( see ( [ eq : meanforceapprox ] ) below ) . in this section , we shall suppose that we have obtained an estimate of @xmath13 at points @xmath14 , and we focus on the reconstruction of the free energy @xmath7 from these data . a specific way to pick these points and compute @xmath15 will be given in sec . [ sec : tamd ] , but it is worth pointing out that the reconstruction method proposed here works with data set collected in any other ways . our reconstruction method uses a radial - basis function representation for the free energy @xmath7 with centers at @xmath16 @xcite : @xmath17 here @xmath18 is a constant used to adjust the overall height of @xmath19 but is otherwise irrelevant , @xmath20 denotes the euclidean norm in @xmath21 , and @xmath22 where @xmath23 is a radial - basis function ; a convenient choice is to use the gaussian packet @xmath24 though other radial - basis functions ( multiquadric , sobolev splines , wendland , etc . @xcite ) can be used as well , see sec . [ sec : ad ] . in ( [ eq : radialbasis ] ) the heights @xmath25 and the radial - basis function width @xmath26 are adjustable parameters which we determine by minimizing over @xmath25 and @xmath27 the following objective function , which measures the discrepancy between the negative gradient of the function @xmath28 in ( [ eq : radialbasis ] ) at the centers @xmath29 , @xmath30 , and the mean force @xmath31 estimated at these centers : @xmath32 before explaining how we perform this minimization , let us give several reasons why the radial - basis representation ( [ eq : radialbasis ] ) for @xmath7 is natural and convenient . first , the centers @xmath29 in ( [ eq : radialbasis ] ) do not have to lie on a regular grid , which permits to use mean force data collected anywhere . second , the representation ( [ eq : radialbasis ] ) can be used in any dimension . third , this representation has very good convergence properties , i.e. a small number of centers gives an accurate representation of @xmath7 . in fact , unlike standard representations based e.g. on linear interpolation on a regular grid , the rate of convergence in @xmath33 of the representation in ( [ eq : radialbasis ] ) can be made independent of @xmath3 ( a feature which the radial - basis representations share with sparse grids @xcite ) . going back to the minimization of @xmath34 , it can be performed as follows . for fixed @xmath27 , the @xmath35 minimizing ( [ eq : objective ] ) solve the following linear algebraic system @xmath36 where @xmath37 and @xmath38 are given by @xmath39 given the centers @xmath29 and the estimates @xmath40 of the mean force at these centers , the coefficients @xmath41 and @xmath42 can be easily computed , and the linear system ( [ eq : linalg ] ) can be solved by any standard technique , e.g. gaussian elimination . once the solution @xmath43 of ( [ eq : linalg ] ) is determined , to find the optimal @xmath44 satisfying @xmath45 we compute the residual @xmath46 for increasing values of @xmath27 starting from the distance between the centers . more sophisticated procedures could be used to minimize @xmath46 over @xmath27 , but the brute force method that we used proved to be efficient enough because computing successive solutions of ( [ eq : linalg ] ) for various @xmath27 is very fast . to measure the error in the approximation , we used the residual per center defined as @xmath47 which reaches its minimum value at the same @xmath44 as @xmath46 . overall , the procedure is simple and inexpensive since the determination of @xmath35 at fixed @xmath27 is computationally straightforward and cheap , and can be easily repeated to perform the one - dimensional minimization over @xmath27 . one caveat that we should mention , however , is that the condition number of the matrix @xmath37 increases rapidly when the number of centers and/or @xmath27 increase . this is a known problem of radial - basis functions @xcite . to avoid any problems , we capped the admissible condition number at @xmath48 and , in situations where this threshold value was reached while @xmath49 was still decreasing , picked for @xmath44 the corresponding value of @xmath27 . these situations only occurred in the two - dimensional example ( see sec . [ sec : app1 ] ) when a lot of centers were used ( 500 or more , i.e. much more than what will be used in the ad example ) , and even in these cases , such a large condition number did not lead to any noticeable loss of accuracy in the results ( even though the coefficients @xmath35 were then very large ) . we also observed that , given a number of centers @xmath29 and a value of @xmath27 , the condition number is typically lower when the dimension of @xmath12 is larger . finally , we observed that the condition number was much lower with the wendland radial - basis function ( see ( [ eq : wend ] ) in sec . [ sec : ad ] ) than with the gaussian radial - basis function ( [ eq : kernel ] ) . it remains to explain how to identify the centers @xmath50 and estimate the mean force at these points . following @xcite , we will do so using the extended system @xmath51 & \quad + \text{thermostat terms at $ \beta^{-1}$ } \end{aligned } \\[20pt ] \displaystyle\gamma \dot \zz = \kappa ( \ttheta(\xx ) - \zz ) + \sqrt{2 \gamma \bar \beta^{-1}}\ , \eeta(t ) \end{cases}\ ] ] where @xmath52 is the mass matrix , @xmath53 is a white - noise , i.e. a gaussian process with mean 0 and covariance @xmath54 , and @xmath55 , the friction coefficient @xmath56 and the artificial inverse temperature @xmath57 are parameters whose role we explain now . the system in ( [ eq : tamd ] ) describes the motion of @xmath2 and @xmath12 over the extended potential @xmath58 as shown in @xcite , by adjusting the parameter @xmath59 so that @xmath60 and the friction coefficient @xmath61 so that @xmath12 moves slower than @xmath2 , one can generate a trajectory @xmath62 in @xmath63-space which effectively moves at the artificial temperature @xmath64 on the free energy computed at the physical temperature @xmath6 . by taking @xmath65 , the @xmath62 trajectory visits rapidly the regions where the free energy is relatively low ( i.e. within a range of a few @xmath64 ) even if these regions are separated by barriers which the system would take a long time to cross at the physical temperature @xmath6 . this gives us a way to determine automatically where are the relevant regions in free energy space . in @xcite , the extended system in ( [ eq : tamd ] ) was proposed to sample the free energy landscape directly . here , we make a different use of ( [ eq : tamd ] ) : we utilize the trajectory @xmath62 to rapidly sweep through @xmath12-space and generate the centers @xmath16 used in the radial - basis representation ( [ eq : radialbasis ] ) . specifically , we start from @xmath66 , then deposit a new center @xmath29 along @xmath62 each time @xmath62 reaches a point which is more than a prescribed distance @xmath67 away from all the previous centers , where @xmath67 is a parameter controlling the density of the covering by the centers ( the smaller @xmath67 , the higher the number of centers deposited ) . at the same time , at each of these centers @xmath29 , we launch a simulation of ( [ eq : tamd ] ) with @xmath68 fixed , i.e we use @xmath69 & \quad + \text{thermostat terms at $ \beta^{-1}$ } \end{aligned}\ ] ] and compute : @xmath70 the calculations of these time averages are independent of each other , and hence they can be distributed , using ( ideally ) at least one processor per center @xmath29 , an approach that optimally fits with the purposes of grid computing @xcite . the estimator in ( [ eq : meanforceapprox ] ) has the advantage of being simple , but it introduces an error due to the finiteness of @xmath71 . this error can be decreased by using @xmath71 in ( [ eq : tamd2 ] ) and ( [ eq : meanforceapprox ] ) larger than @xmath59 in ( [ eq : tamd ] ) , or even eliminated by using constrained instead of restrained simulations and using the blue - moon estimator for the mean force @xcite . once the centers @xmath72 have been deposited and the estimates @xmath73 of the mean force at these centers have been obtained , we use the reconstruction procedure explained in sec.[sec : reconstruct ] and compute the optimal set of coefficients @xmath35 and the optimal @xmath44 to use in the representation ( [ eq : radialbasis ] ) for @xmath7 . we conclude this section by stressing that using the extended dynamics ( [ eq : tamd ] ) to sweep through @xmath12-space and deposit the centers @xmath29 is very different than using it to sample @xmath7 directly , which makes our approach very different from wham or metadynamics . unlike with sampling , revisiting twice a region in @xmath12-space is unnecessary and even undesirable since no new center will be deposited . the accuracy of the reconstruction depends on the number of centers and the accuracy at which the mean force is computed in ( [ eq : meanforceapprox ] ) much more than the precise locations where the centers are deposited . an important practical consequence is that it is rather straightforward to pick the parameters @xmath59 and @xmath61 in ( [ eq : tamd ] ) since the final result is robust against variations in these parameters . ) by forward euler with a time - step @xmath74 for @xmath75 steps ( white curve ) shown above the contour plot of the mueller potential ( with @xmath76 level sets evenly distributed between @xmath77 and @xmath78 in a scale where the minimum of the potential is @xmath77 ) . the red circles are the locations of the centers deposited along the trajectory using @xmath79 . in this run , @xmath80 centers were deposited.,width=312 ] since , given the location of the centers @xmath29 , the mean force estimation at these points is quite standard , as a first illustration we use a two - dimensional example for which @xmath81 and @xmath82 where @xmath5 is the mueller potential @xcite . in this case , there is no need to extend the system as in ( [ eq : tamd ] ) , and the temperature accelerated dynamics simply reduces to ( setting @xmath83 by appropriate rescaling of time ) @xmath84 fig . [ fig:1 ] shows a tamd trajectory generated by solving ( [ eq : tamdex ] ) by forward euler with the initial condition @xmath85 and a time - step of @xmath86 for @xmath75 time - steps at @xmath87 ( for comparison the energy barrier between the two main minima of the mueller potential is about @xmath88 ) . also shown are the centers @xmath89 obtained by depositing a new center along the trajectory each time the trajectory reaches a point which is @xmath79 away for all the previous centers . in this run , @xmath80 centers were deposited . at the centers , we used @xmath90 as estimate of the `` mean force '' ( i.e. there is no sampling error in the present example ) . we then used these data to reconstruct the free energy as explained in sec . [ sec : reconstruct ] . [ fig : muelsweepres ] shows the residual per center @xmath49 defined in ( [ eq : resid ] ) . the optimal @xmath27 for this run was @xmath91 and the condition number at this @xmath44 was @xmath92 . the level sets of the reconstructed potential are shown in fig . [ fig:2 ] and compared to those of the original mueller potential , while fig . [ fig : muelsweeptrj1 ] compares the values of the original and reconstructed mueller potential along the tamd trajectory shown in fig . [ fig:1 ] . as a simple estimate of the error , we used : @xmath93 where @xmath94 denotes the reconstructed potential and @xmath95 is the domain in which the original potential remains less than @xmath96 above its minimum value . the error defined in ( [ eq : error1 ] ) for this calculation was @xmath97 . these results , which are already very good , can be improved by diminishing @xmath67 and thereby increasing the number of centers without having to increase the length of the tamd trajectory . for example , by taking @xmath98 , we obtained @xmath99 centers in a trajectory still @xmath100 steps long . using these centers to reconstruct the mueller potential , we obtained @xmath101 . the level sets of the reconstructed and original potential defined in fig . [ fig:2 ] now overimposed so perfectly that they could not be distinguished on the scale of fig . [ fig:2 ] ( data not shown ) . the optimal @xmath27 for this calculation was @xmath102 , at which value the residual was @xmath103 and the condition number was @xmath104 . finally , we note that the result can also by improved by keeping the same distance @xmath98 between centers but increasing the length of the tamd trajectory . for instance , increasing the number of steps to @xmath105 produced a reconstruction with an error @xmath106 ( data not shown ) . defined in ( [ eq : resid ] ) for the reconstruction of the mueller potential with the @xmath107 steps single - sweep trajectory shown in fig . [ fig:1 ] . the optimal @xmath27 for this run was @xmath91.,width=312 ] ) ( black curve ) . here we use the @xmath80 centers shown in fig . [ fig:1 ] . the optimal @xmath27 is @xmath91 , see fig . [ fig : muelsweepres ] . we show 29 level sets evenly distributed between @xmath77 and @xmath78 . the level sets of the reconstructed potential and the original one are in so close agreement that they can only be distinguished in some localized regions ( e.g. near the saddle point between the two minima in the lower right corner ) . , width=312 ] because metadynamics @xcite also uses an extended dynamical system for @xmath2 and @xmath12 and the gaussian packet ( [ eq : kernel ] ) to represent @xmath7 , the single - sweep method bears similarities with it . yet , there is an essential difference between the two methods . unlike the single - sweep method , metadynamics does not use the mean force , and estimates @xmath7 by direct sampling , which turns out to be a less efficient way to proceed . let us elaborate on this claim . recall that metadynamics uses an extended system like ( [ eq : tamd ] ) but where the equation for @xmath12 is replaced by @xcite @xmath108 here @xmath6 is now the physical temperature of the system , and @xmath59 and @xmath61 are parameters playing the same role as in ( [ eq : tamd ] ) . the integral term in ( [ eq : zzmeta ] ) is a flooding term ( with @xmath109 controlling the flooding rate ) which deposits gaussian packets @xmath110 on the energy landscape wherever @xmath62 goes , thereby progressively leveling the effective free energy landscape felt by @xmath62 . the negative of the integral of the gaussian packets deposited then gives an approximation of the free energy . for a trajectory with @xmath111 time - steps of size @xmath112 , the time - discretized approximation of this integral reads ( compare ( [ eq : radialbasis ] ) ) @xmath113 where @xmath18 is a constant used to adjust the height of @xmath19 . steps single - sweep trajectory shown in fig . [ fig:1 ] . the green line shows the absolute value of the difference between the two.,width=312 ] despite the fact that ( [ eq : freemeta ] ) uses gaussian packets which are radial - basis functions , the representation ( [ eq : freemeta ] ) is very different from the standard radial - basis representation ( [ eq : radialbasis ] ) used in the single - sweep method . in particular , there are no coefficients @xmath25 to adjust in ( [ eq : freemeta ] ) this has the important consequence that , instead of requiring a single sweep across @xmath12-space to get an accurate estimate of the free energy , metadynamics requires that the trajectory revisits many times the same locations in @xmath12-space to deposit centers ( i.e. @xmath111 in ( [ eq : freemeta ] ) must be much larger than @xmath33 in ( [ eq : radialbasis ] ) to achieve the same accuracy ) . this is because the leveling out achieved by the integral term in ( [ eq : zzmeta ] ) and , hence , the convergence of the representation ( [ eq : freemeta ] ) , only occur statistically @xcite ( in contrast , the mean force data used at each center in the single sweep method contains already all the statistical information needed at that center ) . this is consistent with metadynamics being in essence an histogram method , albeit one where the histogram windows are adjusted on - the - fly . steps overimposed on the original mueller potential.,width=312 ] what this entails in terms of efficiency can be illustrated on the two - dimensional mueller example considered before . in this example , to generate the metadynamics trajectory we used ( [ eq : tamdex ] ) with flooding terms added as in ( [ eq : zzmeta ] ) , consistent with what was done in ref . @xcite to test the efficiency of metadynamics in a similar set - up . [ fig : muelmetaxy ] shows the metadynamics trajectory obtained by integrating ( [ eq : zzmeta ] ) for @xmath75 timesteps with a time - step of @xmath114 ( same as in the single - sweep method for the results shown in figs . [ fig:1 ] to [ fig : muelsweepres ] ) . as can be seen in fig . [ fig : muelmetaxy ] , this number of timesteps was enough for the trajectory to visit the important regions of the potential . the reconstructed free energy from this calculation is compared in fig . [ fig:6 ] to the original one . the metadynamics result is also compared in fig . [ fig:6b ] to the one obtained by the single - sweep method with @xmath80 centers . the error ( [ eq : error1 ] ) for this metadynamics calculation was @xmath115 , i.e. almost two orders of magnitude higher than with the single - sweep method . [ fig : muelmetatrj1 ] shows the original and reconstructed mueller potential along the metadynamics trajectory ( blue and red lines , respectively ) , together with the absolute value of their difference ( green line ) . by comparing figs . [ fig : muelsweeptrj1 ] and [ fig : muelmetatrj1 ] , it can be seen that the discrepancy between the original and reconstructed potential is much larger with metadynamics than with the single - sweep method on trajectories of the same length . note that these results clearly indicates that it is not sufficient that the metadynamics trajectory visits once a region on phase space to get an accurate representation of the free energy in this region . this was already noted in refs . @xcite . steps with a time - step of @xmath74 ( same as in figs.[fig:1 ] and [ fig:2 ] ) . we only use 10 level sets evenly distributed between @xmath77 and @xmath78 because the differences between the maps are much bigger than with the single - sweep method and drawing more level sets makes the figure difficult to read . ( there are only seven black level sets in the energy reconstructed by metadynamics because it levels off around @xmath116.),width=312 ] steps and a time - step of @xmath74 . other representation of these contourplots were already shown in figs . [ fig:2 ] and [ fig:6 ] respectively . the map reconstructed by the single - sweep method is very close to the map of the original mueller potential . the colormaps used in both panels are the same and the reconstructed potentials are shifted so that their minimum is @xmath77 ; the white region in the left panel is where the energy is above @xmath96 and is not shown ( the result of metadynamics shown in the right panel levels off around @xmath116 which is why there is no white).,width=312 ] steps and a time - step of @xmath74 . other representation of these contourplots were already shown in figs . [ fig:2 ] and [ fig:6 ] respectively . the map reconstructed by the single - sweep method is very close to the map of the original mueller potential . the colormaps used in both panels are the same and the reconstructed potentials are shifted so that their minimum is @xmath77 ; the white region in the left panel is where the energy is above @xmath96 and is not shown ( the result of metadynamics shown in the right panel levels off around @xmath116 which is why there is no white).,width=312 ] we were able to improve the metadynamics result by extending the simulation to @xmath117 steps . the covering of the important regions in the potential was now extensive ( data not shown ) , and the reconstructed potential ( data not shown ) looked visually better than the one obtained with the shorter trajectory . yet the error ( [ eq : error1 ] ) was @xmath118 , i.e. still two orders of magnitude larger than the highest error we obtained with the single - sweep method using a 10 times shorter trajectory . we did not attempt to go to longer runs with metadynamics because the memory term in ( [ eq : zzmeta ] ) makes such simulations increasingly expensive ( their cost scales as the square of the number of timesteps ) . it should also be stressed that in all these calculations , we optimized the parameters @xmath119 , @xmath27 and @xmath6 the best we could . this optimization , however , turns out to be complicated since there is no systematic way to perform it because , unlike with the single - sweep method , there is no objective function to minimize in metadynamics . the results shown in figs . [ fig : muelmetaxy][fig : muelmetatrj1 ] were obtained with @xmath120 , @xmath121 and @xmath122 . to be fair , we should conclude this comparison by mentioning that the simplicity of the mueller potential example tends to exaggerate the gain that the single sweep method provides over metadynamics . indeed , in realistic situations , the single - sweep method also requires to compute the mean force via ( [ eq : meanforceapprox ] ) , an operation which was unnecessary in the mueller example since the force was readily available . computing the mean force adds an extra cost to the method . it is worth stressing again , however , that the computation of the time averages in ( [ eq : meanforceapprox ] ) can be distributed over several processors . this means that in the ideal situation where the user has at least one processor per center , the effective time to compute all of the mean forces is the same as the one for computing a single one of these forces , i.e. we have perfect scalability . metadynamics can be parallelized per replica as well , as was proposed in refs . @xcite , but not as straightforwardly and not with perfect scalability . indeed , in all of these versions of metadynamics , the simulated replica are never completely independent from each other . steps metadynamics shown in fig . [ fig : muelmetaxy ] . the green line shows the absolute value of the difference between the two . the green line here should be compared with the one in fig . [ fig : muelsweeptrj1 ] for the single - sweep method : the discrepancy between the original and reconstructed potential is always larger with metadynamics than with the single - sweep method.,width=312 ] in this section , we use the single - sweep method to reconstruct the free energy of the solvated alanine dipeptide ( ad ) molecule in two and four torsion angles at @xmath123 k. while ad is not an example of biochemical interest _ per se _ , we study it because it has been extensively used as a benchmark example for free energy calculations in the literature @xcite . on top of this the system is simple enough that we can use it to systematically investigate how the accuracy of the reconstruction method depends on the number of centers and how robust the method is with respect to statistical errors in the input data for the mean forces . another question we investigate in this section is the robustness of the method against the choice of radial - basis functions . specifically , we compare results obtained using the gaussian packet ( [ eq : kernel ] ) and the wendland function @xmath124 where @xmath125 if @xmath126 , and @xmath127 otherwise . ( [ eq : wend ] ) is another well - known example of radial - basis function which has the pleasant property that it is compactly supported . this property is appealing in the calculations since it limits the range over which centers interact in ( [ eq : objective ] ) . all md simulations reported below were performed with a version of the moil code @xcite suitably modified by us , and the amber / opls @xcite force field ( for details of the md set - up see appendix [ sec : md ] ) . we use the standard dihedral angles @xmath128 and @xmath129 . at @xmath123 k , the system is confined in a region of the @xmath130 space with @xmath131 by energy barriers higher than @xmath6 . in order to overcome these barriers and sweep through the whole @xmath132 ^ 2 $ ] space , we generated a trajectory by using ( [ eq : tamd ] ) with @xmath133 , @xmath134 kcal / mol / rad@xmath135 , a friction coefficient @xmath136 kcal@xmath137ps / mol / rad@xmath135 and an artificial temperature @xmath138 kcal / mol . with this choice of the parameters , the important regions of the @xmath132 ^ 2 $ ] space were visited in @xmath139 steps ( @xmath140 ps in the time units of the md variables ) . the time series of @xmath128 and @xmath129 along this trajectory are shown in fig . [ fig : adtrj ] . variations in @xmath59 , @xmath61 and @xmath141 led to qualitatively similar tamd trajectories , indicating that the the method is robust with regard to the choice of these parameters . sets with a different number @xmath33 of centers were deposited along the tamd trajectory afterwards by processing this trajectory using various distances @xmath67 between the centers . specifically , we generated sets of @xmath142 , @xmath143 , @xmath144 , @xmath145 , @xmath146 and @xmath147 centers using , respectively , @xmath148 , @xmath149 , @xmath150 , @xmath151 , @xmath152 , @xmath153 . and @xmath129 along the @xmath140 ps long tamd trajectory for the solvated alanine dipeptide ( ad ) . , width=312 ] given a set of @xmath33 centers @xmath154 , we computed the mean forces via @xmath33 independent md simulations with restraints at @xmath155 , i.e. by simulating ( [ eq : tamd2 ] ) in the isokinetic ensemble at @xmath123 k , and estimated the mean force via ( [ eq : meanforceapprox ] ) with @xmath156 kcal / mol / rad@xmath135 . this value of @xmath157 was high enough since we checked that the reconstructed free energy remained invariant with higher values of @xmath157 ( we did so up to @xmath158 kcal / mol / rad@xmath135 ) . we then used this data in the reconstruction procedure explained in sec . [ sec : reconstruct ] . note that since the free energy in the @xmath130 angle is periodic , we have to periodically extend the centers for the representation . this amounts to changing the representation in ( [ eq : radialbasis ] ) into @xmath159 where @xmath160 is the unit vector in @xmath21 . in practice , only few periodic replica of the centers are needed ( i.e. @xmath161 for @xmath162 ) because the radial - basis functions centered at the centers further away from the cell under consideration make negligible contributions to the result in this cell . we first detail our result with this choice of parameters to pick an example which led to a good balance between accuracy and efficiency . other choices of parameters are discussed below . thus , fig . [ fig:7 ] shows the reconstructed free energy map obtained with @xmath163 ( @xmath145 centers ) and by computing the mean forces from ( [ eq : meanforceapprox ] ) with @xmath164 ps . the optimal @xmath27 in this calculation was @xmath165 . in the figure , the minimum of the free energy is set at @xmath166 kcal / mol , and contour levels are plotted at @xmath167 kcal / mol , @xmath168 kcal / mol and then every @xmath168 kcal / mol . the centers are represented as white circles , and the mean forces at the centers as arrows . and @xmath129 dihedral angles at @xmath123 k calculated with the single - sweep method by using @xmath145 centers deposited at a distance of @xmath163 from each other . units for the free energy are kcal / mol , and contour levels are plotted at @xmath167 kcal / mol , @xmath168 kcal / mol , and then every @xmath168 kcal / mol . the optimal @xmath27 in this reconstruction was @xmath169 . the centers are represented as white circles . at every center , the corresponding mean force vector is also shown . mean forces were calculated by using ( [ eq : meanforceapprox ] ) with @xmath156 kcal / mol / rad@xmath135 and @xmath164 ps.,width=312 ] since the free energy map depends on the force field used , comparison with results in the literature is difficult . to assess the accuracy of our result self - consistently , we compared it with the free energy calculated by computing the pdf of @xmath128 and @xmath129 from a direct md simulation ( dmds ) of about @xmath170 ns . while this trajectory does not cover all the @xmath132 ^ 2 $ ] space , it covers the important regions and allows for an unbiased estimation of the free energy in these regions which can be used as benchmark . the left panel in fig . [ fig:8 ] shows the contour levels of the free energy from the single - sweep ( black lines ) and that from dmds ( red lines ) . the contour levels are plotted at @xmath171 kcal / mol ( dotted lines ) , every @xmath167 kcal / mol from @xmath167 to @xmath172 kcal / mol , and then every @xmath173 kcal / mol . as can be seen , single - sweep results agree remarkably well with those of the dmds . in terms of cost , to generate the result shown in fig . [ fig:7 ] , we had to make one simulation run of @xmath140 ps to generate the tamd trajectory , plus @xmath145 independent runs of @xmath174 ps distributed on different nodes ( the additional cost of estimating the parameters @xmath35 and @xmath44 to use in ( [ eq : radialbasis ] ) is insignificant ) . this makes for a total of @xmath175 ns of absolute simulation time . however , after distribution , the effective simulation time needed is only @xmath142 ps . on top of this , we show below that a good estimate of the free energy can be obtained with as low as 90 centers ( i.e. with an absolute simulation time of @xmath176 ns and the same effective simulation time , @xmath142 ps ) . for comparison , in ref . @xcite ensing _ et al . _ report a @xmath172 ns calculation performed with metadynamics to estimate the free energy of ad in @xmath128 and @xmath129 . it is not clear how well this metadynamics calculation can be parallelized to reduce its effective cost ( it was not parallelized in ref . in addition , the result of this metadynamics calculation is unlikely to be as accurate as the one in fig . [ fig:7 ] ( in ref . @xcite no comparison like the one shown in fig . [ fig:8 ] is provided ) centers , the right panel the one with @xmath147 centers . the contour levels of the free energy are plotted at @xmath171 kcal / mol ( dotted lines ) , from @xmath167 to @xmath177 kcal / mol separated by @xmath167 kcal / mol , and then separated by @xmath173 kcal / mol.,width=325 ] next we analyze how robust are the results with respect to the statistical error in the mean force data and the choice of radial - basis function . we also analyze convergence in function of the number of centers . as reference value , we take the free energy reconstructed with @xmath178 ( @xmath147 centers ) and @xmath179 ps of time averaging in ( [ eq : meanforceapprox ] ) . the map of the free energy calculated with these parameters ( data not shown ) is visually very similar to the one shown in fig . [ fig:7 ] , but it is more accurate . the residual error can be estimated from the right panel of fig . [ fig:8 ] which shows the contour levels of the free energy from the single - sweep ( black lines ) and that from dmds ( red lines ) : these level sets coincide up to statistical errors in the dmds , indicating that the free energy provided by the single sweep method with @xmath147 centers and @xmath179 ps can indeed be taken as an `` exact '' benchmark . and @xmath129 reconstructed with gaussian functions using @xmath147 centers and @xmath179 ps . the figures in the different panels correspond to various number of centers and length of time averaging for the mean forces , as indicated . units are kcal / mol . note that the scale of the colormap is different from the one in fig . [ fig:7 ] . in particular , the differences are mostly below @xmath167 kcal / mol with @xmath144 centers and @xmath174 ps simulations already.,width=312 ] fig . [ fig : diffsad ] shows the differences between the map of the reference free energy reconstructed with @xmath178 ( @xmath147 centers ) and @xmath179 ps and those reconstructed with less centers and shorter restrained simulations . the largest errors are in the regions corresponding to the highest peaks of the free energy ( these are also the regions were the least centers were deposited ) . the differences never exceed @xmath180 kcal / mol with @xmath142 centers and @xmath164 ps and they fall mostly below @xmath167 kcal / mol with @xmath144 centers and @xmath164 ps already . we also compared the quality of the reconstruction of the free energy when using gaussian ( [ eq : kernel ] ) and wendland ( [ eq : wend ] ) basis functions . [ fig : resad ] shows the residual per center versus the number of centers for ad , by using gaussian ( filled symbols ) and wendland ( empty symbols ) basis functions , using @xmath164 ps ( diamonds ) and @xmath179 ps ( circles ) long restrained simulations to estimate the mean forces . at equal values of @xmath33 and @xmath181 , the reconstruction is slightly more accurate with gaussian than with wendland functions , though these differences turn out to be quite small in terms of the free energy maps themseves ( data not shown ) . and @xmath129 angles . data are from calculations with different time averaging length , using gaussian ( g ) and wendland ( w ) basis functions.,width=312 ] using longer simulations for the mean force ( which means a smaller random error on these forces ) also improves the results . with @xmath147 centers and @xmath179 ps long simulations for the mean forces , the maps reconstructed with gaussian and wendland basis functions were almost identical ( data not shown ) , and they were not significantly different from the map shown in fig . [ fig:7 ] . these results , however , were obtained at very different values of optimal @xmath27 : @xmath182 with gaussians and @xmath183 with wendland functions . the condition numbers at the optimal @xmath27 were @xmath184 and @xmath185 , respectively , which is low . note that the same trends here described were observed in a periodic test case for which the potential was known exactly ( data not shown ) . as a second more challenging test , we computed the free energy of ad in the four torsion angles , @xmath128 , @xmath129 , @xmath186 , and @xmath187 . a tamd trajectory of @xmath188 ps was generated by using ( [ eq : tamd ] ) with @xmath189 , @xmath134 kcal / mol / rad@xmath135 , an artificial temperature @xmath190 kcal / mol , friction coefficients @xmath136 kcal@xmath137ps / mol / rad@xmath135 for @xmath128 and @xmath129 and @xmath83 kcal@xmath137ps / mol / rad@xmath135 for @xmath186 and @xmath187 . the md potential keeps the amide planes in _ trans _ configuration , and so @xmath186 and @xmath187 were varying in the range @xmath191 $ ] . in @xmath188 ps , the tamd trajectory covered well the accessible state space for @xmath128 , @xmath129 , @xmath186 and @xmath187 , in the sense that the time series for these angles were similar in their respective state space to those shown in fig . [ fig : adtrj ] ( notice however that extensive coverage of the four - dimensional space is unlikely in so short a run ) . along the tamd trajectory , @xmath192 centers at a distance of @xmath193 were deposited . at these centers , the mean forces @xmath40 were computed by using ( [ eq : meanforceapprox ] ) with @xmath156 kcal / mol / rad@xmath135 and @xmath164 ps ( i.e. the absolute time of simulation was about @xmath194 ns , but the effective time after distribution was @xmath195 ps only ) . we used these @xmath40 in the objective function ( [ eq : objective ] ) to finally get the representation ( [ eq : radialbasis ] ) of the four - dimensional free energy @xmath196 . the optimal @xmath27 in this representation was @xmath197 and the condition number at this value of @xmath27 was @xmath198 . , @xmath129 , @xmath186 angles obtained from the marginal in these angles of the pdf associated with the free energy in four angles @xmath196 . data are represented for @xmath199 $ ] . note that the scale of the colormap is different from the one in fig . [ fig:7],width=312 ] since a full graphical representation of @xmath200 is not possible , we did several tests to validate our result . [ fig:9 ] shows the three dimensional free energy @xmath201 obtained from the marginal in these angles of the pdf associated with @xmath200 . this marginal was calculated _ a posteriori _ by numerical integration over @xmath187 of @xmath202 with the full @xmath200 reconstructed by the single - sweep method . the map is reasonable , and shows nontrivial features in all three directions . [ fig:10 ] shows the two dimensional free energy @xmath203 obtained from the marginal in these angles of the pdf associated with @xmath196 . the map is in remarkably good agreement with the one in fig . [ fig:7 ] . as a further test of accuracy , we re - calculated the mean force using ( [ eq : tamd2 ] ) and ( [ eq : meanforceapprox ] ) using a different set of centers than those used in ( [ eq : radialbasis ] ) . then we estimated the relative error between these mean forces and the ones obtained by taking the negative gradient of the reconstructed @xmath196 using the original set of centers and mean forces : @xmath204 where @xmath205 are the new centers and @xmath206 are the mean forces at these centers . the new centers were 20 points chosen at random in the domains @xmath207 $ ] , @xmath208 $ ] . , @xmath129 angles obtained from the marginal in these angles of the probability density associated with the free energy in four angles @xmath196 . contour levels are as in fig . [ fig:7 ] . note the remarkable agreement between this map and the one shown in fig . [ fig:7].,width=312 ] fig . [ fig : randtest ] , top panel , shows , for each of these centers , the distance from the closest of the @xmath192 centers ( black line ) . data are compared to the minimal distance between the @xmath192 centers ( red dashed line ) . this result shows that , with @xmath193 , these centers fill properly the four dimensional domain in the sense that every new center is always a distance about @xmath67 to one of the original centers . [ fig : randtest ] , middle panel , shows the relative error @xmath209 for @xmath210 ( black solid line ) , when @xmath164 ps long restrained simulations are used to compute the mean forces . the mean value of @xmath209 ( black dashed line ) is @xmath211 , with standard deviation @xmath212 and maximum value @xmath213 . for comparison , the mean value of the relative residual per center ( red dashed line ) is @xmath212 , with standard deviation @xmath214 and maximum value @xmath215 ( red dashed - dotted line ) . [ fig : randtest ] , bottom panel , shows @xmath209 when @xmath179 ps long restrained simulations are used to compute the mean forces . in this case , the mean value of @xmath209 ( black dashed line ) is @xmath216 , with standard deviation @xmath217 and its maximum value is @xmath218 . for comparison , the mean value of the relative residual per center ( red dashed line ) is @xmath212 , with standard deviation @xmath214 and maximum value @xmath219 ( red dashed - dotted line ) . these results show that , in points away from the original centers , the reconstructed free energy is as accurate as it is at the centers , which is clearly the best we can hope for . centers ( black line ) , compared with the minimal distance between the @xmath192 centers ( red line ) . middle and lower panel , relative error @xmath209 defined in ( [ eq : relerror ] ) for mean forces computed respectively from @xmath174 and @xmath220 ps restrained simulations : the error @xmath209 ( black solid line ) and its mean value ( black dashed line ) , compared with the mean value of the relative residual per center for the @xmath192 centers set ( red dashed line ) and its maximum value ( red dashed - dotted line).,width=312 ] centers ( black line ) , compared with the minimal distance between the @xmath192 centers ( red line ) . middle and lower panel , relative error @xmath209 defined in ( [ eq : relerror ] ) for mean forces computed respectively from @xmath174 and @xmath220 ps restrained simulations : the error @xmath209 ( black solid line ) and its mean value ( black dashed line ) , compared with the mean value of the relative residual per center for the @xmath192 centers set ( red dashed line ) and its maximum value ( red dashed - dotted line).,width=312 ] in summary , we have proposed a method for the calculation of free energies which is simple , accurate , and efficient . unlike standard histogram methods such as wham and metadynamics , the single - sweep method uses the mean force computed at a set of centers to reconstruct the free energy . this set of centers is determined using tamd to rapidly sweep through the important regions of the free energy , and the mean forces at these centers are estimated in a standard way via the computation of a conditional expectation using time - averaging along restrained or constrained simulations . from these data , the free energy @xmath7 is then reconstructed globally by minimization of an objective function to determine the coefficients in a radial - basis function representation of @xmath7 . if convenient , this reconstruction step can use data for the centers and the mean forces obtained by other means than tamd . compared with histogram methods and metadynamics , the single - sweep technique combines several advantages : * it does not require _ a priori _ knowledge of the free energy since it uses tamd to find the important regions in the landscape automatically . * the most costly step of the calculation , namely the computation of the mean forces at the centers , can be straightforwardly distributed on different , independent , processors . * the reconstruction step is variational , i.e. the optimal coefficients in the free energy representation are determined automatically , which limits the number of parameters to adjust beforehand . * the results can be easily monitored for convergence , and systematically improved if desired . in particular , new centers can be added on top of previous ones along the same tamd trajectory to increase the accuracy without having to repeat the previous calculation . * the method can be used in more than 2 dimensions and its computational complexity is the same regardless of the dimension . we believe that these features make the single - sweep method appealing to calculate the free energy of systems more complicated but also more interesting than the ones studied in this paper . we thank giovanni ciccotti and david chandler for carefully reading the manuscript ; weinan e for pointing out sparse grid methods which prompted us to test our method in four dimensions ; ron elber and anthony west for their help with the moil code ; sara bonella , simone meloni , michele monteferrante and maddalena venturoli for useful discussions ; and finally , eric darve for suggesting the test using ( [ eq : relerror ] ) . this work was partially supported by nsf grants dms02 - 09959 and dms02 - 39625 , and by onr grant n00014 - 04 - 1 - 0565 . all md simulations were performed with the moil code @xcite , and the amber / opls @xcite force field as implemented in the code . a starting structure for the ad molecule ( ch@xmath221-co - nh - c@xmath222hch@xmath221-co - nh - ch@xmath221 ) was solvated in a box of @xmath223 water molecules of volume @xmath224 . periodic boundary conditions were used . van der waals interactions were truncated at @xmath225 . electrostatic interactions were treated with the particle mesh ewald method @xcite with real space cutoff @xmath225 , a grid of @xmath226 points , and @xmath172-th order _ b_-splines for the interpolation of the structure factor ( in order to be in the high accuracy range @xcite ) . the tip3 model @xcite was used for the water molecules . non - bonded interaction lists were updated every @xmath194 steps . all chemical bonds in the system were kept fixed with the shake algorithm @xcite . amide planes were restrained to be always in _ trans _ configuration . the velocity verlet algorithm was used for the dynamics of the cartesian variables with time - step @xmath168 fs , and all velocities were scaled at every step to keep the temperature at @xmath123 k. in order to obtain the initial configuration for the temperature accelerated md simulation ( tamd ) , the system was first equilibrated for @xmath88 ps by keeping the solute molecule fixed ( i.e. by zeroing forces and velocities of its atoms ) and by assigning to all water atoms at every step velocities sampled from a maxwell distribution at @xmath123 k. then , the whole system was simulated for @xmath227 ps for equilibration . the torsion angles used in the simulations are defined by the quadruplets of atoms ( c , n , c@xmath222,c ) and ( n , c@xmath222,c , n ) for @xmath128 and @xmath129 , and ( o , c , n , c@xmath222 ) and ( c@xmath222,c , n , h ) for @xmath186 and @xmath187 . in the tamd simulation , the equations of motion of the collective variables were integrated with the forward euler scheme with time - step @xmath168 fs , @xmath136 kcal@xmath137ps / mol / rad@xmath135 for @xmath128 and @xmath129 and @xmath83 kcal@xmath137ps / mol / rad@xmath135 for @xmath186 and @xmath187 . the force constant for the restraint potential was @xmath134 kcal / mol / rad@xmath135 , and the effective temperature such that @xmath228 kcal / mol . the cartesian coordinates of water and ad atoms were saved during the tamd simulation . in this way , for every center @xmath29 deposited along the trajectory in collective variables space , there is a corresponding configuration @xmath229 of the system in cartesian space such that @xmath230 . we used these configurations as initial conditions for the restrained simulations at the centers . data for the mean force calculations were accumulated after further relaxation of the system for @xmath231 ps .
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a simple , efficient , and accurate method is proposed to map multi - dimensional free energy landscapes .
the method combines the temperature - accelerated molecular dynamics ( tamd ) proposed in [ maragliano & vanden - eijnden , chem .
. lett . * 426 * , 168 ( 2006 ) ] with a variational reconstruction method using radial - basis functions for the representation of the free energy .
tamd is used to rapidly sweep through the important regions of the free energy landscape and compute the gradient of the free energy locally at points in these regions .
the variational method is then used to reconstruct the free energy globally from the mean force at these points .
the algorithmic aspects of the single - sweep method are explained in detail , and the method is tested on simple examples , compared to metadynamics , and finally used to compute the free energy of the solvated alanine dipeptide in two and four dihedral angles .
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the hadronization of quark gluon plasma ( qgp ) possibly produced in the early universe or expected to be formed in relativistic heavy - ion collisions has been the focus of much attention during the past few years . however , the mechanism of hadronization ( qcd phase transition ) remains an open question . the prediction of lattice qcd on the order of the transition is still unclear , if physical masses for quarks are used @xcite . quenched qcd ( no dynamical quarks ) shows a first - order phase transition , albeit a weak one , with small surface tension and latent heat @xcite . assuming the transition to be first - order , homogeneous nucleation theory @xcite has been invoked extensively to study the dynamics of the quark hadron phase transition both in the context of early universe as well as for the plasma produced during relativistic heavy - ion collisions @xcite . in this picture , the transition is initiated by the nucleation of critical - size hadron bubbles from a supercooled metastable qgp phase . these hadron bubbles can grow against surface tension , converting the qgp phase into the hadron phase as the temperature drops below the critical temperature , @xmath0 . this is indeed the case for a sufficiently strong first order transition , where the assumption of a homogeneous background of qgp is justified at the time when the nucleation begins . however , for a weak enough transition , the qgp phase may not remain in a pure homogeneous state even at @xmath1 , due to pre - transitional phenomena . for temperatures much above @xmath0 , matter is in a pure qgp phase with the effective potential exhibiting one minimum at @xmath2=0 . here @xmath2 is an effective scalar order parameter generally used to model the effective potential describing the dynamics of a phase transition . as the plasma expands and cools to some temperature @xmath3 , an inflection point is developed away from the origin which on further cooling separates into a maximum at @xmath2=@xmath4 and a local minimum at @xmath5 , corresponding to the hadron phase . at @xmath1 , the potential is degenerate with a barrier separating the two phases . `` pre - transitional phenomena '' refers to nonperturbative dynamical effects above @xmath0 in the range @xmath6 . such phenomena are known to occur in several areas of condensed matter physics , as in the case of isotropic to nematic phase transition in liquid crystals@xcite , and are also expected in the cosmological electroweak phase transition leading to large phase mixing at @xmath1@xcite . in such cases , the phase transition may proceed either through percolation @xcite or , if the phase mixing is below the percolation threshold , by the nucleation of critical bubbles in the background of isolated hadronic domains , which grow as @xmath7 drops below @xmath0 . in either case , the kinetics is quite different from what is expected on the basis of homogeneous nucleation @xcite . we will argue that , for a wide range of physical parameters , a large amount of thermal phase mixing at @xmath1 is expected to occur during the quark - hadron phase transition in the early universe , as well for the plasma produced in heavy - ion collisions @xcite . for high enough temperatures and low enough cooling , large - amplitude thermal fluctuations will populate the new minimum at @xmath5 in the range @xmath8 . although these fluctuations which are in the form of subcritical hadron bubbles will shrink and finally disappear , there will always be some non - zero number density of hadron bubbles at a given temperature @xmath7 . in this work , we study the equilibrium density distribution of subcritical hadron bubbles for a wide spectrum of very weak to very strong first order qcd phase transition , using the formalism developed in refs . it is found that the density of subcritical hadron bubbles builds up faster as the transition becomes weaker , leading , in some cases , to complete phase mixing at @xmath1 . further , using reasonable values for the surface tension and correlation length as obtained from lattice qcd calculations , we find that ( although large ) the amount of phase mixing remains below the percolation threshold . therefore , the quark - hadron phase transition will begin with the nucleation of critical - size hadron bubbles from a supercooled and inhomogeneous background of quark - gluon plasma . since the background contains subcritical hadron bubbles , the homogeneous theory of nucleation needs to be modified . in ref . @xcite , an approximate method was suggested to incorporate this inhomogeneity by modelling subcritical bubbles as gaussian fluctuations , resulting in a large reduction in the nucleation barrier . here , we will study inhomogeneous nucleation in the framework of homogeneous theory , but with a reduced nucleation barrier that accounts for the inhomogeneity of the medium . finally , we also briefly discuss possible implications of inhomogeneous nucleation to relativistic heavy - ion collisions and cosmology . the paper is organized as follows . in the next section , we begin with the discussion of a quartic double - well potential used to describe the dynamics of a first - order quark - hadron phase transition . the parameters of the potential are obtained in terms of relevant physical quantities such as critical temperature , surface tension and correlation length . in section iii , we estimate the equilibrium fraction of subcritical hadron bubbles from very weak to strong first - order phase transitions . we also estimate the reduction in the nucleation barrier by incorporating the presence of subcritical bubbles in the medium . using this reduced barrier , we study nucleation and supercooling in section iv . finally , we present our conclusions in section v. we consider a general form of the potential ( or equivalently , the homogeneous part of the helmholtz free energy density ) to study the quark - hadron phase transition in terms of a real scalar order parameter @xmath2 given by @xmath9 where @xmath10 and @xmath11 are positive constants . ignatius et . @xcite use this parameterization to describe the phase transition from a qgp ( symmetric phase ) to a hadron phase ( broken symmetry ) . the meaning of @xmath2 is obvious for a symmetry - breaking transition , but the same description can be used if no symmetry is involved . the order parameter could then be related to energy or entropy density . the parameters @xmath12 , @xmath10 and @xmath11 are determined in terms of surface tension @xmath13 , correlation length @xmath14 and critical temperature @xmath15 . the potential has two minima , one at @xmath16 and the other at @xmath17 , which in our case will represent quark and hadron phases respectively . these phases are separated by a maximum defined by @xmath18 . at @xmath1 , @xmath19 having the required degeneracy . the above condition yields , @xmath20 using these relations , the barrier height at @xmath0 can be written as @xmath21 therefore , if the parameter @xmath11 is kept fixed , @xmath10 can be varied to characterize a wide spectrum of very weak to very strong first - order phase transitions . the transition is strong enough for large @xmath22 and very weak or close to second order as @xmath23 . in the following , we relate the parameters @xmath10 and @xmath11 to the surface tension and the correlation length in the quark phase . the surface tension can be defined as the one dimensional action given by , @xmath24.\end{aligned}\ ] ] under the thin - wall limit , @xmath25 , the surface tension can be expressed as @xcite @xmath26 similarly , the correlation length around the quark phase is obtained using @xmath27 @xmath28 . at the critical temperature , using eq . ( [ relat ] ) , we get @xmath29 from eqs . ( [ sig ] ) and ( [ xi ] ) we get @xmath30 in terms of the values of @xmath31 and @xmath32 at @xmath0 . the barrier height @xmath22 can now be written as @xmath33 thus , the barrier height is proportional to the ratio @xmath34 . the transition becomes very weak as @xmath31 decreases and @xmath32 increases . here , we fix @xmath35 fm at @xmath1 and vary @xmath31 to investigate phase transitions with different strengths . the temperature dependence of @xmath12 is deduced by equating the depth of the second minimum with the the pressure difference @xmath36 between the two phases at all temperatures . this yields an equation @xmath37 which is solved to get the parameter @xmath38 , giving the temperature dependence of @xmath32 . the surface tension will also have small temperature dependence which we ignore , as we are not going too far from the critical temperature . thus , we have parameterized the free - energy density in terms of the surface tension , correlation length , critical temperature and equation of state , which can be obtained from lattice qcd calculations . the bag equation of state which is a good depiction of the lattice results is used to calculate the quark / hadron pressure @xmath39 as follows @xmath40 where @xmath41 is the bag constant . the quark phase is assumed to consist of a massless gas of @xmath42 and @xmath43 quarks and gluons , while the hadron phase contains massless pions . thus , the coefficients @xmath44 and @xmath45 are given by @xmath46 and @xmath47 . the critical temperature is taken as @xmath48 mev . 1 shows the plot of @xmath49 as a function of @xmath2 at three different temperatures for a typical value of @xmath31 = 30 mev/@xmath50 and @xmath51 fm . at @xmath1 , the potential is degenerate with a large barrier that separates the two phases . below @xmath0 , the phase @xmath5 has lower free - energy density , and the qgp phase becomes metastable . above @xmath0 , the potential has a metastable minima at @xmath5 ( hadron phase ) as long as @xmath7 remains below @xmath3 . the temperature @xmath3 [ at which @xmath52 and @xmath53 can be obtained analytically by solving eq . ( [ atemp ] ) as , @xmath54^{1/4 } \ , t_c.\end{aligned}\ ] ] it may be mentioned here that the dynamics of the phase transition has also been studied in ref . @xcite using a different form of the potential which has been parameterized as a fourth order polynomial in the energy density @xcite . this form is unsuitable over a wide range of temperatures due to the persistence of metastability at much above and below @xmath0 . we closely follow the work of refs . @xcite to estimate the equilibrium density distribution of subcritical hadron bubbles by modeling them as gaussian fluctuations with amplitude @xmath55 and radius @xmath56 @xmath57 the amplitude @xmath55 is the value of the field at the bubble s core away from the quark phase . for smooth interpolation between the two phases in the system , @xmath58 . the free energy of a given configuration can then be found by using the general formula @xcite , @xmath59.\end{aligned}\ ] ] using eq . ( [ gau ] ) and eq . ( [ vt ] ) in eq . ( [ fs ] ) we get @xmath60 where @xmath61 , @xmath62 , @xmath63 and @xmath64 are given by @xmath65 \pi^{3/2 } \phi_a^2\end{aligned}\ ] ] and @xmath66 it may be mentioned here that @xmath67 is positive and is much greater than @xmath68 . therefore , the free energy grows linearly for small values of @xmath56 . further , hadron bubbles of all configurations will be subcritical as long as @xmath68 is positive . at @xmath1 , both @xmath62 and @xmath64 are positive for all amplitudes . however , below @xmath0 , @xmath62 may become negative for some values of @xmath55 . for such configurations , the free energy has a maximum at @xmath69 and these bubbles are not strictly subcritical . the same is true for @xmath64 above @xmath0 . we thus restrict the amplitudes @xmath55 to the range where @xmath68 is positive . if not exactly the same , the limits of integration @xmath70 and @xmath71 for @xmath55 are found to be quite close to @xmath4 and @xmath72 respectively . there will be fluctuations from quark to hadron phase and back . to obtain the number density @xmath73 of subcritical bubbles , we define the distribution function @xmath74 where @xmath75 is the number density of bubbles with radius between @xmath56 and @xmath76 and amplitude between @xmath55 and @xmath77 at time @xmath78 . it satisfies the boltzmann equation @xcite @xmath79 the first term on the rhs is the shrinking term . here , @xmath80 is the shrinking velocity , which we assume to be given by the velocity of sound ( @xmath81 ) in a massless gas . the second term is the nucleation term where @xmath82 is the gibbs distribution function defined as @xmath83 . here @xmath84 is the nucleation rate per unit volume of subcritical bubbles from the quark phase to the hadron phase . similarly @xmath85 is the corresponding rate from the hadron phase to the quark phase . the factor @xmath86 is defined as the fraction of volume in the hadron phase and is obtained by summing over subcritical bubbles of all amplitudes and radii within this phase . the gibbs distribution function is defined as @xcite @xmath87 where @xmath88 is of @xmath89 @xcite . if the equilibration time scale is smaller than the expansion time scale of the system , we can obtain the equilibrium number density of subcritical bubbles by solving eq . ( [ boltz ] ) with @xmath90 . since the early universe expands at a much slower rate @xcite , the above assumption is quite reasonable in the context of the cosmological qcd phase transition . however , qgp produced during heavy ion collision may expand at a faster rate as compared to the early universe . in this case , it is possible that the density distribution of the subcritical bubbles will not attain full equilibrium . for simplicity , we will assume an equilibrium situation so that the present results on the fraction of subcritical bubbles and phase mixing can be considered as an upper limit . using the boundary condition @xmath91 , we get the equilibrium distribution given by @xmath92 where @xmath93 the equilibrium fraction @xmath86 of volume occupied by subcritical bubbles is given by , @xmath94 which is solved to get @xmath95 where @xmath96 here , @xmath70 and @xmath71 define the range within which both @xmath62 and @xmath64 are positive . @xmath97 is the smallest radius of the subcritical bubbles taken as @xmath32 , the correlation length of the fluctuations . the @xmath56 integration should be carried out over all bubbles with radii from @xmath98 to @xmath99 . for very weak transitions , both @xmath100 and @xmath101 are very small and the @xmath56 integration may not have good convergence . however , we found that the @xmath86 value is maximized when @xmath102 is about 3 to 4 fm . therefore , we use @xmath102=3.5 fm . this is a reasonable choice as bubbles with @xmath103 will be statistically dominant and larger fluctuations have larger free energy and are exponentially suppressed . 2 shows the plot of the subcritical hadron fraction @xmath86 as a function of @xmath31 at @xmath1 and at a fixed value of @xmath104 fm . the fraction @xmath86 has been estimated ( dashed curve ) assuming that , for a degenerate potential , @xmath105 , as in ref . this assumption is valid only for the configuration for which @xmath106 . however , when we include other configurations in the range @xmath70 to @xmath71 , the integral @xmath107 turns out to be always higher than @xmath108 at @xmath0 . therefore , @xmath86 obtained using @xmath109 is always lower than when the approximation @xmath110 is used . in both cases , the value of @xmath86 increases with decreasing @xmath31 i.e. when the transition becomes weak . it may be mentioned here that as per lattice qcd calculations without dynamical quarks @xcite , @xmath31 lies between 2 mev/@xmath50 and 10 mev/@xmath50 . there would be @xmath111 to @xmath112 phase mixing corresponding to these @xmath31 values , which is still below the percolation threshold ( @xmath113 ) . if @xmath114 , the two phases will mix completely , the mean - field approximation for the potential breaks down , and the phase transition may proceed through percolation @xcite . however , for a surface tension in the range 2 mev/@xmath115 10 mev/@xmath50 , the phase transition will proceed through the formation of critical - size hadron bubbles from a supercooled metastable qgp phase . since the qgp phase is no longer homogeneous , the dynamics of the phase transition will be quite different from what is expected on the basis of homogeneous nucleation theory @xcite . we refer to it as `` inhomogeneous nucleation . '' we would also like to mention here that the present results are in disagreement with the findings of ref . @xcite , where a large fraction of subcritical hadron phase was found at and above @xmath0 . this scenario is highly unrealistic and probably could be due to the choice of the potential parameterization , which shows a metastable hadron phase much above @xmath0 . therefore , the authors of ref . @xcite found a finite fraction of hadron phase at temperatures as high as twice @xmath0 . furthermore , the value of @xmath86 strongly depends on how the shrinking term is incorporated in the calculation . in our case , it is proportional to the gradient @xmath116 that appears in the kinetic equation ( [ boltz ] ) in a natural way , whereas in ref . @xcite , a specific assumption is made to take into account the shrinking of the hadronic volume . the nucleation rate in the standard theory @xcite which neglects phase mixing , is given by @xmath117 here @xmath118 is free energy needed to form a critical bubble in the homogeneous metastable background . for an arbitrary thin - wall spherical bubble of radius @xmath56 and amplitude @xmath119 , the free energy of the bubble takes the well - known form @xmath120 in the above , @xmath121 is defined as the difference in free - energy density between the background medium and the bubble s interior . for a homogeneous background ( metastable ) , we can write , @xmath122 if there is significant phase mixing in the background metastable state , its free energy is no longer @xmath123 . one must also account for the free energy density of the nonperturbative large amplitude fluctuations . following ref . @xcite , we write the free energy density of the metastable state as @xmath124 , where @xmath125 is the extra free energy density which can be estimated from the density distribution of subcritical bubbles as follows : @xmath126 once we know the hadronic fraction @xmath86 and the free energy @xmath127 for a bubble of a given radius @xmath56 and amplitude @xmath55 , we can estimate the free - energy density correction due to the presence of gaussian subcritical bubbles . since , for a critical size bubble , @xmath128 , we can use eq . ( [ fthin ] ) to obtain the free energy needed to form a thin - wall critical bubble in a background of subcritical bubbles , @xmath129 for a very strong first - order phase transition , the subcritical bubbles are suppressed @xmath130 , and both @xmath131 and @xmath132 approach the homogeneous background expression . however , in the presence of subcritical bubbles , extra free energy becomes available in the medium , reducing the nucleation barrier . in other words , the extra background energy enhances the nucleation of critical bubbles . to illustrate this , we have plotted @xmath133 and @xmath86 as a function of @xmath134 in figs . 3 to 5 with @xmath31 values of 50 mev/@xmath50 , 30 mev/@xmath50 and 10 mev/@xmath50 , respectively , which are widely used in the literature . as evident , with decreasing temperature , the nucleation barrier decreases and the subcritical hadron fraction @xmath86 increases . the reduction in barrier height due to @xmath125 ( or due to @xmath86 ) is more significant for lower values of @xmath31 , corresponding to a weaker transition . since the height of the nucletaion barrier decreases , the nucleation rate will also be enhanced , reducing the amount of supercooling further . the time evolution of the temperature and the supercooling are discussed in the next section . as mentioned before , the background metastable state is inhomogeneous due to subcritical hadron bubbles . it is now possible to study the kinetics of the nucleation of critical hadron bubbles using the corrected nucleation rate , as obtained in the previous section . in the present work , the prefactor in the nucleation rate is taken as @xmath135 [ see eq . ( [ rate ] ) ] . in our previous work , @xcite , we have used a prefactor derived by csernai and kapusta @xcite for a dissipative qgp . in ref . @xcite , ruggeri and friedman had derived a prefactor for a non - dissipative qgp . recently , using a more general formalism , we have also derived a prefactor @xcite which has both dissipative and non - dissipative components corresponding to ref . @xcite and ref . @xcite , respectively . however , for consistency with the subcritical formalism , we use a more generic form @xmath136 , with @xmath88 a constant of order unity , as used in many studies of quark - hadron phase transition ( see , for example , refs . the question of how to estimate the prefactor appearing in the nucleation rate of subcritical bubbles remais open . using the nucleation rate @xmath137 , the fraction @xmath138 of space which has been converted to hadron phase due to nucleation of critical bubbles and their growth can be calculated . if the system cools to @xmath0 at a proper time @xmath139 , then at some later time @xmath140 the fraction @xmath138 is given by @xcite , @xmath141 v(\tau',\tau).\end{aligned}\ ] ] here , @xmath142 is the volume of a critical bubble at time @xmath140 which was nucleated at an earlier time @xmath143 ; this takes into account the bubble growth . the factor @xmath144 $ ] accounts for the available space for new bubbles to nucleate . the model for bubble growth is simply taken as @xcite @xmath145 where @xmath146^{3/2}$ ] is the velocity of the bubble growth at temperature @xmath7 @xcite . the evolution of the energy density in 1 + 1 dimensions is given by @xmath147 the energy density @xmath148 , enthalpy density @xmath149 and the pressure @xmath150 in pure qgp and hadron phases are given by the bag model equation of state . in the transition region , the @xmath148 and @xmath149 at a time @xmath140 can be written in terms of the hadronic fraction as @xmath151 h(\tau ) , \nonumber \\ \omega(\tau ) = \omega_q(t ) + [ \omega_h(t)-\omega_q(t ) ] h(\tau).\end{aligned}\ ] ] equations ( [ frac ] ) , ( [ hydro ] ) , and ( [ mixed ] ) are solved to get the temperature as a function of time in the mixed phase @xcite with the initial conditions for temperature @xmath152 mev and proper time @xmath153 fm / c at @xmath0=160 mev . after getting @xmath7 and @xmath138 as a function of time , the density of nucleating bubbles at a time @xmath140 can be obtained in our model as @xmath154.\end{aligned}\ ] ] the density @xmath155 would increase as the temperature drops below @xmath156 and would ultimately saturate as @xmath138 increases . figure 6 shows the temperature variation as a function of proper time @xmath140 at @xmath157 mev/@xmath50 . as the system cools below @xmath0 , the nucleation barrier decreases and also @xmath86 increases . if only homogeneous nucleation ( dashed curve ) is considered , the system will supercool up to 0.945 @xmath0 . at this temperature , the hadronic fraction @xmath86 has reached 10 % ( see fig . 3 ) , which corrects the amount of supercooling ( solid curve ) by about @xmath15810 % ( up to 0.95 @xmath0 ) . figure 7 shows a similar study at @xmath31 = 30 mev/@xmath50 . since the nucleation barrier reduces with decreasing @xmath31 , the system supercools only up to 0.98 @xmath0 under homogeneous nucleation . the hadronic fraction @xmath86 corresponding to this value is @xmath159 % ( see fig . 4 ) which reduces the amount of supercooling by about @xmath160 ( up to 0.984 @xmath0 ) . for @xmath31 around 10 mev/@xmath50 , the supercooling will be reduced further ( up to @xmath161 ) . lattice qcd calculations predict a surface tension even smaller than 10 mev/@xmath50 , indicating a very weak first order transition . although supercooling will be reduced further with decreasing @xmath31 , we do not use very small @xmath31 due to increased numerical inaccuracy . further , it may be mentioned here that , although the fraction @xmath86 grows with decreasing @xmath31 , we never encountered @xmath86 greater than 0.3 : we remained within the sub - percolation regime throughout our analysis . apart from @xmath31 , the amount of supercooling also depends on @xmath139 , the time taken by the system to cool from @xmath162 to @xmath0 . in qgp phase , the solution of eq . ( [ hydro ] ) @xmath163 = constant ) predicts @xmath164 . the choice of @xmath165 = 1fm , @xmath162=250 mev and @xmath0=160 mev results in @xmath139=3.8 fm / c . however , formation of qgp with higher initial temperature ( as high as 3 to 4 times @xmath0 resulting in large @xmath139 ) can not be ruled out at rhic and lhc energies @xcite . therefore , we have also studied the effect of @xmath139 on supercooling , specifically , on the hadronization rate as well as on the density of nucleating bubbles . 8(a ) and 8(b ) show the plots of @xmath166 and @xmath167 as a function of @xmath140 for two typical values of @xmath139 ( 3.8 fm / c and 25 fm / c ) corresponding to @xmath31=10 mev/@xmath168 both with ( solid curve ) and without ( dashed curve ) inhomogeneity corrections . a general observation ( both with and without correction ) is that the amount of supercooling , the rates of hadronization and bubble nucleation are reduced when @xmath139 becomes larger . although supercooling reduces with increasing @xmath139 , the system will get reheated at an earlier temperature and also will encounter a larger nucleation barrier as compared to the case when @xmath139 is small . as a result , the rate of hadronization and also the rate of increase of density of the nucleating hadron bubbles will proceed at a slower rate when @xmath139 is large . however , the reverse happens when the inhomogeneity correction is applied . even though the medium gets heated up earlier , the reduction in the barrier height is quite significant as @xmath7 approaches @xmath0 . another parameter that affects both @xmath166 and @xmath167 is the expansion rate of the medium , _ i.e. _ , the rate of change of temperature between @xmath139 and @xmath140 , which also depends on @xmath139 . the overall effect is that both @xmath166 and @xmath167 rise faster as compared to their homogeneous counterparts ( see fig . 8 for @xmath139=25 fm / c ) , particularly when @xmath139 is very large . ( compare the left and right curves on fig . 8 . ) the increase in rates of @xmath166 and @xmath167 is also larger for small @xmath31 at large @xmath139 . for weak enough transition , the presence of inhomogeneity may also affect several observables which can be detected experimentally . for example , the faster rate of hadronization at large @xmath139 as compared to its homogeneous counterpart will lower the amount of entropy production , which , in turn , will affect the final hadron multiplicity distributions . although not studied here , the bubble size distribution will also be affected by the dynamics of nucleation @xcite . since the nucleating bubble will act as a source of pion emission , the effect of inhomogeneity can also be inferred through interferometry measurements . in a cosmological context , the value of @xmath139 is much larger than what was quoted here . since the presence of inhomogeneities weakens the transition , more critical bubbles will be nucleated per unit volume , decreasing the inter - bubble distance , @xmath169 ; the presence of subcritical bubbles can be thought as seeds for nucleation . as a consequence , the transition will produce smaller fluctuations in baryon number , protecting homogeneous nucleosynthesis . although the present study is indicative enough of the reduction in the mean inter - bubble separation as compared to homogeneous nucleation , a quantitative estimate would require a more detailed analysis , including expansion . however , since the cosmological expansion rate is typically much slower than the subcritical bubble nucleation rate , we believe our results for the inter - bubble distance will carry on in this case as well . we have estimated the amount of phase mixing due to subcritical hadron bubbles from very weak to very strong first - order phase transitions . with a reasonable set of values for the surface tension and correlation length ( as obtained from lattice qcd calculations ) , we found that phase mixing is small at @xmath1 , building up as the temperature drops further . we have shown that the system does not mix beyond the percolation threshold , allowing us to describe the dynamics of the phase transition on the basis of homogeneous nucleation theory with a reduced nucleation barrier . accordingly , we have found an enhancement in the nucleation rate which further reduces the amount of supercooling . although we have not included cosmological expansion in our analysis , we believe that our results indicate that the presence of an inhomogeneous background of subcritical bubbles will decrease the inter - bubble mean distance , and thus the fluctuations in baryon number which could damage homogeneous nucleosynthesis . we have assumed that the equilibration time - scale for subcritical fluctuations is much larger than the cooling time - scale of the system . this may be the case for a quark - hadron phase transition in the early universe , where the expansion rate is quite slow . in the case of qgp produced at rhic and lhc , the cooling rate is much faster than cosmological time - scales , and the subcritical bubbles density distribution may not attain full equilibration . we are presently investigating this issue in more detail . however , the present results should provide an upper bound on the fraction of subcritical hadron bubbles and their effect on supercooling and nucleation rates . 3 the nucleation barrier @xmath133 for critical bubbles with ( solid line ) and without subcritical bubble correction ( dashed curve ) as function of temperature for @xmath157 mev/@xmath50 is shown in upper panel . corresponding subcritical hadron fraction @xmath86 is shown in the lower panel . fig . 8 ( a ) density of nucleating bubbbles as a function of proper time with ( solid curve ) and without subcritical bubble correction ( dashed curve ) for @xmath31= 10 mev/@xmath50 ( b ) the hadronic fraction @xmath86 as a function of @xmath140 . the left curves are for @xmath172 fm / c and the right curves for @xmath173 fm / c .
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the effect of subcritical hadron bubbles on a first - order quark - hadron phase transition is studied .
these subcritical hadron bubbles are created due to thermal fluctuations , and can introduce a finite amount of phase mixing ( quark phase mixed with hadron phase ) even at and above the critical temperature . for reasonable choices of surface tension and correlation length , as obtained from the lattice qcd calculations , we show that the amount of phase mixing at the critical temperature remains below the percolation threshold .
thus , as the system cools below the critical temperature , the transition proceeds through the nucleation of critical - size hadron bubbles from a metastable quark - gluon phase ( qgp ) , within an inhomogeneous background populated by an equilibrium distribution of subcritical hadron bubbles .
the inhomogeneity of the medium results in a substantial reduction of the nucleation barrier for critical bubbles . using the corrected nucleation barrier
, we estimate the amount of supercooling for different parameters controlling the phase transition , and briefly discuss its implications to cosmology and heavy - ion collisions .
+ pacs number(s ) : 12.38.mh , 64.60.qb , 05.70.fh , 25.75-q , 98.80.cq
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one of the most convenient ways to trace magnetic field is related to emission and absorption of polarized radiation by aligned dust . therefore aligned dust is widely used for this purpose both in diffuse interstellar medium and molecular gas . moreover , there is evidence of aligned dust around young stellar objects and evolved stars as well as astrophysical environments . at the same time , the processes that aligned dust and their relation to magnetic field are still the subject of debates and require further studies ( see lazarian 2007 for a review ) . while radiative torques ( dolginov & mytrophanov 1976 , draine & weingartner 1996 , 1997 , weingartner & draine 2003 , cho & lazarian 2005 , lazarian & hoang 2007 , hoang & lazarian 2007 ) are currently seen as the most promising candidate mechanism , the variety of astrophysical conditions may enables other mechanisms to dominate in particular environments . mechanical alignment was pioneered by gold ( 1951 , 1952 ) and then quantified and elaborated by other researchers ( e.g. purcell 1969 , purcell & spitzer 1971 , dolginov & mytrophanov 1976 , lazarian 1994 , 1997 , roberge et al . while the original mechanism could deal with thermally rotating grains only , two modifications of the mechanism introduced in lazarian ( 1995 ) and elaborated later in lazarian & efroimsky ( 1996 ) , lazarian , efroimsky & ozik ( 1997 ) , efroimsky ( 1999 ) enabled the alignment of grains rotating at much higher rates . the latter were introduced to the field by purcell ( 1979 ) ( see also lazarian & draine 1999ab , where the limitations on the size for suprathermally rotating grains are discussed ) . the main shortcoming of the mechanical alignment processes was that they required supersonic gas - dust drift to get any appreciable degree of alignment ( see purcell 1969 ) . although later studies indicated that such drifts can be produced by ambipolar diffusion ( roberge & hanany 1990 , roberge et al . 1995 ) or interactions of charged grains with mhd turbulence ( lazarian 1994 , lazarian & yan 2002 , yan & lazarian 2002 , yan , lazarian & draine 2004 ) , the degree of alignment that is achievable for the mach number drifts of the order of unity is insufficient to explain observations ( see estimates in lazarian 1997 ) . the possibility of subsonic flows to mechanically align grains was mentioned in passing in lazarian ( 2007 ) and lazarian & hoang ( 2007 , henceforth lh07 ) , but no relevant calculations were provided there . this possibility is related to irregular grains demonstrating _ helicity _ , i.e. ability of spin up in a _ regular way _ while interacting with a flow of particles . in lh07 radiative torques that arise from the interaction of photons with a grain were considered . similar torques , however , should emerge when atoms bombard the surface of an irregular grain . in figure 1 we provide a simple model of a helical grain . the grain consists of an oblate spheroid or an ellipsoid with a mirror attached to it at an angle @xmath0 . the difference with lh07 is that we assume that the mirror is reflective to atoms rather than photons . for our purposes in the paper , the actual properties of a spheroid or an ellipsoid do not matter . those geometric shapes are too symmetric and do not produce torques that can spin up a grain in a regular way , i.e. with angular velocity growing in proportion to time . the stochastic , i.e. with angular velocity growing as a square root of the time , spin up is the essence of the gold - type alignment processes and it is ignored here , where we deal with subsonic flows . for the sake of simplicity , we assume that the larmor precession of the grain in the external magnetic field is much faster than precession arising from the interaction of the ellipsoidal body with the gaseous flow . as a result , following arguments similar to those in lh07 , one can disregard the effects of the gaseous flow on the ellipsoidal body altogether . thus , the only torques to consider are those arising from the mirror . as in lh07 , the the major role of the ellipsoidal body is to provide a steady rotation about its axis corresponding to the maximal moment of inertia . this ensures that the model grain is subject to regular torques while it interacts with the gaseous flow . the difference between the calculations in lh07 and those in the current paper stems from the fact that , while in lh07 the photons are coming as a beam from a single direction , for subsonic flow that we deal with , atoms hit the mirror from all directions characterized by the flux of atoms given by equation ( [ ap2 ] ) . this induces averaging of the torques that we implement below . consider a helical grain ( see fig . [ f0 ] ) drifting across a gas chamber with a velocity @xmath1 along the @xmath2 axis . the velocity of an atom with respect to the grain is @xmath3 where @xmath4 is the thermal velocity of the atom . the torque from the perfect reflecting of atoms on a surface area @xmath5 of the mirror is given by @xmath6 a f({\bf s}-{\bf s}_{d } ) d^{3 } s,\label{ap1}\ ] ] where @xmath7 , vectors @xmath8 is the radius vector directed from the spheroid to the mirror , @xmath9 and @xmath5 is the surface area of the mirror , and @xmath10 is the flux of incoming atoms that can collide with the grain . angular momentum element @xmath11 is antiparallel to the normal vector and given by @xmath12 where @xmath13 . here @xmath14 are radius and normal vector which are functions of angles @xmath15 describing the orientation of the grain in the lab system defined by @xmath16 and @xmath17 ( see fig . [ f0 ] ) , as given in lh07 . and @xmath18 are components of @xmath19 in the grain coordinate system . ] substituting equations ( [ ap2 ] ) and ( [ ap3 ] ) into equation ( [ ap1 ] ) we get @xmath20 ( ( { \bf s}-{\bf s}_{d}).{\bf n})^{2 } a e^{-s^{2 } } d^{3}s,\label{ap4}\ ] ] taking into account the contribution from the reflection on the other surface , total torque becomes then @xmath21\int_{({\bf s}-{\bf s}_{d}).{\bf n}<0 } ( ( { \bf s}-{\bf s}_{d}).{\bf n})^{2 } a e^{-s^{2 } } d^{3}s\nonumber\\ -2 n m_{h } v_{th}^{2 } [ { \bf r}\times { \bf n}]\int_{({\bf s}-{\bf s}_{d}).{\bf n}>0 } ( ( { \bf s}-{\bf s}_{d}).{\bf n})^{2 } a e^{-s^{2 } } d^{3}s , \label{ap5}\ ] ] integral ( [ ap5 ] ) can be analytically evaluated and the resulting torques are given by @xmath22\lbrace |n_{1}| n_{1}(2 s_{d } e^{-s_{d}^{2}}+\sqrt{\pi } erf(s_{d})\nonumber\\ + 2 s_{d}^{2 } \sqrt{\pi } erf(s_{d}))+\frac{n_{1}}{|n_{1}|}(n_{2}^{2}+n_{3}^{2})erf(s_{d})\rbrace,\label{eq8}\ ] ] where @xmath23 and @xmath24 are components of @xmath19 in the lab coordinate system . equation ( [ eq8 ] ) can be written as @xmath25 where @xmath26 is the vector torque efficiency consisting of components @xmath27 and @xmath28 on axes @xmath29 and @xmath17 , respectively . two first components for subsonic and supersonic cases are shown in figure [ f1 ] . the third component produced by the mirror is found to be zero . it is clear that the component @xmath28 is responsible for the precession ( lh07 ) . this component is non - zero for a spheroidal body of the grain . in fact , as we discussed above , we disregard this component , as it does not induce either alignment or spin up . in figure [ f1 ] we show torque components for subsonic case @xmath30 and a supersonic case @xmath31 . it can be seen that the component @xmath32 has zero points at @xmath33 and @xmath34 in both subsonic and supersonic case . the component @xmath35 is very symmetric in the latter . however , the torques are distorted due to the thermal collisions of gaseous atoms in the former case . the essential properties ( i.e. symmetry of @xmath35 and zero points of @xmath32 ) of mechanical torques which are similar to those of radiative torques indicate that mechanical torques can align grains in the same ways as radiative torques do , i.e. , grains tend to aligned with long axes perpendicular to magnetic fields . we study this problem in the section below . when the larmor precession of a grain is larger than the precession induced by mechanical torques ( see 11.6 in lh07 ) , the alignment occurs with respect to magnetic field ( either @xmath36 or @xmath37 ) which defines the axis of alignment . to study the grain alignment induced by mechanical torques , similar to lh07 , we solve the equations of motion for angular momentum @xmath38 in time t : @xmath39 where @xmath40 is given by equation ( [ eq8 ] ) and @xmath41 is the gaseous damping time . for simplicity , we assume a perfect internal alignment ; i.e. the axis of major inertia @xmath42 is always parallel to @xmath38 ( dw97 ; lh07 ) . we consider a model grain drifting across magnetic field by an angle @xmath43 acted upon by a subsonic flow with @xmath44 . representing equation ( [ eq10 ] ) in the spherical coordinate system defined by @xmath45 where @xmath46 is the magnitude of @xmath38 , @xmath47 is the angle between @xmath38 and @xmath48 , and @xmath49 is the larmor precession angle , we get three equations for @xmath50 and @xmath49 ( see sec . 7 in lh07 ) . averaging over @xmath49 from @xmath51 to @xmath52 , we obtain only equations for @xmath53 and @xmath47 . we construct trajectory map based on the obtained solution @xmath54 . figure [ f2 ] shows the obtained trajectory map with initial condition @xmath55 and the angle @xmath47 generated from a uniform distribution . we see that grain phase trajectories that start at different angles @xmath47 converge on either high-@xmath46 attractor points a and b corresponding to perfect alignment , or a low-@xmath46 attractor point c. the latter point formally corresponds to @xmath56 . however , as it was explained in lh07 and shown by numerical simulations in hoang & lazarian ( 2007 ) , this is an artifact of our ignoring thermal fluctuations within the grain . if those are taken into account the low-@xmath46 attractor points correspond to the angular momentum of the order of the thermal angular momentum corresponding to the temperature of the grain body ( see also weingartner & draine 2003 , hoang & lazarian 2007 ) . what should be noted is that the attractor points for @xmath44 correspond to the alignment of @xmath42 parallel to @xmath48 ; i.e. the grain gets aligned with long axes perpendicular to magnetic field . this is similar to what the davis - greenstein ( 1951 ) mechanism of paramagnetic relaxation predicts , but the process we discuss here does not invoke paramagnetic relaxation . gaseous bombardment randomizes grain phase trajectories . however , rather than tracing phase trajectories , it is practically convenient to have a criterion for the efficiency of grain alignment . the maximal rotational velocity of grains induced by torques can be used for this purpose . a study in hoang & lazarian ( 2007 ) which solves the lagevin equation in order to account for gaseous bombardment shows that this velocity which us a function of @xmath57 should be approximately 3 times larger than the thermal rotational velocity of the grain . for practical calculations of the rotational velocity we assume that 10% of the atoms impinging on the grain mirror are reflected . figure [ f3 ] shows the angular velocity of a grain as a function of the mach number of the flow . the horizontal line corresponds to @xmath58 for @xmath59 . it is clear from looking at figure [ f3 ] that for our model , helical grains get aligned for relatively low velocities of gas - grain drift . all the earlier mechanisms of mechanical alignment are inefficient for such low subsonic velocities . indeed , drifting velocities of the order of @xmath60 of sound speed should be quite common in the ism ( see yan , lazarian & draine 2004 ) , so potentially the mechanism is widely applicable . however , it requires further studies . one may wonder whether the condition of perfect reflection is absolutely necessary . it can be shown that the torques are changed by a factor of unity if impinging atoms are absorbed on the grain surface , thermalized there , and emitted from the point that they hit the surface . moreover , for regular torques to act on a helical grain , it is sufficient to have a correlation of the place that the atom impinges on the grain surface and is evaporated from it . it is only when there is no correlation at all , that our model grain does not experience _ regular _ torques . therefore we believe that we do not overestimate the torques with our assumption of 10% reflection . in general , to characterize the interaction of grain surface with the gaseous flow , one can introduce a `` reflection efficiency factor '' @xmath61 . a different issue is the degree of helicity of an irregular grain . naturally , most of the grains do not have facets , but have numerous irregularities . an idealized model grain ( see fig . 1 ) , for which the damping arising from the gas interactions with the ellipsoidal body tends to zero , has the maximal possible value of the torques @xmath62 for a given grain size @xmath63 and the mach number @xmath64 and its `` helicity reduction factor '' @xmath65 . this factor , in a general case , can be characterized by @xmath66 . further numerical studies should clarify the value of @xmath67 for actual irregular grains . in figure [ f3 ] we show results for different @xmath68 factors . in our letter we use the same toy model of a helical grain as in lh07 . in lh07 we proved the validity of the model ( which uses a geometric optics approximation ) comparing the functional dependencies of model s torques and those numerically calculated for the actual irregular grains . in comparison , the justification of the model for the gaseous bombardment is self - evident . however , the combination @xmath68 presents a combined uncertainty factor , which characterizes both the uncertainties in grain - gas interactions and grain shape . testing of the former requires laboratory studies , while the latter can be established via the numerical research . what is the relative role of traditional mechanisms of mechanical alignment ? those mechanisms require supersonic motions to be efficient . it is only mechanical alignment of helical grains that allows alignment for the subsonic motions . thus we may claim that mechanical alignment of helical grains constitutes a new class of mechanical alignment processes . in fact , the difference between the mechanical alignment of helical grains and the gold mechanism of grain alignment is similar to the difference between the radiative torque alignment and the harwit ( 1970 ) process . the latter appeals to stochastic torques arising from absorption of photons from a light beam and was shown by purcell & spitzer ( 1971 ) to be inefficient for most interstellar situations . similarly , as the radiative torque alignment is more efficient than the harwit ( 1970 ) alignment , we believe that the mechanical of helical grains is more efficient than the gold processes . therefore , at supersonic velocities we expect the mechanical alignment of the actual irregular grains to be governed by their helicity , rather than the degree of their oblateness of prolateness , which is the case for the gold mechanisms . the consequence of this is that the mechanical alignment of grains will happen with long grain axes perpendicular to magnetic field irrespective on the gas - grain velocities . this is in contrast to the gold alignment , for which the change of alignment happens at the van vleck angle @xmath69 . in hoang & lazarian ( 2007 ) , the effects of gaseous bombardment and uncompensated purcell s torques ( purcell 1979 ) , e.g. h@xmath70 torques , was studied in connection with the alignment of helical grains subjected to radiative torques . that study shows that , in the case when both high-@xmath46 and low-@xmath46 attractor points coexist , gaseous bombardment transfers grains from low-@xmath46 to high-@xmath46 thus increasing attractor points compared to the low-@xmath46 ones . ] , counter - intuitively , the degree of alignment . this is surely true only if @xmath71 is sufficiently high , e.g. higher than @xmath72 . the purcell torques , do not change @xmath46 in the low-@xmath46 attractor points because grains flip fast at those points causing thermal trapping , as discussed in lazarian & draine ( 1999a ) . they , however , can increase the values of @xmath46 at high-@xmath46 attractor points . for most situations , this does not affect the degree of alignment . our considerations above deal with ordinary paramagnetic grains . for those , the influence of paramagnetic torques is mostly negligible for the typical ism conditions . the situation changes if grains are superparamagnetic ( see jones & spitzer 1967 ) . in this situation one can expect always to have alignment with high-@xmath46 attractor points , thus having degree of alignment close to 100% . the assumption of the presence of superparamagnetic inclusions is an extra assumption , however . as we have stressed above , the mechanical alignment we discuss here and the radiative torque alignment are different incarnations of the alignment of helical grains . while the radiative torques have attracted a lot of attention recently , the mechanical alignment of helical grains has only be mentioned in a couple of publications ( lazarian 1995 , 2007 , lazarian et al . 1997 , lh07 ) . the relative role of the mechanisms depends on the yet uncertain factor @xmath68 for the mechanical processes . the observations tends to be in agreement with the radiative torque predictions ( see lazarian 2007 and ref . therein ) . it is encouraging that both mechanisms predict the alignment with long grain axes perpendicular to magnetic field . therefore one may hope that the subsonic alignment of helical grains can reveal via polarimetry the magnetic fields in the situations when the radiative torques fail . the variety of astrophysical circumstances ensures that there are situations when grains are aligned by mechanical subsonic flows . note that aligned grains were reported not only for the ism , but also comets and circumstellar regions . they are also likely to be present in the disks around evolved and young stars . the goal of this letter above is to attract the attention of the community to the possibility of mechanical alignment of helical dust grains . while further studies of the process are necessary , the following points can be made at this moment : 1952 , mnras , 112 , 215 harwit , m. 1970 , nature , 226 , 61 - 63 hoang , t. , & lazarian , a. 2007 , mnras , in press ( astro - ph/0707.3645 ) jones , j.v . , spitzer , l. 1967 , apj , 146,943 lazarian , a. 2007 , journal of quantitative spectroscopy and + radiative transfer , 106,225 lazarian , a. & hoang , t. 2007 , mnras , 378 , 910 ( lh07 ) lazarian , a. , goodman , a.a . & myers , p.c . 1997 , apj , 490 , 273 - 280 ( lgm97 ) lazarian , a. 1994 , mnras , 268 , 713 - 723 , ( l94 ) 2000 , apj , 536 , l15-l18 lazarian , a. , & draine , b.t . , 1997 , apj , 487 , 248 - 258 lazarian , a. , & yan , h. 2002 , apj , 566 , l105-l108 lazarian , a. , & efroimsky , m. , ozik , j , 1996 , apj , 472 , 240 - 244 lazarian , a. , goodman , a.a . & myers , p.c . 1997 , apj , 490 , 273 - 280 ( lgm97 ) purcell , e.m . 1969 , on the alignment of interstellar dust , physica , 41 , 100 roberge , w.g . , & hanany , s. 1990 , b.a.a.s . , 22 , 862 roberge , w.g . , hanany , s. , & mesinger , d.e . 1995 , apj , 453 , 238 yan , h. , & lazarian , a. 2003 , apj , 592 , l33-l36 yan , h. , lazarian , a. , draine , b. 2004 , apj , 616 , 895 weingartner , j. , & draine , b. 2003 , apj , 589 , 289 ( wd03 ) and @xmath74 averaged over the rotation are shown for different mach numbers @xmath64 of the relative grain - gas flow . for large @xmath31 the torque @xmath75 is symmetric and @xmath76 has three zeros at @xmath77 and @xmath34 . for subsonic flows @xmath44 , the torques are modified due to the effect of thermal velocity , but zeros points of @xmath76 are unchanged.,scaledwidth=40.0% ] and @xmath78 , in diffuse gas with @xmath79 , temperature @xmath80 corresponding to the march number @xmath44 for an angle @xmath57 between @xmath1 and magnetic field @xmath48 equal @xmath81 has attractor points a and b of high angular momentum , c with @xmath56 . in the presence of thermal grain wobbling the point c becomes a low-@xmath46 attractor point with @xmath82.,scaledwidth=40.0% ] . the horizontal line corresponds to @xmath83 which presents the threshold for grain alignment obtained in hoang & lazarian ( 2007 ) . the results are obtained for a gaseous flow moving parallel to the magnetic field . for grains with velocity mostly perpendicular to the magnetic field ( see yan & lazarian 2003 ) , the angular momentum @xmath46 can be further decreased by a factor 10 - 100,scaledwidth=40.0% ]
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we show that grains can be efficiently aligned by interacting with a subsonic gaseous flow .
the alignment arises from grains having irregularities that scatter atoms with different efficiency in the right and left directions .
the grains tend to align with long axes perpendicular to magnetic field , which corresponds to davis - greenstein predictions .
choosing conservative estimates , scattering efficiency of impinging atoms and conservative `` degree of helicity '' , the alignment of helical grains is much more efficient than the gold - type alignment processes .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
dust grains produce extinction and reddening of stellar light from the ultraviolet ( uv ) to the infrared ( ir ) ( draine 2003 ) . accurate determination of reddening to a star is vital for reliable derivation of its basic stellar parameters , such as effective temperature and distance . constructing a 3d galactic extinction map plays an essential role in galactic astronomy , particularly in achieving the driving goals of the lamost spectroscopic survey of the galactic anti - center ( lss - gac ; liu et al . this volume ) . the sloan digital sky survey ( sdss ; york et al . 2000 ) has delivered low - resolution spectra for about 0.7 m stars in its data release 9 ( dr9 ; ahn et al . 2012 ) . the lamost galactic surveys ( deng et al . 2012 and this volume ) have obtained over 1 m stellar spectra and will obtain over 5 m in the next four years . with millions of stellar spectra , `` identical '' stars in different environments can be easily paired and compared , which opens great opportunities to a number of studies with the standard pair technique ( stecher 1965 ; massa , savage & fitzpatrick 1983 ) . by comparing the differences in photometric colors from large scale imaging surveys , such as the galaxy evolution explore ( galex ; martin et al . 2005 ) in the uv , the sdss in the optical , the two micro all sky survey ( 2mass ; skrutskie et al . 2006 ) in the near - ir and the wide - field infrared survey explorer ( wise ; wright et al . 2010 ) in the mid - ir , one can measure multi - band reddening for a large number of targets and constrain the reddening laws ( yuan , liu & xiang 2013 ) . by comparing the differences in normalized spectra , one can detect absorption features from the interstellar medium ( ism ) , particularly the diffuse interstellar bands ( dibs ; yuan & liu 2012 ) , as well as abnormal stellar absorption / emission lines from chemically particular or active stars . such method has the advantages that it s straight - forward , model - free and applicable to the majority of stars . combining stellar distances from gaia ( perryman et al . 2001 ) or from spectro - photometry , one can further map the galactic extinction , extinction laws and dibs in 3d . if by chance a diffuse nebula falls in the sightline of some targets , lines emitted by the nebula ( e.g. [ o iii ] @xmath04959 , 5007 ) will also be recorded and can be used to discover new faint emission line nebulae ( e.g. planetary nebulae and supernova remnants ) in the galaxy ( yuan & liu , submitted ) . finally , the technique can also be used to perform photometric calibration for wide field imaging surveys ( yuan et al . in prep ) . in the following sections , we use the cases above to demonstrate the power of the standard pair technique when applied to large scale spectroscopic and photometric datasets including the sdss and on - going lamost surveys . using star pairs selected from the sdss , and combining the sdss , galex , 2mass and wise photometry ranging from the far - uv to the mid - ir , yuan , liu & xiang ( 2013 ) have measured dust reddening in the @xmath1 and @xmath2 colors for thousands of galactic stars . the measurements , together with the @xmath3 values given by schlegel et al . ( 1998 ) , allow us to derive the empirical reddening coefficients for those colors . the results are compared with previous measurements and the predictions of a variety of galactic reddening laws . we find that 1 ) the dust reddening map of schlegel et al . ( 1998 ) over - estimates @xmath3 by about 14% , consistent with the work of schlafly et al . ( 2010 ) and schlafly & finkbeiner ( 2011 ) ; and 2 ) all the new reddening coefficients , except those for @xmath4 and @xmath5 , prefer the @xmath6 = 3.1 fitzpatrick reddening law ( fitzpatrick 1999 ) rather than the @xmath6 = 3.1 ccm ( cardelli et al . 1989 ) and odonnell ( odonnell 1994 ) reddening laws . using the @xmath7-band extinction coefficient predicted by the @xmath6 = 3.1 fitzpatrick law and the observed reddening coefficients , we have deduced new extinction coefficients for the @xmath8 and @xmath9 passbands . we recommend that the new reddening and extinction coefficients should be used in the future and an update of the fitzpatrick reddening law in the uv is probably necessary . lss - gac has obtained about 0.8 m spectra of @xmath10 per pixel and basic stellar parameters for about 0.6 m spectra of @xmath11 per pixel ( liu et al . this volume ) . with the same technique , we measured dust reddening in the @xmath12 and @xmath13 colors for over 0.2 m stars from lss - gac . the reddening coefficients for these colors relative to that for @xmath14 are consistent with the work of yuan , liu & xiang ( 2013 ) , as seen in fig.1 . with spectro - photometric distances , we have constructed a preliminary 3d extinction map in the outer disk of the galaxy . fig.2 shows @xmath3 as a function of distance from the sun and galactic longitude at @xmath15 , 0@xmath16 and @xmath17 . in spite of the limited sky coverage , distinct features , such as the perseus and outer arms , and effects of warps of the outer disk , are clearly visible . given the multi - band reddening of the stars , @xmath6 values are also estimated and their spatial variations are investigated in fig.3 . a median value of @xmath6 = 3.2 is obtained , consistent with previous results . no obvious spatial variations of @xmath6 are detected , indicating that dust properties do not change significantly in the outer disk . the lss - gac will obtain low - resolution spectra and basic stellar parameters for a statistically complete sample of @xmath18 m stars in a large contiguous sky area ( @xmath19 , @xmath20 ) . for @xmath21 , the lss - gac plans to sample 1,000 stars per sq.deg . for @xmath22 , the sampling is doubled . combining spectroscopic data from the lamost and sdss , photometric data from the galex , sdss , the xuyi schmidt telescope photometric survey of the galactic anti - center ( xstps - gac ; liu et al . this volume ) , pan - starrs ( kaiser 2004 ) , 2mass and wise , and spectro - photometric distances and gaia parallaxes in the future , we will produce high spatial resolution ( about 10 arcmin ) , multi - band extinction maps in the galactic anti - center , and then study the distribution of dust and variations of extinction laws . , @xmath23 , @xmath24 and @xmath13 colors versus that of @xmath14 for 225,422 stars from lss - gac that have xstps - gac , 2mass photometry and lamost spectral @xmath25 per pixel . to avoid crowdness , only one - in - five stars are shown . for comparison , relations obtained in yuan , liu & xiang ( 2013 ) are over - plotted . , width=480 ] as a function of distance from the sun ( ranging from 0 10 kpc ) and galactic longitude ( ranging from 150@xmath16 210@xmath16 ) in the galactic outer disk . , width=480 ] ( left ) and its scattering ( right ) . the x - axis and y - axis are galactic longitude ranging from 230@xmath16 120@xmath16 and latitude ranging from @xmath2640@xmath16 50@xmath16 , respectively . , width=336 ] dibs are weak absorption features detected in the spectra of reddened stars from the near uv to the near ir . dibs have been discovered for almost a century , and to date over 400 dibs have been detected in galactic and extragalactic sources ( e.g. hobbs et al . 2008 , 2009 ) , but none of their carriers is identified ( herbig 1995 ; sarre 2006 ) . the nature of dibs remains one of the most challenging problems in astronomical spectroscopy . most recent work to identify and investigate the properties and carriers of dibs concentrates on high - resolution spectroscopy of a small number of selected sight - lines . using a template subtraction method based on the standard pair technique , yuan & liu ( 2012 ) have successfully identified the dibs @xmath05780 , 6283 in the sdss low - resolution spectra of a sample of about 2,000 stars and measured their strengths and radial velocities . the sample is by far the largest ever assembled . the targets span a large range of reddening , @xmath3 @xmath27 0.2 1.0 , and are distributed over a large sky area and involve a wide range of stellar parameters , confirming that the carriers of dibs are ubiquitous in the diffuse ism . the sample is used to investigate relations between strengths of dibs and magnitudes of line - of - sight extinction , yielding results ( i.e. , @xmath28(5780 ) @xmath29 @xmath3 and @xmath28(6283 ) = 1.26 @xmath30 @xmath3 ) consistent with previous studies ( e.g. friedman et al . 2011 ) . dib features have also been detected in the lamost spectra ( fig.4 ) of resolving power similar to that of sdss . in the commissioning spectra of an emission line star , even 9 dibs have been detected ( see fig.5 from yuan & liu 2012 ) . detections of dibs towards hundreds of thousands of stars are expected with lamost . the huge dib database will provide an unprecedented opportunity to study the demographical distribution of dibs . when combined with other data - sets , it will enable us to address questions like : how the properties of dibs ( e.g. dib - dib , dib - extinction , dib - gas relations ) depend on local environment ( e.g. uv radiation field , @xmath6 , extinction in the fuv and the 2175 extinction bump ) ? where are their carriers formed ? can they be formed in the circumstellar environments ? meanwhile , dibs and atomic absorption lines from the ism can act as good tracers to probe the distribution and properties of the ism and dust . @xmath315780 , 6283 detected in the lamost spectra of three f / g dwarfs . basic information of each star is listed at the bottom of each row . for each row , segments of the target and the scaled template spectra are shown in the left two panels . the residuals are plotted in the right two panels , with the line fitting results over - plotted . , width=384 ] using @xmath27 1.7 m spectra from the sdss dr7 ( abazajian et al . 2009 ) , we have undertaken a systematic search for galactic planetary nebulae ( pne ) via detections of the [ o iii ] @xmath04959 , 5007 lines ( yuan & liu , submitted ) . examples of the sdss spectra with well detected [ o iii ] @xmath04959 , 5007 lines are shown in fig.5 . thanks to the excellent sensitivity of the sdss spectroscopic surveys , this is by far the deepest search for pne ever taken , reaching a surface brightness of the [ o iii ] @xmath315007 line s@xmath32 down to about 29.0 magnitude arcsec@xmath33 . the search recovers 14 previously known pne in the galactic caps . most of them are clearly visible on the sdss broad - band images owe to their high surface brightness . in total , about 60 new planetary nebula ( pn ) candidates are identified , including 7 probable candidates of multiple detections . all the probable candidates are extremely large ( between 21 154arcmin ) and faint , located mostly in the low galactic latitude region with a kinematics similar to disk stars , confirming the presence of a significant population of previously undetected , large , nearby , highly evolved pne in the solar neighborhood . four of the candidates have angular sizes between 84 154arcmin , and might well be the largest pne ever reported . based on sky positions and kinematics , 12 of the possible candidates probably belong to the halo population . if confirmed , they will double the numbers of known pne in the galactic halo . most newly identified pn candidates are very faint , with s@xmath32 between 27.0 30.0 magnitude arcsec@xmath33 , and very challenging for previously employed techniques ( e.g. slitless spectroscopy , narrow - band imaging ) . they greatly increase the number of faint pne and may well represent the `` missing '' pn population . the results have demonstrated the power of large scale fiber spectroscopy in hunting for ultra - faint pne and other types of nebulae by detecting nebular emission lines . the detection limits can be further increased by applying the same method which is used to detect dibs to the sdss and lamost stellar spectra . with millions of spectra from the sdss , lamost and other projects , it will provide a statistically meaningful sample of ultra - faint and large pne as well as new supernova remnants to improve their censuses . @xmath04959 , 5007 lines from galactic pne and pn candidates . the png identification , sdss spectral i d , initial target type and redshift are labeled . the fluxes are in unit of 10@xmath34ergs cm@xmath33 s@xmath36 @xmath36 . , width=384 ] uniform photometric calibration plays a central role in the large - scale imaging surveys , such as the sdss , the dark energy survey ( des ; the des collaboration 2005 ) , pan - stars and lsst ( ivezic et al . 2008 ) . the stellar locus regression ( slr ) method ( high et al . 2009 ) , adopted in des , can make one wholesale correction for differences in instrumental response , for atmospheric transparency , for atmospheric extinction , and for galactic extinction by adjusting the instrumental broadband colors of stars to bring them into accord with a universal stellar color - color locus , yielding calibrated colors accurate to a few percent . the slr method assumes the standard stellar locus is universal , which is however not always true , due to varying stellar populations and extinction , especially in the galactic disk region . it also requires a blue filter in addition to at least two of any other filters . to overcome the limitations above , we propose a new method to perform photometric calibration using star pairs from large scale spectroscopic surveys ( yuan et al . in prep ) , such as lss - gac . the star pairs here are composed of target and control stars from uncalibrated and calibrated fields , respectively . this method requires that 1 ) extinction values of the targets are known , which can be from schlegel et al . ( 1998 ) or derived from existing photometric data ; 2 ) the reddening laws do not change in one field ; and 3 ) there are a few calibrated fields to obtain the intrinsic colors . then it makes one wholesale correction by adjusting the instrumental colors of target stars to bring them into accord with their intrinsic colors , and obtain the reddening coefficient simultaneously . if the accuracy of instrumental colors is about 1% and about 100 star pairs can be selected for each field , this method will yield a color calibration accuracy about a few mmag . it is also useful in checking and improving calibration accuracies of existing surveys .
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with modern large scale spectroscopic surveys , such as the sdss and lss - gac , galactic astronomy has entered the era of millions of stellar spectra . taking advantage of the huge spectroscopic database , we propose to use a `` standard pair '' technique to a ) estimate multi - band extinction towards sightlines of millions of stars ; b )
detect and measure the diffuse interstellar bands in hundreds of thousands sdss and lamost low - resolution spectra ; c ) search for extremely faint emission line nebulae in the galaxy ; and d ) perform photometric calibration for wide field imaging surveys . in this contribution
, we present some results of applying this technique to the sdss data , and report preliminary results from the lamost data .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
dynamical evolution of globular clusters is strongly affected by gravitational tidal shocks . when the clusters cross the disk of the galaxy they experience disk shocking ; when the clusters pass near the galactic center , they experience bulge shocking . the effects of the tidal shocks depend on the density of the background stars and are especially pronounced in the inner regions of the galaxy . tidal shocks increase the energy of random motion of stars , reduce the binding energy of the cluster , accelerate core collapse , and lead to the faster overall evolution and destruction of globular clusters ( for example , spitzer 1987 ; weinberg 1994 ; murali & weinberg 1997a , b , c ; gnedin & ostriker 1997a ) . we calculate the rate of destruction of globular clusters as a result of various physical processes : two - body relaxation , evaporation of stars through the tidal boundary , disk shocking and bulge shocking ( gnedin & ostriker 1997a ) . we use a fokker - planck code which includes tidal shocks semi - analytically . we introduce the adiabatic corrections that account for the conservation of adiabatic invariants of the fast moving stars ( gnedin & ostriker 1997b ) . these corrections reduce the energy input due to the shocks in the inner parts of the cluster . the results ( figure 1 ) show that tidal shocks dominate cluster evolution near the galactic center . overall , as many as 50% to 90% of the present sample of globular clusters may be destroyed within the next hubble time . tidal shocks destroy most easily the low - mass , low - density clusters . figure 2 shows the distribution of the inner and outer galactic globular clusters . as expected , there are no low - density clusters in the inner part of the galaxy where the tidal shocks operate most efficiently . removal of those low - mass clusters makes the mean , or the peak , of the luminosity function ( lf ) to shift towards bighter magnitudes . an apparent correlation of the masses and densities of outer clusters allows us to construct an intrinsic distribution of globulars , unaffected by the shocks . by applying the dynamical calculations , we can estimate the amount of brightening of the peak of the lf . assuming that in all galaxies the initial distribution is the same , we can reconstruct the shock history in an external galaxy and infer the peak of the original distribution of globular clusters . comparing that peak with the center of the intrinsic distribution in the galaxy , we obtain a distance estimate to the galaxy . this method is fully independent and makes unnecessary the common assumption that the peak of the lf is a standard candle . applied to the best known samples of m31 and m87 ( ostriker & gnedin 1997 ) , our method gives a distance estimate in very close agreement with that obtained with cepheids and other methods . tidal heating can be simply parametrized to study semi - analytically the evolution of the individual clusters . we derive analytic equations for the first and second order energy changes , @xmath0 and @xmath1 , of stars in the cluster ( gnedin , hernquist & ostriker 1997 ) . these equations are supplemented by the adiabatic corrections depending on the effective duration of the shock . heating on the nearly circular orbits is strongly suppressed because the `` shock '' becomes very slow ( figure 3 ) . the analytical estimates are tested against the self - consistent n - body simulations of the shocking event along a true trajectory of the cluster in the galaxy . we find a remarkable agreement with the simulations . gnedin , o. y. , hernquist , l. & ostriker , j. p. 1997 , , submitted gnedin , o. y. & ostriker , j. p. 1997a , , 474 , 223 gnedin , o. y. & ostriker , j. p. 1997b , , submitted murali , c. & weinberg , m. d. , 1997a , , 288 , 749 murali , c. & weinberg , m. d. , 1997b , , 288 , 767 murali , c. & weinberg , m. d. , 1997c , , 291 , 717 ostriker , j. p. & gnedin , o. y. 1997 , , 487 , 667 spitzer , l. 1987 , dynamical evolution of globular clusters ( princeton : + princeton university press ) weinberg , m. d. 1994 , , 108 , 1414
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the semi - analytic theory of tidal shocks proves to be a powerful tool to study tidal interactions of star clusters and satellite galaxies with their massive hosts .
new models of the globular cluster evolution employ a combination of analytic estimates , solutions of the fokker - planck equation and direct n - body simulations .
the models predict large destruction rates for the galactic globular clusters .
those on the highly eccentric orbits around the galactic center are much more likely to be disrupted than the ones on nearly circular orbits .
the destruction rates are largely increased near the bulge .
disruption of the low - mass clusters changes the luminosity function of the globular cluster system , shifting the peak of the luminosity function to the brighter end .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
calculation of the form factors for semileptonic transitions of @xmath9 mesons has been being a subject discussed intensely . recently , it has been shown that the @xmath10 transition form factor can be consistently analyzed by using the different approaches in the different @xmath11 regions @xcite . the perturbative qcd ( pqcd ) can be applied to the @xmath10 form factor in the large recoil ( small @xmath11 ) region and it is reliable when the involved energy scale is large enough @xcite . the qcd light - cone sum rules ( lcsr ) can involve both the hard and soft contributions to the @xmath10 form factor below @xmath12 @xcite . the lattice qcd simulations of the @xmath10 transition form factor @xcite are available only for the soft region @xmath13 , because of the restriction to the @xmath14 energy smaller than the inverse lattice spacing . thus the results from these three approaches might be complementary to each other . in ref.@xcite we recalculate the @xmath10 form factor in the pqcd approach , with the transverse momentum dependence included for both the hard scattering part and the nonperturbative wave functions(of @xmath14 and @xmath9 ) to get a more reliable pqcd result . by combining the results from these three methods we obtain a full understanding of the @xmath10 transition form factor in its physical region @xmath15 . up to now , in comparison with heavy - to - light cases the calculations on heavy - to - heavy transitions can be done only for a certain specific kinematical range , although there have been a lot of discussions in the literature . in the bsw model @xcite , the relevant form factors at zero momentum transfer are expressed as an overlap of initial and final meson wave functions for which they take the solutions of the bethe - salpeter ( bs ) equation in a relativistic harmonic oscillator potential . then one extrapolates the result at @xmath16 to the whole kinematical region assuming the nearest pole dominance . with the discovering of the heavy quark symmetry , the @xmath17 form factor have been known better at zero recoil . this is because of the fact that in the heavy quark limit the resulting form factors isgur - wise functions @xcite at zero recoil are rigorously normalized . including the leading symmetry breaking corrections , the deviation from this limit can be estimated at an order of a few percent due to luke s theorem and therefore the value of the form factor at this point can be determined within a higher accuracy @xcite . however , the dependence of the form factor on the velocity transfer @xmath18 ( with @xmath19 and @xmath20 being the velocities of the @xmath9 and @xmath1 mesons respectively ) is difficult to get even in the leading order , in view of the arbitrary function @xmath21 @xcite which is introduced to simulate higher - resonances in the heavy quark effective theory ( hqet ) . the lattice qcd , despite a rigorous nonperturbative approach , is just adequate to estimate the behavior of the form factors near the zero recoil @xcite . among the other approaches are the qcd sum rules and pqcd . ref.@xcite applies the traditional @xmath22-point sum rule to calculate the form factor at zero momentum transfer . it is concluded in ref.@xcite that pqcd approach is applicable in the large recoil region and can give a consistent result with the experiment . it is necessary that there is a reliable estimate of @xmath23 transition in the whole kinematically accessible range @xmath24 , in order to account for the data on @xmath0 . for this purpose , it is practical , as shown in @xmath25 case , to combine the result of qcd lcsr with those from the lattice qcd , heavy quark symmetry and pqcd . the lcsr approach @xcite , where the non - perturbative dynamics are effectively parameterized in so - called light - cone wave functions , is regarded as an effective tool to deal with heavy - to - light exclusive processes . although the @xmath17 transition in question is a heavy - to - heavy one , the @xmath26-quark is much lighter compared to @xmath27-quark and so discussing it with lcsr is plausible for the kinematical range where the ope near light - cone @xmath3 is valid . the other problem with our practical calculation is that the higher twist da s of d meson , which are important but less studied , would enter into the sum rule result . however , an effective approach @xcite has been presented to avoid the pollution by some higher - twist da s . this improved lscr method uses a certain chiral current correlator as the starting point so that the relevant twist-@xmath22 wave functions make no contributions and the reliability of calculation can be enhanced to a large extend . its applicability has been examined by a great deal of studies @xcite . in this paper we would like to employ the improved lcsr to discuss the form factor for the @xmath17 transition and try to give a full understanding of qcd dynamics involved in the @xmath28 . this paper is organized as follows . in the following section we derive the lcsr for the form factor for @xmath23 . a discussion of the da models for the @xmath1-meson is given in section iii . section iv is devoted to the numerical analysis and comparison with other approaches . the last section is reserved for summary . the @xmath17 weak form factors @xmath29 and @xmath30 are usually defined as : @xmath31 with @xmath32 being the momentum transfer . on the other hand , when applying the heavy quark symmetry to do discussion the following definition is advisable , @xmath33.\ ] ] if we neglect the masses of leptons in the decay final state of @xmath34 , only @xmath29 is relevant and thus we can confine us to the discussion on @xmath29 . obviously , the following relation is observed between @xmath29 and @xmath35 , @xmath36 where @xmath37 , @xmath38 . using the heavy quark symmetry , the value of form factor @xmath39 at zero recoil could be fixed better . since in heavy quark limit @xmath40 and @xmath41 , the form factor @xmath39 should be close to @xmath42 . a systematic investigation gives @xmath43@xcite , with a less model dependence . this result is also confirmed with lattice calculations @xcite . pqcd analyses are also made in the large recoil region @xmath44 , yielding a result consistent with the data . the lcsr calculation can help to understand @xmath45 in the whole kinematical region in complementary to the lattice qcd with the heavy quark symmetry and pqcd approaches . to achieve a lcsr estimate of @xmath46 , we follow @xcite and use the following chiral current correlator @xmath47 : @xmath48 in the first place , we discuss the hadronic representation for the correlator . this can be done by inserting the complete intermediate states with the same quantum numbers as the current operator @xmath49 . isolating the pole contribution due to the lowest pseudoscalar @xmath9 meson , we have the hadronic representation in the following : @xmath50 note that the intermediate states @xmath51 contain not only the pseudoscalar resonance of masses greater than @xmath52 , but also the scalar resonances with @xmath53 , corresponding to the operator @xmath54 . with eq.([eq : def ] ) and the definition of the decay constant @xmath55 of the @xmath9-meson @xmath56 and expressing the contributions of higher resonances and continuum states in a form of dispersion integration , the invariant amplitudes @xmath57 and @xmath58 read , @xmath59=\frac{2f(q^2)m_b^2f_b}{m_b(m_b^2-(p+q)^2)}+ \int^\infty_{s_0}{\frac{\rho^h(s)}{s-(p+q)^2}ds}+\mbox{subtractions},\ ] ] and @xmath60=\frac{\tilde{f}(q^2)m_b^2f_b}{m_b(m_b^2-(p+q)^2)}+ \int^\infty_{s_0}{\frac{\tilde{\rho}^h(s)}{s-(p+q)^2}ds}+\mbox{subtractions},\ ] ] where the threshold parameter @xmath61 should be set near the squared mass of the lowest scalar @xmath9 meson , the spectral densities @xmath62 and @xmath63 can be approximated by invoking the quark - hadron duality ansatz @xmath64 on the other hand , we need to calculate the corrector in qcd theory to obtain the desired sum rule result . in fact , there is an effective kinematical region which makes ope applicable : @xmath65 for the @xmath66 channel and @xmath67 for the momentum transfer . for the present purpose , it is sufficient to consider the invariant amplitude @xmath68 which contains the desired form factor . the leading contribution is derived easily by contracting the @xmath69quark operators to a free propagator : @xmath70substituting eq.([eq : pro ] ) into eq.([eq : cc ] ) , we have the two - particle contribution to the correlator , @xmath71 an important observation , as in ref.@xcite , is that only the leading non - local matrix element @xmath72 contributions to the correlator , while the nonlocal matrix elements @xmath73 and @xmath74 whose leading terms are of twist @xmath22 , disappear from the sum rule . proceeding to eq.([eq : qq ] ) , we can expand the nonlocal matrix element @xmath75 as @xmath76 where @xmath77 is the twist-2 da of d meson with @xmath78 being the longitudinal momentum fraction carried by the @xmath26-quark , those da s entering the higher - twist terms are of at least twist @xmath79 . the use of eq.([eq : da ] ) yields @xmath80=2f_dm_b\int_0 ^ 1{du\frac{\varphi_d(u)}{m_b^2-(up+q)^2}}+\mbox{higher twist terms}. \label{eq : qq1}\ ] ] invoking a correction term due to the interaction of the b quark with a background field gluon into ( 10 ) , the three - particle contribution @xmath81 is achievable . however , the practical calculation shows that the corresponding matrix element whose leading term is of twist @xmath22 also vanishes . thus , if we work to the twist-3 accuracy , only the leading twist da @xmath82 is needed to yield a lcsr prediction . furthermore , we carry out the subtraction procedure of the continuum spectrum , make the borel transformations with respect to @xmath83 in the hadronic and the qcd expressions , and then equate them . finally , from eq.([eq : f12 ] ) follows the lcsr for @xmath84 , which is applicable to the velocity transfer region @xmath85 , @xmath86 } \varphi_d(u)},\label{eq : ff}\end{aligned}\ ] ] where @xmath87 and @xmath88 has been used . now let s do a brief discussion on an important nonperturbative parameter appearing in the lcsr formula([eq : ff ] ) , the leading twist da of @xmath1-meson , @xmath89 . @xmath1-meson is composed of the heavy quark @xmath26 and the light anti - quark @xmath90 . the longitudinal momentum distribution should be asymmetry and the peak of the distribution should be approximately at @xmath91 . according to the definition in eq.([eq : da ] ) , @xmath89 satisfies the normalization condition @xmath92 which is derived by the leptonic decay @xmath93 . in the pqcd calculations @xcite , a simple model ( we call model i ) is adopted as @xmath94 which is based on the expansion of the gegenbauer polynomials . eq.([eq : da1 ] ) has a free parameter @xmath95 which ranges from @xmath96 to @xmath42 . we will take @xmath97 as input . on the other hand , it was suggested in @xcite that the light - cone wave function of the @xmath1-meson be taken as : @xmath98}\label{eq : wf}\ ] ] which is derived from the brosky - huang - lepage(bhl ) prescription @xcite . @xmath99 can be related to the da by the definition @xmath100 substituting eq.([eq : wf ] ) into eq.([eq : wf - da ] ) , we have a model of the da(model ii ) @xmath101},\label{eq : da2}\ ] ] where the parameters @xmath102 and @xmath103 can be fixed by the normalization([eq : constraint1 ] ) and the probability of finding the @xmath104 fock state in the @xmath1- meson , @xmath105 @xmath106 as discussed in ref.@xcite , @xmath107 is a good approximation for the @xmath1-meson ( as we have checked , change of @xmath105 makes a numerical effect less than @xmath108 ) . then , taking @xmath109 , @xmath110 , @xmath111 and @xmath112 , we have @xmath113 , @xmath114 . furthermore , as argued in ref.@xcite , a more complete form of the light - cone wave function should include the melosh rotation effect in spin space : @xmath115}\ ] ] with the melosh factor , @xmath116 it can be seen from eq.([eq : melosh ] ) that @xmath117 as @xmath118 , since there is no spin interaction between the two quarks in the heavy - flavor meson , ie . , the spin of the heavy constituent decouples from the gluon field , in the heavy quark limit @xcite . however the @xmath26-quark is not heavy enough to neglect the melosh factor . after integration over @xmath119 the full form of @xmath1 meson da can be achieved ( model iii ) : @xmath120 \exp{\left[-b_d^2\frac{(xm_d^2+(1-x)m_c^2-y^2)}{x(1-x)}\right]}\label{eq : da3},\ ] ] where @xmath121 and the error function @xmath122 is defined as @xmath123 . using the same constraints as in eq.([eq : constraint1 ] ) and ( [ eq : constraint2 ] ) , the parameters @xmath102 and @xmath103 are fixed as @xmath124 and @xmath125 . in this paper we will employ the above three kinds of models to do numerical calculation . all these da s of the @xmath1-meson are plotted in fig.([fig : wf ] ) for a comparison . it is shown that they are of similar shape and all of them exhibit a maximum at @xmath126 as expected . apart from the da of @xmath1-meson , the decay constant of @xmath9-meson @xmath55 is among the important nonperturbative inputs . for consistency , we use the following corrector @xmath127 to recalculate it in the two - point sum rules . the calculation should be limited to leading order in qcd , since the qcd radiative corrections to the sum rule for @xmath46 are not taken into account . the value of the threshold parameter @xmath128 is determined by a best fit requirement in the region @xmath129 , where @xmath130 is the corresponding borel parameter . the same procedure is performed for @xmath46 , resulting in different values of the threshold parameter @xmath61 . the result is listed in tab.[tab : s0 ] . choosing @xmath61 and @xmath128 in this way , the dependence of @xmath55 and @xmath46 on the borel parameter is very weak and thus we can simply evaluate them at @xmath131 . the other input parameters are taken as @xmath132 . as we have ignored all the radiation corrections , we do nt expect our values of @xmath55 to be good predictions of that quantity . .parameter sets for @xmath55 and @xmath46 , @xmath128 and @xmath61 for @xmath55 and @xmath46 respectively ; @xmath133 and @xmath55 are given in gev , @xmath61 and @xmath128 in @xmath134 . [ cols="<,<,^,^,^",options="header " , ] with the parameters chosen , it is straightforward to calculate the form factor @xmath46 in the region @xmath135 . the results with different sets of parameters are plotted in fig.([fig : f(mb ) ] ) , where only model ii has been used for simplicity . it is shown that the change of parameters can induce a uncertainty of about @xmath136 if we let @xmath133 vary between @xmath137 . by fitting the data , the behavior of @xmath45 has been known using the parametrization @xmath138,\ ] ] with @xmath139 corresponding to the linear and quadratic fits @xcite , respectively . with the three da models , the resulting dependence of @xmath46 on the velocity transfer @xmath140 , along with that extracted experimentally is illustrated in fig.([fig : f(y ) ] ) . in what follows , we denote the lcsr results for the form factor by @xmath141 and those extracted experimentally by @xmath142 . for comparison , a figure - copy which expresses the pqcd results in @xcite is given in fig.([fig : pqcd ] ) . in the region to which the lcsr method is applicable , the central values of @xmath141 turn out to be a bit smaller than the corresponding those of @xmath143 , using the da models ii and iii as inputs ; however , both of them are in accordance with each other within the error . the situation with model i da is about the same . the central value of the form factor at the largest recoil is @xmath144 versus @xmath145 , depending on the da models . we note that the behavior of @xmath141 is essentially unchanged when the three different da s are used . from the present calculations , therefore it is too early to draw a conclusion which da model is more suitable to reflect the characteristics of qcd dynamics inside the @xmath1 meson . when a comparison is made between the pqcd and lcsr predictions , the consistent results can also be observed at the larger recoil . of course , the two approaches describe the different dynamics in @xmath146 transitions . whereas the use of lcsr approach is to assume that the soft exchanges dominate in the weak decay in question , applying pqcd method to do calculation corresponds to the viewpoint that the hard exchanges do . in fact , the kinematical region we give , which makes lcsr results valid , is a conservative estimate . it is possible to extrapolate the present lcsr calculation to the small recoil region . if it is true , we find that in the whole kinematically accessible range @xmath147 , the yielding lcsr estimates are compatible with the data . for instance , at zero recoil it follows that @xmath148 ( using model iii ) , which is in a good agreement with the evaluation obtained using the heavy quark symmetry : @xmath43 . nevertheless , we have to emphasize that a full understanding of the dynamics involved in @xmath149 transition should be obtained by combining the three different approaches the lattice qcd calculations with the heavy quark symmetry considered , lcsr results and pqcd predictions , which are complementary to each other . the lcsr results with chiral current correlator may act as a bridge connecting those of other approaches . @xmath150 @xmath151 @xmath152 @xmath153 we have discussed the form factor for @xmath17 transitions @xmath46 , using the improved qcd lcsr approach where with the chiral current correlator chosen only the leading twist da of the @xmath1-meson is relevant at twist-3 accuracy . the resulting lcsr s for @xmath46 are available in the velocity transfer region @xmath2 . calculation is done using three different twist-2 da models for d meson . it is shown the numerical results are less sensitive to the choice of da , and are of a central value slight smaller than but within the error in a agreement with those obtained by fitting the data on @xmath4 . in the larger recoil region @xmath5 where pqcd is applicable , the results presented here are consistent with ones of pqcd . from the practical calculations , we find that the present results might be extrapolated to the smaller recoil region so that the @xmath17 transitions are calculable in the whole kinematically accessible range , using the improved lcsr approach . also , we argue that for understanding the form factor for @xmath154 in the whole kinematical range a combined use is necessary of three different methods : the lattice qcd ( with the heavy quark symmetry considered ) , improved lcsr and pqcd approaches , which are adequate to do calculation in different kinematical regions and so could be complementary to each other . the lcsr approach plays a bridge role in doing such calculation . the present findings can be improved once the qcd radiative correction to the lcsr is taken into account and a more reliable twist-2 da of @xmath1 meson becomes available . from the previous discussion in @xcite , however , it is expected that the qcd radiative correction can not change the present results too much . s. j. brodsky , t. huang and g. p. lepage , in _ particles and fields-2 _ , proceedings of the banff summer institute , banff , alberta , 1981 , edited by a. z. capri and a. n. kamal ( plenum , new york , 1983 ) , p143 ; g. p. lepage , s. j. brodsky , t. huang , and p. b. mackenize , _ ibid . _ , p83 ; t. huang , _ in proceedings of xxth international conference on high energy physics _ , madison , wisconsin , 1980 , edited by l. durand and l. g. pondrom , aip conf . 69 ( aip , new york , 1981 ) , p1000 .
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within the framework of qcd light - cone sum rules ( lcsr ) , we calculate the form factor for @xmath0 transitions with chiral current correlator .
the resulting form factor depends on the distribution amplitude ( da ) of the @xmath1-meson .
we try to use three kinds of da models of the @xmath1-meson . in the velocity transfer region @xmath2 , which renders the operator product expansion ( ope ) near
light - cone @xmath3 go effectively , the yielding behavior of form factor is in agreement with that extracted from the data on @xmath4 , within the error . in the large recoil region @xmath5 ,
the results are observed consistent with those of perturbative qcd ( pqcd ) .
the presented calculation can play a bridge role connecting those from the lattice qcd , heavy quark symmetry and pqcd to have an all - around understanding of @xmath0 transitions .
* form factor for @xmath0 in the light - cone sum rules with chiral current correlator * * .3 cm fen zuo,@xmath6 zuo - hong li @xmath7 and tao huang @xmath8 *
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in recent years the field of ultracold atomic physics has attracted the attention of a great many condensed matter theorists , largely due to the prospect of finding novel realizations of many - body systems . part of the appeal doubtless lies in the exquisite experimental control that may be exercised over the parameters of a system in which many of the complicating factors familiar from the solid state ( disorder , phonons , etc . ) are absent . intrinsic _ aspects of ultracold atomic gases not dependent on the specifics of the experimental setting are qualitatively novel , however , and without antecedent in the study of condensed matter . in this latter category we may place the possibility of spontaneous ordering of the spin degrees of freedom in a bose gas . indeed , prior to the ` ultracold revolution ' the only bose superfluid that could be studied in the laboratory was @xmath0he , which has zero spin . with the advent of optical trapping of bose condensates of alkali atoms , which allows for a fully rotationally invariant setting , the experimental study of spin ordering within a hyperfine multiplet came within reach @xcite . the earliest theoretical works motivated by these developments explored possible ordered phases using a mean field description , in both the spin 1 and spin 2 hyperfine multiplets @xcite . in this description the concept of spontaneous symmetry breaking plays a central role . up until very recently , however , there were no experiments in which this spontaneous ordering was apparent . the reason for this is that the simplest experimental protocol for the investigation of the hyperfine state of the gas is to apply a magnetic field gradient to split a gas cloud into different components in a stern - gerlach experiment @xcite . the different components are subsequently imaged to determine their ( relative ) occupancies . this technique naturally imposes a quantization axis , and any information concerning the _ coherence _ between different hyperfine levels is lost . thus magnetic alignment in the plane perpendicular to this axis , which depends on the relative phase of these different levels , can not be observed . the characterization of magnetic ordering in atomic gases has taken a leap forward in the last few years thanks to the work of the berkeley group , who demonstrated _ in situ _ dispersive imaging of the transverse magnetization of a gas of @xmath1rb in the spin 1 multiplet @xcite , and subsequently employed this technique to investigate a number of fascinating aspects of this system , including the dynamics of spontanteous symmetry breaking , defect production , and the role played by magnetic dipole forces @xcite . the above developments illustrate two important needs . firstly , imaging of the spin order was necessary to bring much of this new physics out into the open . secondly , the nonequilibrium character of most experiments requires that the mean field theory of equilibrium ordered states be supplemented with a _ dynamical _ description of the relevant order parameters . it is our hope that the next few years will see the development of imaging techniques capable of detecting some of the spin orders to be discussed in section [ sec : spin ] , of which an average magnetization is only the simplest . the aim of this work is to address the second need : uncovering the low energy description of the order parameter . for the case of the bose ferromagnet , which is appropriate to the spin 1 @xmath1rb system , this description was provided in an earlier paper @xcite . in this work we will focus instead on states with vanishing average magnetization . for reasons that will be become clear in the following sections these cases are qualitatively different . in an ordered phase of matter we expect that the low energy degrees of freedom consist of variations of the order parameter on some manifold of symmetry broken states ( goldstone modes ) , together with any conserved quantities . in our earlier work on the bose ferromagnet @xcite the degrees of freedom were the local magnetic moment @xmath2 and the superfluid velocity @xmath3 . in the long wavelength limit these were found to obey the coupled equations @xmath4 @xmath5 where @xmath6 is the convective derivative . ( [ lle ] ) is a modified landau - lifshitz equation and gives rise to quadratically dispersing spinwaves when linearized on around a solution @xmath7 . the quadratic dispersion is a consequence of the two transverse deviations of the order parameter being canonically conjugate . for the phases with vanishing average magnetization that are the focus of this work , the conjugate variables involve deviations from the order parameter manifold . this results in linearly dispersing goldstone modes . the situation is analogous to the case of spin waves in an antiferromagnet , where the conjugate variables are the difference in magnetization on neighboring sites the nel order parameter and the sum . it also follows from the vanishing of the average magnetization that the order parameter dynamics and superfluid flow are decoupled except for a global topological constraint . again , this is quite different from the case of the ferromagnet , where the two are coupled together in the equations of motion . it follows that we can write a lagrangian for the spin degrees of freedom only . this lagrangian is expressed in terms of a rotation matrix @xmath8 that specifies the local orientation of the spin state relative to some reference state . by expressing the matrix elements of @xmath8 in terms of an orthonormal triad @xmath9 , with @xmath10 , the spin lagrangian may be written @xmath11\ ] ] here @xmath12 are @xmath13 are some constants to be specified later , which depend on the ordered phase in question . the lagrangian eq . ( [ l_genspin ] ) is the main result of this paper . the only other phase for which the spin lagrangian was previously obtained is the polar phase of the spin 1 gas , to be discussed below , in which @xmath14 , @xmath15 , giving the familiar @xmath16 model @xcite . the structure of the remainder of this paper is as follows . in section [ sec : basics ] we review the basic description of a spinor condensate , and in particular the structure of the interaction hamiltonian . section [ sec : spin ] describes in detail the ground state manifolds of the different phases of spinor condensates , beginning with the simplest non - trivial case , spin 1 , before introducing the majorana ( or stellar ) representation that is very useful in visualizing spin states . the global structure and local geometry of the order parameter manifolds are then introduced , as well as a parametrization for the dynamical variables conjugate to the order parameter . after this the derivation of the low energy lagrangian is a relatively simple matter , and is described in section [ sec : low ] along with the derivation of the equations of motion . we have tried to keep the presentation pedagogical throughout , though at various points there are parenthetical technical comments that the casual reader should feel free to ignore . in a related work , barnett _ et al . _ @xcite have derived the full equations of motion of a spinor condensate in terms of the majorana representation and applied a group - theoretical analysis to the determination of all normal modes about an ordered ground state . we pursue a complimentary goal of obtaining the full _ nonlinear _ lagrangian for the goldstone modes only . one could take the point of view that the description of the dynamics of a dilute spinor gas is no different from its spinless counterpart , being governed by the time - dependent gross - pitaevskii equation , with lagrangian density @xmath17 ( we set @xmath18 from now on ) . for a spin @xmath19 gas @xmath20 is a @xmath21 component spinor and the hamiltonian density has the form @xmath22 where the first term is the kinetic energy . the interaction part @xmath23 is quartic in @xmath20 and its form will be given below for @xmath24 and @xmath25 . we will not discuss the influence of the trapping potential , save to assume that it preserves rotational symmetry . for stationary solutions of the form @xmath26 the time - dependent description reduces to the time - independent gross - pitaevskii theory , with @xmath27 the chemical potential . instead of treating the action @xmath28 classically , we can interpret it as the quantum action in a coherent state path integral . little that we will have to say will depend upon this distinction . in fact this superficial similarity between the spinless and spinful problems is quite misleading . the ground state in a uniform system corresponds to some constant @xmath20 . in the spinless case this fixes @xmath20 up to a phase once the density @xmath29 is specified . but in the spinful case we still have to find the correct ` direction ' of @xmath20 in the complex @xmath21 dimensional spinor space , determined by minimizing the interaction hamiltonian @xmath23 . the interactions will be assumed to respect rotational symmetry , so this minimum is only unique up to rotation ( specified by the three euler angles , say ) . choosing this rotation starting from some arbitrary reference state specifies the spontaneous breaking of rotational symmetry in the ground state . the slow variation of this rotation in space and time constitutes the low energy dynamics of the system , the description of which is the subject of this work . to characterize these low energy manifolds we must first specify the form of @xmath30 . the structure of @xmath30 has been discussed in many works , starting with the first papers treating spinor bose condensates . we will therefore keep the following discussion relatively brief . low energy scattering between a pair of bosons occurs in the @xmath19-wave channel only , and can be treated as a @xmath31-function interaction , characterized by a set of interaction constants @xmath32 , @xmath33 for a pair of bosons with total spin @xmath34 . bose symmetry dictates that the interaction vanishes for odd total spin . for spin 1 , the resulting interaction may be presented in the form @xcite @xmath35 where @xmath36 are the spin @xmath19 angular momentum matrices . ( [ spin1hint ] ) is the sum of a density - density and a spin - spin interaction . for spin 2 we have @xcite @xmath37 here @xmath38 is a scalar representing the amplitude of singlet pairs of spin s ( eq . ( [ spin1hint ] ) can be expressed using this operation instead of the spin - spin interaction . in the spin 2 case both terms are needed ) . our use of @xmath39 for two different quantities matches the notation of the works cited above , where one may find explicit expressions for @xmath40 @xmath41 in terms the constants @xmath32 defined above . in the spin 1 case it is fairly clear how to minimize eq . ( [ spin1hint ] ) at fixed density . for the spin 2 case things are less obvious . in the next section we will discuss a method of parametrizing the spinor @xmath42 to make this operation transparent . in this section we will introduce the majorana ( or _ stellar _ ) representation of spin states . this provides a vivid way to picture spin ordering in higher spin condensates in which rotational symmetry is manifest . none of the calculations of section [ sec : low ] depend upon this representation ; its use is rather in providing a concrete way to picture the ground state manifold . before beginning it is worth setting the scene with a more pedestrian discussion of the spin 1 case @xcite . let us minimize eq . ( [ spin1hint ] ) with a spinor @xmath42 normalized to unity ( thus we are adopting units in which the density @xmath43 ) . this is a matter of minimizing ( maximizing ) @xmath44 for @xmath45 ( @xmath46 ) . one way to make the resulting states more clear is to write the complex spinor @xmath47 where @xmath48 and @xmath49 are two real vectors satisfying @xmath50 , and to work in cartesian components . the relationship to the usual components @xmath51 @xmath52 is @xmath53 in this basis the angular momentum matrices take the form @xmath54.then we have @xmath55 for @xmath46 the interaction energy is minimized for @xmath48 and @xmath49 perpendicular and equal in magnitude . this state is termed _ ferromagnetic _ as it corresponds to maximal polarization of the spin . the resulting order parameter manifold corresponds to the set of all configuration of a pair of orthogonal vectors , and is thus identified with the group of rotations @xmath56 for @xmath45 @xmath48 and @xmath49 are aligned . the resulting _ polar _ state ( the name originates from an analogous state in superfluid @xmath57he ) can therefore be written as @xmath58 for @xmath59 a _ real _ unit vector . note that this parametrization has some redundancy in that the points @xmath60 and @xmath61 are identified . the resulting manifold is known as the _ mapping torus _ of the antipodal map of the sphere @xmath62 . the global topology of the above ferromagnetic and polar order parameter manifolds naturally determines the character of the topological defects in the ordered phases , and certain features of the ordering transitions , some of which have already been discussed in the literature @xcite . this is not the focus of the present work and topology will not be further discussed , even though the defect physics of the higher spin condensates promises to be highly non - trivial @xcite . the polar state has @xmath63 . nevertheless the above discussion makes it clear that polar ordering involves a choice of axis : the spinor @xmath64 is the @xmath65 state with respect to the axis @xmath59 . it is natural to ask for an operator that acquires a non - zero expectation value in the polar state . the obvious candidate is the spin 2 quadrupole tensor ( or _ nematicity _ ) @xmath66 with @xmath67 such expressions are familiar in the study of nematic liquid crystals , where the vector @xmath59 is known as the director . in the liquid crystal context the identification of @xmath59 and @xmath68 without the phase factor in eq . ( [ polar_parm ] ) ( the order parameter manifold is then the real projective plane @xmath69 ) makes for very different defect physics , however @xcite . nematic ordering in solid state magnetic systems has been the subject of much experimental and theoretical work in recent years , with a good deal of uncertainty still remaining . the observation of the polar state in the spin 1 bose gas would therefore be an important milestone . searching for higher spin order parameters as the spin of the gas particles increases becomes arduous . we now turn to a more convenient representation of the spin order . the representation of a general spin @xmath19 state that ( sometimes ) bears his name was discovered by majorana in 1932 @xcite , and independently several times since @xcite , though it has antecedents in 19@xmath70 century mathematics @xcite . a very nice discussion can be found in ref . . the result is very simple to state , and represents a generalization of the bloch sphere for spin 1/2 to arbitrary spin . up to normalization and a phase thus in more mathematical terms we are parametrizing the complex projective space @xmath71 an arbitrary spin @xmath19 state can be specified by locating @xmath72 indistinguishable points on the unit sphere ( see fig . [ fig : principal ] ) . such a configuration is sometimes called a _ constellation _ , for reasons that will become clear . spin state , with arrows representing the principal spinors . [ fig : principal],title="fig:",scaledwidth=40.0% ] + there are two steps to understand why this is so . first , imagine forming our spin @xmath19 from @xmath72 spin 1/2 in the totally symmetric subspace . an arbitrary state may then be written as a totally symmetric spinor @xmath73 , where the round brackets denote the operation of symmetrization and each of the @xmath21 indices can take the value @xmath74 or @xmath75 . the relationship between @xmath76 and the corresponding @xmath21 component spinor @xmath42 is @xmath77 next contract every index of @xmath76 with @xmath78 . denoting by @xmath79 the resulting polynomial of order @xmath72 , the fundamental theorem of algebra tells us @xmath80 with @xmath81 some normalization . thus @xmath76 may be written @xmath82 with the _ principal spinors _ @xmath83 , @xmath84 , etc . related to the @xmath85 by @xmath86 relations that are unchanged if we normalize the spinors , in which case they correspond to @xmath72 points on the bloch sphere with coordinates @xmath87 , @xmath88 @xmath89 ( notice that @xmath76 in eq . ( [ root_spinors ] ) is not in general normalized when the principal spinors are ) . then ( minus ) the roots of @xmath79 can be written @xmath90 and correspond to stereographic projection from the north pole to the plane tangent to the sphere at the south pole . the beautiful feature of the majorana representation is that rotations act simply as rotations on the bloch sphere . of course , it is useful to have an explicit expression for the polynomial @xmath79 in terms of the @xmath21 components of the spin @xmath19 state @xmath42 . it is easy to show @xmath91 if spinor indices are raised and lowered using the antisymmetric tensors @xmath92 and @xmath93 ( with @xmath94 and @xmath95 ) @xmath96 then rotational invariance upon contraction of indices is guaranteed ( in fact the result is invariant under the larger group @xmath97 , a result that will be useful later ) . if we denote by @xmath98 the result on @xmath42 of raising all indices of the corresponding symmetric spinor , then one can readily see that @xmath99 , and thus @xmath100 . after raising indices of the principal spinors we have @xmath101 , etc .. under stereographic projection @xmath102 represents the antipodal map on the unit sphere . thus we see that the spinor @xmath103 is represented by a set of points antipodal to those representing @xmath42 . furthermore , the majorana representation of a normalized state with @xmath104 , corresponding to @xmath105 , consists of pairs of antipodal points ( and is thus only possible for integer spin ) . this fact will be useful in minimizing the interaction energy ( recall the form of eq . ( [ spin2hint ] ) ) . the transformation @xmath106 is in fact the ( anti - unitary ) operation of time reversal . the use of the majorana representation to visualize spin ordering in a bose gas was suggested in ref . . let us first see how the phases of the spin 1 gas discussed in section [ sec : spin1phases ] appear in this representation , before moving on to the spin 2 case . as mentioned in section [ sec : hint ] , ( [ spin1hint ] ) , the interaction hamiltonian in the spin 1 case may be written @xmath107 for @xmath45 we should maximize @xmath108 . based on the discussion of the previous section , this corresponds to placing the two points antipodally in the majorana representation . it is evident that the corresponding spin 1 spinor is just the symmetric @xmath65 state with respect to the resulting axis . this is just the polar state described before . for @xmath46 @xmath108 can be set to zero by making the two principal spinors equal since @xmath109 . it is clear that this represents the ferromagnet , being a maximally polarized ( or _ coherent _ ) spin state , a result that generalizes to arbitrary @xmath19 . suppose that in eq . ( [ spin2hint ] ) @xmath110 . the corresponding term in the hamiltonian can be fully satisfied by states with @xmath111 . by inspecting the spin 2 angular momentum matrices @xmath112 it is easy to convince oneself that up to rotation the most general state satisfying these conditions is @xmath113 the principal spinors form a polyhedron with four identical triangular faces known as a _ disphenoid _ ( see fig . [ fig : disphenoid ] ) . , @xmath114 in eq . ( [ disphenoid ] ) . bottom left : rectangular state with @xmath115 , @xmath114 . bottom right : tetrahedron with @xmath116 . [ fig : disphenoid],title="fig:",scaledwidth=30.0% ] + , @xmath114 in eq . ( [ disphenoid ] ) . bottom left : rectangular state with @xmath115 , @xmath114 . bottom right : tetrahedron with @xmath116 . [ fig : disphenoid],title="fig:",scaledwidth=20.0% ] , @xmath114 in eq . ( [ disphenoid ] ) . bottom left : rectangular state with @xmath115 , @xmath114 . bottom right : tetrahedron with @xmath116 . [ fig : disphenoid],title="fig:",scaledwidth=20.0% ] to fix the parameters @xmath117 , @xmath118 in eq . ( [ disphenoid ] ) we turn to the third term in eq . ( [ spin2hint ] ) . if @xmath46 , this term is minimized by placing four points in two antipodal pairs so that @xmath119 . the principal spinors form a rectangle , corresponding to @xmath115 in eq . ( [ disphenoid ] ) . the aspect ratio of the rectangle varies with @xmath117 , with @xmath120 being a square and @xmath121 corresponding to a pair of points at either pole . the fact that the energy is minimized for _ any _ @xmath117 in this parameter regime is rather surprising , and we will return to it briefly below . this state will be referred to as _ rectangular _ in the following ( in ref . it was called nematic since @xmath122 . for @xmath45 @xmath123 should be minimized . this can be done by taking @xmath116 . the result is a regular tetrahedron . note that this state has @xmath124 . for @xmath125 one must compare the energy of the rectangular state with that of the ferromagnet . the resulting phase diagram can be found in refs . . note that while the rectangular state maximizes the magnitude of the quadratic scalar @xmath126 the tetrahedral state maximizes the cubic invariant @xmath127 the inclusion of a sextic term proportional to @xmath128 in the interaction energy would lift the accidental degeneracy in the @xmath117 parameter discussed above . a microscopic derivation of such a term is discussed in refs . , with a positive sign favoring the square state ( @xmath120 ) and a negative sign the uniaxial state in which two pairs of points coincide ( @xmath121 ) . the inclusion of such a term in our formalism is a straightforward matter , and we will not discuss it further . the rectangular and tetrahedral states evidently have certain discrete symmetries that are rather hard to discern by inspection of eq . ( [ disphenoid ] ) , and indeed went unnoticed in the earliest works on the spin 2 condensate @xcite . it appears that the term ` cyclic ' used in several works to describe the tetrahedral phase is a consequence of a misidentification of the symmetry . this illustrates the utility of the majorana representation in the visualization of spin order . in section [ sec : spin1phases ] we identified the order parameter manifolds of the phases of the spin 1 gas . with the help of the majorana representation we can now do the same for the spin 2 case . roughly speaking , we expect the manifold to consist of all configurations related to those of section [ sec : spin2phases ] by rotation . some rotations will leave the configuration of principal spinors unchanged , however , so the manifold can not simply be identified with the rotation group @xmath56 . in mathematical terms the problem is to determine the _ orbits _ of a reference spinor under the action of the spin @xmath19 representation of the rotation group . if we ignore the phase of the spinor for a moment , so that we are considering orbits in @xmath71 , this problem can be solved using the majorana representation by considering the manifold of constellations generated by all possible rotations @xcite . for instance , the ferromagnetic spin 1 state has orbit in @xmath129 equal to @xmath62 ( and this is true in general for any spin @xmath19 coherent state ) , while for the polar state we have @xmath62 with antipodal points identified : the real projective plane @xmath69 . in general if one finds a configuration of principal spinors unchanged under some subgroup of @xmath130 ( the _ stabilizer _ subgroup of a constellation ) , then the orbit is given by @xmath131 . thus in the case of spin 1 , @xmath132 for the ferromagnet and @xmath133 in the polar phase , since in the latter case a parity transformation also leaves the points unchanged . the reader will notice that these are _ not _ the order parameter manifolds identified in section [ sec : spin1phases ] for the spin 1 case . we have neglected the phase of the spinor , which is a real degree of freedom . we might then guess that any spinor on the order parameter manifold can be written @xmath134 where @xmath135 is the spin @xmath19 representation of the rotation @xmath8 and @xmath136 is some reference spinor corresponding to the phase in question . this does not mean that the order parameter manifold is @xmath137 , because the stabilizer subgroups mentioned above leave the spinor corresponding to a particular constellation unchanged up to a phase . we denote these phases as @xmath138 , with @xmath139 . they must form a one - dimensional unitary representation of @xmath140 @xmath141 and allow us to make the identification @xmath142 showing that the order parameter manifold is @xmath143 , where the tilde is to denote the action of @xmath140 on @xmath137 : @xmath144 the simplest case to consider is the polar phase with @xmath133 , for which the only non - trivial phase is a @xmath145 associated with the parity transformation . if @xmath140 were @xmath146 with only trivial phases , the ground state manifold would be @xmath147 , with the first factor coming from the @xmath148 and the second from @xmath149 . the parity transformation and the associated minus sign are responsible for the identification @xmath150 already discussed in section [ sec : spin1phases ] . to turn to a less trivial example , let us see how this works for the case of the tetrahedral phase . in this case @xmath151 , the symmetry group of the tetrahedron . a tetrahedron has 3 orthogonal 2-fold axes , and 4 3-fold axes . for the representative spinor given earlier @xmath152 the @xmath153 axis is aligned with one of the 2-fold axes . one easily verifies that @xmath153 axis rotations through @xmath154 leave the spinor unchanged . alternatively , we can align one of the 3-fold axes with the @xmath153 axis with the choice ( see fig . [ fig:3axis ] ) @xmath155 ) . [ fig:3axis],scaledwidth=40.0% ] ( this is most easily seen by considering the majorana polynomial , which has a root at @xmath156 ) . now a rotation through @xmath157 is seen to reproduce the same spinor but with phase factors @xmath158 . it is not hard to verify that these phases form a one - dimensional representation of @xmath159 ( the other non - trivial one - dimensional representation comes from changing the sense of the 3-fold axes ) . the topological properties of the resulting space @xmath160 , and the implications for superfluid vortices in the tetrahedral phase , were discussed in ref . . for the rectangular phase @xmath161 in general , but @xmath162 for the square case ( @xmath120 ) . here @xmath163 denotes the dihedral group . the only non - trivial phases occur in the latter case , as may be seen by considering the value @xmath164 , when the four points lie on the equator of the bloch sphere . then we have @xmath165 and a @xmath166 rotation about the @xmath153 axis is seen to give rise to a @xmath145 . this construction generalizes readily to other ordered states of arbitrary spin once the corresponding stabilizer subgroups and phases are identified @xcite . note that when @xmath140 is discrete , as for the spin 2 phases other than the ferromagnet , the order parameter manifold is 4 dimensional . next we turn to the local properties of the order parameter manifold . the inner product naturally endows this space with a metric @xcite . using the parametrization eq . ( [ gen_param ] ) consider two states @xmath167 and @xmath168 related @xmath169 and @xmath170 with @xmath171 an infinitesimal rotation corresponding to @xmath172 we find the squared distance between these two states to be @xmath173 is @xmath174 eq . ( [ distance ] ) makes it clear that if @xmath175 , rotations and phase changes are coupled together . there is an arbitrariness in the way the phase is apportioned between the two factors in eq . ( [ gen_param ] ) that amounts to a choice of gauge . this gauge structure was discussed in ref . for the case of the ferromagnet , the only one of the phases discussed in section [ sec : spin ] with @xmath175 . the discussion of the general case is relegated to appendix [ sec : gauge ] . our main interest is in the other phases having @xmath176 , for which eq . ( [ distance ] ) decouples into separate contributions from the change in phase and the rotation , with the latter being characterized by the metric tensor @xmath177 . note that @xmath177 is simply related to the nematicity @xmath178 in eq . ( [ nematicity ] ) . the notion of distance described by the metric tensor is left invariant i.e. preserved if states @xmath167 are mapped by @xmath179 for some @xmath180 , but not right invariant , under which @xmath181 . focusing now on the spin 2 case , we evaluate the metric for the state eq . ( [ disphenoid ] ) @xmath182 for the tetrahedral phase ( @xmath116 ) @xmath183 , showing that the order parameter manifold has a left and right invariant geometry . this is perhaps not surprising given the highly symmetric arrangement of points in the majorana representation . for the rectangular phase ( @xmath115 ) , we see that in the case @xmath121 , @xmath184 . the physical meaning is clear : because we have two points at either pole the stabilizer subgroup is @xmath185 , so that the order parameter manifold is only three dimensional . the same holds true for the spin 1 polar phase . we will see that the metric plays a crucial role in fixing the dynamics on the order parameter manifold . having characterized the order parameter manifold for the spinor condensates , we are almost ready to study the dynamics on that manifold . it remains to identify the _ conjugate variables_. we expect these to be coupled by the first term of eq . ( [ gp_action ] ) , which expresses the conjugacy of @xmath42 and @xmath186 . substituting the parametrization eq . ( [ gen_param ] ) into that term gives @xmath187 where @xmath188 are the components of the angular velocity . by analogy with rigid body dynamics , we refer to this as the ` body frame ' angular velocity : eq . ( [ conj_param ] ) shows that it is conjugate to the ` body frame ' angular momentum @xmath189 , i.e. that of the unrotated state . as is well known , the time integral of the term @xmath190 has an alternative interpretation as the berry phase associated with the time evolution of @xmath20 . in this context the formula eq . ( [ conj_param ] ) appears in ref . . thus in the cases of interest where @xmath191 the variables conjugate to the rotations are nonzero only as one deviates from the order parameter manifold . to account for these deviations we generalize the parametrization eq . ( [ gen_param ] ) to @xmath192 where @xmath193 is a state with @xmath194 . note that @xmath195 @xmath196 is angular momentum in the ` lab frame ' . to find the deviation corresponding to @xmath193 , let us consider the state @xmath197 where @xmath198 is some normalization factor , and @xmath199 it is not hard to see that to quadratic order the normalization takes the form @xmath200 and that to this order @xmath194 , as required . the effect of the @xmath201-distortion on the tetrahedral state is shown in fig . [ fig : distort ] . due to boosts in the @xmath153-direction [ fig : distort],title="fig:",scaledwidth=23.0% ] due to boosts in the @xmath153-direction [ fig : distort],title="fig:",scaledwidth=23.0% ] using the parametrization eq . ( [ gen_param_conj ] ) in eq . ( [ conj_param ] ) gives then @xmath202 to first order in @xmath201 . we make the following aside . the matrix @xmath203 that acts on @xmath136 is a polar decomposition of the @xmath204 representation of an element of @xmath97 @xcite . now @xmath205 is isomorphic to the connected lorentz group . this isomorphism has a beautiful physical counterpart . the lorentz transformations have a natural action on the celestial sphere , the space of light rays on the past ( say ) light cone . further , the elements of @xmath97 @xmath206 have a natural action on the bloch sphere : @xmath207 , corresponding to a mbius transformation on the stereographic coordinates @xmath208 @xmath209 ( the fact that @xmath210 and @xmath211 correspond to the same mbius transformation accounts for the @xmath212 quotient above ) . remarkably , group elements related by the isomorphism mentioned above correspond to identical transformations of the sphere @xcite ! the deformations of eq . ( [ l_deform ] ) are just those generated on the celestial sphere by a boost of the frame of reference , which in particular determines the aberration of the fixed stars . thus the name ` stellar representation ' sometimes used to describe the majorana picture of spin states is more than picturesque . with the parametrization eq . ( [ gen_param_conj ] ) we have accounted for six spin degrees of freedom ( plus the superfluid phase ) : three rotation variables parametrizing the order parameter manifold and three conjugate variables . the remaining @xmath213 variables required to specify the spin state do not correspond to any broken symmetries and must describe gapped modes . for the spin 2 case , the missing two degrees of freedom can be readily found by considering the ratio of the two ( complex ) @xmath97 invariants in eqns . ( [ quad_inv],[cubic_inv ] ) . @xmath214 the powers are chosen so that the normalization in eq . ( [ l_deform ] ) drops out . by construction @xmath27 is unchanged for all @xmath201 and @xmath8 starting from the state eq . ( [ disphenoid ] ) , and could be expressed in terms of @xmath117 and @xmath118 . this concludes our discussion of the spin degrees of freedom in a spinor condensate . with this background , we will see that the derivation of the low energy lagrangian is extremely straightforward . we are going to use the parametrization eq . ( [ gen_param_conj ] ) in the lagrangian eq . ( [ gp_action ] ) . in doing so , we are treating the problem as as _ constrained _ dynamical system , in which deviations from the ground state manifold associated with the gapped modes described in the previous section are assumed to be infinitely stiff . the lagrangian for the spin degrees of freedom then takes the form @xmath215 where @xmath216 @xmath217 , in analogy to the earlier definition of @xmath218 . we could additionally allow for a variation @xmath219 in the density of the gas , which is conjugate to the phase @xmath220 , to describe the density modes , but will not do so here . in eq . ( [ l_exp ] ) we have not included the gradient of the conjugate variables ( including @xmath219 ) in the part arising from the kinetic energy , as such terms can be neglected in the long wavelength limit of interest . the conjugate variables do however appear in the interaction term @xmath221 the notation is of course chosen to emphasize the rigid body analogy . the precise form of the ` inertia tensor ' will depend on the phase under consideration ; we discuss the spin 2 phases for definiteness . in that case the interaction hamiltonian has the form eq . ( [ spin2hint ] ) . computing the quadratic variation of this expression with the conjugate variables is facilitated by the @xmath97 invariance of the third term : its variation is determined solely by the normalization in eq . ( [ norm ] ) . we obtain @xmath222 it is then straightforward to eliminate the @xmath201 degrees of freedom using the equation of motion , obtained from eq . ( [ l_exp ] ) @xmath223 to obtain the final result @xmath224 eq . ( [ l_exp_l_elim ] ) represents the main conclusion of this work , being the low energy lagrangian for the spin degrees of freedom of the condensate . it takes the form of a sigma model in @xmath225 dimensions with target space @xmath56 ( if we ignore the possibility of vortices in the superfluid phase @xmath220 the subtle global structure of the ground state manifold discussed in section [ sec : global ] can be ignored ) . in the case of the tetrahedral phase , the metric tensor @xmath226 , and the lagrangian eq . ( [ l_exp_l_elim ] ) becomes that of the _ principal chiral model _ , having independent left and right @xmath56 symmetries . an alternative form for eq . ( [ l_exp_l_elim ] ) follows from noting that , if @xmath227 @xmath228\ ] ] with @xmath229 thus we have @xmath230\ ] ] by expressing the matrix elements of @xmath8 in terms of an orthonormal triad @xmath9 , with @xmath10 this may be written @xmath231\ ] ] recall that for the spin 1 polar phase and for the special value @xmath121 in the spin 2 rectangular phase the metric tensor has one zero eigenvalue and two equal non - zero eigenvalues . as a result both @xmath232 and @xmath233 have _ two _ vanishing eigenvalues , and eq . ( [ l_alt2 ] ) reduces to the usual @xmath234 sigma model . we find the equations of motion corresponding to eq . ( [ l_exp_l_elim ] ) by writing the variation @xmath235 this gives @xmath236 substitution into eq . ( [ l_exp_l_elim ] ) leads to the equations of motion . @xmath237 it follows from their definition that @xmath238 satisfy the ( maurer - cartan ) equation @xmath239 note that if @xmath238 is interpreted as a non - abelian gauge field , the above condition corresponds to vanishing field strength , and to the absence of topological defects . the equations of motion can be linearized by ignoring the right hand side of eq . ( [ mc_eqn ] ) , allowing us to write @xmath240 . the linear equations of motion following from eq . ( [ eom ] ) are then @xmath241 a wave equation describing the propagation of three spin wave modes with velocities @xmath242 for the spin 2 case eq . ( [ spin2_inertia ] ) gives @xmath243 with @xmath244 given by the diagonal elements of eq . ( [ disphenoid_metric ] ) with @xmath115 . for the square case ( @xmath245 ) we have @xmath246 and @xmath247 these results check with ref . , which also includes the normal phonon mode as well as the mode associated with variations of the @xmath117 and @xmath118 parameters in eq . ( [ disphenoid ] ) . in the case of the rectangular phase this latter mode appears gapless in mean field theory , but as explained in section [ sec : spin2phases ] this is the result of an accidental degeneracy that does not persist in the next order of approximation . we have achieved our goal of providing a framework in which the parameters entering the low energy spin lagrangian of an arbitrary ordered state of a spinor condensate ( with @xmath191 ) may be easily calculated . though we focused on the spin 2 states , any other state can be treated by the same method once the problem of minimizing the mean field energy is solved . the extension of the present formalism to spin ordered mott insulating phases in which the phase variables are quantum disordered does not present any particular difficulties . perhaps the most interesting problem that we have not addressed in detail relates to the character of topological defects in these systems . the occurrence of nonabelian stabilizer subgroups means that vortices have very novel characteristics @xcite . we mention one consequence of our work for the _ quantum _ description of such vortices . the phase factors associated with elements of the stabilizer subgroups that were discussed in section [ sec : global ] will appear in the path integral when vortices are present , as may be seen from eq . ( [ conj_param ] ) . consider an imaginary time path integral with fields obeying the boundary condition @xmath248 if , as we go from @xmath249 , the field at a point @xmath250 is subject to a rotation that evolves from @xmath251 , for @xmath252 , the @xmath220 variable must increase @xmath253 in order to ensure periodicity of the fields , leading to a phase factor @xmath254 in the path integral ( @xmath255 is the density ) . ref . discusses the effect of these phases for the simplest case of the spin 1 polar phase ( or rather the mott insulating phase based upon it ) , where they are @xmath256 and the defects are abelian . the nonabelian case remains unexplored . the support of the nsf under grant dmr-0846788 is gratefully acknowledged , as well as several useful conversations with ryan barnett and gil refael . in the general case the distance between states on the ground state manifold has the form eq . ( [ distance ] ) . as mentioned in the text , there is an arbitrariness in the apportioning of phase between the two factors of eq . ( [ gen_param ] ) that belies a natural gauge structure . in other words , eq . ( [ gen_param ] ) is unchanged under @xmath258 for some arbitrary function @xmath259 . let us define a ( berry ) vector potential @xmath260 where @xmath261 denotes an expectation in the state @xmath136 , @xmath262 , and @xmath263 denotes the exterior derivative ( we find it convenient to use the language of differential forms ) . the vector potential allows us to define a covariant derivative @xmath264 . the metric tensor @xmath265 can then be cast in the form the two terms in eq . ( [ gen_metric ] ) are manifestly gauge invariant under the transformation eq . ( [ gauge_trans ] ) . the gauge invariant metric tensor @xmath267 that appears in the first term takes the form @xcite @xmath268 in the second term , @xmath269 by itself is gauge dependent , but the associated field strength is not @xmath270 where in the second step we have used the maurer - cartan equation eq . ( [ mc_eqn ] ) . ( [ field_strength ] ) has a more familiar form , as may be seen by introducing a unit vector @xmath271 parallel to @xmath272 . then we have @xmath273 where @xmath274 , and in the penultimate line we have used the fact that the determinant of the rotation matrices is unity . as a result @xmath275 which generalizes the mermin - ho relation in eq . ( [ v_fix ] ) to an arbitrary spin state . an important example is provided by the ferromagnet , for which the state @xmath276 is a fully spin polarized ( coherent ) state , and we have @xmath277 then the first term of eq . ( [ gen_metric ] ) takes the form @xmath278 the resulting metric sets the form of the hamiltonian in ref . .
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we present a derivation of the low energy lagrangian governing the dynamics of the spin degrees of freedom in a spinor bose condensate , for any phase in which the average magnetization vanishes .
this includes all phases found within mean - field treatments except for the ferromagnet , for which the low energy dynamics has been discussed previously .
the lagrangian takes the form of a sigma model for the rotation matrix describing the local orientation of the spin state of the gas .
= 1
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negative thermal expansion ( nte ) is a rare and counter - intuitive phenomenon found primarily in low density materials with crystal structures that are networks of linked coordination polyhedra . the study of these materials is not only of fundamental scientific importance , but also has many technological applications such as aerospace technologies @xcite , optics @xcite , and electronics @xcite . while much effort has been put into finding new materials and investigating the origin of nte in them , much less attention has been paid to the change in nte behaviour subject to heating and stress , which holds great importance for the possible applications of the material . for example , due to stresses and heating , problems such as phase transitions of the nte filler and thermal expansion misfit between the nte filler and matrix are always encountered in designed composites with tailored thermal expansion @xcite . in this paper , we conduct a simulation study of zn(cn)@xmath0 focusing on the pressure and temperature effects on its negative thermal expansion . we chose this material for several reasons . firstly , zn(cn)@xmath0 is a well - known representative nte material @xcite . it has a framework structure consisting of tetrahedral groups of atoms linked by diatomic rods of c n and has exceptionally large isotropic nte of @xmath1 mk@xmath2 ( twice as large as that of zrw@xmath0o@xmath3 @xcite ) . secondly , the material shows a variety of exotic properties in experiments @xcite , including reduction of its nte on heating , pressure - enhanced thermal contraction , and pressure - induced softening , none of which are fully understood . thirdly , with previous dft calculations of zn(cn)@xmath0 @xcite explaining the origin of nte of the material in terms of grneisen theory , it would be useful to draw a clear link between the values of the grneisen parameters in energy space and the structural vibrations in real space with full anharmonicity ( which should be important in such an nte system ) based on theoretically reproducing the aforementioned exotic properties . here , we have built a zn(cn)@xmath0 potential model based on first - principles calculations . lattice - dynamic calculations and large - scale molecular dynamics ( md ) simulations were carried out for the material using this model which was justified by comparing against the available experimental data . the results in both energy and real space provide us fundamental clues to understand the nte as well as the related exotic behaviours of zn(cn)@xmath0 . c c c c potential & form of the potential & type of bond & values of parameters + & & zn c & @xmath4 , @xmath5 , @xmath6 + & & zn n & @xmath7 , @xmath8 , @xmath9 + & & c zn c & @xmath10 , @xmath11 + harmonic three - body potential & @xmath12 & c zn n & @xmath13 , @xmath11 + & & n zn n & @xmath14 , @xmath11 + & & zn c n & @xmath15 + & & zn c & @xmath16 + & & c c & @xmath17 , @xmath18 , @xmath19 + buckingham potential & @xmath20 & c n & @xmath21 , @xmath22 , @xmath23 + & & n n & @xmath24 , @xmath25 , @xmath26 + [ table1 ] we started with total - energy calculations for a [ zn(cn)@xmath27@xmath28 cluster . the @xmath29 charge comes from the fact that every zinc is shared by four neighbour atoms ( c or n ) , and that the bond should be highly ionic according to an initial judgment of zinc having much smaller electronegativity than carbon and nitrogen . four cn@xmath30 molecular ligands around each zn@xmath31 would give us a @xmath29 charge on the cluster . geometry optimization of the cluster resulted in a perfect tetrahedral conformation . total energies for different configurations of the cluster ( with bond stretching and angular distortions ) were computed using dft in gamess(us ) with the pbe0 functional @xcite . correlation - consistent basis sets up to aug - cc - pvqz were tested , and an aug - cc - pvtz basis set was found to have sufficient accuracy and no significant basis set superposition errors ( bsse ) at different configurations . various interatomic potential forms were then fitted to the calculated energy curves to obtain the initial potential parameters . the short - range interactions are the morse potential for zn c / n , a harmonic three - body - bond - bending term for c / n zn n / c and a linear - three - body term for the angular distortion of zn n n / c . the long - range van der waals interactions are described by a buckingham potential with parameters from williams @xcite . two types of clusters with zn c n order and zn c order are used . the multipoles on each cluster were calculated by distributed multipole analysis ( dma ) @xcite using cam @xcite . the effective point charges on the atoms were then obtained by fitting to the electrostatic potential from the rank 4 ( hexadecapole ) distributed multipoles using the mulfit program @xcite . [ fig1 ] shows the difference in the electrostatic potential between the point - charge model and the dma result . the root mean square of the difference is less than @xmath32 kj / mol and @xmath33 kj / mol for clusters of zn c n and zn c , respectively , corresponding to about 1% relative difference in the electrostatic potential around the clusters . the averaged effective point charges on each atom are ( in electron units ) @xmath34 for zn , @xmath35 for c and @xmath36 for n. the initial potential model was then refined by refitting to dft energy surfaces of various cluster configurations . the point charges were fixed during this process . this gave us final potential parameters that implicitly incorporate the effects of higher - ranking multipole moments and atomic polarization . due to the high strength of the c n bond @xcite , this group was treated as a rigid rod in all cases . the potentials with their parameters are listed in table [ table1 ] . harmonic lattice dynamics ( hld ) and quasi - harmonic lattice dynamics ( qhld ) calculations were carried out using gulp @xcite . molecular dynamics ( md ) simulations were performed using dl@xmath37poly @xcite for a @xmath38 supercell containing @xmath39 atoms with periodic - boundary conditions . a constant stress constant temperature ( n@xmath40 t ) ensemble with a nos - hoover thermostat @xcite was used . the long - range coulomb interactions were calculated using the ewald method with precision of 10@xmath41 . the equations of motion were integrated using the leapfrog algorithm with a time step of @xmath42 ps . a total of @xmath43 time steps were used to achieve equilibration . at different temperatures and pressures , snapshots of atomic trajectories after equilibration were recorded every @xmath44 ps up to a total of @xmath45 ps for the follow - up analysis . both ordered model with @xmath46 symmetry and disordered model with @xmath47 were used . the latter was constructed by randomly switching c and n atoms in the supercell . we compared some basic quantities obtained from the model against experiment . the optimized structure gave the cell parameter as @xmath48 , and the zn c / n bond length as @xmath49 compared to @xmath50 and @xmath51 from an experiment at @xmath52 k @xcite . the model is found to be stable in lattice - dynamic calculations , and the calculated phonons are in good agreement with the spectroscopy data @xcite , as shown in table [ freqtable ] . the dispersion curves shown in fig . [ fig : freqsoft ] generally agree well with previous dft calculations @xcite . .calculated phonon frequencies at the @xmath53 point for both cn ordered and cn disordered ( virtual crystal ) models . experimental infra - red and raman @xcite data and dft calculated results @xcite are provided for comparison . frequencies are in cm@xmath2 . [ cols= " < , < , < , < , < , < " , ] [ freqtable ] it is worth noting the close similarity between the dispersion relations of wave vectors along @xmath53x r and r m@xmath53 , as well as the near - mirror symmetry of the dispersion curves along @xmath53r ( the middle line of @xmath53r is the mirror line ) . this makes sense given the existence of the two interpenetrating cristoablite - like networks in the material . for the lowest - frequency mode at wave vector r , the two networks would translate like acoustic modes but out of phase with each other , leading to a dispersion relation that has the appearance of an acoustic mode and a positive grneisen parameter but with a non - zero band gap . the frequency of this mode is 0.16 thz , lower than the values of 0.69 thz and 0.48 thz from the dft calculations in ref . and ref . , respectively , suggesting a softer long - range interaction between the two networks in our model . [ fig : freqsoft ] [ fig3 ] [ fig3c ] the calculated nte curves of zn(cn)@xmath0 under different pressures from @xmath55 to @xmath56 gpa with increments of @xmath57 gpa are shown in fig . [ fig3 ] . the pressure - enhanced @xmath58 ( averaged over @xmath45@xmath59 k ) at @xmath55 , @xmath60 and @xmath61 gpa are @xmath62 , @xmath63 and @xmath64mk@xmath2 , respectively , compared to the experimental values @xcite of @xmath65 , @xmath66 and @xmath67mk@xmath2 at the corresponding pressures . the md successfully captured the gradual reduction of nte on heating which has been observed in x - ray scattering @xcite . according to the md , @xmath68 is @xmath69 mk@xmath2 at 300 k , much lower than the value at 25 k , @xmath70 mk@xmath2 . experiment @xcite has confirmed that zn(cn)@xmath0 exists in a disordered form with @xmath47 symmetry , i.e. the carbon and nitrogen atoms are randomly placed on their symmetric sites in the structure , compared to the ordered model with @xmath46 symmetry . nte of the disordered model from the md calculation is displayed as symbol plots in fig . [ fig3c ] compared to that of the ordered model in solid lines . clearly , there is no significant difference between the nte of these two systems . pressure and temperature dependence of the bulk modulus of zn(cn)@xmath0 . results calculated using the @xmath71@xmath72 data from md simulations and standard thermodynamic relations . pressure has far less influence on the bulk modulus than temperature.,width=309 ] the bulk modulus @xmath73 is directly related to the coefficient of thermal expansion @xmath74 according to the classical form @xmath75 where @xmath76 is the specific heat and the sum is over all modes with frequencies @xmath77 . @xmath78^ { - 1}$ ] is the bose - einstein relation with @xmath79 the boltzmann constant . the overall grneisen parameter @xmath80 is calculated by summing over all mode grneisen parameters weighed with their contribution to the specific heat . at 300 k , the values of the bulk modulus @xmath81 and its first derivative with respect to pressure @xmath82 at zero pressure obtained by using the @xmath83rd - order birch - murnaghan equation of states to fit to the isotherms calculated from our md are @xmath84 gpa and @xmath85 . both are in good agreement with the experimental values of @xmath86 gpa and @xmath87 , respectively @xcite . however , the values of the bulk modulus obtained from the previous dft calculations @xcite are much higher than the experiment . we obtained the bulk modulus of zn(cn)@xmath0 at different temperatures and pressures by numerically computing derivatives of the @xmath71@xmath72 data from the md simulations . as shown in fig . [ fig5 ] , on cooling from 300 k to 50 k , @xmath73 increases by 26% compared to the experiment of 15% @xcite . note that @xmath73 does nt change much with pressure , but changes largely with temperature . referring to fig . [ fig3 ] , we found that a @xmath88% decrease in volume caused by heating at zero pressure corresponds to as much as @xmath89% decrease in the bulk modulus , while the same amount of volume decrease caused by compression would only reduce the bulk modulus by less than @xmath88% . this means that the bulk modulus of the material not only depends on the volume change per se , but also on the means of changing the volume by heating or compression . this breaks birch s law of corresponding states @xcite . the same anomaly has been observed experimentally in zrw@xmath0o@xmath3 , where the bulk modulus increases by @xmath90% on cooling from @xmath59 k to @xmath91 k @xcite . pressure and temperature - dependence of zn(cn)@xmath0 volume , calculated using md . the inset shows the change in volume with pressure up to @xmath92 gpa at @xmath59 k. a high - pressure phase settles down after the clear discontinuity at @xmath93 gpa.,width=302 ] according to the thermodynamic expressions of @xmath94 and @xmath95 combined with maxwell relation @xmath96 , the pressure enhanced nte of the material follows naturally from the relation @xmath97 since the temperature dependence of @xmath73 is negative as shown in fig . [ fig5 ] , @xmath98 would become more negative on compression . this is consistent with what seen in the positive expansion materials where @xmath99 so that @xmath98 would decrease on compression . we found that the value of @xmath82 from a lattice - dynamic calculation using gulp @xcite is @xmath100 compared to its negative value at 300 k. this suggests that all the mechanical contributions at @xmath101 are from the zn c / n bonds that become stiffened on compression . with elevated temperature , one can imagine that the zn c / n n / c angle flexing starts to contribute to the change of volume on pressure . this mechanism costs much less energy than compressing the zn c / n bonds as suggested by values of the parameters in the morse potential and the linear - three - body potential of the model ( table [ table1 ] ) . as a result , the bulk modulus decreases on compression hence the negative @xmath82 , i.e. pressure - induced softening of the material . at high temperature , @xmath82 is expected to become less negative on heating due to the rising energy cost of further increasing the zn c / n n / c angle vibrational amplitude on compression . the temperature dependence of @xmath82 is specially discussed in our other work @xcite . in fig . [ fig6 ] , simulations were conducted for zn(cn)@xmath0 at different temperatures in a much broader range of pressure from @xmath102 to @xmath92 gpa , and there is clearly a phase transition of the material caused by compression at each temperature . with elevated temperature , the phase - transition pressure indicated by the discontinuity in the volume change increases , implying that the phase transition may be triggered by some soft modes that can be stabilized on heating due to the anharmonic term in their frequencies . indeed , as shown in fig . [ fig : freqsoft ] , if we compare the dispersion curves calculated at a pressure beyond 1.0 gpa ( in light red ) to that at zero pressure ( in black ) , we find softening of the acoustic modes around the zone boundaries , especially the modes at m and the mid - point of @xmath53r @xmath103 which are the first ones to become unstable . the concurrent softening of the optic modes ( @xmath104 thz ) directly above these acoustic modes suggests a possible hybridization between the acoustic modes and the optic modes , resulting in a @xmath105 energy behaviour @xcite at @xmath106 . same result was found for the disordered model . the inset of fig . [ fig6 ] shows the change of volume with pressure up to @xmath92 gpa at @xmath59 k. the volume discontinuities at @xmath107 and @xmath93 gpa may suggest hysteresis in the phase transition . we found that the new high - pressure phase is orthorhombic with @xmath108 space group ( @xmath109 , @xmath110 , @xmath111 ) . this is compared to an orthorhombic phase with @xmath112 space group found beyond 1.3 gpa in the x - ray diffraction experiment @xcite . the zn(cn)@xmath0 phonon density of states at 297 k. solid lines are data from our md calculations ; filled circles are neutron scattering experimental results@xcite . the two agree well with one another.,width=302 ] the fourier transformation of the atomic velocity auto - correlation function ( vacf ) gives us the phonon density of states ( dos ) @xcite of the material . to obtain the dos at different pressures and temperatures , we used the trajectory data of atoms from md to calculate the vacf of zn(cn)@xmath0 . the correlation function @xmath113 at time @xmath114 can be expressed as , @xmath115 where @xmath116 is the velocity component of the @xmath117th atom . @xmath118 is the time interval of @xmath44 ps . @xmath119 is the total number of time steps . the system was simulated for a total of @xmath45 ps which corresponds to @xmath120 . the correlation function was calculated for each atom with a time length of @xmath121 ps ( @xmath122 ) . a gaussian profile was used before fourier transformation to suppress the ripple effect caused by time cut - off . the vacf of angular velocities of the c n rigid rods rotating about their center of mass was also calculated . the fourier transformation then gave us the dos of the pure rotational modes of these rigid rods . to obtain the phonon spectrum , the calculated dos was first multiplied by a weighting factor @xmath123 containing the scattering length @xmath124 and the atomic mass @xmath125 of the @xmath126th atom . then , in order to mimic experimental resolution @xcite , the dos is convolved with a gaussian with fwhm of @xmath127% of the energy transfer . [ fig7 ] shows the good agreement between the calculated spectrum and the experiment @xcite . vibrational dos calculated using md at 300 k and at 0.026 gpa , 0.20 gpa , 0.46 gpa and 0.65 gpa . ( a ) shows the full dos for the cn - disordered model ; ( b ) shows the full dos for the cn - ordered model ; ( c ) shows the dos for only c n rigid rod rotations in the ordered model . the peaks at around 14 thz , corresponding to pure zn c(n ) bond flexing , increase in frequency on compression.,width=302 ] in order to draw links between the atomic vibrations and the phonon properties , we computed both the overall dos and the angular dos of rigid rod c n , as shown in fig . [ fig9 ] and [ fig8 ] . fig . [ fig9 ] ( a ) and ( b ) shows the dos at ambient temperature ( @xmath59 k ) under different pressures for the disordered and ordered model , respectively . the acoustic peak around 0.5 thz is softened on compression . the optic peaks around @xmath128 thz and @xmath129 thz follow the same trend , which means that all these modes have negative grneisen parameters ( @xmath130 ) and contribute to the nte of the material . the highest energy peak around @xmath131 thz is stiffened under compression , suggesting positive @xmath130 for the zn c / n bond flexing modes . the only difference between the disordered model and the ordered one is that the former has broader peaks , especially the merged two peaks around @xmath129 thz ( with a broadening of @xmath132 thz ) , due to the ` fluffiness ' caused by random positions of c and n atoms . vibrational dos calculated using md at 0.0 gpa and temperatures 162 k , 297 k , 473 k and 603 k. ( a ) shows the full dos ; ( b ) shows the dos for only c n rigid rod rotations . the peaks in the full dos at around 14 thz , corresponding to pure zn c(n ) bond flexing , broaden and decrease in frequency on heating.,width=302 ] by comparing fig . [ fig8](b ) with fig . [ fig8](a ) , we found that about half of the acoustic peak around @xmath133 thz is from vibrations involving rotations of the c n rod around its centre of mass ( bearing in mind that the angular dos in fig . [ fig8](b ) is renormalized , so that only the relative heights of the peaks are indicative ) . optic peaks round 2.0 and 9.0 thz are hardly changed , because much of their motions can be cast onto the rotations of the c the peak with the highest frequency in the overall dos is from pure zn c / n bond flexing it completely disappears in the angular dos . unlike the acoustic peak , this peak is softened on heating due to the thermal expansion of the zn c / n bond , and flattens with elevated temperature due to the finite life time of the corresponding phonon . full dos calculated using lattice - dynamics . ( a ) each bin is coloured according to its average grneisen parameter : red bins have an average @xmath134 value of @xmath135 ; white bins have a positive average @xmath134 . ( b ) each bin is coloured according to its average rum component . red bins are pure rums ; blue bins have zero rum character.,width=302 ] profiles of mode grneisen parameters of zn(cn)@xmath0 calculated at three different pressures from lattice dynamics : black curve is 0.0 gpa ; blue curve is 0.2 gpa ; red curve is 0.4 gpa . the grneisen parameters of the modes around 0.5 thz , 2.0 thz and 9.0 thz become more negative as elevated pressures , resulting in a more negative overall grneisen parameter as pressure is increased.,width=302 ] previously , the peak around @xmath133 thz has been found to be the major contributor to the nte in both experiment @xcite and calculations @xcite due to its large negative @xmath130 . however , if one can further identify the corresponding real - space picture of the vibrations , the reason of why the peak has the most negative @xmath130 compared to the other optic peaks can be revealed . to understand the nature of various peaks in the dos , we decided to categorize the vibrational modes in the material using the rigid unit mode model @xcite . first , we calculated @xmath130 from phonon frequencies of expanded and contracted ( @xmath1360.01% ) unit - cell volumes . we then coloured the dispersion curves according to both magnitudes and signs of @xmath130 , as shown in fig . [ fig : dispersioncuves](a ) . this representation highlights the most important phonon branches responsible for nte of the material , namely the low - lying acoustic modes around @xmath133 thz and the lowest - energy optic branches around @xmath128 thz , which both have the most negative @xmath130 and span the entire brillouin zone . this was also highlighted in the dos in fig . [ fig10](a ) , where we coloured the dos according to the mean values of @xmath130 for each frequency bin . then , we calculated the rigid unit modes ( rums ) of zn(cn)@xmath0 using the crush code @xcite as the set of eigenvectors of the dynamical matrix whose eigenvalues are zero . we took the dot product between the eigenvectors of the rums and the eigenvectors from the lattice - dynamic calculation , and then coloured the dispersion curves in fig . [ fig : dispersioncuves](b ) by the extent to which each mode eigenvector can be described in terms of correlated whole - body translations ( in blue ) and rotations ( in red ) of [ zn(c / n)@xmath27 tetrahedra . the dos in fig . [ fig10](b ) was also coloured accordingly . one can see that all the modes with negative @xmath130 that contribute to the nte of the material are rums . as shown by fig . [ fig : dispersioncuves](c ) , the acoustic modes around @xmath133 thz , like those at m and x , are characterized by translational motions of the rigid tetrahedral units , partly involving angular rotations of the c the optic modes around @xmath128 and 9.0 thz can be seen as neighbouring tetrahedral rotating against each other , as respectively shown by fig . [ fig : dispersioncuves](d ) and ( e ) . the rum nature of these modes guarantees their low frequencies and large negative @xmath130 . the relatively high frequencies of the optic rums is due to the breaking of the zn c / n n / c alignment , and the magnitudes of their negative @xmath130 suffer accordingly . study of the eigenvectors also directly revealed that the non - rum modes around @xmath127 and @xmath131 thz correspond to the pure angular vibration of c / n zn n / c within the tetrahedra and the pure bond flexing of zn c / n , respectively . we also found that , besides the negative @xmath137 , zn(cn)@xmath0 also has negative @xmath138 and @xmath139 . [ fig11 ] shows the profiles of @xmath130 up to @xmath61 gpa calculated in hld . it is clear that @xmath137 becomes more negative on compression due the contributions from the rums around @xmath133 thz , @xmath128 thz and @xmath129 thz , with the translational rums having the most negative @xmath130 ( refer to fig . [ fig10](a ) ) contributing the most . the negativity of the pressure derivatives of @xmath137 would result in negative @xmath140 at non - zero temperatures @xcite . calculated temperature dependence of the nte with both anharmonicity and quantum effects included ( ` md+qe ' , solid curve ) , together with the results from md ( dashed curve ) and qhld ( dash - dot curve ) . the former clearly has a better agreement with the experiment ( empty circle ) @xcite.,width=302 ] we found that the acoustic peak in the dos ( see fig . [ fig10 ] ) with frequencies less than 1 thz ( @xmath141 k ) accounts for half of the nte ( @xmath142 mk@xmath2 is reduced to @xmath143 mk@xmath2 when excluding these modes , calculated by eq . [ eq1 ] at 300 k in qhld ) . this suggests that even at low temperatures these modes will not be ` frozen ' out and can still be excited and contribute to nte and its relevant properties such as pressure - enhanced nte and pressure - induced softening of the material . thus , the classical md results at low temperatures would not have too much difference from the real quantum picture and can give a good qualitative agreement with experiments . ( a ) calculated normalized ( in red solid ) , implicit ( in red dash ) and intrinsic ( in red dots ) anharmonicity using cumulative distributions of dos from the md at 180 and 240 k , compared to the total anharmonicity from the experiment dos at the same two temperatures @xcite . ( b ) calculated normalized anharmonicity ( using cumulative distributions of dos from md at 180 and 240 k ) at different pressures . the important modes around 0.5 , 2.0 and 9.0 thz have positive anharmonicity with the translational rums ( around 0.5 thz ) shows the largest value compared to others . the modes around 15 thz corresponding to zn c(n ) bond flexing show negative anharmonicity . the anharmonicity of all these modes is enhanced by compression.,width=302 ] however , with the following method , we can include the effect of both anharmonicity and quantum effects in the temperature dependence of nte of the material . first , at a certain temperature , we calculate two dos from md for two adjacent volumes ( with @xmath144 difference ) . then we can obtain phonon frequencies and mode grneisen parameters by using the cumulative distributions of these two dos . finally , @xmath98 can be calculated by eq . we then repeat this process at different temperatures up to 600 k , and obtain the temperature dependence of nte of the material . as such , anharmonicity is accounted for by the use of the dos from the md , while quantum effects are included in the formalism of eq . [ eq1 ] to calculate @xmath98 . together with the direct md and qhld results , the temperature dependence of the nte from this method ( ` md+qe ' ) is shown in fig . [ figa1 ] . at low temperatures , the curve acts like the qhld result due to quantum quenching . at high temperatures , the curve becomes less negative like md due to anharmonicity , making the curve in a better agreement with the experiment . from the same analysis , we can also obtain the temperature and pressure dependence of anharmonicity . the normalized anharmonicity @xcite measuring the change of mode frequency with temperature at constant pressure is defined as @xmath145 where , on the right - hand side , the first term is the intrinsic anharmonicity and the second term is the contribution from the contraction of lattice ( implicit ) . by using the dos from the md at 180 and 240 k , the normalized anharmonicity of each mode was calculated , as shown in fig . [ figa2](a ) . the results of the important modes around 0.5 , 2.0 and 9.0 thz , as indicated by the vertical lines , agree quite well with the experiment ( using dos at the 180 and 240 k ) @xcite . the implicit anharmonicity was calculated using the mode grneisen parameters and the ` md+qe ' value of @xmath98 at 180 k. the intrinsic anharmonicity was then obtained from eq . [ eq7 ] . temperature dependence of the normalized anharmonicity at 0.0 gpa . at 25 , 280 and 560 k , calculations were conducted using the cumulative distributions of couples of dos at 25/45 k , 280/300 k and 560/590 k , respectively . the general trend is that the mode frequencies change less rapidly with temperature on heating.,width=302 ] the normalized anharmonicity at different pressures is shown in fig . [ figa2](b ) . as mentioned in the former sections , the modes around 0.5 , 2.0 and 9.0 thz , corresponding to the translational and rotational rums , respectively , are stiffened on heating with positive normalized anharmonicity . among these , the translational rums ( around 0.5 thz ) show the largest normalized anharmonicity of more than @xmath146 k@xmath2 at 0.0 gpa . the modes around 15 thz corresponding to pure bond flexing of zn c(n ) are softened on heating with negative anharmonicity . the normalized anharmonicity of all these peaks is strengthened on compression . we further calculated the normalized anharmonicity at low ( 25 k ) , medium ( 280 k ) and high ( 560 k ) temperatures to see its temperature dependence , as shown in fig . [ figa3 ] . at each temperature , two dos with temperature difference less than 30 k are used . the low - temperature ( 25 k ) value of the anharmonicity of the modes around 0.5 thz is @xmath147 k@xmath2 , in good agreement with the experimental value of @xmath148 k@xmath2 in ref . . the figure also shows that the mode frequency would change less rapidly on heating at high temperatures . the same trend is seen for the 0.5 thz peak in the experiment in ref . . distributions of ( a ) the zn c / n bond length ; ( b ) the cosine of zn c / n n / c angle distortion ; ( c ) the n / c zn c / n angle within the tetrahedral unit at 0.0 gpa.,width=302 ] in real space , variations of geometrical features of the material , such as the zn c / n bond length , the n / c zn c / n angle within the tetrahedral unit , and the zn c / n n / c angle are important for us to build a local picture of the system subject to both compression and heating . distributions of these quantities , as shown in fig . [ fig12 ] , were obtained from the atomic trajectory data of the md . the large spread of the distributions suggests large vibrations of these quantities at high temperature . the slightly expansion - biased broadening of the bond - length distribution on heating indicates an enhanced thermal expansion in the bond . we found that the deviation of the average n / c zn c / n angle in the tetrahedral unit from its equilibrium of @xmath149 is trivially small ( @xmath150 ) even at very high temperature ( @xmath151 k ) and pressure ( @xmath152 gpa ) . averaged zn c / n bond length as a function of temperature , calculated from md at pressures @xmath55 gpa , @xmath60 gpa , @xmath61 gpa and @xmath153 gpa . the bond length increases with elevated temperature . the inset shows superlinear behaviour of the bond length on heating.,width=302 ] the average bond length and the average angle distortion in zn(cn)@xmath0 as functions of both temperature and pressure are shown in fig . [ fig13 ] and [ fig13 - 2 ] , respectively . compression progressively increases the zn c / n n / c angle with elevated temperature . at very low and zero temperature , the trend is that the angle will be hardly changed by pressure . this is exactly what expected in the previous section where we suggested that , at zero temperature , the volume change of the material due to the pressure arises solely from the compression of the zn c / n bonds , resulting in positive @xmath82 . it is when the zn c / n n / c angle starts to increase under compression and contribute to the volume change that the material shows negative @xmath82 , i.e. the pressure - induced softening . the inset of fig . [ fig13 ] shows the increase of bond length with temperature . the superlinear behaviour indicates the softening of the bond at higher temperature due to the thermal expansion . in fig . [ fig13 - 2 ] , the inset shows the increase of the zn c / n n / c angle with elevated temperature . the sublinear behaviour suggests that the angle will become more rigid , which should result in a less negative @xmath82 at high temperatures @xcite . both the superlinearity and the sublinearity in the plots rely on the capture of the anharmonicity of the material . averaged zn c / n n / c angle as a function of temperature , calculated from md at pressures @xmath55 gpa , @xmath60 gpa , @xmath61 gpa and @xmath153 gpa . the angle distortion increases with elevated temperature and pressure . the inset shows the sublinear behaviour of the angle distortion on heating.,width=287 ] to see the real - space picture of rums in zn(cn)@xmath0 , we quantified the proportion of thermal - excited rigid - unit rotations at different pressures . using our gasp code @xcite , we compared ten snapshots of an md simulation , where each snapshot is separated by 2 ps , with the ideal structure . this is repeated for various pressures and temperatures . gasp , using geometric algebra , can partition the atomic displacements for every comparison made into the mean squared rigid - tetrahedron rotations , translational displacements and unit deformations . we then computed the average proportion of rigid - unit rotations at each temperature . to exclude those rigid - unit rotations due to pure topology reasons , i.e. rotations that accidentally maintain the shape of the tetrahedral unit under thermal excitation but are not because of the features of motion , we set up a benchmark calculation using an ideal cristobalite structure without any interactions other than bonds to hold zn c / n and c n . the reason to use the single - framework lattice is to avoid the problem of two interpenetrating frameworks crushing into each other in the md due to the lack of long - distance interactions . [ fig14 ] shows the results at different pressures and temperatures with coloured areas . the proportion of the rigid - unit rotation of a real silica system is given in the plot as a comparison . the figure suggests that compression will enhance the rigid - unit rotations . at ambient temperature @xmath154 k , for example , the average proportions are @xmath155 at @xmath55 gpa , @xmath156 at @xmath60 gpa , @xmath157 at @xmath61 gpa and @xmath158 at @xmath153 gpa , compared to @xmath159 of the benchmark and @xmath160 of the silica system . another important point is that , at certain pressure , the proportion of rigid - unit rotation will decrease with elevated temperature . this trend corresponds to the peaks around @xmath133 , @xmath128 and @xmath129 thz in dos stiffened on heating , as shown in fig . the grneisen parameters of these modes will consequently become less negative , and so does the coefficient of thermal expansion . plot showing the proportion of rotational rums present in zn(cn)@xmath0 as a function of temperature , calculated using gasp and data from md simulations . data is presented for four pressures plus two additional benchmark systems . the green region is the rotational component present in the ideal cristobalite structure ; the yellow region is the additional rotational component present in zn(cn)@xmath0 at 0.0 gpa ; the orange region is the additional component present at 0.2 gpa , the pink region is the additional component at 0.4 gpa ; the red region is the additional component at 0.6 gpa and the dark red region is the additional component present in amorphous silica . the results clearly show the enhancement of the rotational rums in zn(cn)@xmath0 under pressure , as well as the reduction of the rotational rums on heating.,width=302 ] there is a puzzling observation from the x - ray pair distribution function measurements of zn(cn)@xmath0 @xcite that instantaneous zn@xmath161zn distances contract less rapidly on heating than does the cell length , while crystallographically the two should be linked . the explanation relies on the nature of the acoustic modes as translational rums . the acoustic modes around @xmath133 thz are translational rums which count for half of the nte of the material as mentioned in sec . vii . unlike rotational rums around @xmath128 and @xmath129 thz that will reduce the nearest zn@xmath161zn distance , the translational rums correspond to collective translations of the neighbouring rigid units that moves zinc atoms off site and retains the distance of the nearest - neighbour zincs , as seen in fig . [ fig : dispersioncuves](c ) . this kind of vibration involves the rotation of c n rod around its centre of mass , consistent with the previous finding in the dos that part of the acoustic modes are from the c n rod rotations . we calculated the average distance of the nearest zn@xmath161zn using the the trajectory data from the md simulations . [ fig15 ] shows the temperature dependence of the average distance under different pressures . at zero pressure , the ratio between the linear coefficient of thermal expansion ( cte ) of the averaged zn@xmath161zn distance , @xmath162 , and the overall linear cte of the material , @xmath58 , is 0.67 , with @xmath163mk@xmath2 and @xmath164mk@xmath2 . this result agrees well with the experimental ratio @xcite of 0.71 , with @xmath165mk@xmath2 and @xmath166mk@xmath2 . variation of the averaged nearest - neighbour zn@xmath161zn distance , calculated using md . distances have been calculated for the temperature range of 0500 k and at pressures 0.0 gpa , 0.2 gpa , 0.4 gpa and 0.6 gpa . the inset shows the cte of the nearest - neighbour zn@xmath161zn distance as a function of pressure.,width=302 ] this study shows that almost all the modes responsible for the nte of zn(cn)@xmath0 are rums . we managed to categorize these modes in terms of their vibrational motions . the ta modes around 0.5 thz spanning to the lowest energy correspond to collective motions of zn cn zn as a rigid body , which can keep the distance of the nearest - neighbour zincs . the low energy hence the most negative grneisen parameters of these modes are due to this kind of collective motion without involving relatively high - energetic angle bending in the zn cn zn linkage . these modes contribute half of the nte of the material . the optic modes around 2.0 thz and 9.0 thz correspond to rotations of the neighbouring tetrahedral units against each other involving angle bending in the zn cn zn linkage , resulting in higher mode energy and less negative grneisen parameters . although the increase of pressure or temperature would both result in volume contraction in zn(cn)@xmath0 , the pressure and temperature dependence of the nte in zn(cn)@xmath0 are totally different . increasing temperature stiffens the low - frequency peaks and softens the high - frequency peaks in the dos ( fig . [ fig8 ] and fig . [ figa2 ] ) accompanied by the reduction of nte , while compression would soften the low - frequency peaks and stiffen the high - frequency peaks ( fig . [ fig9 ] ) resulting in nte enhancement . raising temperature slows the mode softening caused by compression , and postpones the phase transition . the enormous decrease of the bulk modulus on heating contrasts the small change of the bulk modulus on compression , which disobeys birch s law of corresponding states . the pressure and temperature dependence of the zn c / n bond and the n / c zn c / n angle are also intriguing . the large vibrational amplitude of the n / c zn c / n angle extends the zn c / n bond , which can be seen ( under constant pressure ) in fig . [ fig13 ] . the superlinearity suggests an enhancement of the thermal expansion in the bond with more distorted angle at higher temperature . the sublinearity of the average angle distortion corresponds to the stiffened rums involving the rotational vibrations of the c n rods . the ability to carry out md for zn(cn)@xmath0 using the potential model is vital in this study . it allows us to capture the anharmonicity to reproduce the exotic properties of the material , and to study their pressure and temperature dependence . the origins of various properties are revealed by linking features in both energy and real space . one example is the pressure - enhanced nte , which has been well reproduced in fig . [ fig3](a ) . this behaviour is a natural result followed by softening of the bulk modulus on heating ( eq . [ eq4 ] ) , and is linked to the feature in energy space that the modes around @xmath133 , @xmath128 and @xmath129 thz are softened under compression ( as in fig . [ fig9 ] ) , as well as the rising proportion of the rigid - unit rotations in real space on compression ( as in fig . [ fig14 ] ) . another example is the reduction of nte with elevated temperature . stiffening of the modes around @xmath133 , @xmath128 and @xmath129 thz on heating ( as shown in the dos in fig . [ fig8 ] ) makes their grneisen parameters less negative , hence the reduction of nte in the material . in real space , [ fig14 ] clearly shows the trend of reduction in the proportion of rigid - unit rotations on heating . the third example is the temperature dependence of the bulk modulus and its first derivative @xmath82 . the large decrease of the bulk modulus on heating is due to the involvement of the zn c / n n / c angular vibrations at non - zero temperature and the softening of the zn c / n bond , which can be seen both in the dos ( fig . [ fig8](a ) ) and in the real - space picture of fig . [ fig12 ] , [ fig13 ] and [ fig13 - 2 ] . as shown in fig . [ fig13 - 2 ] , the average zn c / n n / c angle starts to possess an almost linear pressure dependence at medium temperature and contributes to the pressure - induced volume change of the material . according to a simple geometrical relation @xcite , the relative decrease in cell is roughly proportional to the zn c / n n / c angle squared , hence to the pressure squared , resulting in negative @xmath82 . however , this contribution from the pressure - induced change in the zn c / n n / c angle to the volume contraction will be hindered at high temperature due to the anharmonic stiffening of the corresponding modes ( fig . [ fig8](a ) and fig . [ figa2 ] ) . on the other hand , the contribution from zn c / n bond compression will increase due to the thermal softening of the bond . the combined effect of these two trends is expected to result in a less negative @xmath82 at high temperatures @xcite . we gratefully acknowledge financial support from the cambridge international scholarship scheme ( ciss ) of the cambridge overseas trust and fitzwilliam college of cambridge university ( hf ) , nerc and crystalmaker software ltd . the md simulations were performed using the camgrid high - throughput environment of the university of cambridge . the interatomic potential was developed through our membership of the uk hpc materials chemistry consortium , funding by epsrc ( ep / f067496 ) , using the hector national high - performance computing service provided by uoe hpcx ltd at the university of edinburgh , cray inc and nag ltd , and funded by the office of science and technology through epsrc s high end computing programme . m. s. gordon , m. w. schmidt , _ in `` theory and applications of computational chemistry , the first forty years '' _ , 2005 , chapter 41 , 1167 - 1189 , c. e. dykstra , g. frenking , k. s. kim , g. e. scuseria , elsevier , amsterdam . a. j. misquitta and a. j. stone , `` cam : a program for studying intermolecular interactions and for the calculation of molecular properties in distributed form '' , 2012 . http://www-stone.ch.cam.ac.uk/programs.html . g. g. ferenczy , c. a. reynolds , p. j. winn , and a. j. stone _ mulfit : a program for calculating electrostatic potential - fitted charges _ , 1998 . may be obtained by contacting a. j. stone , email address : [email protected] .
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pressure and temperature dependence of the negative thermal expansion in zn(cn)@xmath0 is fully investigated using molecular dynamics simulations with a built potential model .
the advantage of this study allows us to reproduce the exotic behaviours of the material , including the negative thermal expansion ( nte ) , the reduction of nte with elevated temperature , the pressure enhancement of nte and the pressure - induced softening .
results of the study provide us detailed data to link the properties in the energy space and the real space , giving us insights to understand the properties and the connections between them .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
since the celebrated paper by edwards and anderson@xcite , mean - field theory of spin glass ( sg ) has been extensively investigated . the replica theory@xcite is one of the most successful achievement that has revealed the nature of the low temperature phase of mean - field sg models . parisi s pioneering work provided the replica method with implementation of replica symmetry breaking ( rsb ) . originally , @xmath0 step rsb ( @xmath0rsb ) was proposed as `` a sequence of approximated solutions '' to the true solution and the full rsb solution was derived as a @xmath4 limit . this approach has actually proven to be exact recently@xcite for the sherrington - kirkpatrick ( sk ) model@xcite . although this introduction of rsb is motivated by de almeida - thouless ( at ) condition@xcite , which is the instability of replica symmetric ( rs ) solution with respect to replica couplings , it should be noted that at instability is only one of the possible scenario for rsb@xcite and that the origin of rsb is in general model - dependent . in addition , a 1rsb solution for various mean - field sg models@xcite is stable with respect to further rsb perturbation , and @xmath0rsb rarely appears for @xmath5 . these facts suggest that there is another mechanism to break the replica symmetry and it distinguishes 1rsb from full rsb ( frsb ) . recently , the authors have shown@xcite that @xmath6-body sk model , which is a typical model to exhibit a sg transition to 1rsb phase , actually has another reason to break the replica symmetry above the gardner temperature@xcite . it is the monotonicity condition of the cumulant generating function of the free energy @xmath7 , whose limiting value at @xmath8 is the averaged free energy , rather than the at condition that causes rsb@xcite . the relevance of these conditions is reversed at the gardner temperature , where the transition between 1rsb and full rsb takes place . furthermore , it is proved that if the monotonicity is broken in the absence of external field , which ensures the smallest overlap parameter @xmath9 , then the correct 1rsb solution is given by the rs solution at @xmath10 , which is defined as the monotonicity breaking point , @xmath11 , @xmath12 . this has revealed that the continuation of the cumulant generating function @xmath7 to @xmath13 is strongly restricted by a kind of thermodynamic constraints and that it naturally induces the 1rsb solution in the case of a fully connected mean - field sg model . regarding @xmath14 as a fictitious inverse temperature , we can resort to the thermodynamics for extracting high - temperature , or replica , limit(@xmath15 ) from low - temperature behavior(@xmath16 ) . these facts strongly suggest that 1rsb is a consequence of the monotonicity breaking and frsb is that of at stability breaking . finite connectivity sg model has been considered as a first non - trivial extension of the mean - field theory , and challenged in many literatures . as a straight - forward extension from the case of fully connected model , perturbation theories in the region of the large connectivity or near the transition temperature have been studied in the replica formalism@xcite . another replica calculation@xcite has succeeded to derive an exact expression of the free energy under a non - trivial ansatz called factorized ansatz . the difficulty in these works appears in the search for an rsb saddle - point , because rsb is defined using the symmetry of a saddle - point in the theory . in contrast , the cavity method turned out to be an alternative and promising approach to study the finite connectivity models within 1rsb scheme@xcite . the key concept of this method is the complexity@xcite , logarithm of a number of the pure states , which enables one to deeply understand the microscopic structure of configuration space . it is found that the non - negativity condition of the complexity is relevant for the 1rsb cavity scheme , that provides a general procedure for mean - field type models including finite connectivity sg . in this paper , we further examine the possibility of 1rsb scenario suggested in our previous work , which might be important for a better understanding of the sg theory and also the replica method itself . the model discussed is a finite - connectivity ising sg model with @xmath0-body interactions . the reason why this model is considered as a good example is twofold . first our construction of 1rsb solution is applicable to the finite - connectivity sg model , because rs solution can be explicitly obtained . second , we see a direct correspondence between the guiding principle of introducing 1rsb in the replica method and the cavity method@xcite . the organization of this paper is as follows . in sec . [ sec : rep ] , we review our previous work@xcite for complete and detailed instructions of our scheme , in which a construction of a 1rsb solution from rs ansatz is explained . then a sg model defined on a sparse random graph is introduced and the 1rsb solution for the model obtained by our scheme is presented . we also discuss a relationship between our scheme based on the replica theory and the cavity method for the model . in sec.[sec : num ] , we compare the 1rsb solution to the result by mc simulation . finally sec . [ sec : sum ] is devoted to our conclusions and discussions . in this section , we briefly review our previous work@xcite and explain our scheme for the construction of a 1rsb solution in a general manner . for a given hamiltonian @xmath17 , equilibrium statistical mechanics requires to calculate the partition function @xmath18 , where tr denotes the sum over all possible configurations of the dynamical variables and @xmath19 is the inverse temperature . in the case of disordered system , one may evaluate @xmath20 for quenched disorder @xmath21 and take average of @xmath22 over @xmath21 with an appropriate weight . using the replica method@xcite , the averaged free energy @xmath23 $ ] is rewritten as a limit of cumulant generating function @xmath7 of @xmath24 as @xmath25 = \lim_{n\to 0}\left\{-\frac{1}{n\beta n}\log[z^n]\right\ } = : \lim_{n\to 0 } \phi(n ) , \label{repid}\ ] ] where @xmath26 $ ] denotes the average with respect to the quenched disorder . in case where @xmath14 is a real number , to proceed the calculation of the right hand side in eq . ( [ repid ] ) needs some ansatz . a typical one is replica symmetric ( rs ) ansatz , which is considered to be correct only for sufficiently large @xmath14 . we denote the solution based on the rs ansatz as rs solution @xmath27 . thus , the limit of @xmath7 we are interested in becomes nontrivial when we have no alternatives except the rs solution . in general , however , the function @xmath7 is restricted by the following conditions : @xmath28(monotonicity ) , @xmath29(convexity ) , and at stability . the two former conditions , monotonicity and convexity , come from a thermodynamic restriction if the replica number @xmath14 is regarded as a `` temperature '' . in particular , they lead to the following proposition@xcite : if @xmath12 for @xmath30 , + then @xmath31 for @xmath32 . therefore , if the rs solution is valid for @xmath33 , the limit @xmath34 is performed by this proposition . figure [ sch1rsb ] shows how the function @xmath35 is connected to the origin . it is also shown@xcite that the solution @xmath36 corresponds to the 1rsb solution for a wide class of models with @xmath9 , not restricted to the fully connected models . this relationship has already been pointed out in a solvable model@xcite . the proposition provides us a simple construction of a 1rsb solution using only the rs solution . we summarize our procedure for the 1rsb construction as follows : 1 . calculate the rs solution @xmath27 as a function of the finite replica number @xmath14 . 2 . find the value @xmath10 which satisfies @xmath37 . 3 . set @xmath38 while the right hand side of eq . ( [ nackofrep ] ) is analytically tractable but doubtful for @xmath39 because of the rs ansatz , the left hand side is equal to the free energy as stated in eq . ( [ repid ] ) but analytically intractable . one may notice that this procedure is analogous to the original saddle - point method , if one identifies the replica number with the breaking parameter . we consider this correspondence as the reason why we have to maximize with respect to the breaking parameter in literatures . it should be noted that this procedure can apply to any model in which the rs solution is explicitly obtained for any real @xmath14 . our procedure does not require overlap matrix or the introduction of breaking parameter . as a function of the replica number @xmath14 . this shows the construction of a 1rsb solution using monotonicity and convexity condition . the dashed line represents rs solution , which breaks the monotonicity condition at @xmath40 . below @xmath10 , @xmath7 becomes a constant function down to zero , corresponding to the 1rsb solution . ] hereafter we deal with a finite - connectivity ising sg model . the hamiltonian with @xmath0-spin interactions on a regular random graph with connectivity @xmath41 is defined as : @xmath42 where @xmath43 here @xmath44 represents ising spins on the random graph with @xmath45 sites . the interactions @xmath46 take @xmath47 with equal probability which gives the unit of energy and temperature . @xmath48 are quenched variables , satisfying the condition @xmath49 for each site @xmath50 , namely all the sites having the same number of the neighbors @xmath41 . we calculate the cumulant generating function of the model described above within the framework of sec . [ sec21 ] . following the calculation@xcite , @xmath7 under the rs ansatz is evaluated as @xmath51 where @xmath52 differentiating @xmath53 with respect to @xmath54 and @xmath55 , we have the saddle - point equations @xmath56 we solve eqs . ( [ iter1 ] ) and ( [ iter2 ] ) for each @xmath14 numerically and obtain the saddle - point functions @xmath57 and @xmath58 . details for the numerical method we use to solve these equations are shown in appendix [ apppi ] . inserting the saddle - point functions into eq . ( [ eq : phirs ] ) , we evaluate @xmath27 as a function of @xmath14 . [ funcphi ] shows an example of @xmath27 plotted against @xmath14 for @xmath3 and @xmath59 at @xmath60 , which is well below the expected sg transition temperature , @xmath61 . as shown in the figure , @xmath27 violates the monotonicity condition at a certain value @xmath62 which is defined by @xmath63 . following our scheme mentioned above , this is enough to construct a 1rsb solution . the 1rsb free energy per site @xmath64 is given as @xmath65 . it would be interesting to see the information of finite replica number is used to describe the 1rsb free energy . this is a consequence of the thermodynamic construction , with which the rs solution is connected to the physical limit @xmath15 . we have evaluated @xmath27 at @xmath66 for @xmath1 and @xmath3 , which yields temperature dependence of the 1rsb free energy shown later . for comparison , we also evaluate an rs free energy , which is defined as @xmath67 . temperature dependence of @xmath10 for some values of @xmath41 is plotted for @xmath1 and @xmath2 in fig . [ fig - nm ] . we also show the parameter @xmath68 for @xmath1 and @xmath59 in fig . [ fig - nm ] , evaluated in ref . . they are in good agreement with each other . the transition temperature for @xmath1 is derived from the condition that the instability condition of @xmath69 and then @xmath10 begin to deviate from zero . the estimate of @xmath70 is consistent with the known expression @xmath71@xcite considering an appropriate factor @xmath72 . for @xmath3 , @xmath70 is determined by an onset temperature at which the monotonicity breaking point emerges . then , @xmath10 deviates from unity , that is often observed in some models exhibiting 1rsb transition . while the analytic expression of @xmath70 for @xmath3 has not known yet , the estimate for @xmath59 and @xmath73 is consistent with that obtained by the cavity method@xcite . here we compare our scheme to the established cavity method , in particular for the finite connectivity ising sg model@xcite . the saddle - point equations , eqs . ( [ iter1 ] ) and ( [ iter2 ] ) , in our scheme are the same as the recursion equation derived as eqs . ( a.3 ) and ( a.4 ) in ref . , when the functions @xmath54 and @xmath74 are identified as the distribution of cavity field and cavity bias , respectively . while the parameter @xmath14 is determined by the monotonicity condition @xmath75 , the 1rsb parameter @xmath68 in the cavity context is determined by the non - negativity condition of the complexity @xmath76 : @xmath77 within the formalism of monasson@xcite . this means that these two methods are equivalent when the complexity is a well - defined quantity . in the previous works@xcite , it is shown that the result of the cavity method corresponds to that of the replica method with a factorized ansatz for the finite connectivity models . thus , our construction is also equivalent to the replica theory with the factorized ansatz . in the formalism , the replica number @xmath14 is substituted for the breaking parameter @xmath68 in the expression of free energy without taking the limit @xmath15 . then , the maximization of the free energy with respect to the overlap parameter @xmath78 and breaking parameter @xmath68 is equivalent to the monotonicity breaking condition in our scheme . this reasoning does not give a correctness proof of the factorized ansatz ( and also our ) solution , but we convince ourselves that it reveals the reason why the factorized ansatz gives numerically correct solution . dependence of @xmath27 of a finite - connectivity ising sg for @xmath3 and @xmath59 at @xmath60 . ] for @xmath1 ( top panel ) and for @xmath3 ( bottom panel ) . temperature is scaled as @xmath79 . each mark represents @xmath10 for connectivity @xmath59 , @xmath80 and @xmath73 . the solid line represents @xmath10 for @xmath0-body sherrington - kirkpatrick model with gaussian interaction . in the top panel , thermodynamic value of 1rsb parameter for @xmath1 and @xmath59 evaluated in ref is also shown in filled circle . , title="fig : " ] for @xmath1 ( top panel ) and for @xmath3 ( bottom panel ) . temperature is scaled as @xmath79 . each mark represents @xmath10 for connectivity @xmath59 , @xmath80 and @xmath73 . the solid line represents @xmath10 for @xmath0-body sherrington - kirkpatrick model with gaussian interaction . in the top panel , thermodynamic value of 1rsb parameter for @xmath1 and @xmath59 evaluated in ref . is also shown in filled circle . , title="fig : " ] in the previous section , we obtain the 1rsb solution for the ising sg model with @xmath0-body interactions by using our scheme . this is the true solution if the at instability or others would not occur above @xmath10 , but it is difficult to examine the validity of @xmath27 . this situation is similar to the case of the cavity method . instead , here we verify our 1rsb solution by comparing it to monte carlo ( mc ) data . we use exchange mc method@xcite in order to accelerate relaxation time to equilibrium . the number of temperatures is fixed to be 30 and the lowest temperature is down to 0.5 for all the system sizes @xmath45 and @xmath0 . the simulation parameters for @xmath1 and @xmath3 are presented in table [ 24mcs ] and [ 34mcs ] , respectively . equilibration of the mc simulations is confirmed by seeing that the observed quantities are stable within range of error by doubling mc steps . by using the mc simulation we measure the energy @xmath81 per site and calculate the free energy @xmath82 per site by thermodynamic integration : @xmath83 and the entropy @xmath84 per site as @xmath85 through the data at discrete temperatures obtained by the exchange mc method , the energy as a continuous function of @xmath86 is evaluated by reweighting formula@xcite : @xmath87 where @xmath88 denotes the mc average at the inverse temperature @xmath89 . we apply this formula by setting @xmath90 as actually simulated temperature and @xmath89 as required one . we choose @xmath90 as the nearest temperature to @xmath89 from the whole set of simulated temperatures . . parameters of simulation in the case of @xmath1 and @xmath59 . the total number of monte carlo steps @xmath91 and the total number of samples @xmath92 are presented for each size @xmath45 . the first @xmath93 are discarded for equilibration and the subsequent @xmath93 are used in measurement . [ cols="^,^,^",options="header " , ] to see thermodynamic properties , we extrapolate our mc data with finite sizes to the thermodynamic limit @xmath94 . because finite - size correction terms and its exponent are a priori unknown in sg models , an extrapolation method itself should be investigated . we assume that the leading finite - size correction terms for the energy , free energy and entropy are expressed as @xmath95 where @xmath96 , @xmath97 and @xmath98 are the thermodynamic limit of the respective quantities , and the correction exponent @xmath99 is assumed to be independent of the quantities . as shown in the previous work@xcite , the ground - state energy of the ising sg model for @xmath1 defined on a regular random graph is scaled with @xmath100 . thus , we assume that the exponent @xmath101 holds for @xmath1 at finite temperatures and is independent of physical quantities . [ enext24 ] show the thermodynamic quantities as a function of @xmath102 for @xmath1 at @xmath103 , which is the lowest observed temperature . the data are fitted well with the assumption @xmath100 as shown in the figure . the extrapolated values by the best fit and the results by the 1rsb and the rs solutions are shown in table [ bestfit ] . the thermodynamic values by mc results agree with those by the 1rsb solution rather than the rs one . the energy extrapolated in a wide range of temperature is displayed in fig . [ k2temp ] . this also suggests that the 1rsb solution is consistent with numerical results . and @xmath59 . the extrapolated value from mc data is marked by filled square . the 1rsb , rs and paramagnetic solutions are represented by solid , long - dashed and short - dashed lines , respectively . the inset is an enlarged view at low temperatures . ] and the filled squares and cross marks are the extrapolated value from mc data with the extrapolation form including the leading correction term and up to the next correction terms , respectively . the 1rsb , frozen and paramagnetic solutions are represented by solid , long - dashed and short - dashed lines , respectively . long - short - dashed line is the result of the cavity method in ref . . the iso - complexity energy obtained in ref . is also shown by dotted line . the inset is an enlarged view at low temperatures . ] we turn to the case of @xmath3 , where the value of @xmath99 is not known even at zero temperature . although a naive way to suppress higher order corrections is to study the system for large sizes and/or at lower temperatures apart from critical temperature , it has not been feasible to perform the mc simulation below @xmath104 for @xmath105 in moderate cpu time because of extremely slow relaxation especially in the case of @xmath3 . this is contrast to @xmath1 model . however , for relatively smaller systems , the distribution function of the energy is found to be almost a delta function with the weight at the lowest energy . this implies that vthe distribution depends weakly only on temperature @xmath86 below @xmath106 . this fact enables us to obtain the energy at temperatures down to @xmath107 using the reweighting method@xcite . we evaluate the correction exponent @xmath99 for the energy by the least - squares estimation at @xmath108 and @xmath106 with a form of eq . ( [ eqn : efenegy ] ) . the estimate of @xmath99 is not compatible with @xmath100 used in the case of @xmath1 , and is rather close to @xmath109 . this tendency is enhanced by omitting the smallest size @xmath110 from the analysis . these findings suggest that @xmath111 and higher order corrections are not negligible . therefore , we extrapolate the mc result for @xmath3 by assuming the forms of eqs . ( [ eqn : efenegy ] ) , ( [ eqn : ffenegy ] ) and ( [ eqn : sfenegy ] ) for @xmath109 with the next leading correction term @xmath112 . the data for @xmath110 are omitted from the extrapolation analysis . figures [ enext34 ] shows the result of the thermodynamic quantities for @xmath3 and @xmath59 at @xmath108 . the extrapolated values , presented in table [ bestfitk3 ] , are consistent with those of the 1rsb solution by taking into account the next leading correction term . we also show the thermodynamic value of the energy for @xmath3 as a function of @xmath86 in fig . [ k3temp ] . the extrapolated values by the form including the next leading correction term are consistent with those by the paramagnetic solution at @xmath113 and those by the 1rsb solution at low temperatures , though a systematic deviation still remains around @xmath70 because of the critical fluctuation . as shown in the inset of fig . [ k3temp ] , the agreement between the extrapolated value and the value of 1rsb solution is held at very low temperatures and the limiting value of energy at zero temperature coincides with that obtained by zero - temperature calculations@xcite . for the case of @xmath3 , the result of the cavity method is also shown in fig . [ k3temp]@xcite . analytic results are in good agreement with the mc data at low temperatures . these support the validity of the scheme also for @xmath3 . before closing this section , we would like to mention mc algorithm for studying sg models . in recent works@xcite , it is claimed that in annealing simulations a slow annealing limit of the energy often leads to the iso - complexity energy , significantly above the static equilibrium energy in glassy systems . this has been confirmed for @xmath3 by an annealing simulation@xcite . in contrast , as shown in fig . [ k3temp ] , the energy extrapolated to the infinite - volume limit in our exchange mc results is well lower than the iso - complexity energy and is compatible with that of the 1rsb solution at low temperatures . this suggests that the exchange mc is suitable for equilibration of the sg system even when the system have the iso - complexity energy separated from the static one . we have studied a construction of a 1rsb solution for quench disordered systems . our construction is based on thermodynamic conditions for the cumulant generating function @xmath7 of free energy , which are derived as a necessary condition in the replica analysis . the only requirement for our construction is to obtain the replica symmetric solution for @xmath7 as a function of @xmath14 . this is a quite general scheme which may provide an unified way to give a correct solution for 1rsb systems . in fact , our scheme reproduces the well - known 1rsb solution for fully connected mean - field sg models such as @xmath6-spin model@xcite and potts glass model@xcite . as a non - trivial example we have applied our scheme to study a 1rsb solution for finite - connectivity ising sg models with @xmath0-spin interactions . the thermodynamic quantities are explicitly evaluated from numerically obtained rs solution with finite replica number @xmath14 using our scheme . the saddle - point equations to be solved in our scheme are found to be equivalent to recursion equations of the cavity - field distributions in the 1rsb cavity formalism for this model . in a sense , our scheme based on the replica theory can be regarded as a reinterpretation of the 1rsb cavity method . while the cavity method can predict the microscopic detail of a model through complexity , which is an interesting quantity in glassy physics , one can not obtain such a quantity with our scheme at present . this would be discussed as a remaining issue . in contrast , we can construct the 1rsb theory irrespective of details of the model , even non - mean - field model in principle , because our scheme does not rely on the microscopic details , or complexity . since the pure state in finite dimensions is difficult to formulate in a tractable manner , this complexity - independent formalism of 1rsb may be useful to investigate nature of rsb in finite dimensions@xcite . because the replica method itself is originally independent of calculus of spin variables , this theoretical flexibility would give another perspective if rsb is formulated within macroscopic level . therefore , we consider that the cavity method and our method are complementary in order to understand the nature of sg . the correspondence of their results in this model has a significance because they should provide the same result in the intersection of their validity range . unfortunately , the validity of our 1rsb solution could not be established within the scheme because of the lack of at analysis . some at analyses for finite - connectivity models are recently proposed in the previous works@xcite . they are to be resolved for our model and compared with each other in future study . to confirm the validity of our scheme in the present work , equilibrium mc simulations with the help of extended ensemble method have been performed for the model with @xmath1 and @xmath2 . it is shown that for @xmath1 , the resulting thermodynamic quantities by our scheme are in agreement with those obtained by mc simulation within statistical error . for @xmath3 , assuming that the size dependence of the thermodynamic quantities is expressed as a polynomial of @xmath114 , we have concluded that our 1rsb solution is also consistent with those extracted from the finite - size mc data . if we have the correction exponent @xmath99 a priori , we can promote the accuracy of our extrapolation . optimization techniques for ground - state search would be a promising approach for estimating the value of @xmath99 for @xmath3 . as a by - product of the mc simulations , it is found that a coefficient of the first finite - size - correction term is positive . namely , the finite - size data reach their thermodynamic value from above with increasing the system size . this suggests that fluctuations on the positive side of the thermodynamic value is relevant for the finite - size corrections in these models . on the other hand , the probability of large deviations which can be calculated using the replica theory with @xmath115 is the negative side for the free energy in the fully connected sk model@xcite . the replica theory with @xmath116 for the large deviations is required to evaluate the finite - size correction . the key ingredient in our scheme for constructing the 1rsb solution is the thermodynamic constraints as a necessary condition in the replica theory . this is compared to the fact that the standard replica method introduces rsb scheme through the symmetry of the saddle point . another thermodynamic constraint , thermodynamic homogeneity , has been discussed in ref . . one might stress the importance of such a thermodynamical approach which leads to an universal framework irrespective of microscopic models . actually , our scheme is rather general and quite simple . it only needs the function @xmath27 which is constructed in the way of replica symmetric analysis . thus , we can avoid the arbitrariness to introduce breaking parameter in the replica theory . one can find further applications in related statistical - mechanical systems in which the rs solution can be constructed . we would like to thank y. kabashima for helpful comments and discussions , and for explaining the details of algorithm for solving the saddle - point equations@xcite . we are also grateful to f. krzakala for making his numerical data in ref . @xcite available to us . tn also gratefully acknowledges k. mimura for the kind and helpful lecture . he is strongly inspired by the lecture . this work was supported by the grant - in - aid for scientific research on the priority area `` deepening and expansion of statistical mechanical informatics '' ( no . 1807004 ) by ministry of education , culture , sports , science and technology , japan . for @xmath117 and @xmath118 . ] in this appendix , we explain details of the numerical method we used to solve the saddle - point equations ( [ iter1 ] ) and ( [ iter2 ] ) . we use an iteration method , introduced in ref . . the saddle - point functions @xmath57 and @xmath119 are approximated by a large @xmath120 number of samples from @xmath54 and @xmath55 . the algorithm for evaluating the function @xmath57 and @xmath121 is as follows : 1 . give an appropriate array @xmath122 as an initial condition to @xmath54 . 2 . sample @xmath123 independent values of @xmath124 @xmath125 from @xmath57 by generating a random integer @xmath126 uniformly distributed from 1 to @xmath120 and setting @xmath127 , and evaluate @xmath128 . 3 . put the sign chosen with probability 1/2 to @xmath129 and get @xmath130 , which corresponds to a sample of @xmath55 . 4 . repeat the steps 2 and 3 @xmath120 times and obtain the @xmath120 samples of @xmath55 , @xmath131 . sample @xmath132 independent values of @xmath133 @xmath134 by a procedure similar to that of step 2 and evaluate @xmath135 . accept @xmath136 obtained in step 5 with probability @xmath137 and accumulate a new set of @xmath138 of @xmath54 till the number reaches @xmath120 . 7 . return to 2 .
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an one - step replica - symmetry - breaking solution for finite connectivity spin - glass models with @xmath0 body interaction is constructed at finite temperature using the replica method and thermodynamic constraints . in the absence of external fields ,
this construction provides a general extension of replica symmetric solution at finite replica number to one - step replica - symmetry - breaking solution .
it is found that this result is formally equivalent to that of the one - step replica - symmetry - breaking cavity method . to confirm the validity of the obtained solution , monte carlo simulations are performed for @xmath1 and @xmath2 .
the thermodynamic quantities of the monte carlo results extrapolated to a large - size limit are consistent with those estimated by our solution for @xmath1 at all simulated temperatures and for @xmath3 except near the transition temperature .
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qpt is performed out by noting how input states , that span the space of @xmath0 , map to output states . the key insight in what follows is that it is not the input states of @xmath0 that are relevant , rather it is the preparation procedures itself , i.e. , the preparation map @xmath11 . for a @xmath20 dimensional system there are @xmath21 linearly independent states that span its space . however , there are @xmath22 linearly independent operations ( preparations ) that span the space of preparations . if we determine the corresponding output states for a set of linearly independent preparations then by linearity we have can predict the output state for any preparation . let us denote this map as @xmath23-map . the form of @xmath23-map arises naturally when considering the whole process in physical terms : at the beginning of the experiment @xmath2 is in an unknown ( correlated ) state , @xmath24 . the system is prepared into a known input state by the preparation procedure @xmath11 , followed by a joint unitary dynamics . the output is given by tracing over the environmental degrees of freedom : @xmath25 \left({\rho^{\mathcal{se}}}\right ) u^\dag \right].\ ] ] we want a map acting on the preparation map @xmath11 and yielding the output state @xmath12 : @xmath26 . then @xmath23-map is everything on the right hand side of eq . ( [ output ] ) that is not @xmath11 . the expression for @xmath23-map in terms of matrix indices is @xmath27 above a sum over repeated indices is implied . @xmath23-map is a ` super super - operator ' that acts on the super operator @xmath11 . @xmath23-map is a @xmath28 tensor , which is contracted with a preparation @xmath11 , a @xmath29 tensor , yielding the output state @xmath12 , a @xmath30 matrix . in term of matrix indices , the action is as follows : @xmath31 again , a sum over repeated indices is implied . in methods , a full derivation for @xmath23-map in the last equation is given . see fig . 2 for a graphical illustration of @xmath23-map . figure 2 : _ quantum process tomography with @xmath23-map . _ at the beginning of the experiment the system - environment state is correlated . a preparation is made on the system and the corresponding output state @xmath12 is observed . this process is described by the completely positive map @xmath23 , which is a function of the initial system - environment state and the unitary dynamics . the @xmath23-map takes preparations @xmath11 to output states @xmath12 . note that , in standard quantum process tomography state of @xmath1 is a constant of the process , here it is the initial @xmath2 state that is the constant of the process , i.e. , it is a fixed quantity . physically , the constancy of @xmath24 means that the experiment should be initialised in the same manner for every run , and then a preparation on @xmath0 can be made . @xmath23-map contains both @xmath32 and @xmath24 ; however knowing @xmath23 is not sufficient to determine @xmath32 and @xmath24 . as expected , it should not be possible to determine @xmath32 and @xmath24 through measurements and preparations on the system alone without access to the environment . conversely , @xmath23-map contains all information necessary to fully determine the output state for any preparation of @xmath0 . the advantage of dealing with the @xmath23-map is that we have separated the preparation procedure from uncontrollable dynamical elements and the initial conditions . @xmath23-map contains all of the dynamical information for the system and in the next section we will extract some of this information from the @xmath23-map . first let us mention some properties of @xmath23-map derived in methods : its action on a mixture of preparations is linear , it preserves trace , it preserves hermiticity , and it is completely positive . in methods we show that @xmath33map can be experimentally determined by making a set of linearly independent preparation of the system . this is similar to what one has to do in standard qpt . in standard qpt a linearly independent set of states are fed into the process and the corresponding outcomes are observed . knowing the inputs and the outputs the standard process map is determined . the difference here is that a linearly independent set of preparations are fed in to the process . this is of our major result of this paper : we have given a prescription to determine the dynamics of a system in an operational way , i.e. , a mapping from preparations to output states . the @xmath23-map contains the dynamics of the system before any preparation is made on the system . it is a function of the initial state @xmath24 state as well as the @xmath2 unitary transformation . @xmath23-map is a tensor , taking its trace with respect to the indices that belong to the initial state of @xmath0 we can obtain the dynamics of the system as if the initial correlations we absent . using this with the knowledge of the initial state of @xmath0 , in methods we show that from @xmath23 we can derive another matrix , @xmath34 matrix @xmath35 is fully determinable from @xmath33map and the two are the same when there are no initial correlations . we will call the difference between @xmath23 and @xmath36 , @xmath37 , the _ memory matrix _ : @xmath38 since @xmath23 contains @xmath24 and @xmath36 contains @xmath39 , the difference between the two is a function of only @xmath5 . the action of the correlation - memory matrix on a preparation yields @xmath40(\chi^{\mathcal{se } } ) u^\dag],\ ] ] which is the coherence coming into the system from the initial correlations . for non - markovian dynamics the future state of @xmath0 may depend on the initial @xmath2 correlations . this is the non - markovian ` memory ' due to the initial @xmath2 correlations and it is a key feature of non - markovian dynamics @xcite . the correlation - memory matrix is an important result for studying non - markovian systems . it is an operational way of measuring the information that flows into @xmath0 due to correlations at the time of the preparation . once @xmath33map is determined , we have the full knowledge of the dynamics of @xmath0 that is due to the initial correlations . the correlation - memory matrix provides quantitative information about the initial correlation and it is more than a witness for initial correlations @xcite . for the special case , when the preparation is chosen to be the identity map , we get pure dynamics of the correlation - memory matrix @xmath41,\ ] ] which is the reduced dynamics of @xmath2 correlations . this is exactly @xmath42 in eq . [ cpmap ] . from @xmath23-map we can determine matrices @xmath36 and @xmath43 . in turn , from @xmath36 we can get @xmath7 ( see eq . ( [ l - cpmap ] ) ) and from @xmath43 we can get @xmath42 , and together they give us @xmath9 of eq . ( [ ncpmap ] ) , which can be a not - completely positive map . this gives not - completely positive maps an operational meaning . @xmath23-map is the result of a quantum process tomography procedure for initially correlated system - environment states . it is acts on the preparation of the initial state of the system , and only contains dynamical information . we study the properties of @xmath23-map , showing it to be linear , preserving of trace and hermiticity , and completely positive . dynamical information about the evolution of the initial correlations can be retrieved from @xmath33map , in the form of the correlation - memory matrix @xmath43 . @xmath23-map allows us to determine the output state for any preparation of the system , while the correlation - memory matrix @xmath44 provides a quantitative expression for the coherence due to the initial correlations . an important question is when is @xmath23-map relevant ? clearly , when @xmath0 and @xmath1 are initially uncorrelated then @xmath44 will be zero . alternatively , just the presence of initial @xmath2 correlations does not warrant for @xmath23-map . suppose @xmath10 but @xmath45 = 0 $ ] , then the completely positive map of eq . ( [ cpmap ] ) would suffice to describe the dynamics correctly for any preparation of @xmath0 @xcite . one downside to @xmath33map is that it requires a lot of resources to construct . in standard quantum process tomography @xmath46 input states are fed through the process and the corresponding output states are determined . to determine @xmath23 map , @xmath47 preparations are necessary , which is a significant growth over the standard procedure . therefore an efficient way , such as compressed sensing @xcite , to determine this map is desirable . this should be possible , as determining @xmath23-map is equivalent to carrying out @xmath20 standard quantum process tomography procedures . another limitation that faces the procedure is the assumption that the preparation acts only on the system and not on the environment . this assumption is crucial , as we are mapping from the set of preparations on the system to the corresponding output states . if this assumption fails , then we would need to make a set of preparations that span the space of operations on the combined system - environment space . however , the environment can be arbitrarily large and we do not have any control over it . therefore the tools given in this article may not be valid when the preparation affects the environment directly . when the preparation procedure acts on @xmath1 as well as @xmath0 , the positivity of @xmath23-map may be affected . note that , as long the effect of all preparations on @xmath1 is a constant for then our prescription remains valid . lastly , since @xmath23-map contains all dynamical information , we are able to construct @xmath7 of eq . ( [ cpmap ] ) from it . similarly , from the correlation - memory matrix , we can construct @xmath42 of eq . ( [ cpmap ] ) . knowing the two we can determine @xmath9 of eq . ( [ ncpmap ] ) , which can be a not - completely positive map . this gives operational meaning to not - completely positive dynamical map as the descriptor for the dynamics of the system when identity preparation is made . on the other hand , the non - completely positive map is not experimentally determinable without determining @xmath23-map . finally , it remains an open question , when @xmath48 , is @xmath9 not completely positive ? the calculations in this sections are done in terms of matrix indices as @xmath23-map and the correlation - memory matrix @xmath43 are nontrivial tensors . we use the einstein summation notation , i.e. , repeated indices are summed over . bipartite state of @xmath2 is expressed with four indices with the latin indices belong to @xmath0 and greek indices to @xmath1 . for instance , the state in eq . ( [ corrmatrix ] ) has the form @xmath49 . a map acting on a density matrix is written as @xmath50 , where @xmath51 are the sudarshan - kraus operators ( see @xcite ) . @xmath52 is the complex conjugation of @xmath53 and @xmath54 is the transpose ; together they give hermitian conjugation . let us rewrite the generalised process equation , eq . ( [ output ] ) , in terms of matrix indices @xmath55 where the sum over @xmath56 is the trace with respect to the environment . we are interested in the reduced dynamics of @xmath0 as a function of the preparation procedures . thus , we can pull the preparation map out of everything else and regard it all as a map acting on the preparation map : @xmath57 \left ( { \mathcal{a}}_{r'r'';s 's '' } \right)\\ & = & \label{rawprocessequation } { \mathcal{m}}_{rr'r'';ss 's '' } \left ( { \mathcal{a}}_{r'r'';s 's '' } \right).\end{aligned}\ ] ] in the last equation , the matrix @xmath23 is defined as : @xmath58 let @xmath59 be a set of pure states that linearly span the space of @xmath0 . there are @xmath21 such matrices . that is , any state of @xmath0 can be written as a linear sum of these pure states : @xmath60 . a preparation map acting on @xmath0 is a @xmath29 hermitian matrix . therefore , any matrix in this space can be spanned by a tensor product of the basis matrices @xmath61 , which is a basis in for @xmath29 space of maps . there are @xmath22 elements in the basis @xmath61 . we can write action of one of these basis element on a density operator on @xmath0 as @xmath62 it is crucial to note here that @xmath63 , as these vectors are eigenvectors of the basis elements @xmath64 that do not commute . these preparations are can be thought of as a projection followed by a rotation . action of any map on space of @xmath0 acting on the @xmath2 state can be expresses as a linear sum @xmath65\\ & = & \sum_{mn } \alpha^{{(mn)}}p^{{(m)}}p^{{(n)}}\otimes \rho^{{\mathcal{e}}|{{(m)}}},\end{aligned}\ ] ] where @xmath66 are the coefficients that determine @xmath11 in terms of @xmath67 . @xmath16 is the conditional state of the @xmath1 and @xmath68 $ ] is the probability for the outcome @xmath15 . knowing the output states corresponding to each of these inputs , @xmath69,\ ] ] along with the success probabilities @xmath70 , for all @xmath71 , is enough to predict the output state for any preparation : @xmath72\\ & = & \sum_{mn } \alpha^{{(mn)}}p^{{(m)}}{{\rm tr}_\mathcal{e}}[up^{{(n)}}\otimes \rho^{{\mathcal{e}}|{{(m ) } } } u^\dag]\\ & = & \sum_{mn } \alpha^{{(mn)}}p^{{(m)}}q^{{(mn)}}.\end{aligned}\ ] ] @xmath23-map can be determined choosing @xmath73 , followed determining the corresponding @xmath74 and @xmath70 , and standard inversion techniques @xcite . note that any other set of linearly independent preparation can be linearly mapped to the preparations given in eq . ( [ linpreps ] ) , and therefore will suffice . determining all @xmath74 is done by quantum state tomography . this is equivalent to carrying out @xmath20 standard qpt procedures , one each @xmath16 . additionally measuring @xmath70 is equivalent to doing quantum state tomography of @xmath75 . before moving on a simple example may be useful . for one qubit , we may take the following projectors as a linearly independent basis : @xmath76 note that , this is a linear but not a convex decomposition : @xmath77 . the eigenvectors of @xmath78 , @xmath79 , @xmath80 , and @xmath81 are @xmath82 , @xmath83 , @xmath84 , and @xmath85 respectively . using these eigenvectors we can write basis elements for the maps that operate on the space of one qubit . for instance , @xmath86 and so on . the initial state of the system is labeled by indices @xmath87 and @xmath88 . tracing over everything else we can find the initial state of @xmath0 ( before preparation ) from @xmath23-map : @xmath89 this is , of course , attainable by doing state tomography at the beginning of the experiment , by measuring the values of @xmath70 from last section . next , let us the trace over the system indices @xmath87 and @xmath88 @xmath90 the last equation is exactly the dynamical map in the absence of initial correlations , given in eq . ( [ cpmap ] ) . in other words , in the absence of initial correlations , qpt would yield this map . this means , even though the @xmath23-map contains the information about uncorrelated @xmath2 state and the correlations separately . consider the following matrix composed of the matrices in eqs . ( [ initialstates ] ) and ( [ processfromm ] ) @xmath91 the last equation is similar to the expression for the @xmath23-map , except the state of the system and the state of the environment are uncorrelated . writing the state of @xmath2 in @xmath23-map in terms of eq . ( [ corrmatrix ] ) , we get @xmath92 now we can define the correlation - memory matrix as @xmath93 mathematically , @xmath33map acts on the preparation map just as the dynamical map acts on a density operator . in fact , we are not varying the initial state of the system , rather the preparation procedure on that state . therefore the linearity of quantum mechanics is preserved for the @xmath23-map acting on different preparation procedures , i.e. @xmath94 = \alpha_1 { \mathcal{m}}{\mathcal{a}}^{(1 ) } + \alpha_2 { \mathcal{m}}{\mathcal{a}}^{(2)}.\ ] ] this is very much like the dynamical maps action on mixtures of states . furthermore , if we show that the @xmath33map preserves trace , hermiticity , and positivity on its domain then all of these properties will be preserved on the state space . in other words for any preparation , @xmath95 that preserves trace , hermiticity , and positivity , the action of the @xmath33map on it will yield an output state , @xmath96 , that is unit - trace , hermitian and positive . let us start with the trace of @xmath23 with respect to the final indices @xmath97 with @xmath98 : @xmath99 & = & \delta_{rs } { \mathcal{m}}_{rr'r'';ss's''}\nonumber\\ & = & u_{r \epsilon , r ' \alpha } { \rho^{\mathcal{se}}}_{r''\alpha , s''\beta}u^*_{r \epsilon , s ' \beta}.\end{aligned}\ ] ] since @xmath100 , then @xmath101 = \delta_{r 's ' } \delta_{\alpha \beta } { \rho^{\mathcal{se}}}_{r''\alpha , s''\beta } = \mathbb{i } \otimes { \rho^{\mathcal{s}}}.\ ] ] a preparation acting on the above matrix will yield @xmath101({\mathcal{a}})={\mbox{tr}}[{\mathcal{a}}({\rho^{\mathcal{s}}})]=1.\ ] ] the implication being @xmath23 preserves the trace of @xmath102 . as long as the preparation is trace a preserving operation we get a unit - trace matrix for the output state . as with the case of general quantum operations , matrix @xmath23 is hermitian . this is easy to see by taking the complex conjugate of matrix @xmath23 , @xmath103 the complex conjugate of @xmath23 is not only the transpose of @xmath23 , but each element of @xmath23 is also transposed . hence @xmath23 is a hermitian matrix . the @xmath23-map is composed of a unitary matrix operating on a density matrix . then we can take the square root of the density matrix to get @xmath104 where @xmath105 and @xmath106 . we have written the @xmath23-map in operator sum representation , hence it is completely positive . where @xmath107 are the sudarshan - kraus operators @xcite . this means , the @xmath23-map acting on any preparation procedure will lead to a physical state . this was not the case when a standard qpt procedure is carried out on initially correlated @xmath2 states . the action of @xmath23-map can now be written as @xmath108 the properties shown above are precisely the conditions for a generic quantum operation to preserve trace , hermiticity and positivity . therefore @xmath33map preserves the attributes on the preparations , which in return will preserve these attributes on the states . 10 kossakowski , a. on quantum statistical mechanics of non - hamiltonian systems . , 247274 ( 1972 ) . lindblad , g. on the generators of quantum dynamical semigroups . , 119130 ( 1975 ) . gorini , v. , kossakowski , a. & sudarshan , e. c. g. completely positive dynamical semigroups of n - level systems . , 821825 ( 1976 ) . engel , g. et al . evidence for wavelike energy transfer through quantum coherence in photosynthetic systems . , 782786 ( 2007 ) . meiera , c. & tannor , d. j. non - markovian evolution of the density operator in the presence of strong laser fields . , 33653376 ( 1999 ) . geva , e. , rosenman , e. & tannor , d. j. on the second - order corrections to the quantum canonical equilibrium density matrix . , 13801390 ( 2000 ) . rodrguez - rosario , c. a. & sudarshan , e. c. g. non - markovian open quantum systems . , 16171634 ( 2011 ) . wolf , m. m. , eisert , j. , cubitt , t. s. & cirac , j. i. assessing non - markovian dynamics . , 150402 ( 2008 ) . breuer , h .- p . , laine , e .- & piilo , j. measure for the degree of non - markovian behavior of quantum processes in open systems . , 210401 ( 2009 ) . rivas , a. , huelga , s. f. & plenio , m. b. entanglement and non - markovianity of quantum evolutions . , 050403 ( 2010 ) . devi , a. r. u. , rajagopl , a. k. & sudha open system quantum dynamics with correlated initial states , not completely positive maps and non - markovianity . , 022109 ( 2011 ) . laine , e .- m . , piilo , j. & breuer , h .- witness for initial system - environment correlations in open system dynamics . , 60010 ( 2010 ) . smirne , a. , brivio , d. , cialdi , s. , vacchini , b. & paris , m. g. a. experimental investigation of initial system - environment correlations via trace distance evolution . , 032112 ( 2011 ) . rossatto , d. z. , werlang , t. , castelano , l. k. , villas - boas , c. j. , & fanchini , f. f. purity as a witness for initial system - environment correlations in open - system dynamics . , 042113 ( 2011 ) . sudarshan , e. c. g. , mathews , p. m. & rau , j stochastic dynamics of quantum - mechanical systems . , 920924 ( 1961 ) . kraus , k. . springer - verlag , ( 1983 ) . carteret , h. , terno , d. & zyczkoski , k. dynamics beyond completely positive maps : some properties and applications . , 042113 ( 2008 ) . modi , k. , rodrguez - rosario , c. a. & aspuru - guzik , a positivity in the presence of initial system - environment correlation . : 1203.5209 , ( 2012 ) . shaji , a. & sudarshan , e. c. g. who s afraid of not completely positive maps ? , 4854 ( 2005 ) . chuang , i. l. & nielsen , m. a. prescription for experimental determination of the dynamics of a quantum black box . , 24552467 ( 1997 ) . poyatos , j. f. , cirac , j. i. & zoller , p. complete characterization of a quantum process : the two - bit quantum gate . , 390393 ( 1997 ) . nielsen , m. a. , knill , e. & laflamme , r. complete quantum teleportation using nuclear magnetic resonance . , 5255 ( 1998 ) . childs , a. m. , chuang , i. l. & leung , d. w. realization of quantum process tomography in nmr . , 012314 ( 2001 ) . mitchell , m. w. , ellenor , c. w. , schneider , s. & steinberg , a. m. diagnosis , prescription , and prognosis of a bell - state filter by quantum process tomography bell - state filter by quantum process tomography . , 120402 ( 2003 ) . weinstein , y. s. et al . quantum process tomography of the quantum fourier transform . , 61176133 ( 2004 ) . obrien , j. l. et al . quantum process tomography of a controlled - not gate . , 080502 ( 2004 ) . neeley , m. et al . process tomography of quantum memory in a josephson - phase qubit coupled to a two - level state . , 523526 ( 2008 ) . chow , j. m. et al . randomized benchmarking and process tomography for gate errors in a solid - state qubit . , 090502 ( 2009 ) . howard , m. et al . quantum process tomography and lindblad estimation of a solid - state qubit . , 33 ( 2006 ) . myrskog , s. h. , fox , j. k. , mitchell , m. w. & steinberg , a. m. quantum process tomography on vibrational states of atoms in an optical lattice . , 013615 ( 2005 ) . pomyalov , a. , meier , c. & tannor , d. j. the importance of initial correlations in rate dynamics : a consistent non - markovian master equation approach . , 98108 ( 2010 ) . yuen - zhou , j. & aspuru - guzik , a. quantum process tomography of excitonic dimers from two - dimensional electronic spectroscopy . i. general theory and application to homodimers . , 134505 ( 2011 ) . yuen - zhou , j. , krich , j. , mohseni , m. & aspuru - guzik , a. quantum state and process tomography of energy transfer systems via ultrafast spectroscopy . , 17615 ( 2011 ) . kuah , a .- m . , modi , k. , rodrguez - rosario , c. a. & sudarshan , e. c. g. how state preparation can affect a quantum experiment : quantum process tomography for open systems . , 042113 ( 2007 ) . modi , k. & sudarshan , e. c. g. the role of preparation in quantum process tomography . , 052119 ( 2010 ) . modi , k. preparation of states in open quantum mechanics . , 253 ( 2011 ) . nielsen , m. a. & chuang , i. l. . cambridge university press , ( 2000 ) . modi , k. . phd thesis , the university of texas at austin , ( 2008 ) . [ _ arxiv_:0903.2724 ] emerson , j. et al . symmetrised characterisation of noisy quantum processes . , 1893 ( 2007 ) . flammia , s. t. , gross , d. , liu , y .- k . & eisert , j. quantum tomography via compressed sensing : error bounds , sample complexity , and efficient estimators . : 1205.2300 , ( 2012 ) . we are grateful to a. brodutch , a. rezakhani , c. a. rodrguez - rosario , and keith burnett for valuable conversations . we acknowledge the financial support of john templeton foundation , national research foundation , and ministry of education in singapore . part of the work presented here was done while the author was at the university of texas at austin .
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a central aim of physics is to describe the dynamics of physical systems .
schrdinger s equation does this for isolated quantum systems .
describing the time evolution of a quantum system that interacts with its environment , in its most general form , has proved to be difficult because the dynamics is dependent on the state of the environment and the correlations with it . for discrete processes , such as quantum gates or chemical reactions
, quantum process tomography provides the complete description of the dynamics , provided that the initial states of the system and the environment are independent of each other .
however , many physical systems are correlated with the environment at the beginning of the experiment . here
, we give a prescription of quantum process tomography that yields the complete description of the dynamics of the system even when the initial correlations are present .
surprisingly , our method also gives quantitative expressions for the initial correlation .
= 1 there is a rich history to the studies of decoherence of quantum systems due to the interactions with the surrounding degrees of freedom . when the dynamics of the system ( @xmath0 ) is markovian it can be described by a master equation @xcite . nowadays
many researchers are interested in systems that are non - markovian , as there is mounting evidence that some natural systems of importance may be non - markovian @xcite and such features may allow to manipulate and control quantum systems in desired ways .
there is also a great deal of interest in systems that are initially correlated with their environments ( @xmath1 ) because non - markovianity and initial system - environment ( @xmath2 ) correlations are intimately related @xcite .
grasping the mathematical and physical aspects of non - markovian systems , especially with initial @xmath2 correlations , has proved to be a tough road .
nevertheless , there is a great deal of progress on deciding whether a system is non - markovian in the recent years @xcite .
however , avoiding the initial @xmath2 correlations is not always possible in reality @xcite .
working with initial correlations in practice has proved to be much trickier than in theory .
this is because the presence of correlations do not allow for a clear definition of the state @xmath0 independent from the state of @xmath1 and vice versa .
physical systems are complicated and have many additional degrees of freedom that are not of experimental interest . yet
these extra degrees of freedom interact with the degrees of interest leading to correlations .
therefore initially uncorrelated @xmath2 state is often an approximation . in theory of open quantum systems ,
discrete quantum transformations are described by the dynamical map formalism @xcite : @xmath3 .
the dynamical map can be thought of as coming from the contraction of @xmath2 unitary dynamics .
let us write the state of @xmath2 as @xmath4 where @xmath5 is the correlations matrix @xcite .
the dynamical map is the mapping from the initial states of @xmath0 to the final states of @xmath0 , resulting from unitary dynamics of the @xmath2 state @xmath6\label{ncpmap}\\ & = & { { \rm tr}_\mathcal{e}}\left[u { \rho^{\mathcal{s}}}\otimes { \rho^{\mathcal{e}}}u^\dag\right]+ { { \rm tr}_\mathcal{e}}\left[u \chi^{\mathcal{se}}u^\dag\right]\\ & = & { \mathcal{b}}^{\rm cp } ( { \rho^{\mathcal{s } } } ) + { \mathcal{b}}^{\rm aff},\label{cpmap}\end{aligned}\ ] ] where @xmath7 is a completely positive map and @xmath8 is the affine correction term due to the initial @xmath2 correlations .
this means that @xmath9 may not a be completely positive map when @xmath10 , nevertheless it fully describes the dynamics of @xmath0 @xcite .
however , to determine such a map experimentally would require preparing different states of @xmath0 while keeping the @xmath2 correlations fixed .
such preparations are not operationally feasible because altering the state of @xmath0 will also alter the @xmath2 correlations .
therefore , a nonpositive dynamical map is not an operationally meaningful quantity .
the operational approach to quantum dynamics relies on the fact that quantum theory is a theory of preparations and measurements .
the experimental method to determine a dynamical map corresponding to a quantum process is called _ quantum process tomography _ ( qpt ) @xcite .
it is the central tool in determining a discrete quantum process ; e.g. quantum gates @xcite or chemical reactions @xcite . to see the difference between qpt and dynamical maps
let us review the four basic steps necessary to carried out qpt @xcite : * input states that span the space of @xmath0 are prepared .
* the input states are sent through the process . *
the corresponding output states are determined by quantum state tomography . *
the knowledge of input states , the corresponding output states , and assuming linearity completely determines the process .
figure 1 : _ standard quantum process tomography .
_ at the beginning of the experiment the system - environment state is uncorrelated . a preparation ( @xmath11 )
is made on the system and the corresponding output state @xmath12 is observed .
this process is described by the completely positive map of eq .
( [ cpmap ] ) , which is a function of initial state of environment and the unitary dynamics .
it maps the initial states of the system to output states @xmath12 .
let us denote input states as @xmath13 and output states as @xmath12 .
the first step of qpt is state preparation .
a preparation procedure takes an unknown state of @xmath0 to a known state of @xmath0 .
mathematically , it is described by a completely positive map acting on the system @xcite .
for instance , consider a set of preparations that project @xmath0 into pure states : @xmath14 \left ( { \rho^{\mathcal{se}}}\right)= p^{{(m)}}\otimes \rho^{{\mathcal{e}}|{{(m)}}}$ ] . since @xmath15 is a pure state , the post - preparation @xmath2 state is fully uncorrelated , where @xmath16 is the conditional state of @xmath1 .
@xmath17 is the identity operator acting on @xmath1 , as we assume that the preparation procedure only acts on @xmath0 and not @xmath1 .
we will discuss the implications of relaxing this assumption in discussions .
lastly , if the preparation is not trace preserving , it should be divided @xmath18 $ ] for normalisation . the @xmath2 evolution , after the preparation yields the output state : @xmath19({\rho^{\mathcal{se } } } )
u^\dag\right ] \label{preq}\\ & = & { { \rm tr}_\mathcal{e}}\left[u p^{{(m)}}\otimes \rho^{{\mathcal{e}}|{{(m ) } } } u^\dag\right ] .
\label{prepaf}\end{aligned}\ ] ] the key difference between the dynamical map in eqs .
( [ ncpmap ] ) and ( [ preq ] ) is the act of state preparation .
because dynamical maps of do accommodate state preparation , they are not operationally defined . in the presence of initial @xmath2 correlations ,
state preparation affects the state of @xmath1 in a nontrivial manner .
that is , the state of @xmath1 in eq .
( [ prepaf ] ) is conditioned by the choice of the preparations . in deriving the standard qpt procedure
it is implicitly assumed that the initial state of @xmath2 is uncorrelated @xcite , i.e. , the state of @xmath1 is thought to be a constant of the problem . when that is the case , the state of @xmath1 in eq .
( [ prepaf ] ) is not conditioned by the preparation procedure . in this case
the derived map for the process is completely positive and is the same as the completely positive dynamical map in eq .
( [ cpmap ] ) .
see fig . 1 for a graphical illustration .
in the presence on initial @xmath2 correlation , the conditional state of @xmath1 will be different for each preparation , and the assumption of linearity in step ( iv ) of qpt is violated , i.e. the map is a function of the preparation procedure .
such maps are nonpositive , nonlinear , or simply put nonsensical @xcite .
it then begs the question , can we determine the dynamics of a system that is initially correlated with @xmath1 ?
this is an important question for two reasons : first , there may be physical system of interest that may have initial correlations .
is it possible to study their dynamics ?
second , for foundational reasons we may care to know what are the limitations in describing the dynamics of physical systems .
a partial solution to these questions was given in @xcite .
in this article we show that not only complete dynamics of initially correlated system can be determined , we can also determine the contribution due to initial correlations .
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hidden markov models ( hmms ) are stochastic models for systems with a set of unobserved states between which the system hops stochastically , sometimes emitting a signal from some alphabet , with probabilities that depend upon the current state . the situation in which we are specifically interested is human mobility , partially observed , _ i.e. _ , occasional signals about a person s location . for example , consider the cells of a mobile phone network , from which a user can make calls . in this case the states of a hmm are the cells , and the emitted signals are the cell itself , if a call is made by a particular user during each of a sequence of time intervals , or nothing ( 0 ) , if that user does not make a call . in the latter case , the state ( location ) of the user is ` hidden ' , and must be inferred , while in the former case , assuming no errors in the data , the ` hidden ' state is revealed by the call record . since these are data from a _ mobile _ phone network , a user can move from cell to cell . although many analyses of human mobility have estimated no more than rather crude statistics like the radius of gyration , the fraction of time spent at each location , or the entropy of the timeseries of locations @xcite , others have used hmms to describe partially observed human mobility and have estimated their parameters @xcite . with short time steps , however , a standard hmm ( with time - independent parameters ) is not a plausible model , since human mobility behavior changes according to , for example , the time of day @xcite . we would like to create , therefore , a hmm with time-_dependent _ parameters . of course , allowing , for example , arbitrary transition / emission probabilities at each time step , would lead to an extremely underdetermined model . rather , we need a model with only a few additional parameters to capture the time - dependence of human mobility . since the total numbers of trips @xcite and mobile phone calls @xcite vary with time of day and day of week , we develop a time - dependent hmm in which the non - trivial transition and emission probabilities are proportional to _ activity levels _ , _ i.e. _ , to some given functions modeling how active humans are at different times and places . since the transition and emission probabilities in our hmm are not constant in time , it is a _ non - stationary _ hmm . many generalizations of hmms have been considered previously , of course , as more faithful models of various real systems . some of these are non - stationary : deng , for example , considers a class of models in which the emission probabilities are somewhat non - markovian , depending on a number of previous emissions , and also have polynomial - in - time trend components which are to be estimated @xcite . _ duration _ hmms ( dhmms ) , first suggested by ferguson @xcite , allow a sort of non - stationarity in the state transition process by including a randomly chosen duration each time the state changes , _ i.e. _ , a number of time steps without a transition away from that state . this kind of model has been generalized to make the transition probabilities functions of the number of steps the system has been in the current state @xcite . in a different direction , since one can think of transitions between the hidden states with different emission probability distributions as a kind of non - stationarity , _ triplet markov chains _ ( tmcs ) include an auxiliary set of underlying states , each of which corresponds to a different stationary regime for a hmm @xcite . our approach is different than that of dhmms and tmcs in that the time dependence of the transition and emission probabilities is not intrinsic and random , but rather exogenous and deterministic . furthermore , unlike deng s models @xcite , we take the `` trend '' part of the time dependence to be given , not an additional ( set of ) parameter(s ) to be estimated . formally , our model consists of @xmath0 possible hidden states , and we denote by @xmath1^t$ ] the time series of @xmath2 hidden states ( for any @xmath3 , @xmath4=\{1,\ldots , n\}$ ] ) . state transitions happen according to a sequence of matrices giving the conditional probabilities of transitions , @xmath5 at each state we observe an emission that takes a value from the set @xmath6\cup\{0\}$ ] , where @xmath7 denotes `` absence of an emission '' . let @xmath8\cup\{0\})^t$ ] be the series of observed emissions . the probability of emission @xmath9 at time @xmath10 , from state @xmath11 , is @xmath12 we define time varying _ activity levels _ for state @xmath11 by a pair of functions with non - negative values , @xmath13\to [ 0,1]^2 $ ] , the _ activity functions _ , which modulate transitions and emissions from the state , respectively . given a state @xmath11 , the transition probabilities from that state are functions of _ transition parameters _ @xmath14 , @xmath15 $ ] , and the activity level : @xmath16 , i\neq j } \tau_{ij } & \text{if } i = j , \end{array } \right.\ ] ] subject to the constraints that for all @xmath17 $ ] and all @xmath18 $ ] , @xmath19 , i\neq j}\tau_{ij } \leq 1.\ ] ] in practice we may have _ a priori _ knowledge that some transitions do not occur , so that @xmath20 ( and @xmath21 ) have some entries set to @xmath7 . for example , with 10 minute time intervals and states representing cells in a mobile phone network , transitions between sufficiently distant cells are precluded . similarly , we assign emission parameters @xmath22 , @xmath23 $ ] , to each state @xmath11 . the emission probabilities are @xmath24}\epsilon_{s j } & \text{if } s=0 , \end{array } \right.\ ] ] subject to the constraints that for all @xmath17 $ ] and all @xmath18 $ ] , @xmath25}\epsilon_{s j}\leq1.\ ] ] denote the initial distribution over states by @xmath26 , subject to the constraints @xmath27 and @xmath28 } \pi_j = 1.\ ] ] were this a typical hidden markov model , we could estimate its parameters using the baum - welch algorithm @xcite . since it is not , we develop a novel expectation maximization ( em ) algorithm @xcite to estimate the parameters @xmath29 , given @xmath30 , @xmath31 and @xmath32 , as follows . the expectation maximization algorithm maximizes , at each iterative step , the ( expected ) log - likelihood function described below . let @xmath33 be the set of all possible time series of states and let @xmath34 be the estimate of @xmath35 at the @xmath36-th iteration of the algorithm . @xmath37 the algorithm begins by initializing the parameter estimates in the first ( @xmath38 ) iteration . then the @xmath39 iteration consists of two steps : 1 . [ itr1 ] compute the _ expectation value _ of the log - likelihood , using the current ( @xmath40 ) estimate for the parameters : @xmath41 \pr(x\mid y;{\hat\theta}^k),\ ] ] where @xmath42 means @xmath43 in a probability distribution parametrized by @xmath35 . [ itr2 ] find the parameters that _ maximize _ the expected log - likelihood : @xmath44 subject to constraints in the inequalities , and . as in the regular baum - welch algorithm , we express our computations in terms of certain conditional probabilities based on the parameters estimated at the @xmath40 iteration , @xmath45 some reindexing of eq . yields the following expression for @xmath46 in terms of these probabilities : @xmath47}\log(\pi_j)\gamma^k_j(1 ) + \sum_{i , j\in[n]}\sum_{t=1}^{t-1 } \log\bigl(a_{ij}(t)\bigr)\xi^k_{i j}(t ) + \sum_{j\in[n]}\sum_{t=1}^{t } \log\bigl(b_{y_t j}(t)\bigr)\gamma^k_j(t).\ ] ] we iterate steps [ itr1 ] and [ itr2 ] , for which we compute @xmath48 and @xmath49 in eqs . , and @xmath46 in eq . above . we continue until some standard of convergence is achieved . we then output the final @xmath34 as our estimate of @xmath35 . [ mainthm ] there is a constrained expectation maximization algorithm giving a sequence of estimates @xmath50 that converges to a critical point of the likelihood function , which is the maximum likelihood estimate @xmath51 for the observed sequence @xmath30 when the initial guess is sufficiently close . further , to achieve a precision @xmath52 in the estimates , the time complexity of the algorithm is @xmath53 . in particular , for a fixed precision @xmath52 , the time complexity is @xmath54 . suppose we have the estimates @xmath55 defined in eq . from step @xmath36 of the algorithm . we proceed to compute @xmath56 and @xmath57 in eqs . to begin the next iteration . just as in the regular baum - welch algorithm , we apply dynamic programming . denote the @xmath40 estimate of the transition matrix by @xmath58 , where @xmath59 similarly , @xmath60 } \hat\epsilon^k_{sj } & \text{if } s=0 . \end{array } \right .\ ] ] it is convenient to define @xmath61 to be the diagonal matrix with @xmath62 entry @xmath63 . now compute two sequences of ( co)vectors , @xmath64 and @xmath65 , recursively , as follows : @xmath66 then denotes the diagonal matrix whose @xmath62 entry is @xmath67 if @xmath68 and @xmath69 otherwise . ] @xmath70 the estimates for the initial probabilities @xmath71 are the same as in the normal baum - welch algorithm , as is clear from the expression for @xmath46 in eq . . thus @xmath72 now notice that all the constraints in and necessary to define @xmath46 are implied by the strongest constraints : for all @xmath17 $ ] , @xmath73 where @xmath74 } \ : f_{j}(t)$ ] , and @xmath75 } \epsilon_{s j } \leq 1,\ ] ] where @xmath76 } \ : g_{j}(t)$ ] . consider the computation of @xmath77 , for @xmath15 $ ] . it should lie in the domain @xmath78 defined by constraints and the non - negativity of the parameters . since these constraints are independent for different @xmath11s , we can consider each @xmath11 separately , and find the optimal parameters @xmath79 by computing the critical points of @xmath46 relative to @xmath80 . using eqs . and , @xmath81 if the left sum in eq . , @xmath82 , the derivative is nonpositive , so @xmath46 is weakly decreasing and @xmath83 gives its largest value . if the right sum in eq . , @xmath84 , the derivative is nonnegative , so @xmath46 is weakly increasing and takes its maximum value when @xmath79 is as large as possible , _ i.e. _ , when it saturates constraint . we will show how to handle this situation after discussing the generic case which we do next . assuming then that neither sum in eq . is @xmath7 , to find the stationary points of @xmath46 we set eq . to @xmath7 and solve for @xmath79 . specifically , @xmath79 must satisfy @xmath85 we note that if @xmath86 , which makes the transition probabilities , @xmath87 , time independent , then the solution to eq . is the familiar baum - welch solution : @xmath88 for all @xmath89 $ ] . for non - constant activity functions , however , the solution is more complicated . since the right side of eq . is manifestly independent of @xmath90 , the left side must be , too . let @xmath91 where the last expression uses the antiderivative convention that for a function of @xmath10 denoted by a letter in lower case , the corresponding upper case letter is upper case @xmath92 ; @xmath93 is upper case @xmath94 ; @xmath95 is upper case @xmath96 ; @xmath0 is upper case @xmath97 ; @xmath98 is upper case @xmath99 . ] represents its sum over its domain of definition ( @xmath100 to @xmath101 in this case ) . now @xmath102 if we denote the probability of _ moving _ away from state @xmath11 by @xmath103 we can write @xmath104 . substituting in eq . , this gives : @xmath105 or equivalently : @xmath106 we must solve this equation for @xmath107 , whence we can use eq . to solve for each of the @xmath79 . since @xmath107 is nonnegative and constraint must hold , we recast these conditions in terms of @xmath107 as : @xmath108 let @xmath109\,\mid\,\xi^k_{jj}(t)\neq0 } \ : f_{j}(t)\,\,\leq f^*_{j}.\ ] ] each of the terms in the sum on the right side of eq . is strictly decreasing in @xmath110 when it is well - defined ( @xmath111 ) . values of @xmath110 just larger than @xmath112 make the sum arbitrarily large , and as @xmath110 increases from that value , the sum decreases monotonically to 0 , so exactly one value of @xmath113 will satisfy eq . . we can solve the equation numerically to find this value , call it @xmath114 . if @xmath115 , splitting it proportionally to @xmath116 according to eq . gives the _ unique _ critical point @xmath80 of @xmath46 . we can compute explicitly the hessian of @xmath117 with respect to the @xmath79 ; its components are : @xmath118 thus , as a matrix the hessian can be written as the sum of two matrices : @xmath119 where @xmath120 is the vector of all @xmath69s . each of the matrices on the right is negative semi - definite , so the hessian is also . thus the ( unique ) critical point we found in this case is a global maximum of @xmath117 in @xmath121 and hence the choice for @xmath122 . if the solution does not satisfy the original constraint , _ i.e. _ , @xmath123 , or if the right side of eq . is @xmath7 , the maximum will be on the boundary of @xmath121 . thus we maximize @xmath117 subject to the boundary constraint @xmath124 } \tau_{ij } = \frac{1}{f^*_j}.\ ] ] this is in the form of the constraint in the regular baum - welch algorithm with 1 replaced by @xmath125 and the self - transition probability set to 0 . thus the critical @xmath79 can be computed as in the baum - welch algorithm , for all @xmath126 . ] with @xmath127 replaced by @xmath128 and the solution divided by @xmath129 : @xmath130 this is the unique critical point in this case , and the global maximum of @xmath117 in @xmath121 , by the same argument as in the baum - welch algorithm . this becomes the choice for @xmath122 . we turn to the computation of @xmath131 , @xmath132 $ ] . as before , we begin by finding the stationary points of @xmath46 , now relative to @xmath133 defined by constraints and the non - negativity of these parameters . using eqs . and gives : @xmath134}\epsilon_{lj}}\delta_{0,y_t}.\ ] ] as we did for eq . , we must consider the situations when either of the sums in eq . vanishes . when the left sum is @xmath7 , a extreme value is given by @xmath135 , and when the right sum is @xmath7 , constraint is saturated . assuming neither of the sums vanishes , we find the stationary points by solving @xmath136}\epsilon_{lj}}\delta_{0,y_t}.\ ] ] solution to the _ emission _ equations , eqs . , follows using the same steps as for the _ transition _ equations , eqs . . we first denote the probability of emission @xmath132 $ ] from state @xmath11 by : and every state @xmath11 emits either the signal @xmath11 or @xmath7 ; in other words , @xmath137 if @xmath138 $ ] . this simplifies the following expressions : for @xmath10 such that @xmath139 , @xmath140 ( the state is @xmath11 with certainty if the observed emission is @xmath11 ) , _ i.e. _ , @xmath141 , and @xmath142 if @xmath138 $ ] . ] @xmath143 in terms of which we rewrite eq . as @xmath144}\epsilon_{lj}}.\ ] ] we define ( independent of @xmath9 ) @xmath145 we also define the probability of any _ non - zero _ emission from state @xmath11 , @xmath146}\lambda^k_{lj}(t),\ ] ] which , used in eq . , gives @xmath147 let @xmath148\,\mid\,\lambda^k_{0j}(t)\neq0}\:g_{j}(t)\,\,\leq g^*_{j}.\ ] ] as before , there is exactly one solution @xmath149 to eq . . we can find it numerically , and if @xmath150 , we use eq . to find all the @xmath151 , which will then satisfy constraint . again we can compute the hessian explicitly to confirm that this is a global maximum of @xmath117 , now in @xmath152 , and hence the choice for @xmath153 . if @xmath154 , or if the right side of eq . is @xmath7 , we must find instead the critical point on the boundary of @xmath152 : @xmath155 } \epsilon_{sj } = \frac{1}{g^*_j}.\ ] ] again , this is in the form of the constraint in the regular baum - welch algorithm . accordingly , we set the critical @xmath156 to the baum - welch estimate , rescaled by @xmath157 : @xmath158 this is the unique critical point and the global maximum of @xmath117 in @xmath152 , and therefore the choice for @xmath131 in this case . we have shown how to find @xmath159 maximizing @xmath46 in eq . . this algorithm converges as claimed because it is an instance of expectation maximization . to understand its time complexity we must consider the numerical solution of eqs . and . to simplify notation we rewrite eq . in terms of @xmath160 , and a function @xmath161 , @xmath162 as @xmath163 . as an initial estimate for the root , @xmath164 , we can use @xmath165 such that @xmath166 where @xmath167 $ ] satisfies @xmath168 and @xmath169 . @xmath170 since we found it using only one of the nonnegative terms in the sum in eq . . now recall that @xmath171 for @xmath172 $ ] , and @xmath173 } \mu^k_j(t ) = \sum_{t\in [ t-1 ] } \sum_{r\neq j}\xi^k_{rj}(t ) \le \sum_{t\in [ t-1 ] } \gamma^k_{j}(t ) \le t-1.\ ] ] thus each term in the sum in eq . is no more than @xmath174 at @xmath175 this is @xmath176 , so @xmath177 , which implies @xmath178 using newton s method , once we have an initial estimate `` sufficiently close '' to the root of @xmath163 , the time complexity to find it with error less than @xmath52 is @xmath179 , where the @xmath2 comes from the cost of evaluating @xmath161 and @xmath180 at each iteration ; in practice this is how we would find the root . since the length of the interval in is @xmath181 , however , the bisection method gets us to precision @xmath52 with @xmath182 steps , with total cost @xmath183 ; thus this is the total complexity . we need to solve eqs . and @xmath0 and @xmath95 times , respectively , at each iteration , which thus adds @xmath184 to the @xmath185 complexity of the computations for @xmath186 and @xmath187 . thus the time complexity for the whole algorithm is @xmath188 . to demonstrate the effect of the activity functions we consider a simple model with @xmath189 states and the same number of possible emissions ( @xmath190 ) . from any state @xmath11 , we only allow an emission to be either its own label @xmath11 or @xmath7 , _ i.e. _ , @xmath191 for @xmath192 $ ] , so a non - zero emission uniquely identifies the state that emits it . we choose random transition and emission parameters : @xmath193 where the omitted values are the components for which @xmath194 . we generate sequences of length @xmath195 ( we may think of this as @xmath196 weeks , with an observation every @xmath197 minutes ) . we consider activity functions with variations that may approximate observed data , _ i.e. _ , periodic variations with a period of @xmath198 ( one day ) . specifically , our numerical simulations use the following three functions : we generate a random sequence of states , @xmath202 , and resulting emissions , @xmath30 , using the transition and emission parameters above , and a pair of activity functions ( a list of these pairs is shown in table [ table2 ] ) . before computing the sequence of parameter estimates , we need to specify how we compute initial estimates to start the iteration . this can only depend on the observed emission sequence @xmath30 , since in any real scenario @xmath202 is unknown . as a first guess , for this simple model , we interpolate the state sequence @xmath202 as follows : for every pair of successive non - zero emissions , there is a segment of zeros ( no emission ) separating them . we divide each such segment into two subsegments : let @xmath203 $ ] be the emission immediately preceding the segment , and @xmath204 $ ] be the emission immediately following the segment . the second subsegment starts at the first time step after the one where @xmath205 first attains its maximum value on the segment ( _ i.e. _ , a time at which there is the maximum probability of hopping from state @xmath11 to state @xmath90 ) . we assign state @xmath11 to the time steps in the first subsegment and the state @xmath90 to those in the second . if the emission sequence @xmath30 starts with a segment of zeros , then that segment is assigned the value of the first non - zero emission ; similarly a terminal sequence of zeros is given the value of the last non - zero emission . denote the interpolated states by @xmath206 , @xmath18 $ ] . from @xmath207 , we compute the estimate @xmath208 for the initial distribution over the states @xmath209 by their frequencies of occurrence . for the initial @xmath79 estimate , @xmath210 , we use the method described in the proof of theorem [ mainthm ] to solve eq . , using @xmath211 , including @xmath194 . for the initial @xmath212 estimate , @xmath213 , we also use the method described in the proof of theorem [ mainthm ] to solve eq . , using @xmath214 . to understand the performance of the algorithm in theorem [ mainthm ] , we need a measure of the error between the estimates and the real parameter values . the _ relative entropy _ is one measure for a stationary hmm . in our case we need to account for the time variation of the transition and emission probabilities , @xmath215 and @xmath216 . hence we define a modified version of a relative entropy error criterion , the _ averaged relative entropy_. let @xmath217 and @xmath218 , @xmath18 $ ] , be two finite sequences of discrete probability distributions on a finite set @xmath219 . the _ averaged relative entropy _ ( @xmath220 ) of @xmath221 with respect to @xmath222 is @xmath223}\mathsf{re}(p_t , q_t),\ ] ] where the usual relative entropy ( @xmath224 ) is given by @xmath225 thus the error function that we compute for given @xmath21 and estimate @xmath226 is @xmath227 where @xmath215 and @xmath228 are related through @xmath205 to @xmath79 and @xmath229 by eq . and eq . , respectively similarly , for @xmath230 and estimate @xmath231 , the error function is @xmath232 where @xmath216 and @xmath233 are related through @xmath234 to @xmath235 and @xmath236 by eq . and eq . , respectively . ( remember that we are considering the simple case in which the emission @xmath237 when the state @xmath238 . ) to - & a & b & c & d & e & f & g & h + - @xmath205 & @xmath239&@xmath239 & @xmath239 & @xmath240 & @xmath241 & @xmath242 & @xmath240 & @xmath241 + @xmath234&@xmath241&@xmath240&@xmath239&@xmath240&@xmath241&@xmath239&@xmath239&@xmath239 + - for each set of pairs of activity functions in table [ table1 ] we run the algorithm for @xmath243 iterations . figures [ figtrans ] and [ figemis ] plot the averaged relative entropy for the parameter estimates as a function of iteration step . the labels ( a)(h ) to the right of each plot appear in the order of the final error values . we do not provide a plot showing convergence of @xmath244 since the only noticeable trend is that if they converge to an exact state value , it is usually to the initial state of the interpolated sequence @xmath207 . in each case the error for both the transitions and the emissions decreases to small values . since the @xmath220 depends on the activity functions as well as on the parameters and their estimates , we need to compute a baseline error value for each case . for the parameters @xmath21 and a specific choice of @xmath245 it is : @xmath246,\ ] ] where @xmath215 and @xmath247 are related through @xmath205 to @xmath79 and @xmath248 , respectively , by eq . , and where @xmath249 $ ] denotes the expectation over uniformly random @xmath250 . to estimate this expectation value , we compute the average @xmath220 of the parameters with respect to 1000 independently chosen sets of random parameters ( rather than their estimates from our algorithm ) , for each case ( a)(h ) . for the emission parameters we compute baselines the same way , using 1000 uniformly random values @xmath251 to estimate @xmath252,\ ] ] where @xmath216 and @xmath253 are related through @xmath234 to @xmath235 and @xmath254 , respectively , by eq . , and where @xmath249 $ ] denotes the expectation value over uniformly random @xmath251 . the baseline averages thus obtained for function pairs in table [ table1 ] are recorded in table [ table2 ] , where the row labels indicate the parameters being baselined . to - & a & b & c & d & e & f & g & h + - @xmath255&@xmath256 & @xmath257&@xmath258 & @xmath259&@xmath260 & @xmath261&@xmath262 & @xmath263 + @xmath264&@xmath260&@xmath265&@xmath266&@xmath267&@xmath268&@xmath269&@xmath270&@xmath271 + - we plot these baseline errors as horizontal lines in figures [ figtrans ] and [ figemis ] . most of these are too close to be distinguishable ; indeed they are all @xmath272 , in contrast to the estimation errors plots which are almost all smaller by at least an order of magnitude , and in most cases by 3 or 4 , indicating very good parameter estimates . furthermore , the relative quality of the estimates can be understood : case ( c ) is the standard hmm , for which our algorithm reduces to the baum - welch algorithm @xcite . cases ( a ) and ( b ) have greater errors , which is not surprising since they have non - constant emission activity functions , oscillating in value up to 1 . this means that for each of these cases , non - zero emissions are lower probability events , so there is less information in @xmath30 . possibly surprising is the fact that when the transition activity function is non - constant , cases ( d)(h ) , the errors are _ smaller _ than in the standard hmm case . but this happens because state changing transitions are reduced , so that each non - zero emission observed provides more information . and among these cases , those with varying emission activity levels have larger errors than those without . , for successive iterations @xmath36 . labels to the right are of activity function pairs from table [ table1 ] , and are displayed in the order of the final error values . baseline errors , @xmath255 , from table [ table2 ] are shown as horizontal lines . ] , for successive iterations @xmath36 . labels to the right are of activity function pairs from table [ table1 ] , and are displayed in the order of the final error values . baseline errors , @xmath264 , from table [ table2 ] are shown as horizontal lines . ]
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we define a hidden markov model ( hmm ) in which each hidden state has time - dependent _ activity levels _ that drive transitions and emissions , and show how to estimate its parameters .
our construction is motivated by the problem of inferring human mobility on sub - daily time scales from , for example , mobile phone records .
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w ursae majoris ( w uma ) variables are eclipsing overcontact binaries with orbital periods ranging from 0.2 1.0 day . these systems consist of main sequence stars with spectral a - k type sharing a common convective envelope due to filled roche lobes . in some cases these binaries host o or b spectral type component surrounded with a common radiative envelope , whose true physical understanding is still lacking . in low mass overcontact binaries , it has been argued that the secondary component is oversized with respect to its expected zams radius and at an advanced evolutionary stage ( stepien 2006a ) . the role of common envelope is to distribute the energy uniformly over the surface of the stars ( lucy 1968 ) , having similar brightness with a few percent difference exhibiting chromospheric activity ( vilhu & walter 1987 ) . the overcontact binaries are important astrophysical sources as they help to understand the underlying mechanism of the merging process ( eg . v 1309 sco ; tylenda et al . 2011 ) , stellar dynamo process ( eg . qian et al . 2005 ) , contributing in understanding the galactic structure because of their high number density ( 1/500 ms stars ; rucinski 2002 ) , binary evolution theories ( eg . yakut & eggelton 2005 ) and also serve as distance estimators ( rucinski & duerbeck 1997 ) . all the overcontact binaries are classified in three broad categories , a - type , w - type ( binnendijk 1970 ) and b - type ( csizmadia & klagyivik 2004 ) . in the a - type , the less massive component eclipses the massive one causing the primary minimum and an opposite scenario is observed in case of the w - type . in general , a - types often have low mass ratio ( q @xmath3 0.3 ) , relatively long orbital periods ( p @xmath4 0.3 days ) , whereas w - types have mass ratios , q @xmath4 0.3 and short orbital periods ( p @xmath3 0.3 days ) . it * had been suggested earlier that a - types are in an * advanced evolutionary stage compared to the w - types ( eg . hilditch 1989 ) but later was overruled as a - types have more mass and angular momentum ( gazeas & niarchos 2006 ) . but a number of overcontact binaries known to harbor a third component causing the sinking of angular momentum and hence the discrepancy in the evolutionary status could be resolved by constraining their age . the high temperature difference @xmath4 1000 k between the components in overcontact binaries forms the basis for b - type classification ( csizmadia & klagyivik 2004 ) and * systems in this class * are also known as poor thermal overcontact systems ( rucinski & duerbeck 1997 ) . many of the close binaries of several types and overcontact binaries light curves exhibit asymmetry in the brightness of maximum light , known as * oconnell * effect ( * oconnell * 1951 ; milone 1969 ; davidge & milone 1984 ) and is often associated with a dark spot on the primary component . the strong evidence for the presence of the spot comes from the study of h@xmath0 line in overcontact binary systems . the first detailed study was performed by barden ( 1985 ) on four w uma systems showing that the h@xmath0 line is a strong signature of the magnetic - associated activity in these systems . the study of h@xmath0 line is also important as the magnetic field plays a key role in the evolution of overcontact binaries via the magnetic braking process ( stepien 1995 ) . the presence of spots are related to the chromospheric activity causing the filling of the h@xmath0 line and varying the equivalent width along with the orbital period ( kaszas et al . moreover due to this activity overcontact binary systems are also good x - ray emitters ( mcgale et al . 1996 ; stepien 2001 ; chen et al . 2006 ) and the related x - ray emission is connected to the stellar dynamo activity arising from the synchronous fast rotating convective common envelope ( gondoin 2004 ) . an illustrative study of vw cep ( kaszas et al . 1998 ) , ae phe and yy eri ( maceroni et al . 1994 ; vilhu and maceroni 2007 ) clearly suggest that the activity is related to primary / massive component ( as it has deep convective zones ) which is in agreement with the theoretical studies ( rucinski 1992 , 1994 ) . deep ( f @xmath5 50 % ) low mass ratio ( q @xmath6 0.25 ) overcontact binaries ( dlmr ) are considered to be important sources and are possible progenitors for fk com - type and blue stragglers ( qian et al . * although * a different naming / classification * was * adopted , most of them are a - type overcontact binaries . they have a period domain ranging from [email protected] ( j13031 - 0101.9 ) to [email protected] ( kn per ) . qian et al . ( 2006 ) found that few of the systems undergo secular period decrease . the coupled action of angular momentum loss ( aml ) and thermal relaxation oscillation ( tro ; lucy 1976 ; flannery 1976 ; robertson & eggleton 1977 ) in the overcontact binary , leads to increase in the lifetime of overcontact phase . at this stage the binary can meet hut s criteria i.e. j@xmath8 @xmath4 1/3 j@xmath9 ( hut 1980 ) or can encounter dynamical instability ( rasio & shaprio 1995 ) which results in merging of the components . such mergers are rare but v1309 sco can be considered as a prototype for such events . tylenda et al . ( 2011 ) concluded that v1309 sco was a cool overcontact binary system and instabilities caused the secular period decrease of about 24.5 min over a duration of six years . based on the formation models of the cool overcontact binaries ( stepien 2004 , 2006a , b , 2009 ) , stepien ( 2012 ) concluded that the loss of mass and angular momentum through magnetic winds played a crucial role in the merging process . the variability of asas j082243 + 1927.0 ( v1 ) at @xmath0@xmath10=08@xmath11 22@xmath12 [email protected] and @xmath14@xmath10=+19@xmath15 26@xmath16 58@xmath17 was discovered in asas ( asas j082243 + 1927.0 ; pojamanski 2002 ) . later light curves of this system were reported by gettel et al . ( 2006 ) with star i d 10100383 , pepper et al . ( 2008 ; i d kp400793 ) and terrell et al . ( 2012 ) . due to the small orbital period 0.@xmath728 and shape of the light curve , it was classified as an ew type variable . the colours of the variable are b - v = 0.57 ( terrell et al . 2012 ) , j - h = 0.31 and h - k = 0.33 ( gettel et al . 2006 ) . due to the large scatter in the earlier reported light curves , the nature of the eclipses and presence of any oconnell effect were not clearly visible . v1 exhibits total eclipses and hence mass ratio can be accurately estimated for such systems ( terrell & wilson 2005 ) . moreover , study of the h@xmath0 line among overcontact binaries has been rare when compared to photometry . keeping this objective in view , we performed v band ccd photometry and spectroscopy concentrating on the variation of h@xmath0 line at various phases for this variable . the v band photometric observations were made on january 19 , 20 and 22 , 2013 with the 2.0-m telescope of the iucaa - girawali observatory ( igo ) . iucaa faint object spectrograph and camera ( ifosc ) was used which employes a 2k @xmath18 2k , thinned , back - illuminated ccd with a pixel size of 13.5@xmath19 m , gain 1.5e@xmath20 / adu and read out noise 4e@xmath20 . the ccd provides a 10.5@xmath21 @xmath18 10.5@xmath21 image with a plate scale of 0.3 arcsec pixel@xmath22 . the observations were performed in the air mass range 1.01 1.80 . the images were acquired in bessell s v filter with an integration time of 10 s. the extinction corrections were not * applied * as the comparison is close to the variable and of similar brightness ( see figure 1 ) and transformation to the standard system was not done . the figure 1 shows the location of variable ( v1 ) , comparison ( tyc 1386 - 1630 - 1 ) and check ( tyc 1386 - 121 - 1 ) stars . the v magnitudes and colour indexes of variable , comparison and check stars are shown in table 1 . these stars were selected as they are nearby and of similar brightness to v1 . the magnitude difference between variable and comparison stars along with check and comparison stars observed on 19 january 2013 are shown in figure 2 . the magnitude difference between check and comparison stars was found to be constant @xmath1 0.02 @xmath23 0.002 . spectroscopic observations were performed on december 56 , 2013 , with the 2.3-m vainu bappu telescope ( vbt ) of the vainu bappu observatory ( vbo ) equipped with the optomechanics research ( omr ) spectrograph along with a detector of 1k @xmath18 1k ccd . the obtained spectra cover a range of 3000 centered around h@xmath0 line with a 600 lines / mm dispersion resulting in a resolution of 2.6 /pixel . the exposure time was 3540 min for both the variable v1 and a spectrophotometric standard ( bd+08 2015 ) . the _ fene _ arc lamp was observed for wavelength calibrations . we used different packages made available in _ iraf _ to reduce the data . in both photometric and spectroscopic reduction , bias and flat - field correction were made and later _ apphot _ and _ onedspec _ packages were used to measure the magnitudes and to obtain the spectra . later the spectra were normalized for further studies . we determined the period of the variable using _ period04 _ package ( lenz & breger 2005 ) and times of minima using the kwee & van woerden s method ( 1956 ) . the ephemeris determined for the variable is min@xmath24 = 2456312.2997(89 ) + [email protected](2)e . the obtained period is similar to the one reported in previous studies . the v band light curve covering all the phases with 418 data points is shown in figure 3 . visual inspection of the light curve clearly shows the total eclipse at secondary minimum along with an oconnell effect notable at phase 0.25 . based on the colour indices of the variable , i.e. , b - v=0.57 and j - h=0.31 , the primary component temperature was fixed at 5960 k. the photometric solutions were obtained using the latest version of wilson - devinney ( w - d ) code v2013 with an option of non - linear limb darkening via a square root law ( wilson & devinney 1971 ; wilson 1979 ; wilson 1990 ; van hamme & wilson 2007 ; wilson 2008 ; wilson & van hamme 2013 ) . the new code has 60 parameters among which 50 are * adjustable*. in this version , a high precision star spot algorithm is incorporated which allows * one * to develop time varying spots by adjusting times of onset , size , and disappearance which affects the light curve * ; * the new code also permits the spot to drift , independent of star rotation ( wilson 2012 ) . the distance of binaries can be estimated via direct distance estimation ( dde ) process using a control integer ifcgs=1 in the code , but to adjust this parameter , the code requires well calibrated photometry in at least two passbands ( wilson et al . we adopted the following * model and * method ( eg . ravi , sriram & vivekananda rao 2012 ; shanti , sriram & vivekananda rao 2013 ) . a convective outer envelope was assumed for both components ; gravity darkening co - efficients @xmath25 = 0.32 ( lucy 1967 ) and bolometric albedos @xmath26 = 0.5 ( rucinski 1969 ) were taken as fixed parameters . the value of the limb darkening coefficients of components @xmath27 were fixed at 0.64 ( van hamme 1993 ) for v band . the adjustable parameters were the following : temperature of secondary component @xmath28 , orbital inclination ( _ i _ ) , the dimensionless potentials of the primary component @xmath29= @xmath30 and the bandpass luminosity of the primary star @xmath31 . the overcontact configuration i.e. mode 3 option of w - d method was used to determine the parameters after overruling the mode 2 option . + as no spectroscopic mass ratio was determined for this variable , the grid search method was adopted . the total eclipse at secondary minimum suggests that it is a low mass ratio system ( 0.1 @xmath3 q @xmath3 0.2 ) ( terrell & wilson 2005 ; wilson 2006 ) . we searched the parameter space of q in the range of 0.02 @xmath3 q @xmath3 10.0 along with other adjustable parameters . during the computations , the solutions gradually converged from detached ( mode 2 ) to overcontact ( mode 3 ) configuration . the resulting sum of the weighted square deviations , @xmath32 over the selected range of mass ratio q were noted and a minima at q = 0.121 was observed . later , it was made as an adjustable parameter along with others in the differential correction routine . during the computations , high residuals were observed at phase 0.25 and hence a solution with a dark spot over primary / secondary component and group of spots were tested to model the observed oconnell effect . first computation was * done * with an * application of a dark spot * over * the * primary , which resulted in lowering the residuals ( dark 1 in table 2 ; fig . 3 , top panel ) . a relatively dark and large spot over * the * secondary , resulted in a good fit ( dark 2 in table 2 ) . we also attempted a solution by varying the spot longitude over * the * primary ( ifsmv1=1 ) and drift of the spot was made independent * of * the star rotation ( eg . f1@xmath33=0.80 ) but the overall solution remained almost the same except that the longitude and latitude of the spot varied slightly for the fixed size of the spot . if f1@xmath33 is varied below 0.8 then the solution was found to be diverging . we also found that * the * secondary temperature is higher than the primary s although an occultation is observed at secondary minimum ( table 2 ) . based on the solutions , we found that the best value of mass ratio to be q @xmath1 0.106 . to check the consistency of the solution , the temperatures and inclination were varied by 510% and no significant difference in the respective solutions were found . the table 2 shows the photometric elements obtained from the best solutions and the corresponding theoretical fits are shown in figure 3 . the fill - out factor or degree of overcontact parameter , f = @xmath34 was derived for all the solutions and were found to be in the range of @xmath1 6172 % . the obtained values of q and f reveals that the variable is a low mass ratio overcontact binary with a high degree of geometrical contact . + the h@xmath0 line was observed at phases : 0.15 , 0.33 , 0.51 and 0.81 for the variable and its mean equivalent width ( ew ) was found to be 1.60 @xmath23 0.13 . figure 4 shows the variation of the h@xmath0 profiles with respect to the spectrophotometric standard . it is evident that at phase 0.51 , the variable h@xmath0 line is possibly filled - in with respect to the standard star at a significance of about 2.7@xmath35 whereas at other phases , * the * filled - in feature is relatively weak . the * fill - in * , especially at phase 0.5 is possibly caused by chromospheric emission . this is also supported by the observed oconnell effect exhibited in the light curve due to * the * presence of a dark spot on the primary or secondary . as continuous spectroscopic observations are lacking , determining the differences in the profiles arising from individual component and obtaining the h@xmath0 ew for each component were not possible . however at phase 0.5 , when the secondary is being eclipsed , we argue that the ew@xmath36@xmath37 = 1.09 of h@xmath0 line is most probably arising from the primary component . the intrinsic equivalent width of the line at phase 0.5 was calculated using the relation + ew@xmath36@xmath37=a @xmath18 ew@xmath36 and ew@xmath38@xmath37=a @xmath18 ew@xmath38 , where a = 1 + q@xmath39 ( @xmath40)@xmath41 ( webbink 2003 ) , ew@xmath36 , ew@xmath38 are measured equivalent widths and ew@xmath36@xmath37 , ew@xmath36@xmath37 are intrinsic equivalent widths . a small sample of periods and ews for overcontact binaries viz . ae phe , yy eri ( vilhu and maceroni 2007 ) , vw cep ( kaszas et al . 1998 ) suggests a correlation between the two parameters ( fig . 4 ) . since the conclusion of presence of the h@xmath0 filled - in was limited to visual inspection and hence a continues spectroscopic monitoring of the variable is important to unveil the nature of the h@xmath0 line . the v band light curve shows a total eclipse at secondary minimum and exhibits * the oconnell * effect . these features were not clear in the previous published light curves . however the variable oconnell effect is not unusual in overcontact binaries . based on the derived solutions , perhaps the variable can be classified as a w - type w uma system , * as the * secondary temperature was found to be relatively higher ( table 2 ) . however the shape of the light curve , derived mass ratio q @xmath10.106 and f @xmath1 72% indicate that the variable is a low mass ratio overcontact binary with a high degree of geometrical contact , a unique characteristic of a - type w uma systems . the best fit solutions indicate the presence of a dark spot on primary or secondary ( table 2 ) . the presence of spots do not affect the mass ratio of a system and hence an a - type classification is justifiable . in general , the occultation at secondary minimum suggests that the primary is hotter than the secondary component but our solutions indicate that the secondary is slightly hotter . this phenomenon is probably caused due to the migration of the spot on the stellar surface , responsible for causing variable oconnell effect and could * account * for reversed minima , making an a - type to a w - type and vice - versa . qian & yang ( 2005 ) reported a - type to w - type and then w - type to a - type features in the light curves of the overcontact binary fg hya . overall , there are three such low mass ratio overcontact binaries which resembles these features viz . fg hya ( zola et al . 2010 ; qian & yang 2005 ) , v802 aql ( samec et al . 2004 ; yang et al . 2008 ) and v902 sgr ( samec & corbin 2002 ) . if the dark spot / spots are responsible for the ambiguity of v1 s classification ( a - type or w - type ) , a radial velocity spectroscopy is necessary to unveil the true nature of the source . the combination of low mass ratio ( q @xmath6 0.25 ) and high filling factor in overcontact binaries ( f @xmath5 50 % ) are considered to be the progenitors for fk com - type and blue straggler stars . the gradual decrease of mass ratio makes these systems evolve as single fast rotating stars due to the inducement of darwin s instability when the orbital angular momentum is more than three times of spin angular momentum , i.e. , j@xmath9 @xmath5 3 j@xmath42 ( hut 1980 ) . the merging is also possible when the degree of overcontact is high which helps in causing the dynamical instability ( rasio & shapiro 1995 ) . table 3 lists the high filling factor ( f @xmath4 50% ) and low mass ratio ( q @xmath3 0.25 ) overcontact binary systems . for these systems , we studied the correlation between the mass ratio and period ( fig . interestingly , we found systematic trends along the mass ratio with a cut - off . we fitted a broken power law model @xmath43 , where p1 is break ( a value of log q ) , p2 and p3 are power - law indexes before and after the break and p4 is the normalization . we found that the break is at log q = -1.071 @xmath23 0.04 and the error bar indicates the 90 % confidence limit . the line in figure 5 shows the best fit and obtained a cut - off mass ratio q = 0.085 . the mass transfer from primary to secondary results in a short period high mass ratio systems ( i.e. trend below q@xmath2 @xmath6 0.085 ) . however above q@xmath2 @xmath4 0.085 , as a system has to evolve from high to low am state , the simultaneous processes of mass transfer from primary to secondary and mass loss of secondary component and partial transferred mass from lagrangian point l2 can drive the system towards short period low mass system configuration . moreover as period decreases , the critical roche lobes shall be shrinking , making f to increase and finally evolve as a single rapid rotator supported from the sample ( table 3 ) i.e. high f values are associated with low mass ratio systems . theoretically a critical mass ratio has been predicted because till now no overcontact binaries with low q @xmath3 0.07 have been discovered . rasio & shaprio ( 1995 ) constrained a minimum mass ratio of 0.09 and while considering the rotation of the secondary , the critical mass ratio , q@xmath2 lowers to 0.076 ( li & zhang 2006 ) . assuming convective and radiative envelope in both secondary and primary , the q@xmath2 was found to be around 0.0940.109 ( arbutina 2007 ) and taking into account differential rotation of the primary , the q@xmath2 converges at 0.0700.074 . the mass ratio of the variable v1 is found to be @xmath10.106 , close to the theoretically predicted values . we have also plotted the period - colour relation for 110 overcontact binary systems from the data of terrell et al . the line in figure 6 shows the short period blue envelope ( spbe ) theoretical line i.e. b - v@xmath44 = 0.04 @xmath18p@xmath45 ( rucinski 1997 ) and the variable v1 is found to be above it . any overcontact binary above spbe line has to further evolve towards longer period and cooler temperature ( towards lower right in figure 6 ) . overall , the variable v1 is an important overcontact binary source as it has the low mass ratio and high fill - out factor ( see table 3 ) and these kinds of sources are considered to be at a stage of merger , potentially forming a fk - com / blue straggler type star . moreover the solutions suggest that the a dark spot is affecting the occultation at secondary minimum and hence v1 is an interesting source for future observations which may unveil the variable oconnell effect . the h@xmath0 6563 line emission in overcontact binaries is sparsely studied . this line along with ca h and k , fe ii and the mg ii resonance line in the uv are the primary indicators of chromospheric activity in solar type stars.the strong absorption of the h@xmath0 line forms the basis of a zero point for the measure of chromospheric activity , however few stars are found to be associated with the weak level of activity and shows weak absorption . the fill - in effect is generally explained as the consequence of chromospheric activity which decreases the equivalent width of the h@xmath0 absorption line . this scenario was first observed in overcontact binaries v566 oph , ah vir and w uma by barden ( 1985 ) . the h@xmath0 variation in vw cep was studied by barden ( 1985 ) , frasca et al . ( 1996 ) and kaszas et al . ( 1998 ) and concluded that activity is associated with the primary rather than secondary component in overcontact binaries . from fig . 4 it is clear that at phase 0.5 the h@xmath0 is more filled - up suggesting a high level of chromospheric activity associated with the primary . the conclusion was based on the visual inspection , hence a continues spectroscopic monitoring is needed to explain the nature of the h@xmath0 profile . because h@xmath0 line ew varies due to chromospheric activity , we studied the correlation between period and ew for a few overcontact binaries ( fig . it is clear that as the orbital period decreases ( i.e fast rotation ) , the ew of the h@xmath0 line is found to be decreasing ( which is measure of increasing activity ) , leading towards the saturation at lower periods . overcontact binaries are potential sources to estimate the distance of binary systems . rucinski & duerbeck ( 1997 ) introduced * a * calibration scheme to determine the distance of overcontact binaries and the best calibration is given by m@xmath46= -4.44 log p + 3.02 ( b - v)@xmath47 + 0.12 , where m@xmath46 can be predicted with an uncertainty of up to @xmath23 0.25 mag . using this relation and m@xmath46 = 11.61 , the distance was estimated to be @xmath1279 pc . gettel et al . ( 2006 ) also estimated the distance to the variable to be @xmath1306 pc based on the relation log d = 0.2 v@xmath48 - 0.18 log ( p ) -1.60 ( j - h ) + 0.56 . considering the error of @xmath230.25 mag in the value of m@xmath46 , the difference in estimated distances is found to be within the error bars . based on well studied overcontact binaries , gazeas ( 2009 ) obtained three dimensional correlations as stated below : + log @xmath49 = 0.725 log p - 0.076 log q + 0.365 + log @xmath50 = 0.725 log p + 0.924 log q + 0.365 + log @xmath51 = 0.930 log p - 0.141 log q + 0.434 + log @xmath52 = 0.930 log p + 0.287 log q + 0.434 + log @xmath53 = 2.531 log p - 0.512 log q + 1.102 + log @xmath54 = 2.531 log p + 0.352 log q + 1.102 + using these correlations ( gazeas 2009 ) , the absolute parameters obtained are @xmath55= 1.10 @xmath23 0.08 m@xmath56 , @xmath57= 0.17 @xmath23 0.05 m@xmath56 , @xmath58= 1.15 @xmath23 0.04 r@xmath56 , @xmath59= 0.42 @xmath23 0.02 r@xmath56 , @xmath31= 1.66 @xmath23 0.24 l@xmath56 and @xmath60= 0.22 @xmath23 0.02 l@xmath56 . in conclusion , we report the photometric solution and variation of the h@xmath0 line of a overcontact binary asas j082243 + 1927.0 . the high value of f and low q suggests that it is a low mass ratio overcontact binary with high degree of contact , making it as one of the potential sources for merger studies . we found a cut - off mass ratio in mass ratio period plane at q@xmath2 = 0.085 , closely matching with various theoretically predicted ones . the h@xmath0 line varied at different phases showing the relatively fill - in effect at secondary minimum and is arising arguably from the primary component . a small sample of h@xmath0 ew and period of overcontact binaries shows a correlation indicating a saturation towards chromospheric emission . a spectroscopic radial velocity study is essential to resolve the ambiguity of the variable s classification ( a - type or w - type ) , to confirm the photometric mass ratio and to constrain the explicit nature of filled - in h@xmath0 line . moreover a long term photometric study of this variable is necessary to know its secular period variation and explore the variable oconnell effect . we acknowledge the director of iucaa , pune and the director of iia , bengaluru for allocating the time for the observations at igo and vbo . we also thank dr . vijay mohan , iucaa for helping us with the observations . we thank the referee for the valuable comments and suggestions which improved the quality of the paper . k.s acknowledge the support from ugc - bsr research start - up grant , government of india . ccccccc name & v & b - v + + v1 ( variable)&11.62 @xmath230.13&0.57@xmath23 0.01 + tyc 1386 - 1630 - 1 ( comparison ) & 11.58 @xmath230.13&0.48 @xmath230.08 + tyc 1386 - 121 - 1 ( check ) & 11.99 @xmath23 0.21 & 0.54 @xmath230.08 + ccccccc parameters & & unspotted & dark 1 & dark 2 + + a@xmath61 = a@xmath62 & & 0.50 & 0.50 & 0.50 + g@xmath61 = g@xmath62 & & 0.32 & 0.32 & 0.32 + @xmath63(k ) & & 5960 & 5960 & 5960 + @xmath28 ( k ) & & 6080 @xmath2334 & 6078 @xmath23 30 & 6038@xmath2328 + q & & 0.110 @xmath230.004 & [email protected]&[email protected] + i@xmath64 & & 75.60 @xmath231.04 & [email protected]&[email protected] + @xmath65 & & 1.9218 @xmath230.0039 & [email protected]&[email protected] + fill - out factor ( f % ) & & 73.82 @xmath231.32 & [email protected]&[email protected] + @xmath66 & pole & 0.5278 @xmath230.0036 & [email protected] & [email protected] + & side & 0.5865 @xmath230.0055 & 0.6128 @xmath230.0019 & [email protected] + & back & 0.6068 @xmath230.0057 & 0.6349 @xmath230.0020 & [email protected] + @xmath67 & pole & 0.1985 @xmath230.0154 & [email protected] & 0.1879 @xmath230.0090 + & side & 0.2062 @xmath230.0182 & [email protected] & 0.2059 @xmath230.0109 + & back & 0.2365 @xmath230.0348 & [email protected] & 0.2089 @xmath230.0219 + @xmath68 & & [email protected]&[email protected]&[email protected] + spot colatitude ( @xmath69 ) & & & [email protected] & 79.45 @xmath231.37 + spot longitude ( @xmath69)&&&[email protected]&[email protected] + spot radius(@xmath69)&&&[email protected]&30.93 @xmath231.69 + t@xmath70/t@xmath71 & & & 0.74 @xmath230.04 & 0.64 @xmath230.07 + @xmath72w(o - c)@xmath73 & & 0.0088 & 0.0024&0.0022 + acerbi , f. , barani , c. , & martignoni , m. 2014 , new astronomy , 31 , 1 acerbi , f. , barani , c. , & martignoni , m. 2011 , raa , 11 , 843 arbutina b. , 2007 , mnras , 377 , 1635 arbutina b. , 2009 , mnras , 394 , 501 barden , s. c. 1985 , , 295 , 162 binnendijk , l. 1970 , vistas astron . , 12 , 217 broens , e. 2013 , , 430 , 3070 chen , w. p. sanchawala , k. & chiu , m. c. 2006 , , 131 , 190 csak , b. , kiss , l. l. , vinko , j. , & alfaro , e. j. 2000 , a&a , 356 , 603 csizmadia , s. , & klagyivik , p. 2004 , a&a , 426 , 1001 davidge & milone 1984 , apjs , 55 , 571 essam , a. , saad , s. m. , nouh , m. i. , dumitrescu , a. , elkhateeb , m. m. & haroon , a. 2010 , new astronomy , 15 , 227 flannery , b. p. 1976 , apj , 205 , 217 frasca , a. , sanfilippo , d. & catalano , s. 1996 , 313 , 532 gazeas , k. d. , & niarchos , p. g. 2006 , , 370 , 29 gazeas , k. d. 200 , communications in asteroseismology , 159 , 129 gettel , s. j. , geske , m. t. , & mckay , t. a. 2006 , , 131 , 621 goderya , s. n. , leung , k. c. , & schmidt , e. g. 1997a , ap&ss , 246 , 291 goderya , s. n. , leung , k. c. , & schmidt , e. g. 1997b , ap&ss , 254 , 295 gondoin , p. 2004 , a&a , 415 , 1113 hilditch , r. w. 1989 , ssrv , 50 , 289 hut , p. 1980 , a&a , 92 , 167 kaszas , g. , vinko , j. , szatmary , k. , hegedus , t. , gal , j. , kiss , l.l . & borkovits , t. 1998 , a&a , 331 , 231 kilmartin , p. m. , bradstreet , d. h. , & koch , r. h. 1987 , aj , 319 , 334 kwee , k. k. , & van woerden , h. 1956 , ban , 12 , 327 lapasset , e. 1980 , aj , 85 , 1098 labadorf , c. , samec , r. g. , figg , e. r. , fronaulkner , d. r. , & van hamme , w. 2009 , baas , 41 , 212 lenz , p. , & breger , m. 2005 , commun . 144 , 41 li , l. , & zhang , f. 2006 , , 369 , 2001 li , l .- f . , liu , q .- y . , zhang , f .- h . , & han , z .- w . 2001 , aj , 121 , 1091 lucy , l. b. 1967 , z. astrophys . , 65 , 89 lucy , l. b. 1968 , 151 , 1123 lucy , l.b . 1976 , , 205 , 208 maceroni , c. , vilhu , o . , vantveer , f. , & vanhamme , w . 1994 , a&a , 288 , 529 mcgale , p. a. , pye , j. p. , & hodgkin , s. t. 1996 , mnras , 280 , 627 maciejewski , g , czart , k. , niedzielski , a. , & karska , a 2003 , ibvs , 5400 , 1 milone , e. f. 1969 , communication of the konkoly observatory , 65 , 457 nelson , r. h. , terrel , d. , & milone , e. f. 2014 , new astronomy reviews , 59 , 1 oconnell 1951 , pub . riverview college obs . , 2 , 85 pepper , j. , stanek , k. z. , pogge , r. w. , latham , d. w. , depoy , d. l. , siverd , r. , poindexter , s. , & sivakoff , g. r. 2008 , , 135 , 907 pojamanski , g. 2002 , acta astron . , 52 , 397 pribulla , t. , chochol , d. , rovithis - livaniou , h. , & rovithis , p. 1999 , a&a , 345 , 137 pribulla , t. & vanko , m. 2002 , coska , 32 , 79 pribulla , t. , rucinski , s. m. , blake , r. m. , et al . 2009 , aj , 137 , 3655 qian , s. b. & liu , q. 2000 , a&a , 355 , 171 qian , s. b. 2001a , mnras , 328 , 635 qian , s. b. 2001b , mnras , 328 , 914 qian , s. b. & yang , y. g. 2004 , aj , 128 , 2430 qian , s. b. & yang , y. , zhu , l. , he , j. & yuan , j. 2006 , ap&ss , 304 , 25 qian , s. b. , liu , l. , soonthornthum , b. , zhu , l. y. & he , j. j. 2007 , aj , 134 , 1475 qian , s. b. , he , j .- j . , soonthornthum , b. , et al . 2008 , aj , 136 , 1940 qian , s. b. , yang , y .- g . , soonthornthum , b. , et al . 2005 , aj , 130 , 224 qian , s. b. & yang , y. 2005 , mnras , 356 , 765 qian , s. b. , liu , l. , zhu , l. y. , he , j. j. , yang , y. g. & bernasconi , l. 2011 , aj , 141 , 151 rasio , f. a. , & shapiro , s. l. 1995 , apj , 438 , 887 ravi , k. y. , sriram , k. & vivekanada rao , p. 2012 , basi , 40 , 51 robertson , j. a. , & eggleton , p. , , 179 , 359 rucinski , s. m. 1969 , acta astron . , 19 , 245 rucinski , s.m . , 1992 , aj 103 , 960 rucinski , s.m . , 1994 , aj 107 , 738 rucinski , s.m . , 1997 , aj , 113 , 407 rucinski , s. m. , & duerbeck , h. w. 1997 , pasp , 109 , 1340 rucinski , s. m. , & lu , w. 1999 , aj , 118 , 2451 rucinski , s. m. 2002 , pasp , 114 , 1124 samec , r. g. , & corbin , s. 2002 , the observatory , 122 , 22 samec , r. g. , martin , m. , & faulkner , d. r. , 2004 . stars , 5527 samec , r. g. , labadorf , c. m. , hawkins , n. c. , faulkner , d. r. & van hamme , w. 2011 , aj , 142 , 117 senavci , h. v. , nelson , r. h. , ozavci , i. , selam , s. o. , & albayrak , b. 2008 , new astronomy , 13 , 468 shanti priya , d. sriram , k. & vivekananda rao , p. 2013 , raa , 13 , 465 stepien , k. 1995 , , 274 , 1019 stepien , k. , schmitt , j. h. m. m. , & voges , w. 2001 , a&a , 370 , 157 stepien , k. 2004 , in stars as suns : activity , evolution and planets , ed . a. k. dupree , & a. o. benz , asp , iau symp . , 219 , 967 stepien , k. 2006a , acta astron . , 56 , 199 stepien , k. 2006b , acta astron . , 56 , 347 stepien , k. 2009 , mnras , 397 , 857 stepien , k. 2006 , act . , 56 , 199 stepien , k. 2012 , a&a , 531 , a18 szalai , t. , kiss , l. l. meszaros , sz . vinko , j. & csizmadia , sz . 2007 , a&a , 465 , 943 terrell , d. , gross , j. , & conney , w. r. jr . 2012 , , 143 , 99 terrell , d. , & wilson , r. e. 2005 , ap&ss , 296 , 221 tylenda , r. , hajduk , m. , kaminski , t. et al . 2011 , a&a , 528 , a114 ulas , b. , kalomeni , b. , keskin , v. , kose , o. , & yakut , k. 2012 , new astronomy , 17 , 46 van hamme , w. 1993 , , 106 , 209 van hamme , w. & wilson , r. e. 2007 , apj , 661 , 1129 vilhu , o. , & walter , f. m. 1987 , , 321 , 958 vilhu , o. , & maceroni , c. 2007 , iuas , 240 , 719 webbink r.f . 2003 , in turcotte s. , keller s. , keller s.c . , carallo r.m . ( eds . ) , asp conf.ser.vol.293 , 3d stellar evolution contact binaries , astron.soc.pac . , san francisco , p.76 . wilson , r. e. , & devinney , e. j. 1971 , apj , 166 , 605 wilson , r. e. 1979 , apj , 234 , 1054 wilson , r. e. 1990 , apj , 356 , 613 wilson , r. e. , & van hamme , w. 2013 , computing binary star observables wilson , r. e. 2006 , aspc , 349 , 71 wilson , r. e. 2008 , apj , 672 , 575 wilson , r. e. , & van hamme , w. & terrel , d. 2010 , apj , 723 , 1469 wilson , r. e. 2012 , aj , 144 , 73 yakut , k. , kalomeni , b. , & ibanoglu , c. 2004 , a&a , 417 , 725 yakut , k. & eggleton , p. p. 2005 , , 629 , 1055 yang , y .- g . , qian , s .- b . , & zhu , l .- y . 2005 , aj , 130 , 2252 yang , y .- g qian , s .- b . , & zhu , l .- y .. liu , l. , & nakajima , k. 2008 , pasj , 60 , 803 yang , y .- g . , qian , s .- b . , zhu , l .- y . , & he , j .- j . 2009 , aj , 138 , 540 yang , y .- g . , qian , s .- b . & soonthornthum , b. 2012 , aj , 143 , 122 yang , y .- g . , qian , s .- b . , zhang l. y. , dai h. f. , soonthornthum , b. 2013 aj , 146 , 35 zhu , l .- y . , qian , s .- b . , soonthornthum , b. , & yang , y .- g . 2005 , aj , 129 , 2806 zola , s. , gazeas , k. , kreiner , j. m. , ogloza , w. , siwak , m. , koziel - wierzbowska , d , & winiarski m. 2010 , mnras ,
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both high precision ccd photometric and h@xmath0 line studies are presented for an overcontact binary asas j082243 + 1927.0 . the light curve exhibits a total eclipse at secondary minima along with an * oconnell * effect . the light curve was modeled using the wilson devinney code and the best solution provides the mass ratio q @xmath1 0.106 and fill - out factor f @xmath1 72% .
these parameters indicate that the system is a low mass ratio overcontact binary with a high degree of geometrical contact .
the h@xmath0 line equivalent width varied at different phases and it is found that the line is possibly filled - in at secondary minima . from a small sample of overcontact binaries , we found a correlation between the orbital period and h@xmath0 line equivalent width of the primary component .
based on a sample of high filling factor and low mass ratio contact binaries , a mass ratio cut - off is observed at q@xmath2 = 0.085 in mass - ratio period plane . it was observed that below q@xmath2 @xmath3 0.085 , period decreases with an increase in q and above it , period increases as the mass ratio increases .
interestingly , the observed mass ratio cut - off value lies close to the critical mass ratio range as predicted in the literature .
the observational evidence of the cut - off of the mass ratio and its variation with orbital period * are * discussed in terms of mass transfer and angular momentum loss . based on the results
, we suggest that , asas j082243 + 1927.0 is at the verge of merger , eventually forming a fast rotating star .
[ firstpage ] binaries : close binaries ; binaries eclipsing ; stars : individual ( asas j082243 + 1927.0 )
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a prompt , 9-th magnitude , optical flash has recently been detected ( @xcite ) accompanying the gamma ray burst ( grb ) grb990123 . the most natural explanation of this flash is emission from a reverse shock propagating into fireball ejecta shortly after it interacts with surrounding gas ( @xcite ) . although optical - uv emission from the reverse shock accompanying , or following shortly after , the @xmath0-ray emission has been predicted ( @xcite ) based on the simplest fireball models for gamma - ray bursts ( grbs ) , the intensity of optical emission could not have been reliably predicted due to the uncertainty in reverse shock parameters . the observations of grb990123 suggest that the electron and magnetic field energy fractions in the reverse shock are similar to those in the forward shock propagating into the surrounding gas and producing the long term afterglow . this in turn implies that strong optical flashes accompanying @xmath0-ray emission is a generic grb characteristic . this is consistent with previous non - detection of optical flashes ( @xcite ) given the wide grb luminosity function ( @xcite ) and the fact that grb990123 is in the top 0.3% of the batse brightness distribution ( @xcite ) . there is evidence in several grb afterglows for significant dust extinction , which may imply an association of grbs with star forming regions ( @xcite ) . it has been shown that if grbs indeed reside in such environment , the ionizing x - ray and uv afterglow radiation may lead to time dependent ( on hour time scale ) absorption ( @xcite ) and emission ( @xcite ) line features . if h@xmath11 is present near the grb , conspicuous 11101650absorption will be imprinted on the spectrum , followed by uv fluorescence ( @xcite ) . on longer time scales , up to @xmath12 yr , grb photo - ionization may lead to indicative recombination line features , which may allow identification of grb remnants in nearby galaxies ( @xcite ) . here , we discuss dust sublimation by the optical - uv flash accompanying the grb . since grb990123 was an exceptionally bright burst , with exceptionally high intrinsic @xmath0-ray luminosity , we first discuss in 2 the optical - uv flash itself to obtain the scaling of optical - uv luminosity with burst energy , and to estimate the prompt optical - uv emission at energies above the @xmath13 ev energy of the prompt emission observed by akerlof et al . we show that if electron and magnetic field energy fractions in the reverse shock are similar among different bursts , then an optical - uv flash of luminosity @xmath14 , where @xmath2 is the beaming solid angle , is expected for a typical grb . dust destruction by thermal sublimation is discussed in 3 , and other possible destruction mechanism are considered in 4 . it should be emphasized that the analysis of the physics of dust destruction is independent of the model for optical - uv emission , and therefore of the discussion presented in 2 . however , we present numerical results for dust destruction based on the characteristic optical - uv luminosity derived in 2 . in 5 we discuss dust infra - red emission . the implications of our results are discussed in 6 . in fireball models of grbs ( see @xcite for a recent review ) , the energy released by an explosion is converted to kinetic energy of a thin baryonic shell expanding at an ultra - relativistic speed . after producing the grb , the shell impacts on surrounding gas , driving an ultra - relativistic shock into the ambient medium . after a short transition phase , the expanding blast wave approaches a self - similar behavior ( @xcite ) , where the expansion lorentz factor decreases with radius as @xmath15 . the initial interaction of fireball ejecta with surrounding gas produces a reverse shock which propagates into and decelerates the fireball ejecta . transition to self - similar behavior occurs on a time scale comparable to the reverse shock crossing time of the ejecta . the long term afterglow is produced by the forward , expanding shock that propagates into the surrounding gas . this shock continuously heats fresh gas and accelerates relativistic electrons , which produce the observed radiation through synchrotron emission . the most natural explanation of the optical flash is that it is due to synchrotron emission of electrons which are accelerated by the reverse shock , during the transition to self - similar behavior . once the reverse shock crosses the ejecta , the ejecta expand and cool adiabatically . thus , emission from the fireball ejecta is suppressed after the transition to self - similar expansion . since the optical flash is produced when the reverse shock crosses the ejecta , the plasma emitting the radiation expands with a lorentz factor which is close to that given by the blandford - mckee self - similar solution , @xmath16 , where @xmath17 is the fireball energy and @xmath18 is the surrounding gas number density . the characteristic time over which radiation emitted by the fireball at radius @xmath19 is observed by a distant observer is @xmath20 ( @xcite ) . the plasma lorentz factor during optical flash emission @xmath21 where @xmath22 erg , @xmath23 s is the observed duration , @xmath24 , and @xmath25 . transition to self - similar expansion occurs on time scale @xmath26 comparable to the longer of the two time scales set by the initial conditions : the ( observed ) grb duration @xmath27 and the ( observed ) time @xmath28 at which the self - similar lorentz factor equals the original ejecta lorentz factor @xmath29 , @xmath30 . that is , @xmath31 . \label{delta_t}\ ] ] during the transition , the unshocked fireball ejecta propagate at the original expansion lorentz factor , @xmath32 , and the lorentz factor of plasma shocked by the reverse shock in the rest frame of the unshocked ejecta is @xmath33 . if @xmath34 then @xmath35 , the reverse shock is relativistic , and the lorentz factor associated with the random motion of protons in the reverse shock is @xmath36 . if @xmath37 then @xmath38 , and the reverse shock is not relativistic . nevertheless , the following argument suggests that the reverse shock speed is not far below @xmath39 , and that the protons are therefore heated to relativistic energy , @xmath40 . the comoving time , measured in the fireball ejecta frame prior to deceleration , is @xmath41 . the expansion lorentz factor is expected to vary across the ejecta , @xmath42 . such variation would lead to expansion of the ejecta , in the comoving frame , at relativistic speed . thus , at the deceleration radius , @xmath43 , the ejecta width exceeds @xmath44 . since the reverse shock should cross the ejecta over a deceleration time scale , @xmath45 , the reverse shock speed must be close to @xmath39 . we therefore conclude that the lorentz factor associated with the random motion of protons in the reverse shock is approximately given by @xmath46 for both @xmath38 and @xmath35 . for protons shocked by the forward shock @xmath47 , and therefore the ratio between thermal energy per proton in the reverse and forward shocks is @xmath48 . below we use this relation to derive the emission characteristics of the reverse shock by scaling the exact analytic results given for the forward shock emission in @xcite . if the fraction of thermal energy carried by electrons , @xmath49 , and magnetic field , @xmath50 , is similar in the forward and reverse shocks , then the frequency of peak synchrotron emission from the reverse shock is smaller than that of the forward shock by a factor @xmath51 . this is due to the fact that the energy density behind the reverse and forward shocks are similar , so that similar @xmath50 implies similar magnetic field strength in both regions , while similar @xmath49 implies @xmath52 . using eq . ( 10 ) of @xcite for the forward shock peak frequency , we find that the reverse shock emission peaks at a frequency @xmath53 ( measured at the grb redshift ) . here @xmath49 and @xmath50 are the values relevant for the reverse shock , which in general may differ from those of the forward shock . the numerical values we have used are those characteristic of the forward shock ( @xcite ) . as demonstrated below , the cooling time of electrons in the reverse shock , radiating in the optical - uv range , is long compared to the fireball expansion time . in this case , the peak synchrotron intensity is proportional to the product of magnetic field strength and number of radiating electrons ( and independent of the electron lorentz factor ) . the number of radiating electrons in the reverse shock is larger than in the forward shock by a factor @xmath54 . this can be deduced from the following considerations . the proton random lorentz factor in the reverse shock is @xmath54 times smaller than that in the forward shock . since the energy density in both regions is similar , the density of protons , and therefore of electrons , in the reverse shock is higher than that in the forward shock by the same factor . in addition , the width of the shocked fireball ejecta is similar to that of the shell of shocked surrounding gas , since both shocks propagate relativistically in the shocked plasma frame . thus , if @xmath50 is similar in the reverse and forward shock , the peak synchrotron intensity @xmath55 is higher in the reverse shock by a factor @xmath54 . using eq . ( 11 ) of @xcite we find for the reverse shock @xmath56 for a flat universe with zero cosmological constant , and @xmath57 . the decay of grb990123 optical flux implies an electron energy distribution @xmath58 with @xmath59 ( @xcite ) , for which the intensity at @xmath60 is @xmath61 . thus , the optical [ @xmath62 hz ] intensity is @xmath63 for the parameters of grb990123 , @xmath64 s and @xmath65 erg ( based on the @xmath0-ray fluence @xmath66 ) , we obtain @xmath67 jy , approximately twice the peak observed flux . thus , the observed optical flash can be naturally explained by the simplest fireball model , provided the reverse shock parameters @xmath49 and @xmath50 are similar to those implied for the forward shock by afterglow observations , @xmath68 and @xmath69 . if this is typical , than for a typical grb , with @xmath70 erg and @xmath71 s , we find @xmath72 where @xmath73 is the frequency at the redshift of the grb . the form ( [ l_nu ] ) is valid for frequencies @xmath73 larger than the peak frequency @xmath74 , and smaller than the frequency @xmath75 above which emission is dominated by electrons for which the cooling time is shorter than the dynamic time . since the energy density in the reverse and forward shock regions is similar , @xmath75 ( at the grb redshift ) is given by @xmath76 at frequencies @xmath77 the spectrum steepens to @xmath78 . strong optical - uv emission requires , similar to grb @xmath0-ray production , large initial lorentz factor , @xmath79 : eq . ( [ delta_t ] ) implies that @xmath80 for @xmath81 , and the optical - uv luminosity given by eq . ( [ l_nu ] ) drops as @xmath82 . this implies that the plasma emitting the optical - uv photons must also be expanding at high lorentz factor , @xmath83 [ see eqs . ( [ delta_t],[gamma ] ) ] , and strong optical - uv emission may be confined , similar to @xmath0-rays , to a cone around the line of sight of opening angle @xmath84 . this may be the case if , e.g. , the fireball is a jet of opening angle @xmath85 . a jet of finite opening angle expands as if it were a conical section of a spherical fireball , as long as @xmath86 . thus , the analysis presented above is valid for a jet - like fireball . in this case , the energy @xmath17 in the above equations should be understood as the energy that the fireball would have carried if it were spherically symmetric , and the optical - uv , as well as @xmath0-ray , emission is confined to a small solid angle @xmath87 with optical - uv luminosity given by @xmath88 . it should be noted that prompt optical emission has been observed for only 1 grb to date , and therefore may not be typical of grbs . indeed , fireballs in low density environments with @xmath89 would not be expected to produce strong prompt emission [ see eq . ( [ eq : f_v ] ) ] . however , we note from eq . ( [ eq : f_v ] ) that optical flashes are expected for typical grbs with @xmath90 and @xmath91 , and should not be limited to unusually luminous grbs such as grb990123 . as discussed in 3.3 below , in dense regions radiation with @xmath92 ev will largely go into photoionizing h and h@xmath11 , and photons in the 7.513.6 ev range will primarily be absorbed by h@xmath11 , which is rovibrationally excited by ultraviolet pumping ( draine 1999b ) . we therefore first discuss , in 3.1 and 3.2 , sublimation of dust grains under the assumption that dust heating is dominated by @xmath93 ev photons , and discuss the contribution of 7.550 ev photons to dust sublimation in 3.3 . using equations ( [ l_nu ] ) and ( [ nu_c ] ) , we find that for typical grb parameters , the prompt luminosity in the 17.5 ev range is @xmath94 and the 7.550 ev prompt luminosity is @xmath95 . since our simple analysis overestimates the flux of grb990123 by a factor @xmath96 , we will take the typical 17.5 ev luminosity to be @xmath97 , and the 7.550 ev luminosity to be @xmath98 . consider a grain of radius @xmath99 located at a distance @xmath19 from a transient source of radiation radiating a 17.5 ev power @xmath100 into a solid angle @xmath2 . if the radiation from the grb is `` beamed '' into @xmath101 , we will consider only dust grains within the beam . note that we expect the beaming of optical - uv emission from the forward shock to be similar to that of gamma - ray emission . the grain temperature @xmath102 is determined by @xmath103 where @xmath104 is the effective optical depth for attenuation of the optical - uv flash , @xmath105 is the density of the grain material , @xmath106 is the mean atomic mass , @xmath107 is the chemical binding energy per atom , @xmath108 is the usual planck - averaged absorption efficiency , and @xmath109 is the absorption efficiency factor averaged over the 17.5 ev spectrum of the optical - uv flash . for the grain radii @xmath110 expected in dense clouds , we will assume @xmath111 since we are interested in energy depositions large enough to sublime grains , the heat capacity of the grain has been neglected in equation ( [ eq : balance ] ) . the sublimation rate can be approximated by @xmath112 guhathakurta & draine ( 1989 ) have estimated @xmath113 , @xmath114 for mg@xmath11sio@xmath115 , and @xmath116 , @xmath117 for graphite . we adopt @xmath118 , @xmath119 , and @xmath120 as representative values for refractory grains . if we assume the grain temperature @xmath102 is approximately constant over the time @xmath121 , then the condition for the grain to be completely sublimed during this time would be @xmath122 , where @xmath123 } \approx 2300 { \rm k } \left [ 1 + 0.033\ln\left ( \frac{a_{-5 } } { \delta t_{1 } } \right ) \right ] \label{eq : t_c}\ ] ] where @xmath124 . equivalently , the grain survival time at temperature @xmath102 is @xmath125{\rm\ , s } \quad . \label{eq : t_sur}\ ] ] the infrared emissivity is quite different for graphite and silicate materials ( draine & lee 1984 ; draine 1999a ) . for the temperature range of interest for dust sublimation , @xmath126 , we approximate @xmath127 with @xmath128 and 0.3 for astronomical silicate and graphite , respectively . we adopt equation ( [ eq : q_t ] ) with @xmath129 as representative of refractory grain material . for @xmath132 , so that sublimation cooling dominates over radiative cooling , the grain temperature is @xmath133\quad , \label{eq : t_sub}\ ] ] where @xmath134 . for @xmath135 radiative cooling dominates and the grain temperature is @xmath136 where @xmath137 is the instantaneous attenuation of the 17.5 ev flash by intervening absorption . if attenuation by intervening material can be neglected , grains are completely sublimed out to a destruction radius @xmath138 where @xmath139 is the radius where the unattenuated flash can heat grains to the critical temperature @xmath140 . for our nominal parameters , eq . ( [ eq : t_c ] ) and ( [ eq : t_r = s ] ) give @xmath141 , so that radiative cooling dominates at @xmath142 , and the critical distance @xmath139 is given by @xmath143 in the optically - thin limit , then , the optical - uv flash from the grb will destroy dust _ in the beam _ out to a substantial distance . while the 17.5 ev photons from a grb may be capable of destroying dust out to @xmath144 pc in the optically - thin limit , in dusty regions ( such as molecular clouds ) attenuation of the radiation by dust grains before they are destroyed will limit the grain destruction to a smaller radius . the dust optical depth is a function @xmath145 of both space and time , as the grb flash `` burns '' its way through the cloud . the attenuation of @xmath517.5 ev photons is dominated by dust . to estimate the effect of high optical depth , we will assume that only a single dust type is present . let @xmath146 be the dust destruction radius : all grains at @xmath147 are destroyed by the heating effects of the optical - uv flash . we approximate the 17.5 ev emission from the grb as a rectangular pulse . at radii @xmath148 , the _ leading _ edge of the optical - uv pulse is attenuated by the dusty medium through which it propagates , but the _ trailing _ edge of the pulse is unattenuated since it propagates through a dustless medium , and we are neglecting the effects of gas - phase absorption . rather than solve for @xmath145 , we will simplify the problem by assuming that the effects of extinction can be approximated as primarily a _ narrowing _ of the optical pulse , retaining a rectangular profile . the problem then reduces to determination of a function @xmath149 , the fraction of the flash energy which is absorbed by dust interior to radius @xmath19 , and survival of grains at radius @xmath19 when irradiated by a radiation field @xmath150 for a time @xmath151 . the function @xmath149 then satisfies @xmath152\over \delta t}\ ] ] where @xmath153 is the survival time of a grain irradiated by the unattenuated radiation field . in the unattenuated portion of the optical pulse , grains are heated to temperatures given by equations ( [ eq : t_sub],[eq : t_rad ] ) with @xmath154 . the dust destruction radius @xmath146 is then determined by the condition @xmath155\delta t$ ] . figure [ fig : r_d ] shows @xmath146 as a function of cloud density @xmath156 for a standard dust - to - gas ratio @xmath157 , for several different values of the grb 17.5 ev luminosity ( @xmath158 ) , duration ( @xmath159 ) , and characteristic dust grain radius ( @xmath160 ) . the radius @xmath161 for `` typical '' grb parameters is shown as the heavy curve . we see that if a grb occurred in a dusty region , the optical - uv flash from the grb can clear out a substantial amount of dust which lies in the beam . radiation with @xmath92 ev may ionize h and h@xmath11 . neglecting dust opacity , the prompt flash will be able to photoionize a mass @xmath162 or out to a radius @xmath163 where @xmath164 , and we approximate the flash by a rectangular pulse of duration @xmath165 . we have taken the typical 13.650 ev luminosity to be @xmath166 . the dust destruction radius , @xmath161 , is compared with @xmath167 in figure [ fig : r_d ] . at densities for which @xmath168 , typically @xmath169 ( characteristic of star - forming regions ) , our neglect of absorption of @xmath170 photons by dust in estimating @xmath167 is justified , because the dust will be destroyed by absorption of @xmath171 photons prior to arrival of most of the ionizing photons at a given point in the cloud . at these densities , @xmath172 photons are fully absorbed by the gas in a region smaller than the dust destruction zone , thus justifying our neglect of ionizing radiation when estimating @xmath161 . since the number of 7.513.6 ev photons is comparable to the number of @xmath170 photons , this also shows that , for densities @xmath173 , the 7.513.6 ev photons will be mainly absorbed by h@xmath11 , and can be neglected for purposes of dust destruction . at lower densities , where @xmath174 , some fraction of @xmath175 ev radiation would also contribute to dust sublimation . at these densities the grain temperature near @xmath161 is determined by radiative cooling and therefore @xmath176 , where @xmath177 is the uv luminosity available for dust heating . since @xmath178 , the contribution of @xmath175 ev radiation may increase @xmath161 by up to @xmath179 . because of the large fluence of energetic photons , dust destruction could , in principle , also result from extreme ionization of the dust grain . high degrees of ionization could result in fission of the grain , or emission of individual ions from the grain surface by the process known as `` ion field emission '' . for an approximately spherical grain of radius @xmath99 , charged to a potential @xmath180 , the tensile stress averaged over a cross section @xmath181 is @xmath182 . if the maximum tensile stress which the grain material can support is @xmath183 , then the potential gradient and grain charge @xmath184 are limited by @xmath185 @xmath186 we note that if a grain with @xmath187 fissioned into two halves , each of the fragments would have @xmath188 and there would therefore be no further fragmentation unless additional ionization took place . a grain contains @xmath189 electrons . if the mean atomic number is @xmath190 and the photoionization cross section per electron is @xmath191 , then substantial grain destruction by this process would require a fluence @xmath192 . grain fission would therefore occur within a radius @xmath193 where we have taken the fluence @xmath194 . note that this process changes the grain size distribution and therefore affects the optical extinction curve , but grain fission alone would not appreciably reduce the ultraviolet extinction , and might even increase it . ideal materials have tensile strengths @xmath195 , so that a coulomb explosion would not take place until the surface electric field reaches @xmath196 . however , in the laboratory electric fields exceeding @xmath197 are observed to result in `` ion field emission '' , where individual ions are emitted from the sample ( muller & tsong 1969 ) . as a result , if the grain tensile strength @xmath198 , intense irradiation by @xmath199 photons will first cause the grain to charge up to @xmath200 , and each subsequent ionization will result in emission of an ion ( assuming that electron capture is negligible during the @xmath201 s of the gamma ray burst ) . if the mean atomic number is @xmath190 and the photoionization cross section per electron is @xmath191 , then substantial grain destruction by this process would require a fluence @xmath202 . grain destruction by ion field emission would therefore occur only within a radius @xmath203 ion field emission is evidently much less important than sublimation . for clouds of mass @xmath204 extending to @xmath205 pc , most of the optical - uv energy absorbed by sublimated dust is re - radiated in the infra - red , typically around @xmath206 m . this is due to the fact that at distances larger than @xmath207 pc radiative cooling of dust grains dominates over sublimation cooling [ see eqs . ( [ eq : t_rad],[eq : t_r = s ] ) ] , and to the fact that for @xmath204 grains are heated to the temperature @xmath208 k required for complete sublimation out to a distance @xmath209 pc [ see eq . ( [ eq : t_c ] ) , fig . 1 ] ( for @xmath210 , optical photons are completely absorbed at distances @xmath211 pc , where sublimation cooling dominates and only a small fraction of the absorbed energy is re - radiated ) . at radii where radiation cooling dominates , the grain temperature drops approximately as @xmath212 . the strong dependence of grain survival time @xmath213 on temperature , eq . ( [ eq : t_sur ] ) , then implies a sharp transition , i.e. over a distance @xmath214 , between the region at @xmath148 where @xmath213 is much smaller than the flash duration @xmath215 , to the region at @xmath216 , where @xmath217 . since the energy radiated by grains is proportional to @xmath218 , infra - red emission of sublimated dust is dominated by emission from grains just outside @xmath146 . if the optical depth for uv photons due to dust at @xmath219 is @xmath220 , then most of the flash energy would be absorbed by dust and re - radiated in the infra - red . most of the infra - red radiation would escape the cloud , and may therefore be detected , if the infra - red optical depth is not high , @xmath221 . for @xmath222 , the requirements @xmath220 and @xmath221 may be written as @xmath7 . in order to estimate the dust infra - red luminosity in the case where @xmath221 and @xmath220 , let us first assume that optical - uv flash emission is beamed into a small solid angle around the line of sight , @xmath223 . the observed duration of the infra - red emission is then @xmath224 , where @xmath225 pc is the radius out to which grains are heated to @xmath226 k. this may be written in the form @xmath227 day , which is valid for @xmath228 as well as for @xmath229 . the infra - red luminosity is given by the ratio of the flash energy absorbed by dust , @xmath230 erg , and the observed duration @xmath231 , @xmath232 . the luminosity of the prompt optical - uv emission accompanying grb @xmath0-ray emission is given by eqs . ( [ l_nu ] ) and ( [ delta_t ] ) . for typical grb parameters we expect an optical - uv flash with 17.5 ev luminosity @xmath233 , assuming isotropic emission . such a uv flash can destroy dust by sublimation out to an appreciable distance , @xmath209 pc ( see figure [ fig : r_d ] ) , and may clear the dust out of @xmath234 of molecular cloud material , where @xmath2 is the solid angle into which the optical - uv emission is beamed , and where dust is sublimed . if grb sources indeed lie in dusty regions , then the extinction would decrease with time during prompt optical - uv emission , over tens of seconds . detection of such time dependent extinction would provide strong constraints on the grb environment . the destruction of dust implies that existing , or future , observations of not - heavily - reddened fireballs are not inconsistent with grbs being associated with star formation . we have shown in 5 that if the optical depth due to dust beyond @xmath6 pc is of order unity , most of the uv flash energy is absorbed and re - radiated in the infra - red , typically at @xmath8 m . the resulting infra - red luminosity , @xmath9 , extends over an apparent duration of @xmath10 day . for grbs at @xmath235 , therefore , k - band photometry may reveal thermal emission from dust grains . in fact , such emission may already have been observed in grb970228 ( @xcite ) and grb980326 ( @xcite ) . in both cases , a deviation from a power law decline of optical flux , which at early time is consistent with synchrotron emission from shock accelerated electrons , is observed at @xmath236 d delay . as the flux drops below @xmath237jy , a new infra - red emission component is revealed , with a flux @xmath238jy between @xmath239 m and @xmath206 m for grb970228 and @xmath240jy at @xmath241 m for grb980326 . in both cases , the spectrum is modified at this time to @xmath242 at @xmath243 m . the infra - red flux is of the same order of magnitude estimated for dust grain emission , and the spectrum is consistent with dust emission peaking at @xmath244 m ( at the source redshift ) , provided grb980326 is at redshift @xmath245 . we note that for the typical parameters adopted in this paper , dust emission is expected to peak at somewhat longer wavelength , @xmath246 m . however , since the grain properties are not well known , dust emission can not be ruled out as an alternative to the proposal that the `` excess '' emission is due to a supernova ( @xcite ) we note also that the non - detection of optical emission from grb980326 at @xmath247 d , implies , under the dust emission hypothesis , beaming of the optical - uv flash to @xmath248 , consistent with the interpretation that the rapid , @xmath249 , flux decline is due to the fireball being a jet of small opening angle ( @xcite ) . we have shown in 2 that strong optical - uv emission requires , like grb @xmath0-ray production , large initial expansion lorentz factor , @xmath79 , which also implies that the plasma emitting the optical - uv flash must be expanding with @xmath83 . thus , if the fireball is a jet of finite opening angle , @xmath250 , then both @xmath0-ray and optical - uv emission will be confined to a small solid angle @xmath87 . in this case , dust would be evaporated only within a narrow cone around the line of sight . a jet - like fireball expands as a conical section of a spherical fireball , with @xmath251 , as long as @xmath252 . after deceleration to @xmath253 , the jet expands sideways , its opening angle increasing to @xmath254 and @xmath255 ( @xcite ) . at this stage radiation reaching us must travel a distance @xmath256 through gas which was not exposed to the initial flash , and which will still contain dust . for @xmath257 , and using @xmath258 , we have @xmath259 . thus , a significant increase in extinction over time would be observed if @xmath260 . confinement of @xmath0-ray and strong optical - uv emission to a small solid angle is not limited to the case of a jet - like fireball . it may also arise if @xmath29 , the initial expansion lorentz factor , is anisotropic . consider a fireball carrying similar energy per unit solid angle in all directions , with @xmath29 a decreasing function of angle with respect to the line of sight , such that @xmath261 for @xmath262 . in this case , optical - uv ( and @xmath0-ray ) emission would be suppressed at angles @xmath262 . the isotropic fireball energy per unit solid angle implies that , after a transition phase , the fireball would approach spherical expansion , with @xmath251 . at this stage , most of the radiation detected at time @xmath263 by a distant observer is produced by fireball plasma within a narrow ring of radius @xmath264 around the line of sight , where the fireball radius @xmath19 and expansion lorentz factor @xmath265 are related to @xmath263 through @xmath266 ( @xcite ) . thus , here too a significant increase in reddening is expected once @xmath267 drops below @xmath268 . for @xmath269 most of the radiation reaching us must pass through a path length @xmath270 of gas which was not exposed to the initial flash . finally , it should be noted that although optical flashes are expected for typical grbs [ with @xmath90 and @xmath271 , see eq . ( [ eq : f_v ] ) ] , prompt optical emission has been observed for only 1 grb to date and therefore may not accompany all grbs . indeed , fireballs in low density environments with @xmath89 would not be expected to produce strong prompt emission [ see eq . ( [ eq : f_v ] ) ] . in addition , if the fireball initial lorentz factor @xmath272 , reverse shock emission may be shifted to photon energies above 7.5 ev [ see eq . ( [ nu_m ] ) ] , where most photons are absorbed by h and h@xmath11 . akerlof , c. w. , _ et al . _ 1999 , gcn 205 blandford , r. d. , & mckee , c. f. 1976 , phys . fluids 19 , 1130 bloom , j. s. , _ et al . _ 1999 , astro - ph/9905301 bttcher , m. _ et al . _ 1998 , astro - ph/9809156 djorgovski , s. g. , _ et al . _ 1999 , gcn circular no . 289 draine , b.t . 1999a , http://www.astro.princeton.edu/@xmath5draine/dust/dust.diel.html draine , b.t . 1999b , submitted to apj ( astro - ph/99997232 ) draine , b.t . , & lee , h .- 1984 , , 285 , 89 fruchter , a. s. , _ et al . _ 1999 , , 516 , 683 galama , t. j. , _ et al . _ 1999 , astro - ph/9907264 ghisellini , g. _ et al . _ 1998 , astro - ph/9808156 granot , j , piran , t. & sari , r. 1998 , astro - ph/9808007 gruzinov , a. & waxman , e. 1999 , ap . j. 511 , 852 guhathakurta , p. , & draine , b.t . 1989 , , 345 , 230 hogg , d. w. & fruchter , a. s. , submitted to ap . j. ( astro - ph/9807262 ) . kouveliotu , c. _ et al . _ 1999 , nature , submitted krumholtz , m. , thorsett , s. e. , & harrison , f. a. , ap . j. * 506 * , l81 ( 1998 ) . kulkarni , s. r. , _ et al . _ 1998 , nature 395 , 35 loeb , a. & perna , r. 1998 , , 503 , l35 mau , s. & mo , h. j. 1998 , a&a 339 , l1 mszros , p. & rees , m. 1997 , ap . j. 476 , 232 mszros , p. & rees , m. 1999 , submitted to mnras , ( astro - ph/9902367 ) mszros , p. , rees , m. & papathanassiou , h. 1994 , , 432 , 181 metzger , m. _ et al . _ 1997 , iauc 6676 muller , e.w . , & tsong , t.t . 1969 , field ion microscopy ( new york : american elsevier ) paczyski , b. 1998 , ap . j. 494 , l45 panaitescu , a. & mszros , p. 1998 , , 501 , 772 park , h. s. _ et al . _ 1997 , , 490 , l21 perna , r. , & loeb , a. 1998 , , 501,467 perna , r. , raymond , j. , & loeb , a. 1999 , astro - ph/9904181 piran , t. 1998 , to appear in phys . reports ( atro - ph/9810256 ) reichart , d. e. 1999 , astro - ph/9906079 rhoads , j. e. 1999 , submitted to apj ( astro - ph/9903399 ) sari , r. , & piran , t. 1999a , submitted to apj ( astro - ph/9901338 ) sari , r. , & piran , t. 1999b , submitted to apj ( astro - ph/9902009 ) waxman , e. 1997a , ap . j. 485 , l5 waxman , e. 1997b , ap . j. 489 , l33 waxman , e. 1997c , ap . j. 491 , l19 wijers , r. a. m. j. , & galama , t. j. 1998 , astro - ph/98050341
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the prompt optical flash recently detected accompanying grb990123 suggests that , for at least some grbs , @xmath0-ray emission is accompanied by prompt optical - uv emission with luminosity @xmath1 , where @xmath2 is the solid angle into which @xmath0-ray and optical - uv emission is beamed .
such an optical - uv flash can destroy dust _ in the beam _ by sublimation out to an appreciable distance , @xmath3 pc , and may clear the dust out of as much as @xmath4 of molecular cloud material on an apparent time scale of @xmath5 ten seconds .
detection of time dependent extinction on this time scale would therefore provide strong constraints on the grb source environment .
dust destruction implies that existing , or future , observations of not - heavily - reddened fireballs are not inconsistent with grbs being associated with star forming regions . in this case , however , if @xmath0-ray emission is highly beamed , the expanding fireball would become reddened on a @xmath51 week time scale .
if the optical depth due to dust beyond @xmath6 pc from the grb is @xmath7 , most of the uv flash energy is converted to infra - red , @xmath8 m , radiation with luminosity @xmath9 extending over an apparent duration of @xmath10 day .
dust infra - red emission may already have been observed in grb970228 and grb980326 , and may possibly explain their unusual late time behavior .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
balanced detection provides a unique tool for many physical , biological and chemical applications . in particular , it has proven useful for improving the coherent detection in telecommunication systems @xcite , in the measurement of polarization squeezing @xcite , for the detection of polarization states of weak signals via homodyne detection @xcite , and in the study of light - atom interactions @xcite . interestingly , balanced detection has proved to be useful when performing highly sensitive magnetometry @xcite , even at the shot - noise level , in the continuous - wave @xcite and pulsed regimes @xcite . the detection of light pulses at the shot - noise level with low or negligible noise contributions , namely from detection electronics ( electronic noise ) and from intensity fluctuations ( technical noise ) , is of paramount importance in many quantum optics experiments . while electronic noise can be overcome by making use of better electronic equipment , technical noise requires special techniques to filter it , such as balanced detection and spectral filtering . even though several schemes have been implemented to overcome these noise sources @xcite , an optimal shot - noise signal recovery technique that can deal with both technical and electronic noises , has not been presented yet . in this paper , we provide a new tool based both on balanced detection and on the precise calculation of a specific pattern function that allows the optimal , shot - noise limited , signal recovery by digital filtering . to demonstrate its efficiency , we implement pattern - function filtering in the presence of strong technical and electronic noises . we demonstrate that up to 10 db of technical noise for the highest average power of the beam , after balanced detection , can be removed from the signal . this is especially relevant in the measurement of polarization - rotation angles , where technical noise can not be completely removed by means of balanced detectors @xcite . furthermore , we show that our scheme outperforms the wiener filter , a widely used method in signal processing @xcite . the paper is organized as follows . in section [ sec : pattern ] we present the theoretical model of the proposed technique , in section [ sec : experiment ] we show the operation of this tool by designing and implementing an experiment , where high amount of noise ( technical and electronic ) is filtered . finally in section [ sec : conclusions ] we present the conclusions . to optimally recover a pulsed signal in a balanced detection scheme , it is necessary to characterize the detector response , as well as the `` electronic '' and `` technical '' noise contributions @xcite . we now introduce the theoretical framework of the filtering technique and show how optimal pulsed signal recovery can be achieved . to model a balanced detector , see fig . [ fig : expsetup ] , we assume that it consists of 1 ) a polarizing beam splitter ( pbs ) , which splits the @xmath0 and @xmath1 polarization components to two different detectors 2 ) the two detectors pd@xmath2 and pd@xmath3 , whose output currents are directly subtracted , and 3 ) a linear amplifier because the amplification is linear and stationary , we can describe the response of the detector by impulse response functions @xmath4 . if the photon flux at detector @xmath5 is @xmath6 , the electronic output can be defined as @xmath7 where @xmath8 is the electronic noise of the photodiodes , including amplification . here , @xmath9 stands for the convolution of @xmath10 and @xmath11 , i.e. , @xmath12 . for clarity , the time dependence will be suppressed when possible . it is convenient to introduce the following notation : @xmath13 , @xmath14 , @xmath15 and @xmath16 . using these new variables , eq . takes the form @xmath17 from this signal , we are interested in recovering the differential photon number @xmath18 , where @xmath19 is the time interval of the desired pulse , with minimal uncertainty . more precisely , we want to find an estimator @xmath20 $ ] , that is unbiased @xmath21 , and has minimal variance @xmath22 . in order to make @xmath23 unbiased , we realize that it must linearly depend on @xmath24 . this because @xmath25 and @xmath24 are linear in both @xmath26 and @xmath27 . therefore , the estimator must have the form @xmath28 in eq . , @xmath29 refers to as _ pattern function _ , which describes the most general linear estimator . in this work , we will consider three cases : 1 ) a raw estimator , @xmath30 for @xmath31 and 0 otherwise ; 2 ) a wiener estimator , which makes use of a wiener - filter - like pattern function , @xmath32 , where @xmath33 represents the wiener filter in the time domain @xcite , and 3 ) a model - based pattern function estimator @xmath34 . notice that both @xmath33 and @xmath35 are defined in @xmath36 , allowing to properly choose a desired pulse . in what follows , we explicitly show how to calculate the model - based pattern function estimator @xmath35 . we assume that @xmath37 have known averages ( over many pulses ) @xmath38 , and similarly the response functions @xmath39 have averages @xmath40 . then the average of the electronic output reads as @xmath41 and @xmath42 . in writing eq . , we have assumed that the noise sources are uncorrelated . from this we observe that if a balanced optical signal is introduced , i.e. @xmath43 , the mean electronic signal @xmath44 is entirely due to @xmath45 . in order that @xmath23 correctly detects this null signal , @xmath35 must be orthogonal to @xmath45 , i.e. @xmath46 our second condition derives from @xmath47 which is in effect a calibration condition : the right - hand side is a uniform - weight integral of @xmath48 , while the left - hand side is a non - uniform - weight integral , giving preference to some parts of the signal . if the total weights are the same , the above gives @xmath49 . we note that this condition is not very restrictive . for example , given @xmath50 , and given @xmath35 up to a normalization , the equation simply specifies the normalization of @xmath35 . notice that the condition given by eq . may still be somewhat ambiguous . if we want this to apply for all possible shapes @xmath51 , it would imply @xmath52 const . , and would make the whole exercise trivial . instead , we make the physically reasonably assumption that the input pulse , with shape @xmath53 is uniformly rotated to give @xmath54 , @xmath55 . similarly , it follows that @xmath56 . we note that this assumption is not strictly obeyed in our experiment and is a matter of mathematical convenience : a path difference from the pbs to the two detectors will introduce an arrival - time difference giving rise to opposite - polarity features at the start and end of the pulse , as seen in fig . [ fig : restech](a ) . a delay in the corresponding response functions @xmath10 is , however , equivalent , and we opt to absorb all path delays into the response functions . in our experiment the path difference is @xmath57 , implying a time difference of less than 0.2 ns , much below the smallest features in fig . [ fig : restech](a ) . absorbing the constant of proportionality into @xmath35 , we find @xmath58 which is our calibration condition . we consider two kinds of technical noise : fluctuating detector response and fluctuating input pulses . we write the response functions in the form @xmath59 , for a given detector @xmath5 , where the fluctuating term @xmath60 is a stochastic variable . similarly , we write @xmath61 , where @xmath62 is @xmath63 or @xmath64 . by substituting the corresponding fluctuating response functions into eq . , the electronic output signal becomes @xmath65 where @xmath66 is the summed technical noise from both @xmath67 and @xmath68 sources . we note that the optical technical noise , in contrast to optical quantum noise , scales as @xmath69 , so that @xmath70 . in passing to the last line we neglect terms @xmath71 on the assumption @xmath72 , @xmath73 . we further assume that @xmath74 and @xmath75 are uncorrelated . we find the variance of the model - based estimator , @xmath76 , is @xmath77 with the first term describing technical noise , and the second one electronic noise . to compare against noise measurements , we transform eq . to the frequency domain . using parseval s theorem , see eq . ( [ eq : simp1mwm ] ) , we can write the noise power as @xmath78 our goal is now to find the @xmath79 that minimizes @xmath80 satisfying the conditions in eqs . and , which in the frequency space are @xmath81 @xmath82 the specific form of the solution is given in appendix [ sec : pattesolution ] . at 150 mhz ( blue dashed line ) and the amplified one @xmath83 at 5 mhz ( green solid line ) . for the sake of comparison , both pulses are normalized.,width=226 ] [ cols="^ " , ] to illustrate the performance of our technique when filtering technical noise , we introduce a high amount of noise about @xmath84 db above the shot noise level at the maximum optical power to the light pulses produced by the aoms . after balancing a maximum of 10 db remains in the electronic output , which is then filtered by means of the optimal pattern function technique . we have verified the correct noise filtering by comparing the results with shot - noise limited pulses . for this purpose , we compute @xmath85 , the variance of the optimal estimator for each power , and for each data set , the shot - noise limited and the noisy one . figure [ fig : res3 ] shows the computed noise estimation as function of the optical power for both . notice that the two noise estimations are linear with the optical power . moreover , we observe that both curves agree at @xmath86 , using the ratio of the slopes , which allows us to conclude that , by using this technique , we can retrieve shot - noise limited pulses from signals bearing high amount of technical noise . the experimental setup that we have implemented , see fig . [ fig : expsetup ] , can perform also as a pulsed signal polarimeter . for instance , it is possible to determine a small polarization - rotation angle @xmath87 from a @xmath88 linear polarized light pulse . along these lines , we make use of three estimators @xmath89 and @xmath90 to determine the amount of noise on the estimation of the polarization - rotation angle . from the obtained results , we show that the model - based estimator outperforms the other two . we proceed to calculate the noise on the polarization - rotation angle @xmath87 estimation , for this determination we calculate the variance of @xmath87 . we notice that the taylor approximation of the variance of @xmath91 is @xmath92 for small angles @xmath87 , the function @xmath91 is approximately linear on @xmath87 , so the contribution from higher order terms can be disregarded . therefore , the noise on the angle estimation is @xmath93 we can then compute this expression using the three before mentioned estimators . for such task we use the experimental data together with an analytical approximation of the derivative , that takes as input the measured data . figure [ fig : resangle ] depicts the noise angle estimation , showing that the optimal pattern function performs better than the other estimators when eliminating the technical noise and reducing the electronic noise . in particular , the based - model estimator surpasses the wiener estimator , which is a widely used method in signal processing @xcite . we have studied in theory and with an experimental demonstration , the optimal recovery of light pulses via balanced detection . we developed a theoretical model for a balanced detector and the noise related to the detection of optical pulses . we minimized the technical and electronic noise contributions obtaining the optimal ( model - based ) pattern function . we designed and implemented an experimental setup to test the introduced theoretical model . in this experimental setup , we produced technical noise in a controlled way , and retrieved shot - noise limited signals from signals bearing about 10 db of technical noise after balanced detection . finally , we compare against nave and wiener filter estimation for measuring rotation angles , and confirm superior performance of the model - based estimator . the results presented here might lead to a better polarization - rotation angle estimations when using pulses leading to probe magnetic atomic ensembles in environments with technical noise @xcite . this possibility is especially attractive for balanced detection of sub - shot - noise pulses @xcite , for which the acceptable noise levels are still lower . we note the inner - product form of parseval s theorem @xmath94 where the functions @xmath95 are the fourier transforms of @xmath96 , respectively . for any stationary random variable @xmath97 , @xmath98 ( if this were not the case , there would be a phase relation between different frequency components , which contradicts the assumption of stationarity ) . from this , it follows that @xmath99 we will minimize the noise power @xmath80 ( see eq . ) with respect to the pattern function @xmath79 using the two conditions ( see eq . and eq . ) . we solve this by the method of lagrange multipliers . for this , we write @xmath100 and then solve the equations @xmath101 the first equation reads @xmath102 with formal solution @xmath103 the second and third equations from eq . are the same as eq . and eq . above . the problem is then reduced to finding @xmath104 , @xmath105 which ( through the above ) , make @xmath79 satisfy the two constraints . substituting eq . into eq . and eq . , we find @xmath106 and @xmath107 where @xmath108 @xmath109 @xmath110 @xmath111 with @xmath112 . the solution to the set of eqs . and is then given by @xmath113 it should be noted that quantum noise is not explicitly considered in the model . rather , it is implicitly present in @xmath114 which may differ from their average values @xmath115 due to quantum noise . note that the point of this measurement design is to optimize the measurement of @xmath116 including the quantum noise in that variable . for this reason , it is sufficient to describe , and minimize , the other contributions . the wiener filter estimator @xmath117 can be derived from the frequency domain wiener filter output @xmath118 @xcite define as we thank f. wolfgramm , f. martn ciurana , j. p. torres , f. beduini and j. zieliska for helpful discussions . this work was supported by the european research council project `` aqumet '' , the spanish mineco project `` mago '' ( ref . fis2011 - 23520 ) , and by fundaci privada cellex barcelona . y. a. de i. a. was supported by the scholarship bes-2009 - 017461 , under project fis2007 - 60179 . bach , `` ultra - broadband photodiodes and balanced detectors towards 100 gbit / s and beyond , '' in `` optics east 2005 , '' ( international society for optics and photonics , 2005 ) , pp . 60,140b60,140b13 . youn , _ measurement of the polarization state of a weak signal field by homodyne detection _ ( intech , available from : http://www.intechopen.com/books/photodetectors/ , from the book @xcite , 2012 ) , chap . 17 , pp . 389404 . m. kubasik , m. koschorreck , m. napolitano , s. r. de echaniz , h. crepaz , j. eschner , e. s. polzik , and m. w. mitchell , `` polarization - based light - atom quantum interface with an all - optical trap , '' phys . rev . a * 79 * , 043,815 ( 2009 ) . v. g. lucivero , p. anielski , w. gawlik , and m. w. mitchell , `` shot - noise - limited magnetometer with sub - pt sensitivity at room temperature , '' arxiv * quant - ph * , 1403.7796 ( submitted to phys . a ) ( 2014 ) . n. behbood , f. m. ciurana , g. colangelo , m. napolitano , m. w. mitchell , and r. j. sewell , `` real - time vector field tracking with a cold - atom magnetometer , '' applied physics letters * 102 * , 173,504 ( 2013 ) . h. hansen , t. aichele , c. hettich , p. lodahl , a. i. lvovsky , j. mlynek , and s. schiller , `` ultrasensitive pulsed , balanced homodyne detector : application to time - domain quantum measurements , '' opt . * 26 * , 17141716 ( 2001 ) . p. j. windpassinger , m. kubasik , m. koschorreck , a. boisen , n. kj , e. s. polzik , and j. h. mller , `` ultra low - noise differential ac - coupled photodetector for sensitive pulse detection applications , '' measurement science and technology * 20 * , 055,301 ( 2009 ) . v. ruilova - zavgorodniy , d. y. parashchuk , and i. gvozdkova , `` highly sensitive pump probe polarimetry : measurements of polarization rotation , ellipticity , and depolarization , '' instruments and experimental techniques * 46 * , 818823 ( 2003 ) . t. ezaki , g. suzuki , k. konno , o. matsushima , y. mizukane , d. navarro , m. miyake , n. sadachika , h .- j . mattausch , and m. miura - mattausch , `` physics - based photodiode model enabling consistent opto - electronic circuit simulation , '' in `` electron devices meeting , 2006 . international , '' ( 2006 ) , pp . 14 . r. j. sewell , m. koschorreck , m. napolitano , b. dubost , n. behbood , and m. w. mitchell , `` magnetic sensitivity beyond the projection noise limit by spin squeezing , '' phys . * 109 * , 253,605 ( 2012 ) .
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we demonstrate a new tool for filtering technical and electronic noises from pulses of light , especially relevant for signal processing methods in quantum optics experiments as a means to achieve the shot - noise level and reduce strong technical noise by means of a pattern function .
we provide the theory of this pattern - function filtering based on balance detection .
moreover , we implement an experimental demonstration where 10 db of technical noise is filtered after balance detection . such filter can readily be used for probing magnetic atomic ensembles in environments with strong technical noise .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the link between a certain type iip supernova ( sn iip ) and its main - sequence progenitor is poorly known despite the widely accepted view that these supernovae arise from the initial mass range of @xmath5 ( e.g. , * ? ? ? the primary reasons for the uncertainty in this field are the small number of presupernova ( pre - sn ) mass determinations and the uncertain amount of matter lost through winds , presumably dominated by the red supergiant ( rsg ) wind . while the former is becoming well - studied , the latter uncertainty is related to our poor knowledge of the complicated physics of mass loss and the unsatisfactory situation with the empirical measurements of mass loss from rsgs . even for the most studied close massive rsg , betelgeuse , the range of observational estimates is large : @xmath6 yr@xmath2 @xcite . the situation for sne iip has some promise because the dominant mass loss at the rsg stage may be observed by the detection of circumstellar ( cs ) interaction at x - ray and radio wavelengths . the type iip sn 1999em was detected in x - rays with _ chandra _ @xcite , leading to the mass loss estimate @xmath7 yr@xmath2 ( assuming a wind velocity of 10 km s@xmath2 ) . a recent study of available x - ray and radio data for sne iip produced a range of mass loss rates of pre - sn iip @xmath8 yr@xmath2 @xcite . the application of these rates to the full rsg stage ( @xmath9 yr ) suggests the loss of @xmath10 for pre - sne iip . the wide range of estimates emphasizes the need for the individual determination of the wind density for each particular sn iip under consideration . unfortunately , this is not always possible since x - ray and radio observations of sne iip are often not available . here , we propose two new diagnostics for the wind density in sne iip that could help . both rely on spectroscopic observations of h@xmath0 and hei 10830 at the photospheric epoch . the first one is based on the fact that the interaction of sn ejecta with the wind results in the emission of x - rays from both forward and reverse shocks . the x - rays cause ionization and excitation of sn ejecta that may be revealed , e.g. , through specific emission lines @xcite . unfortunately , in sne iip the wind density is low and emission lines caused by cs interaction are extremely weak and can not be detected . we find , however , that the excitation of h and he produced by x - rays in sn iip ejecta turns out to be sufficient to be detected as high velocity ( hv ) absorption features in h@xmath0 and hei 10830 lines against the bright sn iip photosphere . this is the core of our proposed diagnostic for the wind in sne iip . the second proposed probe for cs interaction exploits the possibility that a cool dense shell ( cds ) might form at the sn / cs interface because of radiative cooling . the cds excited by x - rays could become visible as narrow hv absorption in h@xmath0 . the velocity of this absorption would provide a direct measure of the expansion velocity of the sn / cs interface , a valuable dynamical characteristic of the cs interaction . the identification of the expected hv lines in observed spectra is complicated by the presence of weak metal lines @xcite . however , we argue that hv lines of h@xmath0 and h@xmath11 have been observed in spectra of sne iip , as previously discussed by @xcite for sn 1999em . the paper is organized as follows . we start with a description of the interaction model and cs interaction effects in the h@xmath0 and hei 10830 absorption lines formed in the unshocked ejecta during the photospheric epoch ( [ sec - mod ] ) . we then compare our cs interaction models with the available spectra of sne iip in the h@xmath0 and hei 10830 lines and estimate the wind density for particular sne iip ( [ sec - inter ] ) . in [ sec - narrow ] we address the issue of the h@xmath0 absorption in the cool dense shell at the sn / cs interface of sne iip . we discuss implications of our models in the last section . the model for ejecta - wind interaction effects in h@xmath0 and he i 10830 lines formed in the sn ejecta consists of three major parts : ( i ) an interaction model that provides the dynamical evolution of the sn / wind interface and the x - ray emission from the reverse and forward shocks ; ( ii ) a model for the ionization and excitation of h and he in the unshocked sn ejecta irradiated by x - rays ; and ( iii ) the calculation of line profiles . we perform ( iii ) using either a standard sobolev approximation or the direct integration of the radiation transfer equation , depending on the validity of the sobolev approximation . the interaction of sn ejecta with cs wind leads to a canonical double - shock structure @xcite with the forward shock propagating in the cs gas and the reverse shock in the sn ejecta . we treat the cs interaction of ejecta in the thin shell approximation @xcite in which the double - shock layer is reduced to an infinitely thin shell . we assume that the freely expanding ( @xmath12 ) sn envelope has a sharp boundary at the velocity @xmath13 and begins to interact with a wind starting some moment ( @xmath14 day ) which corresponds roughly to the shock breakout phase . free expansion is expected to take several doubling times of the initial radius , or several days , to be set up , but the observations we are modeling are at later times . the maximum velocity @xmath13 is set by the escape of radiation from the shock wave at shock breakout . for the density distribution @xmath15 in the sn iip envelope we use an analytical expression , @xmath16 that closely approximates the combination of an inner plateau and outer power law tail found in hydrodynamic models ( e.g. , * ? ? ? the parameters @xmath17 and @xmath18 are determined by the kinetic energy @xmath19 , ejecta mass @xmath20 , and @xmath21 : @xmath22 where @xmath23 the power law index @xmath21 lies in the range @xmath24 and is henceforth set to be @xmath25 . we adopt the boundary velocity for a typical sn iip of @xmath26 km s@xmath2 , which is consistent with models of shock breakout @xcite . also , the blue edge of the h@xmath11 emission in sn 1999gi on day 1 was observed to be at 15,000 km s@xmath2 @xcite . hydrodynamical modeling predicts that at the shock breakout phase a thin dense shell forms at the outer boundary @xcite . the physics of the ` boundary shell ' formation is in the transition from the adiabatic to the radiative regime of shock wave propagation in the outermost layers of the exploding star . simple considerations of radiative diffusion @xcite give an estimate for the shell mass : @xmath27 where @xmath28 is the progenitor star radius and @xmath29 is the velocity at shock breakout , before free expansion has been established . during the acceleration phase the boundary shell is subject to the rayleigh - taylor ( rt ) instability @xcite and therefore could be corrugated or even fragmented . however , it is not clear whether the rt instability results in full fragmentation ; 2d or 3d hydrodynamic modeling of the phenomenon is lacking . we will treat the boundary shell in our model as an intact thin spherical shell which is already in place at the initial moment . the mass of the boundary shell is presumably equal to the mass of the outer envelope ( @xmath30 ) with an extrapolated power law @xmath31 . this provides a good estimate of the shell mass for sn 1999em , @xmath32 , in agreement with hydrodynamical modeling @xcite . the boundary shell is considered below as a seed for the cds that may , or may not , grow further due to radiative cooling at the reverse shock . to calculate the x - ray emission from the reverse and forward shocks we assume that the postshock layers are uniform and their densities are 4 times larger than the corresponding preshock density . we allow the electron temperature to be partially equilibrated , assuming the following interpolation : @xmath33 where @xmath34 is the electron - ion temperature equilibration time @xcite and @xmath35 is the equilibrium shock temperature with @xmath36 . the lower limit @xmath37 is indicated by observational data on supernova remnants with shock velocities of @xmath38 km s@xmath2 @xcite . the reverse shock is equilibrated during the photospheric phase for a mass loss rate of @xmath39 yr@xmath2 with a wind velocity of @xmath40 km s@xmath2 , while the forward shock is not . since the reverse shock is equilibrated and dominates the x - ray luminosity , the uncertainty in @xmath41 does not affect our results . the shock x - ray luminosity is expressed in terms of the kinetic luminosity as @xmath42 . the efficiency parameter @xmath43 is smoothly interpolated between extreme cases , @xmath44 , where @xmath45 is the cooling time and is calculated for each shock using the cooling function of @xcite . the wind is assumed to be steady with the density @xmath46 . since all the interaction effects for low wind velocities ( @xmath47 ) are determined by the wind density @xmath48 , we use hereafter the dimensionless wind density parameter @xmath49 , where @xmath50 is the mass loss rate in units of @xmath3 yr@xmath2 and @xmath51 is the wind velocity in units of @xmath40 km s@xmath2 . the interaction model for a sn iip with the wind density @xmath52 is illustrated in fig . [ f - dyn ] . the two cases shown have the same ejecta mass @xmath53 but different kinetic energy : @xmath54 erg and @xmath55 erg . the high energy case corresponds presumably to sn 1999em @xcite , while the low energy case illustrates sn 2004dj , as we will see below . accordingly , we adopt @xmath56 km s@xmath2 in the high energy case and @xmath57 km s@xmath2 in the low energy case . the model with parameters for sn 1999em reproduces the observed ( absorbed ) _ chandra _ x - ray luminosity of sn 1999em in the @xmath58 kev band @xcite for a distance @xmath59 mpc @xcite quite satisfactorily ( fig . [ f - dyn ] ) , which indicates that @xmath52 is a reasonable choice for the wind density . the estimated galactic absorption , @xmath60 @xmath61 @xcite , is small and does not significantly affect the observed x - ray luminosity . the x - ray luminosity from the reverse shock dominates in both cases . the typical electron temperature for the reverse shock at @xmath62 d is @xmath63 kev , while for the forward shock it is @xmath64 kev . the early behavior of the velocities differs from the self - similar evolution @xcite because of the ejecta density cut - off in our model and the presence of the boundary shell at the initial moment . the mass of the boundary shell is @xmath65 and @xmath66 in the high and low energy models , respectively . the total mass of the cds attained at the time of the termination of the radiative regime in the reverse shock ( @xmath67 ) is @xmath68 and @xmath69 with @xmath70 d and @xmath71 d for the high and low energy models , respectively . remarkably , the reverse shock is adiabatic in the self - similar models for the same parameters through this epoch . this fact is related to the different dynamics of the self - similar model at an early stage , which results in a lower preshock density at the reverse shock . the spectrum of x - rays is described below using a standard approximation @xmath72 . we adopt @xmath73 for the forward shock , a reasonable approximation for temperature @xmath64 kev @xcite , while for the low temperature reverse shock we take @xmath74 . the latter choice qualitatively describes the calculated energy distribution of x - rays with a significant contribution of emission lines for the electron temperature @xmath75 kev @xcite . the hydrogen and helium ionization in a sn iip atmosphere at the photospheric epoch is time - dependent @xcite . to take this effect into account , we use a hydrogen atom represented by four levels and continuum . all the essential radiative and collisional processes for hydrogen are taken into account , including non - thermal ionization and excitation induced by x - ray absorption . the excitation of the helium @xmath76s level is determined by a simple model with two excited levels ( 2@xmath77s and 2@xmath78s ) . we consider only major processes for the population of the 2@xmath78s level : nonthermal excitation and ionization with subsequent recombination into triplet states , depopulation by the collisional deexcitation transition @xmath76s@xmath79s , penning ionization @xcite and continuum absorption in the 10830 line ( 2@xmath78s2@xmath78p transition ) with the subsequent spontaneous transition 2@xmath78p1@xmath77s terminated by photon escape . the energy balance includes the energy deposition due to the absorption of x - rays and the hydrogen photoionization from excited levels by photospheric radiation . the cooling includes adiabatic loss ( @xmath80 work ) , free - free , free - bound , bound - bound emission of hydrogen , and cooling in the [ oi ] 6300 , 6364 and mgii 2800 lines . the fe ii line cooling is approximately included by multiplying the mgii 2800 cooling rate by a factor of two . given the possibility of a high temperature regime ( @xmath81 k ) in the outer layers , the cooling at this range of temperatures is treated using the cooling function of @xcite . the distribution of the ionization and excitation of h and hei in the atmosphere at one time is calculated by tracking the evolution of a lagrangian zone after its emergence from the photosphere . the initial ionization and excitation at the photosphere is set assuming saha - boltzmann equations for the effective temperature . subsequent ionization and excitation are computed by solving the system of time - dependent kinetic equations together with the time - dependent energy balance . the average intensity of the photospheric radiation is assumed to be @xmath82 , where @xmath83 is the dilution factor and @xmath84 is the photospheric brightness . the luminosity evolution at the photospheric epoch ( @xmath85 d ) is described by the expression @xmath86\,.\ ] ] with @xmath87 erg s@xmath2 and @xmath88 d , this corresponds to the bolometric luminosity of sn 1999em @xcite , assuming a distance of 11.7 mpc . the photospheric radius is determined by the velocity at the photosphere and the age ( @xmath89 ) , where @xmath90 is deduced from observational data of @xcite . the spectrum of the photospheric continuum is taken to be blackbody with @xmath91 in the wavelength region longward of the balmer jump , while the ultraviolet ( uv ) radiation shortward of the balmer continuum edge is suppressed by a factor @xmath92 which was assumed to monotonically drop after the explosion from unity to 0.03 on a timescale of 10 d. this evolution was deduced from the behavior observed in sn 1987a @xcite . we found that the results are only weakly sensitive to the specific form of the variation of @xmath93 . the energy deposition rate in the atmosphere due to the absorption of x - rays is calculated with an absorption coefficient @xmath94 @xmath95 g@xmath2 ( where @xmath19 is in kev ) . the deposited energy is shared between heating , ionization , and excitation according to the recipes of @xcite and @xcite . the solution of the kinetic equations for each lagrangian zone provides populations of the hydrogen levels @xmath96 and @xmath97 , and the @xmath76s level of hei in the atmosphere . these values are then used to calculate line profiles of h@xmath0 and hei 10830 . we compared our model in the case of zero wind density with the detailed model of @xcite and found that the present model results in slightly weaker h@xmath0 absorption ; at the relative intensity of 0.9 , the radial velocity in the simple model is lower by @xmath98 km s@xmath2 . this drawback of the simple model does not noticeably affect the amplitude of the calculated hv absorption formed in the external layers , where the ionization unaffected by x - ray absorption is small . the model line profiles of h@xmath0 and hei 10830 are computed for different wind densities , ages , ejecta mass and energy ( table 1 ) . the standard model is characterized by @xmath53 , @xmath54 erg , @xmath25 . in all cases @xmath56 km s@xmath2 with the exception of the very low energy model ( ew ) for which @xmath99 km s@xmath2 . we designate a model at a particular time by adding the age in days to the model name ; e.g. , w50 stands for model w on day 50 . the effect of the wind density on h@xmath0 on day 50 is shown in fig . [ f - modha]_a _ for the wind density parameters @xmath100 , 1 , 2 , and 4 . without a wind , the h@xmath0 absorption on day 50 forms because of the time - dependent ionization effect ; a high ionization is maintained due to ly@xmath0 trapping and reionization from excited levels of hydrogen . the cs interaction results in the ionization and excitation of the outer recombined layers of unshocked ejecta and , as a result , a hv absorption feature appears in the blue wing of the undisturbed profile . fig . [ f - cart1 ] illustrates how h@xmath0 absorption is modified by the cs interaction . for @xmath100 , the cs interaction effect is small ( fig . [ f - modha]_a _ ) and the wind density @xmath100 seems to be the least that could be detected by hv absorption in h@xmath0 . for @xmath101 , the strength of the hv absorption increases with the wind density roughly as @xmath102 and for @xmath103 becomes saturated . the decrease of the velocity of the blue absorption edge is due to the deceleration caused by the dense wind . the fact that for @xmath104 the hv absorption merges with undisturbed h@xmath0 absorption implies that in the case of a very dense wind it would be difficult to infer the density parameter @xmath105 using only the h@xmath0 absorption . the evolution of the hv absorption ( fig . [ f - modha]_b _ ) shows that there is an optimal phase at about the middle of the plateau ( @xmath106 d ) when the cs interaction effect is most pronounced . at an early stage ( e.g. , on day 20 ) the cs interaction effect is not clearly seen because of the merging of hv absorption with the strong undisturbed absorption , while at a late stage ( e.g. , 80 d ) hv absorption becomes very faint . the sensitivity of hv absorption to the ejecta density is shown by the model mw50 with a mass @xmath107 and energy @xmath108 erg . the energy is chosen to match the blue wings of the main absorption of models mw50 and w50 . the hv absorption in the low mass case turns out the same as in the model w50 , i.e. , the effect of reducing the ejecta density by a factor of 2 is negligible . we also consider the cs interaction effect for sne iip with a very low kinetic energy , @xmath109 erg , which presumably applies to the low - luminosity sne iip , e.g. , sn 1999br @xcite . this model with @xmath99 km s@xmath2 and @xmath52 predicts strong hv absorption which is merged with the undisturbed absorption ( fig . [ f - modha]_d _ ) . for all the models when hv absorption is unsaturated , the sobolev optical depth of this feature for @xmath110 is @xmath111 . this implies that in the h@xmath11 line the optical depth @xmath112 , so we do not expect that hv absorption in this line will be pronounced ( but see [ sec - narrow ] ) . we examined the sensitivity of hv absorption to the power law index @xmath113 of the x - ray spectrum of the reverse shock . we found that a softer spectrum ( @xmath114 ) and harder spectrum ( @xmath115 ) produce slightly deeper hv absorption , so the preferred case @xmath74 corresponds approximately to a minimal effect of the cs interaction on h@xmath0 . we checked the sensitivity of hv absorption to the adopted sn bolometric luminosity as well . the hv absorption strength anticorrelates with the bolometric luminosity , but the dependence is weak . the mechanism for this dependence is the depopulation of excited levels by photoionization . the cs interaction effect in the hei 10830 line substantially differs from that in h@xmath0 in one important respect : in the absence of a wind the model does not predict noticeable absorption in hei 10830 (fig . [ f - modhe]_a _ ) , except for a very early stage ( @xmath116 d ) . the wind density @xmath100 produces weak hv absorption , on the verge of detectability , while the case @xmath52 yields hv absorption with a relative depth of @xmath117 and could be easily detected . the absorption gets markedly stronger for larger @xmath105 , showing the sensitivity of hei absorption to the wind density in this particular range of the @xmath105 parameter . remarkably , while in h@xmath0 the contribution of the interaction effect for @xmath104 is difficult to discern ( fig . [ f - modha]_a _ ) , the hei 10830 hv absorption provides a probe for such a dense wind . the strength of hv absorption in hei 10830 increases between days 20 and 50 but gets weak on day 80 ( fig . [ f - modhe]_b _ ) , resembling the behavior of hv absorption in h@xmath0 . the effects of the ejecta density ( fig . [ f - modhe]_c _ ) are similar to those in h@xmath0 . the hv absorption for the low energy case is very strong and easily discernable , unlike in h@xmath0 . spectra of sn 1999em @xcite on days 41 and 54 ( assuming the explosion date jd 2451476 ) show a depression in the blue wing of the h@xmath0 line ( fig . [ f - ha99em ] ) at the predicted position of hv absorption ( @xmath118 to @xmath119 km s@xmath2 ) . to model the h@xmath0 profiles we adopted the standard parameters @xmath53 , @xmath54 erg , @xmath25 , and @xmath56 km s@xmath2 . we find the best value of the wind density parameter is @xmath52 with an uncertainty of @xmath120 , assuming the other parameters to be fixed . the indicated error reflects only the accuracy of estimating this parameter from the best fit and does not include errors in the observed spectrum and uncertainties in the model . also on day 54 , the model reproduces the general strength of the observed hv absorption ( fig . [ f - ha99em ] ) . yet we find that the observed hv absorption on day 54 seems to contain a notch component that is deeper than in the model . the question arises of why the model fails to describe in detail the undisturbed h@xmath0 . the model emission is weaker , while the absorption is stronger , than is observed ( fig . [ f - ha99em ] and * ? ? ? the weak model emission is probably related to our crude description of the uv continuum in the sn atmosphere ( cf . * ? ? ? however , this can not explain why the observed absorption is shallow . in fact , a check of available spectra of sne iip indicates that the h@xmath0 absorption seems to be shallow in any normal sn iip ( e.g. , sn 2004dj , see below ) . in contrast , for the peculiar sn 1987a related to the explosion of a blue supergiant , the h@xmath0 absorption is deep and well reproduced by the time - dependent model @xcite . we speculate that this is a strong indication that the shallowness of the h@xmath0 absorption in normal sne iip is related to the red supergiant structure of the presupernova . for instance , convection in the red supergiant atmosphere could produce large amplitude density perturbations that are significantly amplified during the blast wave propagation . as a result , the outer layers of ejecta would acquire a clumpy structure responsible for the shallowness of the h@xmath0 absorption . returning to the hv absorption , in fig . [ f - ha04dj ] we show the model fit to the observed h@xmath0 in sn 2004dj @xcite on days 55 and 64 ( for an explosion date jd 2453185 ) . the model parameter set is @xmath53 , @xmath25 , as for sn 1999em , but @xmath121 erg . one could retain the energy @xmath122 erg used for sn 1999em and vary @xmath20 . however , in this case one requires @xmath123 to attain a comparable fit . since the ejecta mass unlikely exceeds @xmath124 , the lower energy case for sn 2004dj is preferred . in accord with the lower energy we take @xmath57 km s@xmath2 and lower velocities at the photosphere by a factor 0.9 compared to sn 1999em . the derived wind density ( @xmath125 ) is determined by the strength of the hv absorption and its position , while the lower energy is needed to reproduce the correct position of the hv absorption . to check the effect of the adopted boundary velocity @xmath13 , we modeled the case @xmath126 km s@xmath2 . in this case the wind density is lower , @xmath127 . we prefer @xmath57 km s@xmath2 on the basis of the arguments presented below in sec . [ sec - conduct ] . on day 55 the hv absorption in the observed spectrum has the appearance of a relatively broad shallow depression similar to sn 1999em on day 41 . a bit later , on day 64 , a notch ( `` v - shaped '' narrow absorption ) appears at a radial velocity @xmath128 km s@xmath2 ( fig . [ f - ha04dj ] ) . the feature dominates the hv absorption until at least day 102 in spectra of sn 2004dj reported by @xcite . neither the strength nor the shape of the notch at the late photospheric stage @xmath129 d can be produced by a model in which the hv absorption forms only in the unshocked ejecta . this suggests that the notch in h@xmath0 has a _ different origin _ which is addressed in [ sec - narrow ] . the hv absorption in sn 2004dj has a markedly lower velocity compared to sn 1999em at a similar epoch ( 8200 vs. 11,500 km s@xmath2 ) . this fact is significant for the origin of the depression in the blue wing of h@xmath0 since it rules out the possibility that the feature might be produced by some unidentified metal line . photospheric velocities at this epoch are similar in both sne within several @xmath130 km s@xmath2 , so the expected positions of a metal line absorption should coincide in both sne within the same range of velocities . since this is not the case , we conclude that hv absorption in these supernovae _ can not be _ produced by an unidentified line . an additional argument against the ` metal line ' option is that sn 1987a does not show hv absorption in h@xmath0 at all . this would be difficult to understand in the context of the metal line option , but is easily explained by the very low density of the pre - sn wind ( @xmath131 , * ? ? ? * ) that is incapable of producing detectable interaction effects . the hv absorption of hei 10830 in sn 1999em on day 20 was identified by @xcite and attributed originally to radioactive excitation by external @xmath132ni . here we abandon this explanation and propose excitation of he in the external layers by cs interaction . the hei 10830 line of sn 1999em on day 20 @xcite is shown in fig . [ f - he99em ] together with models for @xmath52 and the best value @xmath133 , which has an uncertainty of @xmath134 . the wind density parameter derived from the hv absorption strength in hei 10830 agrees well with the value @xmath52 found from the h@xmath0 line . the clear - cut observed profile and the sensitivity of hv absorption of hei 10830 to the wind density makes this line a valuable diagnostic tool for probing the cs wind in sne iip . interestingly , based on a time - dependent model for a sn iip spectrum without cs interaction , @xcite predict strong hv absorption in hei 10830 at a relatively late epoch ; on day 50 , the relative depth of hei 10830 is @xmath135 for an enhanced he abundance @xmath136 . this suggests a relative depth of @xmath137 for a solar he abundance . the absorption lies in the range of @xmath119 to @xmath138 km s@xmath2 , which corresponds to the blue wing of hei 10830 in sn 1999em on day 20 ( fig . [ f - he99em ] ) . the hei 10830 line in sn 1999em on day 20 does not show evidence for the additional he absorption in the blue wing . yet it is possible that in a more detailed model the time - dependent effects for he lines might be larger than in our simple model and would substantially contribute to the hv absorption of the hei 10830 line at an age @xmath62 d. however , for sn 1999em on day 20 , we note the close correspondence between the @xmath105 values deduced from the h@xmath0 and hei 10830 lines . spectra of sn 1999em covering the hei 10830 line are also reported by @xcite for three epochs : 8 , 24 , and 34 d. the first spectrum shows a strong hei 10830 line with a p cygni profile . this line , related to the early high temperature phase , rapidly disappears and is not seen in the next two spectra . instead , hv absorption appears in this line . on day 24 , the hv absorption of hei 10830 is much the same as on day 20 in fig . [ f - he99em ] , and it also appears on day 34 . a steady hv absorption feature with time is expected in our model ( [ f - modhe]b ) . on day 34 , an additional redder absorption at @xmath139 appears separate from the hv absorption . the transformation of the hei 10830 line from a single hv absorption into a double absorption structure is also observed between days 22 and 44 in another type iip supernova , sn 1995v @xcite . we believe that the low velocity absorption is the result of he excitation by gamma - ray photons leaking from the inner radioactive decays of @xmath132ni and @xmath132co @xcite . to produce the required excitation of he about @xmath140 of @xmath132ni should be mixed out to a velocity @xmath141 km s@xmath2 , which does not contradict the conclusion that the majority of the @xmath132ni resides in deeper layers with velocity @xmath142 km s@xmath2 @xcite . we now address the issue of the notch that emerged in h@xmath0 in both sn 1999em and sn 2004dj at about day @xmath143 . we propose that the notch is produced by h@xmath0 absorption of the photospheric radiation in the cds . we also consider an alternative mechanism : additional excitation of hydrogen in the unshocked ejecta by ly@xmath0 photons emitted from the cds . we first study the preferred explanation for the notch , h@xmath0 absorption in the cds , and consider the clearly observed case of sn 2004dj . in modeling the h@xmath0 absorption produced by the cds , the principal parameters are the column density of the cool gas , and its excitation temperature and turbulent velocity . the undisturbed or weakly disturbed cds is characterized by a turbulent velocity of the order of the thermal velocity of the cool gas , @xmath144 km s@xmath2 . however , with this turbulent velocity the equivalent width of the hv absorption turns out to be too small . a possible solution is to increase the turbulent velocity , as expected by the action of the rayleigh - taylor ( rt ) instability of the decelerating thin shell . a significant deceleration of the model cds of sn 2004dj , about 20% during the first 10 days ( see low energy case in fig . [ f - dyn ] ) , favors the development of the rt instability of the cds followed by mixing of the cool dense gas with hot forward shock gas @xcite . the mixed cool gas thus can acquire a significant velocity dispersion and turbulent velocities up to @xmath145 of the shell velocity . both components could contribute to the notch formation : absorption in the weakly disturbed cds and in the mixed cds material . we therefore consider two components : ( 1 ) a weakly perturbed spherical shell with a turbulent velocity of @xmath146 km s@xmath2 and ( 2 ) mixed cds gas with a large turbulent velocity @xmath147 km s@xmath2 . the mass fraction of the second component is denoted @xmath11 . the corresponding spectral components of the hv absorption related to the cds can be dubbed narrow notch and broad notch . these components should be distinguished from the ` ejecta component ' a broader shallow hv absorption formed in the unshocked ejecta and addressed in the previous section . a cartoon ( fig . [ f - cart ] ) illustrates the origin the hv absorption produced by the ejecta and the cds . hydrogen excitation of the cds gas is presumably maintained by the absorption of x - rays emitted from the reverse shock . the absorbed energy of x - rays is assumed to be homogeneously distributed through the cds material , as expected for the modest shell column densities for our situation . the number density in the cds is determined by pressure equilibrium @xmath148 ( where @xmath149 is the pre - shock wind density ) for the typical temperature of the cool gas of @xmath150 k implied by the energy balance . on day 64 , the cds mass in our model for sn 2004dj is @xmath69 . the other input parameters provided by the interaction model are : cds radius , @xmath151 cm , velocity @xmath152 km s@xmath2 , x - ray luminosity of the reverse shock @xmath153 erg s@xmath2 , and electron temperature in the reverse shock @xmath154 kev . the population of the second level of hydrogen is computed using a steady - state approximation ( justified since @xmath155 @xmath156 ) , and assuming a two level plus continuum model for the hydrogen atom . both non - thermal excitation and ionization by secondary electrons lead to the excitation of the second level , while depopulation proceeds by the two - photon transition , ly@xmath0 escape and collisional deexcitation . the calculated profiles of hv absorption related to the cds are overplotted on the model h@xmath0 profile with ejecta hv absorption and displayed in fig . [ f - nhva ] along with the observed spectrum of sn 2004dj @xcite on day 64 . shown are three cases : one without the cds absorption and two cases with the cds absorption that differ by the mass fraction of the broad cds component ( @xmath157 and 0.5 ) . for the quoted values , the optical depths of the narrow components are 520 and 175 , while these values for the broad component are 0.2 and 0.45 , respectively . the absolute value of the radial velocity of the hv absorption produced by the cds ( @xmath158 ) is slightly lower than the velocity of the cds , @xmath159 , owing to the sphericity effect . assuming uniform brightness of the photosphere ( lambert s law ) it is straightforward to show that the average absolute value of the radial velocity of the absorption is @xmath160\ , , \label{eq - hva}\ ] ] where @xmath161 is the ratio of the photospheric to the cds velocity . according to this expression @xmath162 and , in the limit @xmath163 , @xmath164 with high accuracy . the sphericity effects are responsible also for the additional broadening of the absorption over the radial velocity range @xmath165 , which is @xmath166 km s@xmath2 for the model of sn 2004dj on day 64 . the alternative mechanism is that the notch could form as a result of an enhancement of hydrogen excitation in the unshocked sn ejecta due to the scattering of ly@xmath0 photons emitted from the cds that in turn is irradiated by x - rays from the reverse shock . a resonance condition implies that ly@xmath0 photons can scatter in the unshocked ejecta only in the narrow velocity range @xmath167 , where @xmath168 is the boundary velocity of the unshocked sn ejecta . the expected flux of ly@xmath0 from the cds is @xmath169 where @xmath170 is the efficiency of ly@xmath0 emission and @xmath171 is the x - ray luminosity absorbed by the cds . the average excitation rate in the ejecta within the velocity width @xmath172 is balanced by ly@xmath0 escape ; other depopulation mechanisms are negligible . the population of the second level ( @xmath96 ) thus is approximately determined by the rate equation @xmath173 where @xmath174 is the sn age , @xmath175 is the ly@xmath0 photon energy , and @xmath176 is the sobolev escape probability for ly@xmath0 . for the model of sn 2004dj , the notch in h@xmath0 produced by ly@xmath0 scattering in the ejecta is calculated on day 64 ( fig . [ f - nhva ] ) . two cases , with efficiencies @xmath177 and 0.5 , are plotted . one sees that even for the unacceptably high efficiency @xmath178 the notch produced by the ly@xmath0 scattering is significantly weaker than that predicted by the absorption in the cds . we conclude that the latter mechanism is the most likely explanation of the notch in h@xmath0 . we examined whether the absorption in the cds might produce some effect in the hei 10830 line . for the above model , the optical depth in the hei 10830 line turns out to be a factor of @xmath179 lower than in h@xmath0 . therefore , it is unlikely that the notch can be seen in hei 10830 . the main reason is the efficient depopulation of the 2@xmath78s level in the cds by penning ionization of hydrogen . given the relatively high h@xmath0 optical depth of the cds , a notch at the same velocity could be seen in h@xmath11 as well . we inspected spectra of sn 1999em @xcite and found the signature of a notch in h@xmath11 at the right position ( @xmath180 km s@xmath2 ) between days 39 and 81 . in fact , @xcite already discussed the presence of the hv absorption in h@xmath0 and h@xmath11 in spectra of sn 1999em and concluded that these features can not be caused by metal line absorptions . we also find a notch in h@xmath11 in spectra of sn 2004dj obtained by @xcite . since the hv absorption components of h@xmath0 and h@xmath11 have not been previously mentioned for this supernova , we show parts of the sn 2004dj spectra @xcite retrieved from the sn archive suspect ( fig . [ f - obs04dj ] ) . the spectra show conspicuous notches at about @xmath181 km s@xmath2 both in h@xmath0 and h@xmath11 between days 67 and 102 . below we will model hv absorption in both lines in more detail . here we present a model that can account for the strength and shape of the notch in h@xmath0 of sn 2004dj at the late photospheric epoch . preliminary modeling suggests that the simple two - component model of the cds ( narrow and broad ) introduced above is not able to describe the strength of the notch at late times ( @xmath182 d ) : the excitation produced by the x - ray absorption is insufficient . this suggests the presence of a more efficient excitation mechanism or an additional source of heat . another problem is the relative depth of notches in the h@xmath0 and h@xmath11 lines : they are comparable and shallow . this indicates that some veiling is present , possibly due to clumpiness . both reasons force us to modify the two - component model of the cds absorption . an important outcome of the rt mixing of the cds material in the forward shock is the growth of the area of the contact surface ( @xmath183 ) between cool and hot gas . the development of mixing may be thought as the progressive decrease of the minimum linear scale of fragments ( @xmath184 ) . this process is demonstrated in numerous experiments on the rt instability and in the case of sn / cs interaction is clearly seen in 2d and 3d hydrodynamical computations @xcite . a fragmentation cascade of a fixed amount of cool gas leads to the scaling @xmath185 , while for a stationary mixing process @xmath186 @xcite . the growth of @xmath183 and , accordingly , of the surface - to - volume ratio for the mixed cool gas , should lead to a greater role of thermal conduction in the heating of the cool gas at late times . we speculate that this is the missing factor that could resolve the excitation problem . to describe the effect of thermal conduction , we introduce a third cds component : an additional broad cds component powered predominantly by thermal conduction . the mass fraction of the third component is @xmath187 , while the cumulative area of the contact surface for the third component , @xmath188 , can be large , with an area ratio @xmath189 . we assume that the third component is homogeneously heated by thermal conduction , which is a sensible approximation for a small thickness of sheets of cds gas , @xmath190 cm , comparable to the mean free pass length of hot electrons and protons in the cool dense gas . the heat flux is presumably a small fraction of the saturated flux @xmath191 ; we adopt @xmath192 to allow qualitatively for the possible suppression of thermal conduction by a magnetic field . the saturated flux is taken of the form @xmath193 @xcite . the forward shock is not equilibrated ( @xmath194 ) , so the ion ( proton ) conduction could be significant . in our case , the ratio of proton - to - electron thermal flux is @xmath195 and this fact is taken into account . ionization and thermal balance are solved for the dense hydrogen cooled by ly@xmath0 and two - photon radiation . the rt mixing generally produces an inhomogeneous ( lumpy ) distribution of mixing regions . this is the result of the existence of a dominant angular scale of the rt instability , about @xmath196 @xcite and of the intermittence of the turbulent mixing . we describe the distribution of mixed regions as an ensemble of isolated lumps which occupy a fraction @xmath197 of the volume ; @xmath198 corresponds to a homogeneous mixture of cool and hot phases . the fraction @xmath199 should not be confused with the filling factor of the mixed cool dense gas , which is of the order of @xmath200 . observationally , the lumpiness of mixing zones could be manifested as a veiling effect , i.e. , a comparable moderate depth of absorption lines with different optical depth ( viz . , h@xmath0 and h@xmath11 as in our case ) . to facilitate the treatment of the radiation transfer in the lumpy medium we use an occultation optical depth , @xmath201 , i.e. , the average number of lumps along the radius . this value is defined by @xmath199 and a ratio of the total thickness of the mixing layer ( forward post - shock layer @xmath202 ) to the mean intercepted length of the lump @xmath203 ( for a sphere , @xmath204radius ) : @xmath205 . generally , @xmath201 could be different for each of the three components . however , we assume the same value of @xmath201 for all three components . the effective optical depth in the line ( @xmath206 ) then can be expressed through the average optical depth of homogeneously distributed cds material @xmath207 and @xmath201 : @xmath208 $ ] @xcite . we thus consider a three component model of the cool dense gas which includes : ( i ) a narrow cds component with turbulent velocity @xmath209 km s@xmath2 , ( ii ) a broad cds component with @xmath210 km s@xmath2 , both powered by x - rays , and ( iii ) a broad cds component powered by the thermal conduction . the cds notches in h@xmath0 and h@xmath11 in sn 2004dj are calculated on the basis of the previous sn 2004dj model assuming @xmath211 km s@xmath2 for narrow and @xmath212 km s@xmath2 for broad cds components . the adopted fraction of the broad cds component is @xmath213 at the early epoch ( @xmath214 d ) , with a subsequent increase proportional to the age , while the fraction of the third component is @xmath215 ( table 2 ) . the cds absorption in h@xmath0 is superimposed on the model line profile with the ejecta hv absorption . for the h@xmath11 line , the cds hv absorption is superimposed on the background spectrum taken as a linear interpolation between the blue and red wings of hv absorption in the observed profile . the occultation optical depth @xmath216 ( table 2 ) is chosen to satisfy the ratio h@xmath0/h@xmath11 , while the area ratio @xmath217 is determined from the strength of the absorption . ( [ f - hvaev ] ) shows that a three - component model with the area ratio @xmath218 ( table 2 ) describes the hv absorption of both h@xmath0 and h@xmath11 lines satisfactorily . on day 102 the ejecta hv absorption is somewhat stronger , which is responsible for the red - blue asymmetry in the model profile . however , this does not affect the intensity of the cds absorption feature . the comparison to the two - component model and the behavior of @xmath217 show that the role of thermal conduction increases with time , which is the expected result . interestingly , the result depends rather weakly on @xmath219 . this is related to the canceling of effects of the variation of @xmath219 in the thermal excitation of hydrogen and the column density of the third component . we performed modeling of the cds hv absorption adopting the lower boundary velocity , @xmath126 km s@xmath2 , and found that in this case deceleration of the cds is too slow , which leads to a larger velocity of the cds hv absorption on day 102 than is observed . the value @xmath57 km s@xmath2 is about the least that provides the required deceleration rate of the cds . although the parameter values used for the third component are reasonable , the modeling can not be considered as proof that thermal conduction actually dominates the excitation of the cds material at a late epoch the model contains many assumptions . yet the need for an additional source of excitation of hydrogen responsible for the hv absorption is an established result . thermal conduction is a natural mechanism strongly suggested by the inevitable rt instability and mixing of the cds matter with hot gas from the forward shock . on the other hand , the additional excitation might at least partially be related to enhanced soft x - ray radiation from hot dense inhomogeneities produced by the rt mixing and heated by thermal conduction up to @xmath220 k. the identification of the features produced by h@xmath0 and h@xmath11 absorption in the cds provides an efficient tool for the direct determination of the velocity of the sn / wind interface in sne iip . with this value in hand we are able to obtain the wind parameter @xmath105 using the thin shell model of the deceleration . of course , the method assumes that the ejecta parameters @xmath20 , @xmath19 , and @xmath21 are known . to facilitate the application of this method we use the self - similar solution . this description is valid at the late photospheric epoch , when the initial conditions related to the structure of the outermost ejecta layers are already ` forgotten . ' the self - similar evolution of the cds radius suggests @xmath221 , with @xmath222 @xcite . the factor @xmath223 for the adopted density distribution ( eq . [ [ eq - den ] ] ) is @xmath224^{1/(k-2)}\,,\ ] ] were @xmath225 is defined by equation ( [ eq - cmce ] ) and @xmath18 by equation ( [ eq - v0 ] ) . the parameters for sn 1999em are as before : @xmath53 , @xmath54 erg , and @xmath25 . the radial velocity of hv absorption is calculated both for the lambert photosphere ( eq . [ [ eq - hva ] ] ) and with limb darkening described by the first chandrasekhar approximation . the evolution of the photospheric velocity is set according to @xcite . the observational estimates of the radial velocity of hv absorption of h@xmath0 are obtained using the spectra reported by @xcite . the best fit for the wind density parameter is found to be @xmath226 ( fig . [ f - cds ] ) . this estimate is in a good agreement with the value derived from the ejecta hv absorption of h@xmath0 ( @xmath227 ) and the hei 10830 line ( @xmath228 ) . uncertainties in the limb darkening law and in the photospheric velocity do not affect the derived value of the wind density . in the case of sn 2004dj we use the same model as before : @xmath53 , @xmath229 erg , and @xmath25 . the radial velocities of hv absorption are measured from spectra obtained by @xcite . the best fit wind parameter is @xmath230 ( fig . [ f - cds ] ) . this agrees with the value @xmath231 obtained for sn 2004dj from the ejecta hv absorption of h@xmath0 . the coincidence of @xmath105 values found by diagnostics based on the hv absorption in ejecta and in the cds is remarkable because the first probe is based on the model of x - ray emission by the reverse shock and the model of the ionization and excitation of ejecta , while the second is based only on the self - similar dynamics of the thin shell . our goal was to study signs of cs interaction in the h@xmath0 and hei 10830 absorption lines of sn iip during the photospheric stage . we developed a model for the ionization and excitation of h and he in the unshocked ejecta taking into account time - dependent effects and the irradiation of ejecta by x - rays . the principal result is a prediction that for a typical rsg wind ( @xmath101 ) , the effects of cs interaction in a normal sn iip are significant and can be detectable at the photospheric stage in both lines . these effects consist of the emergence of : ( i ) hv absorption in the blue wing of the undisturbed h@xmath0 line ( absorption ` shoulder ' ) at @xmath232 d , and ( ii ) hv absorption in hei 10830 at an age @xmath233 days at a radial velocity of @xmath234 km s@xmath2 . we identify the hv absorption in sn 1999em in both h@xmath0 and hei 10830 and derive a wind density parameter @xmath235 . hv absorption in h@xmath0 is identified in sn 2004dj as well and the value of the wind density parameter is again found to be @xmath235 . the second important finding is that , in addition to the ejecta , absorption in the cds formed at the sn / wind interface plays an important role in the hv feature , especially at the late photospheric phase . this component of the hv absorption is manifested as a notch in the hv absorption of h@xmath0 in sn 1999em and sn 2004dj after about day 60 . a similar notch is seen in the h@xmath11 line of these sne . we developed a model that produces the notch in both h@xmath0 and h@xmath11 at the different epochs of sn 2004dj which suggests a growing role for thermal conduction in the heating of the mixed cds gas . the wind densities estimated from the observed velocities of hv absorption produced by the cds in sn 1999em and sn 2004dj are consistent with the densities derived from the hv absorption produced by unshocked ejecta . the identification of hv absorption in h@xmath0 and h@xmath11 in sn 1999em has been discussed by @xcite . here , we identified hv absorption in h@xmath0 and h@xmath11 in sn 2004dj at a velocity @xmath236 km s@xmath2 lower than in sn 1999em . the difference provides an additional argument against a possible metal line origin for these features . @xcite claim that a similar component is also seen in the nai doublet . indeed , there is a weak dip with a depth of @xmath237 at the right velocity , @xmath238 km s@xmath2 , if it is associated with the nai 5890 doublet . we calculated the optical depth of the nai 5890 doublet for sn 1999em on day 60 in the model of absorption by the cds and found that the narrow cds component in this line has an optical depth of @xmath239 , which means that after broadening by about 10 times the depth of absorption should become equal to @xmath240 , comparable to the observed notch . however , we did not find a similar notch in the nai line of sn 2004dj , or in any other sn iip . at present , the reality of the hv absorption in the nai 5890 doublet remains uncertain . we examined other published spectra of sne iip at the photospheric epoch and found evidence for hv absorption in the h@xmath0 blue wing of sn 1985p @xcite , in both h@xmath0 and hei 10830 of sn 1995v @xcite , in h@xmath0 of sn 2003gd , and sn 2006ov @xcite . available spectra of sn 1999gi do not cover the range @xmath241 days when the most pronounced cs interaction effects should be observed . in noisy spectra on day 39 and 89 reported by @xcite , we do not find unequivocal evidence for hv absorption in h@xmath0 . we checked spectra of sn 2004et @xcite and did not find convincing evidence for hv absorption in h@xmath0 or h@xmath11 . early stage ( @xmath242 d ) spectra of sn 2004et show a shallow absorption in the h@xmath0 blue wing rapidly evolving towards the red . however the siii 6347 , 6371 doublet may be responsible for this absorption . at a later epoch , a weak notch at 6282 is present but unfortunately it coincides with the telluric absorption 6281.7 line and the identification with hv absorption is doubtful . @xcite consider the notch in h@xmath0 of sn 2004et to be real and remark that it might be related to cs interaction . if the hv absorption in sn 2004et is absent , the question arises why the dense wind suggested by a relatively high radio luminosity of this sn compared to sn 2004dj and sn 1999em @xcite is not revealed optically . the answer may be that the wind is so dense that the hv absorption is strong and gets merged with the main absorption as in the model case of @xmath104 ( fig . [ f - modha ] ) . this conjecture is consistent with the strong h@xmath11 absorption observed in sn 2004et @xcite . a direct test of the proposed explanation might be the detection of strong hei 10830 absorption in sn 2004et . unfortunately , to our knowledge , observations of the hei 10830 region in sn 2004et are lacking . interestingly , sn 2006my @xcite , which does not show hv absorption , also has strong h@xmath11 like sn 2004et . if the same mechanism ( dense wind , @xmath243 ) explains the absence of the hv absorption in this case , then sn 2006my should demonstrate high radio and x - ray luminosities for a sn iip . we did not find signatures of hv absorption in h@xmath0 observations of the low - luminosity sn 1999br in spectra taken during the first 42 days after the explosion @xcite . in the spectrum on day 42 , the h@xmath11 line is rather weak and the explanation that has been invoked for sn 2004et ( strong cs interaction effect ) is not applicable in this case . the likely reason for the absence of hv absorption in sn 1999br is a low wind density , @xmath244 . this conjecture has an interesting implication . since the mass loss rate increases with the stellar mass according to the scaling law of @xcite , we expect that the main - sequence mass of pre - sn 1999br should be lower than that of sn 1999em . an interesting result , although model dependent , is that at the late photospheric stage of sn 2004dj ( @xmath245 d ) thermal conduction may be the dominant heating mechanism of the mixed cds gas in the forward shock . in this regard , we recall the h@xmath0 problem in sn 1979c : at an age of @xmath63 yr the h@xmath0 luminosity was @xmath246 of the total reverse shock luminosity @xcite , which is beyond reasonable values of the h@xmath0 emission efficiency . a natural solution to the problem of the high h@xmath0 luminosity in sn 1979c could be thermal conduction in the mixing layer of the forward shock @xcite . another possibility , however , is that the mixing process could bring about enhanced x - ray emission from the hot dense mixed component , which may be another source of additional excitation of hydrogen in the cds . this soft x - ray component might explain the softening of the x - ray emission observed in sn 1999em @xcite . using _ chandra _ x - ray data , @xcite conclude that the wind density parameter for sn 1999em is in the range of @xmath247 ( or @xmath248 for the distance 11.7 mpc ) . our present value ( @xmath235 ) is within a factor of 1.5 of the lower limit . this should be considered as good agreement for independent estimates . for sn 2004dj the x - ray data are missing . the interpretation of radio data for sn 1999em and sn 2004dj in the model of wind free - free absorption indicates that the wind density in sn 2004dj is twice as low as in sn 1999em @xcite . this is somewhat discrepant with the present results which indicate similar wind densities for these supernovae . the wind density of sn 1999em may be overestimated compared to sn 2004dj from radio data because of the uncertain maximum in the 8.47 ghz light curve or a possible contribution of synchrotron self - absorption in sn 1999em @xcite . the wind density around sn 1999em and sn 2004dj suggests that the amount of the pre - sn material lost at the rsg phase ( @xmath9 yr ) is moderate in these cases . assuming the wind velocity of a typical rsg , @xmath249 km s@xmath2 , we find with @xmath52 that at the rsg stage the mass lost by the rsg wind is @xmath250 . with the ejecta mass of @xmath251 @xcite , @xmath252 enclosed in the neutron star , and @xmath253 lost by the main - sequence wind , the initial mass of sn 1999em progenitor turns out to be about @xmath254 . the mass loss rate by massive stars is often estimated using the formula of @xcite in which @xmath255 is defined through the stellar luminosity ( @xmath256 ) , mass ( @xmath20 ) , and radius ( @xmath28 ) . for the rsg stage , this formula can be modified using the relation between the mass and luminosity @xmath257 obtained from evolution calculations of @xcite in the range @xmath258 for moderate / no rotation . with this relation , the mass loss rate by @xcite becomes @xmath259 yr@xmath2 . since the luminosity at the rsg stage is determined by the mass of the he core which in turn is determined by the initial mass , we should use the initial mass to estimate @xmath255 . for sn 1999em with @xmath260 and @xmath261 @xcite this formula predicts a mass loss rate @xmath262 yr@xmath2 , i.e. , eight times larger than our value . a similar result is expected for sn 2004dj . the disparity between our estimate of the pre - sn mass loss rate and the popular scaling law requires an explanation . the disparity can not be attributed to a lower metallicity , because the metallicities of ngc 1637 ( host galaxy for sn 1999em ) and ngc 2403 ( host galaxy for sn 2004dj ) are approximately solar @xcite . the difference can be reduced by a factor of 1.5 if we adopt a higher velocity for the pre - sn wind , @xmath263 km s@xmath2 instead of 10 km s@xmath2 . the choice of @xmath264 km s@xmath2 is supported by the wind velocity of the massive rsg betelgeuse , @xmath265 km s@xmath2 @xcite . this , however , does not completely resolve the problem . we admit the possibility that the hydrodynamical model of the light curve @xcite might significantly overestimate the mass of ejecta for some unknown reason . in that case , to obtain the wind @xmath266 ( for @xmath263 km s@xmath2 ) the initial mass of the presupernova must be @xmath267 with an ejecta mass of @xmath268 instead of @xmath269 . with the sn energy reduced accordingly , the depth and position of the hv absorption remain the same ( fig . [ f - modha]_c _ ) . to conclude , we have found three new probes of the wind density in sne iip that rely on spectroscopic observations during the photospheric epoch 20 - 100 d of ( 1 ) the ejecta hv absorption in h@xmath0 ; ( 2 ) the ejecta hv absorption in hei 10830 ; and ( 3 ) the hv absorption ( notch ) produced by the cds . the third method is more practical than the other two because it does not require modeling of the absorption lines ; it uses velocity measurements and the self - similar model for the velocity fit . of course , this method , as well as the other two , assumes that we know the ejecta parameters ( @xmath20 , @xmath19 , @xmath21 ) . however , the application of all three methods may be used to constrain ejecta parameters as well . the first two methods are sensitive to winds with a density parameter @xmath270 . however in h@xmath0 cs interaction effects saturate for large density @xmath271 . therefore , only hei 10830 remains a useful diagnostic tool for a high density wind with @xmath103 . for the third method the range of accessible wind density is less certain , because a cds formed by the initial boundary shell could exist in principle even in the case of a low density wind , although the hydrogen excitation and shell density depend on the wind density . further understanding of the relation between the strength of deep hv absorption and the wind density requires 3d hydrodynamical modeling of the shock breakout phase in the presence of a wind and new spectroscopic observations in both h@xmath0 and hei 10830 lines . we thank daniela korcakova for sending us spectra of sn 2004dj , jozsef vinko for informing us about sn 2004dj spectra in the archive suspect , d. k. sahu for sending us spectra of sn 2004et , and the referee for helpful comments on the manuscript . this research was supported in part by nsf grant ast-0307366 .
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we propose new diagnostics for circumstellar interaction in type iip supernovae ( sne iip ) by the detection of high velocity ( hv ) absorption features in h@xmath0 and hei 10830 lines during the photospheric stage . to demonstrate the method
, we compute the ionization and excitation of h and he in supernova ejecta taking into account time - dependent effects and x - ray irradiation .
we find that the interaction with a typical red supergiant wind should result in the enhanced excitation of the outer layers of unshocked ejecta and the emergence of corresponding hv absorption , i.e. a depression in the blue absorption wing of h@xmath0 and a pronounced absorption of hei 10830 at a radial velocity of about @xmath1 km s@xmath2 .
we identify hv absorption in h@xmath0 and hei 10830 lines of sn 1999em and in h@xmath0 of sn 2004dj as being due to this effect .
the derived mass loss rate is close to @xmath3 yr@xmath2 for both supernovae , assuming a wind velocity 10 km s@xmath2 .
we argue that , in addition to the hv absorption formed in the unshocked ejecta , spectra of sn 2004dj and sn 1999em show a hv notch feature that is formed in the cool dense shell ( cds ) modified by the rayleigh - taylor instability . the cds results from both shock breakout and radiative cooling of gas that has passed through the reverse shock wave .
the notch becomes dominant in the hv absorption during the late photospheric phase , @xmath4 d. the wind density deduced from the velocity of the cds is consistent with the wind density found from the hv absorption produced by unshocked ejecta .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the solar corona is often populated by peculiar objects , dense clouds of cold plasma inexplicably floating tens of thousands of kilometres above the photosphere . such objects are routinely seen during solar eclipses , when they can be easily distinguished by their red glow , but they can also be unveiled with the help of filters , such as h@xmath1 , devised to observe the chromosphere ( fig . [ fig1 ] ) . to put it in simple words , these objects ( usually called prominences or filaments ) are like chunks of chromospheric gas defying the downward pull of gravity and staying in a place higher than the one that apparently corresponds to their large density . this is not the only enigma around solar prominences . in contrast with the mk temperature of the surrounding corona , prominences remain at a comparatively cool 10,000 k , which prompts one to ask what prevents the mechanisms that heat the corona from also raising the temperature of prominences and consequently dispersing them . other pieces of the prominence puzzle concern their beginning and end : first , one may wonder not just how prominences form but also why they are born in an adverse environment . second , despite their internal dynamics , prominences that have been stable and healthy for weeks suddenly die in an spectacular eruption . hence , like all living beings these creatures are born , grow , ( do not reproduce ) and die , but unlike many living beings the processes shaping the lifetime of prominences are largely unknown . nevertheless , the intervention of a decisive element is quite clear : the magnetic field , that is central to all the mentioned processes . the reason of our limited insight into the nature of prominences probably has three causes @xcite : there is no such thing as a canonical prominence , but a wide range of parameters is observed in different objects ; no prominence has a uniform structure , but they are made of thin threads ( or fibrils ) and , in addition , different parameter values can be detected in different parts of a prominence ; and no structure is really isolated , so it is necessary to understand the physics of the prominence - corona interface , the effect of the coronal radiation field ( e.g. * ? ? ? * ) and to trace the magnetic fields permeating the prominence to their origin at the sun s surface ( e.g. * ? ? ? our knowledge about prominences has been well reviewed by @xcite , where ample information on the topic can be found . one of the first studies about prominence oscillations was carried out by @xcite , who noted that in a sample of 68 non - active region prominences , 31% of the objects presented no significant velocity change along the line - of - sight , 28% showed apparently random line - of - sight velocity variations and 41% presented a definite oscillatory behaviour . similar results were obtained for a set of 45 active region prominences . there are several reasons that may lead to the absence of periodic variations in some prominences , e.g. the velocity amplitude or its projection along the line - of - sight are too small to stand above the instrumental noise level ; or the prominence material does not actually oscillate at the time the observations are performed ; or the light emitted or absorbed by various plasma elements along the line - of - sight and having different oscillatory properties results in a noisy signal . after this leading work the subject of prominence oscillations remained dormant for about 15 years and revived in the 1980s , when a great amount of observations about prominence oscillations started being collected . according to the oscillatory amplitude , these events have been classified into two groups @xcite : * _ large amplitude oscillations _ , that arise when the whole prominence is shaken by a moreton wave @xcite impinging on it @xcite . as a consequence , the prominence gas undergoes a large displacement from its equilibrium position and the prominence as a whole vibrates with considerable velocity amplitude ( of the order of 20 km s@xmath0 or higher ) . these phenomena are reviewed in a separate work of this volume ( tripathi et al . * _ small amplitude oscillations _ , with velocity amplitudes much smaller than those of large amplitude oscillations have also been frequently observed . the detected peak velocity ranges from the noise level ( down to 0.1 km s@xmath0 in some cases ) to 23 km s@xmath0 , although larger values have also been reported @xcite . a distinguishing property of small amplitude oscillations , which gives them a distinction from large amplitude ones , is that they seldom influence the entire prominence . thus , when spectrographic observations are carried out with a slit it is usually found that only a few consecutive points along the slit present time variations with a definite period , while all other points lack any kind of periodicity ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? @xcite made a comprehensive study of the two - dimensional spatial distribution of doppler velocity oscillations in a very large polar crown prominence and showed that , in their particular observation , oscillations are concentrated in a restricted area ( 54,000 @xmath2 40,000 km in size ) . both propagating waves and standing oscillations are detected in this region and their wavelength and phase speed are determined . using hinode sot observations of a quiescent prominence , @xcite also report on oscillations that do not affect the whole prominence body . these oscillations have periods from 20 to 40 minutes and typically last for only one or two cycles . they correspond to waves propagating vertically along threads with a projected phase speed of @xmath3 km s@xmath0 . it has been mentioned that solar prominences are constituted by many thin , parallel magnetic threads filled with cold plasma , and as a consequence the dynamics of these components can form the basis of the dynamics of small amplitude oscillations . early works @xcite already noted the possible link between small amplitude prominence oscillations and the fibril structure . unfortunately , the spatial resolution of the data analysed by @xcite is not good enough to distinguish the prominence threads . it was necessary , hence , to wait until the advent of telescopes with much better spatial resolution to have observations in which the prominence fine structure is well resolved @xcite . in the analysis of the doppler velocity in two threads belonging to the same filament , @xcite finds a clear sign of propagating waves and determines their period , wavelength and phase speed . this study is followed by a much more profound one in which the two - dimensional motions and doppler shifts of 328 features ( or `` blobs '' ) of different threads are examined . these features are observed to flow along the filament axis while oscillating at the same time . to simplify the examination of oscillations , @xcite computed average doppler signals for each fibril and found that groups of adjacent threads oscillate in phase with the same period . this has two consequences : first , since the periodicity is outstanding in the averaged signal for each thread , the wavelength of oscillations is larger than the length of the thread . second , fibrils have a tendency to vibrate bodily , in groups , rather than independently . h@xmath1 observations conducted with the swedish 1-m solar telescope by @xcite lead to similar results concerning the collective dynamics of fibrils , although propagating doppler velocity signals with various periods and wavelengths in other threads of the same filament are also detected . all these observations seem to indicate that prominence fibrils sometimes support collective oscillations and sometimes oscillate on their own . this topic deserves a more detailed observational study and , given the simplicity of fibrils compared to the full filament structure , theoretical investigations can give rise to fruitful comparisons with observations . during the 1980s some observational works , e.g. those by @xcite , provided evidence about a temporal decrease of the oscillatory amplitude , which suggests that prominence oscillations are attenuated in time . a very clear example of this phenomenon can also be found in figure 5a of @xcite . here the integrated line intensity of the he d@xmath4 line displays oscillations with a period around 25 min and with an amplitude that decreases in time . these oscillations are present for a few cycles but , unfortunately , a precise value of the attenuation rate is not computed in this case ( figure 3 of * ? ? ? * also presents similar results , although with a much better spatial and temporal resolution ) . more detailed observational analyses of damped prominence oscillations can be found in @xcite , who studied the spatial and temporal features of doppler velocity oscillations in limb prominences . in the second of these works , already mentioned before , a sinusoidal function multiplied by a factor @xmath5 is fitted to the doppler series , which allowed these authors to obtain values of the damping time , @xmath6 , which are usually between 1 and 3 times the corresponding oscillatory period . in spite of the lack of similar studies , the existing evidence suggests that small - amplitude oscillations in prominences are excited locally and are damped in a few periods by an unknown mechanism . many previous observational and theoretical results on prominence oscillations have been reviewed by e.g. @xcite , so our purpose here is to extend these review works by summarising other papers not discussed by these authors . thus , we describe the first seismological analysis of prominence oscillations ( section [ seismology ] ) and the theoretical studies on the attenuation of small amplitude oscillations ( sections [ leakage ] , [ nonideal ] and [ resabs ] ) . the plausible theoretical explanations of this effect are reviewed here and the relevance of different mechanisms is assessed on the basis of the ratio of the damping time to the period ( @xmath7 ) , that must be in the range 110 to agree with the observational results . works that invoke wave leakage from the prominence as the cause for the observed damping are described in section [ leakage ] , attenuation caused by non - ideal effects is discussed in section [ nonideal ] and wave damping by resonant absorption is presented in section [ resabs ] . finally , section [ conclusions ] contains the concluding remarks . observations with the hinode solar optical telescope by @xcite display an active region prominence composed of many fine horizontal threads that oscillate independently in the vertical direction as they move along a path parallel to the photosphere . six different threads are studied and in each of them several points are considered . all points in each thread are found to oscillate in phase . the relevant parameters for a seismological analysis of these events are flow velocities in the range 1546 km s@xmath0 , oscillatory periods in the range 135250 s and thread lengths in the range 170016,000 km . a prominence thread is a cold plasma condensation that occupies a segment of a much longer magnetic tube . in the present case it is not easy to directly measure the length of such magnetic tubes , although @xcite estimate that the wavelength of the oscillations is at least 250,000 km and so the minimum length of the magnetic tubes is 125,000 km . @xcite provide another estimation of this quantity , which they argue must be at least 100,000 km . these authors perform an independent seismological study of the six events by assuming that the thread is a dense plasma moving along a horizontal and straight magnetic tube which is tied to the dense photosphere at its ends . additionally the low-@xmath8 and linear approximations are also imposed . for each thread three different studies are done ; in increasing order of complexity they are : ( a ) thin tube approximation and no flow ; ( b ) thin tube approximation and flow ; and ( c ) full ideal mhd equations with flow . while the thin tube approximation is well supported by the thinness of the threads compared to their length , ignoring the effect of the flow does not seem a good idea because the detected flow velocities are rather large . nevertheless , as we describe below the results prove just the contrary . in the absence of flow , case ( a ) , transverse motions of the kind observed by @xcite can only be produced by the kink mode @xcite . the dispersion relation of the kink mode in a magnetic tube containing prominence material can be deduced from equation ( 27 ) of @xcite . @xcite conclude that , if no assumptions about the magnetic field , density and magnetic tube length ( @xmath9 ) are made , then this dispersion relation contains too many unknowns and can not be uniquely solved . but if a value of @xmath9 is assumed ( with the restriction @xmath10 km from observations ) , then a dependence of the prominence versus the coronal alfvn speed can be obtained . figure [ dr_hinode ] displays such a dependence for two of the six threads and several magnetic tube lengths . this figure shows that , unless the ratio of prominence to coronal density is unrealistically low ( e.g. below 50 ) , the alfvn speed in the thread reaches a minimum value that is not modified by that density ratio . this lower limit of @xmath11 is around 120 km s@xmath0 for thread # 6 and lies between 200 and 350 km s@xmath0 for the other five threads . to test the validity of this seismological result , @xcite consider case ( b ) , which involves solving a simple partial differential equation . their numerical simulations of a thread that moves bodily along the magnetic tube yield in - phase oscillations of the whole thread , so a kink - like behaviour is recovered even in the presence of flow . these results are in excellent agreement with the observations of @xcite . perhaps surprisingly , for all six threads the period is only slightly smaller than that of case ( a ) , the difference being below 5% , i.e. smaller than the error bars of observations ( of the order of or larger than 6.8% ) . hence , the period of these transverse oscillations is almost insensitive to the thread flowing motions . finally , @xcite also investigate case ( c ) and recover the results of case ( b ) , which implies that the thin tube approximation is quite reasonable , as expected . after these two studies the lower bounds for the alfvn speed derived from case ( a ) are firmly established and although this investigation does not lead to particular values of physical parameters , such alfvn speeds agree with the intense magnetic fields and large densities usually found in active region prominences . wave leakage is a common phenomenon by which the energy imparted by a disturbance to a given structure ( e.g. a coronal loop or a solar prominence ) is not confined as standing oscillations of the structure , which emits waves carrying energy into the external medium . this situation has been studied at length in relation with coronal loop oscillations ; see @xcite and references therein . most of the works described in this section make no mention of wave leakage , but our interpretation is that the obtained attenuation of oscillations is simply a consequence of the emission of waves from the prominence into the corona . in the celebrated model of @xcite a prominence is treated as a horizontal line current suspended in the corona . such an approximation is justified by the fact that if the prominence plasma is supported against gravity by the lorentz force , then a current runs along the filament . in this work the photosphere is substituted by a rigid , perfectly conducting plane with a surface current distribution caused by the coronal magnetic field generated by the filament current . such induced currents in turn give rise to a coronal magnetic field whose effect on the prominence is to provide the lifting force necessary to counteract gravity . the full magnetic field arrangement in the corona is determined by both the line current and the photospheric currents and can be of two types : either of normal polarity ( np ) or of inverse polarity ( ip ) ; see @xcite for more details . this magnetic configuration was used by @xcite to study the oscillations of a filament . it is worth to remark that in these works the prominence is treated as an infinitely thin and long line , so that it has no internal structure , although the interplay of the filament current with the surrounding magnetic arcade and photosphere is taken into account . in addition , both np and ip structures were considered ; see figure [ fig_np_ip ] , in which the poloidal magnetic field is displayed ( the toroidal magnetic field component being zero ) . the fundamental ingredient of these papers is that if a disturbance causes a displacement of the whole line current , that remains parallel to the photosphere during its motion , then the coronal magnetic field is reshaped and the photospheric surface current is modified . this , in turn , influences the magnetic force acting on the filament current . such a force can either strengthen the initial perturbation , so that the original equilibrium is unstable , or diminish it , so that the system is stable against the initial disturbance . this simple description becomes more complicated when one takes into account that perturbations travel at a finite speed ( namely the alfvn velocity ) which causes time delays in the communication between the line current and the photosphere . @xcite investigated the effect of these time delays on the filament dynamics . exponentially growing or decaying solutions were found both for np and ip prominence models ; see figure 4 of @xcite and figure 5 of @xcite as examples . an important conclusion that can be extracted here is that the attenuation is very efficient for many parameter values , and consequently the ratio of the damping time to the period is between 1 and 10 ( i.e. in agreement with observations ) . this point is well illustrated by figure [ fig_leakage]a , in which the quality factor is depicted as a function of the alfvn speed . different curves in this diagram correspond to different sets of parameters , with solid and dashed lines used to distinguish between ip and np configurations . all the ip curves and one of the np curves in this figure take values below @xmath12 , equivalent to @xmath13 , which is an indication of the efficiency of the damping of oscillations . an ip magnetic configuration similar to that used by @xcite was taken by @xcite , although in this study the prominence is not infinitely thin and instead is represented by a current - carrying cylinder . @xcite carried out numerical simulations of the isothermal magnetohydrodynamic ( mhd ) equations and this implies that the equilibrium temperature is set to a constant value ( @xmath14 k ) everywhere . the authors mention that , despite the large temperature difference between solar prominences and the corona , this isothermal assumption can only have a minor effect . in addition , the photosphere is described as a perfectly conducting boundary , as in the papers examined above . the inner part of the filament is disturbed by a perturbation with a velocity amplitude of 10 km s@xmath0 and at an angle of @xmath15 to the photosphere . this causes the prominence to move like a rigid body in the corona , both vertically and horizontally , and to undergo exponentially damped oscillations ; see figure 7 in @xcite . it is found that the horizontal and vertical motions are decoupled from one another ( and so can be investigated separately ) and that the period and damping time of horizontal oscillations are much larger than their respective counterparts for vertical oscillations . again , strong damping can be achieved for some parameter values ; see figure [ fig_leakage]b . it turns out that vertical oscillations are very efficiently attenuated for all the parameters considered in this work and that the same happens with horizontal oscillations for coronal densities above @xmath16 kg m@xmath17 . these contrasting properties of damped horizontal and vertical oscillations contain some potential for performing seismology of prominences . now we turn our attention to the interpretation of the results presented so far . none of the works cited in the first part of this section , i.e. @xcite , mention wave leakage as the cause of the attenuation , but instead these papers connect this behaviour with the effect of the distant photosphere and the presence of time delays in the communication of disturbances between the filament and the photosphere . on the other hand , @xcite link the damping of oscillations to the emission of waves by the prominence : the damping of horizontal motions is attributed to the emission of slow waves , whereas fast waves are invoked as the cause of the damping of vertical motions . given that the main difference between this work and the other ones lies essentially in the cross section of the filament , there is no reason to believe that the physics involved are much different regardless of the prominence being modeled either as a straight and infinitely thin current or as a current - carrying cylinder . this issue must be examined in more detail to better understand the dynamics of solar prominences . a final remark about the exact nature of the wave leakage found by @xcite must be made . in this work ( see their figure 6a ) the plasma-@xmath8 in the prominence ranges from @xmath18 in its central part to @xmath19 at its boundary . hence , waves emitted by the prominence into the corona propagate in a @xmath20 environment in which magnetic field lines are closed . under these conditions , slow modes propagate along magnetic field lines and are unable to transfer energy from the prominence into the corona and so wave leakage in the system studied by @xcite is only possible by fast waves . then , it is hard to understand how the prominence oscillations can be damped by the emission of slow waves in this particular model , in which the dense , cool plasma is only allowed to emit fast waves . it must be mentioned , however , that the plasma-@xmath8 in the corona increases with the distance from the filament , which implies that the emitted fast waves can transform into slow waves when they traverse the @xmath21 region . this effect has been explored by @xcite , but see also references therein for similar studies . @xcite took into account some dissipative mechanisms and explored , through some order - of - magnitude calculations , their importance as possible damping agents . in this work we are warned about the simplicity of the calculations and about the neglect of the solar corona , whose role on the damping of prominence oscillations is beyond this simple study . dissipative mechanisms are separated into isotropic and anisotropic . amongst the first ones , viscosity and magnetic diffusion are found to be rather inefficient in attenuating perturbations in only a few periods , as demanded by observations . moreover , radiative losses ( here modeled by means of newton s cooling law , that gives an extremely simplified account of radiation in a solar prominence ) , may be of some importance , although their effect is difficult to establish unambiguously because of the presence of the unknown radiative relaxation time - scale in newton s cooling law . therefore , radiation by the prominence plasma requires a more specific treatment to determine its significance . regarding anisotropic mechanisms , in @xcite it is determined that , because of the very large density and very low temperature of prominences , viscosity can be considered isotropic and thus irrelevant as the cause of the observed damping of oscillations . in addition , thermal conduction is dominated by the presence of a magnetic field and is essentially parallel to the field direction . this mechanism is very efficient as a damping agent for very short wavelengths . since ballai s work has been extended in other papers discussed below , the meaning of `` very short wavelengths '' will become clear later . although other damping mechanisms are not contemplated in @xcite , e.g. wave leakage , ion - neutral collisions or resonant absorption , this work is important since it allowed subsequent investigations to discard the irrelevant mechanisms and to concentrate only in the rest . the efficiency of thermal conduction ( parallel to the magnetic field ) and radiation in transporting heat , and thus in causing the attenuation of oscillations , has been addressed in a number of papers ; see @xcite . these works are concerned with the spatial and temporal damping of fast and slow waves propagating in a uniform medium . radiative losses are simulated by equation ( 7 ) of @xcite , which contains two parameters that can be tuned to represent optically thin or thick radiation ; see @xcite . different sources of plasma heating are also included , although this mechanism has a negligible consequence on the properties of the waves . to understand the results of these studies it is worth to consider the characteristic time - scales of thermal conduction and radiative losses ( @xmath22 and @xmath23 ) , that following e.g. @xcite can be defined as @xmath24 @xmath25 in these expressions @xmath26 , @xmath27 and @xmath28 are the plasma pressure , density and temperature , @xmath9 is a characteristic length - scale ( the wavelength of oscillations , for example ) , @xmath29 the parallel thermal conductivity and @xmath30 and @xmath1 are the two parameters in hildner s radiative loss function . we see that @xmath23 is independent of @xmath9 and so the rate at which the plasma looses energy through radiation does not vary with the wavelength or frequency of the waves . on the other hand , the conduction time - scale , @xmath22 , decreases linearly with @xmath9 and so conduction becomes more effective at transporting heat for short wavelengths , just as noticed by @xcite , for example . in the limit of very short wavelengths , conduction is so efficient that the isothermal regime is reached . the critical wavenumber , @xmath31 , at which this transition occurs is given by @xcite as @xmath32 for slow and fast waves , respectively . here @xmath33 is the number density , @xmath34 boltzmann s constant , @xmath35 and @xmath36 the sound and alfvn speeds and @xmath37 the angle between the wavenumber and the unperturbed magnetic field . to derive these formulas it is necessary to assume @xmath38 , together with @xmath39 for the slow mode and @xmath40 for the fast mode , where @xmath41 is the real part of the frequency and @xmath42 the component of the wavenumber parallel to the equilibrium magnetic field . figure [ fig_nonadiab ] presents a plot of the characteristic conduction and radiative times together with the critical slow and fast mode wavenumbers marking the transition from the adiabatic to the isothermal regime . now , let us consider a perturbation with tunable wavelength travelling in a magnetised , uniform medium in which thermal conduction and radiation can transport heat . let us start with an extremely short wavelength , so that the wavenumber is larger than @xmath31 and the perturbation is isothermal ( hence , we are at the right of figure [ fig_nonadiab ] ) . let us now increase the wavelength until the wavenumber becomes just smaller than @xmath31 . now perturbations are no longer isothermal , but given that thermal conduction is still more efficient than radiation , since it works in a very short time - scale ( i.e. @xmath43 ) , the perturbation is dominated by conduction . next the wavelength is allowed to increase further and consequently the conduction time - scale increases proportionally to @xmath9 . under these conditions , conduction becomes less and less efficient and there is eventually a wavenumber ( or length - scale ) at which the conduction and radiation time - scales become equal ; it corresponds to the position in figure [ fig_nonadiab ] in which the solid and dashed lines cross . this length - scale , @xmath44 , is obtained by imposing @xmath45 and with the help of equations ( [ tau_c ] ) and ( [ tau_r ] ) is @xmath46 finally , for wavelengths larger than @xmath44 radiative losses dominate the energy budget of the plasma . these formulas are useful to explain some features of the damping time ( in the case of temporal damping of oscillations ) or the damping length ( in the case of spatial damping ) obtained by @xcite . consider , for example , the damping time and the damping per period ( @xmath47 ) of the fast mode modified by radiative losses and thermal conduction in a uniform medium . their variation with the wavenumber is displayed in figure [ fig_nonadiab2 ] , in which each of the three curves originates from a particular pair of values of @xmath30 and @xmath1 , the two parameters in the radiative loss function . in our discussion we concentrate on the solid line , although similar conclusions can be deduced for the other two ( and also for the slow mode , which is not examined here ) . all curves in figure [ fig_nonadiab2 ] display three changes of slope whose origin can be partly explained on the basis of the transitions between different regimes studied above . for the paramater values used in this work , we have @xmath48 m@xmath0 and @xmath49 m , that corresponds to the wavenumber @xmath50 m@xmath0 . the right - most change of slope in figure [ fig_nonadiab2 ] is around @xmath51 and coincides roughly with the value of @xmath52 , so it is caused by the transition between the isothermal and the conduction - dominated regimes . in addition , the intermediate change of slope is around @xmath53 and its agreement with @xmath54 confirms that it corresponds to the transition from the conduction - dominated to the radiation - dominated regimes . therefore , the shift from one regime to another leaves its imprint in the variation of @xmath6 and @xmath7 with respect to the wavenumber . we also note that , since the difference between the three curves of figure [ fig_nonadiab2 ] lies in the radiative loss function used , they overlap in the conduction dominated regime . the main conclusions of @xcite regarding the damping of prominence oscillations are that the slow mode is strongly attenuated for wavenumbers within the range of observed values ( i.e. for @xmath55 in the range @xmath56 m@xmath0 ) , while the fast mode is not so strongly damped and its attenuation rate is too small compared with the values from observations . moreover , the considered approximations for optically thin or thick plasmas can give very different attenuation properties and all the considered heating mechanisms are of no relevance . the influence of the surrounding corona on the damping of magnetoacoustic oscillations via thermal conduction and radiative losses has been studied by @xcite . in @xcite the filament is taken as a plasma slab embedded in the solar corona and with a uniform magnetic field parallel to the filament axis ; see figure [ fig_struct_along]a . both the prominence plasma and the coronal plasma are uniform and threaded by the same magnetic field . to understand the results of @xcite it is necessary to describe the main properties of the linear , adiabatic normal modes of such a configuration , that have been studied by @xcite . in the latter work the solutions are classified in three types ( figure [ fig_modes_along ] ) : fast , internal slow and external slow modes and it is concluded that the perturbations associated to internal slow modes are essentially confined to the prominence slab , so that these waves are hardly influenced by the coronal environment . on the other hand , fast modes have tails that penetrate in the corona and as a result coronal conditions are important in determining the features of these waves . moreover , the confinement of fast modes becomes poorer for small values of the wavenumber in the direction of the filament axis . regarding external slow modes , they produce almost negligible perturbed amplitudes inside the cold slab and thus seem uninteresting in the context of prominence oscillations . @xcite included the thermal conduction , radiative losses and plasma heating terms in the energy equation and studied the damping properties of the fast and internal and external slow modes . they obtained values of @xmath7 in agreement with observations both for the internal slow and the fast modes ; this last result implies that the presence of the surrounding corona modifies the damping properties of fast waves . such as expected , the damping rate of the internal slow mode is not modified by the inclusion of the corona in the model and it only depends on the value of the wavenumber in the direction parallel to the magnetic field . the attenuation of this mode can be recovered by inserting the parallel wavenumber , @xmath42 , in the expression @xmath57 and by substituting @xmath35 by a `` modified sound speed '' , @xmath58 , whose square is given by @xmath59 where @xmath60 and @xmath61 are determined from the partial derivatives of the heat - loss function with respect to density and temperature , respectively . the damping time of the fast kink mode , however , is severely influenced by the addition of the coronal medium . this can be clearly appreciated by comparing the damping time in a uniform medium ( figure [ fig_nonadiab2]a ) and in a structured medium with longitudinal magnetic field ( figure [ fig_fast_mode_along ] ) . the reason for such a strong modification of the fast kink mode damping time is , firstly , that this mode couples with the external slow mode , whose properties are essentially dictated by the coronal plasma . and secondly , apart from this coupling the coronal medium has a direct influence because of the presence of coronal tails in the perturbed variables . in @xcite a close examination of the role of the corona in the damping of the fast mode was done by removing one of the important non - adiabatic mechanisms ( thermal conduction or radiation ) either in the filament or in the corona and then computing the period and damping time . the relevance of each mechanism is revealed by a discrepancy between the damping time computed with and without this mechanism ( figure [ fig_fast_mode_along ] ) . the conclusion of this study is that coronal mechanisms govern the attenuation of the fast modes for wavenumbers @xmath62 m@xmath0 , whereas prominence mechanisms prevail for @xmath63 m@xmath0 . in each of these ranges radiation is dominant in the low-@xmath42 range and thermal conduction is dominant in the high-@xmath42 range , such as found in section [ non - adiab - unif - medium ] for a uniform medium . more specifically , coronal radiation controls the attenuation for @xmath64 m@xmath0 , coronal conduction does so for @xmath65 m@xmath0 @xmath66 m@xmath0 , while prominence radiation and conduction are the most important non - adiabatic mechanisms for @xmath67 m@xmath0 @xmath68 m@xmath0 and @xmath69 m@xmath0 , respectively . hence , in the range of observed wavelengths both thermal conduction in the corona and radiation from the prominence gas are important to explain the reported damping times . in section [ non - adiab - unif - medium ] the characteristic time - scales of thermal conduction and radiation in a uniform medium were invoked to explain the transition from the radiation - dominated wavenumber range to the conduction - dominated wavenumber range . the same idea can be applied to explain this transition in the present , structured configuration both for the prominence and for the corona . in the case of the prominence , the transition value is the same as before , namely @xmath70 m@xmath0 , whereas for the corona it is @xmath71 m@xmath0 . these numbers coincide well with the transitions at which different mechanisms dominate in figure [ fig_fast_mode_along ] . moreover , the wavenumber at which the transition between the ranges of dominance of coronal conduction and prominence radiation takes place can be derived from the equality of @xmath22 in the corona and @xmath23 in the prominence . this calculation yields @xmath72 m@xmath0 , also in good agreement with the value from figure [ fig_fast_mode_along ] . more details can be found in @xcite . in @xcite a prominence - corona configuration with an equilibrium magnetic field perpendicular to the prominence slab is considered ; see figure [ fig_struct_along]b . such a structure can support fast and slow modes of internal or external character , depending on whether their properties are governed mainly by the prominence or the corona , respectively . moreover , there is another type of fast and slow mode , called string by @xcite and hybrid by @xcite , that is simultaneously internal and external . @xcite find that slow modes are efficiently damped by the considered non - adiabatic mechanisms ( thermal conduction and radiative losses ) , such as found by @xcite for a perpendicular orientation of the magnetic field . it is also shown that prominence radiation dominates the attenuation of internal slow modes and that prominence radiation and coronal conduction together dominate that of the hybrid slow mode . this result for the hybrid mode is coherent with the fact that its perturbations achieve large amplitudes both in the prominence and the corona , so one expects that the most relevant damping mechanisms of each medium govern together the attenuation of the hybrid mode . on the other hand , fast modes are very poorly damped in the present configuration , which implies that the assumed magnetic field orientation is decisive for these modes . a remarkable feature of fast modes is that they can become unstable under certain circumstances . thermal conduction allows energy transfer between the prominence slab and the coronal medium . prominence radiation has an essential role in dissipating the extra heat injected from the corona and stabilises the oscillations . nevertheless , thermal instabilities appear if radiative losses from the prominence plasma are omitted or significantly reduced ( e.g. caused by an increase of the optical thickness ) since the plasma can not dissipate the extra injected heat in an efficient way . given that different parametrisations of the radiative loss function have been given in the literature @xcite and that they correspond to different optical thicknesses , the relevance of the optical thickness in the stability of fast waves is not firmly established yet . material flows are common features in solar prominences @xcite . @xcite investigated the attenuation of the slow and thermal waves by radiative losses and thermal conduction in a uniform medium with flow . in this work the fast mode is ignored because these non - adiabatic effects do not contribute in a significant way to its attenuation @xcite ; slow and thermal waves are investigated in the limit of field - aligned propagation . @xcite find that the period and damping per period can show a strong dependence on @xmath73 , the flow speed , for certain values of this parameter ; figure [ fig_nonadiab_flow ] presents their results for the slow mode . they also find that the greatest period and damping per period of slow waves is obtained for flow speeds close to the real part of the non - adiabatic sound speed , which in the presence of flow is deduced from equation ( [ nonadiab_sound_speed ] ) with @xmath74 substituted by @xmath75 . regarding the thermal mode , which in the absence of flow does not propagate , it is found that it becomes propagating for @xmath76 . therefore , when flows and time damped oscillations are observed in a prominence / filament , this wave should be also taken into account as potentially responsible of the observed damped oscillations . these authors remark that it may not be possible to determine whether an observed period and damping time are associated to a slow wave or a thermal wave in a flowing material and that a possibility to determine the origin of such wave is to resort to the temperature perturbation , which should be rather large for the thermal wave and much smaller for the slow wave . instead of considering the attenuation of perturbations in a full - prominence configuration , as in section [ nonadiab_slab ] , @xcite restrict themselves to a single thread . their model consists of a homogeneous , infinite cylinder representing the thread embedded in an unbounded corona ; see figure [ fig_nonadiab_thread]a . two additional ingredients of the model are a uniform magnetic field both inside and outside the thread and flow motions parallel to the magnetic field in both regions . thermal conduction and radiative losses are taken into account as damping mechanisms , and the combined effect of these non - ideal physics and the steady flow on the attenuation of oscillations is assessed . in this work it is found that , in the absence of flow , slow modes are efficiently damped by non - adiabatic effects , while fast kink waves are in practice not attenuated , since their damping times are much larger than typical lifetimes of filament threads . therefore , the damping by non - adiabatic mechanisms of kink oscillations is much less efficient in this cylindrical case than in the slab geometry ( figure [ fig_nonadiab_thread]b ) . in the presence of flow it is well - known that the symmetry between waves propagating parallel or antiparallel to the flow is broken . the thermal mode behaves as a wave that propagates parallel to the flow , and its motions are mainly polarized along the longitudinal direction . nevertheless , this oscillatory behaviour is not likely to be observed in practice due to its quick damping . the damping time of both slow and thermal waves is not affected by the flow . on the contrary , for realistic values of the flow velocity , the larger the flow , the larger the attenuation of parallel fast kink waves , whereas the contrary occurs for antiparallel fast kink solutions . nevertheless , this effect is not enough to obtain realistic damping times in the case of fast kink modes . chromospheric and prominence plasmas are not fully ionised and so several new effects are present in comparison to a fully ionised plasma . for example , electric charges are frozen to magnetic field lines , but neutrals are not and thus the neutral and ionised fractions of the plasma behave differently . collisions between neutrals , on one hand , and electrons and ions , on the other hand , arise and consequently a modified ohm s law is obtained . as a consequence , the main outcome of the interaction between neutrals and charged particles is the presence of enhanced joule heating and enhanced magnetic diffusion . these ideas were presented by @xcite in connection with the heating of the solar chromosphere and corona and have also been invoked to explain the dynamics of chromospheric spicules ( e.g. * ? ? ? * ; * ? ? ? * ) and of flux tubes emerging from the solar interior @xcite . the influence of partial ionisation of the prominence plasma on the properties of the fast and slow modes has been investigated by @xcite . in this work the one - fluid mhd equations for a partially ionised plasma made of electrons , protons and neutral hydrogen are derived . next , a uniform medium with a straight magnetic field is considered . the linear regime is assumed and this results in the vanishing of the joule heating term in the energy equation , so that the corresponding physical effect is absent . one may conclude that in more realistic configurations dissipation may be stronger than that found by @xcite . the main result of this work is that the fast mode can be strongly damped ( in agreement with observations ) for small values of the ionisation fraction ( figure [ fig_pip ] ) , whereas the slow mode is hardly attenuated . the mechanism responsible for the damping of the fast mode is cowling s diffusion , that vanishes in a fully ionised plasma . an analytical approximation for the fast mode damping rate , @xmath77 is derived by @xcite , with @xmath78 the relative density of neutrals , @xmath27 the gas density , @xmath79 the magnetic field strength and @xmath55 the modulus of the wavenumber . such a simple expression can be of practical use in the interpretation of observations . it indicates that the damping is stronger , i.e. @xmath7 is smaller , for almost neutral plasmas with low density , strong magnetic field and for short wavelengths . a natural extension of @xcite , concerned with the attenuation of fast and slow waves by thermal conduction and radiation , and @xcite , concerned with the attenuation of the same waves by ion - neutral effects , is to consider together a partially ionised plasma with thermal conduction and radiative losses . this task was performed by @xcite , who derived the equations governing the dynamics of such medium . a particularly interesting issue is the different conduction properties of electrons ( mainly along magnetic field lines ) and neutrals ( isotropic ) , which implies that the contribution of the neutral species to the thermal conduction must be taken into account separately following @xcite . @xcite find that the period of the magnetoacoustic waves remains basically the same as in the ideal case and that their model gives values of the ratio of the damping time to the period similar to the ones obtained in observations for the three wave modes ( fast , slow and alfvn ) . in addition , the inclusion of non - adiabatic terms in the partially ionised set of equations increases the damping of fast and slow waves in the interval of observed wavelegths as compared with the results obtained for a non - adiabatic fully ionised plasma @xcite and an adiabatic partially ionised plasma @xcite . for slow waves , the minima of the ratio of @xmath7 , corresponding to attenuation maxima , are displaced towards longer wavelengths as compared to when only thermal conduction and radiative losses are included in the model . an increase of the neutral portion in the plasma produces a displacement of these ranges of maximum damping to longer wavelengths ( figure [ fig_pip_nonadiab]a ) . radiative losses are dominant for long wavelengths ( just as found by * ? ? ? * ) , while the rest of the wavenumber interval is dominated by thermal conduction and ion - neutral collisions . regarding fast waves , radiation is the dominant damping mechanism for long wavelengths , while in the rest of the considered wavenumber interval the attenuation is dominated by the effect of ion - neutral collisions . an important result is that fast waves only exist for wavenumbers smaller than a critical value that depends on the ionisation fraction ( see figure [ fig_pip_nonadiab]b ) . in spite of this , the critical wavenumber is large in comparison with the typical wavenumbers of waves in prominences . finally , for typical prominence temperature values , the contribution of electrons to thermal conduction is negligible in front of the contribution of neutrals . the plasma density varies by about two orders of magnitude between a filament thread and the surrounding corona . in such a highly inhomogeneous configuration fast kink modes can be efficiently damped by transferring their energy to alfvn modes through resonant absorption . @xcite examine the relevance of this mechanism in the attenuation of small amplitude prominence oscillations by modeling the magnetic and plasma configuration of an individual fibril as an infinitely long , straight , cylindrically symmetric flux tube of mean radius @xmath80 surrounded by a coaxial layer of thickness @xmath81 ( figure [ fig_thread_model ] ) . the @xmath82 approximation is used . the coronal and thread densities are denoted by @xmath83 and @xmath84 , respectively , and their ratio as @xmath85 . this parameter is much larger than unity and probably of the order of 100 or larger . in this simple model , the magnetic tube representing the thread only contains cool material , i.e. the presence of hot material inside the tube is ignored . taking into account that the observed propagating waves in threads have a wavelength , @xmath86 , much larger than the tube radius , @xcite start by using the long - wavelength approximation to make some analytical progress . next , they use the so - called thin boundary approximation ( @xmath87 ) to derive the following expression for the ratio of the damping time to the period @xmath88 where @xmath89 is a numerical factor that depends on the particular variation of the density in the nonuniform layer : for a linear variation @xmath90 @xcite , while for a sinusoidal variation @xmath91 @xcite . a simple substitution in this formula of a density ratio @xmath92 and @xmath93 , which corresponds to a transitional layer whose thickness is 10% the thread radius , yields @xmath94 , in good agreement with observations . then , this analytical approximation seems to indicate that the process of resonant absorption is an efficient damping mechanism for fast mhd oscillations propagating in these structures . next , @xcite solve numerically the full set of linear , resistive , small - amplitude mhd wave equations for the transverse kink oscillations . in this manner , the approximations that lead to equation ( [ eq_res_abs ] ) are avoided . these numerical calculations confirm the analytical estimate that , for the typical large filament to coronal density contrast , the mechanism produces rapid damping on timescales of the order of a few oscillatory periods only . they also reveal that for thin layers ( @xmath95 ) the inaccuracy of the long - wavelength approximation produces differences of up to @xmath96% for the combination of short wavelength with high - contrast thread . for thick layers , differences of the order of 20% are obtained ( in agreement with * ? ? ? moreover , the authors conclude ( as shown by figure [ fig_res_abs ] ) that the obtained damping rates are only slightly dependent on the wavelength of perturbations and that the damping rate becomes independent of the density contrast for large values of this parameter . this last result has two seismological consequences . first , the observational determination of the density contrast is less critical than it is in the low - contrast regime , the one corresponding to coronal loops . second , according to seismic inversion results that combine the theoretical and observed periods and damping times @xcite , high - density thread models would be compatible with relatively short transverse inhomogeneity length scales . in a subsequent work , @xcite note that in a @xmath97 plasma with prominence conditions embedded in the much hotter corona the fast kink mode can also undergo a resonant coupling to slow continuum modes . hence , it is argued , the combined effect of the two resonant couplings may lead to an enhanced attenuation rate of the kink mode . the structure considered by @xcite is a thread identical to that of figure [ fig_thread_model ] , although the @xmath82 assumption is removed from their calculations and , consequently , the plasma pressure is finite . such as in @xcite , in this work a combination of analytical expressions and numerical analysis is used . the analytical formulas not only allow to predict the positions of the slow and alfvn resonances , but they also provide the contribution of each of the two mechanisms to the damping time , @xmath98 and @xmath99 , whose ratio is @xmath100 with @xmath101 and @xmath102 the azimuthal and longitudinal wavenumbers and @xmath35 and @xmath36 the sound and alfvn speeds . using @xmath103 and a typical value for the wavelength ( @xmath104 ) results in @xmath105 , which means that the resonant coupling of the kink mode with the alfvn continuum produces an attenuation rate that is @xmath106 times smaller than that coming from the coupling with the slow continuum . these results are also verified from the numerical computations ( see figure [ fig_slow_cont ] ) , which confirm that resonant absorption with the slow continuum is irrelevant in the damping of the observed transverse thread oscillations . although the study of small - amplitude waves and oscillations in prominences constitutes a discipline that might complement the direct determination of prominence parameters by providing independent values based in the comparison between observations and theory , so far prominence seismology is more a promise than a reality because of the large gap between observation and theory . the analysis presented in section [ seismology ] is a good example of a seismological application to prominences , but , unfortunately , it is the only one of its kind . such a gap arises because of the little restrictions imposed by observational works ( which sometimes reduce to reporting the period of the detected oscillations ) and the simplicity of theoretical studies ( which neglect the intricate nature of prominences and substitute it by a very simplified physical model ) . nevertheless , these two sides are coming together as the complexity of works increases . in this paper the possible role of several physical mechanisms in the attenuation of prominence oscillations has been reviewed . it has been shown that both the fast and slow waves can undergo strong damping under a variety of physical conditions and with the intervention of different effects . nevertheless , the list of the presumably relevant mechanisms has not been totally explored and other effects should be investigated . in addition , the geometry of the models is too simplified , which implies that the results derived so far only give a rough guide to the damping features in actual prominences . the efficiency of wave leakage and coronal conduction in draining oscillatory energy from the prominence and transferring it to the corona calls for the study of complex configurations in which the structure of the full prominence - corona system is treated more realistically . most of the works are based on the linear approximation , whose validity seems quite robust in view of the small amplitude of prominence oscillations in many cases . nevertheless , apart from the so - called small - amplitude prominence oscillations , that in general affect small regions of a prominence and have doppler velocity peaks typically below 12 km s@xmath0 , there is a different kind of phenomenon , the large - amplitude oscillations ( tripathi et al . 2009 , this volume ) . they are characterised by a much higher amplitude ( with oscillatory velocities up to 90 km s@xmath0 ) and disturb the whole filament ; see @xcite . these prominence vibrations also damp very rapidly , in a few periods , so it is in order to ascertain whether the mechanisms at work in the damping of small - amplitude oscillations are also the main ones for the large - amplitude counterparts or whether non - linear effects can also cause the attenuation of the latter . the author acknowledges the international space science institute for the financial support received to attend a meeting of the team `` coronal waves and oscillations '' in bern . financial support received from the spanish micinn and feder funds , under the grant mec aya2006 - 07637 , and from the conselleria deconomia , hisenda i innovaci of the government of the balearic islands , under grant pctib-2005-gc3 - 03 , is also acknowledged .
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quiescent prominences are thin slabs of cold , dense plasma embedded in the much hotter and rarer solar corona .
although their global shape is rather irregular , they are often characterised by an internal structure consisting of a large number of thin , parallel threads piled together .
prominences often display periodic disturbances mostly observed in the doppler displacement of spectral lines and with an amplitude typically of the order of or smaller than 23 km s@xmath0 , a value which seems to be much smaller than the characteristic speeds of the prominence plasma ( namely the alfvn and sound velocities ) .
two particular features of these small amplitude prominence oscillations is that they seem to damp in a few periods and that they seem not to affect the whole prominence structure .
in addition , in high spatial resolution observations , in which threads can be discerned , small amplitude oscillations appear to be clearly associated to these fine structure constituents .
prominence seismology tries to bring together the results from these observations ( e.g. periods , wavelengths , damping times ) and their theoretical modelling ( by means of the magnetohydrodynamic theory ) to gain insight into physical properties of prominences that can not be derived from direct observation . in this paper
we discuss works that have not been described in previous reviews , namely the first seismological application to solar prominences and theoretical advances on the attenuation of prominence oscillations .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
black holes are perhaps the most tantalizing objects in general relativity . recently , the study of black holes in a background anti - de sitter spacetime has been well motivated from developments in string / m - theory , which naturally incorporate black holes as solitonic d - branes , or simply branes as the higher - dimensional progenitors of black holes . an intriguing example of this is the conjectured duality @xcite between string theory on @xmath2 background and @xmath3 super yang - mills theory in four dimensions , and in particular , witten s interpretation @xcite of the hawking - page phase transition between thermal ads and ads black hole @xcite as the confinement - deconfinement phases of the dual gauge theory defined on the asymptotic boundaries of the ads space . much effort has been put into the weak ads gravity regime , analyzing the implications of ads black holes on dual ( gauge ) theories at non - zero temperature , using the conjectured ads / cft correspondence . in this context , the most interesting black hole solutions are presumably the five dimensional kerr - ads solutions for a stationary black hole @xcite . the thermodynamics of ads quantum gravity has been extensively used to infer the thermodynamics of quantum field theory in the large @xmath4 ( or weak field ) limits , with an ads gravity dual , such as schwarzschild - ads @xcite , kerr - newman - ads @xcite and hyperbolic - ads @xcite black holes . the kerr metric @xcite is a simple explicit exact solution of the einstein vacuum equations describing a rotating black hole in a four - dimensional flat spacetime . shortly after kerr s discovery , carter @xcite provided an elegant generalization of the kerr solution in four - dimensional de - sitter and anti - de sitter backgrounds . a higher dimensional generalization of kerr metric in a flat background was given by myers and perry @xcite . but its generalization to five and higher dimensions with a non - zero cosmological constant was given , only recently , by hawking _ et al . _ @xcite and gibbons _ et al . _ there has been recent interest in constructing the analogous charged rotating solutions in gauged supergravity in four , five and seven dimensions @xcite , and also on non - uniqueness @xcite of those solutions in five and higher dimensions . the study of non - charged rotating ( kerr ) black holes is interesting at least for two reasons . firstly , the thermodynamics of kerr black holes , in a background ads space , can give rise to interesting descriptions in terms of cfts defined on the ( conformal ) boundary of ads space , leading to a better understanding of the ads / cft correspondence @xcite . secondly , astronomically relevant black hole spacetimes are , to a very good approximation , may be described by the kerr metric . as not much is known about the stability of kerr - ads black holes in higher dimensions , in this paper we study the thermodynamic stability of these black holes in five and higher dimensions . we also investigate the gravitational stability of a background kerr-(a)ds spacetime under metric perturbations . the layout of the paper is as follows . we begin in sec . ii by outlining the ( anti)-de sitter background metrics in @xmath5 dimensions and their generalizations to kerr - ads solutions . in sec . iii we pay special attention to the thermodynamic stability of kerr - ads black holes by studying the behavior of hawking temperature , free energy and specific heat in various dimensions . in section iv we study the gravitational stability of background kerr-(a)ds metrics under linear tensor perturbations . the stability of a rotating anti - de sitter background spacetime in dimensions higher than four was not previously studied . our linearized perturbation equations have other interesting applications . in particular , they allow us to study the stability of background ads metrics with non - trivial rotation parameters . separability of hamilton - jacobi and klein - gordon equations in the kerr ( anti)-de sitter backgrounds was discussed in @xcite , especially in the limit when all rotation parameters take the same value , see @xcite for a discussion in five dimensions . an earlier work on separability of the hamilton - jacobi equation and quantum radiation from a five dimensional kerr black hole with two rotation parameters , but in an asymptotically flat background , can be found in @xcite . however , our analysis is different . it corresponds not to a separability of the wave equations for a particle but rather to a separability of radial and angular wave equations under linear tensor perturbations . one of the interesting features of the kerr metric in ( anti)-de sitter spaces is that it can be written in the so - called kerr - schild form , where the metric @xmath6 is given exactly by its linear approximation around the ( anti)-de sitter metric @xmath7 as follows @xcite : @xmath8 where @xmath9 is a null geodesic with respect to both the full metric @xmath6 and the ( a)ds metric @xmath7 . moreover , the ricci tensor of @xmath6 is related to that of @xmath10 by @xmath11 where @xmath12 , with @xmath13 and @xmath14 being the parameters proportional to the mass and gravitational potential of a kerr black hole respectively . thus , the stability of a kerr metric under metric perturbation is specific to the stability of the background metric , which is given by @xmath15 . let us begin with a five dimensional ( anti)-de sitter metric in the standard form : @xmath16 which satisfies @xmath17 , with @xmath18 in ads space . the apparent singularities at @xmath19 are merely coordinate singularities . by defining @xmath20 , one sees that the coordinate @xmath21 has range @xmath22 while @xmath23 have a period @xmath24 , so @xmath25 parameterizes ( topologically ) a @xmath26-sphere , while @xmath27 parameterizes an @xmath28-fiber . the metric ( [ 5dads ] ) is easily generalized to six and higher dimensions . in six dimensions , one has @xmath29 where @xmath30 @xmath31 and @xmath32 . the apparent singularities at @xmath19 are again merely coordinate singularities . by defining @xmath33 one easily sees that @xmath34 parameterize ( topologically ) a @xmath35-sphere . in fact , the generalized ( anti)-de sitter metric can be written in a more compact form : @xmath36 satisfying @xmath37 where @xmath38 , @xmath39 ( if @xmath5 is odd ) , or @xmath40 , @xmath41 ( if @xmath5 is even ) . both in odd and even dimensions , there are @xmath4 azimuthal coordinates @xmath42 , each with period @xmath24 . but when @xmath5 is even there is an extra coordinate @xmath43 , which lies in the interval @xmath44 . in @xmath45 spaces the rotation group is @xmath46 and the number of independent rotation parameters for a localized object is equal to the number of casimir operators , which is the integer part of @xmath47 . thus in four dimensions the metric of a kerr black hole can have only one casimir invariant of the rotation group @xmath48 , which is uniquely defined by an axis of rotation , while in five dimensions it can have two independent rotation parameters associated with two possible planes of rotation . one may introduce to ( [ compact1 ] ) @xmath4 rotation parameters , for example , using the following coordinate transformation : @xmath49 where @xmath50 . the constants @xmath1 which are introduced in ( [ first - co - trans ] ) merely as parameters in a coordinate transformation may be interpreted as genuine rotation parameters after one adds to ( [ compact1 ] ) the square of an appropriate null vector , as in ( [ kerr - schild1 ] ) . using the following coordinate transformations @xcite : @xmath51 and combining the expressions ( [ kerr - schild1 ] ) , ( [ compact1 ] ) , ( [ first - co - trans ] ) , one would obtain the kerr ( a)ds metrics in boyer - lindquist coordinates . we are not going into details of this construction but refer to ref . @xcite for an elegant discussion . in five dimensions , the metric of kerr - ads solution is @xmath52 where @xmath53 , @xmath54 , @xmath55 in the limit @xmath56 , one recovers the standard schwarzschild - ads metric . as we see shortly , black holes with non - zero rotation parameters , or , in general , kerr - ads black holes , enjoy many interesting properties distinct from schwarzschild - ads black holes . using the standard technique of background subtraction , gibbons @xcite have recently calculated the regularized ( euclidean ) actions for the kerr - ads black holes in arbitrary @xmath57 dimensions . the results are @xmath58 for odd @xmath59 , and @xmath60 for even @xmath61 , where @xmath62},\ ] ] is the volume of the unit @xmath63-sphere . in the above we have defined @xmath64 , with @xmath65 being the curvature radius of the ( bulk ) ads space . the dimensionless parameters are : @xmath66 and @xmath67 . , where , as usual , @xmath68 is the radius of the horizon , which occurs at a root of @xmath69 , and @xmath70 . the hawking temperature , which is the inverse of euclidean period , @xmath71 , is given by @xmath72,\ ] ] where @xmath39 for odd @xmath5 and @xmath73 for even @xmath5 . the calculation of total energy in an asymptotically ( a)ds background is somewhat trickier ( see e.g. @xcite ) , mainly because the analogous komar integral for the relevant time - like killing vector diverges , which then requires a regularization , see also ref.@xcite which presents a general analysis for the conserved charges and the first law of thermodynamics for the four dimensional kerr - newman - ads and the five dimensional kerr - ads black holes . in this context , the conserved charges ( energies ) @xmath74 and @xmath75 associated with different killing vectors , respectively , @xmath76 and @xmath77 are different . however , the calculation of free energy itself is unambiguous . in fact , one can always identify the free energy of a kerr - ads black hole as @xmath78 , and hence @xmath79 this result is modified from that of a schwarzschild - ads black hole by certain terms in the product which are now functions of @xmath80 and the rotation parameters @xmath81 . in four spacetime dimensions black holes are stable ( see e.g. @xcite ) , but the issue of stability may be raised in five and higher dimensions . the five - dimensional kerr - ads solutions are particularly interesting as these could be embedded into iib supergravity in ten dimensions . from ( [ free - energy - gen ] ) , it is readily seen that a phase transition between the background ads space and the black hole is set by the scale @xmath82 , so that @xmath83 corresponds to ads black hole ( @xmath84 ) and @xmath85 to a thermal ads space ( @xmath86 ) . this behavior may be seen also in terms of the charge or potential if present . in general , when the values of the rotation parameters @xmath81 are decreased , the free energy lowers towards zero at low temperature . for @xmath87 , in the small @xmath80 region , @xmath88 nearly approaches but never touches the @xmath89 axis ( see figs . @xmath90 and @xmath26 ) . that is , the free energy curve crosses the @xmath89 axis only once , namely when @xmath82 , which usually corresponds to the hawking - page phase transition point . but , in dimensions @xmath91 , this alone does not mean that a first order phase transition of hawking - page type is essentially present . in five dimensions , with @xmath92 , there is a minimum @xmath80 below which the temperature appears to be negative and also diverges as @xmath93 ( see fig . [ figure1 ] ) , which is clearly unphysical . in fact , there is a minimum in temperature below which the kerr black holes simply do not exit . nevertheless , the plots in fig . [ figure3 ] show the free energy can be a well defined function of temperature . we also note that the specific heat is a monotonically increasing function of temperature when @xmath94 . the hawking temperature of a kerr - ads black hole with a non - vanishing rotation parameter approaches zero as @xmath80 goes to zero . the free energy is still a smooth function of both the horizon size and the temperature ( see figs . [ figure2 ] and [ figure4 ] ) . these all imply a thermodynamic stability of a small kerr black hole in @xmath95 space . the thermodynamic behavior above is essentially the opposite in six dimensions , where the temperature always diverges at @xmath96 . as the plots in figs . [ figure5][figure8 ] show the thermodynamics of single parameter solutions are quite different from those with equal rotation parameters . we also note that the @xmath84 region in figs . [ figure9][figure12 ] corresponds to @xmath83 , while the region @xmath97 corresponds to @xmath98 . when @xmath91 , in the case of equal rotation parameters , only small black holes are globally preferred and locally stable , while in the case of single rotation parameter , a thermal ads phase is more preferred . the behavior in five dimensions is special in that there is no high temperature thermal ads phase regardless of the choice of rotation parameters . one of the simplest ways of calculating the energy in an asymptotically ads background is to integrate the first law of ( bulk ) thermodynamics : @xmath99 where the entropy @xmath100 and angular momenta ( of a rotating black hole ) @xmath101 are defined via @xmath102 where @xmath103 in ref . @xcite , the mass ( energy ) of kerr ads black hole was evaluated , by demanding _ as a priori _ that entropy of the black hole is one - quarter the area , @xmath104 , in order to satisfy ( [ firstlaw ] ) . the results are @xmath105 @xmath106 this result differs from the expression of energy suggested by hawking _ et al . _ in @xcite , both in odd and even dimensions , @xmath107 the reason for this is that the energy ( [ energy - hawking ] ) is measured in a frame rotating at infinity with the angular velocities : @xmath108 instead of ( [ ang - velo ] ) . since the angular velocities differ by @xmath109 , the two results above , ( [ energy - odd - d ] ) or ( [ energy - even - d ] ) , ( [ energy - hawking ] ) , agree only in the limit @xmath110 ( i.e. @xmath111 ) . a remark is in order . the energy of background ads spacetime ( i.e. @xmath112 ) is expected to be the same as the casimir energy of a dual field theory in one dimension lower , up to a conformal factor . but from the above result one finds @xmath113 when @xmath112 . to understand this apparent discrepancy , it should be noted that the adm mass @xmath13 is only a local definition of black hole energy , while the total energy of a localized object in a curved background normally takes into account the asymptotic value of the background itself ( which is non - zero in the ads space ) . and , in general , one can write @xmath114 , where @xmath115 is an integration constant . for example , for a schwarzschild - ads black hole with hyperbolic symmetry ( @xmath116 ) , @xmath115 may be given by @xmath117 , where @xmath118 is the black hole mass at the extremal limit , see e.g. @xcite . a black hole as a thermodynamic system is unstable if it has negative specific heat . as it known , small schwarzschild - ads black holes ( i.e. with @xmath119 ) have negative specific heat but large size black holes have positive specific heat . while there also exists a discontinuity of the specific heat as a function of temperature at @xmath120 , and so small and large black holes are found to be somewhat disjoint objects . however , this is essentially not the case when some of @xmath1 are non - trivial , and especially , when @xmath121 , e.g. , the small kerr black holes in @xmath95 space also have positive specific heat . to this end , we shall study the thermodynamic stability of a kerr - ads black hole by evaluating its specific heat , which is given by @xmath122 figures [ figure13 ] and [ figure14 ] show the plots of energy and temperature differentials as functions of the horizon size @xmath80 . in the @xmath123 case , with equal rotation parameters , there is clearly a minimum @xmath80 , below which the temperature diverges . there is also a minimum value of rotation parameter below which @xmath124 can be negative , which is @xmath125 in five dimensions . above this value , both the temperature differential and the specific heat are positive , see figs . [ figure15 ] and [ figure16 ] . when plotted as a function of temperature , the minimum of energy corresponds to the minimum in temperature , and hence the specific heat is a monotonically increasing function of hawking temperature . this is also the case with a single rotation parameter ( see fig . [ figure17 ] ) , but now the critical value is @xmath126 . it should also be noted that , with @xmath123 and @xmath127 , the thermodynamic behavior of a kerr - ads black hole at high temperature can be very different from that at low temperature ( see fig . [ figure17 ] ) . a similar behavior is found in the case of a single non - vanishing rotation parameter , though up to a slightly larger value of @xmath128 ( @xmath129 ) . put another way , in the @xmath95 background , small rotating black holes are unstable only for rotation parameters of order @xmath130 or less ; the precise limit is dimension dependent and black holes with larger angular velocities are thermodynamically stable . similarly , in dimensions @xmath91 , the kerr - ads black holes become unstable below some critical values of rotation parameters , for which a new branch would appear . when @xmath131 , with equal rotation parameters , the critical value is @xmath132 ( see figs . [ figure18][figure20 ] ) and it is slightly higher in the case of a single rotation parameter . looking at the behavior of free energy and specific heat as functions of horizon position @xmath80 , we remark that a five dimensional kerr - ads black hole with a single rotation parameter is thermodynamically more stable over two ( equal ) rotation parameter solutions . but this is essentially not the case in dimensions six or more . in all odd dimensions , the specific heat has a single branch at high rotation but two branches at low rotations : the critical value of @xmath128 which distinguishes these two cases increases with the number dimensions , and also with number of non - trivial rotation parameters . a similar behavior is observed in all even dimensions @xmath91 , but in this case an interesting difference is that the specific heat can never be zero with @xmath133 . it seems relevant to ask what happens at the critical angular velocity limit , @xmath134 . apparently , the action as well as the entropy is divergent in this limit . nevertheless , as discussed in @xcite ( see also @xcite ) , there exists a scaling of the mass parameter @xmath135 which makes the physical charges of the configuration finite . with equal rotation parameters , when @xmath136 , a kerr - ads black hole is more preferred than a thermal ads phase even at low temperature . in fact , in all dimensions @xmath91 , small kerr - ads black holes with a single non - vanishing rotation parameter are unstable . in our plots we have used the energy expressions suggested by gibbons _ et al . _ @xcite , which differ from those suggested by hawking _ _ @xcite by some overall constant factors . this itself does not introduce any significant difference in the behavior of specific heat and hence the thermodynamic stability of kerr - ads solutions . at any rate , the energy measured in a non - rotating frame appeared more suggestive to be used because it can be derived using various other methods @xcite ; the energy ( or total mass ) expressions given in @xcite , however , disagree with those in @xcite in odd spacetime dimensions . it was shown recently in @xcite that at fixed entropy , the temperature of a rotating black hole is bounded above by that of a non - rotating black hole , in four and five dimensions , but not in six or more dimensions . we verify this claim by plotting temperature as a function of entropy , in various dimensions ; some of the plots are depicted in figs . [ figure21]-[figure23 ] . in dimensions six or more , the minimum of entropy is not always the minimum of temperature , it actually depends upon the choice of rotation parameters . this is precisely the case where the inequality @xmath137 may be realized with a very small entropy . but in this limit the temperature actually diverges , so the effect like this might be absent in a physical picture . at fixed entropy , but @xmath138 , the hawking temperature of a rotating black hole is always suppressed relative to that of a non - rotating black hole and the inequality @xmath139 holds in all dimensions . this result , presumably , holds with various charges and classical matter fields ( such as gauge fields , dilaton , etc ) and is in accord with the earlier observation made by visser while studying a static spherically symmetric case in four dimensions with no cosmological term @xcite . a five dimensional kerr - ads black hole with a single non - vanishing rotation parameter possesses an interesting ( and perhaps desirable ) feature ; in this case the entropy vanishes when the temperature becomes zero . a similar feature is present in seven dimensions , but with two equal rotation parameters : @xmath140 , @xmath141 . in a recent work @xcite , on the evolution of a five dimensional rotating black hole via the scalar field radiation , maeda observed that , in a flat background ( @xmath142 ) , the asymptotic state of a five dimensional rotating black hole with a single non - vanishing parameter is described by @xmath143 . it would be interesting to know a similar result in an anti - de sitter background , @xmath18 . following @xcite , one would expect the partition function of a kerr - ads black hole to be related to the partition function of a cft in a rotating einstein universe on the ( conformal ) boundary of the ads space . a curious observation in ref . @xcite is that the cardy - verlinde entropy formula works more naturally using the bulk thermodynamic variables defined by hawking _ et al . _ this seems to indicate that the energy expression ( [ energy - hawking ] ) is still relevant in a dual cft . the killing vector is then given by @xmath144 where @xmath145 are the angular coordinates . this property normally allows the thermal radiation to rotate with black hole s angular velocity all the way to conformal infinity . one could ask whether or not the bulk thermodynamic variables suggested by gibbons _ et al . _ @xcite , which were measured with respect to a frame that is non - rotating at infinity , can be mapped onto the boundary cft variables by using the usual scaling argument . this does not seem to be the case as long as the cft is assumed to be on a surface of large @xmath80 in boyer - linquist coordinates . however , such a mapping might exist when the cft is assumed to be on a large spherical surface , that is one for which the coordinate @xmath146 at large @xmath147 . that is to say , it is possible that the set of bulk variables for kerr - ads black holes given by gibbons _ et al._@xcite , in some ( modified ) form , match onto the boundary cft variables that satisfy the first law of thermodynamics . this was indeed shown to be the case in @xcite . let us briefly discuss the role of non - trivial rotation parameters on the existence of an equilibrium between kerr - ads black hole and rotating thermal radiation around it . for this the requirement of a positive specific heat is a necessary condition . in five dimensions , the specific heat is always positive and also a monotonically increasing function of temperature when one ( or both ) of the rotation parameters takes a value at least one - quarter the ads length scale @xmath65 . this means , unlike in minkowski ( infinite ) space , the rotating kerr - ads black holes can be in equilibrium with rotating thermal radiation around it , when @xmath148 ; that is , the rotation parameter is not significantly smaller than the ads curvature radius , so as to attain a stable equilibrium . in this section we study the gravitational stability of kerr - ads background metrics ( with @xmath15 ) in dimensions five and higher . for this purpose , it is sufficient to consider the following @xmath5-dimensional ( time - independent ) metric _ ansatz _ : @xmath149 where the metric @xmath150 is effectively separated into two parts : a diagonal `` bulk '' line element and @xmath151 , which is the metric on an @xmath152-dimensional base manifold whose curvature has not been specified , ( so @xmath153 or @xmath154 ) , and hence can be replaced by any einstein - khler metric with the same scalar curvature . however , in the present work we study only the @xmath155 case , and hence the base @xmath156 may be viewed as an @xmath28 fiber over @xmath157 ( for odd @xmath152 ) or as @xmath158 ( for even @xmath152 ) . for example , in the @xmath123 ( or @xmath159 ) case , the event horizon is @xmath160 . under a small linear metric perturbation @xmath161 with @xmath162 , the variation in the ricci tensor is given by @xmath163 where the spin-2 lichnerowicz operator @xmath164 is defined by ( see , for example , @xcite ) @xmath165 the stability of background metrics of the form ( [ metric - gen ] ) with @xmath166 , under certain metric perturbations , is specific to tensor perturbations . we therefore would like to restrict our analysis here to the tensor mode fluctuations that satisfy @xmath167 unless @xmath168 , where the indices @xmath169 run from @xmath170 and the indices @xmath171 will run from @xmath172 for the @xmath152-dimensional base . the variation of the ricci tensor on the base @xmath173 must then satisfy @xmath174 where @xmath175 is the @xmath5-dimensional cosmological constant , with @xmath176 h_{i j}\nonumber \\ & + & \sum_{\nu=1}^{d - n}\left[\partial^\sigma g_{\sigma \nu } -\frac{1}{2}g^{\sigma\rho}\partial_\nu g_{\sigma\rho } + ( 4-n)\frac{\partial_\nu\gamma}{\gamma}\right]\partial^\nu h_{ij}\nonumber \\ & - & \frac{4}{\gamma^2}\left[g^{\mu\nu}\partial_\mu \gamma(x ) \partial_\nu \gamma(x)\right ] h_{i j},\end{aligned}\ ] ] where @xmath177 is the spin-2 lichnerowicz operator acting on the base @xmath173 . the lichnerowicz operator @xmath178 is compatible with the transverse , trace - free ( de donder ) gauge for @xmath179 : @xmath180 , see e.g. @xcite . let us first consider a background spacetime where @xmath181 , such that we can write the metric as @xmath182 we can write the lichnerowicz operator as @xmath183 h_{i j } \nonumber \\ & + & \left[\frac{\beta_r}{\beta}-\frac{\alpha_r}{\alpha } + ( 4-n)\frac{\gamma_r}{\gamma}\right ] \frac{\partial_r h_{i j}}{\beta^2}\nonumber\\ & - & \frac{4}{\gamma^2 } \frac{\gamma_r^2}{\beta^2 } \,h_{i j},\end{aligned}\ ] ] where the subscripts @xmath184 , @xmath185 denote derivatives w.r.t . @xmath184 , @xmath185 respectively . in this case we find it convenient to choose @xmath186 such that @xmath187 where @xmath188 are coordinates on @xmath173 and @xmath189 is the eigenvalue of the lichnerowicz operator on @xmath156 . we want to write the perturbed equations ( [ perturbed1 ] ) in the form : @xmath190 to facilitate this we introduce two transformations : @xmath191 with @xmath192 we then find ( see appendix for details ) @xmath193 where , @xmath194.\end{aligned}\ ] ] the above potential correctly reproduces the result in @xcite ( cf . ( 41 ) with @xmath195 and @xmath196 ) , see also @xcite . apparently , the case @xmath197 is special . while we believe the stability analysis of kerr - ads background metrics can be generalized to non - equal rotation parameters ( or angular momenta ) , we shall focus on the case with equal rotation parameters . in the case of an odd number of spacetime dimensions @xmath198 , the zero - mass ( @xmath15 ) kerr ( anti)-de sitter background metric may be given by @xmath199 where the rotation parameters are set equal ( i.e. , @xmath200 ) . the base space @xmath173 , which is topologically @xmath201 , may be parameterized by the metric @xmath202 where @xmath203 is the canonically normalized fubini - study metric on an @xmath204 dimensional complex projective space @xmath205 , and @xmath206 is a local potential for the khler form @xmath207 on @xmath205 . for example , in five dimensions , the metric on base @xmath208 is @xmath209 . in the above background , the linear tensor perturbations satisfy @xmath210 where @xmath211 h_{i j } + \frac{1-ca^2}{r^2+a^2 } ( \tilde{\delta}_l h)_{i j}.\end{aligned}\ ] ] in terms of the regge - wheeler type coordinate @xmath212 , which may be defined by @xmath213 and using eqs . ( [ trans1]),([tranf2 ] ) the differential equation is cast in the standard form : @xmath214 where the potential is @xmath215 this potential is well behaved around @xmath216 unlike for the ads - schwarzschild metric ( i.e. @xmath217 ) . there exists a criterion for stability ( e.g. the schrdinger equation not possessing a bound state with @xmath218 ) , in terms of the minimum lichnerowicz eigenvalue , @xmath219 , on the base manifold @xmath173 . in the case of a vanishing cosmological constant ( @xmath142 ) , this criterion is the same as that for a schwarzschild - ads background @xcite : @xmath220 a requirement that @xmath221 constrains the spacetime dimensions to @xmath222 ( or @xmath223 ) . the stability of a potential depends on the eigenvalue @xmath189 , ensuring that the potential is positive everywhere and bounded from below . defining @xmath224 , with @xmath225 , we require @xmath226 for @xmath18 and we find @xmath227 the lower bound on @xmath228 required for gravitational stability of the background metric is found to be stronger than that for thermodynamic stability . in the de sitter case ( i.e. @xmath229 ) , there is a mass gap , so @xmath189 starts from a finite value , and @xmath230 is unbounded from below . instead of solving the schrdinger equation directly in terms of @xmath212 , one can solve the radial part of equation ( [ lich - eq ] ) by expressing it as a hypergeometric equation , whose solution is given by a linear combinations of @xmath231 where @xmath232 , and @xmath233 we note that reality of @xmath234 immediately gives the stability condition ( [ instab ] ) . reality of the solution also requires @xmath235 which implies that there are no exponentially growing ( unstable ) modes . requiring the solution to be bounded as @xmath236 fixes one arbitrary constant which leaves @xmath237 decaying as @xmath238 . given that @xmath239 we find that @xmath240 decays as @xmath241 . by considering the large @xmath185 limit of potential ( [ potr1 ] ) we also see that @xmath242 so that eq ( [ schro r ] ) remains bounded as required to make the total energy finite . the radial equation is easily solved to yield @xmath255 however , regularity of the radial solution at @xmath256 requires @xmath257 and hence as @xmath258 the radial solution behaves as @xmath259 equation ( [ angular - eq ] ) , together with boundary conditions of regularity at @xmath260 and @xmath261 , constitute an eigenvalue problem for the separation constant @xmath254 . for @xmath262 , the solution is @xmath263,\ ] ] where @xmath264 the criterion for gravitational stability , in terms of the minimum lichnerowicz eigenvalue @xmath219 on the base manifold @xmath156 , namely @xmath265 , now translates into the requirement that @xmath266 . however we note that @xmath267 in this case . in ads space , since @xmath18 , for reality of the solution we also require , @xmath268 for real @xmath269 , @xmath270 and hence @xmath271 is imaginary , but this is not allowed by the radial wave equation . therefore there are no normalisable solutions with @xmath272 . for @xmath273 , one requires @xmath274 . a useful inequality for stability of the background ads metric ( [ background - d - even ] ) is therefore , @xmath275 instead of considering the large @xmath185 limit in ( [ main - eq - r - theta ] ) , let us now consider the special case where the angular velocity approaches the critical limit , @xmath276 ( or @xmath277 ) . the eigenfunctions are then the associated legendre polynomials @xmath278 , @xmath279 , where , @xmath280 an interesting case is @xmath281 , which allows one to study supergravity solutions in @xmath282 . it would be interesting to know what the limit @xmath283 corresponds to in a dual field theory . we leave this issue to future work . in this paper we have studied the thermodynamics and stability of higher - dimensional ( @xmath284 ) rotating black holes in a background ( anti)-de sitter spacetime . the thermodynamic quantities for kerr - ads black hole solutions suggested by gibbons _ et al . _ @xcite have been used to study the behavior of the free energy and specific heat ( which are defined unambiguously in all spacetime dimensions @xmath285 ) as functions of temperature and horizon positions . the two apparently different expressions of energy in the kerr - ads background suggested by hawking _ et al . _ @xcite and gibbons _ et al . _ @xcite do not introduce any significant difference in the behavior of bulk thermodynamic quantities ( such as entropy , free energy , specific heat , etc ) and therefore the stability of kerr - ads solutions . nevertheless , the gibbons _ _ bulk variables are more suggestive to be used as they map onto the boundary variables with the natural definition of conformal boundary metric , that is the one for which the coordinate @xmath146 for large @xmath147 , and satisfy the first law of thermodynamics . as for thermodynamic stability , rotating black holes are found to be stable down to a critical value of the rotation parameter , below which the specific heat becomes negative . for example , a five dimensional kerr - ads black hole is thermodynamically stable when the rotation parameters take values @xmath286 ; larger angular velocities usually stabilize the black hole . similarly , a zero - mass kerr - ads background is gravitationally stable down to a critical eigenvalue , below which the schrdinger equation may involve growing tensor mode perturbations . again larger angular velocities stabilize the background kerr - ads spacetimes , although the bound on the rotation parameters required for the gravitational stability of rotating black holes is not directly related to that of thermodynamic stability . an obvious extension to study , for completeness , would be the inclusion of non - zero charges and all of the possible rotation parameters in dimensions five and higher . a particularly interesting problem would be a study of the gravitational stability of _ massive _ kerr - ads black hole spacetimes in @xmath95 and @xmath287 spaces . some of the problems will be discussed in a follow - up paper @xcite . * acknowledgements * it is a pleasure to thank roy kerr and david wiltshire for discussions . i.p.n . also wishes to thank gary gibbons for useful discussions and chris pope for helpful remarks . this work was supported in part by the mardsen fund of the royal society of new zealand . consider a second order differential equation of the form @xmath288 where @xmath289 are functions of @xmath185 only . we find it is convenient to choose @xmath290 , such that @xmath291 . we then have @xmath292 + for non - zero fluctuations , @xmath293 , this implies that @xmath294 where @xmath295 . we would like to write this in the form @xmath296 to facilitate this we introduce two transformations : @xmath297 the differential equation then takes the form @xmath298 \varphi^\prime \nonumber \\ & & ~~+\left[\frac{a}{r_x^2}\frac{\chi^{\prime\prime } } { \chi}+\left(\frac{b}{r_x}-\frac{a\,r_{xx } } { r_x^3}\right)\frac{\chi^\prime}{\chi}+ \tilde{c}\right]\varphi=0,\nonumber \\\end{aligned}\ ] ] where @xmath299 . let us define @xmath300 this implies @xmath301 the differential equation then takes the standard form : @xmath302 where @xmath303 where @xmath304 . j. m. maldacena , adv . * 2 * , 231 ( 1998 ) ; s. s.gubser , i. r. klebanov and a. m. polyakov , phys . lett . b * 428 * , 105 ( 1998 ) ; e. witten , adv . * 2 * , 253 ( 1998 ) . e. witten , adv . theor . math . * 2 * , 505 ( 1998 ) . s. w. hawking and d. n. page , commun . phys . * 87 * , 577 ( 1983 ) . s. w. hawking , c. j. hunter and m. m. taylor - robinson , phys . d * 59 * , 064005 ( 1999 ) . s. w. hawking and h. s. reall , phys . d * 61 * , 024014 ( 2000 ) . r. b. mann , phys . d * 61 * , 084013 ( 2000 ) ; m. m. caldarelli , g. cognola and d. klemm , class . * 17 * , 399 ( 2000 ) . a. m. awad and c. v. johnson , phys . d * 61 * , 084025 ( 2000 ) . k. landsteiner and e. lopez , jhep * 9912 * , 020 ( 1999 ) . d. birmingham , class . * 16 * , 1197 ( 1999 ) . r. g. cai , phys . d * 63 * , 124018 ( 2001 ) . i. p. neupane , phys . d * 69 * , 084011 ( 2004 ) . r. p. kerr , phys . * 11 * , 237 ( 1963 ) . b. carter , commun . * 10 * , 280 ( 1968 ) . r. c. myers and m. j. perry , annals phys . * 172 * , 304 ( 1986 ) . g. w. gibbons , h. lu , d. n. page and c. n. pope , j. geom . phys . * 53 * , 49 ( 2005 ) . g. w. gibbons , h. lu , d. n. page and c. n. pope , phys . * 93 * , 171102 ( 2004 ) . m. cvetic , h. lu and c. n. pope , phys . b * 598 * , 273 ( 2004 ) ; phys . rev . d * 70 * , 081502 ( 2004 ) ; m. cvetic , g. w. gibbons , h. lu and c. n. pope , arxiv : hep - th/0504080 . o. madden and s. f. ross , class . * 22 * , 515 ( 2005 ) . m. vasudevan , k. a. stevens and d. n. page , class . quant . * 22 * , 339 ( 2005 ) . h. k. kunduri and j. lucietti , phys . d * 71 * , 104021 ( 2005 ) . v. p. frolov and d. stojkovic , phys . d * 67 * , 084004 ( 2003 ) ; v. p. frolov and d. stojkovic , phys . d * 68 * , 064011 ( 2003 ) . g. w. gibbons , m. j. perry and c. n. pope , arxiv : hep - th/0408217 . i. papadimitriou and k. skenderis , arxiv : hep - th/0505190 . r. gregory and r. laflamme , phys . lett . * 70 * , 2837 ( 1993 ) . m. cvetic , p. gao and j. simon , arxiv : hep - th/0504136 . m. henneaux and c. teitelboim , commun . phys . * 98 * , 391 ( 1985 ) . a. ashtekar and s. das , class . * 17 * , l17 ( 2000 ) . s. das and r. b. mann , jhep * 0008 * , 033 ( 2000 ) . n. deruelle and j. katz , class . * 22 * , 421 ( 2005 ) [ arxiv : gr - qc/0410135 ] . s. deser , i. kanik and b. tekin , arxiv : gr - qc/0506057 . g. w. gibbons , m. j. perry , and c. n. pope , arxiv : hep - th/0506233 . m. visser , phys . rev . d * 46 * , 2445 ( 1992 ) . h. nomura , s. yoshida , m. tanabe and k. i. maeda , arxiv : hep - th/0502179 . r. g. cai , l. m. cao and d. w. pang , arxiv : hep - th/0505133 . i. p. neupane , class . quant . grav . * 19 * , 1167 ( 2002 ) . g. gibbons and s. a. hartnoll , phys . d * 66 * , 064024 ( 2002 ) . h. kodama and a. ishibashi , prog . phys . * 110 * , 701 ( 2003 ) . carter and i. p. neupane ( unpublished ) .
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we study the thermodynamic and gravitational stability of kerr anti - de sitter black holes in five and higher dimensions .
we show , in the case of equal rotation parameters , @xmath0 , that the kerr - ads background metrics become stable , both thermodynamically and gravitationally , when the rotation parameters @xmath1 take values comparable to the ads curvature radius . in turn
, a kerr - ads black hole can be in thermal equilibrium with the thermal radiation around it only when the rotation parameters become not significantly smaller than the ads curvature radius .
we also find with equal rotation parameters that a kerr - ads black hole is thermodynamically favored against the existence of a thermal ads space , while the opposite behavior is observed in the case of a single non - zero rotation parameter .
the five dimensional case is however different and also special in that there is no high temperature thermal ads phase regardless of the choice of rotation parameters .
we also verify that at fixed entropy , the temperature of a rotating black hole is always bounded above by that of a non - rotating black hole , in four and five dimensions , but not in six and more dimensions ( especially , when the entropy approaches zero or the minimum of entropy does not correspond to the minimum of temperature ) . in this last context ,
the six dimensional case is marginal .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
although the general picture of synchrotron emission at low energies and inverse compton at high energies is well established , important aspects of blazars are not well understood . in particular , the location of the gamma - ray emission region is not clearly established . there are models and observations that argue for a location close to central engine ( @xmath0 pc , e.g. , * ? ? ? * ; * ? ? ? * ) , and also another set of models and observations pointing to a location further down the jet , at tens of parsecs ( e.g. , * ? ? ? * ; * ? ? ? this problem would have a straightforward answer if imaging in gamma - rays was possible , but this is not the case with current technology where we can get angular resolutions of @xmath1 at the highest gamma - ray energies observable with the large area telescope onboard the fermi gamma - ray space telescope ( _ fermi_-lat , * ? ? ? nonetheless , we have a good idea about the origin of the radio emission , thanks to the submilliarcsecond resolution achievable with very long baseline interferometry ( e.g. , * ? ? ? we also know that blazars are extremely variable at most wavelengths and one alternative is to use that property to put constraints on the relative location of the radio and gamma - ray emission . the basic idea of our programme is that if the radio and gamma - ray emission are produced in the same region , we expect to see correlated variability between the light curves of these two bands , while delayed emission will tell us about the relative location of these two emission regions . this observational programme requires simultaneous monitoring of a large sample of objects in the radio and gamma - ray bands . for the gamma - ray band we can use the excellent capabilities of _ fermi _ , that provide continuous monitoring of the sky with complete coverage every 3 hours . the complementary resources needed in the radio band are provided by the ovro 40 m monitoring programme described below . since mid-2007 we have carried out a dedicated long - term monitoring programme at 15 ghz using the owens valley radio observatory 40 meter telescope . we are currently observing about 1800 blazars twice per week . the sample includes all the _ fermi_-lat detected blazars north of declination @xmath2 . a detailed description of the programme can be found in @xcite , and a list of publications and data in our website http://www.astro.caltech.edu / ovroblazars/. one of the main advantages of continuously monitoring a large number of sources is that we can study the variability properties of a large sample , and their variation over different source populations . the results of such studies are presented in @xcite and @xcite . similar studies including optical data are presented in @xcite . a question that is closely related to the existence of correlated radio / gamma - ray variability is the correlation of their mean fluxes . this question has been studied using single epoch surveys , but monitoring data can overcome several of the difficulties of those studies . detailed discussions about this subject , and a look over different source classes is presented in @xcite and @xcite . besides those results , we discovered a 120 - 150 d quasi periodic oscillation in a blazar , the shortest one ever identified @xcite . we regularly contribute data for multiwavelength campaigns , which makes this programme a valuable result for the community . variability is one the main characteristics of blazars , so we need to understand it in order to incorporate it in emission models . efforts to incorporate this characteristic in models are starting to appear in the literature ( e.g. , * ? ? ? * ; * ? ? ? * ) , but they still require further development to provide testable observational predictions . a detailed understanding of the multiwavelegth variability properties of blazars is also essential for our understanding of the significance of multiwavelength cross - correlations as explained below . we can use the power spectral density ( psd ) , but there are other models that we do not consider here but that are discussed elsewhere and in this conference ( e.g. , * ? ? ? even for the simple psd case , the characterization is complicated by the uneven sampling of the light curves . we refer the reader to our recent paper @xcite , which describes the monte carlo approach used to for the psd fit . we characterize the variability using a simple power law psd ( psd @xmath3 ) . this model has a single parameter , the exponent @xmath4 . using 4 years of radio data on 1593 sources we can find good constraints for @xmath4 in 238 sources . we emphasize that the number of sources with good constraints is only a fraction of the total number of sources . this is because in most cases we need longer time series , or because the amplitude of the variability is not large enough to discriminate between different values of @xmath4 . the lesson to learn is that getting errors for the psd is as important as finding a best fit value , because it has a direct impact of the uncertainty in the estimation for the significance of a correlation as described in @xcite and briefly below . for the sources with constraints we found a range of values of @xmath4 , with the property that they cluster around 2.3 . using our large sample we can look at variations of the psd over different source populations ( e.g. , bl lac versus fsrq ) , but we did not find any variations of the psd between classes , probably due to the large uncertainties in @xmath4 that we expect to reduce with longer radio light curves ( max - moerbeck et al . in preparation ) . with a characterization of the variability , we can look at the significance of cross - correlations . but before , we look at how important is the variability for the to study of the significance . figure 12 of @xcite illustrates the different characteristics of the variability for time series with @xmath5 and @xmath6 . for @xmath7 we see that no clear flares are seen in the light curve , but for @xmath8 there appear clear flares similar to the ones observed in blazars . the problem arises when we look at correlations between those independent light curves . in this case ( see figure 13 of * ? ? ? * ) , we see that @xmath7 does not produce large amplitude correlations , but that @xmath8 produce large amplitude peaks in the cross - correlation . the lesson is that , whenever we correlate @xmath9 noise time series we will see correlations , so the most important question is how significant they are . to evaluate the significance of a cross - correlation we use monte carlo simulations . these simulation take into account the psd , the sampling and also the observational noise . in figure 28 of @xcite we can see how the results of the significance of correlations vary when we change the model used for the simulated light curves . this figure shows how important is to have a good characterization of the variability for the significance estimate . having large errors in the psd power - law exponent is equivalent to having fuzzy significance contours . unfortunately this is the situation with most sources , and should be indicated when presenting cross - correlation significance results , as it one of the largest sources of uncertainty in these estimates . we studied the cross - correlation significance between 15 ghz and gamma - ray light curves . we used 4 years of radio data and 3 years of fermi data for 86 sources @xcite . of those 86 sources only 41 are variable , have no long term trends and thus a significance analysis is possible . of those 41 only 1 has a larger than @xmath10 significance and 2 more above @xmath11 significance , the level at which we expect to see one chance cross - correlation in a sample of 41 . from this we find two main results : * only a minority of the sources have significant correlations * in all the cases with correlations , the radio emission lags the gamma - ray emission . these results demonstrate very clearly the difficulty of measuring statistically robust multiwavelength correlations and the care needed when comparing light curves even when many years of data are used . we use a simple model to estimate the location of the gamma - ray emission site . in this model the emission is coming from the moving components we observe with vlbi . the peak of the gamma - ray flare occurs at @xmath12 , and some time later @xmath13 , when the component propagates to optically thin part of the jet , beyond the radio core , we see the peak of the radio flare . our cross - correlation result , gives @xmath14 , vlba measurements and variability constraints provide the geometry of the jet and speed of the moving components . we have enough ancillary data for two sources . for ao 0235 + 164 we consider the two peaks above @xmath10 seen in the correlation and we get a very weak constraint . for pks 1502 + 106 , the constraint is better and locates the gamma - ray emission a few parsecs away the central engine , but the errors are very large ( 22@xmath1515 pc for a conical jet , and 12@xmath159 pc for the collimated jet case ) . the emission mechanism in the radio band is synchrotron radiation , and the variations can be due to changes in electron density or magnetic field , or to relativistic beaming effects . in this case polarization observations provide data on the evolution of the magnetic fields in the emission region . the ovro 40 meter telescope has a completely new receiver , all the way from the front to the backend , that has been working since june 2014 . this new receiver covers a broader band and will soon provide polarized flux density light curves . this programme provides the largest data base of single frequency light curves and has produced many results by itself and for the community . we have used this data to learn about the variability of blazars in the radio band , and how it relates to the gamma - ray variations . our main result is that only a minority of sources show significant correlations between the radio and gamma - ray bands , with radio lagging the gamma - ray emission .
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since mid-2007 we have carried out a dedicated long - term monitoring programme at 15 ghz using the owens valley radio observatory 40 meter telescope ( ovro 40 m ) .
one of the main goals of this programme is to study the relation between the radio and gamma - ray emission in blazars and to use it as a tool to locate the site of high energy emission . using this large sample of objects we are able to characterize the radio variability , and study the significance of correlations between the radio and gamma - ray bands .
we find that the radio variability of many sources can be described using a simple power law power spectral density , and that when taking into account the red - noise characteristics of the light curves , cases with significant correlation are rare .
we note that while significant correlations are found in few individual objects , radio variations are most often delayed with respect to the gamma - ray variations .
this suggests that the gamma - ray emission originates upstream of the radio emission .
because strong flares in most known gamma - ray - loud blazars are infrequent , longer light curves are required to settle the issue of the strength of radio - gamma cross - correlations and establish confidently possible delays between the two .
for this reason continuous multiwavelength monitoring over a longer time period is essential for statistical tests of jet emission models .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the study of superconducting and superfluid systems has recently attracted wide attention , among other things because of the realization of ultracold fermi gasses in optical lattices @xcite . these systems can be considered as quantum simulators that can be used for probing fundamental problems in condensed - matter physics @xcite , for instance the search for exotic new phases in strongly magnetized superconductors . ultracold fermi gasses offer important advantages over conventional superconductors , mainly because of their extensive tunability . in a superconductor , the number of spin - up and spin - down electrons is equal and the interaction strength is fixed . in ultracold fermi gasses , one can not only tune the interaction strength by use of feshbach resonances @xcite , but also the population imbalance can be freely adapted . this experimental freedom has led to the study of a variety of new phenomena in imbalanced ultracold fermi gasses @xcite . one fundamental question , that is still not settled , concerns the nature of the ground state of an imbalanced fermi gas . when population imbalance between the spin - up and spin - down components is introduced into these systems , complete pairing is no longer possible . clogston and chandrasekhar suggested that above a critical imbalance , the superfluid system would undergo a transition into the normal state @xcite . this effect has been observed experimentally by the mit @xcite and rice @xcite groups . however , their observations were not in exact agreement , and there still exists some controversy @xcite about the exact nature of the phases of the superfluid system at high levels of imbalance . in 1964 fulde and ferell @xcite and independently larkin and ovchinnikov @xcite proposed that a superfluid system can accommodate population imbalance , by making a transition into a state with a finite center - of - mass momentum ( and thus a spatially modulated order parameter ) . this state is the so - called fulde - ferell - larkin - ovchinnikov - state ( fflo - state ) . recently , there has been an ongoing theoretical search for this exotic state @xcite . in 1d , the fflo - state was predicted to exist in superconducting systems @xcite . this work was confirmed numerically @xcite and elaborated further through theoretical studies of the ground state and the phase diagram of a 1d fermi gas @xcite . furthermore it has been shown that the 1d analogue of the fflo - state is stable in a large section of the bcs - bec crossover phase diagram @xcite , compared to the case of a 3d fermi gas . although this 1d fflo - state has not yet been observed directly , a recent paper reports the experimental observation of density profiles that agree quantitatively with theoretical predictions at low temperature @xcite . in three dimensions however , the experimental observation of the fflo - state has so far remained elusive . one of the main reasons for this is that the fflo - state in three dimensions only occurs in a tiny section of the bcs - bec crossover phase diagram @xcite . this then begs the question , is there a way to stabilize the fflo - state in a 3d fermi gas ? the purpose of this paper is twofold : first we develop a path - integral description for a superfluid fermi gas which can accommodate the fflo - state , and second we propose a method to stabilize the fflo - state through an optical potential . in two recent papers it was suggested to stabilize the fflo - state by the use of a 3d optical lattice @xcite . in this paper we investigate the stabilizing effect of a 1d optical potential in order to investigate the interplay between the wavevector of the fflo - state and the wavevector of the laser which creates the optical potential . in the present work , the 1d optical potential provides a periodic modulation in one direction . we emphasize that we do not look at the fflo - state in a one - dimensional gas @xcite , but in a 3d gas with a superimposed one - dimensional periodic potential . in section [ the free fermi gas ] we describe the fflo state in an imbalanced fermi gas in 3d within the path - integral framework . as a test for the correctness of this description , we calculate the zero temperature phase diagram for this system and compare our findings with recent theoretical results @xcite . in section [ modelling a one - dimensional external potential ] we investigate two models to account for the effect of a one - dimensional optical potential . we show that the presence of such a potential leads to a substantial increase of the stability region of the fflo - state . finally in section [ conclusions ] we draw conclusions . the partition sum of an imbalanced fermi gas in 3d can be written as a path integral over the fermionic fields @xmath0 and @xmath1 : @xmath2{l}\mathcal{z}={\textstyle \int } \mathcal{d}\bar{\psi}\mathcal{d}\psi \text { } \exp \left ( -{\displaystyle \sum \limits_{\mathbf{k},n } } { \displaystyle \sum \limits_{\sigma } } \bar{\psi}_{\mathbf{k},\omega_{n},\sigma}\left ( -i\omega_{n}+\mathbf{k}^{2}-\mu_{\sigma}\right ) \psi_{\mathbf{k},\omega_{n},\sigma}\right . \\ \qquad \left . -\dfrac{g}{\beta l^{3}}{\displaystyle \sum \limits_{\mathbf{k},n } } { \displaystyle \sum \limits_{\mathbf{k}^{\prime},n^{\prime } } } { \displaystyle \sum \limits_{\mathbf{q},m } } \bar{\psi}_{(\mathbf{q}/2)+\mathbf{k},\omega_{m}+\omega_{n},\uparrow}\bar { \psi}_{(\mathbf{q}/2)-\mathbf{k},\omega_{m}-\omega_{n},\downarrow}\psi_{(\mathbf{q}/2)-\mathbf{k}^{\prime},\omega_{m}-\omega_{n^{\prime}},\downarrow}\psi_{(\mathbf{q}/2)+\mathbf{k}^{\prime},\omega_{m}+\omega_{n^{\prime}},\uparrow}\right ) . \end{array } \label{toestandssom begin}\ ] ] here @xmath3 is the wavevector , @xmath4 are the fermionic matsubara frequencies and @xmath5 denote the two different hyperfine states . furthermore , @xmath6 is the inverse temperature given by @xmath7 , @xmath8 is the lateral size of the system , the chemical potential of a particle with spin @xmath9 is denoted by @xmath10 and @xmath11 is the renormalized interaction strength . we use units such that @xmath12 . this partition sum ( [ toestandssom begin ] ) can be made more tractable by introducing the hubbard - stratonovic transformation , which decouples the fourth - order interaction term into second order terms by introducing two auxiliary bosonic fields @xmath13 and @xmath14 , interpreted as the pair fields . as a first approximation , only the saddle - point is taken into account in the path integral over the bosonic fields . to describe the fflo - state , we propose to use a saddle point at which the atomic pairs have a finite wavevector @xmath15:@xmath16 by using this particular form of the saddle - point , we choose to describe the ff - state , which has an order parameter given by a plane wave @xmath17 . it is also possible to describe the lo - state , which is a superposition of two plane waves with wavevector @xmath15 and @xmath18 . in this paper , we will not consider the lo - state . for the remainder of the article , the ff - state is referred to as the fflo - state . when @xmath15 is set equal to zero in ( [ zadelpunt ] ) , the description of the normal superfluid is recovered @xcite . in expression ( [ zadelpunt ] ) , the prefactor @xmath19 ensures that @xmath20 has units of energy . using ( [ zadelpunt ] ) the fermionic fields can be integrated out in expression ( [ toestandssom begin ] ) for the partition function , leading to an effective action @xmath21 , through @xmath22 + \dfrac{\beta l^{3}}{g}\left \vert \delta \right \vert ^{2}\right ) = \exp \left ( -\mathcal{s}_{sp}\right ) , ~ \label{saddle - point partition sum}\ ] ] with @xmath23 the inverse nambu propagator which is given by@xmath24{cc}-i\omega_{n}+(\mathbf{q}/2+\mathbf{k})^{2}-\mu_{\uparrow } & -\delta \\ -\delta^{\ast } & -i\omega_{n}-(\mathbf{q}/2-\mathbf{k})^{2}+\mu_{\downarrow}\end{array } \right ) .\ ] ] expression ( [ saddle - point partition sum ] ) can be simplified further by performing the sum over the matsubara frequencies . also , it is useful to express the results as a function of the total chemical potential @xmath25 and the imbalance chemical potential @xmath26 . furthermore , the interaction between particles is modeled with a two - body contact potential @xmath27 . the renormalized interaction strength @xmath11 can then be written as follows @xcite : @xmath28 where @xmath29 is the 3d s - wave scattering length . as a final step the continuum limit is taken , and a thermodynamic potential is associated with the effective action @xmath30 . this then results in@xmath31 -\xi_{\mathbf{q},\mathbf{k}}-\frac{\left \vert \delta \right \vert ^{2}}{2k^{2}}\right ) -\frac{\left \vert \delta \right \vert ^{2}}{8\pi \left ( k_{f}a_{s}\right ) } , \label{thermodynamic potential free fermi gas}\ ] ] where the following notations were introduced @xmath32{l}\xi_{\mathbf{q},\mathbf{k}}=k^{2}-\left ( \mu-\dfrac{q^{2}}{4}\right ) \\ e_{\mathbf{k}}=\sqrt{\xi_{\mathbf{q},\mathbf{k}}^{2}+\left \vert \delta \right \vert ^{2}}\\ \zeta_{\mathbf{q , k}}=\zeta+\mathbf{q\cdot k}\end{array } \right . . \label{korte notaties vrij fermi gas}\ ] ] the resulting form of the thermodynamic potential ( [ thermodynamic potential free fermi gas ] ) has a similar form as the original result for the homogeneous 3d fermi gas derived by iskin and s de melo @xcite and coincides with it for @xmath33 . in order to test the correctness of ( [ thermodynamic potential free fermi gas ] ) , the thermodynamic potential is used to calculate the zero temperature phase diagram of an imbalanced fermi gas in 3d . this can be done for a fixed number of particles or for fixed chemical potentials . to transform between these two descriptions , the number equations , given by@xmath34 have to be solved . as an example , the phase diagram is calculated for a fixed density @xmath35 and for a fixed imbalance chemical potential @xmath36 . to do this , the first number equation ( [ first number equation ] ) is solved , given values of @xmath37 and @xmath38 , to determine the chemical potential @xmath39 . the set of values @xmath40 is then substituted in the free energy @xmath41 . the minima in the free energy landscape determine which state is the ground state of the system , for a given imbalance chemical potential @xmath36 and a given interaction strength @xmath42 . there are three local minima that can be identified in the free energy landscape : the bcs - state ( spin - balanced superfluid ) with @xmath43 , @xmath44 , the fflo - state with @xmath43 , @xmath45and the normal state with @xmath46 figure [ contours_geenzeta.eps ] shows that the fflo - state can indeed be the ground state of an imbalanced fermi gas in three dimensions ( at zero temperature ) . in this figure , the free energy of the system is shown , as a function of the bandgap @xmath20 and the wavevector @xmath15 , relative to the fermi energy @xmath47 and the fermi wavevector @xmath48 respectively . [ h ] contours_geenzeta2.eps for relatively small values of the imbalance chemical potential @xmath36 , the system is in the bcs ground state ( a ) . when @xmath36 increases , the system undergoes a first order transition into the fflo - state ( b ) . when the imbalance chemical potential increases further , the system continuously goes over into the normal state ( c ) . figure [ contours_geenzeta.eps ] only shows these phase - transitions for one specific value of the interaction strength , near the bcs - limit @xmath49 . however , since the first number equation is used to calculate the value of the chemical potential @xmath39 , the description is also valid for the complete bcs - bec crossover regime . it must be noted that the mean field approximation breaks down in the unitarity limit . however , the fflo - state is expected to form only in the bcs region of the bcs - bec crossover , where we expect the mean field description to give qualitatively correct results . the phase diagram of the system is shown in figure [ fd_vrijfermigas ] . this diagram shows that , theoretically , the fflo - state can be formed in an imbalanced fermi gas in 3d , but since it only occurs on a tiny section of the phase diagram , it may be hard to observe this state experimentally . our results coincide with recent theoretical results @xcite . [ h ] fd_vrijfermigas.eps as shown in section [ the free fermi gas ] , the problem with detecting the fflo - state in an imbalanced fermi gas in 3d is that it exists only in a relatively small section of the bcs - bec phase diagram . in this section , we describe the 3d imbalanced fermi gas in a 1d optical potential . there are two main reasons why such a potential can stabilize the fflo - state . the first reason is that in an imbalanced fermi gas in 3d , the fflo - state can have a wavevector @xmath15 in an arbitrary direction . this freedom of choice leads to low - energy bosonic excitations , or goldstone modes , which render the fflo - state unstable . in the presence of a 1d optical potential however , it will be energetically favorable for the fflo - state to form in the direction of the optical potential . this will limit the choice for the wavevector @xmath15 to just one value , thus suppressing the goldstone modes , which is expected to stabilize the fflo - state . the second reason is that the optical potential will enhance the 1d modulation of the fflo - order parameter . we therefore expect the enhancement to be largest when the wavevector of the fflo - state is equal to the wavevector of the optical potential . the present mean - field treatment does not include the effect of excitations such as the goldstone modes , but it does include the energy lowering of the modulated order parameter due to the optical potential . in this section we propose two approaches of modeling a 3d imbalanced fermi gas in a 1d optical potential . in both approaches , the optical potential is described by using a modified dispersion relation . in section [ anisotropic masses ] we model the optical potential by introducing an anisotropic effective mass in the direction of the potential @xcite ( from here on this is supposed to be the @xmath50-direction ) . this approximation is valid when the fermi energy of the system lies near the bottom of the lowest bloch band , i.e. in the case of low density or a short - wavelength optical potential . in section [ bloch - dispersion ] , we model the optical potential by treating the full lowest bloch band in the tight binding approximation @xcite . this model is valid when the fermi energy lies in the lowest bloch band ( otherwise more bands have to be considered ) , but contrary to the first case , it does not need to lie at the bottom of the band . in both section [ anisotropic masses ] and section [ bloch - dispersion ] the optical potential is supposed not to forbid tunneling , as this would inhibit the formation of the fflo - state . this implies that we will treat the imbalanced fermi gas in 3d in a 1d optical potential as a three dimensional system with a one - dimensional periodic modulation . in this section , the 1d optical potential is modeled through the use of a modified effective mass of the fermionic particles in the direction of the optical potential @xmath51 where @xmath52 is the effective mass of the particles in the @xmath50-direction . here and for the remainder of the paper , @xmath53 denotes the magnitude of the in - plane wavevector . this is the wavevector which lies perpendicular to the laserbeam which creates the 1d optical potential . the derivation of the thermodynamic potential for this case is analogous to the derivation in section [ the free fermi gas ] . in the present derivation however , it is assumed that the fflo - state will form in the @xmath50-direction @xmath54 , because this is energetically favorable to other directions , due to the anisotropy introduced through ( [ dispersie anisotroop ] ) . the resulting thermodynamic potential , in the limit for temperature going to zero , is given by@xmath55 -\xi_{\mathbf{k}}-\frac{\left \vert \delta \right \vert ^{2}}{2\left ( k^{2}+\dfrac{k_{z}^{2}}{2m_{z}}\right ) } \right ) -\frac{\left \vert \delta \right \vert ^{2}}{8\pi \left ( k_{f}~a_{s}\right ) } , \label{thermodynamic potential anisotropic masses}\ ] ] with the modified notations @xmath32{l}\xi_{\mathbf{q},\mathbf{k}}=k^{2}+\dfrac{1}{2m_{z}}\left ( k_{z}^{2}+\dfrac{q^{2}}{4}\right ) -\mu \\ e_{\mathbf{k}}=\sqrt{\xi_{\mathbf{q},\mathbf{k}}^{2}+\left \vert \delta \right \vert ^{2}}\\ \zeta_{\mathbf{q , k}}=\dfrac{1}{2m_{z}}k_{z}~q-\zeta \end{array } \right . .\ ] ] the number equations are still given by ( [ first number equation ] ) and ( [ second number equation ] ) , but the density @xmath35 has changed to @xmath56 because of the modified dispersion relation ( [ dispersie anisotroop ] ) . in the limit @xmath57 the thermodynamic potential ( [ thermodynamic potential anisotropic masses ] ) and the density ( [ dichtheid ] ) converge to the corresponding expressions in the case of an imbalanced fermi gas in 3d , described in section [ the free fermi gas ] . expression ( [ dichtheid ] ) implies that when the effective mass @xmath52 changes , the density @xmath35 changes with it . it would be interesting however , to compare the phase diagrams for fermi gasses with different effective masses at equal density . this can in fact be achieved , because we have found that the thermodynamic potential of the system with an effective mass @xmath52 can be rescaled to the thermodynamic potential of the system with effective mass @xmath58 ( [ thermodynamic potential free fermi gas ] ) , using the following scaling relation@xmath59 from a theoretical point of view , this rescaling property is time - saving for calculations and gives a deeper insight into the role of the effective mass @xmath52 . the main advantage is however , that all physical properties can be studied at the same density . this property relates to experiment , because when an external potential is turned on , the effective mass is altered , but the average density will remain the same . the effect of changing @xmath52 on the bcs - bec crossover phase diagram is shown in figure [ mz_ok.eps ] . this figure shows the fflo phase boundaries of the imbalanced fermi gas in 3d for different values of the effective mass @xmath52 , before rescaling according to ( [ herschaling ] ) ( and hence at different densities ) . [ h ] mz_ok.eps there is a tilting of the fflo - region about a fixed point at unitarity . furthermore , there is an increase in the width of the fflo - region ( relative to the abscissa ) as the effective mass increases . by using the scaling - relation ( [ herschaling ] ) , the phase diagrams in figure [ mz_ok.eps ] can be rescaled to equal density . after rescaling , the phase diagrams for the different effective masses maps onto the phase diagram of the imbalanced fermi gas with isotropic effective mass ( @xmath58 ) , described in section [ the free fermi gas ] . from this we conclude that the fflo - state is not fundamentally influenced by an optical potential in which the fermi energy lies near the bottom of the first bloch - band . this can be explained by the fact that no fundamental anisotropy is introduced into the system by altering the effective mass , because independent of the effective mass , the system can be scaled back to the case of the imbalanced fermi gas where the effective mass equals 1/2 . in section [ anisotropic masses ] it was shown that a more fundamental anisotropy is needed , in order for the optical potential to have an effect on the fflo - state . in this section , we model a 1d optical potential in the tight - binding approximation , using the first bloch - band . for this purpose , the quadratic dispersion in the @xmath50-direction is replaced by a periodic dispersion @xcite @xmath60 ~. \label{bloch - dispersie}\ ] ] here @xmath61 is the wavevector of the optical potential and @xmath62 is a prefactor with units of energy , given by @xcite@xmath63 with @xmath64 the depth of the potential and @xmath65 the recoil energy given by @xmath66 , with @xmath67 the wavelength of the optical potential and @xmath68 the mass of the fermionic particles . in the limit for small @xmath69 expression ( [ bloch - dispersie ] ) simplifies to ( [ dispersie anisotroop ] ) , with@xmath70 given the new dispersion ( [ bloch - dispersie ] ) , the thermodynamic potential for this system can be calculated . the result is@xmath71 -\xi_{\mathbf{k}}-\frac{\delta^{2}}{2\left \ { k^{2}+\delta \left [ 1-\cos \left ( \frac{\pi k_{z}}{q_{l}}\right ) \right ] \right \ } } \right ) -\frac{\delta^{2}}{8\pi \left ( k_{f}~a_{s}\right ) } \label{thermodynamic potential bloch dispersion}\ ] ] with the following notations@xmath32{l}\xi_{\mathbf{q},\mathbf{k}}=k^{2}+\delta \left [ 1-\cos \left ( \frac{\pi}{2}\frac{q}{q_{l}}\right ) \cos \left ( \frac{\pi k_{z}}{q_{l}}\right ) \right ] -\mu \\ e_{\mathbf{k}}=\sqrt{\left \ { k^{2}+\delta \left [ 1-\cos \left ( \frac{\pi}{2}\frac{q}{q_{l}}\right ) \cos \left ( \frac{\pi k_{z}}{q_{l}}\right ) \right ] -\mu \right \ } ^{2}+\delta^{2}}\\ \zeta_{\mathbf{q , k}}=\zeta-\delta \sin \left ( \frac{\pi}{2}\frac{q}{q_{l}}\right ) \sin \left ( \frac{\pi k_{z}}{q_{l}}\right ) \end{array } \right.\ ] ] it can easily be shown that expression ( [ thermodynamic potential bloch dispersion ] ) is equal to the corresponding thermodynamic potential ( [ thermodynamic potential anisotropic masses ] ) of the anisotropic effective mass case , in the limit @xmath72 , @xmath73 with @xmath52 held constant , according to ( [ relatie mx ql delta ] ) . the two number equations again are given by ( [ first number equation ] ) and ( [ second number equation ] ) and the density @xmath35 can be calculated using the general expression@xmath74 \right \ } \right)\ ] ] which yields@xmath75{c}\dfrac{q_{l}}{2\pi^{2}}\left ( 1-\delta \right ) ~\left ( 1\geq2\delta \right ) \\ \dfrac{q_{l}}{2\pi^{3}}\left [ \left ( 1-\delta \right ) \arccos \left ( \dfrac{\delta-1}{\delta}\right ) + \delta \sqrt{1-\left ( \dfrac{\delta -1}{\delta}\right ) ^{2}}\right ] ~\left ( 1<2\delta \right ) \end{array } \right . ~. \label{n}\ ] ] here it must be noted that our derivation is only exact if @xmath76 , because otherwise more than one bloch band has to be considered . as in the previous sections , the phase diagram for this system can be constructed by studying the local minima of the free energy . figure [ fd_mettekst.eps ] shows a comparison between the phase diagram of an imbalanced fermi gas in 3d and the phase diagram of an imbalanced fermi gas in 3d subject to a 1d optical potential , modeled in the tight - binding approximation using the first bloch band . [ h ] fd_mettekst.eps this figure shows that the fflo - region is enlarged by a factor 3 to 6 , due to the stabilizing effect of the 1d optical potential . figure [ fd_mettekst.eps ] further shows that at @xmath77 a transition point occurs , where the fflo - region reaches a maximum width ( relative to the abscissa ) , and narrows quickly for larger values of @xmath36 . this effect finds its origin in the magnitude of the wavevector of the fflo - pairs @xmath78 . when the imbalance chemical potential @xmath36 increases , @xmath78 increases likewise to accommodate for the widening gap between the fermi surfaces of the two spin - species . at a certain level of imbalance ( in the case of figure [ fd_mettekst.eps ] at @xmath77 ) @xmath78 equals the wavevector of the optical potential @xmath61 . at this point , the fflo - state is optimally enhanced , because the spatial modulation of the fflo - state is equal to the spatial modulation of the optical potential . this results in a maximal width of the fflo - region . when @xmath36 increases further , @xmath78 retains the constant value @xmath61 , and is not able to grow any further . this effect is shown in figure [ deltas_ql12.eps ] . [ h ] deltas_ql12.eps hence , we can conclude that , although the imbalance has increased further , the optical potential forces the fflo - state into a state where the form of the fflo - order parameter matches the form of the optical potential . this results in a narrowing of the fflo - region , because the wavevector of the fflo - state is not sufficiently large anymore to bridge the gap between the fermi - surfaces of the spin - up and spin - down particles . the value of the transition point , where @xmath78 becomes equal to @xmath61 , rougly increases linearly with the value of @xmath62 , as shown in figure [ delta050607.eps ] . [ h ] delta050607.eps qualitatively , this is because the rate of change of the fflo - wavevector @xmath78 with increasing imbalance chemical potential @xmath36 , decreases when @xmath62 becomes larger . this means that @xmath36 has to be larger ( compared to the case of lower @xmath62 ) for @xmath78 to reach the limiting wavevector of the optical potential @xmath61 . the advantage of this resonant enhancement of the fflo - state is that , for a given level of imbalance , an optimal stability region for the fflo - state can be created , simply by tuning the wavelength of the 1d optical potential . it should be noted that , when considering an imbalance chemical potential @xmath36 smaller than @xmath79 , @xmath62 has to become smaller than @xmath80 and more bands have to be taken into account in order for our description to be exact . we do not treat this case in the present paper . to obtain a more direct link with experimental parameter values , we convert our units back to si units for the three situations depicted in figure [ delta050607.eps ] . a possible choice of atoms which we considered is @xmath81 atoms in a one - dimensional harmonic trap , with a density of @xmath82 . for instance , the case with @xmath83 and @xmath84 corresponds to the case of an optical potential with wavelength equal to @xmath85 , a recoil energy of @xmath86 and an optical potential depth of @xmath87 . the numerical values for these experimental parameters in the case of @xmath88 and of @xmath89 are depicted in figure [ delta050607.eps ] panels ( a ) and ( c ) respectively . for the illustrative cases of figure [ delta050607.eps ] we use @xmath84 , but theoretically , any choice of @xmath61 was possible because we found that the wavevector of the optical potential @xmath61 acts as a scaling parameter , according to the following scaling relation:@xmath90 in principle this means that we can vary @xmath61 from zero to infinity . however , there exist some limitations on this parameter . first there is a lower limit for @xmath61 because below a certain value of @xmath61 no value of the optical potential depth @xmath64 can satisfy equation ( [ tight binding delta ] ) , given values for @xmath62 and for the recoil energy @xmath65 . second , when the depth of the optical potential becomes too large , particles will be confined in the direction of the optical potential , thus inhibiting the formation of fflo - states . this sets an upper limit for the ratio of @xmath91 and subsequently for the value of @xmath61 . during the course of our work , loh and trivedi published their results on the lo - state in a 3d cubic lattice @xcite . they found that in a 3d cubic lattice , the lo - state was more stable than the ff - state . since we already find a substantial increase in the ff - state using a 1d optical potential , we expect that the effect on the lo - state will be similar or larger . it would be interesting to apply our 1d - potential scheme also to the lo - case . we have described the fflo - state in an imbalanced fermi gas in 3d within the path - integral framework , by choosing a suitable saddle - point at which the atomic pairs have a finite centre - of - mass momentum . as a platform to address the case of a 3d imbalanced fermi gas in a 1d optical potential and to validate our path - integral description , we rederived the zero - temperature phase diagram for an imbalanced fermi gas in 3d . for this case , our results coincide with recent theoretical results . as a proposal to stabilize the fflo - state we have studied an imbalanced 3d fermi gas in a 1d optical potential . this potential was modeled in two different ways . for the first model , where we considered anisotropic effective masses , we have found that this model is a rescaling of the case of a 3d imbalanced fermi gas with isotropic effective mass . in the second model , we described the effect of the 1d optical potential using the first bloch band in the tight - binding approximation . in this case we have found a substantial increase in the stability region of the fflo - state , as compared to the case of the 3d fermi gas without the 1d optical potential . related results were recently found in the case of a 3d cubic optical lattice @xcite . the advantage of our 1d optical potential scheme , compared to a 3d cubic optical lattice , is that it allows to find an optimal stability configuration for the fflo - state with a given level of imbalance , by tuning the wavelength of the optical potential . this resonant enhancement of the fflo - region occurs when the wavevector of the fflo - pairs is equal to the wavevector of the optical potential . this tunability makes a 1d optical potential a suitable experimental configuration for the stabilization of the fflo - state . we therefore propose that this concept can facilitate the experimental observation of the fflo - state in an imbalanced fermi gas in 3d . t. mizushima , k. machida , and m. ichioka , phys . lett . * 94 * , 060404 ( 2005 ) ; d. e. sheehy and l. radzihovsky , ibid . * 96 * , 060401 ( 2006 ) ; j. kinnunen , l. m. jensen , and p. trm , ibid . * 96 * , 110403 ( 2006 ) ; k. machida , t. mizushima , and m. ichioka , ibid . * 97 * , 120407 ( 2006 ) ; p. castorina , m. grasso , m. oertel , m. urban , and d. zappal , phys . rev . a * 72 * , 025601 ( 2005 ) ; n. yoshida and s .- k . yip , ibid . * 75 * , 063601 ( 2007 ) ; w. zhang and l .- m . duan , ibid . * 76 * , 042710 ( 2007 ) ; t. k. koponen , t. paananen , j .- martikainen , m. r. bakhtiari , and p. trm , new j. phys . * 10 * , 045014 ( 2008 ) . k. b. gubbels , m. w. j. romans , and h. t. c. stoof , phys . lett . * 97 * , 210402 ( 2006 ) ; t .- dao , m. ferrero , a. georges , m. capone , and o. parcollet , ibid . * 101 * , 236405 ( 2008 ) ; c .- c . chien , q. chen , y. he , and k. levin , ibid . * 97 * , 090402 ( 2006 ) ; * 98 * , 110404 ( 2007 ) .
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we describe an imbalanced superfluid fermi gas in three dimensions within the path - integral framework . to allow for the formation of the fulde - ferell - larkin - ovchinnikov - state ( fflo - state ) , a suitable form of the saddle - point
is chosen , in which the pairs have a finite centre - of - mass momentum . to test the correctness of this path - integral description
, the zero - temperature phase diagram for an imbalanced fermi gas in three dimensions is calculated , and compared to recent theoretical results .
subsequently , we investigate two models that describe the effect of imposing a one - dimensional optical potential on the 3d imbalanced fermi gas .
we show that this 1d optical potential can greatly enlarge the stability region of the fflo - state , relative to the case of the 3d fermi gas without 1d periodic modulation .
furthermore it is show that there exists a direct connection between the centre - of - mass momentum of the fflo - pairs and the wavevector of the optical potential .
we propose that this concept can be used experimentally to resonantly enhance the stability region of the fflo - state .
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a majority of stars are in binaries , and a substantial fraction of binaries have short enough orbital periods that they are likely to interact during either their main sequence or post - main sequence evolution . many of the most interesting phenomena in astronomy can be directly traced to the interaction of close binaries ; an incomplete list would include binary neutron stars and white dwarfs , supernovae ia , cataclysmic variables , and blue stragglers . there is a vast literature on the subject ( e.g. , paczynski 1971 ; wellstein & langer 1999 ; hurley , tout & pols 2002 ; belczynski , kalogera & bulik 2002b ) . although there are many ingredients that must be considered in interacting binaries , an implicit assumption in much theoretical work has been that the lifetimes of the stars are almost always quite different . this assumption arises naturally from two considerations . first , the single star initial mass function ( imf ) is a steep function of mass , with low mass stars being far more numerous than high mass stars ( e.g. salpeter 1955 ) , and strong mass - lifetime relationship for all but the most massive stars implies a large lifetime difference unless the masses are very close . second , a flat mass ratio spectrum ( see for example kuiper 1935 ) for binaries that are likely to interact is adopted in most population synthesis studies , leading to very few `` equal '' component mass binaries . pinsonneault & stanek ( 2006 ) have argued that observations indicate the existence of a substantial population of nearly equal mass binaries ( `` twins '' ) . in such systems a strong inequality in lifetime is not present , so there might be important qualitative differences in their evolution compared to unequal mass binaries . survey of astronomical literature strongly suggests binary twins are a general feature of close binary population , as a peak near @xmath10 was reported by a number of investigators . for example , halbwachs et al . ( 2003 ) studied a large sample of spectroscopic binaries type f7 to k ( masses from about 1.7 down to @xmath11 ) , including binaries in open clusters . they find that the mass ratio has a broad peak from @xmath12 to @xmath13 , and a sharp peak for @xmath14 . as they discuss , the strength of the peak for @xmath14 gradually decreases with the increasing orbital period , which is to be expected . the fraction of twins can be as high as @xmath15 for periods @xmath16days and it is still significant ( as high as 35% ) for much longer periods of up to 1000 days . a much earlier study by lucy & ricco ( 1979 ) also finds a strong and narrow peak of binaries with @xmath17 , again using a sample of spectroscopic binaries corrected for various observational errors and biases . tokovinin ( 2000 ) confirms that finding using additional data and in fact also calls this population `` twins '' , arguing that they constitute 10 - 20% of the total binary population in the @xmath18 days regime . additional , although perhaps more anecdotal support for the significant twin population comes from the realm of very high mass stars found in eclipsing binaries . the most massive binary known , wr 20a ( rauw et al . 2004 ; bonanos et al . 2004 ) , is an eclipsing system , so the masses of both components can be measured accurately . the masses are @xmath19 and @xmath20 ( rauw et al . 2005 ) , giving a mass ratio of @xmath21 . given that @xmath22 stars are extremely rare ( both due to the steepness of the mass function and their short lifetime ) , having such extremely massive secondary would be quite unlikely unless the twin phenomenon is involved . there are also some theoretical considerations that seem to indicate that double neutron star binaries form _ only _ from twins ( bethe & brown 1998 ; chang - hwan , hong - jo & brown 2007 ) . if this is the only double neutron star formation scenario , the twin fraction must be high to explain the observed rates of these binary systems . however , not all evidence points towards a large population of twins . first , there are some loopholes to the arguments pushing toward the theoretical requirement of twins to make double neutron star systems . in addition , the existence of low - mass x - ray binaries requires some systems with very different masses ( kalogera & webbink 1998 ; fryer , burrows & benz 1998 ) . even with the intermediate - mass progenitors of these low - mass x - ray binaries ( podsiadlowski , rappaport & pfahl 2002 ) , a large twin fraction coupled on top of a otherwise flat mass ratio distribution would have trouble explaining low - mass x - ray binaries . finally , not all the observational evidence points toward a twin fraction . kobulnicky & fryer ( 2007 ) argue that for their dataset of 120 o and early b stars , the twin fraction must be less than 25% . their study used one of the largest datasets of o and early b stars focusing on a single stellar association - cygnus ob2 ( kiminki et al . 2007 ) . with observations and theory arguing both for and against twins , we investigate the effect of twin binaries on population of close ( coalescing within hubble time ) double compact objects , focusing on observations that might allow us to distinguish a twin population of stars from the more standard stellar mass ratio distributions in this study we present the population synthesis study of double neutron star ( ns - ns ) , black hole neutron star ( bh - ns ) and double black hole ( bh - bh ) progenitors . we employ two basic calculations ; one with the usually adopted flat mass ratio distribution and one that includes a very large ( @xmath15 ) population of twins . the results are discussed in context of double compact object mergers that are expected to be the prime sources of gravitational radiation for ground based observatories like ligo or virgo ( e.g. , kalogera et al . 2007 ) , and are also considered as very likely short - hard gamma ray burst progenitors ( nakar 2007 ) . in a forthcoming paper ( belczynski & pinsonneault , in prep . ) we will study the influence of twins on lighter compact object binaries with white dwarfs and their connection to type ia supernovae and formation of blue stragglers . binary population synthesis is used to calculate the merger rates and properties of double compact objects . the population synthesis code employed in this work , startrack , was initially developed for the study of double compact object mergers in the context of gamma - ray burst ( grb ) progenitors ( belczynski , bulik & rudak 2002a ) and gravitational - wave inspiral sources ( belczynski et al . in recent years startrack has undergone major updates and revisions in the physical treatment of various binary evolution phases , and especially mass transfer phases . the new version has already been tested and calibrated against observations and detailed binary mass transfer calculations ( belczynski et al . 2007a ) , and has been used in various applications ( e.g. , belczynski , bulik & ruiter 2005 ; belczynski et al . 2006 ; belczynski et al . the physics updates that are most important for compact object formation and evolution include : a full numerical approach to orbital evolution due to tidal interactions , calibrated using high mass x - ray binaries and open cluster observations , a detailed treatment of mass transfer episodes fully calibrated against detailed calculations with a stellar evolution code , updated stellar winds for massive stars , and the latest determination of the natal kick velocity distribution for neutron stars ( hobbs et al . 2005 ) . for helium star evolution , which is of a crucial importance for the formation of double neutron star binaries ( e.g. , ivanova et al . 2003 ; dewi & pols 2003 ) , we have applied a treatment matching closely the results of detailed evolutionary calculations . if the helium star fills its roche lobe , the systems are examined for the potential development of a dynamical instability , in which case they are evolved through a common envelope ( ce ) phase , otherwise a highly non - conservative mass transfer issues . we treat ce events using the energy formalism ( webbink 1984 ) , where the binding energy of the envelope is determined from the set of helium star models calculated with the detailed evolutionary code by ivanova et al.(2003 ) . the progenitor evolution and the roche lobe overflow episodes are now followed in much greater detail . we note significant differences from our earlier studies . for a detailed description of the revised code we refer the reader to belczynski et al . ( 2007a ) . the most recent and important changes in the context of double compact object formation reflect the treatment of the dynamically unstable mass transfer and evolution into the ce phase . first , it was pointed out that there is only ( if any ) a small chance of survival of ce phase if a donor star is on the hertzsprung gap ( hg ) , simply because there is no clear entropy jump between core and envelope so once ce inspiral is initiated it does not stop until the two binary components coalesce ( see belczynski et al . second , we limit accretion onto compact objects during ce phase to @xmath23 of the bondi - hoyle rates based loosely on estimates of outflows ( armitage & livio 2001 ) . we have also slightly modified our input physics in context of rejuvenation , black hole spin ( belczynski et al . 2007c ) evolution and debugged the entire code . two separate evolutionary models for massive star binaries are calculated . they differ only in common envelope treatment . in one calculation ( model a ) that we will refer to as our reference model we do not allow for common envelope survival in case the donor star is crossing hg . this is in effect for h - rich hg stars as well of helium hg stars . the former reduces drastically formation ( and merger ) rates of bh - bh binaries , while the later reduces moderately rates for ns - ns systems as discussed in detail by belczynski et al ( 2007b ) . in alternative common envelope model ( model b ) we allow for ce survival for all donors ( hg included ) . it does not mean that system can survive every ce phase . the regular standard energy balance ( e.g. webbink 1984 ) is performed to check for a potential survival . in both models we vary an assumption on the initial mass ratio ( lower - mass over higher - mass binary component ) of binaries that we evolve . we either employ flat mass ratio distribution and we will refer to these populations as `` flat '' binaries or we impose `` twin '' distribution in which we require that @xmath15 binaries have mass ratio distributed uniformly in range @xmath24 while the remaining @xmath15 have flat distribution for @xmath25 . for each models we evolve @xmath26 binaries with solar metallicity ( @xmath27 ) . we require that the primary mass is drawn from power - law imf with slope @xmath28 , while secondary mass is obtained through a given mass ratio distribution . we additionally require that the primary initial mass is @xmath29 while secondary initial mass is @xmath30 . the range of masses was chosen such that it encompasses entire possible mass range for double compact object formation . in particular , low - mass ends take into account potential rejuvenation of stars through binary accretion . to initialize our populations we first draw a primary mass , then mass ratio is drawn from a given distribution , and then mass of a secondary is calculated from @xmath31 . if @xmath32 is smaller than required minimum mass ( @xmath33 ) we repeat the drawing . such a scheme , although it uses underlying flat mass ratio distribution results in skewed ( toward high-@xmath4 ) distribution . the resulting initial mass ratio distributions are presented in figure 1 ( top panel ) . for calibration and galactic compact object merger rate calculation we use binary fraction of @xmath34 , and we assume that star formation rate ( sfr ) was constant in galaxy through last 10 gyr at the level of @xmath35 . to calculate the synthetic sfr we extend our imf down to hydrogen burning limit ( @xmath36 ) , with a three component broken power - law imf with slopes of @xmath37 and corresponding breaks at @xmath11 and @xmath38 ( kroupa & weidner 2003 ) . for our twin populations we assume that twin binaries are formed independent of binary properties ( like period ) and that they form in entire mass range ( @xmath39 ) . the mass of entire underlying stellar population ( all single and binary stars ) that corresponds to our simulations ( @xmath40 ) is @xmath41 and @xmath42 for flat and twin populations , respectively . since the star forming mass in galaxy is @xmath43 it results in calibration boost factors ( @xmath44 ) of @xmath45 and 57 for flat and twin populations , respectively . after evolution of massive primordial binaries ( @xmath46 ) we obtain population of double compact objects in each model . then in a given model we initiate each double compact object @xmath47 times at different starting time . starting times are chosen from the uniform distribution within the range of @xmath48 gyr ( constant sfr ) . the starting time is then increased by an evolutionary time that was needed for a progenitor binary to form a given double compact object ( @xmath49 myr ) . the double compact objects are then evolved with angular momentum losses due to emission of gravitational radiation until they merge . merger times are denoted as @xmath50 and they can span a wide range of values . the entire lifetime of a given binary is then @xmath51 . we record the time at which they merge . then we calculate an average merger rate in period @xmath48 gyr . these are our predicted galactic merger rates . it is worth to note three things . first , it may seem counter - intuitive that the boost factor is larger for ( more massive ) twin population . however , one needs to realize that in the population of stars of a given mass there is a larger number of high mass binaries ( @xmath29 and @xmath52 ) in twin population as compared with flat population . simply , it is easier to form both binary components with high masses in a population with mass ratio peaked at high values ( twins ) as opposed to population with flatter mass ratio distribution . as we have evolved the same number of twins and flat high mass binaries , it means that the number of stars in an entire underlying stellar population ( @xmath53 ) is smaller for twins than for flat binaries . finally , since the most mass is contained in primaries and single stars , and not in secondaries ( that are heavier in twin population ) , it translates into a smaller mass of underlying stellar population containing twin binaries . smaller the simulated mass ( @xmath40 ) higher the boost factor . second , we have employed an optimistic ( pro - twin ) approach , as we do not put any period constraints on twin formation ( see 1 discussing evidence that twins may form preferentially at short periods ) in addition to adopting a very high fraction of twins ( @xmath15 ) . had we limited population of twins , the boost factor @xmath47 for twins would decrease , making the differences between twin and flat calculations less pronounced . third , since , we also consider population of ultracompact ( extremely short - lived ) double neutron star binaries it is important to notice their increased contribution to merger rates . if at formation there are similar in size populations of short- and long - lived double compact objects , the short - lived systems will dominate merger rates . long - lived systems merge beyond our counting time of 10 gyr ( age of the disk ) unless they happen to form early on , while short - lived systems contribute to merger rate independent of their formation time ( provided that their merger times are much shorter than the age of the disk ) . first we have calculated galactic merger rates for the two models and we have translated them into advanced ligo detection rates using method presented in belczynski et al . ( 2007b ) . the results are presented in table 1 . the galactic merger rates of double compact objects ( combined for ns - ns / bh - ns / bh - bh ) for flat populations are @xmath54 myr@xmath55 while for twin populations @xmath56 myr@xmath55 . the range of the rates corresponds to our different assumption on ce evolution and formation ( or lack of thereof ) of ultracompact ns - ns systems as was discussed in detail by belczynski et al . ( 2007b ) . the factor of @xmath2 increase in rates from flat to twin dominated populations is equally connected to _ ( i ) _ the difference in underlying star population that gives boost factor @xmath57 larger for twins than for flat binaries ( see sfr calibration 2.4 ) and _ ( ii ) _ the slightly higher ( @xmath58 ) formation efficiency of double neutron stars from massive twin binaries . the small magnitude of the later effect may be somewhat surprising , as one would intuitively expect that with the twin population production of double compact objects would significantly increase . in the following we explain this surprising finding . first , we examine the mass ratio distribution of flat population for model a. there is a significant fraction ( @xmath8 ) of massive binaries that we have evolved with low mass ratios ( @xmath59 ; fig.1 ; top panel ) . on the other hand , binaries that actually produce double compact objects ( fig.1 ; middle panel ) are found predominantly with high mass ratios but in a rather wide range ( @xmath60 ) . note that there is no significant peak for double compact object progenitors at high-@xmath61 . second , if we go from flat to twin population we shift half of the initial binaries from the entire mass ratio range to the very high mass ratios ( fig . 1 ; top panel ) . binaries that are shifted from the low-@xmath4 range ( @xmath58 of the population ) will become an extra component in formation of double compact object in twin population . binaries that are shifted from the high-@xmath4 range will produce double compact objects but will not increase the overall production rate since they would have formed double compact objects anyway . therefore , the rate increase factor from the shift of binaries from standard to twin population is only @xmath58 . the above finding is a direct result of the shape of the mass ratio distribution of double compact object progenitors . in model a for the flat population mass ratio is found within range @xmath62 and it falls slowly with the decreasing @xmath61 . the lack of progenitors below @xmath63 is connected to fact that below that value the progenitor binary evolves through common envelope ( rather than stable mass transfer phase ) after primary evolved of main sequence , and the ce leads most often to a merger . this is especially true since most of the donors will start mass transfer on hertzsprung gap as during this phase stars experience maximum radial expansion ( belczynski et al.2007b ) . the slope of the distribution is explained by the narrow range of masses in which double neutron stars form ( and since they dominate double compact object population they set the distribution ) . if a primary is chosen within a range for ns formation ( @xmath64 ) it is easier to find potential secondary that can form ns if mass ratio is higher . if mass ratio is too small , the primary have a greater chance to have mass below ns formation mass , and therefore mass ratio distributions falls off with decreasing @xmath61 . intrinsically , once two stars have masses within ns formation range and their mass ratio is over 0.5 , there is no preference for ns - ns formation at higher @xmath4 . in other words , we do not note any significant evolutionary effects that make it easier to make ns - ns at high mass ratio . advanced ligo detection rates are listed in table 1 . as for galactic merger rates there is a range of values for flat population : @xmath65 yr@xmath55 and for twin population @xmath66 yr@xmath55 . and as before the range corresponds to the change on assumption on ce evolution . however , the increase in rates from model a to b is now due to the increased formation of bh - bh binaries in model b. these binaries , although a small contributor to galactic merger rates , are most important for ligo as they can be detected from much larger distances ( much higher chirp masses ) as compared to ns - ns mergers and therefore they dominate detection rates ( see also belczynski et al . we note that the change of the detection rates for ligo from flat to twin population is rather small ( factor of @xmath2 ) and is much smaller than other model uncertainties ( e.g. ce evolution ) . in figure 2 we show the distribution of chirp mass for coalescing double compact objects . as the population of double compact objects is dominated by double neutron stars we see that the distributions peak at @xmath67 that corresponds to the typical chirp mass of a @xmath68 and @xmath69 ns - ns binary ( see also belczynski et al . we also note that the distributions are almost the same for the flat and twin populations . this is the result of the underlying initial final - mass relation ( see belczynski et al . 2007a for details ) . this relation shows that neutron stars form with the similar mass ( @xmath70 ) for a wide range of progenitor masses @xmath71 and only in the narrow range @xmath72 heavier neutron stars ( @xmath73 ) are formed . such the initial - final mass relation leads to a rather narrow distribution of neutron star masses ( somewhat widened by accretion and mass loss in binaries ) that is obtained in both populations . if the two populations are compared in context of the flat initial - final mass relation it becomes obvious why the two distributions peak at the same value . for flat population two neutron star progenitors are found ( on average ) farther apart in mass than for twins but still they need to fall within the narrow mass limits that allow neutron star formation ( @xmath72 ) . for twin model the two progenitors are closer in mass , but still are within the same mass limits . since the mass of a neutron star does not depend strongly on the initial mass of progenitor the masses of neutron stars in both models are similar . there are other heavier compact objects , namely black hole neutron star systems and double black hole systems with chirp masses reaching all the way to @xmath74 , both for twin and flat populations . in particular we find many more heavier systems in model b as in this model black hole systems form with much higher efficiency as compared to model a ( belczynski et al . 2007b ) . merger time distributions for flat and twin double compact object binaries are presented in figure 3 . in the top panel we show calculations with our reference evolutionary model , while the bottom panel demonstrates results for the alternative common envelope evolution . for the reference model the two distributions are very similar , and the number of mergers falls off rapidly with the decreasing merger time . however , we still predict quite a significant number of double compact objects : @xmath75 with merger times shorter than 100 myr . most of these short lived systems are double neutron stars that have formed along evolutionary channels that end in the stable mass transfer episode with a helium star donor ( e.g. , ivanova et al . 2003 ) . in the model with alternative evolution we allow for common envelope survival even if donor stars are crossing hertzsprung gap . although this may appear not to be supported by the current understanding of inspiral process ( see 2 and belczynski et al . 2007b for more through discussion ) the common envelope evolution and the associated inspiral is not yet fully understood . distributions are similar for flat and twin binaries for high merger times . however , for small merger times there is a an additional component in both distributions as compared to the standard calculations . moreover , this additional component is much more pronounced in twin population than in flat population . in particular we find that in flat population this component ( @xmath76 myr ) contains @xmath77 of mergers while in twin population it reaches @xmath15 . accounting for the shape of distribution and the larger number of mergers in twin population it translates to @xmath78 times as many short - lived systems in twin population as compared to flat population . the systems with very short merer times ( @xmath79 myr ) are so called `` ultracompact '' double neutron stars , that form through one extra common envelope phase ( additional orbit contraction ) as opposed to standard model binaries with larger merger times ( e.g. , belczynski et al . 2002b ; ivanova et al . 2003 ; belczynski et al . 2006 ) . in table 1 we list fractions of prompt double compact object mergers . these will include potential short - hard grb progenitors : ns - ns and bh - ns mergers . although , it is noted again that these fractions are almost completely dominated by ns - ns mergers . fractions are given for binaries that have lifetimes ( @xmath80 ) shorter than 100 ( @xmath81 ) and 1000 myr ( @xmath82 ) . we find that in the reference model @xmath75 of the mergers are expected to occur in young hosts ( with stellar populations as young as 100 myr ) both for flat and twin models . however , if alternative evolution is included in calculations , then the fraction increases to @xmath83 for flat population and to @xmath5 for twin population . this is a direct result of merger time distribution that is similar for twin and flat population in the reference model ( see fig . 3 top panel ) and different for alternative ce model , in particular twins producing many more ultracompact ns - ns binaries ( see fig . 3 bottom panel ) . the fractions of the mergers are also given in table 1 for significantly older ( but still rather young ) hosts : 1000 myr . it is found that great majority of the mergers @xmath84 and @xmath85 for reference and alternative ce models is then expected to take place in hosts of this age . at this age ( or lifetime of the double compact object population ) the ultracompacts are not so important as the classical long - lived systems play an important role in overall population and the fractions are rather independent of whether twin of flat populations are considered ( see fig . 3 ) . in the following we explain the more effective production of ultracompacts in the twin population than in the flat population for alternative ce evolution ( see fig . 3 ; model b ) . we will limit the discussion to double neutron stars and their progenitors as they constitute the vast majority ( @xmath86 ) of double compact object ( dco ) systems with ultrashort merger times ( i.e. , @xmath87 myr ) . the distribution of initial mass ratio for progenitors of ultracompact dcos is presented in figure 4 . for the flat population ( see fig . 4 , top panel ) we notice that _ ( i ) _ in model a there are rather few progenitors with high mass ratios ( @xmath88 ) in contrast to model b in which we find a prominent peak of the distribution at high mass ratios . therefore for model b , redistribution of progenitors from the flat to twin mass ratio distribution is enhancing the production of ultracompacts . in fact , for twin population ( see fig . 4 , bottom panel ) we observe an increase of ultracompact systems by a factor of @xmath89 in model b ( see the significant increase of these systems at high-@xmath90 ) . note that the change of the mass ratio distribution from flat to twin increases number of progenitors due to the calibration ( see 2 ) but this increase factor is only @xmath91 . the additional increase is solely due to the peak in the number of high-@xmath90 systems for model b ultracompacts . the shape of the mass - ratio distribution for progenitors of ultracompact systems is understood in the framework of evolution of massive stars leading to the formation of double neutron stars . in general , classical ( long - lived ) ns - ns binaries form from progenitors that experience only two mass transfer episodes . the ultracompact systems progenitors usually go through an additional mass transfer episode . this third mass transfer episode is encountered just before second ns formation . a low - mass helium star overfills its roche lobe and starts transferring he - rich material to the first born ns . most often such a transfer occurs when a helium star is crossing hertzsprung gap ( large radial expansion ) . depending on the mass ratio and the evolutionary stage of the helium donor ( where on hg ) the mass transfer is either stable or it evolves into ce phase . since the most of the neutron stars in our simulations have mass @xmath92 the mass ratio is set by the mass of helium donor . for very light helium stars ( @xmath93 ) , stable mass transfer is predicted while , for more massive donors ( @xmath94 ) , a ce develops . the mass of the helium star is set predominantly by the initial mass of the progenitor star ( in addition to mass gain and loss in earlier binary interactions ) ; the lower the mass of the progenitor , the lower mass of helium star it forms . since the helium star is formed out of secondary ( most cases ) and @xmath61 was defined as the ratio of the mass of secondary to primary , it is expected that systems going through the stable mass transfer ( lower mass helium star progenitor ) have initially a lower mass ratio than systems developing ces ( higher mass helium star progenitor ) . in model a we do not allow for ce survival if donor is on hg and therefore progenitors with very high mass ratios are disfavored . in model b , that allows for survival of the ce phase with the hg donor , the high mass ratio progenitors are abundant and they contribute to formation of ultracompacts . additionally , the higher mass ratio systems are more likely to survive initial mass transfer episodes in the evolution of progenitor binary ( e.g. , closer in mass components so lower the probability of a merger during ce phase ) bethe & brown ( 1998 ) proposed a scenario of double neutron star formation from twin binaries . in this scenario , because the two stellar components of the binary are nearly equal mass , the system undergoes both common envelope phases prior to the collapse of either star . in such a scenario , the neutron stars formed in collapse need not undergo a ce phase , and hence avoid accreting additional material . this model provides a natural explanation for double neutron star systems in which both neutron stars had nearly equal masses . but it only works when the two binary components have nearly equal mass , and hence , strongly depends on the number of twins . bethe & brown ( 1998 ) and subsequent work by lee et al . ( 2007 ) argue that this scenario can explain all of the ns - ns binary systems observed if a large twin population exists . they argue that any formation scenario that forces a neutron star to go through a common envelope phase will produce a low - mass black hole , not a neutron star . the bulk of the simulations by fryer , woosley & hartmann ( 1999 ) also made this assumption , and came up with similar conclusions : with appropriate choices of the other free parameters , one can match the observed ns - ns systems . but whether or not this scenario is the dominant formation path for double neutron star binaries hinges on the fact that the accretion onto a neutron star in a common envelope system is equal to the bondi - hoyle rate . recall that it was realized that neutrino cooling would allow the neutron star to accrete beyond the eddington rate , causing the neutron star to accrete as much material as is fed it . if one assumes this rate is equal to the bondi - hoyle rate , the accretion can be up to a solar mass . but the actual accretion rate may be much less . first , how much one accretes depends sensitively on the evolution of the common envelope phase ( fryer et al . very few simulations have focused particularly on neutron stars in stellar mergers with giant companions . most examples have very rough boundary conditions and/or do not model the inspiral of a neutron star in a massive companion ( e.g. ruffert 1999 ; armitage & livio 2000 ; zhang & fryer 2001 ; ricker & taam 2007 ) . this preliminary work has yet to solve the actual merger process . but these studies have determined a few key issues with the bondi - hoyle assumption in neutron star accretion in stellar inspiral : density / velocity gradients can alter the bondi - hoyle accretion at large scales , density / velocity gradients can lead to disk formation and outflows . ruffert & anzer ( 1995 ) , ruffert ( 1999 ) and taam & ricker ( 2007 ) have focused on the deviation at large scales of the bondi - hoyle accretion rate . at issue here is that angular momentum in the accreting material can provide pressure support for the infalling material , slowing the accretion . fryer et al . ( 1996 ) calculated values for the density and velocity gradients and compared these results with those of ruffert & anzer ( 1995 ) and found that this pressure support on the global scale was minimal ( @xmath95% level for quite large velocity gradients ) . since this time , ruffert ( 1999 ) studied the same effect under density gradients . again , if we use the estimates from fryer et al . ( 1996 ) for the density gradients in 10,20m@xmath96 supergiants , we expect 10 - 20% variations away from the accretion rate predicted by the bondi - hoyle formalism . this only changes as the neutron star spirals near the inner edge of the hydrogen envelope , where density gradients can become quite large . fryer , benz & herant ( 1996 ) assumed that as long as bondi - hoyle accretion were unaffected , the angular momentum would somehow be transported outward , allowing the material to accrete onto the neutron star . but subsequent studies are showing that this assumption may well be incorrect ( armitage & livio 2001 ; fryer et al . 2006 ; fryer 2007 ) . armitage & livio ( 2001 ) showed that the angular momentum in the inflow would lead to disk formation , and ultimately , an outflow that could halt accretion . if the material is unable to get rid of the energy produced by viscous interactions , an outflow is bound to occur ( blandford & begelman 1999 ) . fryer et al . ( 2006 ) and fryer ( 2007 ) have specifically studied accreting neutron star systems and found that even a small amount of angular momentum would lead to outflows . in these low - angular momentum flows , the outflows decreased the accretion rate by 50% , but for the high - angular momentum flows in ce phases , the outflow could decrease the accretion by more than an order of magnitude ( blandford & begelman 1999 ) . because of such results , we estimate our mass accretion by assuming an accretion rate of 10% the bondi - hoyle rate . it could even be an order - of - magnitude lower . this reduced accretion rate allows additional scenarios for forming double neutron star systems which can also be shown to match the current observations of double neutron star systems ( this study ; belczynski et al . it also avoids any problems over - producing ( or hiding ) the number of massive ns - ns systems . brown & bethe ( 1998 ) turned these systems into low - mass bh - ns systems by requiring a maximum neutron star mass between 1.7 - 1.8m@xmath96 . however , observations may indicate that the maximum neutron star mass is @xmath97 ( e.g. , ransom et al . 2005 ; barziv et al . 2001 ) while some theoretical work allows for equation of states with @xmath98 ( e.g. , morrison , baumgarte & shapiro 2004 ) . if these systems form black holes , we also do not see the low - mass black holes in the galaxy - all black holes in the galaxy have masses above @xmath93 ( orosz 2003 ; cesares 2007 ) , although fryer & kalogera ( 2001 ) argued that this is more - likely the result of observational biases . additionally , this new accretion estimate agrees very well with the amount of matter that is needed to mildly recycle a pulsar ( zdunik , haensel & gourgoulhon 2002 on theoretical calculations ; jacoby et al . 2005 on observational estimate ) . this is not to say that the brown scenario requiring twin binaries does not contribute to the double neutron star population . but in our scenario with the reduced accretion rate , it is simply not the dominant formation scenario . because of this , our results are much less sensitive to the size of the twin population . our calculations discussed in this paper involved a very simple twin scenario , i.e. half of all the binaries were postulated to be equal mass ( @xmath99 ) . in reality , the true fraction of twins is likely to depend on the mass of the primary , most certainly on the orbital separation of the two stars , and also possibly on the metallicity of stars . indeed , even the actual binary fraction is likely a function of primary mass ( e.g. , lada 2006 ) . there was a recent report that the fraction of b - type binaries in the lmc might be significantly lower than in our galaxy ( mazeh , tamuz & north 2006 ) , indicating a possibility of strong metallicity dependence in the efficiency of binary formation . from the absolute rates point of view , we show that effect of including twins is relatively minor : although the merger rate does indeed increase when twins are considered , the rate increase is fairly small ( @xmath2 ) . this is the direct result of evolutionary calculations that provide numerous channels of ns - ns formation without a strong preference for the high mass ratio of progenitor binaries . the same calculations recover the empirically estimated rates of double neutron star mergers as some of their observed properties ( belczynski et al . also , chirp mass distribution for double compact objects formed with or without twins are almost indistinguishable . if double compact object are short - hard grb progenitors , including twins in population synthesis calculations does not alter significantly the earlier rate predictions for the event rate . nevertheless , there are some interesting changes when we include significant twin population . for one channel of binary evolution that allows ultracompact binaries , introducing twins doubles the rate of very prompt ns - ns mergers ( time to merger less than @xmath3 years ) compared to models with the `` flat '' @xmath4 distribution . in that specific case , @xmath84 of all ns - ns binaries would merge within @xmath100years after their formation ( see table 1 ) , indicating a possibility of a very significant population of short - hard gamma - ray bursts associated with star forming galaxies . we should mention that twins are not necessary to have a significant prompt population of ns - ns binaries . even using most conservative assumptions , @xmath75 of all ns - ns binaries merge within @xmath100years after their formation ( see table 1 ) . this is very interesting because of the well localized short - hard bursts , roughly 40% occured in young hosts , 15% in old hosts , and location of roughly 45% is still unknown ( berger et al . 2007 ; e. berger , private communication ) . so not only are the `` prompt '' short - hard grbs very common , they might turn out to be the majority of this class of bursts , as the rough current limits are from 40% to even 85% . the fact that so many short - hard burst are found in star - forming galaxies may have far reaching consequences for our understanding of binary evolution ( if indeed short - hard grbs are connected to double compact object mergers ) . if further observations find that even higher fraction of short - hard grbs is in young galaxies , that will indicate strongly that the ultracompact channel of the binary evolution does indeed lead to double compact object formation , something that otherwise is very hard to resolve observationally . if that fraction is higher still , i.e. 70% and above , we will not only need the ultracompact channel to be allowed , we might also need a significant binary twin population to explain such a high rate ( see table 1 ) . needless to say , more observational constraints on short - hard grbs and their hosts are needed here . -values ( for details see 2.3 ) . note the change of vertical scale from the top panel to bottom panels . the numbers correspond to the entire galactic population of double compact objects and their progenitors . ] ) have merger times shorter than 100 myr . model b with an alternative approach to common envelope evolution in which we allow survival of systems with hertzsprung gap donors leads to formation of ultracompact double neutron stars that contribute to the short merger time peak ( @xmath101 myr ) . we note that in this model even more : @xmath102 and @xmath84 dcos form with short merger times for flat and twin mass ratio distributions , respectively . the short - lived systems are natural candidates for prompt short - hard grbs observed in young host galaxies . ]
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we investigate the effect of including a significant `` binary twin '' population ( binaries with almost equal mass stars , @xmath0 ) for the production of double compact objects and some resulting consequences , including ligo inspiral rate and some properties of short - hard gamma - ray bursts .
we employ very optimistic assumptions on the twin fraction ( @xmath1 ) among all binaries , and therefore our calculations place an upper limits on the influence of twins on double compact object populations . we show that for ligo the effect of including twins is relatively minor : although the merger rates does indeed increase when twins are considered , the rate increase is fairly small ( @xmath2 )
. also , chirp mass distribution for double compact objects formed with or without twins are almost indistinguishable .
if double compact object are short - hard grb progenitors , including twins in population synthesis calculations does not alter significantly the earlier rate predictions for the event rate .
however , for one channel of binary evolution , introducing twins more than doubles the rate of `` very prompt '' ns - ns mergers ( time to merger less than @xmath3 years ) compared to models with the `` flat '' @xmath4 distribution . in that case , @xmath5 of all ns - ns binaries merge within @xmath6 years after their formation , indicating a possibility of a very significant population of `` prompt '' short - hard gamma - ray bursts , associated with star forming galaxies .
we also point out that , independent of assumptions , fraction of such prompt neutron star mergers is always high , @xmath7 .
we note that recent observations ( e.g. , berger et al . ) indicate that fraction of short - hard grbs found in young hosts is at least @xmath8 and possibly even @xmath9 .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
since the discovery of gamma - ray burst ( grb ) afterglows there has been growing evidence linking grbs to massive stars : the host galaxies of grbs are star - forming galaxies and the position of grbs appear to trace the blue light of young stars @xcite ; some of the host galaxies appear to be dusty with star - formation rates comparable to ultra - luminous infrared galaxies @xcite . on smaller spatial scales , there is growing evidence tying grbs to regions of high ambient density @xcite and the so - called dark grbs arise in or behind regions of high extinction @xcite . however , the most direct evidence linking grbs to massive stars comes from observations of underlying supernovae ( sne ) and x - ray lines . the presence of x - ray lines would require a significant amount of matter on stellar scales ( e.g. @xcite ) , as may be expected in models involving the death of massive stars . however , to date , these detections ( e.g. @xcite ) have not been made with high significance . if grbs do arise from the death of massive stars , then it is reasonable to expect associated sne . the grb - sn link was observationally motivated by two discoveries : the association of grb 980425 with the peculiar type ic sn 1998bw @xcite and an excess of red light superposed on the rapidly decaying afterglow of grb 980326 @xcite . however , these two discoveries were not conclusive . the sn association would require grb 980425 to be extra - ordinarily under - energetic as compared to all other cosmologically located grbs and the case for grb 980326 is weakened by the lack of a redshift for the grb or the host galaxy . nonetheless , the two discoveries motivated searches for similar underlying sn components . as summarized in section [ sec : conclusions ] , suggestions of similar red `` bumps '' in the light curves of various other grb afterglows have been made ( to varying degrees of confidence ) . however , there is little dispute that the well - studied red bump in the afterglow of grb 011121 is most easily explained by an underlying supernova @xcite . furthermore , from radio and ir observations of the afterglow @xcite , there is excellent evidence that the circumburst medium was inhomogeneous with ambient density @xmath2 , as expected from a massive star progenitor @xcite ; here , @xmath3 is the distance from the progenitor . these developments are in accordance with the expectation of the `` collapsar '' model @xcite . in this model , the core of a rotating massive star collapses to a black hole which then accretes matter and drives a relativistic jet . internal shocks within this jet first cause bursts of @xmath4-rays and then subsequently result in afterglow emission as the jet shocks the ambient medium . it is important to appreciate that the sn light is primarily powered by radioactive decay of the freshly synthesized @xmath5ni whereas the burst of @xmath4-rays are powered by the activity of the central engine . in the current generation of collapsar models , there is sufficient flexibility to allow for a large dispersion of @xmath5ni and the energy of the engine . thus , the next phase of understanding the grb - sn connection will benefit from ( and require ) observational measures of these parameters . motivated thus , we have an ongoing program of searches for sne in grb afterglows with the _ hubble space telescope _ ( hst ) . here , we present a systematic search for a sn underlying grb 010921 . in [ sec : observations ] we present our observations and the details of photometry in [ sec : subphot ] . we fit afterglow models and constrain the brightness of an underlying sn in [ sec : discussion ] . we then present an overview of previous such efforts and conclude in [ sec : conclusions ] . grb 010921 was detected by the high energy transient explorer ( hete-2 ) satellite at 2001 september 21.219 ut @xcite and the position was refined by the interplanetary network error - box @xcite . using the 5-m hale telescope and the very large array we discovered the afterglow of this event as well as the redshift of the host galaxy @xcite . the low redshift of this event , @xmath6 , made it a prime candidate for a search for an underlying sn . accordingly , as a part of our large _ hubble space telescope _ ( hst ) cycle 9 program ( go-8867 , p. i. : kulkarni ) , we triggered a series of observations with the wide field planetary camera 2 ( wfpc2 ) aboard hst . owing to the lateness in identifying the afterglow candidate , the first observation was on day 35 , slightly after the expected peak of the sn . at each of epochs 13 we obtained @xmath7 s exposures in each of five filters ( f450w , f555w , f702w , f814w and f850lp ) with a single diagonal dither by 2.5 pixels to recover the under - sampled point - spread function ( psf ) . the fourth epoch was optimized for photometry of the host galaxy and , accordingly , we increased the exposure time to @xmath8 s. we used `` on - the - fly '' pre - processing to produce debiased , flattened images . the images were then drizzled @xcite onto an image with pixels smaller than the original by a factor of 0.7 using a pixfrac of 0.8 . after rotation to a common orientation the images were registered to the first epoch images using the centroids of common objects in the field . the typical r.m.s . registration errors were less than 0.15 drizzled pixels . the host galaxy of grb 010921 has an integrated magnitude of @xmath9 mag or about 5@xmath10jy @xcite . consequently great care has to be taken to properly photometer the fading afterglow . below , we review various photometric techniques . * total magnitudes : * the simplest technique is to perform aperture photometry ( e.g. @xcite ) . the afterglow flux is obtained by subtracting the host flux estimated from a very late time measurement . a major concern is that the host flux is dependent upon the choice of aperture ( both center and size ) . thus , if different images have different seeing then it is possible to obtain an artificial bump in the light curve . * host subtraction : * the above concern can be alleviated by subtracting a late - time image from the earlier images . the afterglow may then be easily photometered in the host - subtracted images . this method has been used with considerable success by those observing sne ia ( e.g. @xcite ) . * @xmath11 subtraction : * in this technique , each image is subtracted from every other image and the afterglow residual photometered . the flux at each epoch can be fit through least - squares , assuming the flux at the final epoch is zero ( novicki and tonry , personal communication ) . this method makes use of the fact that the host galaxy has not been observed only once at late times , but at each epoch and thus better s / n can be obtained from the over - constrained system . we employed the @xmath11 subtraction technique to photometer the grb 010921 afterglow in our hst images . the images were subtracted using a modified version of isis @xcite and photometered using the analytic psf - fitting routine within vista ( j. tonry , personal communication ) . we used the synphot package within iraf to calculate the response of the instrument and filter combination to a source with constant flux of 1 mjy ; the resulting values are ab magnitudes @xcite , expressed as fluxes . corrections were made for charge - transfer ( in)efficiency ( cte ) using the prescription of @xcite and aperture - corrected to infinity . we have also re - analyzed and photometered ground - based images @xcite of the afterglow , applying @xmath11 subtraction . since this technique assumes that the flux of the afterglow in the final epoch is zero , which may not be correct for these images , we subtracted the appropriate fourth - epoch hst observation ( which we have assumed contains no afterglow ) from the final ground - based images , measured the flux of the afterglow and added this value to the fluxes derived from the @xmath11 subtraction . the results of the photometry are host - subtracted fluxes for the afterglow in each of the images , under the assumption that the afterglow flux in the final hst image ( 2001 dec 21 ) is zero ( or negligible ) . these values are presented in tables [ tab : hst ] and [ tab : ground ] . the values in table [ tab : ground ] supersede the corresponding measurements presented in @xcite and @xcite . we plot the afterglow light curves in figure [ fig : lc ] . the light - curves are monotonically decreasing ( i.e. do not level off ) , and hence we deduce that our assumption of negligible flux in the final hst image is justified . temporal breaks in optical light - curves have been seen in many afterglows and are usually attributed to a `` jet '' geometry ( see @xcite ) . we adopt a standard optical afterglow model , consisting of a broken power - law temporal decay with power - law indices @xmath12 and @xmath13 , and a power - law spectral index , @xmath14 @xcite . each of these indices are functions of the electron energy distribution index , @xmath15 , dependent upon the location of the cooling break relative to the optical bands , and so we consider two cases : the cooling break is redward of the optical ( hereafter , case r ) ; and the cooling break is blueward of the optical ( case b ) . we consider , in addition to a constant circumburst medium , an inhomogeneous circumburst medium , @xmath16 ( see @xcite and @xcite ) . we apply the parametric extinction curves of @xcite and @xcite using the interpolation calculated by @xcite . these extinction curves are characterized by two values , the magnitude of the extinction in the rest - frame of the host galaxy , @xmath17 , and the slope of the uv extinction curve , @xmath18 ( see @xcite ) . following @xcite , we adopt @xmath19 , corresponding to an lmc - like extinction curve . adopting other extinction curves ( e.g. mw , smc ) yields similar , but more - constraining results ( i.e. any underlying sn must be even fainter than the upper limit we derive below ) ; see @xcite . the main purpose of this analysis is to determine whether the light curves contain an sn component . to this end , we use the observations of sn 1998bw for an sn template since it is one of the well observed bright ib / c sne which may be related to a grb @xcite . specifically , we used the @xmath20 photometry of @xcite and derived the flux distribution of sn 1998bw , using the zero - points and filter curves of @xcite . the resulting low resolution spectrum ( consisting of 5 points at the effective wavelength of each broadband filter ) , is redshifted to @xmath6 @xcite , assuming a flat lambda cosmology with @xmath21 and @xmath22 km s@xmath23 mpc@xmath23 . the redshifted spectrum , which represents what sn 1998bw would look like at cosmological distances , is integrated with the appropriate filter curve to derive the apparent brightness at this redshift . sn 1998bw at @xmath6 would peak in the rest - frame @xmath24-band at approximately 4 @xmath25jy . it is evident from figure [ fig : lc ] that the afterglow is much fainter than this , and , further , that there is no clear bump in the afterglow light curve . we therefore allow the sn component to be scaled by @xmath26 magnitudes in our model . the sn is placed behind the same foreground ( i.e. milky way ) and host galaxy extinction as the afterglow ( which can be inferred by demanding that the temporal and intrinsic spectral slopes , which both depend on the electron distribution index , @xmath15 , be consistent ; see e.g. @xcite ) . to calculate the sn detection limit of our observations , we fit the model by minimizing @xmath27 . the afterglow was not detected in any of the f450w images , and so we exclude them from our analysis . subtracting the host f450w image from our ground - based @xmath28 image left a large residual at the position of the host galaxy ( not of the ot ) . this poor subtraction is likely due to the filter mis - match , and so we do not include this point in our analysis . our analyses are summarized in table [ tab : fit ] . in short , we find no evidence for an underlying sn . in order to calculate the formal limits , we re - fit the data for a range of values of the sn brightness and computed the probability distribution from the resultant @xmath29 . as can be seen from table [ tab : sn ] , the least constraining limit comes from the case where the afterglow evolves in a wind - stratified medium with the cooling break redward of the optical band , and even in this case , a sn brighter than @xmath30 mag is excluded at 99.7% confidence , and a sn as bright as sn 1998bw ( @xmath31 mag ) is ruled out at greater than 99.999% confidence . the peak brightness and the time scales for sne ib / c are generally correlated such that fainter sne may peak earlier @xcite . it may be important to take this into account for our analysis , since the observations most sensitive to the presence of an underlying sn are all after the peak . to do this , we shifted the @xmath20 photometry of the ( intrinsically-)fainter type ic sn 1994i @xcite to @xmath0 , and derived the transformation between the redshifted sn 1998bw and 1994i light curves using a similar method as @xcite . this method is analogous to the `` stretch '' method for sne ia @xcite . if we use this transformation in our model to transform the redshifted sn 1998bw light curve to the light curve of a sn fainter than sn 1998bw by @xmath26 magnitudes , then our least - constraining limit on an underlying sn becomes @xmath32 mag fainter than sn 1998bw ( at 99.7% confidence ) . the agreement with the above limit indicates that the uncertainty in our knowledge of the the light - curve shape and luminosity scaling light - curve is not important for this analysis . leaving aside the sn issue , our fits provide a jet - break time of approximately 35 days . from the fregate 8 400 kev fluence of @xmath33 erg @xmath34 , we calculate the @xmath35-corrected isotropic - equivalent energy release @xcite in the @xmath4-ray band , @xmath36 erg . applying the geometric correction from our measurement of the jet break ( using the formulation and normalization of @xcite ) , we obtain a jet opening angle of @xmath37 . thus the true energy release is @xmath38 erg consistent with the clustering of energy releases around @xmath39 erg @xcite . here we report the search for an underlying sn in the afterglow of grb 010921 . thanks to the superb photometric stability of hst and the @xmath11 subtraction technique , we have been able to trace the light curve of the afterglow of grb 010921 over two months . the resulting photometry is unbiased by aperture effects that are so prevalent in simple aperture and psf - fitting photometry . we report two results . first , we find a jet break time of 35 days , using only optical data . second , we find no evidence for an sn . a sn , if present , must be fainter than sn 1998bw by @xmath40 mag at 99.7% confidence . to our knowledge , to date , this is the most stringent limit for an underlying sn associated with a cosmologically located grb . as noted in [ sec : introduction ] , the collapsar model as currently understood has little power in predicting the dispersion in the amount of @xmath5ni synthesized as compared to the energy in relativistic ejecta . underlying sne are directly powered by the former whereas the grb is powered by the latter . observations are needed to start mapping the distribution in these critical explosion parameters . progress can be expected with such observational inputs accompanied by further refinements in the model . motivated thus , we summarize in table [ tab : previous ] the status of sn searches for all table [ tab : previous ] all known grbs with redshift less than 1.2 . the most secure case for an sn is that for grb 011121 @xcite . grb 980326 shows a strong red excess at about a month but unfortunately a redshift is lacking . grb 970228 shows a less clear excess but benefits from a known redshift . stated conservatively , a sn as bright as that of sn 1998bw can be ruled out in grb 000911 . in all cases , save that of grbs 980326 and 011121 , the presence of a host with a magnitude comparable to the brightness of the peak of the sn , makes it difficult to identify an sn component . as noted in [ sec : subphot ] , `` bumps '' can arise from host contamination . combining hst and ground based measurements ( as is the case for grb 970228 ) is prone to considerable errors ( [ sec : subphot ] ) . in summary , there is good evidence for an sn comparable in brightness to sn 1998bw in grb 011121 @xcite . for grb 010921 , using the hst observations reported here , we constrain any putative underlying sn to be 1.34 mag fainter than sn 1998bw . in the collapsar framework , this absence could be most readily attributed to the well known dispersion of the peak luminosity of type ib / c sne . an alternative possibility is that there may be more than one type of progenitor for long duration grbs . along these lines we note that @xcite claim that some afterglows ( e.g. grb 990123 ) are incompatible with a @xmath16 inhomogeneous circumburst distribution whereas other afterglows ( e.g. grbs 970228 and 970508 ) are better explained by invoking an inhomogeneous circumburst medium . progress requires both searches for underlying sne as well as characterizing the circumburst medium via modeling of the early - time afterglow ( e.g. grb 011121 , see @xcite ) . finally , we note that the afterglow of grb 010921 ( and any coincident sn ) was extincted by @xmath41 mag of dust in the foreground , and @xmath42 mag of dust in the host galaxy ( table [ tab : fit ] ) . thus , in the future , using acs aboard hst it should be possible to extend sn searches to at least 3 mag fainter than sn 1998bw , at which point it will be possible to detect more typical sne ib / c coincident with grbs . we thank pete challis for helpful discussions about wfpc2 reduction , and megan novicki and john tonry for an advance copy of their @xmath11 subtraction paper . srk and sgd thank nsf for supporting our ground - based grb observing program . bps and pap thank the arc for supporting australian grb research . support for proposal number hst - go-08867.01-a was provided by nasa through a grant from space telescope science institute , which is operated by the association of universities for research in astronomy , incorporated , under nasa contract nas5 - 26555 . kh is grateful for support under grant hst - go-09180.07-a . , s. g. _ et al . _ 2001b , in gamma - ray bursts in the afterglow era , proceedings of the international workshop held in rome , cnr headquarters , 17 - 20 october , 2000 . edited by enrico costa , filippo frontera , and jens hjorth . berlin heidelberg : springer , 218 + . , v. v. 2001 , in gamma - ray bursts in the afterglow era , proceedings of the international workshop held in rome , cnr headquarters , 17 - 20 october , 2000 . edited by enrico costa , filippo frontera , and jens hjorth . berlin heidelberg : springer , 136 . ccccc oct 26.731 & f450w & -0.031 @xmath43 0.022 + nov 06.956 & f450w & 0.001 @xmath43 0.028 + nov 24.990 & f450w & 0.067 @xmath43 0.029 + oct 26.791 & f555w & 0.157 @xmath43 0.015 + nov 07.015 & f555w & 0.087 @xmath43 0.017 + nov 25.121 & f555w & 0.063 @xmath43 0.018 + oct 26.859 & f702w & 0.231 @xmath43 0.013 + nov 07.149 & f702w & 0.096 @xmath43 0.015 + nov 25.203 & f702w & 0.045 @xmath43 0.015 + oct 26.932 & f814w & 0.433 @xmath43 0.024 + nov 08.359 & f814w & 0.209 @xmath43 0.024 + nov 25.621 & f814w & -0.003 @xmath43 0.025 + oct 26.992 & f850lp & 0.471 @xmath43 0.092 + nov 08.418 & f850lp & 0.207 @xmath43 0.088 + nov 25.687 & f850lp & 0.030 @xmath43 0.096 + ccccc oct 19.178 & @xmath28 & 0.671 @xmath43 0.097 & p200 + sep 22.144 & @xmath44 & 46.104 @xmath43 0.722 & p200 + sep 22.148 & @xmath44 & 44.995 @xmath43 0.661 & p200 + sep 27.354 & @xmath44 & 2.13 @xmath43 1.223 & p200 + oct 17.145 & @xmath44 & 0.086 @xmath43 0.379 & p200 + oct 18.088 & @xmath44 & 0.189 @xmath43 0.382 & p200 + oct 19.109 & @xmath44 & 0.256 @xmath43 0.285 & p200 + oct 17.165 & @xmath45 & 0.560 @xmath43 0.197 & p200 + oct 18.110 & @xmath45 & 0.523 @xmath43 0.191 & p200 + oct 19.130 & @xmath45 & 0.649 @xmath43 0.153 & p200 + oct 19.149 & @xmath46 & 1.293 @xmath43 4.273 & p200 + sep 22.3038 & @xmath47 & 11.319 @xmath43 0.981 & nofs1.0 + oct 19.253 & @xmath47 & 0.623 @xmath43 0.675 & p60 + sep 22.2976 & @xmath48 & 24.727 @xmath43 1.078 & nofs1.0 + oct 19.206 & @xmath48 & 0.229 @xmath43 0.720 & p60 + sep 22.2930 & @xmath49 & 39.116 @xmath43 5.072 & nofs1.0 + sep 22.3210 & @xmath49 & 36.135 @xmath43 4.486 & nofs1.0 + oct 19.272 & @xmath49 & 0.916 @xmath43 4.284 & p60 + nov 17.151 & @xmath49 & 0.470 @xmath43 4.238 & nofs1.0 + sep 22.2893 & @xmath24 & 84.688 @xmath43 5.778 & nofs1.0 + sep 22.795 & @xmath24 & 40.277 @xmath43 3.950 & ts + sep 22.825 & @xmath24 & 47.281 @xmath43 7.230 & ts + sep 22.878 & @xmath24 & 50.926 @xmath43 3.671 & ts + sep 22.954 & @xmath24 & 41.321 @xmath43 3.636 & ts + nov 17.093 & @xmath24 & 1.229 @xmath43 1.057 & nofs1.0 + lrrrr ism / wind , b & 2.67 @xmath43 0.06 & 33.0 @xmath43 6.5 & 0.95 @xmath43 0.08 & 19.9 + ism , r & 3.03 @xmath43 0.04 & 37.5 @xmath43 4.9 & 1.16 @xmath43 0.07 & 19.2 + wind , r & 2.33 @xmath43 0.10 & 30.3 @xmath43 9.5 & 1.35 @xmath43 0.08 & 23.1 + lrcrrll 970228 & 0.695 & @xmath49 & 25.5 & 25.2 & both & plausible but aperture , color effects with hst . ( 1 ) + 970508 & 0.835 & @xmath24 & 23.6 & 24.0 & ground & aperture effects . ( 2 ) + 980326 & ? ? ? & @xmath49 & 25 & @xmath51 27 & ground & plausible . ( 3 ) + 980613 & 1.096 & @xmath49 & & 24.0 & ground & faint afterglow , no search . ( 4 ) + 980703 & 0.966 & @xmath49 & 24 & 22.6 & ground & consistent with no sn . ( 5 ) + 990705 & 0.840 & @xmath49 & & 22.8 & ground & no search . + 990712 & 0.433 & @xmath48 & 23.8 & 21.2 & ground & aperture effects ? ( 6 ) + 991208 & 0.706 & @xmath49 & 23.9 & 24.4 & ground & bad afterglow fit . ( 7 ) + 991216 & 1.020 & @xmath49 & & 24.85 & ground & no search , consistent with no sn . ( 8) + 000418 & 1.119 & @xmath49 & & 23.8 & ground & consistent with no sn . ( 9 ) + 000911 & 1.058 & @xmath24 & 24.7 & 24.4 & ground & 2@xmath50 detection , sn @xmath52 0.9 @xmath43 0.3 @xmath53 sn1998bw . ( 10 ) + 011121 & 0.365 & @xmath49 & 23 & 26 & hst & secure .
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grb 010921 was the first hete-2 grb to be localized via its afterglow emission .
the low - redshift of the host galaxy , @xmath0 , prompted us to undertake intensive multi - color observations with the _ hubble space telescope _ with the goal of searching for an underlying supernova component .
we do not detect any coincident supernova to a limit 1.34 mag fainter than sn 1998bw at 99.7% confidence , making this one of the most sensitive searches for an underlying sn .
analysis of the afterglow data allow us to infer that the grb was situated behind a net extinction ( milky way and the host galaxy ) of @xmath1 mag in the observer frame .
thus , had it not been for such heavy extinction our data would have allowed us to probe for an underlying sn with brightness approaching those of more typical type ib / c supernovae .
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the creation of directional states of molecules represents an important tool to control and tailor the rotational degree of freedom . when a molecule is oriented the molecular fixed axes are confined along laboratory fixed axes and its dipole moment is pointing in a particular direction . experimentally , the availability of oriented molecules provides a wealth of interesting applications in a variety of molecular sciences , such as in chemical reaction dynamics @xcite , photoelectron angular distributions @xcite , or high - order harmonic generation @xcite . due to this broad interest , special efforts have been undertaken to create samples of oriented molecules and techniques based in the application of inhomogeneous @xcite , and homogeneous @xcite electric fields as well as homogeneous magnetic fields @xcite have been used . a major breakthrough came with the proposal by friedrich and herschbach @xcite of enhancing the orientation of polar molecules by exposing them to combined weak electrostatic and strong non - resonant radiative fields . this theoretical prediction was done within an adiabatic picture assuming that the switching on time of the laser pulse is larger than the molecular rotational period . for linear molecules , a linearly polarized laser field produces a double - well potential along the polarization direction . in the pendular limit , this double - well potential contains nearly degenerate pairs of states with opposite parity forming tunneling doublets . if the molecules possess a permanent electric dipole moment , a strong pseudo - first - order stark effect is induced by coupling the tunneling doublets with an additional electrostatic field . due to this coupling , the two levels in a pendular doublet are efficiently oriented but with their effective electric dipole moments pointing in opposite directions . as a consequence of this oriented and antioriented states pairing , the orientation is small in a molecular ensemble with the population thermally distributed . therefore , the first experimental measures of the orientation of a molecular beam were indeed reduced to small values @xcite . a significant improvement was gained by using a quantum - state selected molecular beam , which allowed the creation of unprecedented degree of orientation for complex asymmetric tops @xcite . a first theoretical study of the mixed - field orientation experiment of asymmetric top molecules , pointed out that a fully adiabatic description of the process does not reproduce the experimental observations @xcite . recently , we have experimentally and theoretically investigated the mixed - field orientation of the carbonyl sulfide molecule @xcite . our analysis has proven that a time - dependent description of the mixed - field orientation process is required to explain the experimental results . we have shown how the non - adiabatic coupling of the levels forming the quasi - degenerate doublets as the laser intensity is increased , gives rise to the reduction of the orientation and , therefore , to the disagreement with the predictions of the adiabatic theory @xcite . herein , we provide a detailed theoretical analysis on the dynamics of a linear molecule exposed to an electrostatic field combined with a non - resonant laser pulse . in the framework of the rigid rotor approximation , we solve the time - dependent schrdinger equation using experimental field configurations , i.e. , a gaussian laser pulse and a weak electrostatic field that is turned on at constant speed . as prototype example , we consider the carbonyl sulfide molecule ( ocs ) . for several rotational states , we investigate the mixed - field orientation dynamics under different field - configurations by varying either the laser peak intensity , the duration of the gaussian pulse , the dc field strength or the angle between both fields . hence , we demonstrate that for some field configurations , the field - dressed dynamics is non - adiabatic and provide a detailed account of the sources of non - adiabaticity and the field regimes at which they appear . for parallel fields , the dynamics is characterized by the population transfer between adiabatic states when the pendular doublets are formed . whereas for non - parallel fields , we encounter additional non - adiabatic effects when the states from the same @xmath0-manifold , having now the same symmetry , are driven apart as the laser intensity is increased on the weak field regime . for different field configurations , we identify and discuss the experimental conditions needed to achieve an adiabatic molecular dynamics . the paper is organized as follows : in we describe the hamiltonian of the system and its symmetries for various field configurations . the results for the energy , alignment and orientation predicted by the adiabatic theory are analyzed in . in , we focus on the molecular dynamics when the fields are parallel . in particular , we explore how the time - dependent orientation varies as the field parameters are modified , and indicate the experimental conditions under which an adiabatic orientation would be achieved . a similar study is performed for tilted fields in , where we show that the conditions for an adiabatic mixed - field orientation are more difficult to fulfill . in , we assume that once the pulse is turned on its peak intensity is kept constant , and investigate the dynamics in this regime . in , first the laser pulse is switched on and then an electric field is applied . in this field configuration , we analyze the orientation of the ground state and provide the field parameters for an adiabatic orientation . the conclusions are given in . we consider a polar linear molecule exposed to an homogeneous static electric field and a non - resonant linearly polarized laser pulse . the field configuration is illustrated in : the polarization of the laser field lies along the @xmath1-axis of the laboratory fixed frame ( lff ) @xmath2 , and the dc field is contained in the @xmath3-plane forming an angle @xmath4 with the @xmath1-axis . the @xmath5-axis of the molecule fixed frame @xmath6 is defined by the permanent dipole moment @xmath7 of the molecule . these two frames are related by the euler angles @xmath8 , cf . the description of this system is done within the rigid rotor approximation , assuming that the vibrational and electronic dynamics are not affected by the fields . thus , the rigid rotor hamiltonian reads @xmath9 where @xmath10 is the field - free hamiltonian @xmath11 with @xmath12 being the total angular momentum operator and @xmath13 the rotational constant . the terms @xmath14 and @xmath15 stands for the interactions with the static and laser fields , respectively . the dipole coupling with the static field reads @xmath16 with @xmath17 , and @xmath18 being the electrostatic field strength . the angle between the dipole moment @xmath7 and this field is @xmath19 , cf , and @xmath20 . we consider a non - resonant laser field linearly polarized along the @xmath1-axis of the lff , @xmath21 , with @xmath22 being its frequency , @xmath23 the peak field strength , and @xmath24 the pulse envelope . assuming that @xmath25 is much shorter than the pulse duration and the rotational period , we average over the rapid oscillations of the non - resonant field . this causes the coupling of this field with the permanent dipole moment to vanish @xcite . thus , the non - resonant laser field molecule interaction can be written as @xmath26 where @xmath27 is the polarizability anisotropy , @xmath28 is the intensity of the laser , @xmath29 is the speed of light and @xmath30 is the dielectric constant . note that we have neglected the term @xmath31 , which represents only a shift in the energy . in this work , the field configurations are chosen based on the mixed - field orientation experiments @xcite . initially , the molecule is in field - free space , then the electrostatic field is switched on , and its strength is increased linearly with time . at @xmath32 the maximum strength is achieved and kept constant afterwards . this time @xmath33 is chosen long enough to ensure the adiabaticity of this turning - on process . for the laser pulse , we use a linearly polarized gaussian pulse with a full width half maximum ( fwhm ) @xmath34 on the nanosecond range . the intensity is given by @xmath35 , with @xmath36 being the peak intensity , and @xmath37 is related with the fwhm @xmath38 . numerically , the non - resonant laser field is turned on in such a way that the interaction due to this field is much weaker than coupling with the dc field . .action of the symmetry operations on the euler angles . [ cols="<,<,<",options="header " , ] the time - dependent schrdinger equation associated to the hamiltonian is solved by means of a second - order split - operator technique @xcite , combined with the discrete - variable and finite - basis representation methods for the angular coordinates @xcite . for reasons of addressability , we will label the time - dependent states as @xmath39 and @xmath40 for @xmath41 and @xmath42 , respectively , with @xmath43 and @xmath44 indicating even or odd parity with respect to the @xmath3-plane . the labels @xmath0 and @xmath45 refer to field - free quantum number to which they are adiabatically connected . note that the labeling of the states depends on the way the fields are turned on @xcite . the time - dependent wave function depends on the time @xmath46 , the peak intensity @xmath36 , the fwhm @xmath34 , the electrostatic field strength @xmath47 , and the angle @xmath4 . for the sake of simplicity , we have not made explicit these dependences , but the field configuration is clearly indicated through the text . to get a better physical insight on the field - dressed dynamics , the time - dependent results will be compared to those from the adiabatic theory . for this system , we take the adiabatic limit by using a constant electrostatic field @xmath47 and constant laser intensity @xmath48 in the hamiltonian . the time - independent schrdinger equation associated to this hamiltonian is solved by expanding the wave function in a basis that respects the symmetries . the adiabatic states are labeled as @xmath49 and @xmath50 for @xmath41 and @xmath42 , respectively , and we have not made explicit their dependence on the field parameters . the field - dressed eigenfunctions of this time - independent hamiltonian form a basis , which is used to analyze the time - dependent wave function @xmath51 by means of the following expansion @xmath52 with @xmath53 , @xmath54 and @xmath55 including all the labels identifying these levels . for computational reasons , we have only considered the lowest - lying @xmath56 adiabatic levels , and always ensured that the contributions of highly excited states are negligible . let us remark that for each time @xmath46 , the expansion of the wave function is performed in a different adiabatic basis obtained by solving the time - independent schrdinger equation using the static field strength and laser intensity at time @xmath46 , i.e. , @xmath18 and @xmath28 . in this work , we use the ocs molecule ( see fig . [ fig : fig_1 ] ) as benchmark to illustrate our results . the rotational constant of ocs is @xmath57 @xmath58 , the permanent dipole moment @xmath59 d and the polarizability anisotropy @xmath60 @xmath61 . and ( c ) @xmath62 as a function @xmath48 of the adiabatic states @xmath63 ( red thick solid ) , @xmath64 ( gold thin solid ) , @xmath65 ( orange thick short - dashed ) , @xmath66 ( dark blue long - dashed ) , @xmath67 ( blue dot - short - dashed ) , @xmath68 ( purple dotted ) , @xmath69 ( green thin short - dashed ) , and @xmath70 ( pink dot - long - dashed ) . the insets show the relevant energy and intensity ranges where the formation of the near - degenerate doublets occurs . ( d ) polar plots of the square of their wave functions at @xmath71 . @xmath72 and @xmath41 for all data . ] we start by analyzing the adiabatic limit . we restrict this study to the following eight states : @xmath63 , @xmath64 , @xmath65 , @xmath66 , @xmath67 , @xmath68 , @xmath69 , and @xmath70 . for @xmath41 , they adiabatically correspond to the states forming the four first doublets . note that they well represent the main physical features observed in the overall molecular dynamics , and similar behavior and properties are , therefore , obtained for states in other irreducible representations . for @xmath72 and @xmath41 , the energies and the expectation values @xmath73 and @xmath62 of these levels are plotted versus the laser intensity in ( a ) , ( b ) and ( c ) , respectively . the weak static field breaks the field - free degeneracy in the magnetic quantum number and , as the laser intensity is increased , these states become high field seekers . in the strong laser field regime , once the pendular regime is reached , pairs of quasi - degenerate states with the same symmetry are formed . the insets in this panel show how these doublets appear . the gap in energy in a doublet goes as @xmath74 , where @xmath75 is the effective dipole moment of the state @xmath76 in the doublet , which is of opposite sign for @xmath77 . within a doublet , the two levels are characterized by the same hybridization of the angular motion @xmath78 and alignment @xmath73 , see ( b ) . in contrast , they possess opposite orientation @xmath62 , one being oriented and the other antioriented , cf . ( c ) . this opposite orientation is also illustrated in ( d ) by the polar plots of the square of their wavefunctions for @xmath71 . the larger is the field - free rotational quantum number of the levels , i.e. , their field - free energy , the stronger is the laser intensity needed to achieve a significant orientation . indeed , the states in the third and fourth doublets are not aligned for @xmath79 and , therefore , not oriented . once the pendular regime is achieved , the orientation of these states @xmath80 approaches to @xmath81 as @xmath48 is enhanced . if the laser field is sufficiently strong , this adiabatic orientation is independent of the dc field strength , and of the angle between both fields . in this section , we investigate the rotational dynamics in a parallel configuration : a dc - field of @xmath82 and a gaussian pulse with fwhm @xmath83 ns and several peak intensities . for the ground state @xmath84 , the expectation value @xmath62 is presented in ( a ) as a function of @xmath28 up until the peak intensity @xmath36 is reached . for comparison , the adiabatic results are also shown . since the fwhm is 125.31 times larger than the rotational period , one would expect that the rotational dynamics follows the adiabatic limit . however this is not the case , and there are significant discrepancies between the time - dependent and adiabatic results . in contrast to what is predicted by the adiabatic theory , the final orientation decreases as the peak intensity of the laser pulse is increased . for @xmath85 , @xmath62 initially resembles the adiabatic behavior , but it achieves a maximum value @xmath86 for @xmath87 . for @xmath88 , @xmath89 , and @xmath90 , the orientation shows a qualitatively similar but quantitatively different behavior : in the weak laser field regime , @xmath62 monotonically increases following the adiabatic limit , but for @xmath91 it reaches a plateau behavior being the orientation smaller than the adiabatic value . for instance , @xmath92 for @xmath93 whereas the adiabatic value is @xmath94 . as a function of @xmath28 for gaussian pulses of @xmath83 ns and peak intensities @xmath85 ( red thick solid ) , @xmath95 ( orange dashed ) , @xmath89 ( gold dotted ) and @xmath90 ( green dot - dashed ) . the adiabatic results for @xmath62 ( thin solid ) are also included . ( b ) the squares of the projections of the time - dependent wave functions onto the adiabatic pendular state @xmath63 . ( c ) adiabatic criteria @xmath96 as a function of @xmath28 . the field configuration is @xmath72 and @xmath41 . ] a first physical insight into the non - adiabatic dynamics could be gained by analyzing the characteristic times of the molecule . when the states in a pendular doublet are quasi - degenerated , the energy gap between them , @xmath97 , defines a time scale of this system @xcite . note that we have assumed @xmath98 , which holds in the strong laser field regime . for @xmath72 and @xmath41 , the energy separation within the first doublet formed by @xmath63 and @xmath64 is @xmath99 @xmath58 giving a time scale of @xmath100 ps , which is larger than the rotational period @xmath101 ps . thus , only long enough pulses compared to this pendular time would ensure an adiabatic orientation of the molecule . since the static field strength is so weak , its impact on the rotational dynamics is very small , and before the pulse the levels could be considered as field - free rotor states . as the laser intensity increases , the states are hybridized by the combined action of the both fields , and the doublets of nearly - degenerate states are formed in the strong laser field regime , as it is shown in ( a ) . when the energy splitting of this pendular doublet approaches the coupling of the two sublevels due to the pseudo - first - order stark interaction , these states can mix because they have the same symmetry for @xmath102 . as a consequence , there is a population transfer between the oriented and anti - oriented states , which results in a decrease of the final orientation compared to the adiabatic limit . for this field configuration , the dynamics can be analyzed by means of the adiabatic states forming this pendular doublet , because their couplings to states in neighboring doublets is much smaller than the energy difference between them . note that these adiabatic states are the eigenstates of the hamiltonian at fixed time @xmath46 . under a time - dependent interaction , i.e. , in our case the interaction with the laser field @xmath103 , the dynamics could be considered as adiabatic if and only if the following condition @xcite @xmath104{\left\langle i \left|\cfrac{\partial h_\textup{l}(t)}{\partial t}\right| j\right\rangle}{_{\textup{p } } } \right|}{\left|e_i - e_j\right|^2}\ll 1\ ] ] is fulfilled , with @xmath105 and @xmath106 being the eigenstates of the hamiltonian in the adiabatic limit , and @xmath107 and @xmath108 their energies . according to this criterion , the probability for mixing , corresponding to a transfer of population from one state of the doublet to the other , is determined by the rate of change of the laser field interaction and the energy separation between the states . thus , as the laser intensity is increased the population transfer between the two states in a doublet takes place because the criterion is not satisfied . to illustrate this phenomenon , we show the contribution of the adiabatic ground state @xmath109 to the time - dependent wave function of @xmath84 , ( b ) , and the adiabatic parameter @xmath96 when @xmath110 , ( c ) . note that @xmath111 . in these four cases , the dynamics is initially adiabatic , i.e. , @xmath109 remains equal to @xmath81 and @xmath112 . as @xmath28 is increased , the energy splitting of the doublet decreases and , moreover , it becomes comparable or even larger than the rate of turning - on the pulse ; thus , @xmath96 is close to @xmath81 , and the population transfer takes place . this region where @xmath96 is not negligible corresponds to the formation of the quasi - degenerate doublet . once the doublet is formed , @xmath113 reaches a small value and slowly decreases as @xmath28 is enhanced ; but the two states are oriented in opposite directions and their wave functions do not overlap . therefore , the coupling due to the alignment laser is much smaller than @xmath114 , @xmath115 and the population transfer does not take place any longer , i.e. , @xmath109 remains constant as @xmath28 is enhanced . the larger is this population transfer , the smaller is the orientation compared to the adiabatic prediction . since these adiabatic states contributing to the dynamics are quasi - degenerated and have very close values of the alignment and hybridization of the angular motion , the lack of adiabaticity is not reflected on the time evolution of the energy , @xmath73 or @xmath116 . for this field configuration , the molecular dynamics of excited states present analogous features for @xmath62 , @xmath73 and @xmath116 as those encountered here for the ground state . the adiabaticity of the field - dressed dynamics is determined by the rate of change of the laser field interaction compared to the largest time scale of the system . in the pendular regime , the energy splitting in a doublet goes as @xmath117 ; then , the population transfer decreases if @xmath47 is increased . on the other hand , by increasing the fwhm augments the time scale on which the pendular doublets are formed , and facilitates the adaptation of the molecule to this field . that is , the mixed - field orientation will be more adiabatic when either longer pulses or stronger static electric fields are used . let us remark that with the expression _ being the dynamics more adiabatic _ we mean that for a certain state , the weight of its corresponding adiabatic state in the time - dependent wave function is closer to one during the dynamics . here , we investigate the orientation at the maximum of the laser pulse , as it is done in most of the experiments @xcite . the rate of change of the laser field and the adiabatic parameter depend linearly on @xmath36 . then , for a gaussian pulse with fixed fwhm , the dynamics will be more diabatic if @xmath36 is increased . in this section , we consider the states @xmath84 , @xmath118 , @xmath119 , @xmath120 , @xmath121 , @xmath122 , @xmath123 , and @xmath124 . their orientation at @xmath125 , i.e. , @xmath62 for @xmath126 , is plotted as a function of @xmath36 in ( a ) and ( b ) for @xmath72 and @xmath127 , respectively . the fields are parallel , and the fwhm of these pulses is @xmath128 ns . at @xmath125 as a function of the peak intensity @xmath36 for the states @xmath84 ( red thick solid ) , @xmath118 ( gold thin solid ) @xmath119 ( orange thick short - dashed ) , @xmath120 ( dark blue long - dashed ) @xmath121 ( blue dot - short - dashed ) , @xmath122 ( purple dotted ) , @xmath123 ( green thin short - dashed ) and @xmath124 ( pink dot - long - dashed ) , for @xmath41 and ( a ) @xmath72 and ( b ) @xmath127 . the fwhm of the laser pulses is @xmath128 ns . ] for @xmath72 , the orientation of the low - lying level in a doublet increases as @xmath36 is enhanced , reaching a maximum and smoothly decreasing thereafter . this is counterintuitive to what is expected in the adiabatic limit ; namely , a larger orientation when the laser intensity is increased . the maximum in the orientation is achieved with an alignment pulse that already gives rise to a non - adiabatic dynamics . however , due to the coupling between the populated adiabatic states in the pendular pair @xmath129 , the orientation is enhanced compared to what happens at the adiabatic limit . by further increasing @xmath36 , the population transferred between the two states is enhanced , but now the coupling between them is very small or even zero due to their opposite orientation . as a consequence , the final orientation decreases as @xmath36 is increased . for a certain pendular doublet , the upper state is antioriented , and @xmath62 shows the opposite behavior as a function of @xmath36 . regarding the third and fourth doublets , the states are not oriented nor aligned for ac pulses with @xmath130 . compared to low - lying states , their orientation is smaller and the maximum of @xmath62 appears at larger peak intensities . by increasing the static field strength to @xmath127 the energy gap of the pendular pair is also increased , being the characteristic time scale of the system reduced . thus , for the same laser pulse , the dynamics is more adiabatic , i.e. , less population is transferred , and the final orientation is increased , see ( b ) . the orientation ( antiorientation ) of the pendular states also achieves a maximum ( minimum ) , but it is so shallow that it is hardly appreciated on the scale of this figure . for the states @xmath84 ( red thick solid ) , @xmath119 ( orange thick short - dashed ) , @xmath121 ( blue dot - short - dashed ) and @xmath123 ( green thin short - dashed ) , for ( a ) @xmath72 and ( b ) @xmath127 . we use @xmath128 ns laser pulses and @xmath41 . ] to illustrate the field - dressed dynamics , we plot in the weights of the adiabatic states associated to the oriented levels in these pendular doublets . for the corresponding antioriented levels , the contributions of its associated adiabatic state are identical to the one presented here , e.g. , for the ground state , we present the contribution of the adiabatic ground state @xmath131 , which is identical to the weight @xmath132 for @xmath133 . in an adiabatic molecular dynamics , these coefficients are equal to one . note that in the considered regime only the two adiabatic states of the pendular doublet contribute to the dynamics . for all these levels , @xmath134 decreases , i.e. , the dynamics is less adiabatic , as @xmath36 is enhanced . by increasing @xmath47 , @xmath114 is increased ; thus , under the same gaussian pulse the population transfer is reduced , i.e. , @xmath134 is closer to one , and the range of peak intensities that could be considered as adiabatic is increased . the duration of the gaussian pulse plays an important role in the molecular dynamics . it has been shown that an alignment pulse of @xmath128 ns is not enough to achieve an adiabatic mixed - field orientation for molecules such as ocs , benzonitrile and iodobenzene @xcite . it has been pointed out the need of increasing the rising time of the laser pulses to achieve the highest possible orientation @xcite . by increasing the fwhm , the time needed to form the pendular doublets is also increased . for a given field configuration , at the point where the pulse reaches a certain intensity the adiabatic parameter is reduced if @xmath34 is increased . hence , the molecular dynamics becomes more adiabatic and , therefore , the population transfer to other pendular states is reduced . here , we investigate how the directional properties of ocs depends on the laser - pulse fwhm . for the same set of states as in the previous section , shows the orientation at @xmath125 as a function of @xmath34 . the fields are parallel , and we consider the peak intensities @xmath85 and @xmath89 , and a dc field of @xmath72 . at @xmath125 as a function of @xmath34 for ( a ) @xmath85 and ( b ) @xmath135 . the fields are parallel and @xmath72 . the states and their labels are the same as in . ] the degree of orientation of the two states in a given pendular pair shows the same behavior as a function of @xmath34 , but with their dipole moment pointing in opposite directions . in the first two doublets and @xmath85 , @xmath80 increases with @xmath34 till it reaches a plateau - like behavior . the second pair satisfies that @xmath136 for @xmath137 ns . since the states in the third and fourth doublets have not achieved the pendular regime for @xmath85 , increasing the pulse duration does not have any impact on their orientation , and @xmath62 keeps a constant value close to zero as @xmath34 is increased . for @xmath135 , the degree of orientation of all the states increases and approaches the adiabatic limit as @xmath34 is enhanced . for the ground state and @xmath138 ns , we obtain @xmath139 , which is very close to the adiabatic value @xmath140 . these results show that for parallel fields , the mixed - field orientation dynamics of any state could be adiabatic if a sufficiently long pulse and sufficiently strong fields are used . for a @xmath141 ns gaussian pulse with @xmath135 , and @xmath72 , the dynamics can be considered as adiabatic for all these states , with @xmath142 . since the energy splitting in a pendular doublet is proportional to the static field strength , the degree of adiabaticity in the molecular orientation should increase if @xmath47 is enhanced , i.e. , the characteristic time scale of the system is reduced . in we present the final orientation at @xmath125 of these eight states versus @xmath47 . we have considered two laser pulses of @xmath83 ns with peak intensities @xmath85 and @xmath89 , and @xmath41 . at @xmath125 as a function of @xmath47 for ( a ) @xmath85 and ( b ) @xmath135 . we use a @xmath128 ns laser pulses and @xmath41 . the states and their labels are the same as in . ] for the lowest laser intensity , the orientation of the @xmath84 and @xmath118 states is constant and independent of the static field for @xmath143 with @xmath144 . the orientation of the levels @xmath119 and @xmath120 monotonically increases as @xmath47 is enhanced , and we obtain @xmath145 for @xmath146 . this peak intensity is not large enough for the states in the third and fourth doublets to be in the pendular regime . thus , these pairs are weakly oriented even if a strong dc field is used , e.g. , for @xmath146 , @xmath147 and @xmath148 for the third and fourth doublets , respectively . for the strong peak intensity , all the states are in the pendular regime , and their @xmath80 increases as @xmath47 is increased reaching a constant value for sufficiently strong static fields . their orientation approaches the adiabatic limit and for @xmath149 , @xmath150 for the states of the first doublet , and @xmath151 of their population is on the corresponding adiabatic pendular state . for @xmath146 , the states in the fourth doublet satisfy @xmath152 and @xmath153 of their population is on the corresponding adiabatic level . in conclusion , by combining sufficiently strong electrostatic fields with standard gaussian pulses , i.e. , with experimentally accessible peak intensities of @xmath89 and @xmath128 ns fwhm , a significant orientation is obtained even for excited rotational levels . it is worth remarking that the fields have to be parallel ; then , techniques such us the ion imaging method @xcite could not be used to measure the degree of orientation ; whereas techniques , such as time of flight @xcite are feasible . in this section we investigate the rotational dynamics when the electrostatic field forms an angle @xmath42 with the linearly polarized laser pulse . the azimuthal symmetry is lost , and the number of irreducible representations is reduced to two , see . thus , states with different field - free magnetic quantum numbers are now coupled by the electrostatic field . the field - free wave function of the initial state is constructed as an eigenstate of the operators @xmath154 and @xmath155 , see ; i.e. , @xmath156=@xmath157@xmath156@xmath158 , where @xmath157 is the rotation operator of an angle @xmath4 around the lff @xmath159-axis @xcite . this ensures that this wave function has the correct symmetries , and that its time evolution corresponds , in the adiabatic limit , to an eigenstate of the field - dressed hamiltonian at any time . before the pulse is turned on , an important feature of the ground state is that its energy gap to the next state with the same symmetry is proportional to the rotational constant @xmath13 , which is much larger than the coupling in the weak laser field regime . in this regime , it evolves as an isolated state , and its interaction to the neighboring levels could be considered negligible . hence , analogously to the parallel field configuration , the formation of the doublets in the pendular regime is the only source of non - adiabatic effects in its field - dressed dynamics . note that the lowest lying level of the odd irreducible representation will show the same behavior . and ( b ) @xmath160 as a function of @xmath28 of a @xmath128 ns gaussian pulse . the field configurations are @xmath161 ( thick ) and @xmath162 ( thin ) with peak intensities @xmath85 ( dashed ) , @xmath95 ( dot - dashed ) , and @xmath89 ( dotted ) . the static electric field is fixed to @xmath72 . the adiabatic results ( solid ) are also included . ] for the ground state , the time evolution of the expectation values @xmath62 and @xmath160 are presented as a function of @xmath28 till the peak intensity is reached in ( a ) and ( b ) . the gaussian pulse has @xmath128 ns fwhm and peak intensities @xmath85 , @xmath95 , and @xmath89 . we consider the inclination angles @xmath161 and @xmath162 , and a dc field of @xmath72 . for comparison , the adiabatic results are also included , with @xmath62 being independent of @xmath4 . for a certain laser pulse , increasing the inclination angle towards @xmath163 implies a decrease of the energy splitting in the first doublet , @xmath164 , and , therefore , an increase of the adiabatic parameter . note that in the energy splitting we have not considered the component of the dc field along the lff @xmath165-axis . compared to the @xmath41 configuration , the dynamics could be considered as less adiabatic being characterized by a larger population transfer to the other adiabatic state in this pendular pair . the final orientation is significantly decreased as @xmath4 is increased , e.g. , for @xmath135 , @xmath166 and @xmath167 , and the contribution of the adiabatic ground state is @xmath168 and @xmath169 for @xmath161 and @xmath162 , respectively . for a certain angle @xmath4 , the orientation achieved at @xmath125 decreases as @xmath36 is increased , cf . . since the molecular dynamics of the ground state is restricted to the two lower pendular adiabatic states , its time dependent results for its energy and expectation values @xmath73 and @xmath116 resemble the adiabatic ones . to illustrate the rotational dynamics of excited states , we show in ( a ) the orientation cosine @xmath62 as a function of @xmath28 for @xmath170 . the field configurations are the same as in . the adiabatic model predicts a sharp wrong - way orientation . in contrast , this state presents a weak or even zero orientation , and in addition , a larger peak intensity does not imply a larger orientation . when the peak intensity is reached , this level shows a weak right - way orientation for @xmath161 : @xmath171 and @xmath172 , for @xmath88 and @xmath89 , respectively . for @xmath173 and the peak intensities @xmath88 and @xmath89 , due to the non - adiabatic dynamics @xmath170 is not oriented let us analyze in detail these results . for highly excited states , the dynamics is more complicated . apart from the doublet formation , there is another physical phenomenon at weak laser intensities which causes loss of adiabaticity . in the presence of only a weak static field , the @xmath45-degeneracy of the states with the same field - free @xmath0 is broken due to the quadratic stark effect , i.e. , the splitting goes as @xmath174 . as the pulse is switched on , the energy gap between two states of this @xmath0-manifold is much smaller than the rate of their coupling due to the laser field , i.e. , @xmath96 is larger than one . for @xmath161 , the adiabatic parameter @xmath96 between the states @xmath175 and @xmath176 , both contributing to the dynamics of @xmath170 , is presented in ( b ) , and it achieves large values for @xmath177 . as the states in this @xmath0-manifold are driven apart by the laser field , the process is non - adiabatic and there is a population transfer between them . the projections of the time - dependent wave function in terms of the adiabatic states @xmath178 , @xmath175 , @xmath176 , and @xmath179 is presented in ( d ) for @xmath161 and @xmath135 . under these diabatic conditions , @xmath180 decreases as @xmath28 is increased , whereas @xmath181 increases . by further increasing @xmath28 , the coupling between these states becomes very small or even zero and their energy separation increases , so that @xmath96 decreases and the population transfer is stopped . this process is so diabatic that the wave function does not change , but its projections on the adiabatic basis are modified because the basis varies with time . for instance , the field - free state is @xmath170@xmath182@xmath65@xmath183@xmath64 , which belongs to the proper irreducible representation . after swichting on the static field , its wave function could be approximated by the same expression because this field is very weak . once the splitting of this @xmath0-manifold is finished , i.e. , for @xmath184 , the contributions of the states @xmath175 and @xmath176 are approximately @xmath185 and @xmath186 , respectively . note that the states @xmath187 and @xmath188 are not related adiabatically . state , expectation value @xmath62 as a function of @xmath28 of a @xmath128 ns gaussian pulse . the field configurations are @xmath161 ( thick ) and @xmath162 ( thin ) , with @xmath85 ( dashed ) , @xmath95 ( dot - dashed ) and @xmath89 ( dotted ) . the adiabatic results ( solid line ) are also included . for @xmath135 and @xmath161 , ( b ) square of the projections of the time - dependent wave function on the adiabatic pendular states @xmath175 ( dot - dashed ) , @xmath176 ( dotted ) , @xmath178 ( long - dashed ) , and @xmath179 ( short - dashed ) , and ( c ) adiabatic parameter between the pendular states @xmath175 and @xmath176 ( dot - dashed ) , @xmath175 and @xmath178 ( dashed ) , and @xmath176 and @xmath179 ( dotted ) . the dc field is fixed to @xmath72 . ] in contrast to the ground state , the wave function of any excited level has contributions from adiabatic states which correspond to different pendular doublets . as the laser intensity is increased , the molecular dynamics is affected by the formation of these pendular doublets . thus , the final orientation could be significantly reduced compared to the parallel fields result . for instance , the time - dependent @xmath170 state has contributions from the adiabatic levels @xmath175 and @xmath176 , which correspond to the first and second pendular doublets , respectively . in ( b ) we show how the adiabatic parameters @xmath96 between the pairs @xmath178-@xmath175 and @xmath176-@xmath179 , which form the first and second doublets , respectively , achieve values close to @xmath81 . the final population of the state @xmath170 is @xmath189 , @xmath190 , @xmath191 , and @xmath192 , which gives rise to a small orientation as a consequence of this population redistribution to other pendular doublets , features of the system such as energy , alignment and hybridization of the angular motion do not resemble the adiabatic results . in particular , since the levels on the second pendular doublet possess a smaller alignment , the adiabatic result is larger than the time - dependent one . for instance , for @xmath161 and @xmath135 , once the time evolution is finished the alignment of this state @xmath170 is @xmath193 , compared to @xmath194 obtained for the adiabatic level @xmath175 . for @xmath173 , despite the fact that the @xmath170 level is significantly aligned , @xmath195 , it is not oriented with @xmath196 for @xmath135 . this state does not gain any orientation if stronger peak intensities are used . this is a consequence of the population redistribution explained above . indeed , this level could be considered as a _ dark state _ for the mixed - field orientation dynamics . this physical phenomenon is not restricted to this state and field configuration . we show below that other levels also behave as _ dark states_. it is worth noting that if in a mixed - field orientation experiment these dark states form part of the molecular beam , the degree of orientation will be diminished . the population redistribution to other pendular doublets significantly affects the expectation value @xmath160 . to @xmath160 contribute terms which mix up adiabatic states with different magnetic quantum numbers since their wave functions could spatially overlap , their coupling matrix elements do not vanish , and @xmath160 oscillates as @xmath46 is increased . the phenomenon of population redistribution at weak laser intensities also occurs for highly excited rotational levels , and for them , more adiabatic states would be involved in it . before the gaussian pulse is turned on , the stark separation of the states in a certain @xmath0-manifold is increased if the electrostatic field strength is enhanced . then , the adiabatic parameter @xmath96 is reduced , and the process of splitting of this @xmath0-manifold becomes less diabatic . indeed , for sufficiently strong dc - fields , the dynamics would be adiabatic without population transfer between the states with the same field - free @xmath0 . for instance , the mixed - field dynamics of the @xmath170 level can be considered as adiabatic on the weak laser field regime for @xmath197 and @xmath161 . let us remark that the excited states could also suffer avoided crossings with adjacent levels , having different field - free magnetic quantum number @xmath45 , as the pulse intensity is varied . the rotational dynamics in most of these crossings will be non - adiabatic @xcite . analogously to the parallel - field configuration , we investigate now the impact of the laser peak intensity on the orientation . to do so , we restrict this study to the following eight states : @xmath198 , @xmath133 , @xmath170 , @xmath199 , @xmath200 , @xmath201 , @xmath202 , and @xmath203 . note that they are related to the ones analyzed in the parallel fields configuration , by a rotation of @xmath4 around the lff @xmath159 axis . their orientation at @xmath125 , i.e. , @xmath62 for @xmath126 , is plotted as a function of @xmath36 in for @xmath161 , panels ( a)-(b ) , @xmath173 , panels ( c)-(d ) , and @xmath204 , panels ( e)-(f ) , and @xmath72 and @xmath127 , respectively . the fwhm of these gaussian pulses is fixed to @xmath83 ns . at @xmath125 as a function of the peak intensity @xmath36 for @xmath198 ( red thick solid ) , @xmath133 ( gold thin solid ) @xmath170 ( orange thick short - dashed ) , @xmath200 ( dark blue long - dashed ) @xmath199 ( blue dot - short - dashed ) , @xmath202 ( purple dotted ) , @xmath201 ( green thin short - dashed ) and @xmath203 ( pink dot - long - dashed ) . the field configurations are ( a)-(b ) @xmath161 , ( c)-(d ) @xmath173 and ( e)-(f ) @xmath204 , with @xmath72 and @xmath127 , respectively . the fwhms of the gaussian pulses are fixed to @xmath128 ns . ] let us start analyzing the results for the ground state . for all field configurations , @xmath62 shows a qualitatively similar behavior as a function of the peak intensity : initially increases , reaches a maximum and decreases thereafter . at the peak intensity where the maximum of @xmath62 takes place , the dynamics of this state is non - adiabatic , but due to the coupling of both states the orientation increases with respect to the adiabatic result . for a fixed peak intensity and electric field strength , @xmath62 decreases as @xmath4 is increased towards @xmath163 . for @xmath204 and @xmath205 , the ground state achieves a moderate maximal orientation , @xmath206 and @xmath207 for @xmath72 and @xmath127 , respectively . the population transfer taking place at weak and strong laser intensities leaves its finger - prints in the dynamics of the excited states . compared to the parallel field results , cf , their orientation is reduced for any inclination angle @xmath4 and the pendular pairs are not any longer formed by a right- and wrong - way oriented states . whereas for most of the field configurations , the ground state possesses the largest orientation , the levels @xmath133 or @xmath208 could achieve a similar or even larger orientation , e.g. , for @xmath173 and @xmath209 , @xmath210 and @xmath72 . for @xmath161 @xmath211 , the degree of orientation is moderate for most of the states . several dark states are found for @xmath173 : @xmath170 , @xmath133 , @xmath202 , and @xmath203 , see ( c)-(d ) . for instance , the levels @xmath170 and @xmath133 are strongly aligned with @xmath212 for @xmath213 and @xmath211 , whereas they are not orientated with @xmath214 and @xmath215 , respectively . for @xmath204 , when the peak intensity of the gaussian pulse is reached most of the states present a weak orientation , only the levels @xmath198 and @xmath170 have a large orientation for small values of @xmath36 . these results indicate that with a @xmath128 ns alignment pulse , strong dc fields and small inclination angles are required to reach a moderate orientation for excited states . for the same set of states as in the previous section , we analyze here how their directional properties depend on the fwhm of the gaussian pulse . in ( a ) and ( b ) we show @xmath62 at @xmath125 as a function of @xmath34 for @xmath161 and @xmath162 , respectively . the static electric field is fixed to @xmath72 , and the peak intensity to @xmath135 . at @xmath125 as a function of @xmath34 . the field configurations are ( a ) @xmath161 and ( b ) @xmath173 , with @xmath135 and @xmath72 . the states and their labels are the same as in . ] the orientation of the ground state increases approaching the adiabatic limit as @xmath34 is increased , and it reaches it with a @xmath141 ns pulse . we encounter several excited states presenting a moderate or weak orientation , and their @xmath80 monotonically increases as @xmath34 is enhanced , e.g. , for @xmath161 the levels @xmath133 , @xmath170 , @xmath199 , @xmath200 , @xmath201 and @xmath202 and for @xmath173 @xmath199 , @xmath200 and @xmath201 . for all of them , a @xmath216 ns pulse is not enough to achieve the adiabatic regime . in contrast , other excited levels present a very small , almost zero , orientation independently of the pulse duration . some of these levels behave as dark states being strongly aligned but not oriented independently of the pulse duration , e.g. , the @xmath203 state has @xmath217 and @xmath218 for @xmath219 and any value of @xmath34 . an analogous behavior is found for the levels @xmath133 , @xmath170 , @xmath202 and @xmath203 at @xmath173 . as described above , this phenomenon is due to the non - adiabatic dynamics at weak laser intensities when the levels of the @xmath0-manifold are driven apart , and it takes places even for @xmath141 ns pulses . in this section , we consider two inclination angles , and investigate the impact of the electrostatic field on the mixed - field orientation dynamics of the same states . figures [ fig : fig_12 ] ( a ) and ( b ) illustrate the behavior of @xmath62 at @xmath125 as a function of @xmath47 for @xmath161 and @xmath162 , respectively . the laser pulse has @xmath83 ns and @xmath135 . at @xmath125 as a function of @xmath47 for the field configurations @xmath83 ns , @xmath135 , and ( a ) @xmath161 and ( b ) @xmath173 . the labeling of the states is done as in . ] the ground state presents the largest orientation , which increases as @xmath47 is enhanced , being strongly oriented for sufficiently large fields , e.g. , @xmath220 for @xmath221 and @xmath161 . regarding the excited states , their orientation strongly depends on the inclination angle . for @xmath161 , @xmath80 monotonically increases till it reaches a plateau - like behavior , and they show a moderate orientation . indeed , for @xmath161 , @xmath135 and @xmath146 , we obtain at the maximum of the gaussian pulse @xmath222 and @xmath223 for the states @xmath133 and @xmath170 , respectively . for @xmath173 , the level @xmath200 presents a large orientation : @xmath224 for @xmath225 . there are some darks states for @xmath173 , which are not oriented even when dc fields of @xmath226 are used , e.g. , @xmath202 and @xmath203 . for non - parallel fields , a strong dc field does not ensure a large orientation for excited rotational states . if the aim is a strongly oriented molecular ensemble , then this should be as pure as possible in the ground state . in the hamiltonian , the term @xmath227 is responsible for the mixing of states with different field - free magnetic quantum numbers . on the weak dc field regime , the mixing between these states is so small that @xmath45 could be considered as conserved , and this term could be neglected . by increasing @xmath47 , this coupling between levels with different field - free @xmath45 becomes important , and this should affect the molecular dynamics . thus , the questions that remain open is how important is the @xmath165-component of the electrostatic field to the dynamics , and for which electric field regime , we could only consider its @xmath1-component @xmath228 . as indicated above , even for tilted fields , the dynamics of the ground state can be described by a two state model . its energy separation to the next state with @xmath229 is of the order of @xmath13 and larger than the dc field coupling to these levels . thus , for @xmath230 , the dynamics considering the dc field is equal to the one obtained when only its @xmath1-component is included . for the excited states , the answer to these questions depends on how the initial wave function , before the fields are switched on , is constructed . the first option is to proceed as indicated at the beginning of this section ; the field - free @xmath231 and @xmath41 wave functions are related by a rotation of @xmath4 around the @xmath159-axis @xmath156=@xmath157@xmath156@xmath158 . in this case , for the level @xmath170 , some differences in its orientation are observed for @xmath149 with @xmath80 being larger if the two components of @xmath232 are considered . these differences are augmented as @xmath47 is increased , e.g. , for a @xmath128 ns laser pulse with @xmath233 , @xmath173 and @xmath234 , we obtain at @xmath125 @xmath235 compared to @xmath236 if only the @xmath1-component of @xmath232 is included . by increasing @xmath47 this state will achieve an adiabatic dynamics only if both components of the static field are present . the second option is to construct the field - free @xmath231 wave function equal to the field - free @xmath41 one . in this case , the results resemble those of the parallel field configuration taking into account @xmath237 as scaling factor for the static field strength . the symmetries of the rotational hamiltonian , see , and , therefore , the rotational dynamics strongly depend on the angle between the fields . in this section , we investigate in detail the impact of the inclination angle in the mixed - field orientation dynamics . for the ground state @xmath198 , the orientation cosines @xmath62 and @xmath160 are plotted in , as a function of @xmath4 , together with the adiabatic results . for a weak dc field and strong laser field , the following relation @xmath238 is satisfied within the adiabatic limit . in @xmath160 the term @xmath239 has been neglected , which can be done as far as the mixing between states with different field - free @xmath45 is very small . by increasing the electrostatic field strength , a regime would be encountered where this approximation does not hold any longer . an analogous relation is satisfied between the time - dependent orientation cosines of the ground state . for @xmath85 , its orientation @xmath62 shows a plateau - like behavior till @xmath240 , which is very close to the adiabatic limit . by further increasing @xmath4 , @xmath62 decreases and approaches to zero . for @xmath241 , the states in a pendular doublet have different symmetry and are not coupled by the dc field , thus they might be strongly aligned but not oriented . for @xmath135 , @xmath62 monotonically decreases as @xmath4 is increased towards @xmath242 , and its value is always smaller than for @xmath85 . for both laser fields , @xmath243 decreases as @xmath4 is increased . ( thick solid line ) and @xmath160 ( thin solid line ) at @xmath125 as a function of @xmath4 for the ground state . the peak intensities are ( a ) @xmath205 and ( b ) @xmath244 . the adiabatic results for @xmath62 ( thick dashed line ) and @xmath160 ( thin dashed line ) are also presented . the fwhm of the laser pulse is fixed to @xmath83 ns and the dc field to @xmath72 . ] in figs . [ fig : fig_14 ] ( a ) , ( b ) , ( c ) and ( d ) , we present the orientation cosine of the pairs @xmath198-@xmath133 , @xmath170-@xmath200 , @xmath201-@xmath203 and @xmath199-@xmath202 , respectively , as a function @xmath4 . the static field strength is @xmath72 and we consider two gaussian pulses of @xmath128 ns fwhm and peak intensities @xmath205 and @xmath89 . due to the complicated field - dressed dynamics of excited states for @xmath42 with contributions from several pendular pairs , in @xmath160 the term @xmath245 can not be neglected . then , the simple relation @xmath246 does not hold for these levels . at @xmath125 as a function of @xmath4 for the states ( a ) @xmath198 ( black ) , and @xmath133 ( blue ) , ( b ) @xmath170 ( black ) , and @xmath200 ( blue ) , ( c ) @xmath199 ( black ) , and @xmath202 ( blue ) , ( d ) @xmath201 ( black ) , and @xmath203 ( blue ) . the field configuration is @xmath72 and @xmath205 ( solid lines ) and @xmath89 ( dashed lines ) . the fwhm of the laser pulse is fixed to @xmath83 ns . ] based on the adiabatic theory , the ground state and the level @xmath133 , should present the same orientation but with opposite directions . however , this is only satisfied for @xmath41 . due to the non - adiabatic effects at weak laser intensities , its @xmath80 is smaller than the corresponding value of @xmath198 for @xmath247 . for the second doublet , cf . ( b ) , @xmath62 oscillates as @xmath4 is varied , and the orientation even changes its direction . both states could present a moderate orientation at a certain value of @xmath4 . the pendular regime is not achieved by the third and fourth pairs with a @xmath128 ns laser pulse and @xmath85 , and their orientation is either zero or very small independently of @xmath4 . for @xmath135 and @xmath41 , these four states show a moderate orientation , which is reduced for any other angle , being small for @xmath248 . at the strong peak intensity @xmath135 , in all pendular doublets one of the two levels presents the dark behavior with respect to the mixed - field orientation dynamics at a certain angle @xmath4 . these results show that if the molecular beam is rotationally cold , a small inclination angle will optimize the degree of orientation observed in the experiment . let us investigate the dynamics for @xmath249 assuming that the laser peak intensity , reached at @xmath125 , and the dc field strength are kept constant for @xmath249 ; i.e. , @xmath250 and @xmath251 for @xmath249 . at @xmath125 , the time - dependent wave function can be expressed in terms of the corresponding adiabatic basis . since the hamiltonian is time - independent for @xmath249 , the contribution of each adiabatic state remains constant as @xmath46 is increased . for a certain state @xmath252 , the expectation value of an operator @xmath253 in this adiabatic basis reads as @xmath254{\langle \gamma_j|\hat{a}|\gamma_j\rangle}{_{\textup{p}}}\\ \label{eqn : oscillate_cosine } & + & 2\sum_{j < k}|c_j(0)||c_k(0)|\tensor[_{\textup{p}}]{\langle\gamma_j|\hat{a}|\gamma_k\rangle}{_{\textup{p}}}\cos\left(\cfrac{\delta e_{jk}t}{\hbar}+\delta_{jk}\right ) \nonumber , \end{aligned}\ ] ] with @xmath255 being the weight at @xmath125 of the adiabatic state @xmath256 to the wave function of @xmath252 , @xmath257 the energy splitting between the adiabatic levels @xmath256 and @xmath258 , and @xmath259 the phase difference of @xmath255 and @xmath260 . based on the results presented above , the time - dependent wave function could have contributions from : i ) only the adiabatic levels forming a pendular doublet , or ii ) several adiabatic levels from at least two pendular doublets . all the states for @xmath41 , and the ground state for @xmath261 could belong to the first case . whereas , the second one refers to all excited states when @xmath42 , unless the static field is very strong . . for the state @xmath170 , ( a ) expectation value @xmath262 with @xmath85 and @xmath41 ( solid ) , @xmath263 ( dashed ) and @xmath162 ( dotted ) ; and ( b ) @xmath243 for @xmath161 and @xmath85 ( solid ) , @xmath95 ( dashed ) and @xmath89 ( dotted ) . the dc field is @xmath264 . ] let us first analyze the case when the dynamics takes place within a pendular doublet . if the adiabatic states are not fully oriented , the coupling term in is non - zero and this expectation value oscillates for @xmath249 with the frequency equal to the energy splitting of the corresponding pendular doublet . for the @xmath170 state , this behavior is shown for the time evolution of @xmath62 in ( a ) , with @xmath85 , @xmath265 and @xmath41 . an analogous behavior is obtained for the ground state and @xmath42 . by further increasing the peak intensity , the orientation of the adiabatic states increases , the coupling terms are reduced approaching zero , and these regular oscillations will disappear . when two pendular doublets participate in the dynamics , this oscillatory behavior becomes irregular , because the frequencies associated with the energy separations within each pendular doublet and between two of them do not form a commensurable set . as an example , we show in ( a ) these irregular oscillations of @xmath62 for @xmath170 with @xmath161 and @xmath162 , @xmath265 and @xmath85 . by increasing @xmath36 , the dynamics of this state still has contributions from different pendular doublets , but the two states in a pendular pair are not populated . as a consequence , the coupling terms are reduced and the oscillation decreases or even disappears . for @xmath42 , the time evolution of @xmath160 is dominated by the couplings of adiabatic levels from doublets with @xmath266 . this is illustrated in ( b ) , for the state @xmath170 , @xmath85 , @xmath95 and @xmath89 and @xmath161 . independently of @xmath36 , in this time scale @xmath160 oscillates with the largest frequency given by the energy gap between of the two pendular doublets involved , which is similar for the three peak intensities . on a larger time scale , the frequencies due to the states in a doublet will modulate the oscillations of @xmath243 in the weak field regime . in previous sections , the field configuration was based on the mixed - field orientation experiments @xcite . here , we investigate the molecular dynamics when the temporal order of the fields is inverted : the gaussian pulse is switched on first , its peak intensity is kept constant , and then the static electric field is turned on . while the laser field is switched on , the pendular doublets of quasi - degenerate states with opposite parity are formed . this process is adiabatic @xcite , and for a sufficiently large peak intensity , these two levels are strongly aligned but not oriented . by turning on the static field , these states have the same symmetry and they should be oriented due to their interaction with this field . for this field configuration , we check now the validity of the adiabatic predictions @xcite by comparing them to a time dependent analysis . the peak intensity of the gaussian pulse is reached at @xmath125 and kept constant afterwards . at this point , if @xmath36 is large enough , the energy gap between the states in a pendular doublet is much smaller than the energy gap with the neighboring doublet . then , for a certain pendular level , its rotational dynamics for @xmath249 , i.e. , when the static field is switched on , could be approximated by a two - state model involving the two levels forming the corresponding pendular doublet @xcite . at @xmath125 , i.e. , @xmath267 and @xmath126 , the pendular states are @xmath268 with @xmath43 and @xmath44 indicating even or odd parity . under this approximation , we assume that the levels @xmath269 and @xmath270 are right- and wrong - way oriented , respectively . the two - state - model hamiltonian yields as @xmath271 where we have taken @xmath272 with @xmath273 and @xmath33 being the switching on speed and time , respectively . this time @xmath33 is chosen so that if these states are exposed only to this field , the turning - on process is adiabatic . note that we have taken @xmath274 , @xmath275 , and @xmath276 . the time - dependent schrdinger equation associated to this hamiltonian admits a scaling factor . that is , when the dynamics is adiabatic using @xmath277 for a pendular doublet with energy splitting @xmath114 at @xmath126 , then , for a peak intensity @xmath278 and @xmath279 , the dynamics is adiabatic for @xmath280 . and ( b ) weight of the adiabatic ground state on its time - dependent wave function as a function of @xmath18 , for turning on speeds @xmath281 ( orange dashed ) , @xmath282 ( red solid ) , @xmath283 ( blue dotted ) and @xmath284 v / scm ( pink dot - dashed ) , and adiabatic results ( thin solid ) the fields are parallel and the gaussian pulse has @xmath83 ns and @xmath85 . ] for the sake of simplicity , we focus on the ground state in a parallel - field configuration . for several switching on speeds , ( a ) and ( b ) display the directional cosine and the population of the adiabatic ground state , respectively , as a function of @xmath18 . the gaussian pulse has @xmath128 ns fwhm and peak intensity @xmath85 . before the dc field is turned on , the alignment of the ground state is @xmath285 , the energy separation within this pendular pair is @xmath286 @xmath58 , and there are @xmath287 @xmath58 to the next pendular doublet . for @xmath288 , the coupling term is @xmath289 @xmath58 with @xmath290 . in an adiabatic picture , the energy gap @xmath114 can not be neglected , and as @xmath18 is increased the energy of the ground state does not increase linearly with @xmath47 @xcite . for @xmath281 v / scm , the adiabatic parameter @xmath291 , the rotational dynamics is non - adiabatic and there is a population transfer between the two states in this doublet . we note that for this process , the adiabatic parameter @xmath96 is defined as in but replacing the laser field interaction @xmath103 by the dipole term @xmath14 . the ground state presents a moderate orientation bellow the adiabatic limit due to the contributions of the adiabatic states @xmath178 and @xmath176 , @xmath109 decreases until a minimum value showing a smooth oscillation afterwards , cf . ( b ) . due to the coupling term , @xmath62 oscillates as @xmath18 is increased , and its frequency is equal to the energy separation between the adiabatic levels @xmath178 and @xmath176 . a similar behavior is observed for @xmath292 v / scm , but the orientation of the ground state oscillates around a value closer to the adiabatic limit because the process is more adiabatic and @xmath293 for @xmath294 . for @xmath295 and @xmath296 v / scm , the dynamics can be considered as adiabatic , with @xmath109 being larger than @xmath297 . however , for @xmath295 , @xmath62 still oscillates around the adiabatic value . by increasing the peak intensity of the laser pulse , the energy splitting of the levels in a pendular doublet is decreased , but their coupling due to the dc field is not significantly modified . thus , the rotational dynamics becomes more diabatic , and larger turning - on times are needed to achieve the adiabatic limit . for @xmath88 , the ground state is separated by @xmath298 @xmath58 from @xmath176 , and by @xmath299 @xmath58 from the next pendular doublet . the coupling due to the dc field is @xmath300 @xmath58 for @xmath288 and with @xmath301 . according to the scaling law of the time - dependent schrdinger equation , this process would be adiabatic for a speed of @xmath302 v / scm . for @xmath135 and the ground state , we find @xmath303 @xmath58 , @xmath304 @xmath58 to the second doublet , and @xmath305 @xmath58 for @xmath288 and with @xmath306 . within an adiabatic framework , as @xmath47 is increased the ground state energy can be approximated by the pseudo - first - order stark linear effect @xmath307@xmath308@xmath309@xmath176@xmath310 @xcite . note that @xmath114 is smaller than the dc filed coupling even for @xmath311 . based on the scaling law of the two - state model schrdinger equation , the dc field should be turned on very slowly , @xmath312 v / scm , to achieve the adiabatic limit . for larger turning - on speeds , the dynamics is so diabatic that the @xmath198 wave function does not change , and its projections on the adiabatic states @xmath178 and @xmath176 are close to the field - free values even for @xmath313 v / scm . in this work , we have investigated the mixed - field orientation dynamics of linear molecules . the richness and variety of the field - dressed rotational dynamics has been illustrated by analyzing in detail the directional properties of several low - lying states . in particular , we have explored the degree of orientation as the peak intensity and fwhm of the gaussian pulse , the electrostatic field strength and the angle between both fields are varied . by considering prototypical field configurations used in current mixed - field orientation experiments , we have proven that the assumption of a fully adiabatic dynamics is incorrect . for parallel fields , a non - adiabatic transfer of population takes place when the quasi - degenerated pendular doublets are formed as the laser intensity is increased . as a consequence , the time - dependent results for the degree of orientation are smaller than the predictions of the adiabatic theory . using current available experimental peak intensities , longer laser pulses or stronger static fields will increase the degree of orientation even for highly excited states . in particular , we have provided the field parameters under which the mixed - field orientation dynamics will be fully adiabatic . we have also shown that the field - dressed dynamics is more complicated if both fields are tilted . apart from the non - adiabatic effects when the pendular doublets are formed , at weak laser intensities there is also population transfer due to the splitting of the states within a @xmath0-manifold having now the same symmetry . for non - parallel fields , we have shown that the ground state is strongly oriented , whereas excited states might only present a moderate orientation , and , furthermore , some of them could behave as dark states to the mixed - field orientation dynamics . the requirements for an adiabatic dynamics are now more difficult to satisfy for excited levels than for the ground state . again , we have indicated the field configuration that will give rise to an adiabatic mixed - field - orientation . if the peak intensity is kept constant after turning on the pulse , we have shown that the orientation of the states might oscillate with time due to the non - adiabatic dynamics . finally , we have investigated the molecular dynamics when the temporal order of the fields is inverted . we have shown that once the ground state is adiabatically aligned , the switching on of the dc field has to be very slow to achieve a significant orientation . although our study is restricted to the ocs molecule , we stress that the above - observed physical phenomena are expected to occur in many other polar molecules . indeed , the hamiltonian can be rescaled , and the above results used to describe another polar linear molecule . in addition , due the complexity of their rotational level structure of asymmetric tops , these non - adiabatic effects should have a negative impact in their mixed - field orientation experiments @xcite . we would like to thank b. friedrich , j. kpper , j. h. nielsen , p. schmelcher and h. stapelfeldt for fruitful discussions . financial support by the spanish project fis2011 - 24540 ( micinn ) , the grants fqm-2445 and fqm-4643 ( junta de andaluca ) , and andalusian research group fqm-207 is gratefully appreciated . j.j.o . acknowledges the support of me under the program fpu .
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we present a theoretical study of the impact of an electrostatic field combined with non - resonant linearly polarized laser pulses on the rotational dynamics of linear molecules . within the rigid rotor approximation
, we solve the time - dependent schrdinger equation for several field configurations . using the ocs molecule as prototype
, the field - dressed dynamics is analyzed in detail for experimentally accessible static field strengths and laser pulses .
results for directional cosines are presented and compared to the predictions of the adiabatic theory .
we demonstrate that for prototypical field configuration used in current mixed - field orientation experiments , the molecular field dynamics is , in general , non - adiabatic , being mandatory a time - dependent description of these systems .
we investigate several field regimes identifying the sources of non - adiabatic effects , and provide the field parameters under which the adiabatic dynamics would be achieved .
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spin - orbit splitting of the energy dispersion relations for electrons in nonsymmetric quantum heterostructures has been theoretically considered during the past decades ( see ref . for a review ) . in bulk materials spin - orbit interaction appears both due to a slow - variable potential ( related to the lattice constant ) @xcite and due to cubic 3 and linear @xcite spin - dependent contributions to the effective hamiltonian . turning to the two - dimensional ( 2d ) case , we can reduce the cubic contribution to a linear one after the replacement of the squared momentum by the quantized value due to confinement @xcite . it is still more important the fact that we have to take into account an additional spin - orbit splitting of the energy spectrum due to the interaction with abrupt heterojunction potentials ( see ref . and discussion in refs . ) . such contribution is of a radically different kind with respect to the listed above because contributions from both sides of a slow confinement potential compensate each other @xcite and , therefore , the spin - splitting of 2d states can not be obtained without a short - range potential contribution . to the best of our knowledge , the relative contributions from the bulk - induced mechanisms @xcite and from the heterojunctions have not been clarified experimentally in spite of a set of existing theoretical calculations @xcite . in this paper we have examined the effect of an in - plane magnetic field on the electron energy spectrum and density of states and we have found that magneto - induced modifications of these characteristics are essentially different from the 2d spin - orbit interaction and zeeman splitting . in principle , this fact makes possible an experimental verification of the above discussed contributions of the spin - orbit interaction . ) and quantum well ( @xmath0 ) . dashed lines show electron ground state energy levels and thin curves correspond to wave functions ; @xmath1 is the qw s width , @xmath2 are the band offsets , and @xmath3 is the effective mass in the narrow - gap region , while @xmath4 and @xmath5 are the barrier effective masses . ] the effect of in - plane magnetic field on the energy spectrum in non - symmetric heterostructures occurs due to the zeeman term in the electron hamiltonian . this fact has been found in ref . , shortly after initial considerations of the spin - orbit splitting in non - symmetric heterostructures ( see refs . ) . a number of peculiarities in transport phenomena for 2d systems under an in - plane magnetic field were also discussed @xcite and more complicated cases , such as quasi-1d transport or spin hall effect ( see refs . in @xcite or @xcite , respectively ) were recently considered . all this papers only take into account the mix between the zeeman contribution and the effective 2d spin - orbit interaction . here we perform the calculations based on the three - band kane model@xcite with non - symmetric boundary conditions , which is valid for narrow - gap heterostructures , and with a slow potential described self - consistently ( see typical band diagrams in fig . the analysis of magneto - induced modifications of 2d energy spectra and corresponding density of states is presented for typical parameters of ingaas / inalas - selectively - doped heterojunctions , and ingaas and insb - based quantum wells ( qws ) . the paper is organized as follows . in the next section we discuss the conduction @xmath6-band eigenstate problem and describe the electronic states in non - symmetric narrow - gap heterostructures using the boundary conditions for the wave functions at the interfaces in the parabolic approximation . in sec . iii we solve the 2d - eigenstate problem using the averaged transverse field approach . numerical self - consistent calculations of the magneto - induced ( caused by a magnetic field ) modifications of the energy spectra and the density of states are presented in sec . conclusions are given in the last section . we start from the formulation of the eigenstate problem for the electronic state of the conduction @xmath6-band localized in the selectively - doped heterojunction or in the non - symmetric qw . all the above listed spin - dependent contributions are taken into account in the analysis . we also present the density of states which is connected to the photoluminescence excitation ( ple ) intensity . the electronic states in narrow - gap heterostructures are described by the three - band kane matrix hamiltonian @xcite @xmath7where the kinetic momentum , @xmath8 , contains the vector potential @xmath9 , @xmath10 is an in - plane magnetic field and @xmath11 is written in the @xmath12-representation through the 2d momentum @xmath13 . here we have also introduced the diagonal energy matrix @xmath14 whose elements determine the positions of the band extrema ( the energy values @xmath15 , @xmath16 , and @xmath17 correspond to electron , heavy- and light - hole extrema ) and the interband velocity matrix @xmath18 . the @xmath19 hermitian matrix @xmath20 is determined by the following non - zero matrix elements @xcite : @xmath21and @xmath22 is the characteristic interband velocity for the kane model . in such definitions eq . ( 3 ) may be rewritten as the spinor eigenvalue problem : @xmath36 _ { z}+\frac{g_{z}}{2}\mu _ { \scriptscriptstyle b}h\hat{\sigma}_{y}\right\ } \psi _ { \mathbf{p}z } & = & 0,\end{aligned}\]]where spinor @xmath37 is determined by the components @xmath38 , @xmath39 is bohr magneton , and @xmath40 is the pauli matrix . the energy values of @xmath6- and @xmath23-band extrema for the narrow - gap ( @xmath41 ) and wide - gap ( @xmath42 ) regions take the forms : @xmath43and the corresponding band diagrams for qw and selectively - doped heterojunction were shown in fig.1 . here the energy is counted from the bottom of @xmath6-band , @xmath44 is the gap , @xmath45 and @xmath46 are the band offsets for @xmath6- and @xmath23-bands , correspondingly . here the slow potential @xmath47 should be determined from a self - consistent procedure . thus , after substitution of ( 6 ) in eq . ( 5 ) we have formulated the eigenstate problem for the spinor @xmath37 . since @xmath48 in narrow - gap heterostructures , then the weak underbarrier penetration of wave function takes place for the electronic states with @xmath49 ( parabolic band approximation ) . because of this , one can neglect the longitudinal motion for the underbarrier region , @xmath50 for the single heterojunction case shown in fig.1@xmath51 , where the solutions take the form : @xmath52here @xmath53 and @xmath54determines the scale of the underbarrier penetration of wave functions . we have also used in eq . ( 7 ) the continuity conditions for eigenfunctions @xmath55 at heterojunctions . since @xmath32 in eq . ( 4 ) is proportional to @xmath56 , then the integration ( 5 ) over heterojunction produces additional contributions to the boundary condition for flows . such contributions are proportional to the band offset at the heterojunction @xmath46 . eliminating the underbarrier contributions from such equation by the use of the explicit expression ( 7 ) we obtain the third kind boundary conditions @xcite : @xmath57 _ { z}\psi _ { % \mathbf{p}z=0}=0.\]]here the momenta @xmath58 characterizes the underbarrier penetration and the parameter @xmath59 determines the spin - orbit coupling due to the abrupt potential of the heterojunction @xmath60and the right - side part is written for the approximation @xmath61 in the parabolic approximation , we describe the @xmath6-band electronic states using the above introduced spinor @xmath37 . below we consider not very strong magnetic fields ( @xmath62 , where @xmath63 is the fermi velocity and @xmath64 is the cyclotron frequency ) when @xmath65 in eq . ( 5 ) is replaced by @xmath66 and the isotropic kinetic energy is given by @xmath67 and includes the effective mass @xmath3 . considering the low - doped structure case ( if @xmath68 ) we rewrite the schrdinger equation ( 5 ) for the narrow - gap region in the form : @xmath69 _ { z}+\frac{g_{z}}{2}\mu _ { \scriptscriptstyle b}h\hat{\sigma% } _ { y}\end{aligned}\]]where @xmath47 is the self - consistent potential , as given in eq . ( 14 ) below , and the effective mass is @xmath70-independent . the spin velocity @xmath32 and the @xmath70-dependent contribution to the @xmath33-factor are proportional to the transverse electric field @xmath71 : @xmath72for the selectively - doped heterojunction case we consider eq . ( 10 ) for the @xmath73 region with the boundary conditions ( 8) and with @xmath74 . for the case of a qw of width @xmath1 , one can eliminate the underbarrier contributions in analogy to eqs . ( 7 - 9 ) and , in addition to eq . ( 10 ) for the region @xmath75 , the following third kind boundary conditions should be used : @xmath76 _ { z}\psi _ { \mathbf{p},\pm \frac{d}{2}}=0.\]]here momenta @xmath77 determine the scale of the underbarrier penetration of wave functions ; the different values of @xmath78 and @xmath5 take into account the differences of band offsets and effective masses , as it is shown in fig . 1@xmath79 . the parameters @xmath80 determine the spin - orbit coupling due to the abrupt potential of the heterojunction according to eq . when @xmath81 , which corresponds to the symmetric case , we will take @xmath82 and @xmath83 , as we can see from ( 9 ) . thus , both interface potentials , which determine the spin - dependent contributions to ( 12 ) , and the intrawell field , which determines the spin velocity @xmath32 in ( 11 ) , are responsible for the spin - splitting of energy spectra . since the zeeman spin - splitting term only appears in the eq . ( 10 ) , the mentioned magneto - induced modifications of electron states due to these two contributions are different . the self - consistent numerical procedure for the eigenstate problem ( 10 ) involves the potential @xmath47 , which is obtained from the poisson equation in the following form : @xmath84\]]here @xmath85 is the 3d concentration of donors and @xmath86 is the dielectric permittivity that we have supposed as uniform across the heterostructure . the electron density distribution is introduced through the electron dispersion relations @xmath87 according to : @xmath88where @xmath89 denotes the hermitian conjugate of @xmath90 and @xmath91 refers to the two possible spin orientations . the heaviside function , @xmath92 , appears here for the zero - temperature case . the fermi energy , @xmath93 , is expressed through the total electron density , @xmath94 , defined as @xmath95 thus , @xmath96 depends on @xmath97 for the fixed concentration case . the density of states is given by the standard formula : @xmath98 in order to analyze @xmath99 one needs to solve the above - formulated eigenstate problem and to perform the integrations in eq . the density of states is connected to ple intensity for the case of near - edge transitions , @xmath100 . since the interband matrix element @xmath101 do not depend on the in - plane quantum numbers , one obtains @xcite : @xmath102where @xmath103 is the polarization vector , @xmath104 and @xmath105 is the gap energy , which is renormalized due to the confinement effect before the numerical consideration we perform a simplified calculation of the eigenstate problem using the uniform transverse field approximation @xmath106 which is valid for non - doped qws under an external modulating field @xmath107 ; for heavy - doped structures @xmath107 implies an averaged self - consistent transverse field . the two @xmath70-dependent fundamental solutions @xmath119 ( labeled below by @xmath120 ) are determined from the equation @xmath121written for the narrow - gap region . the general spinor solution @xmath37 is expressed through these fundamental solutions according to @xmath122where the coefficients @xmath123 are determined from the boundary conditions ( 12 ) or from ( 8) and the requirements @xmath74 . after the substitution of the solution ( 21 ) for @xmath124 , which corresponds to the boundary condition at @xmath125 , into ( 8) and the multiplication of this system by @xmath112 on the left side we rewrite the boundary condition at @xmath126 as follows : @xmath127 \psi _ { \mathbf{p}z=0 } & = & 0 , \nonumber \\ \psi _ { \mathbf{p}}^{\scriptscriptstyle(-1)+ } { } \cdot \left [ \begin{array}{ll } 0 & -p_{+ } \\ p_{- } & 0% \end{array}% \right ] \psi _ { \mathbf{p}z=0 } & = & 0,\end{aligned}\]]with @xmath128 . in order to calculate the proportional to @xmath59 contributions we use here the solutions of the spin - dependent eigenstate problem ( 19 ) . thus , eq . ( 22 ) can be transformed with the use of the relations @xmath129 \psi _ { \mathbf{p}}^{(\sigma ) } & = & \sigma \frac{\overline{\mathrm{v}}% p^{2}+w_{\scriptscriptstyle h}p_{x}}{iw_{\mathbf{p } } } \nonumber \\ \psi _ { \mathbf{p}}^{(\sigma ) } { } ^{\scriptscriptstyle + } \cdot \left [ \begin{array}{ll } 0 & -p_{+ } \\ p_{- } & 0% \end{array}% \right ] \psi _ { \mathbf{p}}^{(-\sigma ) } & = & -i\frac{w_{\scriptscriptstyle % h}p_{y}}{\mathrm{v}p_{\sigma } + \sigma w_{\scriptscriptstyle h}}\end{aligned}\]]to a simple linear system for @xmath130 . the dispersion relation , @xmath131 , is determined from the zero determinant requirement . a similar transformation of the boundary conditions ( 12 ) for qws , after substitution on ( 21 ) , permit us to rewrite the boundary condition at @xmath132 in the form : @xmath133 \psi _ { \mathbf{p},\pm \frac{d}{2 } } & = & 0 , \nonumber \\ & & \left ( \hat{p}_{z}\mp ip_{\pm } \right ) \left ( a_{-}\varphi _ { z}^{% \scriptscriptstyle(a,-1)}+b_{-}\varphi _ { z}^{\scriptscriptstyle% ( b,-1)}\right ) \left\vert _ { z=\pm \frac{d}{2}}\right . \nonumber \\ + \chi \psi _ { \mathbf{p}}^{\scriptscriptstyle(-1)+ } { } \cdot \left [ \begin{array}{ll } 0 & -p_{+ } \\ p_{- } & 0% \end{array}% \right ] \psi _ { \mathbf{p},\pm \frac{d}{2 } } & = & 0.\end{aligned}\]]thus , we have obtained a linear system for @xmath130 and @xmath134 and , therefore , @xmath87 is obtained from the solvability condition for this system . first , let us consider the case of a heterostructure without spin - orbit contributions from heterojunctions , @xmath135 , when the dispersion relation is given by eq . for the zero magnetic field case , @xmath136 , the energy @xmath137 is replaced by @xmath138 and @xmath139 in eq . ( 19 ) is transformed to the isotropic dispersion relation @xmath140 . for this case the density of states was considered in ref . . if @xmath141 , the dispersion relation ( 19 ) becomes anisotropic as it is shown in fig . 2 @xmath51 . we have used in this figure the dimensionless magnetic field @xmath142 . in order to represent a general case , valid for any structure , we have also used dimensionless energy axis . values . solid , dashed and dot - dashed curves corresponds to the dimensionless momenta @xmath1430 , 0.25 , and 1 , respectively . ( b ) density of states vs dimensionless energy for magnetic fields @xmath1440 ( solid ) , 0.5 ( dashed ) , 1 ( dotted ) , 2.5 ( dash - dotted ) , and 5 ( dash - dot - dotted ) . , title="fig : " ] values . solid , dashed and dot - dashed curves corresponds to the dimensionless momenta @xmath1430 , 0.25 , and 1 , respectively . ( b ) density of states vs dimensionless energy for magnetic fields @xmath1440 ( solid ) , 0.5 ( dashed ) , 1 ( dotted ) , 2.5 ( dash - dotted ) , and 5 ( dash - dot - dotted ) . , title="fig : " ] for the case of the strong magnetic field , when @xmath137 is replaced by @xmath145 , the zeeman splitting effect appears to be dominant in the dispersion relation : @xmath146 . the spin - orbit splitting is mainly manifested as a shift of the dispersion paraboloids towards higher ( lower ) @xmath147 values for @xmath148 , respectively . on the other hand , zeeman splitting is shown as a displacement to higher ( lower ) energy values depending on @xmath149 . the anisotropy of the dispersion relation @xmath150 appears to be essential for the region around the cross - point , @xmath151 . after the shift @xmath152 and the integration over @xmath153 with the use of the @xmath154-function , the density of states is transformed from eq . ( 15 ) into the integral with respect to the cosine of the in - plane angle , @xmath155 : @xmath156here @xmath157 and @xmath158 . a straightforward integration for the above - cross - point region @xmath159 gives as the exact relation : @xmath160 , while for the below - cross - point region @xmath161 we plot @xmath162 versus dimensionless energy @xmath163 for different fields @xmath164 . figure 2@xmath165 shows this density of states for different dimensionless magnetic fields . two points deserve special attention . first , the typical @xmath166 singularity around @xmath167 , which occurs for @xmath168 , tends to disappears with increasing field . second , abrupt steps corresponding to the filling of each level are shifted with the field intensity indicating a delay in the filling caused by the zeeman splitting . it should be noted that the lowest @xmath169 level seems to be overfilled for low magnetic fields , i.e. , @xmath170 . electrons equally distributes between levels for magnetic fields higher than @xmath171 . here we perform numerical calculations based on the simplified consideration outlined in sec . ii@xmath42 , as well as the self - consistent solution of the eigenstate problem of sec . ii@xmath41 . the correspondent level - splitting at the fermi energy level in a selectively - doped heterojunction and the densities of states in qws are described . let us consider the 2d approach , when @xmath181 is around the ground energy level , @xmath182 , determined by the equation : @xmath183 . expanding @xmath184 $ ] over @xmath185 one can transform eq . ( 27 ) into a quadratic equation and the dispersion relation takes the form @xmath186 , where the splitting @xmath187 of the energy spectrum is given by @xmath188with the heterojunction - induced spin velocity , @xmath189 , and the internal spin velocity , @xmath190 , introduced in eq . ( 17 ) . in fig . 3 we plot these dispersion relations for the same parameters used in fig . 2@xmath51 . as in fig . 2@xmath51 , we have used dimensionless energy and momenta to represent a general case . comparing fig . 2@xmath191 with fig . 3 we can see the effect of the interface contribution as an enhancement of the splitting between parabolas @xmath91 . for different magnetic fields @xmath192 solid , dashed and dot - dashed curves corresponds to the dimensionless momenta @xmath1430 , 0.25 , and 1 , respectively . ] for high electron concentration the approach of eq . ( 28 ) is no longer valid . in order to obtain @xmath181 for this case we need to solve eq . ( 10 ) together with eq . ( 13 ) by means of a self - consistent approach . to do that we have used the transfer matrix method@xcite . figure 4 shows dispersion relations obtained in this way for in@xmath193al@xmath194as / in@xmath195ga@xmath196as with the electron density @xmath197 @xmath198 , which corresponds to a fermi energy @xmath199 mev . using the standard parameters @xcite one obtains @xmath200 and @xmath201 cm / s , so that dimensionless field @xmath202 corresponds to @xmath203 t. thus , we can compare panel for @xmath202 in fig.3 with panel for @xmath203 t in fig . 4 . we can see that numerically calculated results show a similar behavior than approximation ( 28 ) , although , for the material under consideration , splitting is qualitatively less and shift along momentum is larger in fig . 4 . and @xmath204 @xmath205 solid line : @xmath206 . dashed line : @xmath207 @xmath208 dot - dashed line : @xmath209 . ] the spin splitting of the dispersion relations at fermi energy is directly connected to the shubnikov - de haas ( sdh ) oscillations . since @xmath210 is nearly isotropic over @xmath13-plane , we have also calculated @xmath211 for different magnetic fields , see fig . 5 . following refs . the spin splitting is related to the modulation of the sdh oscillation amplitude according to @xmath212 . for the structure under consideration , experimental value@xcite is @xmath213 mev for @xmath214 t and @xmath197 @xmath198 , which corresponds to @xmath215 mev , calculated numerically for the same magnetic field and density . figure 5 shows a very slight dependence of @xmath216 on the magnetic field for @xmath200 being stronger this dependence for the case @xmath135 . thus , interface contributions should be detectable through the sdh oscillations amplitude . vs magnetic field for @xmath217 and @xmath218 . note that the upper @xmath216 is almost @xmath97-independent . ] ( solid line ) and @xmath229 ( dashed line ) . ( b ) density of states for @xmath26 @xmath230 ( solid line ) , @xmath231 @xmath230 ( dashed line ) , @xmath232 @xmath230 ( dot line ) , and @xmath233 @xmath230 ( dot - dashed line).,title="fig : " ] ( solid line ) and @xmath229 ( dashed line ) . ( b ) density of states for @xmath26 @xmath230 ( solid line ) , @xmath231 @xmath230 ( dashed line ) , @xmath232 @xmath230 ( dot line ) , and @xmath233 @xmath230 ( dot - dashed line).,title="fig : " ] once again , after the expansion near the ground energy , @xmath182 , one can transform the determinant in a similar way to transformations of eqs . ( 26 , 27 ) . numerical solutions of these equations for ingaas- and insb - based qws are represented in figs . 6@xmath191 and 7@xmath234 . we have considered @xmath235 wide ingaas- and insb - based qws using the parameters of refs . . we have also applied a field of @xmath236 @xmath237 , which corresponds to spin velocity @xmath238 cm / s , for the ingaas qw , and a field of @xmath235 @xmath237 , spin velocity@xmath239 cm / s , for the insb . comparing figs . 6@xmath191 and 7@xmath191 with fig . 4 we can see that , for similar spin velocities and magnetic fields , the splitting between levels is bigger whereas the shift in the @xmath147 direction is smaller for the qw case with respect to the selectively - doped structure . thus , the effect of interface contribution seems to be more pronounced for the qw case . this effect is enhanced as the width of the wells diminishes . due to the greater spin velocities of ingaas and insb - based qws , caused by the different effective masses and energy gaps , it is necessary the use of lesser magnetic fields to get a similar effect . both for qws and selectively doped structures , interface contributions are opposite to the intrinsic spin - orbit coupling effect . ( solid line ) and @xmath229 ( dashed line ) . ( b ) density of states for @xmath26 @xmath230 ( solid line ) , @xmath240 @xmath230 ( dashed line ) , @xmath231 @xmath230 ( dotted line ) , and @xmath232 @xmath230 ( dot - dashed line).,title="fig : " ] ( solid line ) and @xmath229 ( dashed line ) . ( b ) density of states for @xmath26 @xmath230 ( solid line ) , @xmath240 @xmath230 ( dashed line ) , @xmath231 @xmath230 ( dotted line ) , and @xmath232 @xmath230 ( dot - dashed line).,title="fig : " ] since the density of states is proportional to the ple intensity@xcite , as shown in eq . ( 16 ) , it is interesting to study the shape of @xmath241 , shown in figs . 6@xmath242 and 7@xmath242 . the effect of the interfaces is manifested in @xmath99 as a delay in the quenching of the @xmath243-singularity corresponding to the zero - field case . although the singularity no longer exists for @xmath244 , a peak still remains at the energy value of the bands anticrossing . for the case which includes interface contributions , this peak gets wider and @xmath99 tends slower to @xmath245 for high energy limit . thus , the ple technique is of great interest to analyze the interface contributions@xcite . it should be notice that @xmath91 levels becomes equally filled for magnetic fields beyond 5 t. in this paper we have examined the electron states in narrow - gap non - symmetric heterostructures under an in - plane magnetic field . the eigenstate problem was formulated in the framework of the three - band kane model with non - symmetric boundary conditions . we have found that the mechanisms of mixing between the zeeman term in the hamiltonian and the two kinds of spin - orbit coupling contributions ( from a slow field and from heterojunctions ) are essentially different . numerical estimates for typical parameters of ingaas / inalas and insb / inalsb structures demonstrate the essential magneto - induced modifications of the energy spectra under magnetic field strength of the order of tesla . let us discuss some possibilities for the experimental verification of the energy spectra modifications obtained here . the magnetotransport measurements of shubnikov - de haas oscillations under nearly in - plane magnetic fields@xcite ( when a quasi - classic quantization of the dispersion relations for the transverse component of the magnetic field is possible ) provide a direct information about in - plane magneto - induced modifications of electron energy spectra . the results of sec . iv@xmath41 are in agreement with ref . but more measurements ( for different in - plane fields and concentrations ) are necessary in order to separate the intrinsic and junction - induced contributions . another way of looking for these peculiarities of energy spectra is the mid - infrared ple spectroscopy@xcite when interband transitions are modified under in - plane magnetic field . ple intensity provides direct information of the energy spectra because it is directly connected to the density of states , as mentioned above . a comparison between sdh oscillations and ple measurements with a precision about 1 mev ( it should be possible for a high - quality structure ) provides more data on mechanisms of spin - orbit interaction in narrow - gap structures . thus , we have shown the essential modifications of the electron energy spectra in non - symmetric narrow - gap qws under in - plane magnetic fields . we suggest that measurements of the magneto - induced contributions to optical and transport properties would be an useful method for an experimental verification of the heterojunction - induced contributions to the spin - orbit interaction in the narrow - gap heterostructures under investigation . a. darr , j.p . kotthaus and t. ando , _ proc . f.g . fumi ( north - holland , amsterdam , 1976 ) , p. 774 ; t. ando , a.b . fowler and f. stern , rev . mod * 54 , * 437 ( 1982 ) ; d.a . romanov , phys . solid state * 35 , * 717 ( 1993 ) . j. luo , h. munekata , f. f. fang , and p. j. stiles , phys . b * 38 * , 10142 ( 1988 ) ; b. das , d.c . miller , s. datta , r. reifenberger , w.p . hong , p.k . bhattacharya , j. singh , and m. jaffe , phys . b * 39 * , 1411 ( 1989 ) . the final schrdinger eq . ( 10 ) and corresponding boundary conditions ( 8 , 12 ) do not change under taking into account the stress - induced splitting of @xmath23-band extrema and such a revision of the parameters changes the numerical estimation only slightly but the main results are the same . following refs . 11 - 13 and i. vurgaftman , j.r . meyer , and l.r . ram - mohan , j. appl . phys . * 89 * , 5815 ( 2001 ) , we use the data for in@xmath246al@xmath194as / in@xmath195ga@xmath196as structure : @xmath247 , @xmath248 , where @xmath249 is the electron mass , factor @xmath250 , permittivity @xmath251 , @xmath252 mev , @xmath253 mev , and @xmath254 mev . national compound semiconductor roadmap : insb , http://www.onr.navy.mil/sci_tech/information ( 2005 ) . data for insb / in@xmath255al@xmath194sb are @xmath256 , @xmath257 , @xmath258 mev , @xmath259 mev , @xmath260 mev , @xmath261 , @xmath262 .
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modifications of spin - splitting dispersion relations and density of states for electrons in non - symmetric heterostructures under in - plane magnetic field are studied within the envelope function formalism .
spin - orbit interactions , caused by both a slow potential and the heterojunction potentials ( which are described by the boundary conditions ) are taken into account .
the interplay between these contributions and the magnetic field contribution to the spin - splitting term in the hamiltonian is essential when energy amount resulting from the zeeman and spin - orbit coupling are of the same order .
such modifications of the energy spectra allow us to separate the spin - orbit splitting contributions due to a slow potential and due to the heterojunctions .
numerical estimates for selectively - doped heterojunction and quantum well with narrow - gap region of electron localization are performed .
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one of the fundamental problems in the classification of complex surfaces is to find a new family of complex surfaces of general type with @xmath0 . in this paper we construct new simply connected _ numerical campedelli surfaces _ with an involution , i.e. simply connected minimal complex surfaces of general type with @xmath0 and @xmath1 , that have an automorphism of order @xmath5 . there has been a growing interest for complex surfaces of general type with @xmath0 having an involution ; cf . j. keum - y . lee @xcite , calabri - ciliberto - mendes lopes @xcite , calabri - mendes lopes - pardini @xcite , y. lee - y . shin @xcite , rito @xcite . a classification of _ numerical godeaux surfaces _ ( i.e. minimal complex surfaces of general type with @xmath0 and @xmath2 ) with an involution is given in calabri - ciliberto - mendes lopes @xcite . it is known that the quotient surface of a numerical godeaux surface by its involution is either rational or birational to an enriques surface , and the bicanonical map of the numerical godeaux surface factors through the quotient map . however , the situation is more involved in the case of numerical campedelli surfaces , because the bicanonical map may not factor through the quotient map ; cf . calabri - mendes lopes - pardini @xcite . in particular it can happen that the quotient is of general type . more precisely , let @xmath6 be a numerical campedelli surface with an involution @xmath7 . if @xmath7 has only fixed points and no fixed divisors , then the minimal resolution @xmath8 of the quotient @xmath9 is a numerical godeaux surface and @xmath7 has only four fixed points ; cf . barlow @xcite . conversely , if @xmath8 is of general type , then @xmath7 has only four fixed points and no fixed divisors ; calabri - mendes lopes - pardini @xcite . there are some examples of numerical campedelli surfaces @xmath6 with an involution @xmath7 having only four fixed points . barlow @xcite constructed examples with @xmath10 . barlow @xcite also constructed examples with @xmath11 whose minimal resolution of the quotient by the involution is the first example of a _ simply connected _ numerical godeaux surface . also all catanese s surfaces @xcite have such an involution and @xmath12 . recently calabri , mendes lopes , and pardini @xcite constructed a numerical campedelli surface with torsion @xmath13 and two involutions . frapporti @xcite showed that there exists an involution having only four fixed points on the numerical campedelli surface with @xmath14 constructed first in bauer - catanese - grunewald - pignatelli @xcite . it is known that the orders of the algebraic fundamental groups of numerical campedelli surfaces are at most @xmath15 and the dihedral groups @xmath16 and @xmath17 can not be realized . recently , the existence question for numerical campedelli surfaces with @xmath18 was settled by the construction of examples with @xmath19 ; frapporti @xcite and h. park - j . shin @xcite . hence it would be an interesting problem to construct numerical campedelli surfaces having an involution with @xmath20 for each given group @xmath21 with @xmath22 . especially we are concerned with the simply connected case because the fundamental groups of all the known examples with an involution have large order : @xmath23 . furthermore the first example of _ simply connected _ numerical campedelli surfaces is very recent ( y. lee - j . park ) , but we have no information about the existence of an involution in their example . the main theorem of this paper is : there are simply connected minimal complex surfaces @xmath6 of general type with @xmath24 and @xmath25 which have an involution @xmath7 such that the minimal resolution @xmath8 of the quotient @xmath9 is a simply connected minimal complex surface of general type with @xmath26 and @xmath27 . we also show that the minimal resolution @xmath8 of the quotient @xmath9 has a local deformation space of dimension @xmath28 corresponding to deformations @xmath29 of @xmath8 such that its general fiber @xmath30 is the minimal resolution of a quotient @xmath31 of a numerical campedelli surface @xmath32 by an involution @xmath33 ; theorem [ theorem : invariant - part ] . in addition , we show that the resolution @xmath8 should be always simply connected if the double cover @xmath6 is already simply connected ; proposition [ proposition : simply - connected=>simply - connected ] . conversely barlow @xcite showed that if the resolution @xmath8 is a simply connected numerical godeaux surface then the possible order of the algebraic fundamental group of the double cover @xmath6 is @xmath34 , @xmath35 , @xmath36 , @xmath37 , or @xmath15 . as far as we know , the example in barlow @xcite was the only one whose quotient is simply connected . it has @xmath38 as mentioned earlier . here we find an example with @xmath39 . hence it would be an intriguing problem in this context to construct an example with @xmath40 . in order to construct the examples , we combine a double covering and a @xmath3-gorenstein smoothing method developed in y. lee - j . park . first we build singular surfaces by blowing up points and then contracting curves over a specific rational elliptic surface . these singular surfaces differ by contracting certain @xmath41-curves . if we contract all of the @xmath41-curves , we obtain a stable surface @xmath42 in the sense of kollr shepherd - barron @xcite , and we prove that the space of @xmath3-gorenstein deformations of @xmath42 is smooth and @xmath43 dimensional ; proposition [ propsotion : stable - godeaux ] . a ( @xmath3-gorenstein ) smoothing of @xmath42 in this space produces simply connected numerical godeaux surfaces . in particular , the smoothing of @xmath42 gives the existence of a two dimensional family of simply connected numerical godeaux surfaces with six @xmath41-curves ; corollary [ corollary : six ] . we also prove that a four dimensional family in this space produces simply connected numerical godeaux surfaces with a @xmath5-divisible divisor consisting of four disjoint @xmath41-curves ; theorem [ theorem : q - smoothing - of - y ] and theorem [ theorem : invariant - part ] . these numerical godeaux surfaces are used to construct the numerical campedelli surfaces with an involution . the desired numerical campedelli surfaces are obtained by taking double coverings of the numerical godeaux surfaces branched along the four disjoint @xmath41-curves ; theorem [ theorem : campedelli ] . on the other hand we can also obtain the campedelli family explicitly from a singular stable surface @xmath44 . it comes from blowing up points and contracting curves over a certain rational elliptic surface ; proposition [ proposition : campexplicit ] . the @xmath3-gorenstein space of deformations of @xmath44 is smooth and @xmath45 dimensional ; proposition [ proposition : stable - campedelli ] . in both godeaux and campedelli cases we compute @xmath46 to show no local - to - global obstruction to deform them ; theorem [ theorem : h2(ty)=0 ] and theorem [ theorem : h2(tx)=0 ] . this involves a new technique ( theorem [ theorem : burns - wahl ] ) which generalizes a result of burns - wahl @xcite describing the space of first order deformations of a singular complex surface with only rational double points . a cyclic quotient singularity ( germ at @xmath47 of ) @xmath48 , where @xmath49 with @xmath50 a @xmath51-th primitive root of @xmath34 , @xmath52 , and @xmath53 , is denoted by @xmath54 . @xmath55 means @xmath56 . a @xmath57-curve ( or @xmath41-curve ) in a smooth surface is an embedded @xmath58 with self - intersection @xmath59 ( respectively , @xmath60 ) . throughout this paper we use the same letter to denote a curve and its proper transform under a birational map . a singularity of class @xmath61 is a quotient singularity which admits a @xmath3-gorenstein one parameter smoothing . they are either rational double points or @xmath62 with @xmath63 and @xmath64 ; see kollr shepherd - barron @xcite . for a normal variety @xmath6 its tangent sheaf @xmath65 is @xmath66 . the dimension of @xmath67 is @xmath68 . the authors would like to thank professor yongnam lee for helpful discussion during the work , careful reading of the draft version , and many valuable comments . the authors also wish to thank professor jenia tevelev for indicating a mistake in an earlier version of this paper , and the referee especially for the remark on the proof of proposition 3.7 which makes it simpler . heesang park was supported by basic science research program through the national research foundation of korea ( nrf ) grant funded by the korean government ( 2011 - 0012111 ) . dongsoo shin was supported by basic science research program through the national research foundation of korea ( nrf ) grant funded by the korean government ( 2010 - 0002678 ) . giancarlo urza was supported by a fondecyt inicio grant funded by the chilean government ( 11110047 ) . in this section we construct a family of simply connected numerical godeaux surfaces having a @xmath5-divisible divisor consisting of four disjoint @xmath41-curves by smoothing a singular surface @xmath69 ; theorem [ theorem : q - smoothing - of - y ] . this is the key to construct numerical campedelli surfaces with an involution . in addition , we describe the explicit stable model of the singular surface @xmath69 . in fact , we construct a rational normal projective surface @xmath42 with four singularities @xmath70 , @xmath70 , @xmath71 , @xmath72 and @xmath73 ample . hence @xmath42 is a stable surface ( cf . kollr - shepherd - barron @xcite , hacking @xcite ) . we will prove that the versal @xmath3-gorenstein deformation space @xmath74 ( cf . hacking @xcite ) is smooth and @xmath43 dimensional , and that the @xmath3-gorenstein smoothings of @xmath42 are simply connected numerical godeaux surfaces . in particular , this shows that there are simply connected numerical godeaux surfaces whose canonical model has precisely two @xmath70 singularities ; corollary [ corollary : godeaux ] . furthermore a four dimensional family in @xmath74 produces the above simply connected numerical godeaux surfaces with a @xmath5-divisible divisor consisting of four disjoint @xmath41-curves ; theorem [ theorem : q - smoothing - of - y ] and theorem [ theorem : invariant - part ] . we start with a rational elliptic surface @xmath75 with an @xmath76-singular fiber , an @xmath77-singular fiber , and two nodal singular fibers . in fact we will use the same rational elliptic surface @xmath75 in the papers h. park - j . shin @xcite . however , we need to sketch the construction of @xmath75 to show the relevant curves that will be used to build the singular surfaces @xmath69 and @xmath42 . let @xmath78 , @xmath79 , @xmath80 , and @xmath81 be lines in @xmath82 and let @xmath83 be a smooth conic in @xmath82 given by the following equations . they intersect as in figure [ figure : pencil ] . @xmath84 we consider the pencil of cubics @xmath85 \in \mathbb{cp}^1 \}\ ] ] generated by the two cubic curves @xmath86 and @xmath87 . this pencil has four base points @xmath88 , @xmath89 , @xmath90 , @xmath91 , and four singular members corresponding to @xmath92=[1:0 ] , [ 0:1 ] , [ 2:3\sqrt{3 } ] , [ 2:-3\sqrt{3}]$ ] . the latter two singular members are nodal curves , denoted by @xmath93 and @xmath94 respectively . they have nodes at @xmath95 $ ] and @xmath96 $ ] , respectively . in order to obtain a rational elliptic surface @xmath75 from the pencil , we resolve all base points ( including infinitely near base - points ) of the pencil by blowing - up @xmath15 times as follows . we first blow up at the points @xmath88 , @xmath89 , @xmath90 , @xmath91 . let @xmath97 , @xmath98 , @xmath99 , @xmath100 be the exceptional divisors over @xmath88 , @xmath89 , @xmath90 , @xmath91 , respectively . we blow up again at the three points @xmath101 , @xmath102 , @xmath103 . let @xmath104 , @xmath105 , @xmath106 be the exceptional divisors over the intersection points , respectively . we finally blow up at each intersection points @xmath107 and @xmath108 . let @xmath109 and @xmath110 be the exceptional divisors over the blown - up points . we then get a rational elliptic surface @xmath111 over @xmath58 ; see figure [ figure : e ] . the four exceptional curves @xmath100 , @xmath106 , @xmath109 , @xmath110 are sections of the elliptic fibration @xmath75 , which correspond to the four base points @xmath91 , @xmath90 , @xmath88 , @xmath89 , respectively . the elliptic fibration @xmath75 has one @xmath76-singular fiber @xmath112 containing all @xmath113 ( @xmath114 ) : @xmath115 , @xmath116 , and @xmath117 ; cf . figure [ figure : e ] . we will use frequently the sum @xmath118 , which will be shown to be @xmath5-divisible . the surface @xmath75 has also one @xmath77-singular fiber consisting of @xmath81 and @xmath83 , and it has two more nodal singular fibers @xmath93 and @xmath94 . ] there is a special bisection on @xmath75 . let @xmath119 be the line in @xmath82 passing through the point @xmath120 $ ] and the two nodes @xmath95 $ ] and @xmath96 $ ] of @xmath93 and @xmath94 . since @xmath119 meets every member in the pencil at three points but it passes through only one base point , the proper transform of @xmath119 is a bisection of the elliptic fibration @xmath121 ; cf . figure [ figure : e ] . note that @xmath122 in @xmath75 . let @xmath123 be the class of the pull - back of a line in @xmath82 . we denote again by @xmath124 the class of the pull - back of the exceptional divisor @xmath125 . we have the following linear equivalences of divisors in @xmath75 : @xmath126 let @xmath127 . note that the divisor @xmath118 is @xmath5-divisible because of the relation @xmath128 in the construction of @xmath130 , we use only one section @xmath131 . we first blow up at the two nodes of the nodal singular fibers @xmath93 and @xmath94 so that we obtain a blown - up rational elliptic surface @xmath132 ; figure [ figure : w ] . let @xmath133 and @xmath134 be the exceptional curves over the nodes of @xmath93 and @xmath94 , respectively . we further blow up at each three marked points @xmath135 in figure [ figure : w ] , and we blow up twice at the marked point @xmath136 ( that is , we first blow - up @xmath136 and then again on the intersection point of the section and the exceptional curve ; see figure [ figure : z ] ) . we then get @xmath137 as in figure [ figure : z ] . there exist two linear chains of the @xmath58 in @xmath130 whose dual graphs are given by : @xmath138 where @xmath139 consists of @xmath81 , @xmath8 , @xmath93 , @xmath133 , and @xmath140 contains @xmath94 , @xmath134 , @xmath119 . ] ] we construct rational singular surfaces which produce under @xmath3-gorenstein smoothings simply connected surfaces of general type with @xmath0 and @xmath2 . we first contract the two chains @xmath139 and @xmath140 of @xmath58 s from the surface @xmath130 so that we have a normal projective surface @xmath69 with two singularities @xmath141 , @xmath142 of class @xmath61 : @xmath143 . denote the contraction morphism by @xmath144 . let @xmath145 be the surface obtained by contracting the four @xmath41-curves @xmath146 in @xmath69 . we denote the contraction morphism by @xmath147 . then @xmath145 is also a normal projective surface with singularities @xmath141 , @xmath142 from @xmath69 , and four @xmath148 s ( ordinary double points ) , denoted by @xmath149 . we finally contract @xmath139 , @xmath140 , @xmath150 and @xmath151 in @xmath130 to obtain @xmath42 . it has the singularities @xmath152 , and two @xmath153 s . let @xmath154 be the contraction . in section [ section : obstruction ] we will prove that the obstruction spaces to local - to - global deformations of the singular surfaces @xmath69 , @xmath145 and @xmath42 vanish . that is : [ theorem : h2(ty)=0 ] @xmath155 , @xmath156 , and @xmath157 . the singular surface @xmath42 is the stable model of the singular surfaces @xmath69 and @xmath145 : [ propsotion : stable - godeaux ] the surface @xmath42 has @xmath158 , @xmath159 , and @xmath73 is ample . the space @xmath74 is smooth and @xmath43 dimensional . a @xmath3-gorenstein smoothing of @xmath42 is a simply connected canonical surface of general type with @xmath0 and @xmath2 . for a surface @xmath42 with only singularities of type @xmath61 we have @xmath160 where @xmath161 is the number of exceptional curves over @xmath88 and @xmath162 is the milnor number of @xmath88 . in our case , @xmath163 . we have @xmath164 because of the rationality of the singularities . we now compute @xmath165 in a @xmath3-numerically effective way . let @xmath166 be the general fiber of the elliptic fibration in @xmath130 . let @xmath167 , @xmath168 , @xmath169 be the exceptional curves over @xmath170 , @xmath171 , @xmath172 respectively . let @xmath173 , @xmath174 be the exceptional curves over @xmath175 with @xmath176 . then , @xmath177 . we also have @xmath178 . writing @xmath179 in @xmath180 and adding the discrepancies from @xmath141 and @xmath142 , we obtain @xmath181 we now intersect @xmath165 with all the curves in its support , which are not contracted by @xmath182 , to check that @xmath73 is nef . moreover , if @xmath183 for a curve @xmath184 not contracted by @xmath182 , then @xmath184 is a component of a fiber in the elliptic fibration which does not intersect any curve in the support . this is because @xmath93 , @xmath133 , @xmath167 , and @xmath168 belong to the support , and they are the components of a fiber . one easily checks that @xmath184 does not exist , proving that @xmath73 is ample . therefore any @xmath3-gorenstein smoothing of @xmath42 over a ( small ) disk will produce canonical surfaces ; cf . kollr - mori @xcite . to compute the fundamental group of a @xmath3-gorenstein smoothing we use the recipe in y. lee - j . we follow the argument as in y. lee - j . consider the normal circles around @xmath119 and @xmath133 . we can compare them through the transversal sphere @xmath167 . since the orders of the circles are @xmath185 and @xmath186 , which are coprime , we obtain that both end up being trivial . the smoothness of @xmath74 follows from theorem [ theorem : h2(ty)=0 ] and hacking @xcite . to compute the dimension , we observe that if @xmath187 is a @xmath3-gorenstein smoothing of @xmath188 and @xmath189 is the dual of @xmath190 , then @xmath189 restricts to @xmath191 as @xmath192 ( tangent bundle of @xmath191 ) when @xmath193 , and @xmath194 with cokernel supported at the singular points of @xmath195 ; cf . wahl @xcite . then the flatness of @xmath189 and semicontinuity in cohomology plus the fact that @xmath196 gives @xmath197 for any @xmath198 . but then , since @xmath191 is of general type , hirzebruch - riemann - roch theorem says @xmath199 this proves the claim . [ corollary : godeaux ] there is a two dimensional family of simply connected canonical numerical godeaux surfaces with two @xmath70 singularities . [ corollary : six ] we consider the sequence @xmath200 at the end of hacking @xcite . we just proved that @xmath201 is @xmath43 dimensional , and we know that @xmath202 is @xmath43 dimensional , since each @xmath70 gives @xmath35 dimensions and each @xmath203 gives @xmath34 dimension . therefore @xmath204 . to produce the claimed family we need to smooth up at the same time @xmath141 and @xmath142 . a simply connected numerical godeaux surfaces with a @xmath5-divisible divisor consisting of four disjoint @xmath41-curves is obtained from a @xmath3-gorenstein smoothing of the singular surface @xmath69 : [ theorem : q - smoothing - of - y ] a. there is a @xmath3-gorenstein smoothing @xmath205 over a disk @xmath206 with central fiber @xmath207 and an effective divisor @xmath208 such that the restriction to a fiber @xmath209 over @xmath210 @xmath211 is @xmath5-divisible in @xmath209 consisting of four disjoint @xmath41-curves and @xmath212 . b. there is a @xmath3-gorenstein deformation @xmath213 of @xmath145 with central fiber @xmath214 such that a fiber @xmath215 over @xmath216 has four ordinary double points as its only singularities and the minimal resolution of @xmath215 is the corresponding fiber @xmath209 of @xmath217 . we apply a similar method in y. lee - j . park @xcite . since any local deformations of the singularities of @xmath145 can be globalized by theorem [ theorem : h2(ty)=0 ] , there are @xmath3-gorenstein deformations of @xmath145 over a disk @xmath206 which keep all four ordinary double points and smooth up @xmath141 and @xmath142 . let @xmath218 be such deformation , with @xmath145 as its central fiber , and @xmath219 ( @xmath216 ) a normal projective surface with four @xmath148s as its only singularities . we resolve simultaneously these four singularities in each fiber @xmath219 . we then get a family @xmath217 that is a @xmath3-gorenstein smoothing of the central fiber @xmath69 , which shows that a @xmath3-gorenstein smoothing of @xmath145 can be lifted to a @xmath3-gorenstein smoothing of the pair @xmath220 , i.e. the @xmath5-divisible divisor @xmath221 on @xmath69 is extended to an effective divisor @xmath208 . we finally show that the effective divisor @xmath222 is 2-divisible in @xmath209 for @xmath216 . according to manetti ( * ? ? ? * lemma 2 ) , the natural restriction map @xmath223 is injective for every @xmath210 and bijective for @xmath224 . here we are using that @xmath225 . since the divisor @xmath221 is nonsingular , @xmath226 is also nonsingular . since @xmath227 in , it follows that @xmath228 , where @xmath229 is extended to a line bundle @xmath230 and @xmath231 is the corresponding restriction . the main purpose of this section is to construct simply connected numerical campedelli surfaces with an involution . along the way , we will introduce a rational normal projective surface @xmath44 with @xmath45 singularities ( two @xmath148 , two @xmath232 , and two @xmath72 ) and @xmath233 ample . a certain four dimensional @xmath3-gorenstein deformation of @xmath44 will produce numerical campedelli surfaces with an involution . recall that the rational surface @xmath130 has a @xmath5-divisible divisor @xmath118 ; cf . . let @xmath234 be the double cover of @xmath130 branched along the divisor @xmath221 , where the double cover is given by the data @xmath227 , @xmath235 . we denote the double covering by @xmath236 . the surface @xmath234 has two @xmath139 s and two @xmath140 s . on the other hand the surface @xmath234 can be obtained from a certain rational elliptic surface by blowing - ups , as we now explain . the morphism @xmath237 blows down to a double cover @xmath238 branched along @xmath221 . the ramification divisor @xmath239 consists of four disjoint @xmath57-curves @xmath240 . we blow down them from @xmath241 to obtain a surface @xmath242 ; cf . figure [ figure : e ] . in figure [ figure : e ] the pull - back of the @xmath243 are the @xmath244 , of the curve @xmath81 is @xmath245 , of the section @xmath8 is @xmath246 , and of the double section @xmath119 is @xmath247 . each @xmath77 in @xmath242 is the pull - back of each @xmath248 in @xmath75 . ] note that the surface @xmath242 has an elliptic fibration structure with two @xmath249-singular fibers and two @xmath77-singular fibers . in fact , the surface @xmath242 can be obtained from the pencil of cubics in @xmath82 @xmath250 \in \mathbb{cp}^1 \}\ ] ] where the @xmath249-singular fibers come from @xmath251 and @xmath252 , and the @xmath77-singular fibers come from @xmath253 and @xmath254 . the two double sections @xmath255 and @xmath256 are defined by the lines @xmath257 and @xmath258 . in summary : [ proposition : campexplicit ] the surface @xmath241 is the blow - up at four nodes of one @xmath249-singular fiber of the rational elliptic fibration @xmath259 . hence the surface @xmath234 can be obtained from @xmath241 by blowing - up in the obvious way . let @xmath260 be the double cover of the singular surface @xmath69 branched along the divisor @xmath221 . note that the surface @xmath260 is a normal projective surface with four singularities of class @xmath61 whose resolution graphs consist of two @xmath139 s and two @xmath140 s . the ramification divisor in @xmath260 consists of the four disjoint @xmath57-curves @xmath261 . let @xmath262 be the double covering . on the other hand the surface @xmath260 can be obtained from the rational surface @xmath234 by contracting the two @xmath139 s and two @xmath140 s . let @xmath263 be the contraction morphism . let @xmath6 be the surface obtained by blowing down the four @xmath57-curves @xmath240 from @xmath260 . we denote the blowing - down morphism by @xmath264 . then there is a double covering @xmath265 branched along the four ordinary double points @xmath149 . finally , let @xmath44 be the contraction of the @xmath41-curves @xmath266 and @xmath267 in @xmath6 . let @xmath268 be the contraction . we then get a double covering @xmath269 . to sum up , we have the following commutative diagram : @xmath270 \ar[d]^{\psi ' } & v \ar[l ] \ar[d]^{\psi } \ar[r]^{\widetilde{\beta } } & \widetilde{x } \ar[d]^{\widetilde{\phi } } \ar[r]^{\beta } & x \ar[d]^{\phi } \ar[r ] & x ' \ar[d]\\ & e(1 ) & \ar[l ] z \ar[r]^{\widetilde{\alpha } } & \widetilde{y } \ar[r]^{\alpha } & y \ar[r ] & y'}\ ] ] we will show in section [ section : obstruction ] the obstruction spaces to local - to - global deformations of the singular surfaces @xmath260 , @xmath6 and @xmath44 vanish : [ theorem : h2(tx)=0 ] @xmath271 , @xmath272 , and @xmath273 . the singular surface @xmath44 is the stable model of @xmath260 and @xmath6 : [ proposition : stable - campedelli ] the surface @xmath44 has @xmath274 , @xmath275 , and @xmath233 ample . the space @xmath276 is smooth and @xmath45 dimensional . a @xmath3-gorenstein smoothing of @xmath44 is a simply connected canonical surface of general type with @xmath0 and @xmath1 . the proof goes as the one for @xmath42 in proposition [ propsotion : stable - godeaux ] , using the explicit model we have for @xmath234 by blowing - up @xmath242 in proposition [ proposition : campexplicit ] . one can check that an intersection computation as in proposition [ propsotion : stable - godeaux ] verifies ampleness for @xmath233 . the proof of the next main result follows easily from theorem [ theorem : q - smoothing - of - y ] . [ theorem : campedelli ] there exist @xmath3-gorenstein smoothings @xmath277 of @xmath260 and @xmath278 of @xmath6 that are compatible with the @xmath3-gorenstein deformations of @xmath279 of @xmath69 and @xmath218 of @xmath145 in theorem [ theorem : q - smoothing - of - y ] , respectively ; that is , the double coverings @xmath280 and @xmath265 extend to the double coverings @xmath281 and @xmath282 between the fibers of the @xmath3-gorenstein deformations . by theorem [ theorem : h2(tx)=0 ] , the obstruction @xmath283 to local - to - global deformations of the singular surface @xmath6 vanishes . the point of the above theorem is that there is a @xmath3-gorenstein smoothing of the cover @xmath6 that is compatible with the @xmath3-gorenstein deformation of the base @xmath145 . [ corollary : campedelli - with - involution ] a general fiber @xmath32 of the @xmath3-gorenstein smoothing @xmath278 of @xmath6 is a simply connected numerical campedelli surface with an involution @xmath33 such that the minimal resolution of the quotient @xmath284 is a simply connected numerical godeaux surface . let @xmath6 be a minimal complex surface of general type with @xmath0 and @xmath1 . suppose that the group @xmath4 acts on @xmath6 with just @xmath28 fixed points . let @xmath285 be the quotient and let @xmath286 be the minimal resolution of @xmath145 . barlow ( * ? ? ? * proposition 1.3 ) proved that if @xmath287 then @xmath288 . conversely : [ proposition : simply - connected=>simply - connected ] if @xmath6 is simply connected , then so is @xmath8 . let @xmath289 be the quotient map . then @xmath290 is a double covering which is branched along the four ordinary double points of @xmath145 . let @xmath291 be the complement of the four branch points ( i.e. , the four @xmath148-singularities ) of @xmath145 and let @xmath292 , that is , @xmath293 is the complement of the four fixed points of the involution @xmath7 . then we get an tale double covering @xmath294 . since @xmath295 , we have @xmath296 . note that the boundary @xmath297 of an arbitrary small neighborhood @xmath298 of one of the four nodes of @xmath145 is a lens space @xmath299 . let @xmath300 $ ] be a generator of @xmath301 represented by a loop @xmath302 contained in @xmath297 . since the lifting of @xmath302 by the covering @xmath294 is not a closed path and @xmath303 , @xmath304 is generated by @xmath300 $ ] . then it follows by van kampen theorem that @xmath305 is trivial . hence @xmath306 is trivial because @xmath8 is obtained from @xmath145 by resolving only four @xmath148-singularities . in this section we prove theorem [ theorem : h2(ty)=0 ] which says that the obstruction spaces to local - to - global deformations of the singular surfaces @xmath69 , @xmath145 , and @xmath42 vanish . that is , we will prove that @xmath307 . at the end , we also prove the analogues , theorem [ theorem : h2(tx)=0 ] , for @xmath260 , @xmath6 , and @xmath44 . at first the vanishing of the obstruction spaces of a singular surface can be proved by the vanishing of the second cohomologies of a certain logarithmic tangent sheaf on the minimal resolution of the singular surface : [ proposition : lee - park - log ] if @xmath308 be the minimal resolution of a normal projective surface @xmath8 with only quotient singularities , and @xmath309 is the reduced exceptional divisor of the resolution @xmath310 , then @xmath311 . [ proposition : flenner - zaidenberg ] let @xmath61 be a nonsingular surface and let @xmath309 be a simple normal crossing divisor in @xmath61 . let @xmath312 be the blow - up of @xmath61 at a point @xmath88 of @xmath309 . let @xmath313 . then @xmath314 . we can add or remove disjoint @xmath57-curves . [ add - delete ] let @xmath61 be a nonsingular surface and let @xmath309 be a simple normal crossing divisor in @xmath61 . let @xmath315 be a @xmath57-curve in @xmath61 such that @xmath316 is again simple normal crossing . then @xmath317 . we can also add or remove disjoint exceptional divisors of rational double points . the following theorem may give a new general way to prove unobstructedness for deformations of surfaces . [ theorem : burns - wahl ] let @xmath8 be a normal projective surface with only rational double points @xmath318 as singularities . let @xmath319 be the minimal resolution of @xmath8 with exceptional reduced divisor @xmath320 . let @xmath321 be a simple normal crossing divisor such that @xmath322 . then @xmath323 . let @xmath324 and @xmath325 be the prime decompositions of @xmath119 and @xmath321 . we have three short exact sequences : @xmath326 we then have the following commutative diagram of cohomologies : @xmath327 & 0 \ar[d ] & & \\ 0 \ar[r ] & h^1(\sheaf{t_{\widetilde{s}}}(-\log(c+m ) ) ) \ar[r ] \ar[d ] & h^1(\sheaf{t_{\widetilde{s}}}(-\log{c } ) ) \ar[r]^{\phi } \ar[d]^{\xi } & \oplus h^1(\sheaf{n_{m_i/{\widetilde{s } } } } ) \ar@{=}[d ] & \\ 0 \ar[r ] & h^1(\sheaf{t_{\widetilde{s}}}(-\log{m } ) ) \ar[r ] \ar[d ] & h^1(\sheaf{t_{\widetilde{s } } } ) \ar[r]^{\psi } \ar[d]^{\zeta } & \oplus h^1(\sheaf{n_{m_i/{\widetilde{s } } } } ) \ar[r ] & 0\\ & \oplus h^1(\sheaf{n_{c_i/{\widetilde{s } } } } ) \ar@{=}[r ] & \oplus h^1(\sheaf{n_{c_i/{\widetilde{s } } } } ) & & } } \ ] ] here all horizontal and vertical sequences are exact . especially the second row is a short exact sequence , which we explain now briefly : it is shown in burns - wahl @xcite ( see also wahl @xcite ) that the composition @xmath328 is an isomorphism because the @xmath329 s are _ rational double points _ ; hence , one has a direct sum decomposition @xmath330 and an isomorphism @xmath331 . therefore the second row is exact . in order to prove the assertion , it is enough to show that @xmath332 is surjective . let @xmath333 . since @xmath334 is surjective , we have @xmath335 for some @xmath336 . by we have @xmath337 for some @xmath338 and @xmath339 such that @xmath182 is mapped to @xmath340 under the composition @xmath341 . since @xmath182 is supported on @xmath119 and @xmath342 , its image @xmath343 under @xmath344 vanishes . therefore @xmath345 , and so @xmath346 for some @xmath347 ; hence , @xmath348 , which shows that @xmath349 is surjective . [ proposition : e - v - sequence ] let @xmath350 be a simple normal crossing divisor on a smooth surface @xmath61 . then one has the following exact sequences : a. @xmath351 . b. @xmath352 . we first claim that @xmath353 by duality , we have to show that @xmath354 since @xmath355 , it follows by proposition [ proposition : e - v - sequence ] that @xmath356 hence it suffices to show that @xmath357 . on the other hand , we obtain a long exact sequence from proposition [ proposition : e - v - sequence ] : @xmath358 since @xmath359 , it is enough to show that the connecting homomorphism @xmath360 is injective . note the map @xmath360 is the first chern class map . but @xmath93 and @xmath133 are linearly independent in the picard group of @xmath361 ; hence , the map @xmath360 is injective . therefore the claim follows . let @xmath362 . by theorem [ theorem : burns - wahl ] we have @xmath363 we use propositions [ add - delete ] and [ proposition : flenner - zaidenberg ] to obtain @xmath364 where @xmath365 . in this way , it follows by the above claim that @xmath366 then , by proposition [ proposition : lee - park - log ] , we have @xmath367 . notice we can modify @xmath368 to obtain vanishing for @xmath369 and @xmath370 as well . we now prove that @xmath371 . we will use our explicit model of @xmath44 in proposition [ proposition : campexplicit ] . the proof goes along the same lines as the proof of the above theorem [ theorem : h2(ty)=0 ] . we may only need to mention that we start with the elliptic fibration @xmath242 , and @xmath5 @xmath77 fibers ( instead of @xmath5 @xmath248 s ) . the involution of a general fiber @xmath32 induced by the double covering @xmath282 extends to a @xmath4-action on the deformation space of @xmath32 . we will count the dimension of the subspace of @xmath372 which is fixed by the @xmath4-action ; theorem [ theorem : invariant - part ] . let @xmath373 be the minimal resolution and let @xmath374 be the blowing - up at the four ramification points . we then have the following commutative diagram where the vertical morphisms are double covers : @xmath375^{\beta_t } \ar[d]_{\widetilde{\phi}_t } & x_t \ar[d]^{\phi_t } \\ \widetilde{y}_t \ar[r]^{\alpha_t } & y_t } \ ] ] recall that the branch divisor @xmath222 of the double covering @xmath376 consists of four disjoint @xmath41-curves @xmath377 and the corresponding ramification divisor @xmath378 consists of four disjoint @xmath57-curves @xmath379 . as before the involution of @xmath380 induced by the double covering @xmath281 extends to a @xmath4-action on the deformation space of @xmath380 . [ lemma : invariant ] @xmath381 . by pardini ( * lemma 4.2 ) , the invariant part of @xmath382 under the @xmath4-action is @xmath383 . therefore we have @xmath384 we know that @xmath385 , where @xmath386 defines the double cover @xmath387 . therefore @xmath388 by theorem [ proposition : stable - campedelli ] . then we have @xmath389 . hence , by proposition [ proposition : e - v - sequence ] , there is a short exact sequence @xmath390 since @xmath222 consists of four disjoint @xmath41-curves , we have @xmath391 . by proposition [ propsotion : stable - godeaux ] , we know that @xmath392 . therefore @xmath393 . [ theorem : invariant - part ] the subspace of the deformation space of @xmath32 invariant under the @xmath4-action is four dimensional . we apply a similar strategy in werner @xcite . we have the exact sequence @xmath394 since each @xmath395 is a @xmath57-curve , we have @xmath396 . on the other hand , it follows by catanese ( * ? ? ? * lemma 9.22 ) that @xmath397 where @xmath398 is the ideal sheaf of the four points in @xmath32 obtained by contacting the exceptional divisors @xmath399 . let @xmath400 be the set of these four points . from the ideal sequence , we have @xmath401 therefore the invariant parts of each space satisfies : @xmath402 according to werner @xcite , we have @xmath403 . therefore it follows by and lemma [ lemma : invariant ] that @xmath404 we briefly describe another rational surface @xmath130 which makes it possible to construct simply connected numerical campedelli surfaces with an involution as before . the associated godeaux surfaces come from a rational surface @xmath42 with @xmath73 ample having three @xmath148-singularities , one @xmath70-singularity , and only one singularity of class @xmath61 . the elliptic fibration @xmath75 is the one in section [ section : godeaux ] . in the construction of @xmath130 , we will use the sections @xmath100 , @xmath106 , @xmath109 among the four sections of @xmath75 . we denote the sections @xmath100 , @xmath106 , @xmath109 by @xmath406 , @xmath407 , @xmath408 , respectively . we first blow up at the two nodes of the nodal singular fibers @xmath93 and @xmath94 so that we obtain a blown - up rational elliptic surface @xmath409 ; figure [ figure : w-2 ] . let @xmath133 and @xmath134 be the exceptional curves over the nodes of @xmath93 and @xmath94 , respectively . we further blow up at each two marked points @xmath135 and blow up four times at the marked point @xmath136 in figure [ figure : w-2 ] . we then get a rational surface @xmath410 ; figure [ figure : z-2 ] . there exists one linear chain of @xmath58s in @xmath130 whose dual graph is @xmath411 notice that the @xmath57-curve @xmath408 is contracted in the way down , which fixes the configuration so that we obtain one singular point of class @xmath61 whose resolution graph is given by @xmath412 the divisor @xmath413 is the @xmath5-divisible one as before . the @xmath414 and the @xmath41-curves @xmath415 , @xmath416 , @xmath151 , and @xmath119 are contracted to obtain a singular surface @xmath42 . one can use again theorem [ theorem : burns - wahl ] to show that the space @xmath74 is smooth ( of dimension @xmath43 ) .
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we construct a simply connected minimal complex surface of general type with @xmath0 and @xmath1 which has an involution such that the minimal resolution of the quotient by the involution is a simply connected minimal complex surface of general type with @xmath0 and @xmath2 . in order to construct the example
, we combine a double covering and @xmath3-gorenstein deformation . especially , we develop a method for proving unobstructedness for deformations of a singular surface by generalizing a result of burns and wahl which characterizes the space of first order deformations of a singular surface with only rational double points .
we describe the stable model in the sense of kollr and shepherd - barron of the singular surfaces used for constructing the example .
we count the dimension of the invariant part of the deformation space of the example under the induced @xmath4-action .
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the level of the sun s magnetic activity is observed to vary on an 11-year time scale and we are currently in the minimum between cycles 23 and 24 . the current solar minimum is attracting a great deal of attention as it is proving to be quite unusual . observations of surface and atmospheric effects , such as the number of visible sunspots , the rate of occurrence of solar flares and the strength of the solar wind , highlight just how quiet the sun is . the sun s activity cycle influences everyday life on the earth . the rate of occurrence of solar flares is dependent on the number of spots on the sun s surface , and large solar flares can disrupt satellite communications and cause power outages . coronal mass ejections ( cmes ) , which are another source of radiation that can disrupt life on the earth , are still being observed regularly on the sun , even in this unusual solar minimum ( see the stereo cor1 cme catalog ) . cosmic rays , which are a significant space radiation hazard , are anticorrelated with the solar cycle . the level of solar activity is possibly correlated to the earth s climate ( see , for example , * ? ? ? * and references therein ) . space - weather groups use predictions of solar cycles to anticipate the amount of orbital drag experienced by satellites . the next solar cycle has already proven difficult to predict as the current solar minimum is lasting significantly longer than expected . @xcite reviews 50 predictions for the size and timing of cycle 24 and finds a wide range of results , especially in comparison to predictions made before the previous solar cycle @xcite . for example , predictions of when cycle 24 will reach its maximum range from 2009 december @xcite to 2014 december @xcite . in fact , the official noaa , nasa and ises solar cycle 24 prediction panel , which studied the predictions collated by @xcite , failed to reach a consensus on when the peak of cycle 24 will occur and how active the upcoming cycle will be . meanwhile , the number of predictions for cycle 24 is ever increasing . surface measures of the sun s activity , such as the number of sunspots , which are used to aid cycle predictions , indicate that we are still in an extended solar - cycle minimum . we ask the question : can we learn anything about this unusual solar minimum from the sun s interior ? to answer this question we investigate the variation with the solar cycle of the frequencies of the sun s natural resonant oscillations , which are known as @xmath0 modes . solar @xmath0 modes are trapped in cavities below the surface of the sun and their frequencies are sensitive to properties , such as temperature and mean molecular weight , of the solar material . it has been known since the mid 1980s @xcite that @xmath0-mode frequencies vary throughout the solar cycle with the frequencies being at their largest when the solar activity is at its maximum . by examining the changes in the observed @xmath0-mode frequencies throughout the solar cycle , we can learn about solar - cycle - related processes that occur beneath the sun s surface . the birmingham solar - oscillations network ( bison ; * ? ? ? * ) makes sun - as - a - star ( unresolved ) doppler velocity observations , which are sensitive to the @xmath0 modes with the largest horizontal scales ( or the lowest angular degrees , @xmath1 ) . consequently , the frequencies measured by bison are of the truly global modes of the sun . these modes travel to the sun s core but their dwell time at the surface is longer than at the solar core because the sound speed inside the sun increases with depth . consequently , the low-@xmath1 modes are most sensitive to variations in regions of the interior that are close to the surface and so are able to give a global picture of the influence of near - surface activity . bison is a network of autonomous ground - based observatories that are strategically positioned at various longitudes in order to provide as continuous coverage as possible of the sun . bison is in a unique position to study the changes in oscillation frequencies that accompany the solar cycle as it has now been collecting data for over 30 years . however , when the network was first established the quality of the data was relatively poor , in comparison to recent years , because of the sporadic coverage of the observations . here , we have been able to analyze the mode frequencies observed during the last two solar cycles in their entirety . the precision with which @xmath0-mode frequencies can be determined is directly related to the length of time series under consideration . consequently , @xmath0-mode frequencies are often determined from time series whose lengths are of the order of years . however , a compromise must be made here regarding the appropriate length of time series for study so that changes in the solar cycle can be resolved . before 1985 april the observed data are sparse because the early bison data were collected from just one or two sites and , initially , just in summer months in the northern hemisphere . therefore , when considering the data collected before 1985 april the @xmath0-mode frequencies were obtained from seven time series of different lengths , which reflected the availability of data . these data are included here for completeness but are not used in the later analysis as the data are too sparse to provide reliable results . after the third bison site , at carnarvon , western australia , was established in 1985 the duty cycle of the bison data increased significantly . therefore , after 1985 april 22 the time series were truncated to 54 days in length , which corresponds to approximately twice the rotation period of the solar surface . the analysis performed here concentrates on the data obtained after this date . a standard likelihood maximization method was used to fit the power spectra of these time series ( e.g. , * ? ? ? * ) , enabling the observed mode frequencies to be determined . we have concentrated on the strongest modes of oscillation , which are in the frequency band from 2100 to 3500@xmath2 . to maintain consistency with previous work ( e.g. , * ? ? ? * ) a minimum activity reference set was determined by averaging the frequencies from time series observed during the minimum activity epoch at the boundary between cycle 22 and cycle 23 . we then defined the solar - cycle frequency shifts , @xmath3 , as the differences between frequencies given in the minimum activity reference set and the frequencies of the corresponding modes observed at different epochs and , consequently , different levels of activity . the size of a frequency shift has a well - known dependency on frequency and mode inertia and these dependencies were removed in the manner described in @xcite . this allowed a weighted average of the frequency shifts observed for each time series to be determined , which provided a mean frequency shift for each epoch , @xmath4 . figure [ figure[flux shifts ] ] shows the frequency - shift data that were observed during cycles 21 , 22 , and 23 . the data are shown up to 2009 april . after 1985 april the data plotted in figure [ figure[flux shifts ] ] have been smoothed over five points , as the mean frequency shifts were obtained from time series of the same length . one of the most obvious manifestations of the sun s solar cycle is the variation in the number of sunspots on the solar surface , formally recorded as the international sunspot number ( isn ) . however , the sunspot number is not the only variable that can be used as a proxy for the solar activity . for example , the radio flux emitted from the sun at a wavelength of 10.7 cm ( hereafter @xmath5 ) also responds to changes in the solar cycle . many authors ( e.g. , * ? ? ? * ; * ? ? ? * and references therein ) have determined and commented upon the good correlations between shifts in the @xmath0-mode frequencies and activity proxies such as the @xmath5 and the isn . therefore , for comparison purposes , the average 10.7 cm flux has been plotted on top of the observed frequency shifts in figure [ figure[flux shifts ] ] . after the first seven points plotted in figure [ figure[flux shifts ] ] the @xmath6 has been smoothed as per the frequency shifts . to determine the scale on which the proxy axis was plotted , a linear least - squares fit between the observed frequency shifts and the activity proxy was performed . the rise and decline of cycle 23 appears to be slow in comparison to cycle 22 . the oscillation frequencies observed in the current unusual solar minimum are significantly lower than the previous minimum in 1996 and the most recent data indicate that we are still on a downward trend . notice that the shapes of the previous and current solar minima are very similar , with both appearing to show a double minimum . this implies that we may still have to wait some time before the rising phase of cycle 24 begins . it is also interesting to note that , if account is taken of the lower quality of the data observed before 1990 , the minimum between cycles 21 and 22 is also deeper than the last solar minimum . this could be because both the minimum between cycles 21 and 22 and the current solar minimum ( between cycles 23 and 24 ) correspond to the same phase of 22-year magnetic hale cycles . it is clearly apparent from figure [ figure[flux shifts ] ] that there is an unusually large difference between the activity proxy and the frequency shifts on the declining phase of cycle 23 . notably , the frequency shifts are still observed to be changing throughout the solar minimum , which is in contrast to the activity proxy data that show little structure in its variation with time since 2006 . + + [ figure[flux proxies ] ] figure [ figure[flux proxies ] ] allows a comparison between the bison frequency shifts and two proxies of the sun s activity , other than the @xmath6 . as noted above the isn is a measure of the number of active sunspot groups and individual spots on the visible disk . the mgii core - to - wing ratio @xcite is determined from space - based uv spectral irradiance measurements . the mgii h and k line cores originate in the chromosphere and the ratio of the core size to that of the more stable background gives a robust indication of the chromospheric activity . the @xmath6 and the mgii core - to - wing ratio appear to show better agreement with the frequency shifts than the isn . this behavior has been observed before @xcite and occurs because the frequency shifts are sensitive to both the strong _ and _ weak components of the sun s magnetic flux . the @xmath5 and the mgii core - to - wing ratio show similar relative sensitivity to the different components of the sun s magnetic flux , whereas , the isn is predominantly sensitive to the strong component of the sun s magnetic field . for the purposes of this letter it is most important to notice that there are large discrepancies between , respectively , all three of the proxies plotted in figures [ figure[flux shifts ] ] and [ figure[flux proxies ] ] and the frequency shifts in the declining phase of cycle 23 . furthermore , the structure that is observed in the frequency shifts in the current solar minimum is not replicated by any of the proxies . it is also important to note that the seismic minimum is decidedly lower than the minimum predicted by the three proxies . the oscillations are sensitive to the conditions beneath the solar surface , whereas the activity proxy is a measure of the magnetic field that is present on the surface . we therefore assume that there are processes happening in the solar interior , which have yet to manifest themselves at the surface . we now turn our attention to the shorter - term variable structure in the frequency shifts visible on top of the general 11-year trend , which has a period of around two years . similar `` quasi - biennial '' variation of activity proxies has been noted before in data from the green coronal emission line at 530.3 nm at high solar activity @xcite . these authors used variations near the solar equator and poles and evidence was found of quasi - biennial variability at both low and high latitudes . an explanation of such quasi - biennial behavior has been put forward in terms of two different types of dynamo operating at different depths @xcite . elsewhere @xcite , it has been possible to identify shorter secondary periods in the activity cycles of some stars and it is conceivable that asteroseismology may be able to detect such effects in data sets which will become available , for example , from the recently launched @xmath7 satellite @xcite . in spite of the known behavior of the proxies , this is the first time that the quasi - biennial variability has been noted in seismic data . furthermore , the signature of the two - year signal in the activity index was previously restricted to times of moderate to high solar activity . for the first time , we are seeing this short - term variability at low solar activity . to investigate this further , a sine wave was fitted to the frequency shifts observed after 1985 april ( see figure [ figure[flux shifts ] ] ) . the sine wave took the form @xmath8 } s(\delta t)=a_0\sin\left(2\pi a_1\delta t+a_2\right)+a_3\delta t+a_4,\ ] ] where @xmath9 is the time in years after 1985 april 22 , which is the start date of the first 54d time series . the @xmath10 coefficients are given in table [ table[a coefficients ] ] . the sine wave was calculated using a weighted least - squares fit to the unsmoothed frequency shifts . in equation [ equation[sine wave ] ] , the first collection of terms accounts for the sine wave structure of the frequency shifts . the second and third terms account for an offset which linearly decreases with time . @xmath10 coefficients used in calculating equation [ equation[sine wave ] ] . [ cols="^,^",options="header " , ] [ table[a coefficients ] ] we have determined the residuals between the best - fitting sine wave and the observed frequency shifts and the results are plotted in figure [ figure[residuals ] ] . flux residuals were also determined between a sine wave , which was scaled in amplitude using the linear fit between the flux and the frequency shifts , and the @xmath5 . clearly , there is a lot of structure in both sets of observed residuals and since @xmath11 the frequency - shift residuals are quite substantial in size . large discrepancies between the two sets of residuals are clearly evident . a periodogram of the data from the last two cycles shows significant periods of about two and three years with greater than 99% confidence , but the phase of the signal is not locked to that of the underlying 11-year cycle and several incidences of phase jumps are evident , meaning that periods are not easy to determine . the three - year periodicity is not visible in a periodogram of the activity proxy residuals . the solar - cycle shifts that are observed in @xmath0-mode frequencies are usually well correlated with proxies of the sun s activity , such as the 10.7 cm radio flux . however , in the declining phase of cycle 23 and the current solar minimum we find unusually large differences between the frequencies observed in bison data and the activity levels . the current cycle minimum indicated by the helioseismic data is significantly deeper than the minima observed by the activity proxies and the structure that is clearly evident in the frequency shifts is not replicated in the proxy data . we also observe a quasi - biennial signal in the @xmath0-mode frequencies at _ all _ activity levels . interestingly , this signal is _ only _ visible at _ high_-activity levels in the proxy data . as the frequency shifts respond to conditions beneath the surface of the sun whereas the proxies respond to changes at or above the surface , we suggest that these differences may be caused by changes in the magnetic flux that have yet to manifest at the surface . it is also possible that the magnetic flux responsible for the discrepancies between the frequency shifts and the activity proxies will never reach the sun s surface . the analysis presented here was based on averages made over groups of modes ( @xmath12 , @xmath13 ) . further work on individual modes may allow one to isolate the location of the variability because each mode shows a different sensitivity to the latitudinal distribution of the surface activity . such an analysis is currently in progress . in the seismic data , the previous solar minimum exhibited a double minimum and it appears that the current solar minimum is showing a similar structure . furthermore , the most recent @xmath0-mode frequencies indicate that the current minimum is still declining . therefore , it may still be some time before we observe the rising phase of cycle 24 . there have been suggestions that this recent strange behavior of the sun is indicative that the current grand maximum is about to end @xcite . that would indeed be an occurrence of great significance . although our results can not predict whether this is true they do indicate that the next solar cycle should be observed with great interest . this letter utilizes data collected by the birmingham solar - oscillations network ( bison ) . we thank the members of the bison team , both past and present , for their technical and analytical support . we also thank p. whitelock and p. fourie at the south african astronomical observatory ( saao ) , the carnegie institution of washington , the australia telescope national facility ( australian commonwealth scientific and research organization , csiro ) , e.j . rhodes ( mt . wilson , californa ) and members ( past and present ) of the instituto de astrofisica de canarias ( iac ) , tenerife . bison is funded by the science and technology facilities council ( stfc ) . the authors also acknowledge the financial support of stfc . abreu , j.a . , beer , j. , steinhiber , f. , tobias , s.m . & weiss , n.o . , 2008 , , 35 , 20109 benevolonskaya , e. e. , 1998a , , 509l , 49 benevolonskaya , e. e. , 1998b , , 181 , 479 chaplin , w. j. , elsworth , y. , isaak , g. r. , miller , b. a. , & new , r. , 1999 , , 308 , 424 chaplin , w. j. , elsworth , y. , miller , b. a. , verner , g.a . & new , r. , 2007 , , 659 , 1749 chaplin , w. j. et al . , 1996 , , 168 , 1 christensen - dalsgaard et al . , 2008 , commun . in asteroseismol . , 157 , 266 elsworth , y. et al . , 1990 , , 345 , 536 howe , r. , 2008 , adv . space res . , 41 , 846 joselyn , j. a. et al . , 1997 , eos trans . , 78 , 205 lockwood , m. & frhlich , c. , 2007 , proc . r. s. , 463 , 2447 maris , g. & oncica , a. , 2006 , sun geosphere , 1 , 1 mccomas , d. j. et al . , 2008 , , 35 , 18103 pesnell , w. d. , 2008 , , 252 , 209 saar , s. h. , & brandenburg , a. , 2002 , astron . nachr . , 323 , 357 tsirulnik , l. b. , kuznetsova , t. v. & oraevsky , v. n. , 1997 , adv . s. res . , 20 , 2369 vecchio , a. & carbone , v. 2008 , , 683 , 536 viereck , r. et al . , 2001 , , 28 , 1343 woodard , m. f. & noyes , r. w. 1985 , , 318 , 449
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the sun is a variable star whose magnetic activity and total irradiance vary on a timescale of approximately 11 years .
the current activity minimum has attracted considerable interest because of its unusual duration and depth .
this raises the question : what might be happening beneath the surface where the magnetic activity ultimately originates ?
the surface activity can be linked to the conditions in the solar interior by the observation and analysis of the frequencies of the sun s natural seismic modes of oscillation - the @xmath0 modes .
these seismic frequencies respond to changes in activity and are probes of conditions within the sun .
the birmingham solar - oscillations network ( bison ) has made measurements of @xmath0-mode frequencies over the last three solar activity cycles , and so is in a unique position to explore the current unusual and extended solar minimum .
we show that the bison data reveal significant variations of the @xmath0-mode frequencies during the current minimum .
this is in marked contrast to the surface activity observations , which show little variation over the same period .
the level of the minimum is significantly deeper in the @xmath0-mode frequencies than in the surface observations .
we observe a quasi - biennial signal in the @xmath0-mode frequencies , which has not previously been observed at mid- and low - activity levels .
the stark differences in the behavior of the frequencies and the surface activity measures point to activity - related processes occurring in the solar interior , which are yet to reach the surface , where they may be attenuated .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the results reported here derive from a numerical analysis of orbits in the triaxial dehnen potentials , where ( _ cf . _ merritt & fridman 1996 ) @xmath0 with @xmath1 assuming fixed axis ratios @xmath2 and @xmath3 but allowing for a variable cusp index @xmath4 and a variable black hole mass @xmath5 . different segments of the same chaotic orbit can be extremely different in terms of their visual appearance and their degree of exponential sensitivity ( siopis & kandrup 2000 ) . these differences can be quantified in terms of the sizes of short time lyapunov exponents ( _ cf . _ kandrup & mahon 1994 ) or the _ complexity _ of their fourier spectra , _ i.e. , _ the degree to which the power in an orbit is concentrated near a few special frequencies ( _ cf . _ kandrup , eckstein , & bradley 1997 , siopis , eckstein , & kandrup 1998 ) . chaotic orbits typically have continuous spectra , but when they look ` nearly regular ' most of the power is concentrated near a few special frequencies . in agreement with intuition , there is a strong correlation between the complexity of an orbit segment and the value of its largest short time lyapunov exponent : chaotic orbit segments which look ` nearly regular ' and have less complex spectra tend also to exhibit comparatively small exponential sensitivity . if two chaotic orbits in the same connected phase space region are integrated for a sufficiently long time , it appears that they will eventually share the same statistical properties . however , the time required for this can be extremely long . if , for example , a single chaotic initial condition is integrated into the future , it can take as long as @xmath6 dynamical times @xmath7 , or even longer , before the short time lyapunov exponent exhibits a reasonable convergence towards the true lyapunov exponent @xmath8 , as defined in a @xmath9 limit . the overall rate of convergence can be quantified through an examination of distributions of short time lyapunov exponents , @xmath10 $ ] , generated for ensembles of chaotic orbit segments of varying length @xmath11 . in the absence of any significant stickiness , the dispersion associated with @xmath12 $ ] scales as @xmath13 with @xmath14 . for very sticky orbits , @xmath15 . because chaotic orbits are so sticky , one might anticipate that they could be used as building blocks for the construction of self - consistent near - equilibria which , albeit not strictly time - independent , behave as nearly time - independent entities over time intervals long compared with @xmath16 , the age of the universe . ( in the language of merritt & fridman [ 1996 ] , these would be ` quasi - equilibria ' involving stochastic building blocks that are only ` partially mixed . ' ) however , this supposition relies crucially on the assumption that the statistical properties of chaotic orbit segments are relatively insensitive to the effects of weak perturbations of the form which act on real galaxies . in point of fact , this does not appear to be the case . orbits in the unperturbed triaxial dehnen potential were perturbed to mimic various effects to which real stars in real galaxies are typically exposed . discreteness effects , _ i.e. , _ gravitational rutherford scattering between individual stars , were modeled as dynamical friction and white noise , _ i.e. , _ near - instantaneous kicks . the effects of one or two companion objects or satellite galaxies were modeled as nearly periodic perturbations . the effects of a dense cluster environment were modeled as coloured noise , _ i.e. , _ random kicks of finite duration . internal oscillations of the form that might , _ e.g. , _ be triggered by a close encounter were treated as a superposition of normal , or pseuo - normal , modes that induced a periodic driving and an incoherent combination of more irregular excitations modeled as coloured noise . the basic conclusion of this investigation ( siopis & kandrup 2000 , kandrup & siopis 2000 ) , consistent also with analyses of motions in other two- and three - dimensional potentials ( pogorelov & kandrup 1999 , kandrup , pogorelov , & siopis 2000 ) , is that low amplitude irregularities can have a surprisingly large effect both @xmath17 by accelerating diffusion within a given nearly disjoint phase space region ; and @xmath17 by accelerating diffusion along an arnold web or through cantori connecting nearly disjoint chaotic phase space regions . some of the topological obstructions associated with the arnold web are extremely robust , so that weak perturbations have a comparatively minimal effect . however , in general such perturbations tend to accelerate dramatically the rate of phase space transport throughout the entire chaotic phase space . the perturbations act via a resonant coupling between the characteristic frequencies of the perturbations and the frequencies of the orbits . that periodic driving works in this way should be obvious . that noise also involves a resonant coupling can be understood if one recalls ( _ cf . _ van kampen 1981 ) that a superposition of periodic forces combined with random phases is equivalent mathematically to ( in general coloured ) noise with a nonzero autocorrelation time @xmath18 . within a nearly disjoint phase space region , the perturbations allow microscopic motions which , in a strictly time - independent potential , are prohibited by liouville s theorem . e.g. , _ it becomes possible for phase space trajectories to cross , which helps a collection of orbits to ` fuzz out ' on short scales . the perturbations facilitate diffusion through cantori or along the arnold web by ` jiggling ' orbits in such a fashion as to help them find phase space holes . the details of the perturbation appear largely immaterial : all that seems to matter is the amplitude of the perturbations and their characteristic time scales . even the dependence on amplitude and time scale is comparatively weak . this implies that the details associated with realistic perturbations which might be difficult to extract from observations are largely irrelevant . the overall efficacy of the perturbations scales logarithmically in the amplitude . for time scales @xmath19 the perturbations have almost no effect ( adiabatic limit ) . for somewhat shorter time scales , the dependence on @xmath18 is again logarithmic . but what amplitude is required to have a significant effect , _ e.g. , _ by destabilising nearly time - independent building blocks ? very weak white noise corresponding to relaxation times @xmath20 can have appreciable effects within a time as short as @xmath21 , a period which , in the inner regions of a cuspy triaxial galaxy would be short compared with @xmath16 . alternatively , coloured noise and/or periodic driving corresponding to perturbations of fractional amplitude as small as @xmath22 and a characteristic time scale @xmath18 as long as @xmath23 can prove important on a time scale @xmath24 . making @xmath18 shorter facilitates a stronger resonant coupling between the perturbation and the orbits , thus making the perturbation even more effective . these results suggest the possibility that galaxies could settle down towards quasi - stationary states which , albeit not true collisionless equilibria , could exist as nearly time - independent entities for times @xmath25 , at least in the absence of irregularities . this seems especially likely , given the recognition that , for triaxial systems , true equilibria will in general be substantially more complex than the equilibria associated with spherical and axisymmetric configurations . for a generic triaxial system , there is only one global integral , namely the energy @xmath26 or jacobi integral @xmath27 , but it well known that equilibria @xmath28 and @xmath29 can not be used to model triaxial systems with a strong central condensation . unless the system is assumed to be characterised by a very special potential , _ e.g. , _ an integrable staeckel potential , it can not be in a true equilibrium unless that equilibrium involves an intricate balance of ` local integrals ' ( kandrup 1998 ) . the obvious point , then , is that even if such an intricate balance is hard to achieve , the system could evolve towards an approximate balance involving nearly time - independent building blocks . more pragmatically , these results would also suggest that , when constructing equilibria using schwarzschild s method or any analogue thereof , it would be strongly advisable to work with an orbit library constructed from orbits that have been evolved in the presence of weak noise or some other low amplitude perturbations . orbits evolved in the presence of such perturbations are more likely to constitute nearly time - independent building blocks and , as such , would seem less likely to be destabilised by weak irregularities associated with discreteness effects and/or a perturbing external environment . numerical computations demonstrate that much of the behaviour associated with chaotic orbit ensembles evolved in the triaxial dehnen potentials especially those associated with orbits which , in the absence of a cusp and black hole , would correspond to regular box orbits can be reproduced by the very simple potential @xmath30 given as the sum of an anisotropic oscillator and a plummer potential . orbits in this potential exhibit the same remarkable stickiness , yield comparable distributions of short time lyapunov exponents , and again manifest a strong susceptibility towards even very weak perturbations ( kandrup & sideris 2000 ) . that this simple toy model can reproduce the qualitative features of the more complicated potential ( 1 ) suggests strongly that _ the results derived for chaotic orbits in the triaxial dehnen potentials are generic for cuspy triaxial potentials_. that the potential is so simple makes it comparatively easy to understand what exactly is going on . as noted , _ e.g. , _ by merritt ( 1998 ) , supermassive black holes in real galaxies are seldom if ever larger than @xmath31 the mass of the entire galaxy and the central cuspy region typically corresponds to only a small fraction of the total mass . it follows that , for @xmath32 , @xmath33 , and @xmath34 of order unity , the physically relevant choices of @xmath35 entail @xmath36 . however , the qualitative behaviour in this regime is easily understood by a combination of perturbation theory and common sense . for @xmath36 and @xmath37 in eq . ( 2 ) , it appears that , except for the very lowest energies ( where the potential is essentially keplerian ) , essentially all of the orbits are chaotic , but that they tend to behave in a nearly regular fashion nearly all of the time . as the orbit evolves it will usually find itself in a region where @xmath38 , so that the potential in which it is moving is very nearly integrable and the short time lyapunov exponents are extremely small occasionally , however , the orbit will move comparatively close to the center of the galaxy , so close that @xmath39 becomes comparable to @xmath40 . when this happens the orbit feels the competing influences of two different potentials of comparable magnitude with very different symmetries and the values of the positive short time lyapunov exponents increase precipitously . the fact that , for small @xmath41 , almost all the orbits are chaotic is not difficult to understand . in the limit that @xmath42 , the orbits all reduce to boxes which densely fill a region in configuration space that includes the origin . one might expect that , for small but nonzero @xmath41 , the orbits can still pass arbitrarily close to the origin but , for any nonzero @xmath41 there is a minimum radius @xmath43 inside of which @xmath39 becomes large compared with @xmath40 . that significant chaos is only triggered when the trajectory passes relatively close to the black hole is illustrated in figure [ fig-1 ] , which exhibits a segment of a chaotic orbit evolved in the potential ( 2 ) . curve in the top panel exhibits the phase space separation @xmath44 between the original orbit and a perturbed orbit displaced originally by a distance @xmath45 and periodically renormalised in the usual way ( _ cf . _ lichtenberg & lieberman 1992 ) . the dashed curve shows an analogous plot of @xmath44 for a regular orbit . the lower panel plots @xmath46 , the distance from the origin . most of the time the perturbed and unperturbed orbits remain very close together , with comparatively little systematic exponential divergence . only when @xmath47 becomes as small as @xmath48 , so that @xmath39 becomes as large as @xmath49 , do the orbits tend to diverge significantly . this toy model is particularly simple since , in the limit @xmath50 , all the orbits are regular boxes . for generic triaxial potentials , in the absence of a cusp or black hole one would expect both centrophilic box orbits and centrophobic tubes . because the tubes are centrophobic , they should not in general be impacted all that much by the introduction of a cusp or a central black hole . what , _ does _ , however , seem to be true is that many of the orbits which , in the absence of a cusp , behave as regular boxes can , in the presence of a cusp or black hole , be converted into orbits which , albeit formally chaotic , behave in a nearly regular fashion much of the time . we are pleased to acknowledge useful collaborations with brendan bradley , barbara eckstein , salman habib , and elaine mahon . this research was supported in part by nsf ast-0070809 and by the institute for geophysics and planetary physics at los alamos national laboratory . kandrup , h. e. 1998 , , 299 , 1139 kandrup , h. e. , eckstein , b. l. , bradley , b. o. 1997 , a&a , 320 , 65 kandrup , h. e. , mahon , m. e. 1994 , a&a , 290 , 762 kandrup , h. e. , pogorelov , i. v. , sideris , i. v. 2000 , , 311 , 719 kandrup , h. e. , sideris , i. v. 2000 , , submitted kandrup , h. e. , siopis , c. 2000 , in preparation lichtenberg , a. j. , lieberman , m. a. 1992 , regular and chaotic dynamics . springer : berlin merritt , d. 1998 , comments astrophys . , 19 , 1 merritt , d. , fridman , t. 1996 , , 460 , 136 pogorelov , i. v. , kandrup , h. e. 1999 , phys . rev . e , 60 , 1567 siopis , c. , kandrup , h. e. 2000 , , in press siopis , c. , eckstein , b. l. , kandrup , h. e. 1998 , ann . n. y. acad . , 867 , 41 van kampen , n. g. 1981 , stochastic processes in physics and chemistry . north holland : amsterdam
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this talk provides a progress report on an extended collaboration which has aimed to address two basic questions , namely : should one expect to see cuspy , triaxial galaxies in nature ? and can one construct realistic cuspy , triaxial equilibrium models that are robust ?
three technical results are described : ( 1 ) unperturbed chaotic orbits in cuspy triaxial potentials can be extraordinarily sticky , much more so than orbits in many other three - dimensional potentials .
( 2 ) even very weak perturbations can be important by drastically reducing , albeit not completely eliminating , this stickiness .
( 3 ) a simple toy model facilitates a simple understanding of why black holes and cusps can serve as an effective source of chaos .
these results suggest that , when constructing models of galaxies using schwarzschild s method or any analogue thereof , astronomers would be well advised to use orbital building blocks that have been perturbed by ` noise ' or other weak irregularities , since such building blocks are likely to be more nearly time - independent than orbits evolved in the absence of all perturbations .
a contributed talk at:_the international conference on stellar dynamics : from _ .2 in _ from classical to modern _ , sobolev astronomical institute , august 2000 .2 in
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You are an expert at summarizing long articles. Proceed to summarize the following text:
two - dimensional planar billiards are nonlinear systems with rich and interesting dynamical properties . a point particle , moving with constant velocity within a closed boundary and exhibiting specular reflections on the walls , can have regular , mixed or fully chaotic dynamics , in strong dependence on details of the boundary shape . in physics , two - dimensional billiards offer good examples of coexistence of regular , mixed and chaotic dynamics in hamiltonian systems . this type of behavior , illustrated by the standard map and explained by means of the kam - theorem , is present in many realistic phenomena , such as planetary systems and various types of coupled oscillators@xcite . chaotic billiards were first introduced by sinai@xcite who considered the defocusing effects of circular scatterers in the two - dimensional lorentz gas . after the important discovery by bunimovich@xcite that also the focusing circular arcs can lead to a fully chaotic behavior , many investigations were devoted to billiards with circular arcs and , in a smaller extent , to other types of curved boundaries . the systematic mathematical description of chaotic billiards and an extended list of references can be found in the book by chernov and markarian@xcite . rigorous investigations were concentrating on methods for producing fully chaotic billiards and on specific properties ( bernoulli , k - property , mixing and hyperbolicity ) expressing differences between chaotic systems@xcite . various aspects of billiard dynamics have been extensively examined during last decades@xcite . in recent years , properties of classical billiards and their quantum - mechanical counterparts were used to explain and improve performances of devices in microelectronics and nanotechnology , especially of optical microresonators in dielectrical and polymer lasers@xcite . we are stressing the fact that notable regions of full chaos have been discovered in billiards with elliptical arcs and piecewise flat boundaries , indicating that such billiards deserve further attention@xcite . in our previous work we analyzed several types of billiards with noncircular arcs ( parabolic , hyperbolic , elliptical and generalized power - law ) , exhibiting mixed dynamics@xcite . next we investigated , in the full parameter space@xcite , the elliptical stadium billiards ( esb ) , first introduced by donnay@xcite . here we extend the same type of analysis to the truncated elliptical billiards ( teb ) , which although similar in appearance , have different dynamical properties . the truncated elliptical billiard ( teb ) is defined by a two - parameter planar domain constructed by truncating an ellipse on opposite sides ( fig . 1 ) . a symmetrical stadium - like shape thus obtained consists of a rectangle with two elliptical arcs added at its opposite ends . the corresponding billiard has been introduced by del magno@xcite who , investigating a restricted part of the parameter space and applying the mathematical method of invariant cones , determined the region of hyperbolic behavior and presented an estimate of the region where such billiard could be ergodic . in the present paper we investigate numerically and analitically the truncated elliptical billiard ( teb ) in the full parameter space , by using two shape parameters @xmath0 and @xmath1 . this description of the billiard geometry and dynamics is consistent with our previous analysis of the elliptical stadium billiard ( esb)@xcite , which is a two - parameter generalization of the bunimovich stadium billiard@xcite and is a special case of the mushroom billiard@xcite . it has been confirmed by analysis and numerical computation@xcite that this billiard is fully chaotic ( ergodic ) for a sizeable but strictly limited region in the parameter space , defined by the stable two - bounce horizontal periodic orbit on one and the pantografic orbits on the other side . our investigations of the esb and teb billiards confirm the suggestion by del magno@xcite that in spite of apparently similar stadium - like shapes , these two billiards have essentially different dynamical properties . in the present paper we describe our analytical and numerical investigation of the truncated elliptical billiard and compare the obtained results with those for the elliptical stadium billiard . in section ii we define the teb billiard boundary and describe its geometrical properties . in section iii the existence and stability of selected orbits are discussed and illustrated by poincar ' e plots and orbit diagrams . in section iv the poincar ' e sections are used to estimate , by means of the box - counting numerical method , the degree of chaoticity for a given boundary shape . the results are shown in the parameter - space diagram and compared with the same type of diagram for the elliptical stadium billiard . in section v we briefly discuss the possible generalization of the truncated elliptical billiard providing a transition between two types of the stadium - like elliptical billiards . finally , in section vi we summarize the obtained results and propose further investigations . in our parametrization the truncated elliptical billiard ( teb ) is defined in the @xmath2 plane by means of the two parameters @xmath0 and @xmath1 , satisfying conditions @xmath3 and @xmath4 . the billiard boundary is described as @xmath5 the horizontal diameter is normalized to 2 , so that the horizontal semiaxis of the ellipse is 1 . the vertical semiaxis of the ellipse is @xmath6 , and the possible height @xmath7 of the billiard extends from @xmath8 to @xmath9 . the horizontal length of the central rectangle is @xmath10 . in special cases , for @xmath11 the shape is a full ellipse , for @xmath12 it is rectangular , for @xmath13 it is a square and for @xmath11 and @xmath14 a full circle . for @xmath15 one obtains a set of truncated circle billiards , which separates two distinct billiard classes , one for @xmath16 with elongated elliptical arcs and the other with @xmath17 and flattened elliptical arcs . 1 shows three typical shapes of the truncated elliptical billiard with circular , flattened and elongated elliptical arcs . the coordinates of the focal points are @xmath18\ ] ] for @xmath16 , and @xmath19\ ] ] for @xmath17 . they contain the important term @xmath20 which is negative for @xmath16 , positive for @xmath17 , and zero for @xmath15 ( circular arcs ) . this limit is shown as the thick circular line in fig . 2 presenting the structure of the @xmath21 parameter space . for @xmath22 the curvature radius is @xmath23^{3/2 } } { \gamma(1-\delta^2)}\ ] ] for @xmath24 the boundary is flat and the curvature radius is @xmath25 , but for @xmath26 has a discontinuity and drops to @xmath27^{3/2 } } { \gamma(1-\delta^2)}\ ] ] at the endpoints of the horizontal axis of the ellipse ( @xmath28 ) the curvature radius is @xmath29 which reduces to @xmath30 for circular arcs . for full ellipses with @xmath11 the curvature radius at @xmath31 is @xmath32 . as explained in @xcite , the symbols @xmath33 , @xmath34 and @xmath35 , respectively , denote the angles which the normal , the incoming path and the outcoming path make with the x - axis . the angle between the incoming ( or outcoming ) path and the normal to the boundary is @xmath36 . the angle between the tangent to the boundary at the point t@xmath37 of impact and the incoming ( or outcoming ) path , needed in the computation of the orbit stability , is @xmath38 . the angles @xmath33 , @xmath34 and @xmath35 are connected by the relation @xmath39 the expression ( 7 ) is the basis for finding the existence criteria for particular periodic orbits@xcite . in our further description we refer to the impact points t@xmath37 in the first quadrant , with no loss of generality for the obtained results . in the poincar ' e sections the points p@xmath40 are obtained by plotting the slope of the velocity direction @xmath41 versus the x - coordinate of the intersection point with the x - axis , as explained in @xcite . the poincar ' e diagrams obtained in this way are area preserving . as described in @xcite , the stability of a periodic orbit is assured if the absolute value of the trace of the stability matrix @xmath42 is smaller than 2 , thus if @xmath43 such orbits are elliptic , and those with @xmath44 are neutral ( parabolic ) . the stability matrix of the closed orbit of period @xmath45 can be written as @xmath46 , where the @xmath47 matrix @xmath48 for two subsequent impact points t@xmath49 and t@xmath50 , connected by a rectilinear chord of the length @xmath51 , is @xmath52 this subfamily of truncated elliptical billiards has elongated elliptical arcs . in fig . 3(a - d ) we show poincar ' e sections for @xmath53 and different values of @xmath54 . similar results for @xmath55 and @xmath56 are shown in fig . 4(a - d ) . these pictures reveal a highly chaotic behavior . there are no elliptic islands , however , flights of points typical for neutral orbits can be discerned . this is remarkably different from the corresponding results for the elliptical stadium billiards@xcite , where in the same parameter region there were many fixed points and elliptic islands due to stable pantographic and other orbits . we investigate the existence and stability of the bow - tie orbit ( the lowest pantographic orbit ) , shown in fig . this orbit exists if the coordinates @xmath57 and @xmath58 of the impact point and the derivative @xmath59 of the boundary at this point satisfy the equation ( 7 ) , which now reads @xmath60 giving as solution the coordinates of the point of impact @xmath61 and @xmath62 the condition @xmath63 that this point should lie on the elliptical part of the boundary leads to the requirement @xmath64 this limit is shown in fig . 2 and is denoted with the letter a. if we denote the points with positive @xmath57 by 1 and the points on the negative side by -1 , the deviation matrix can be calculated as @xmath65 the corresponding angle @xmath66 needed in the matrix ( 9 ) is given by @xmath67 the chords are @xmath68 and @xmath69 the curvature radius at the impact point is obtained by substituting ( 11 ) into ( 4 ) and reads @xmath70 if we define @xmath71 the trace of the deviation matrix is @xmath72\ ] ] the left - hand side of the stability condition ( 8) is valid automatically , but the right - hand side is fulfilled only if @xmath73 by substituting ( 15 ) , ( 16 ) , ( 17 ) and ( 18 ) into ( 19 ) , one obtains @xmath74 for all allowed cases . the conclusion is that the bow - tie orbit is neutral for all parameter values satisfying the existence condition . further we investigate properties of the rectangular orbit shown in fig . according to ( 7 ) , this orbit exists if the derivative on the boundary is @xmath75 . corresponding solutions for the impact point are @xmath76 and @xmath77 the condition @xmath63 leads to the requirement @xmath78 this limit is shown in fig . 2 denoted by letter f. stability is calculated with equation ( 9 ) and the matrix ( 14 ) , where the angle @xmath66 is given by @xmath79 . the chords are @xmath80 and @xmath81 and the curvature radius is @xmath82^{3/2}\ ] ] again , the trace is given by ( 20 ) , and for this case one obtains @xmath83 . the conclusion is that also this orbit is neutral for all shapes , both flattened and elongated , allowed by ( 24 ) . the elongated truncated elliptical billiards were discussed in @xcite . their boundary shapes were described by means of two parameters @xmath84 and @xmath85 , related to our parameters @xmath0 and @xmath1 as follows : @xmath86 in @xcite the billiards with @xmath87 and @xmath88 have been analysed and the hyperbolic behavior has been identified in the region @xmath89 . in our parameters , this corresponds to the quasi - triangular region in the parameter space , denoted by a and b in fig . 2 , delimited by curves @xmath90 ( denoted in fig . 2 by letter e ) and @xmath15 and by the straight line @xmath91 . the region @xmath92 is rigorously proved to be ergodic@xcite . written with our parameters , it obeys the conditions @xmath93 @xmath94 the corresponding part of the parameter space in fig . 2 is the one denoted by a. the comparison with our results shows that the limit ( 27 ) or ( 28 ) is identical to the limit ( 24 ) in the parameter space , where the parabolic rectangular orbits emerge . this part of the parameter space , with flattened elliptical arcs , had not been investigated previously . in fig . 3(e - h ) we show poincar ' e sections for @xmath53 and different values of @xmath95 . in this parameter region dynamics is following the kam scenario . similar behavior is noticed for values @xmath55 and @xmath96 ( fig . 4(e - h ) ) . elliptic islands corresponding to the horizontal two - bounce and some other orbits are visible , similarly to the corresponding results for the elliptical stadium billiards@xcite . we investigate the existence and stability criteria for these orbits . the horizontal two - bounce orbit ( fig . 5(c ) ) obviously exists for all combinations of @xmath0 and @xmath1 , but according to @xcite the stability condition @xmath97 takes the form @xmath98 so that bifurcations giving birth to stable diametral orbits appear at the values @xmath15 , corresponding to circular arcs . in the poincar ' e diagrams this orbit and the surrounding quasiperiodic orbits are visible as two large bands near @xmath99 . according to ( 7 ) a tilted two - bounce orbit ( fig . 5(d ) ) exists at the point t@xmath37 on the billiard boundary with derivative @xmath59 if @xmath100 this is realized for any @xmath63 provided that @xmath101 thus only for the truncated circle . since in this case the chord in ( 29 ) is @xmath102 and the radius is @xmath103 , these orbits are neutral . the diamond orbit of period four , shown in fig . 5(e ) , exists for any parameter choice . it has two bouncing points at the ends of the horizontal semiaxis , and the other two on the flat parts on the boundary . to assess its stability , one should calculate the stability matrix @xmath104 . the angles contained in the matrix are given as @xmath105 where @xmath106 the curvature radius at @xmath107 is given by ( 6 ) . this leads to the trace @xmath108\ ] ] and to the condition @xmath109 or @xmath110 , thus the stable diamond orbit appears when @xmath111 this limit is shown in fig . 2 as the line denoted by letter c. the multidiamond orbit of order @xmath112 is the orbit of period @xmath113 , which has two bouncing points at the ends of the horizontal axis and @xmath114 bouncing points on the flat parts of the boundary ( fig . 4 ( f ) ) . such orbit exists if @xmath115 as explained for a similar case in @xcite , the chord @xmath116 in ( 35 ) should be replaced by @xmath117 where , for the truncated elliptical billiard , @xmath118 the trace of the stability matrix is then @xmath119\ ] ] with @xmath120 given by ( 6 ) . the resulting condition for the stability of the multidiamond orbit is @xmath121 the limiting curves ( 41 ) are plotted in fig . 2 . the line with @xmath122 is denoted by letter d , and above it there are several lines for @xmath123 . for @xmath124 the minimal values of @xmath0 above which the multidiamond orbits appear are @xmath125 the emergence of multidiamond orbits can be followed by observing the poincar ' e sections for a set of shapes with @xmath126 ( fig . the values of this parameter for which an orbit of new @xmath112 appears are given as intersections of the straight line @xmath126 with curves ( 41 ) , and obey the equation @xmath127 for the diamond orbit ( @xmath128 ) this equation reads @xmath129 and the orbit appears for @xmath130 for the same type of boundary the stable two - bounce orbit appeared at @xmath131 the hour - glass orbit ( fig . 5(g ) ) looks like the bow - tie orbit rotated by @xmath132 . it exists if the coordinates @xmath57 and @xmath58 of the impact point and the derivative @xmath59 of the boundary at this point satisfy the equation @xmath133 giving as solution the coordinates of the impact point @xmath134 and @xmath135 the condition @xmath63 that this point should lie on the elliptical part of the boundary leads to the requirement @xmath136 @xmath137 these limits define the region shown in fig . 2 denoted by letter b. if we denote the points with positive @xmath58 by 1 and the points on the negative side by -1 , the deviation matrix can be calculated from ( 14 ) . the angle @xmath66 needed in the calculation is given as @xmath138 the curvature radius at this point is given as @xmath139 the chords are @xmath140 and @xmath141 if we define @xmath142 as in ( 19 ) , the trace of the deviation matrix is again given by ( 20 ) and the orbit is stable if @xmath143 . when we substitute the calculated values of @xmath120 , @xmath116 , @xmath144 and @xmath145 into ( 19 ) , we obtain @xmath74 for all allowed shapes and conclude that the hour - glass orbit is neutral . this means that in the truncated elliptical billiards ( teb ) there is no stable hour - glass orbit , at variance with the elliptical stadium billiard ( esb ) , where such an orbit having interesting properties was stable in a large fraction of the parameter space@xcite . besides the diamond and multidiamond orbits , in fig . 6 one discerns the presence of another , `` 8-shaped '' , stable orbit , shown in fig . in this section we return to the question of limits within which the truncated elliptical billiard is fully chaotic . here we test these limits numerically , with the help of the box - counting method@xcite . we calculate the poincar ' e sections for a chosen pair of shape parameters , starting with @xmath146 randomly chosen sets of initial conditions and iterating each orbit for @xmath147 intersections with the x - axis , thus obtaining @xmath148 points in the poincar ' e diagram . then we divide the first quadrant of the phase plane into a grid of @xmath149 squares ( boxes ) , count the number of boxes which have points in them and calculate the ratio of this number to the total number of boxes . the obtained ratio is denoted by @xmath150 . in this way also certain points belonging to invariant curves within the regular islands are included . but since our main aim is to examine the onset of full chaos , this method gives satisfactory results , providing that the appropriate values of @xmath146 , @xmath147 and @xmath112 are used . detailed testing has shown that reliable results are obtained for values @xmath151 , @xmath152 and @xmath153 used in our present calculation@xcite . in fig . 7 we plot in the @xmath154 plane the points representing the pairs of shape parameters . points are plotted in different colors , depending on the corresponding value of @xmath155 . the full chaos , corresponding to @xmath156 , is depicted by black points . colored points denote shapes within intervals between 0 and 0.99 . this diagram confirms that for the truncated elliptical billiards ( teb ) , in the region below the onset of the stable two - bounce horizontal orbit , dynamics is practically completely chaotic . this is in strong contrast with the behavior of the elliptical stadium billiard for which the similar diagram is shown in fig . 8 . for the esb billiards the region of chaos was strictly bounded also from the lower side and determined by emergence of stable pantographic orbits . to examine the possible mechanism for this difference , we assume that the teb and the esb billiards are two extreme cases and search for a possible transition between them . in this section we propose a new large class of stadium - like billiards which we call generalized truncated elliptical stadium - like billiards ( gtesb ) . such a billiard depends on three shape parameters @xmath0 , @xmath1 and @xmath157 . the allowed values of the shape parameters are @xmath158 for the limiting values of @xmath157 we obtain the two billiard families considered before : for @xmath159 gtesb reduces to the elliptical stadium billiard ( esb ) , and for @xmath160 gtesb becomes the truncated elliptical billiard ( teb ) . the new gtesb billiard boundary is obtained by adding elliptical arcs symmetrically at the two opposite ends of a rectangle with sides @xmath10 and @xmath7 . elliptical arcs are cut out from the two identical but generally detached ellipses by two horizontal straight lines at @xmath161 ( fig . the two ellipses have centers at the points @xmath162 the distance between the two centers is @xmath163 . the horizontal and vertical semiaxis are given , respectively , as @xmath164 and @xmath165\ ] ] and the equation of the two ellipses reads @xmath166 the horizontal diameter of the billiard is 2 . for @xmath159 and @xmath167 the gtesb becomes the bunimovich stadium billiard . in fig . 10 the poincar ' e sections are shown for @xmath168 and @xmath169 with @xmath157 assuming different values between -1 and 1 . the four islands typical for the bow - tie orbit are present for all @xmath157 except for @xmath160 ( teb ) , where this orbit becomes neutral and the island reduces to a caracteristical flight of points . the limits separating chaotic from mixed behavior are determined by the onset of the stable horizontal 2-bounce orbit and are given by ( 29 ) . since the curvature radius at @xmath28 is @xmath170}{2(1-\delta)(1+\delta\kappa)}\ ] ] the upper limit of chaos is determined by the condition @xmath171 in fig . 11 the chaotic fraction @xmath155 is shown for the special case @xmath126 for different @xmath157 , in dependence on @xmath1 . it is noticed that in the case @xmath159 ( esb ) there is a narrow , strictly limited region of full chaos , outside of which the values of @xmath155 are low . for @xmath160 ( teb ) the fully chaotic region is much larger and extends practically over all values @xmath0 and @xmath1 below the chaotic limit . between these two limits , the regions of full chaos ( @xmath172 ) are shorter and limited , but there are many shapes with chaotic parameter close to 1 ( between 0.90 and 0.99 ) . this corresponds to a selection of narrow islands in the poincar ' e plots , as seen in fig . in conclusion , our investigation of the elliptical stadium - like billiards has revealed a rich variety of integrable , mixed and chaotic behavior , which is connected with the character of the two elliptical arcs and with their mutual position . this strong dependence on parameters @xmath0 and @xmath1 is confirmed for the truncated elliptical billiards , but is even more enhanced when a third shape parameter @xmath157 is added . analysis shows , however , that among all considered shapes the truncated elliptical billiard ( teb ) , created by cutting a single ellipse with two parallel straight lines , has exceptional properties , notably that it is chaotic practically in the whole region of elongated elliptical arcs . notable is the presence of many neutral orbits in this region , consistent with the fact that these orbits actually can be identified as orbits in an ellipse . for the flattened arcs , the stable islands due to the two - bounce horizontal orbit and to the diamond and multidiamond orbits occupy an important part of the phase plane . our investigations can be useful for the experimental application of billiards in the laser technology , where properties and directional intensities of the optical microresonators depend strongly on the boundary shape . they can also be applied in designing the semiconducting optical devices and in the technology of microwave and acoustic resonant cavities . with this purpose in mind , we propose further analysis of the stadium - like billiards with elliptical arcs and an extension of the present investigation to different types of open billiards . authors are thankful to a. b " acker , m. lebental , n. pavin , t. prosen , m. robnik and t. tanaka for useful discussions and comments and to v. danani ' c and d. radi ' c for help with numerical methods and computation .
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chaotic properties of symmetrical two - dimensional stadium - like billiards with elliptical arcs are studied numerically and analytically . for the two - parameter truncated elliptical billiard the existence and linear stability of several lowest - order periodic orbits are investigated in the full parameter space .
poincar ' e plots are computed and used for evaluation of the degree of chaoticity with the box - counting method .
the limit of the fully chaotic behavior is identified with circular arcs . above this limit , for flattened elliptical arcs , mixed dynamics with numerous stable elliptic islands
is present , similarly as in the elliptical stadium billiards . below this limit
the full chaos extends over the whole region of elongated shapes and the existing orbits are either unstable or neutral .
this is conspicuously different from the behavior in the elliptical stadium billiards , where the chaotic region is strictly bounded from both sides . to examine the mechanism of this difference , a generalization to a novel three - parameter family of boundary shapes
is proposed and suggested for further evaluation .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
asymptotic giant branch ( agb ) carbon stars are produced following @xmath1 dredge - up in thermally pulsing stars ( e.g. iben & renzini 1983 ) . the star changes from oxygen- to carbon - rich when sufficient carbon has been mixed - in with the stellar mantle to yield an abundance ratio c / o@xmath2 . the chemistry of the dust in the circumstellar envelope ( cse ) changes accordingly . the change occurs at smaller core - mass or lower luminosity for lower metallicity stars . clear evidence for this comes from observations of clusters in the large magellanic cloud ( lmc ) that contain both carbon and m - type stars ( lloyd evans 1984 ; marigo et al . 1996 ) . surprisingly , silicate emission from oxygen - rich dust was discovered in the iras low resolution spectra of several galactic carbon stars ( little - marenin 1986 ; willems & de jong 1986 ) . willems & de jong interpreted these `` silicate carbon stars '' as direct evidence for a fast transition of m - type agb stars into carbon stars , but timescales of decades for the silicate emission from an expanding detached oxygen - rich cse to fade away are difficult to reconcile with the lifetimes of silicate carbon stars ( little - marenin et al . 1987 ; le bertre et al . hence the oxygen - rich material must be stored in a stationary component . many galactic silicate carbon stars are @xmath3c - enhanced , j - type , carbon stars ( lambert et al . 1990 ) . unlike genuine , n - type , carbon stars that form on the agb , j - type carbon stars are thought to have become carbon - enriched as a result of binary evolution . the presence of a mass - losing oxygen - rich companion star has been ruled out observationally for a number of galactic silicate carbon stars ( noguchi et al . 1990 ; engels & leinert 1994 ) . the presently most supported explanation for the silicate carbon star phenomenon is that of a keplerian disk of oxygen - rich material , surrounding a binary including a faint companion ( lloyd evans 1990 ) . the oxygen - rich dust may originate from mass loss at a time when the carbon star was still oxygen rich ( lloyd evans 1990 ) . the dust - enshrouded agb star iras04496@xmath06958 was recently discovered to be a luminous carbon star in the lmc by van loon et al . ( 1998 , 1999 ) on the basis of ground - based ( ctio ) 3 @xmath4 m spectroscopy , after having been selected and confirmed to be an agb star by loup et al . ( 1997 ) and zijlstra et al.(1996 ) , respectively . the carbon star nature of this object has been confirmed by groenewegen & blommaert ( 1998 ) using optical spectroscopy . at @xmath5 mag it is the brightest known magellanic carbon star and very close to the maximum agb luminosity ( @xmath6 mag ) . we here present compelling evidence for the presence of oxygen - rich dust close to this remarkable carbon star , making it the first known extra - galactic silicate carbon star . [ cols="<,<,<,<,<,<,<,<,<,<,<",options="header " , ] the spectral energy distribution ( sed ) of this object peaks in the infrared . therefore in order to properly model the spectrum we obtained photometric and spectro - photometric observations with the european infrared space observatory ( iso , see kessler et al . 1996 ) , using the isocam ( cesarsky et al . 1996 ) and isophot ( lemke et al . 1996 ) instruments . the photometric observations at 12 @xmath4 m ( using isocam filter lw10 ) and 25 @xmath4 m ( using isophot ) were obtained on april 22 , 1996 . the 12 @xmath4 m observation was done using @xmath7 pixels , and a total on - source integration time of 50 s split in 25 two - s integration intervals . the 25 @xmath4 m isophot observation was done using the p2 detector , a @xmath8 aperture , and triangular chopping with a chopper throw of @xmath9 . the on - source integration time was 64 s. the 60 @xmath4 m photometry was obtained using the phot - c100 detector using two different methods . one observation ( april 22 , 1996 ) was done using chopping mode with triangular chops and a chopping angle of @xmath10 and an on - source integration time of 64 s. another observation ( april 1 , 1998 ) was done with a @xmath11 raster map with @xmath12 raster steps and an integration time per pointing of 128 s , giving an effective on - source integration time of @xmath13 s. spectro - photometric observations of the source were obtained with isophot - s and the isocam cvf . the phot - s spectrum ( april 22 , 1996 ) was done in staring mode with an on - source integration time of 512 s. two isocam cvf spectra were obtained . the first isocam spectrum ( june 5 , 1997 , hereafter `` spectrum a '' ) spans the wavelength range from 7 to 14 @xmath4 m , the second ( april 1 , 1998 , hereafter `` spectrum b '' ) from 5 to 17 @xmath4 m . the @xmath14 pixel field of view was used , with an integration time per spectral point of 50 and 70 s , respectively . the data was processed using standard processing routines in the phot interactive analysis ( pia ) and cam interactive analysis ( cia ) software . the cam - cvf spectra were constructed using a @xmath11 pixel@xmath15 software aperture and applying a correction for the wavelength dependence of the point spread function . we corrected the phot - s spectrum for the background as derived from the cam - cvf data , accounting for the annual modulation of the zodiacal light using cobe / dirbe weekly all - sky maps ( see also trams et al . the photometric observations are listed in table 1 . the iso observations are supplemented with ground - based j , h , k , and l - band observations made at the south african astronomical observatory ( saao ) , interpolated to the same epochs as the various iso observations . the spectra are presented in fig . 1 . also plotted is a spectrum around 3 @xmath4 m obtained at ctio ( van loon et al . 1999 ) , after scaling to match the approximate continuum level in the phot - s spectrum . we believe that the phot - s spectrum longward of @xmath16 @xmath4 m has been under - estimated , and possibly distorted , due to difficulties in determining the stabilised signal at such low flux density levels . the phot - s and ctio spectra show the strong 3 @xmath4 m feature from hcn and c@xmath17h@xmath17 ( fig . 1 ) , but the long wavelength part of the phot - s spectrum is rather noisy . the cam - cvf spectra , however , show a prominent emission feature between 9 and 12 @xmath4 m with a small dip near 11 @xmath4 m . for comparison we also plot in fig . 2 the iso sws spectrum ( @xmath18 ) of the galactic silicate carbon star v778 cyg , taken from yamamura et al . ( 1997 ) , which shows strong silicate emission from oxygen - rich dust around 10 @xmath4 m . the feature in iras04496@xmath06958 extends to longer wavelengths than in v778 cyg , and closely resembles that of another galactic silicate carbon star , cs1003 ( hen 83 , iras08002@xmath03803 ; see little - marenin 1986 and willems & de jong 1986 ) . we also plot in fig . 2 the ground - based ukirt spectrum ( @xmath19 ) of the galactic carbon star afgl2368 , taken from speck et al . ( 1997 ) , which shows a prominent silicon carbide ( sic ) emission feature around @xmath20 m that is common in carbon stars ( see e.g. little - marenin 1986 ; yamamura et al.1997 ) . the shape of the 9 - 12 @xmath4 m emission feature in iras04496@xmath06958 may be explained by assuming that the feature is a composition of the silicate and sic features . alternative explanations include large silicate grains ( forrest et al . 1975 ; papoular & pgouri 1983 ) , crystalline olivines ( koike et al . 1981 ) and corundum ( alo ) grains ( onaka et al . 1989 ) . we note that similarly shaped emission is observed in the spectra of a wide variety of objects : the s star rt sco and ms stars ( little - marenin & little 1988 ) , the sil@xmath21 and sil@xmath22 classes of m - type mira variables ( little - marenin & little 1990 ) , @xmath23 pictoris ( knacke et al . 1993 ; fajardo - acosta & knacke 1995 ) , inter - planetary dust particles ( sandford 1988 ) and comet halley ( campins & ryan 1989 ) . absorption against the photosphere by hcn and c@xmath17h@xmath17 is seen at 3.1 , 3.8 and 8 @xmath4 m ( fig . 1 ) . on the long - wavelength end of the emission feature the 13.7 @xmath4 m absorption due to c@xmath17h@xmath17 is seen , which is commonly observed in the spectra of galactic carbon stars ( yamamura et al . this absorption is seen against the dust continuum that dominates over the photospheric continuum at these long wavelengths indicating that the molecules are abundant throughout the dusty cse . it is absent in v778 cyg . this may be understood if the molecules - to - dust ratio is larger at lower metallicity , possibly because depletion of molecules into dust grains is less severe at smaller dust - to - gas ratios . iras04496@xmath06958 is a long period variable with a period of @xmath24 d and a k - band amplitude of @xmath25 mag ( whitelock et al . , in preparation ) . the two cvf spectra are taken at different phases in the lightcurve , with spectrum a closer to maximum light . their ratio is plotted in fig . the maximum difference is reached between 9 and 10 @xmath4 m , and no difference is seen for wavelengths @xmath26 @xmath4 m . this suggests that a significant part of the variability around 10 @xmath4 m is due to a variable emission feature , whilst beyond 13 @xmath4 m non - variable dust continuum emission dominates . the cvf ratio around 10 @xmath4 m is 1.2 , which equals the ratio of l - band flux densities at these two epochs ( the k - band flux density ratio is 1.5 ) . this may be compared to n- ( 10 @xmath4 m ) and l - band amplitudes observed in galactic ir - bright carbon stars , @xmath27 mag ( standard deviation 0.23 ) ( le bertre 1992 ) , and oxygen stars , @xmath28 mag ( standard deviation 0.20 ) ( le bertre 1993 ) . the difference results from the fact that in carbon stars the l - band includes variable hcn+c@xmath17h@xmath17 absorption , whereas in oxygen stars the n - band includes variable silicate emission . the @xmath29 mag of iras04496@xmath06958 suggests a contribution of silicate emission to the variability in the n - band . the 10 @xmath4 m variability of iras04496@xmath06958 is additional evidence for the silicate carbon star nature of this star , and 10 @xmath4 m variability might provide a new means for finding or confirming silicate carbon stars . all ir colours between 1 and 25 @xmath4 m of iras04496@xmath06958 are similar to those of carbon stars ( van loon et al . 1998 ; trams et al . 1999 ) , whereas v778 cyg has colours ( chen et al . 1998 ) more similar to oxygen - rich stars . hence the oxygen - rich dust component represents only a minor fraction of the total dust mass that is contained in the cse of iras04496@xmath06958 . its cse is considerably thicker than that of known galactic silicate carbon stars , judged from its very red near - ir colours ( lloyd evans 1990 ; chan & kwok 1991 ) . hence the observation that all galactic silicate carbon stars are of j - type ( lambert et al . 1990 ) may be an observational bias against n - type carbon stars with massive carbon - rich cses for which an optical spectrum to determine the @xmath3c/@xmath30c ratio is relatively difficult to obtain . [ [ the - origin - of - the - oxygen - rich - dust - around - the - carbon - star - iras04496 - 6958 ] ] the origin of the oxygen - rich dust around the carbon star iras04496@xmath06958 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ iras04496@xmath06958 is special because with @xmath31 mag it is a very luminous carbon star ( van loon et al . 1998 , 1999 ) . the popular scenario for the formation of silicate carbon stars in which a j - type carbon star evolves from a less massive r - type star ( lambert et al . 1990 ) implies that silicate carbon stars should not be very luminous , which has some observational support ( barnbaum et al . therefore , iras04496@xmath06958 could be a different class from the galactic silicate carbon stars . nuclear burning at the base of the convective envelope ( `` hot bottom burning '' , hbb ) reduces the carbon abundance of the mantle by cycling it into nitrogen ( iben 1981 ; iben & renzini 1983 ; wood et al . theoretical models show that it occurs for the most massive agb stars ( blcker & schnberner 1991 ) . boothroyd et al . ( 1993 ) predict that hbb prevents the occurence of carbon stars above m@xmath32 mag , consistent with the observed absence of optically bright carbon stars more luminous than m@xmath33 mag ( iben 1981 ; costa & frogel 1996 ) . the existence of luminous dust - enshrouded carbon stars ( van loon et al . 1999 ) like iras04496@xmath06958 in the lmc and iras00350@xmath07436 in the smc ( @xmath34 mag ; whitelock et al . 1989 ) is explained by mass loss reducing the stellar mantle to below a critical mass required for the pressure and temperature at the lower convective boundary to be sufficiently high to support hbb ( boothroyd & sackmann 1992 ) . if such a star experiences another thermal pulse and accompanying dredge - up of carbon to its surface , it may become a carbon star , after all ( frost et al . 1998 ; marigo et al . 1998 ) . hence , iras04496@xmath06958 has been an oxygen - rich star not longer than a thermal pulse interval of @xmath35 yr ago ( see vassiliadis & wood 1993 ) . this does not exclude the possibility that emission from the oxygen - rich dust is still observable around the recently formed carbon star , but the massive carbon - rich cse around iras04496@xmath06958 suggests a relatively long lapse of time since the mass loss was oxygen rich . although the isophot 60 @xmath4 m photometry is rather inaccurate , the high 60 @xmath4 m flux density of iras04496@xmath06958 suggest that its mass - loss rate was higher in the past , some @xmath36 yr ago , which would be consistent with an episode of increased mass loss during a thermal pulse followed by a considerable period of mass loss at a more moderate rate . hence it remains to be seen whether the silicate carbon star nature of iras04496@xmath06958 requires a companion star to have captured the oxygen - rich material in a circumbinary disk , or whether it resulted from single star evolution of a massive agb star . barnbaum c. , kastner j.h . , morris m. , likkel l. , 1991 , a&a 251 , 79 blcker t. , schnberner d. , 1991 , a&a 244 , l43 boothroyd a.i . , sackmann i .- , 1992 , apj 393 , l21 boothroyd a.i . , sackmann i .- j . , ahern s.c . , 1993 , apj 416 , 762 campins h. , ryan e. , 1989 , apj 341 , 1059 carter b.s . , 1990 , mnras 242 , 1 cesarsky c.j . , abergel a. , agnse p. , et al . , 1996 , a&a 315 , l32 chan s.j . , kwok s. , 1991 , apj 383 , 837 chen p .- s . , xiong g .- z . , wang x .- h . , 1998 , acta astron . sin . 39 , 202 costa e. , frogel j.a . , 1996 , aj 112 , 2607 engels d. , leinert ch . , 1994 , a&a 282 , 858 fajardo - acosta s.b . , knacke r.f , 1995 , a&a 295 , 767 forrest w.j . , gillett f.c . , stein w.a . , 1975 , apj 195 , 423 frost c.a . , cannon r.c . , lattanzio j.c . , wood p.r . , forestini m. , 1998 , a&a 332 , l17 groenewegen m.a.t . , blommaert j.a.d.l . , 1998 , a&a 332 , 25 iben i. , 1981 , apj 246 , 278 iben i. , renzini a. , 1983 , ara&a 21 , 271 kessler m.f . , steinz j.a . , anderegg m.e . , et al . , 1996 , a&a 315 , l27 knacke r.f . , fajardo - acosta s.b . , telesco c.m . , et al . , 1993 , apj 418 , 440 koike c. , hasegawa h. , asada n. , hattori t. , 1981 , ap&ss 79 , 77 lambert d.l . , hinkle k.h . , smith v.v . , 1990 , aj 99 , 1612 le bertre t. , 1992 , a&as 94 , 377 le bertre t. , 1993 , a&as 97 , 729 le bertre t. , deguchi s. , nakada y. , 1990 , a&a 235 , l5 lemke d. , klaas u. , abolins j. , et al . , 1996 , a&a 315 , l64 little - marenin i.r . , 1986 , apj 307 , l15 little - marenin i.r . , little s.j . , 1988 , apj 333 , 305 little - marenin i.r . , little s.j . , 1990 , aj 99 , 1173 little - marenin i.r . , benson p.j . , dickinson d.f . , 1987 , apj 330 , 828 lloyd evans t. , 1984 , mnras 208 , 447 lloyd evans t. , 1990 , mnras 243 , 336 loup c. , zijlstra a.a . , waters l.b.f.m . , groenewegen m.a.t . , 1997 , a&as 125 , 419 marigo p. , girardi l. , chiosi c. , 1996 , a&a 316 , l1 marigo p. , bressan a. , chiosi c. , 1998 , a&a 331 , 564 noguchi k. , murakami h. , matsuo h. , et al . , 1990 , pasj 42 , 441 onaka t. , de jong t. , willems f.j . , 1989 , a&a 218 , 169 papoular r. , pgouri b. , 1983 , a&a 128 , 335 sandford s.a . , 1988 , in : dust in the universe , eds . bailey & d.a . williams , cambridge university press , p193 speck a.k . , barlow m.j . , skinner c.j . , 1997 , mnras 288 , 431 trams n.r . , van loon j.th . , waters l.b.f.m . , et al . , 1999 , submitted to a&a van loon j.th . , zijlstra a.a . , whitelock p.a.w . , et al . , 1998 , a&a 329 , 169 van loon j.th . , zijlstra a.a . , groenewegen m.a.t . , 1999 , a&a in press vassiliadis e. , wood p.r . , 1993 , apj 413 , 641 whitelock p.a . , feast m.w . , menzies j.w . , catchpole r.m . , 1989 , mnras 238 , 769 willems f.j . , de jong t. , 1986 , apj 309 , l39 wood p.r . , bessell m.s . , fox m.w . , 1983 , apj 272 , 99 yamamura i. , de jong t. , justtanont k. , cami j. , waters l.b.f.m . , 1997 , in : first iso workshop on analytical spectroscopy , eds . a.m. heras , k.j . leech , n.r . trams & m. perry , esa sp-419 , p313 zijlstra a.a . , loup c. , waters l.b.f.m . , et al . , 1996 , mnras 279 , 32
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we describe iso observations of the obscured asymptotic giant branch ( agb ) star iras04496@xmath06958 in the large magellanic cloud ( lmc ) .
this star has been classified as a carbon star .
our new isocam cvf spectra show that it is the first carbon star with silicate dust known outside of the milky way .
the existence of this object , and the fact that it is one of the highest luminosity agb stars in the lmc , provide important information for theoretical models of agb evolution and understanding the origin of silicate carbon stars .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
exo 0748676 is an intensively studied low - mass x - ray binary that was initially discovered with the european x - ray observatory satellite ( _ exosat _ ) in 1985 february @xcite . however , in retrospect the source already appeared active in _ exosat _ slew survey observations several times beginning 1984 july @xcite , whereas the earliest detection dates back to 1980 may , when exo 0748676 was serendipitously observed with the _ einstein _ satellite @xcite . the system exhibits irregular x - ray dips and displays eclipses that last for @xmath4 min and recur every 3.82 hr , which allow the unambiguous determination of the orbital period of the binary @xcite . the detection of type - i x - ray bursts ( e.g. , * ? ? ? * ) conclusively identify the compact primary as a neutron star . a few x - ray bursts have been observed that exhibited photospheric radius expansion ( pre ) , which indicates that the eddington luminosity is reached near the burst peak and allows for a distance estimate towards the source @xcite . for a helium - dominated photosphere , a distance of @xmath5 kpc can be derived , while assuming solar composition results in a distance estimate of @xmath6 kpc @xcite . the rise time and duration of the pre bursts observed from exo 0748676 suggest pure helium ignition , rendering 7.4 kpc as the best distance estimate @xcite , although this value is subject to several uncertainties @xcite . at the time of its discovery , exo 0748676 was detected at 210 kev luminosities of @xmath7 @xcite . however , during the _ einstein _ observation of 1980 , several years prior to the _ exosat _ detections , it displayed a 0.510 kev luminosity of @xmath8 @xcite . the source can therefore be classified as a transient x - ray binary . nevertheless , such systems typically exhibit accretion outbursts that last only weeks to months ( e.g. , * ? ? ? * ) , whereas exo 0748676 was persistently detected at luminosities of @xmath9 by various satellites for over 24 years . similar prolonged accretion episodes continuing for years to decades have been observed for a few other systems , which are termed quasi - persistent x - ray binaries ( e.g. , * ? ? ? * ) . in 2008 september , observations with the proportional counter array ( pca ) onboard the _ rossi x - ray timing explorer _ ( _ rxte _ ) and _ swift _ s x - ray telescope ( xrt ) indicated that the x - ray flux of exo 0748676 was declining @xcite . optical and near - ir observations of the optical counterpart , uy vol , performed in 2008 october showed that the optical emission had also faded compared to the brighter x - ray state @xcite . these events indicated that the accretion was ceasing and that the system was transitioning from outburst to quiescence . this is also illustrated by fig . [ fig : asm ] , which displays the x - ray lightcurve of exo 0748676 as observed with the all - sky monitor ( asm ) onboard _ rxte _ since 1996 . the decrease in source activity is clearly seen around @xmath10 days . chandra _ observations carried out in 2008 mid - october ( i.e. , after the transition to quiescence started ) revealed an x - ray spectrum composed of a soft , thermal component joined by a hard powerlaw tail that dominates the spectrum above @xmath11 kev ( * ? ? ? * see also section [ subsec : spectraldata ] ) . this is frequently seen for neutron star x - ray binaries in quiescence ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? the non - thermal component is usually well - fitted by a simple powerlaw with index 12 ( e.g. , * ? ? ? the fractional contribution of the hard powerlaw tail to the 0.510 kev x - ray flux widely varies amongst sources and possibly also with changing luminosity @xcite . the physical process that is responsible for the powerlaw spectral component remains elusive ( see e.g. , * ? ? ? * ; * ? ? ? although the soft spectral component has been ascribed to low - level accretion @xcite , it is most often interpreted as thermal surface radiation from the cooling neutron star @xcite . according to this model , the accretion of matter compresses the neutron star crust , which induces a series of electron captures , neutron emissions and pycnonuclear fusion reactions ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the heat energy released in these processes is spread over the neutron star via thermal conduction . the neutron star cools primarily via neutrino emissions from the stellar core , as well as photon radiation from the surface . the former depends on the equation of state of cold nuclear matter and the central density of the neutron star ( e.g. , * ? ? ? * ; * ? ? ? the neutron star core reaches a thermal steady state in @xmath12 years , yielding an incandescent emission from the neutron star surface set by the time - averaged accretion rate of the system , as well as the rate of neutrino emissions from the stellar core ( e.g. , * ? ? ? * ; * ? ? ? when combined with estimates of the outburst history , observations of quiescent neutron stars can constrain the rate of neutrino emissions , thereby providing insight into the interior properties of the neutron star ( e.g. , * ? ? ? once the steady state is reached , the neutron star core temperature will not change appreciably during a single outburst , but the temperature of the crust can be dramatically altered . in regular transients that have a typical outburst duration of weeks to months , the crustal heating processes will only cause a slight increase in the crust temperature @xcite . however , in quasi - persistent x - ray binaries the prolonged accretion episodes can cause a significant temperature gradient between the neutron star crust and core . once the accretion ceases , the crust is expected to thermally relax on a time scale of years , until equilibrium with the core is re - established @xcite . during the initial stages of the quiescent phase the thermal emission will therefore be dominated by the cooling crust , whereas eventually a quiescent base level is reached that is set by the thermal state of the core @xcite . this provides the special opportunity to separately probe the properties of the neutron star crust @xcite . in 2001 , the neutron star x - ray binaries ks 1731260 and mxb 165929 both made the transition to quiescence , following accretion episodes of 12.5 and 2.5 years , respectively @xcite . more recently , in 2007 , the @xmath13-year long outburst of xte j1701462 came to a halt @xcite . all three systems were subsequently monitored with _ chandra _ and _ xmm - newton _ , which revealed that thermal flux and neutron star temperature were gradually decreasing over the course of years ( see also section [ sec : discussion ] ) . this can be interpreted as cooling of the neutron star crust that has been heated during the prolonged accretion outburst . successful modelling of the observed quiescent x - ray lightcurves with neutron star thermal evolution models supports this hypothesis and provides important constraints on the crust properties , such as the thermal conductivity @xcite . along these lines we have pursued an observational campaign of exo 0748676 to study the time evolution of the quiescent x - ray emission following its long accretion outburst . in @xcite , we discussed _ chandra _ and _ swift _ observations obtained between 2008 september 28 and 2009 january 30 . we found a relatively hot and luminous quiescent system with a temperature of @xmath14 kev and a thermal 0.01100 kev luminosity of @xmath15 . no clear decrease in effective temperature and thermal bolometric flux was found over the five - month time span . in this paper we report on continued _ swift _ and _ chandra _ observations of exo 0748676 during its quiescent state . in addition , we include an archival _ xmm - newton _ observation performed @xmath16 months after the cessation of the outburst . chandra _ and _ swift _ observations discussed by @xcite were re - analysed in this work in order to obtain a homogeneous quiescent lightcurve . table [ tab : obs ] gives an overview of all new observations of exo 0748676 discussed in this paper . a list of earlier _ chandra _ and _ swift _ observations obtained during the quiescent phase can be found in @xcite . .observation log . [ cols= " < , < , < , < " , ] [ tab : spec ] note . the observations marked by a dagger were already discussed in @xcite , but re - fitted in this work . these results were obtained by using a combined absorbed nsatmos and powerlaw model , where @xmath17 , @xmath18 , @xmath19 km , @xmath20 kpc and @xmath21 were kept fixed . the quoted errors represent 90 percent confidence levels . @xmath22 represents the 0.510 kev total model flux and @xmath23 gives the 0.01100 kev nsatmos flux ; both are unabsorbed and in units of @xmath24 . @xmath25 gives the 0.01100 kev luminosity of the nsatmos model component in units of @xmath26 and assuming a source distance of d=7.4 kpc . @xmath27 represents the time interval of the observations in days and the fractional powerlaw contribution is given in a percentage of the total unabsorbed 0.510 kev flux . as discussed in section [ subsec : spectraldata ] , the quiescent spectrum of exo 0748676 can be described by a combination of a neutron star atmosphere model and a non - thermal powerlaw tail . we fitted the _ chandra _ and _ xmm - newton _ data simultaneously within xspec to a combined nsatmos and powerlaw model subject to interstellar absorption , to explore the best - fit values for the neutron star mass and radius , source distance and hydrogen column density . we include the first set of _ chandra _ observations obtained in 2008 october ( discussed in * ? ? ? * ) in the analysis . as before , we use the phabs model with the default xspec abundances and cross - sections to take into account the neutral hydrogen absorption along the line of sight . the powerlaw index is fixed to @xmath21 ( the best fit - value obtained from _ xmm - newton _ observations ; see section [ subsec : spectraldata ] ) , because there are not sufficient counts at higher energies in the _ chandra _ spectra to allow this component to vary . the powerlaw normalisation is left as a free parameter . if the neutron star mass and radius are fixed to canonical values of @xmath18 and @xmath28 km , and in addition the source distance is fixed to @xmath20 kpc , the hydrogen column density pegs at its lower limit ( @xmath29 ) . when the distance is left to vary freely , the best - fit value is @xmath30 kpc , which is just outside the range obtained from x - ray burst analysis ( 58.3 kpc ; * ? ? ? therefore , we choose to keep the distance fixed at 7.4 kpc , and instead allow the neutron star radius to vary . this way , we obtain best - fit values of @xmath31 and @xmath32 km . if additionally the neutron star mass is left free to vary in the fit , this parameter is not strongly constrained ( @xmath33 ) . in the final fits we choose to fix the neutron star mass to @xmath34 , because otherwise the uncertainty in this quantity will dominate the errors of the other parameters . for the final spectral analysis , we fit all _ xmm - newton _ , _ chandra _ and _ swift _ data with an absorbed nsatmos plus powerlaw model , where @xmath17 , @xmath18 , @xmath19 km , @xmath20 kpc and @xmath21 are fixed , while the neutron star effective temperature is left as a free parameter . the powerlaw normalisation is left to vary freely for the _ chandra _ and _ xmm - newton _ observations , but fixed for the _ swift _ data ( so that this component contributes 10 percent to the total unabsorbed 0.510 kev flux ) . we fit all data in the 0.510 kev energy range and deduce the absorbed and unabsorbed fluxes in this band . the thermal model fit is extrapolated to the energy range of 0.01100 kev to estimate the thermal bolometric flux . the results from fitting the x - ray spectra in this way are presented in table [ tab : spec ] . the effective temperatures and thermal bolometric fluxes derived from _ chandra _ , _ swift _ and _ xmm - newton _ data are displayed in fig . [ fig : temp ] . examination of fig . [ fig : temp ] suggests that there is a small but discernible offset in the thermal flux and neutron star temperature as deduced from the different satellites . this is briefly discussed in section [ subsec : crosscal ] . ) . , width=302 ] fig . [ fig : temp ] clearly reveals a decaying trend in thermal flux and temperature . to investigate the decay shape , we fit the temperature curve with an exponential decay function of the form @xmath35 , where @xmath36 is a normalisation constant , @xmath37 is the start time of the cooling curve and @xmath38 the e - folding time . given the apparent offset between the different instruments ( see section [ subsec : crosscal ] ) , we perform different fits to the _ chandra _ and _ swift _ data . we fix @xmath37 to 2009 september 5 ( mjd 54714 ) , which is in between the first non - detection by _ rxte_/pca and the first _ swift_/xrt observation of the source @xcite . the simple exponential decay , represented by the dotted lines in fig . [ fig : temp_chanfit ] , yields an e - folding time of @xmath39 days for the _ chandra _ data , but does not provide a good fit ( @xmath40 for 2 d.o.f . ) . for the _ swift _ lightcurve we find @xmath41 days ( @xmath42 for 12 d.o.f . ) . if we include a constant offset ( i.e. , @xmath43 ; solid lines in fig . [ fig : temp_chanfit ] ) , we obtain a better fit for the _ chandra _ data , yielding a normalisation of @xmath44 ev , an e - folding decay time of @xmath45 days and a constant offset of @xmath46 ev ( @xmath47 for 1 d.o.f . ) . for the _ swift _ data we find @xmath48 ev , @xmath49 days and @xmath50 ev ( @xmath51 for 11 d.o.f . ) , which is consistent with the _ chandra _ fit . although an exponential decay provides an adequate description of the data of exo 0748676 , as has been found for other sources ( e.g. , * ? ? ? * ; * ? ? ? * ) , mathematically a neutron star crust is expected to cool via a ( broken ) powerlaw @xcite . if we fit a single powerlaw of the form @xmath52 to the _ chandra _ data , we find an index of @xmath53 and a normalisation of @xmath54 ev ( @xmath55 for 2 d.o.f . ) . for the _ swift _ observations we find @xmath56 and @xmath57 ev ( @xmath58 for 12 d.o.f . ) . these powerlaw fits are indicated by the dashed lines in fig . [ fig : temp_chanfit ] . a broken powerlaw also yields an acceptable fit to the _ swift _ data ( @xmath59 for 10 d.o.f . ) . we find a normalisation of @xmath60 ev , a break at @xmath61 days and decay indices of @xmath62 and @xmath63 before and after the break , respectively . this fit is indicated by the dashed - dotted curve in fig . [ fig : temp_chanfit ] . there are not sufficient _ chandra _ observations to fit a broken powerlaw decay . we note that the shape of the decay curve of exo 0748676 is not strongly affected by our choice of spectral parameters ( @xmath64 , @xmath65 , @xmath66 , and @xmath67 ) or assumed distance ( see also previous studies by e.g. , * ? ? ? * ; * ? ? ? the quiescent lightcurve presented in fig . [ fig : temp ] shows indications that the thermal flux and temperature inferred from the _ chandra _ observations lie below the trend of the _ swift _ data points . this possible shift ( @xmath68 percent for the flux lightcurve ) may be due to cross - calibration issues between the two satellites . a study of the crab nebula indeed revealed an offset between _ chandra _ and _ swift _ , whereas such a discrepancy was not found between _ swift _ and _ xmm - newton _ this might be reflected in our results as well , since the _ xmm - newton _ data point appears to line up with the trend indicated by the _ swift _ data . however , our _ chandra _ and _ swift _ data points may also be ( partly ) offset due to the fact that we can not constrain the powerlaw component in the _ swift _ data , which we therefore fixed to contribute @xmath69 percent of the total 0.510 kev unabsorbed flux ( see section [ subsec : spectraldata ] ) . we discuss _ chandra _ , _ swift _ and _ xmm - newton _ observations obtained after the cessation of the very long ( @xmath7024 year ) active period of exo 0748676 . fitting the spectral data with a neutron star atmosphere model nsatmos , did not reveal clear indications of a changing thermal spectrum during the first five months of the quiescent phase @xcite . however , now that the quiescent monitoring has extended to 19 months ( 1.6 years ) , we find a significant decrease in neutron star effective temperature from @xmath71 to @xmath3 ev . the thermal bolometric flux was observed to decay from @xmath72 to @xmath73 . in addition to a soft , thermal component , the _ chandra _ and _ xmm - newton _ observations show evidence for a hard powerlaw tail with index @xmath21 . the fractional contribution of the hard spectral component to the total unabsorbed 0.510 kev flux initially decreased from @xmath74 percent in 2008 october to @xmath75 percent in 2009 june . however , observations carried out in 2010 april suggest that the powerlaw fraction increased again to @xmath76 percent . similar behaviour has been observed for several other quiescent neutron star systems @xcite , although others show more irregular behaviour @xcite . in cen x-4 , the powerlaw tail in the quiescent spectrum shows variations that appear to be linked to changes in the thermal component , possibly caused by low - level accretion @xcite . the gradual decrease in thermal flux and neutron star temperature observed for exo 0748676 can be interpreted as the neutron star crust cooling down in quiescence after it has been heated during its long accretion outburst . [ fig : sources ] compares our data of exo 0748676 with the crust cooling curves observed for the neutron star x - ray binaries ks 1731260 , mxb 165929 and xte j1701462 . this plot shows that the amount of cooling following the end of the outburst is markedly smaller for exo 0748676 than for the other three sources . we have observed our target over the first 19 months after the cessation of the outburst and during this time the thermal bolometric flux has decreased by a factor of @xmath77 . in a similar time span , the thermal bolometric fluxes of ks 1731260 , mxb 165929 and xte j1701462 had decreased by a factor of @xmath78 , 6 and 2.5 , respectively ( see * ? ? ? * ; * ? ? ? * ) . the effective neutron star temperature of exo 0748676 has decreased by about 10 percent , compared to @xmath79 , 40 and 20 percent for ks 1731260 , mxb 165929 and xte j1701462 . although the observed fractional changes in neutron star temperature and thermal bolometric flux are smaller for exo 0748676 than for the other three sources , the decay itself may not be markedly different . the quiescent lightcurves of ks 1731260 , mxb 165929 and xte j1701462 can be fit with an exponential decay function levelling off to a constant value , yielding e - folding times of @xmath80 , @xmath81 and @xmath82 days , respectively @xcite . for the _ chandra _ data of exo 0748676 , we find an e - folding time of @xmath83 days ( see section [ subsec : decay ] ) . these decay times provide a measure of the thermal relaxation time of the neutron star crust , which depends on the composition and structure of the lattice , the distribution of heating sources and the thickness of the crust ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? @xcite and @xcite calculate theoretical cooling curves for ks 1731260 , assuming different physics for the crust and core . these authors present simulations for both an amorphous crust and an ordered crystalline lattice . for the latter , the spread of nuclide charge numbers ( @xmath84 ) in the crust matter is small , which is referred to as a low level of impurities and results in a highly conductive crust . a large number of impurities gives an amorphous structure , which affects the thermal properties of the crust and results in a low conductivity . in addition , @xcite explore standard ( i.e. , slow ) and enhanced neutrino cooling mechanisms , yielding different core temperatures . comparing our results on exo 0748676 with the decay shapes resulting from those calculations suggests that the neutron star has a highly conductive crust , similar to what has been inferred for the other three sources @xcite . the fact that the decay curve of exo 0748676 is rather shallow may be explained in terms of a relatively small temperature gradient and thus lower thermal flux across the core - crust boundary ( cf . the model curves for a highly conductive crust and different core temperatures presented by * ? ? ? . this can be due to a combination of a warm neutron star core and a relatively low mass - accretion rate during outburst . the exponential decay fit to the _ chandra _ data of exo 0748676 indicates that the neutron star crust might already be close to restoring equilibrium with the core . the fit results in a quiescent base level of @xmath85 ev , while we found a temperature of @xmath86 ev for the observation performed in 2010 april . prior to its last outburst , exo 0748676 was observed in quiescence with the _ einstein _ observatory , displaying a 0.510 kev unabsorbed flux of @xmath87 @xcite . our _ chandra _ observations of 2010 april detected exo 0748676 at a 0.510 kev unabsorbed flux of @xmath88 ( see table [ tab : spec ] ) . assuming that the _ einstein _ detection caught exo 0748676 at its quiescent base level , this supports the idea that the crust has nearly cooled down . this would imply that the neutron star core in exo 0748676 is relatively hot ( cf . * ) , suggesting that either standard cooling mechanisms are operating and that the neutron star is not very massive , or that the time - averaged mass - accretion rate of the system is very high due to a short recurrence time ( see below ) . the energy deposited during outburst is given by @xmath89 ( e.g. , * ? ? ? * ; * ? ? ? , @xmath90 mev is the nuclear energy deposited per accreted baryon @xcite , @xmath91 is the atomic mass unit and @xmath92 is the time - averaged accretion rate of the system . the latter can be expressed as @xmath93 , where @xmath94 is the average accretion rate during outburst episodes , @xmath95 is the outburst duration and @xmath96 is the system s recurrence time . the factor @xmath97 represents the duty cycle of the system . the neutron star core is expected to be in a steady state , in which the energy radiated during quiescence balances the heat deposited during outburst . we can thus obtain an estimate of the duty cycle of exo 0748676 by equating the heating and cooling rates . a neutron star cools primarily via photon radiation from the surface and neutrino emissions from the stellar core . if the lightcurve of exo 0748676 has indeed ( nearly ) levelled off , the bolometric luminosity emitted as photons is thus @xmath98 ( as measured during the _ chandra _ observation of 2010 april ) . the rate of neutrino emissions depends on the temperature of the neutron star core , which can be estimated from the effective surface temperature once the crust has thermally relaxed . a quiescent base level of @xmath99 ev ( as suggested by exponential decay fits to the _ chandra _ data ) , implies an effective surface temperature in the neutron star frame of @xmath100 ev ( @xmath101 k ) , for a canonical values of @xmath18 and @xmath28 km ( i.e. , @xmath102 ) . using the relation between the effective surface temperature and the interior temperature calculated by @xcite , yields @xmath103 k. for such a core temperature , the minimum energy escaping the neutron star as neutrino s ( i.e. , assuming standard core cooling ) is @xmath104 @xcite . equating the energy losses via photon radiation from the neutron star surface ( @xmath105 ) and neutrino emissions from the stellar core ( @xmath106 ) with the energy gained via crustal reactions during outburst ( @xmath107 ) , suggests that exo 0748676 must have a time - averaged mass - accretion rate of @xmath108 . during outburst , exo 0748676 displayed an average bolometric luminosity of @xmath109 @xcite . assuming that the accretion luminosity is given by @xmath110 , this translates into a mass - accretion rate during outburst of @xmath111 for a canonical neutron star with @xmath112 and @xmath113 km . if the crust has indeed thermally relaxed , the above estimates show that exo 0748676 must have a duty cycle of @xmath114 percent to explain the observed quiescent bolometric luminosity of @xmath115 in terms of thermal emission from the cooling neutron star ( i.e. , opposed to continued accretion ) . the outburst of exo 0748676 started between 1980 may and 1984 july and the system returned to quiescence in 2008 september , i.e. , @xmath116 years . if the observed outburst is typical for the long - term behaviour of this source , the expected recurrence time is thus @xmath117 years . in case the neutron star cools via more efficient core neutrino emission processes , the recurrence time required to explain the observed quiescent luminosity is shorter ( i.e. , the duty cycle is higher ) . although the above calculation is only a crude approximation ( e.g. , there is a significant uncertainty in the relation between the surface- and interior temperature of the neutron star , depending on the atmospheric composition and the depth of the light element layer ; * ? ? ? * ) , it illustrates that exo 0748676 must have a high duty cycle if the cooling curve has indeed reached its quiescent base level . @xcite , @xcite and @xcite have suggested that exo 0748676 continues to accrete in quiescence , because the quiescent luminosity inferred from the 1980 _ einstein _ observation is higher than predicted by standard cooling models . however , these conclusions are based on an assumed duty cycle of @xmath118 percent , but we have no a priori knowledge about this . although we can not exclude that the system is indeed accreting in quiescence , the above estimates show that a duty cycle of @xmath114 percent can explain the observed quiescent level of exo 0748676 as being due to thermal emission from the cooling neutron star . a duty cycle of @xmath114 percent is high , although not unprecedented for neutron star transients ( e.g. , * ? ? ? * ; * ? ? ? recently , @xcite demonstrated that the cooling of a neutron star crust is expected to follow a broken powerlaw decay . a break is predicted to occur due to a transition in the crystal structure of the crust matter , and the slope before the break reflects the heat flux from the outer crustal layers . therefore , we also fitted the neutron star temperatures obtained for exo 0748676 to a powerlaw and found decay indices of @xmath119 and @xmath120 for the _ chandra _ and _ swift _ data sets , respectively . the _ swift _ observations indicate that a possible break in the quiescent lightcurve may have occurred @xmath121 days after the cessation of the outburst ( see section [ subsec : decay ] ) . by fitting a broken powerlaw function , we obtain a decay index of @xmath62 before the break , which steepens to @xmath63 thereafter . however , since these slopes are consistent with being equal , further observations are required to confirm whether a break has indeed occurred . the decay parameters that we find for exo 0748676 are comparable to that obtained by @xcite for xte j1701462 . these authors found that the quiescent lightcurve breaks @xmath122 days post - outburst and report decay indices of @xmath123 and @xmath124 before and after the break , respectively . @xcite note that possible cross - calibration effects between _ chandra _ and _ xmm - newton _ might introduce small shifts that also allow a single powerlaw decay with slope @xmath125 . the cooling curves of ks 1731260 and mxb 165929 appear to have steeper decays with indices of @xmath126 and @xmath127 , respectively @xcite . due to the scarcity of data points it is unclear whether a break occurred in the quiescent lightcurves of the latter two sources @xcite . the powerlaw fits show no indications that the quiescent lightcurve of exo 0748676 is levelling off . thus , it is also possible that the neutron star temperature continues to decay further and that the core is cooler than suggested by the exponential decay fits and the 1980 _ einstein _ detection . the relatively slow decrease of exo 0748676 might then reflect that the crust has a high conductivity , albeit lower than that of the neutron stars in ks 1731260 and mxb 165929 . further observations are thus required to determine whether the neutron star crust in exo 0748676 has nearly cooled down and to be able to draw firm conclusions on the crust and core properties . this work was supported by the netherlands organisation for scientific research ( nwo ) and made use of the _ swift _ public data archive . we acknowledge _ swift _ pi n. gehrels and the _ swift _ planning team for their help in carrying out the too campaign . emc was supported by nasa through the chandra fellowship program . mtw , psr and ksw acknowledge the united states office of naval research . jh and whgl acknowledge support from chandra grant go8 - 9045x . p. a. , beardmore a. p. , page k. l. , tyler l. g. , osborne j. p. , goad m. r. , obrien p. t. , vetere l. , racusin j. , morris d. , burrows d. n. , capalbi m. , perri m. , gehrels n. , romano p. , 2007 , , 469 , 379 g. p. , bautz m. w. , ford p. g. , nousek j. a. , ricker jr . g. r. , 2003 , in j. e. truemper & h. d. tananbaum ed . , society of photo - optical instrumentation engineers ( spie ) conference series vol . 4851 of presented at the society of photo - optical instrumentation engineers ( spie ) conference , advanced ccd imaging spectrometer ( acis ) instrument on the chandra x - ray observatory . pp 2844 p. g. , 2008 , in c. bassa , z. wang , a. cumming , & v. m. kaspi ed . , 40 years of pulsars : millisecond pulsars , magnetars and more vol . 983 of american institute of physics conference series , constraining the neutron star equation of state using quiescent low - mass x - ray binaries .
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in late 2008 , the quasi - persistent neutron star x - ray transient and eclipsing binary exo 0748676 started a transition from outburst to quiescence , after it had been actively accreting for more than 24 years . in a previous work
, we discussed _
chandra _ and
_ swift _ observations obtained during the first five months after this transition . here
, we report on further x - ray observations of exo 0748676 , extending the quiescent monitoring to 1.6 years .
_ chandra _ and _ xmm - newton _ data reveal quiescent x - ray spectra composed of a soft , thermal component that is well - fitted by a neutron star atmosphere model . an additional hard powerlaw tail is detected that changes non - monotonically over time , contributing between 4 and 20 percent to the total unabsorbed 0.510 kev flux .
the combined set of _ chandra _ , _ xmm - newton _ and _ swift _
data reveals that the thermal bolometric luminosity fades from @xmath0 to @xmath1 , whereas the inferred neutron star effective temperature decreases from @xmath2 to @xmath3 ev .
we interpret the observed decay as cooling of the neutron star crust and show that the fractional quiescent temperature change of exo 0748676 is markedly smaller than observed for three other neutron star x - ray binaries that underwent prolonged accretion outbursts .
[ firstpage ] accretion , accretion disks - binaries : eclipsing - stars : individual ( exo 0748676 ) - stars : neutron - x - rays : binaries
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recently the evidences of neutrino oscillations are strongly supported by both of the atmospheric @xcite and the solar neutrino experiments @xcite . the former suggests an almost maximal lepton flavor mixing between the 2nd and the 3rd generations , while the favorable solution to the solar neutrino deficits is given by large mixing angle solution between the 1st and the 2nd generations ( lma , low or vo ) @xcite . neutrino oscillation experiments indicate that the neutrinos have tiny but finite masses , with two mass squared differences @xmath1 . since the neutrino mass is much smaller than other quarks and leptons , the origin of the neutrino mass is expected to be different from that of others . the most popular mechanism to induce the small neutrino masses is the seesaw mechanism @xcite , which needs heavy right - handed neutrinos . however , it is important to consider also other possible scenarios which can explain the small neutrino masses , especially low - energy extensions of the standard model . the zee model @xcite is such an alternative , which induces small neutrino masses by radiative corrections , where the lepton number is violated explicitly . generally , if there are lepton number violating interactions , care has to be taken for these interactions not to be too strong at the temperatures above the electroweak scale , since the baryon asymmetry generated in the early universe might be washed out by the `` sphaleron '' process @xcite , which violates a linear combination of baryon ( @xmath2 ) and lepton ( @xmath3 ) number , @xmath4 . if the rate of the lepton number violating process becomes faster than the hubble expansion rate @xmath5 during the epoch when the sphaleron process is in thermal equilibrium , the baryon asymmetry generated at higher temperature would be washed out , and there would be no matter anti - matter asymmetry in the present universe , which conflicts with the observation . in this letter , we analyze the cosmological condition in order for the baryon asymmetry generated in the early universe not to be washed out in the zee model . ( we will not discuss the production mechanism of the baryon asymmetry . instead , we just assume that the desired amount of the baryon asymmetry is generated in the early universe and discuss whether or not this baryon asymmetry can be preserved against the lepton number violating interaction in the zee model together with the sphaleron process . ) in most part of the analysis , we take the low solution to the solar neutrino problem . ( we shall note on the cases of other solutions in the last section . ) we find that the baryon asymmetry is _ not _ washed out , although it has been said that the zee model can not preserve the baryon asymmetry @xcite . this can be easily shown from the view point of a new lepton number @xmath0 . let us give a brief overview of the zee model @xcite at first . the zee model is a simple extension of the standard model , which has two higgs doublets @xmath6 ( @xmath7 ) and one @xmath8-singlet charged higgs field ( zee singlet ) @xmath9 . the zee model has the following interactions in addition to the standard model ones ; @xmath10 where @xmath11 denote the left - handed lepton doublets with flavor indices @xmath12 . notice that the coupling @xmath13 is anti - symmetric for the flavor index . for simplicity , we have omitted the higgs potential @xmath14 which are irrelevant to the following discussion . as for the yukawa interaction , we assume only @xmath15 couples to lepton fields as , @xmath16 where @xmath17 . as can be seen from eqs . ( [ zee_lag ] ) and ( [ l - yukawa ] ) , the lepton number @xmath3 is explicitly violated in the zee model . and @xmath18 , @xmath19 , in @xmath20 . ] in the charged higgs sector , the zee singlet @xmath21 is mixed with the charged higgs boson @xmath22 through the @xmath23 interaction in eq . ( [ zee_lag ] ) ; @xmath24 here , @xmath25 is the charged higgs boson which is orthogonal to the would - be goldstone boson after the neutral higgs fields acquire vacuum - expectation values ( vevs ) @xmath26 , where @xmath27/@xmath28 . we denote the mass eigenstates of the charged higgs sector as @xmath29 and their mass @xmath30 . ( 5,5 ) ( -2,0 ) ( -2,1.7)@xmath31 ( 0,.5)@xmath32 ( 2,-.5)@xmath33 ( 1.2,.7)@xmath34 ( 3.1,.7)@xmath35 ( 4.2,.5)@xmath36 ( 5.7,1.7)@xmath37 ( 2.3,1.5)@xmath36 ( 2,4.5)@xmath38 ( .5,3)@xmath39 ( 3.8,3)@xmath40 ( 2.3,2.7)@xmath41 in the zee model , neutrino masses are generated by radiative corrections , as shown in fig . [ fig : zee_mass ] , and hence this model could provide an explanation of the smallness of neutrino masses . the element of the mass matrix , generated by radiative correction at one loop level , is given by @xmath42 here @xmath43 are the charged lepton masses . in eq . ( [ zee_mass ] ) , we have used @xmath44 . since the coupling constants @xmath45 are antisymmetric for the indices @xmath46 , the mass matrix eq . ( [ zee_mass ] ) is traceless . the above zee mass matrix has been analyzed @xcite in the light of recent neutrino - oscillation experiments , and it was shown that it must be the following bi - maximal form to explain the experimental results ; @xmath47 where @xmath48 , @xmath49 , @xmath50 and @xmath51 . the eigenvalues and the mns matrix induced from the neutrino mass matrix eq . ( [ zee_mass_matrix ] ) is given by @xmath52 and @xmath53 the neutrino mass hierarchy is shown in fig . [ fig : mass ] . the mixing angle between the 1st and the 2nd generations is maximal as well as the mixing angle between the 2nd and the 3rd generations ( @xmath54 ) . from oscillation experiments , the eigenvalues of the mass matrix have to satisfy the following relations , @xmath55 ( 6,5 ) ( 0,0)@xmath56 ( 1,.1)(1,0)4 ( 0,3.1)@xmath57 ( 1,3.2)(1,0)4 ( 0,3.7)@xmath58 ( 1,3.8)(1,0)4 ( 5.3,3.2)(5.5,3.5)(5.3,3.8 ) ( 6.5,3.5)(7.2,1.8)(6.5,.1 ) ( 5.5,3.4)@xmath59 ( 7.2,1.7)@xmath60 then , the relations among @xmath61 , @xmath62 and @xmath63 should be @xmath64 from eq . ( [ zee_mass ] ) and explicit values of @xmath65 , @xmath66 , @xmath60 and @xmath67 . here we take the low solution to the solar neutrino problem . these relations induce the ratio of @xmath68 and @xmath69 as @xmath70 the phenomenological constraints on @xmath45 from various experimental bounds @xcite are given by @xmath71 where @xmath72 is fermi constant and @xmath73 equations ( [ f_emu ] ) and ( [ other ] ) mean that @xmath74 can not be of @xmath75 unless the charged higgs boson masses are of order 10 tev . notice that the constraints for @xmath76 and @xmath77 in eq . ( [ other ] ) are automatically satisfied as long as the relation in eq . ( [ hi-2 ] ) and the constraint for @xmath61 in eq . ( [ f_emu ] ) are satisfied . actually , the conditions eqs . ( [ hi-2 ] ) and ( [ f_emu ] ) give rise to the following constraint on the smallest coupling @xmath76 ; @xmath78 which is much severer constraint than that of eq . ( [ other ] ) . now let us turn to discuss the cosmological constraint on the zee model . a crucial point is that a linear combination of the lepton flavors , @xmath79 , is approximately conserved in the zee model . this is because the @xmath80-number is violated only through the coupling @xmath63 , which is much smaller than the other couplings , as can be seen in eq . ( [ hi-2 ] ) . ( here , we assign the @xmath80 number of the zee singlet @xmath21 to be zero . ) notice that the sphaleron preserves not only the total @xmath81 but each @xmath82 ( @xmath83 ) . thus , the number @xmath84 is also conserved under the sphaleron process , and is violated only by the tiny coupling @xmath63 . when the temperature @xmath85 is higher than the mass of @xmath21 , @xmath86 , the relevant interactions which violate @xmath80 are ; @xmath87 , @xmath88 , and @xmath89 . we consider the three - body process @xmath90 , since it gives the severest constraint on the coupling @xmath63 . , the four - body process @xmath91 can give a severer constraint on @xmath63 than that of the above three - body process . in this case , we must take care the possibility that the condition @xmath92 might make the physical higgs masses , @xmath93 , be negative . if the condition @xmath94 and @xmath95 is satisfied , we can analyze the out - of - equilibrium condition of this four - body process in a similar way to the three - body case . however , the following conclusion does not change , since the coupling @xmath63 is suppressed by @xmath96 for @xmath97 [ see eq . ( [ f_mutau ] ) ] and hence the out - of - equilibrium condition is satisfied more easily than the three - body case . ] the rate of this process is given by @xmath98 this @xmath80-violating interaction is out of equilibrium if the above rates is slower than the hubble parameter of the expanding universe , @xmath99 . ( @xmath100 gev is the reduced planck scale and @xmath101 is the number of relativistic degrees of freedom . ) namely , the out - of - equilibrium condition is given by @xmath102 on the other hand , as discussed in the previous section , the coupling @xmath63 is very small in order to explain the neutrino oscillation experiments . hereafter , we assume @xmath103 for simplicity . in this case , we could consider @xmath104 . then , from eqs . ( [ zee_mass]),([mass_atm ] ) , and ( [ mass_sol ] ) , the coupling @xmath63 is given by @xmath105 where we take @xmath106 and @xmath107 . here we take the explicit values of @xmath41 and @xmath108 , for example , as @xmath109 and @xmath110 . in this case ( [ ooe - condition ] ) and ( [ f_mutau ] ) show that the @xmath80-violating interactions are really out of equilibrium during @xmath111 . this means that the baryon asymmetry is _ not _ washed out in the case of low solution . this result is not changed , unless @xmath86 is much heavier than @xmath112 or @xmath41 is extremely small . at the lower temperature @xmath113 , the rate of the three - body process is reduced because the number density of the @xmath21 particle is suppressed by a boltzmann factor @xmath114 , and the four - body interactions are also suppressed by a factor @xmath115 . as we have seen in this section , the @xmath80-violating interactions have been out of equilibrium since the birth of our universe . this is because @xmath63 must be small in order to explain the neutrino oscillation experiments . actually , the analysis of chemical potentials in the high - temperature phase gives the following relation between the baryon asymmetry @xmath2 and the number @xmath116 ; @xmath117 therefore the baryon asymmetry in our universe can be preserved , once the asymmetry of @xmath116 is produced in the early universe . recent neutrino experiments indicate that neutrino masses are tiny and there are two mass squared differences as @xmath118 . it is well known that the zee model induces small neutrino masses by radiative corrections , where the lepton number is violated explicitly . if the lepton number violating interaction becomes faster than the hubble parameter before the electroweak phase transition , the baryon asymmetry generated at higher temperature ( in the early universe ) would be washed out . therefore , in order not to destroy the baryon asymmetry , the lepton number violating interaction must be out - of - equilibrium . in this letter we have analyzed the cosmological condition in order for the baryon asymmetry generated in the early universe not to be washed out in the zee model , taking the low solution to the solar neutrino problem . in this case , we find the baryon asymmetry is _ not _ washed out , although it has been said that the zee model can not preserve the baryon asymmetry @xcite . this can be easily shown from the view point of the new lepton number @xmath0 . the lepton number @xmath80 is almost conserved quantity which is violated only by the tiny coupling @xmath76 . from the viewpoint of @xmath80 , the sphaleron process preserves @xmath84 . the @xmath80 violating interaction through @xmath63 is out - of - equilibrium in the zee model when the results from neutrino - oscillation experiments and natural mass scales of higgs masses are used for the input parameters . therefore , we can conclude that once the asymmetry of @xmath119 is produced , the baryon asymmetry is preserved in our universe . we have shown that the baryon asymmetry is preserved against the lepton - number violating interaction in the zee model . however , it is very difficult to explain the origin of the asymmetry of @xmath84 within the zee model . therefore , some mechanism which can generate a @xmath84 asymmetry is necessary . finally , we comment on the cases of other solar neutrino solutions ( lma and vo ) . first , we consider the lma case . it has been pointed out @xcite that the zee model could not explain the lma solution , since the zee model induces a nearly maximal mixing of solar - neutrino oscillation ( @xmath120 ) , which is in poor agreement with the observed data @xcite . if one would still apply the present analysis to the lma solution , the out - of - equilibrium condition of @xmath80 violating interaction is marginally satisfied [ see eqs . ( [ ooe - condition ] ) and ( [ f_mutau ] ) ] , although it requires a more detailed analysis including a numerical solution of the boltzmann equation . the second case is the vo solution . as can be seen in eq . ( [ f_mutau ] ) , the coupling @xmath63 which violates @xmath80 is proportional to the @xmath59 . therefore , in the case of the vo solution , where @xmath59 is much smaller than that of the low solution , the rate of the @xmath80-violating interaction is also small enough , and the baryon asymmetry is preserved as discussed in previous section . we would like to thank t. yanagida for the suggestion of this work . we also thank t. moroi for the collaboration in the early stage of this work , and thank m. tanimoto , y. koide and s. kanemura for useful discussions . nh is supported in part in part by the grant - in - aid for science research , ministry of education , science and culture , japan ( no . 12740146 ) . the work of kh was supported by the japanese society for the promotion of science . y. fukuda _ et al . _ [ kamiokande collaboration ] , phys . rev . * 77 * ( 1996 ) 1683 . y. fukuda _ et al . _ [ kamiokande collaboration ] , phys . b * 335 * ( 1994 ) 237 ; + y. fukuda _ et al . _ [ super - kamiokande collaboration ] , phys . * 81 * ( 1998 ) 1562 [ hep - ex/9807003 ] ; + t. kajita [ super - kamiokande collaboration ] , in _ neutrino physics and astrophysics _ , proceedings of the xviiith international conference on neutrino physics and astrophysics ( neutrino 98 ) , june 4 - 9 , 1998 , takayama , japan , edited by y. suzuki and y. totsuka , ( elsevier science b.v . , amsterdam , 1999 ) page 123 ; nucl . suppl . * 77 * , 123 ( 1999 ) [ hep - ex/9810001 ] . y. fukuda _ et al . _ [ super - kamiokande collaboration ] , + phys . rev . lett . * 81 * ( 1998 ) 1158 [ hep - ex/9805021 ] ; erratum - ibid . * 81 * ( 1998 ) 4279 ; + phys . * 82 * ( 1999 ) 2430 [ hep - ex/9812011 ] ; + phys . * 82 * ( 1999 ) 1810 [ hep - ex/9812009 ] . q. r. ahmad _ et al . _ [ sno collaboration ] , nucl - ex/0106015 . v. barger , d. marfatia and k. whisnant , hep - ph/0106207 . + g. l. fogli , e. lisi , d. montanino and a. palazzo , hep - ph/0106247 ; + j. n. bahcall , m. c. gonzalez - garcia and c. pena - garay , hep - ph/0106258 ; + a. bandyopadhyay , s. choubey , s. goswami and k. kar , hep - ph/0106264 . t. yanagida , _ `` horizontal symmetry and masses of neutrinos '' _ , prog . * 64 * ( 1980 ) 1103 , and in proceedings of the _ `` workshop on the unified theory and the baryon number in the universe '' _ , tsukuba , japan , feb 13 - 14 , 1979 , eds . o. sawada and a. sugamoto , kek report kek-79 - 18 , p. 95 ; + m. gell - mann , p. ramond and r. slansky , _ in _ `` supergravity '' ( north - holland , amsterdam , 1979 ) _ eds . _ d.z . freedman and p. van nieuwenhuizen , print-80 - 0576 ( cern ) .
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it is well known that the zee model induces small neutrino masses by radiative corrections , where the bi - maximal flavor mixing is possible .
we analyze the cosmological condition in order for the baryon asymmetry generated in the early universe not to be washed out in the zee model .
since the lepton number is violated explicitly in the zee model , the baryon asymmetry might be washed out through the sphaleron processes together with the lepton - number violating interactions . in this letter , we will show that the baryon asymmetry is _ not _ washed out , although it has been said that the zee model can not preserve the baryon asymmetry generated in the early universe .
this can be seen by considering an approximately conserved number , @xmath0 .
hep - ph/0108013 + dpnu-01 - 18 + ut-956 +
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the r - process nucleosynthesis is called for to explain the origin of about half the elements heavier than iron observed in nature . its astrophysical origin remains a mystery . the r - process is one of the most complex nucleosynthesis process to explore because of the numerous difficulties still affecting the description of both the explosive astrophysical conditions believed to host the process and the nuclear properties of the exotic neutron - rich nuclei involved . from the nuclear physics point of view , the major difficulty lies in the determination of nuclear data for the thousands of nuclei far from the @xmath0-stability , for which essentially no experimental data exist nowadays . these concern mainly nuclear structure properties , @xmath0-decays , neutron captures , photodisintegrations as well as fission processes . in particular , mass predictions for neutron - rich nuclei play a key role since they affect all the nuclear quantities of relevance in the r - process , the @xmath0-decay , neutron capture and photodisintegration rates , as well as the fission probabilities . future ri - beam facilities , which are now under construction or planning , place their first priority to measure masses and half - lives of neutron - rich nuclei which have not been observed yet and are relevant to the r - process studies . in the coming experiments , it is clearly meaningful to measure such masses and half - lives of unknown neutron - rich nuclei . in addition , we emphasize that information on their experimental errors is crucial to promote theoretical studies on mass and @xmath0-decay half - life . the present paper aims at guiding such future experiment in defining _ how far _ from the stability line and _ how much precisely _ these physical quantities should be measured . different approaches can be followed to answer such questions . however , one major fact that should be kept in mind is that the r - process astrophysical site remains totally unknown to date . although the solar system signature clearly shows that the nuclear mechanisms responsible for the production of r - process nuclei concern neutron captures and beta - decay in the exotic neutron - rich region , no astrophysics model can nowadays consistently predict the neutron densities required for a successful r - process . the `` hot bubble '' scenario or the postexplosion outflows expected from protoneutron stars in the seconds after successful core - collapse supernovae are thought to be a likely candidate site for the r - process ( e.g. , meyer 1992 ; woosley 1994 ) . however , recent models of spherical `` neutrino - driven winds '' from protoneutron stars ( e.g. takahashi 1994 ; thompson 2001 ) fail to produce robust r - process nucleosynthesis up to and beyond the third ( @xmath5 ) r - process peak for `` canonical '' neutron stars with @xmath6 = 1.4 @xmath7 and @xmath8 = 10 km . the other proposed sites include such scenarios as `` neutron star mergers '' ( freiburghaus 1999 ) , weak r - process by the shock processing of the helium and/or carbon shells of core - collapse supernovae ( truran & cowan 2000 ) , magnetic protoneutron star winds ( thompson 2003 ) , prompt explosions from collapsing o - ne - mg cores ( wanajo 2004 ) , or even interestingly , some settings with rapid ejection of high - entropy but nearly symmetric matter to produce the r - process nuclei without excess neutrons ( meyer 2002 ) . each of them , however , faces severe problems and can not at the present time explain the production and galactic enrichment of the r - process nuclei observed in the universe . moreover , recent observational studies ( e.g. sneden 2000 ) of the relative abundance pattern of the r - process elements in very metal poor stars and also analysis ( e.g. , wasserburg , busso & gallino 1996 ) based on the isotopic abundances for the early solar system measured in meteorites have suggested that different r - process sites are responsible for the lighter ( @xmath9 135 - 140 ) and heavier ( @xmath9 135 - 140 ) r - process nuclei . this makes the determination of the physical characteristics for the r - process environment further complicated . for the above reason , it remains extremely difficult to estimate the precision required for mass and @xmath0-decay half - life measurements on the basis of r - process abundance calculations . in order to answer the objective questions treated in the present paper , we have therefore chosen to consider criteria independently of any `` realistic '' astrophysics calculations . even when the future experiments are performed , it is clear that theoretical predictions will still have to fill the experimental gaps for the thousands of nuclear data required in r - process simulations . in the first step , these future measurements will therefore mainly help in improving the theoretical models by constraining them further on nuclei closer to the one involved by the r - process , or even directly involved . they might bring new insights on nuclear physics phenomena at large neutron excesses as well as improve the present parametrizations of mass formulas . although most of the recent mass formulas show fits to experimental masses of similar quality , the mass extrapolations far from the valley of @xmath0-stability can differ from each other quite significantly ( for a recent review , see lunney 2003 ) . we have therefore chosen to estimate the nuclei to be involved and the required precision of future measurements by considering arguments on simple astrophysics considerations and existing nuclear model predictions as explained below . when the future experiments supply with information on new masses and half - lives with a reasonable precision , model predictions will tend to converge if their parameters are updated to fit the newly measured masses . in this regard , mass formula studies would not benefit if the experimental errors do not resolve the differences between the model predictions for the most exotic neutron - rich nuclei accessible ( or ideally directly involved in the r - process nuclear flow ) . accordingly , as a first rough guide for the required precision in detector developments , we put a rather simple requirement to mass and half - life measurements as follows : experimental errors subsidiary to the r - process nuclei need to be less than _ half _ the difference between the masses ( or half - lives ) predicted by the different nuclear models . we stress that the total length of the error bar obtained in such a procedure corresponds to @xmath10 . in the following , we discuss such a criterion on neutron - rich nuclei at the @xmath3=50 and @xmath3=82 shell closures . these regions are expected to become accessible in near - future experiments and are known to be of first importance in the development of nuclear structure studies , as well as in our understanding of the r - process nucleosynthesis . we consider here three mass formulas , known as hfb-2 ( goriely 2002 ; samyn 2001 ) , frdm ( @xcite ) , and kuty ( @xcite ) , available for a wide - range use in the nuclear chart and hence at this moment appropriate for r - process abundance calculations . the three mass formulas predict the 2135 measured masses with a root - mean - square deviation of about @xmath11680 kev ( see @xcite ) , although they were derived from quite different leading principles . the hfb-2 model is taken as representative of the microscopic mass formulas recently derived within the hartree - fock - bogoliubov framework based on an effective nuclear force of the skyrme type . on the other hand , the kuty mass formula corresponds to a semi - empirical approach making use of an empirical gross term for the macroscopic properties of spherical nuclei and spherically - based shell terms for the microscopic corrections . here the deviation from the gross properties is explained microscopically as shell and deformation effects . the use of a large number of parameters to describe the single - particle potential and nuclear gross properties enables the kuty model to reproduce relatively well all experimentally known masses as well as the single - particle energy levels . the frdm model is also of the semi - empirical type and was derived from the finite - range droplet model for the macroscopic part , and from a deformed single - particle potential for the microscopic part . -stability valley of its neutron - rich side in the n=82 plane . the values of mass excess from three theoretical mass formulas are shown against proton number , z. the open circles connected with the dashed line depict the mass excess from hfb-2 , the triangles connected with the solid line from kuty , and the crosses from frdm . the origin ( z=54 , n=82 ) is the stable @xmath12xe while the edge of the abscissa corresponds to the hfb-2 neutron drip line . ( b ) predicted mass differences are plotted against z for the n=82 isotones . the filled circles show the mass difference between hfb-2 and kuty , and the crosses between frdm and kuty . the scale of the abscissa is the same as in ( a ) . , title="fig : " ] -stability valley of its neutron - rich side in the n=82 plane . the values of mass excess from three theoretical mass formulas are shown against proton number , z. the open circles connected with the dashed line depict the mass excess from hfb-2 , the triangles connected with the solid line from kuty , and the crosses from frdm . the origin ( z=54 , n=82 ) is the stable @xmath12xe while the edge of the abscissa corresponds to the hfb-2 neutron drip line . ( b ) predicted mass differences are plotted against z for the n=82 isotones . the filled circles show the mass difference between hfb-2 and kuty , and the crosses between frdm and kuty . the scale of the abscissa is the same as in ( a ) . , title="fig : " ] in fig . 1(a ) , we illustrate a global feature of the @xmath0-stability valley given by the three above - mentioned models . in the figure a sectional diagram of the neutron - rich side of the valley is shown for the n=82 isotones . the microscopic mass formula is seen to give a steeper slope of the @xmath0-stability valley and hence predicts larger masses compared with those predicted from semi - empirical formulas . this can be seen in fig . 1(a ) especially for low z , at large neutron excesses . figure 1(b ) depicts the mass differences between hfb-2 and kuty and between frdm and kuty for the n=82 isotones . in particular , it is seen that the mass difference between hfb-2 and kuty is prominent and increases with decreasing proton numbers , when approaching the neutron - drip line . namely , both models predict significantly different masses for the neutron - rich nuclei far from the stability line . when applied to r - process calculations , such mass differences inevitably lead to different r - abundance patterns ( motizuki 2004 ; wanajo 2004 ) . in the following , we will focus on the two mass formulas , hfb-2 and kuty , for which the mass differences are seen to be the most significant ones . the r - process is believed to reach the neutron richness of @xmath13 in dynamical simulations ( e.g. motizuki 2004 ) as well as in the simple parametrized site - independent model ( e.g. goriely & arnould 1996 ) . for example , within the canonical model prediction making use of the kuty masses and the half - lives calculated by the second version of the gross theory ( see below ) , the r - process path determines @xmath14nb as the polestar neutron - richest nuclide among the n=82 isotones . this result is obtained assuming the so - called waiting - point approximation at a temperature of @xmath15 k and a neutron density of @xmath16 in order to reproduce the location and width of the @xmath17 130 peak observed in the solar r - process abundance distribution . the nuclide @xmath14nb is characterized by a ratio of @xmath18 . we accordingly analyze the mass differences between the kuty and hfb-2 models for the @xmath3=82 isotones down to @xmath14nb . as seen in fig . 2(a ) , the mass difference at @xmath14nb is about 3 mev . as mentioned in sect . 1 , the required total error bar @xmath10 is set to _ half _ the mass difference , so that the detectors for mass measurements must have a precision of 1@xmath1 250 kev at the neutron richness of @xmath2 = 3.0 . = 82 isotones is plotted against @xmath19 . the vertical dashed line indicates @xmath20 , down to which the r - process is believed to be extended . ( b ) beta - decay half - life difference between the values calculated by the second version of the gross theory ( gt-2 ) with hfb-2 masses and with kuty for the odd - z n=82 nuclei . both in ( a ) and ( b ) , the total length of the error bars are indicated so as to satisfy the requirement explained in sect . 1.,title="fig : " ] = 82 isotones is plotted against @xmath19 . the vertical dashed line indicates @xmath20 , down to which the r - process is believed to be extended . ( b ) beta - decay half - life difference between the values calculated by the second version of the gross theory ( gt-2 ) with hfb-2 masses and with kuty for the odd - z n=82 nuclei . both in ( a ) and ( b ) , the total length of the error bars are indicated so as to satisfy the requirement explained in sect . 1.,title="fig : " ] similarly , we calculate the @xmath0-decay half - life of the n=82 isotones , and more particularly of @xmath14nb , within different approaches to estimate the precision required in half - life measurements . one of the widely used models used for astrophysics applications is the second version of the gross theory , known as gt-2 ( @xcite ) . we apply this model using both the hfb-2 and kuty @xmath21 predictions . figure 2(b ) shows the half - life difference between both models for the odd - z n=82 nuclei . we observe that the half - lives calculated from the microscopic mass formula are shorter than those from the semi - empirical mass formula . this can be understood by the steeper slope of the @xmath0-stability valley for the microscopic mass formula : the hfb-2 model leads to essentially larger @xmath21-values than the kuty model . in particular , for @xmath14nb , the half - life is 3 ms for gt-2 with hfb-2 masses and 5 ms with kuty . however , the uncertainties in @xmath0-decay predictions stem not only from mass predictions , but also from the theoretical model used to describe the weak interaction . mean field and shell models have been applied in recent years to the calculation of the @xmath0-decay rates of nuclei of astrophysics interest . in the particular case of @xmath14nb , these models predict a half - life of about 4 ms for the df3 density functional plus continuum qrpa approximation of borzov ( 2003 ) including only the allowed transitions and about 3 ms if the first forbidden transitions are also included . a shorter half - life of about 2 ms is obtained by the shell model of martinez - pinedo & langanke ( 1999 ) . considering such half - life differences for @xmath14nb , we find that 1@xmath1 0.15 ms at @xmath18 is required for the half - life measurements . a similar procedure can be followed in the @xmath3=50 region . we find that similar precisions ( 1@xmath1 250 kev for masses and 1@xmath1 0.15 ms for half - lives ) are required from gt-2 calculations at @xmath2 = 2.9 on the @xmath3=50 shell closure , , for @xmath22fe . however , if we consider the doubly magic nuclide @xmath4ni which has been observed but for which the mass and the half - life remains experimentally unknown , the same criterion leads us to a precision of 1@xmath1 300 kev for mass and of 1@xmath1 5 ms for half - life measurements . we have derived the required precision of 1@xmath1 250 kev and 1@xmath1 0.15 ms , respectively , for mass and half - life measurements at the neutron richness of @xmath2 = 3.0 at the @xmath3=82 shell closure and at the @xmath2 = 2.9 at the @xmath3=50 shell closure . for the doubly magic nuclide @xmath4ni , we have found that the detectors must have a precision of 1@xmath1 300 kev for mass and of 1@xmath1 5 ms for half - life measurements . note that not only statistical but also systematic errors should be included in the above discussion . it should also be kept in mind that the precision estimate presented here is based on simple arguments due to our ignorance of the astrophysical site for the r - process . future development in nucleosynthesis models ( takahashi , this volume ) will hopefully bring new insight on the nuclear flow followed by the r - process and consequently on the nuclei involved and the major nuclear quantities of relevance . experiments at riken ri - beam factory will start in 2007 . here ri - beams are planned to be produced by fragmentation and uranium fission methods . the intensity of the ri - beams will be strong enough to reach @xmath4ni ( @xmath2=2.8 ) and @xmath22fe ( @xmath2=2.9 ) at the @xmath23 shell closure to measure these masses and half - lives with the suggested precisions . however , the expectations of the ri - beam intensity created with the _ fragmentation _ method at present come down to one particle per @xmath24 sec at the @xmath2=3.0 ( @xmath14nb ) region at the @xmath25 shell closure . this means that the measurements with the required precisions might be difficult for the present technology : it is indispensable to contrive new type of detectors to overcome this difficulty . future measurements with better precision are strongly encouraged in order to develop theories of nuclear masses and half - lives . progress in these theories and above all in microscopic approaches , as well as further developments of astrophysics models will help us to solve the long - standing mystery that the r - process nucleosynthesis still represents . we would like to thank k. takahashi for useful comments . y.m . would like to acknowledge y. ishida , t. suda , and y. yano for information on experimental status at the ri - beam factory . s.g . is fnrs research associate .
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in order to understand the r - process nucleosynthesis , we suggest precision required for mass and @xmath0-decay half - life measurements planned at future ri - beam facilities . to satisfy a simple requirement that we put on nuclear model predictions ,
it is concluded that the detectors for the mass measurements must have a precision of 1@xmath1 250 kev , and that the detectors for the half - life measurements demand a precision of 1@xmath1 0.15 ms .
both the above precisions are required at the neutron richness of @xmath2 = 3.0 at the @xmath3=82 shell closure and @xmath2 = 2.9 at the @xmath3=50 shell closure . for the doubly magic nuclide @xmath4ni ,
a precision of 1@xmath1 300 kev and 1@xmath1 5 ms are required , respectively , for mass and half - life measurements .
this analysis aims to provide a first rough guide for ongoing detector developments .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in every major scenario for physics above the @xmath1 scale non - perturbative chiral gauge theory dynamics is expected to play an important role , yet our understanding of this dynamics is very limited . recently we proposed a non - perturbative formulation of chiral gauge theory on the lattice , in the hope that the important features of these theories will be calculable in future computer simulations @xcite . we briefly review it here , focussing on a lattice gauge field interpolation procedure which is the crucial feature of our construction . the fermions are taken to live on a euclidean lattice with spacing @xmath2 . fermion doublers are eliminated by the rome group method of using a gauge non - invariant wilson term @xcite . ( the nielsen - ninomiya theorem tells us that we must break chiral gauge invariance to eliminate the fermions doublers . ) the central problem is then to recover gauge invariance in the continuum limit . before summing over gauge fields this can be arranged in a fairly simple way for any @xmath3-@xmath4 theory @xcite . however , once gauge fields are integrated over , new divergences can lead to uncontrollable violations of gauge invariance . a solution to this problem is to cut off the gauge field momenta by a scale @xmath5 ( @xmath6 in the continuum limit . ) @xcite gives detailed lattice power - counting arguments to show that this can be achieved by obtaining the @xmath2-lattice gauge fields as an interpolation of gauge fields living on a lattice of spacing @xmath7 , which are summed over using the standard wilson action . in practice it is possible that @xmath8 may not have to be too large for computing the properties of low - lying states @xcite . ( for another possible way of cutting off gauge field momenta see ref . @xcite . ) it is sufficient for the interpolation procedure to satisfy the following properties @xcite , most easily stated by imagining interpolating the @xmath9-lattice link variables , @xmath10 , all the way to the continuum to give gauge fields , @xmath11 $ ] . ( i ) transverse continuity : the interpolation describes a differentiable continuum gauge field @xmath12 each @xmath9-lattice hypercube , whose transverse components are continuous across hypercube boundaries ( the longitudinal components can jump ) . ( ii ) lattice spacetime symmetries should be respected . ( iii ) gauge covariance : a @xmath9-lattice gauge transformation changes the interpolation only by a continuous gauge transformation . ( iv ) locality : the gauge - invariant behavior of the interpolation should depend locally on @xmath10 , in particular it is sufficient if the trace of any continuum wilson loop depends only on the @xmath10 lying on @xmath9-hypercubes through which the wilson loop passes . we have detailed such an interpolation procedure for non - abelian gauge fields in four dimensions in ref . below , we describe the more transparent case of interpolating @xmath0 gauge fields in @xmath13 dimensions , following a procedure which readily generalizes to the non - abelian case . for simplicity we deal with the case @xmath14 ( the continuum ) and work in units where @xmath15 . in order to maintain transverse continuity it is helpful to build the interpolation from the lowest dimensional sublattices up . we therefore begin by interpolating the link variables , @xmath16 along the points of each plaquette edge . ( we are neglecting the measure - zero set of lattice fields where at least one of the link variables equals exactly @xmath17 . ) the simplest such interpolation is @xmath18 note that parallel transport along the links agrees between the lattice and the continuum fields . we now @xmath19 to interpolate the lattice field into a plaquette interior in such a way as to agree with the above interpolation on the plaquette edges . in order to satisfy locality we try to do this interpolation for each plaquette , using as input only its bounding link variables . in order to satisfy gauge covariance the strategy is to do a @xmath20 gauge transformation on the bounding links of the plaquette which put the link variables into a complete axial gauge . thus all gauge equivalent lattice fields on the plaquette edges are taken to the same gauge - fixed field ( ` almost ' the same in the non - abelian case ) , which we denote by @xmath21 . this lattice configuration will then be smoothly interpolated to the plaquette interior to give a continuum field @xmath22 . we will then try to find a smooth gauge transformation inside the plaquette which makes the result agree with the one - dimensional edge interpolation , eq . ( 2 ) . at this last stage we will fail , but in a way which we can understand and then correct . in detail , let us fix some plaquette and use @xmath23 coordinates @xmath24 for the vertices of the plaquette . the lattice gauge transformation , @xmath25(z ) = u_1(s)^{z_1 } u_2(s + z_1 \hat{1})^{z_2},\ ] ] takes @xmath26 to @xmath27 this lattice field is easily interpolated into the plaquette interior , @xmath28 where @xmath29 are local continuum coordinates for the plaquette interior . the problem is now to find a @xmath30 gauge transformation @xmath31 which takes @xmath22 to a gauge field agreeing with eq . ( 2 ) , so that we can be assured of transverse continuity across plaquette boundaries . in fact it is not hard to see that this demand essentially fixes @xmath31 on the plaquette boundary to be @xmath32 thereby specifying a map from the plaquette boundary ( topologically a circle ) to @xmath0 ( topologically also a circle ) . we can therefore associate a topological winding number ( integer ) to this map for each plaquette . unless this winding is zero , @xmath31 can not be continuously extended from the plaquette boundary to the interior , and we are stuck . it is simple to show that the winding number associated to the plaquette at @xmath33 , @xmath34 , is equal to @xmath35,\ ] ] ( @xmath36 \equiv$ ] nearest integer to @xmath37 ) and is generically non - zero . the way out of this impasse is to generalize the edge interpolation to @xmath38 where the @xmath39 are integer - valued and do not affect agreement of parallel transport between the link variables and the interpolation . ( in the notation of ref . @xcite , @xmath40 . ) if these integers are chosen to satisfy @xmath41 it is easy to show that this new definition differs from the original by a gauge transformation defined on the plaquette boundary with winding number @xmath42 . therefore the gauge transformation which makes @xmath22 agree with eq . ( 9 ) , @xmath43 , has winding number zero and can be smoothly extended to the plaquette interior , allowing us to get an interpolation @xmath44 . a simple choice for this extension of @xmath43 yields for @xmath45 , @xmath46 if @xmath47 then in fact there is no consistent solution to eq . ( 9 ) the reason is that the boundary conditions for the interpolation then correspond to a continuum configuration with topological charge @xmath47 , which can not be represented by a single smooth periodic gauge field . while configurations with non - zero topological charge are physically important , we do not need them in our proposal for lattice chiral gauge theory , because their effects can be inferred from the sector with zero topological charge using cluster decompostion for the full theory . for @xmath48 eq . ( 9 ) has many solutions and it is simple to pick one @xcite . let us check the four central requirements for a successful interpolation . ( i ) it is straightforward to see that eq . ( 10 ) defines a transversely continuous gauge field . ( ii ) even though we had to pick the axes of our complete gauge fixing somehow , the @xmath49-@xmath50 behavior of our interpolation is covariant under lattice translations and rotations , though the gauge dependent form is not . to see this in the @xmath0 case is easy , since from eqs . ( 10 , 9 ) the continuum field strength is a constant in each plaquette and is just the logarithm of the plaquette field strength ( with absolute value less than @xmath51 ) . for the same reason ( iii ) and ( iv ) are also obvious , the non - locality in choosing the @xmath52 does not infect the gauge invariant part of the interpolation ( ie . the field strength for the @xmath0 case ) . while non - abelian interpolation in four dimensions is technically more complicated , the basic steps are the same . the topological obstruction to making higher dimensional interpolations agree with lower dimensional ones now occurs in four dimensions , because the smallest non - abelian group , @xmath53 , is topologically the 3-sphere as is a hypercube boundary . the resolution of the problem generalizes the 2-d @xmath0 case . one extra complication not seen in two dimensions is that the choice of axes for the complete gauge fixing ( important for maintaining gauge covariance ) @xmath54 lead to a breaking of lattice rotational covariance in the gauge - invariant behavior of the interpolation . to repair this one needs to allow the orientation of the axes in each hypercube to be different and to determine this orientation by the gauge - invariant behavior of the link field @xmath10 itself . then if @xmath10 is rotated , so do the axes . @xcite . the interpolation in ref . @xcite differs in that the authors directly interpolate @xmath31 instead of @xmath43 , thus getting singular gauge fields whenever any @xmath34 is non - zero . such singular gauge fields are unsuitable for our formulation of chiral gauge theories . their interpolation also breaks lattice rotational covariance in the non - abelian case . 9 p. hernndez and r. sundrum , nucl . b455 ( 1995 ) 287 . p. hernndez and r. sundrum , nucl . phys . b472 ( 1996 ) 334 . a. borrelli et . al . , nucl . b333 ( 1990 ) 335 . m. gockeler , g. schierholz , nucl.phys . b(proc.suppl ) 29b , c ( 1992 ) 114 ; nucl.phys . b(proc.suppl ) 30 ( 1993 ) 609 ; g.t . bodwin and e.v . kovacs , nucl.phys . b(proc.suppl ) 30 ( 1993 ) 617 . s.a . frolov and a.a . slavnov , nucl . b411 ( 1994 ) 647 ; a.a . slavnov , phys.lett . b319 ( 1993 ) 231 . m. gckeler , a. kronfeld , g. schierholz and u.j.wiese , nucl . b404 ( 1993 ) 839 .
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the importance of lattice gauge field interpolation for our recent non - perturbative formulation of chiral gauge theory is emphasized .
we illustrate how the requisite properties are satisfied by our recent four - dimensional non - abelian interpolation scheme , by going through the simpler case of @xmath0 gauge fields in two dimensions .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
among the nearly 2000 exoplanets discovered so far , more than half are transiting systems . due to the rotation of the host star , the observed stellar radial velocity ( rv ) is expected to change as the planet transits different parts of the rotating stellar surface . this is called the rossiter - mclaughlin ( rm ) effect and was initially measured for eclipsing binary stars @xcite . after the first observation of the exoplanetary rm effect by @xcite , there are now more than 80 exoplanet systems with this effect observed , e. g. @xcite . this effect is normally used to measure the projected angle between the stellar spin and the planetary orbit . @xcite was the first to propose using the wavelength - dependent rm effect to probe the atmospheres of transiting exoplanets . this method exploits the fact that the planetary effective radius is wavelength - dependent due to differential atmospheric absorptions . @xcite theoretically modeled the rm effect for the na absorption of giant exoplanets . the advantage of the rm method compared to the traditional spectrophotometry is that it does not demand the use of photometric reference stars . the rm method promises to become a powerful technique for future transmission spectrum measurements , especially as the next generation of very large ground - based telescopes are likely to have a small field - of - view , making it difficult to employ suitable reference stars . with more and more terrestrial exoplanets being discovered , the characterization of their atmospheres will become a major goal for exoplanet research in which the rm method can play an important role . although it is currently difficult to observe the rm effect of these terrestrial exoplanets due to instrumental limitations , a lunar eclipse provides us with an opportunity to observe the earth transiting the sun and so explore the effectiveness of the rm method . the concept of regarding the earth as an exo - earth using lunar eclipses has been applied in the past by @xcite . these studies obtained the transmission spectrum directly from the ratio spectrum before and during a lunar eclipse . here we present for the first time the use of the rm effect of an earth transit during a lunar eclipse to retrieve the transmission spectrum . we observed the 15-april-2014 lunar eclipse with the high accuracy radial velocity planet searcher ( harps ) mounted on the eso la silla 3.6 m telescope @xcite . a consecutive sequence of observations over one entire night covered all of the eclipse stages , i.e. the penumbral and umbral eclipse and out of eclipse ( called hereafter the bright moon ) . the classical fiber spectroscopy mode was used with the fiber located at the center of the tycho crater ( see fig . [ nasa - tycho ] for the trajectory ) . the telescope used non - sidereal tracking and the tracking velocities were updated every few minutes . in addition , manual guiding on crater structures was performed . this worked reasonably well and limited the pointing coordinate drift to @xmath0 1.5 arcmin . we used varying exposure times during the eclipse because the lunar surface brightness changed dramatically . in total , 382 lunar spectra were obtained ( table [ observation ] ) . [ cols="^,^,^,^,^",options="header " , ] data reduction was performed using the harps pipeline . each exposure frame comprises 72 spectral orders that cover the wavelength range from 378 nm to 691 nm with a spectral resolution of @xmath1 @xmath2 115,000 . radial velocities are measured , for each spectral order as well as for the overall spectral range , using a cross - correlation with a g2 stellar spectral template . the overall measured and theoretical rv curves are shown in fig . [ rme - curve ] . the theoretical rv is the combination of two components : the motion of the sun with respect to tycho and the motion of tycho with respect to the observer . these velocities are calculated with the jpl horizon ephemeris , which considers the orbital motions , rotations of the earth and the moon , and the light travel time corrections . there is an offset between the theoretical rv and the measured rv , which probably originates from the rv zero - point of the spectral template used by the harps pipeline @xcite . we correct this offset by using the data points taken during the bright moon as a baseline . [ rme - curve ] shows the measured rv with the motion rv corrected and the baseline offset subtracted . this is the final rv curve of the rm effect for the earth transiting the sun . at the bright moon stage , the rv is essentially corrected to zero , but there is still a small slope with an amplitude of about 4 m / s which is probably due to the instrumental drift and the non - perfect telescope guiding when observing the moon . since the rm amplitude is on the order of 2 km / s , this small residual slope does not significantly affect our analysis . when the moon enters the 1st penumbra , the observed rv becomes negative ( blueshift ) as the earth begins to transit the redshifted rotating part of the solar disk . at the umbral stage , the rv gradually changes from a blueshift to a redshift . when the moon enters the 2nd penumbral stage , the rv is redshifted since the earth obscures mainly the blueshifted solar region . the rv gradually decreases during this stage as the moon moves out of the penumbral shadow . the umbral rv has a relatively complex structure and its details are determined by a combination of both the refracted part of the solar disk and the properties of the earth s atmosphere refracting the sunlight . in this work , the umbral part is not studied since we focus only on the penumbral parts for the rm effect . to retrieve the wavelength - dependent rm effect due to the earth s atmospheric absorption , we firstly establish an rm model to take into account the wavelength - dependent solar parameters . our model follows the method described in @xcite and @xcite . we divide the solar disk into elements with a size of @xmath3 and the radial velocity due to solar rotation for each element is : @xmath4 where @xmath5 is the rotation angular velocity , @xmath6 is the inclination of the solar spin axis towards tycho , x and y are in the sky - plane coordinate system of which the origin is at the projected solar center and the y - axis is along the projected solar rotation axis . the coordinate values are in units of the solar radius @xmath7 . the rv anomaly @xmath8 due to the rm effect is then calculated by integrating the intensity - weighted rvs of the visible solar disk : @xmath9 the rm curve for white - light ( i.e. the entire wavelength range of the harps spectrum ) is generated from this model ( the black line in fig . [ white - light - model ] ) . the parameters are further described as follows : 1 . the coordinates of the earth s center and the solar center as seen from tycho are generated using the horizon ephemeris . we use the earth radius of @xmath10 km and the solar radius of @xmath11 km . these data are used to determine the visible part of the solar disk at a given time . the inclination @xmath6 is calculated using the solar north pole position angle given by horizon . the mean value of @xmath6 is @xmath12 and changes by about @xmath13 during the eclipse . the solar differential rotation used is @xcite : @xmath14 where @xmath15 is the heliographic latitude and is calculated with : @xmath16 the coefficients adopted are a = 13.46 , b = -2.99 @xcite . the quadratic limb - darkening coefficients @xcite are used for the white - light model to calculate the intensity @xmath17 . we adopt the coefficients used by @xcite , i.e. @xmath18 and @xmath19 . for an exoplanet transit , the projected angle between the stellar spin and the planetary orbit is calculated by fitting the observed rv with the rm model . however , we adopt the actual data from the horizon ephemeris and so the model curve in fig . [ white - light - model ] is not the result of such a fit . the projected spin - orbit " angle for this observation is @xmath20 calculated using the trajectory of the earth as seen from tycho . because this trajectory is the combination of the earth and lunar orbits , the angle is not the real spin - orbit angle of the sun - earth system of @xmath21 . the stellar convective motion results in a blueshift of the lines which is called the convective blueshift ( cb ) @xcite . as the cb radial component varies across the stellar disk , the observed stellar rv changes when a planet transits the stellar surface . @xcite discusses the cb during an exoplanet transit and its effect on the rm effect . we follow the cb model in @xcite and adopt a typical cb value of the sun @xmath22 m / s for the white - light rv calculation . the blueshift value for each element on the solar disk is @xmath23 , where @xmath24 . the rv change caused by cb ( blue line in fig . [ white - light - model ] ) is then calculated in a similar way to the rm model and added to the rv value caused by the rm effect to give the red line in fig . [ white - light - model ] . the effects of atmospheric refraction are important for interpreting the transits of terrestrial planets @xcite and affect the rm effect . as the atmosphere refracts the light from the stellar disk region that should be obscured by the planet , the apparent effective radius of the planet will appear smaller than without refraction . thus in general , refraction would result in a smaller rm amplitude than without refraction . in our rm model , we do not include atmospheric refraction , which means the modelled rm amplitude should be larger than the actual observed rm amplitude . this is consistent with the result in fig . [ white - light - model ] in which the modelled rv value without refraction ( red line ) is generally larger than the observed rv ( green points ) . in the rm model described above , we assume the line profile is dominated by the stellar rotation and use the first - moment approximation @xcite . this assumption gives us the rv anomaly expressed by equation [ equ - deltarv ] . however , other mechanisms like the microturbulence , macroturbulence and instrumental broadening also affect the line profile and can change the shape of the rv curve @xcite . this may contribute to the discrepancy between the modelled rv and the observed rv in fig . [ white - light - model ] . the harps pipeline produces directly a rv value for each of the 72 spectral orders . three orders do not have useful rv data due to the lack of stellar lines in the corresponding spectral mask . after correcting for the earth - moon - sun motion and the baseline offset described in section 2 , we obtain 69 rm curves each representing a different wavelength range . to obtain the rm effect differences caused by the earth s atmosphere , the wavelength - dependent solar parameters need to be considered . the first aspect concerns the stellar convection blueshift . the actual rv caused by cb depends on where the line forms , e.g. the strong lines or low - excitation lines may have a small cb because they usually form high in the stellar atmosphere where the granulation is weak @xcite . as each of the harps spectral orders comprises stellar lines with different cb values , the @xmath25 of each spectral order varies . therefore , unlike the white - light cb model for which we can assume @xmath22 m / s , the actual cb effect for a given harps order is difficult to model . instead , we use two symmetric parts on the rv curve to cancel this effect utilizing the fact that the convection induced rv anomaly during transit is symmetric with respect to the mid - transit as shown by the blue line in fig . [ white - light - model ] . two regions of the rm curve ( labelled as pm1 and pm2 in fig . [ white - light - model ] ) are then chosen . the pm1 and pm2 are symmetric regions of the eclipse , i.e. they are at the same distance from mid - eclipse . [ rme - amplitude]a shows the rv values of pm1 and pm2 and the rv difference between them ( i.e. pm2 - pm1 ) . we use this pm2 - pm1 value to represent the rm effect amplitude for each spectral order . from fig . [ rme - amplitude]a , it can be seen that there is a correlated systematic difference between pm1 and pm2 . we believe that this correlated pattern results from the measured effect of the cb depending on the type and number of spectral lines present in each spectral order but is well - cancelled by using pm2 - pm1 . the second wavelength - dependent solar parameter is the limb - darkening . here we use the empirical power - law limb - darkening coefficients from @xcite . for each of the harps orders , we interpolate a limb - darkening coefficient at the corresponding wavelength and calculate a value of pm2 - pm1 with a fixed earth radius ( 6378 km ) using our rm model . the model values of pm2 - pm1 are shown in fig . [ rme - amplitude]b . the final rm effect at different wavelength ranges , after the correction of the limb - darkening , is presented in fig . [ rme - amplitude]c . the limb - darkening correction made here is mainly for the continuum , however , since the solar lines have different limb - darkening compared to the adjacent continuum @xcite , this could introduce some extra noise in the final rm amplitudes . the rv curve of the wavelength - dependent rm amplitudes in fig.[rme - amplitude]c results from the different effective radius at different wavelengths , and it can be regarded as a mapping of the low resolution transmission spectrum of the earth s atmosphere into effective radius and hence radial velocity space . to interpret the observed atmospheric features , we build a transmission spectral model following the methods of @xcite and @xcite . we calculate the overall transmission spectrum of the earth atmosphere from 0 to 80 km altitude considering the ozone absorption and rayleigh scattering extinction ( blue line in fig . [ rme - amplitude]c ) . this transmission spectrum is overlaid with the rm amplitude curve to compare their shapes . we emphasize here that the transmission spectrum model is not used to fit the observed rm amplitude curve but to demonstrate the presence of the atmospheric features mapped into it . by comparing the spectral shapes in fig . [ rme - amplitude]c , we interpret the rm amplitude curve as follows : the rm amplitude is larger towards the blue due to rayleigh scattering extinction that makes the atmosphere more opaque and the atmospheric effective thickness larger at shorter wavelengths . the broad peak around 600 nm results from the ozone chappuis band absorption while the rm amplitude towards the red becomes smaller since both the ozone absorption and the rayleigh scattering become weaker , rendering the atmosphere more transparent . the rm model we have used for this work suffers from several defects and incompleteness . the first is observational in that there is a slight , but barely significant , drift in the telescope guiding on the lunar surface . the second is that other line broadening mechanisms like the stellar micro / macro - turbulence are not included in the rm model ( cf . subsection [ sec - turbulence ] ) . the third is more fundamental and its full solution is beyond the scope of this letter . this is the determination of the detailed mapping between the rm amplitude and the atmospheric exctinction as a function of wavelength . this depends on the effects of atmospheric refraction in the exoplanetary atmosphere which will be influenced not only by the absorption coefficients of its gaseous constituents but also by the effects of screening due to clouds and aerosols . the combination of these effects can explain the slight deviation of the modelled rv from the observed rv values . it can also explain why the modelled rm amplitudes for the limb darkening correction are larger than the observed rm amplitudes ( shown in fig . [ rme - amplitude]b ) . however , this has a very limited effect on our retrieval of the final rm amplitude curve since the limb - darkening correction produces only a small rv variation with wavelength so that the shape of the final rm amplitude curve is little affected by the rm model . future modelling work containing a proper treatment of refraction and stellar turbulence will allow us to combine the transmission spectral model with the rm model , and to fit the observed rm amplitude curve directly instead of just comparing its shape with the transmission spectrum as presented in the current work . we have observed for the first time the rossiter - mclaughlin effect of the earth transiting the sun using a lunar eclipse . the rm effect curve has been obtained using high accuracy rv observations and an rm effect model is built to analyze the observed result . separate rm curves at different wavelengths are obtained from 69 harps spectral orders . after the correction of the wavelength - dependent limb - darkening of the sun and the convective blue - shift of the solar lines , we retrieve the wavelength - dependent rm amplitudes due to the transmission of the earth s atmosphere . the ozone chappuis band and the rayleigh scattering signatures are clearly detectable . the rm method can be used to detect broad features , such as the rayleigh scattering , in exoplanet atmospheres . the advantage is that no reference stars are needed in contrast to the requirements for the traditional spectrophotometric method . since the next - generation ground - based telescopes such as e - elt will have a relatively small field - of - view , limiting the access to nearby reference stars , the rm method will provide a promising technique for the characterization of planetary atmospheres . thus , in the future , this method can be applied to detect the atmospheres of terrestrial planets and particularly to search for the bio - signature gas ozone . extending the observation to near - infrared ( nir ) wavelengths will yield more absorption features . however , the ability of the rm method is limited by the number of suitable spectral lines of the parent star . the sun , for example , does not have sufficient lines in the nir . however , for m - type stars which are expected to be promising targets for exoplanet atmosphere studies , there are many stellar molecular lines in the nir which can be exploited . we greatly appreciate the excellent support from the lasilla / paranal science operations team . in particular , we thank valentin ivanov for performing the observations on site and his and lorenzo monaco s support during the preparation of the observing run . we would also like to thank gaspare lo curto and andrea chiavassa for helpful discussions and the referee for useful suggestions . the study is supported by the national natural science foundation of china under grants nos . 11390371 and 11233004 . _ facilities : _ .
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due to stellar rotation , the observed radial velocity of a star varies during the transit of a planet across its surface , a phenomenon known as the rossiter - mclaughlin ( rm ) effect .
the amplitude of the rm effect is related to the radius of the planet which , because of differential absorption in the planetary atmosphere , depends on wavelength .
therefore , the wavelength - dependent rm effect can be used to probe the planetary atmosphere .
we measure for the first time the rm effect of the earth transiting the sun using a lunar eclipse observed with the eso harps spectrograph .
we analyze the observed rm effect at different wavelengths to obtain the transmission spectrum of the earth s atmosphere after the correction of the solar limb - darkening and the convective blueshift . the ozone chappuis band absorption as well as
the rayleigh scattering features are clearly detectable with this technique .
our observation demonstrates that the rm effect can be an effective technique for exoplanet atmosphere characterization .
its particular asset is that photometric reference stars are not required , circumventing the principal challenge for transmission spectroscopy studies of exoplanet atmospheres using large ground - based telescopes .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
extra neutral gauge bosons , known in the literature as @xmath0 , appear in many proposals for beyond - the - standard model ( bsm ) physics ; for a review , see for instance @xcite . here we focus on _ minimal _ @xmath0 , previously studied in @xcite , which stand out both for their simplicity , and because they could arise in several of the above mentioned bsm scenarios , such as , _ e.g. _ , grand unified theories ( guts ) and string compactifications . following @xcite , we consider a minimal extension of the sm gauge group that includes an additional abelian factor , labeled @xmath4 , commuting with @xmath5 . the fermion content of the sm is augmented by one right - handed neutrino per family . we require anomaly cancellation , as this allows us to write a renormalizable lagrangian . if family - universality is imposed , then the anomaly cancellation conditions yield a unique solution : @xmath6 , where @xmath3 and @xmath7 are baryon and lepton number respectively . , with @xmath8 arbitrary coefficients . however , the @xmath9 component can be absorbed in the kinetic mixing in the class of models we consider . ] however , if the requirement of family - universality is relaxed , it can be shown that the following set of family - dependent charges satisfy the anomaly cancellation conditions : @xmath10 , where @xmath11 are the lepton flavours , and @xmath12 are arbitrary coefficients . we will consider a specific example of such _ non - universal _ @xmath0 in the following . in the basis of mass eigenstates for vectors , and with canonical kinetic terms , the neutral current lagrangian reads @xmath13 where @xmath14 is the photon field coupled to the electromagnetic current , while @xmath15 are the massive states , which couple to the currents @xmath16 respectively , obtained from @xmath17 \ , \overline{f } \gamma^\mu f \,$ ] and @xmath18 \ , \overline{f } \gamma^\mu f \,$ ] via a rotation of the @xmath19 mixing angle @xmath20 . the explicit expression of the latter reads @xmath21 . thus , under our minimal assumptions , only three parameters beyond the sm ones are sufficient to describe the @xmath0 phenomenology : the physical mass of the extra vector , @xmath22 , and the two coupling constants @xmath23 . in the following discussion , we normalize these couplings to the sm @xmath24 coupling , namely @xmath25 . . the yellow band represents the region of couplings compatible with a gut , whereas dots and lines correspond to specific benchmark models or full supersymmetric gut models , see @xcite for details.,scaledwidth=70.0% ] because guts are one of the motivations for considering minimal @xmath0 , it is interesting to give an estimate of the constraints that a gut would imply on the weak - scale couplings @xmath26 . for choosing the boundary conditions at unification scale @xmath27 , we normalize all charges as in @xmath28 , and take @xmath29 gev . we allow the @xmath0 coupling at unification scale to vary within the interval @xmath30 , and using the rge of the model we obtain the gut - favoured region of weak - scale couplings , shown in fig.[gut ] for the universal case @xmath31 . since the boundary conditions at scale @xmath27 are symmetric under the reflection @xmath32 , it is evident from fig . [ gut ] that mixing effects in the rge ( due to the non - orthogonality of the generators @xmath9 and @xmath2 ) are important . the gut - favoured regions for non - universal models , computed along similar lines , can be found in @xcite . the measurements providing constraints on minimal @xmath0 can be divided into two classes : electroweak precision tests and direct searches at the tevatron . measurements performed at lep1 and at low energy mainly constrain @xmath19 mixing , whereas data collected at lep2 ( above the @xmath33 pole ) constrain effective four - fermion operators . to compute the bounds from ewpt on minimal @xmath0 , we integrate out the heavy vector and use the effective lagrangian thus obtained to perform a global fit to the data . the results are shown in fig . [ chi model ] , for the universal ` @xmath34 model ' , corresponding to a particular direction in the @xmath26 plane often considered in the literature . the cdf and d0 collaborations have derived , from the non - observation of discrepancies with the sm expectations , upper limits on @xmath35 ( @xmath36 ) , @xcite . to extract bounds on minimal @xmath0 , we compute the same quantity at nlo in qcd , and compare it with the limits published by the experimental collaborations . the comparison between bounds from ewpt and from the tevatron is most clear if we plot them in ( coupling _ vs. _ mass ) , for a chosen direction in the @xmath37 plane , as it is done in fig . [ chi model ] for the @xmath34 model . we see that bounds from ewpt have a linear behaviour , because all the effects due to the @xmath0 in the low - energy effective lagrangian depend on the ratio @xmath38 , whereas bounds from the tevatron become negligible above a kinematic limit , which is of the order of 1 tev . thus for low masses the tevatron data give the strongest limits , while above a certain value of @xmath22 ( which is of the order of 500 gev for the @xmath34 model ) , bounds from ewpt are stronger . in particular , for models compatible with guts the strongest bounds are those given by ewpt . at 7 tev , and 400 pb@xmath39 at 10 tev ) for the @xmath34 model ( right ) present bounds and discovery prospects of the lhc at 7 tev and 50 pb@xmath39 for the _ muonphilic _ model with @xmath40 . for @xmath41 , both the bounds from the tevatron and the lhc reach are indeed weaker , because of finite - width effects not included in the figure , but the general message is unaffected . the yellow bands correspond to the gut - favored region , see section [ sect : gut].,title="fig:",scaledwidth=50.0% ] at 7 tev , and 400 pb@xmath39 at 10 tev ) for the @xmath34 model . ( right ) present bounds and discovery prospects of the lhc at 7 tev and 50 pb@xmath39 for the _ muonphilic _ model with @xmath40 . for @xmath41 , both the bounds from the tevatron and the lhc reach are indeed weaker , because of finite - width effects not included in the figure , but the general message is unaffected . the yellow bands correspond to the gut - favored region , see section [ sect : gut].,title="fig:",scaledwidth=50.0% ] the present schedule foresees that in 2010/2011 the lhc will run at 7 tev in the center of mass , collecting up to 1 fb@xmath39 of integrated luminosity @xmath7 . therefore , it is interesting to ask whether there are any minimal @xmath0 which are both allowed by present constraints and accessible for discovery in such early phase . to answer this question , we have performed a nlo analysis similar to the one used in extracting bounds from tevatron data , requiring the @xmath0 signal to be at least a @xmath42 fluctuation over the sm - drell yan background . the results are displayed for the @xmath34 model in fig . [ chi model ] , where a comparison with present bounds is made . we see that for @xmath43 pb@xmath39 ( the luminosity approximately foreseen at the end of 2010 ) , no discovery is possible . on the other hand , for @xmath44fb@xmath39 some unexplored regions become accessible ; however , @xmath0 compatible with guts are still out of reach , and more energy and luminosity will be needed to test them . we have seen that universal models are strongly constrained by present data . on the other hand , when we consider non - universal couplings to leptons , the bounds can be significantly altered . in particular , let us consider the case where @xmath45 , which we called ` muonphilic @xmath46 ' . let us further assume that kinetic mixing is negligible , _ i.e. _ @xmath47 . in this case , the @xmath0 has no coupling to the first and third leptonic families , in particular it has no coupling to the electron . as a consequence , bounds from ewpt are strongly relaxed , the only surviving constraints coming from @xmath48 and @xmath49-@xmath50 scattering ( nutev ) . on the other hand , the tevatron reach is limited , as already noted in section [ bounds ] , to @xmath51 tev : therefore the lhc has access to a wide region of unexplored parameter space already with a very low integrated luminosity at 7 tev , as shown in fig . [ chi model ] . we have discussed the present experimental bounds and the early lhc reach on minimal @xmath0 models , showing that present constraints can not be neglected when assessing the discovery potential of the early lhc . in particular , we have found that exploration of universal models , coupled to @xmath2 , may need more energy and luminosity than those foreseen for 2010/2011 , in particular for values of the couplings compatible with guts . on the other hand , some non - universal models which are weakly constrained by present data , such as the _ muonphilic _ @xmath0 , could be discovered at the lhc with very low integrated luminosity . i am indebted to my collaborators f. zwirner , g. villadoro , and a. strumia . i would also like to thank the organizers of the 2@xmath52 young researchers workshop _ physics challenges in the lhc era _ for giving me the possibility to present this work . 99 p. langacker , rev . * 81 * ( 2008 ) 1199 , arxiv:0801.1345 [ hep - ph ] . t. appelquist , b. a. dobrescu and a. r. hopper , phys . d * 68 * ( 2003 ) 035012 , arxiv : hep - ph/0212073 . e. salvioni , g. villadoro and f. zwirner , jhep * 0911 * ( 2009 ) 068 , arxiv:0909.1320 [ hep - ph ] ; e. salvioni , a. strumia , g. villadoro and f. zwirner , jhep * 1003 * ( 2010 ) 010 , arxiv:0911.1450 [ hep - ph ] . t. aaltonen _ et al . _ [ cdf collaboration ] , phys . * 102 * ( 2009 ) 031801 , arxiv:0810.2059 [ hep - ex ] ; t. aaltonen _ et al . _ [ cdf collaboration ] , phys . * 102 * ( 2009 ) 091805 , arxiv:0811.0053 [ hep - ex ] ; [ d0 collaboration ] , d0 note 5923-conf ( july 2009 ) , http://www-d0.fnal.gov/run2physics/www/results/np.htm .
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we consider a class of minimal extensions of the standard model with an extra massive neutral gauge boson @xmath0 .
they include both family - universal models , where the extra @xmath1 is associated with @xmath2 , and non - universal models where the @xmath0 is coupled to a non - trivial linear combination of @xmath3 and the lepton flavours . after giving an estimate of the range of parameters compatible with a grand unified theory , we present the current experimental bounds , discussing the interplay between electroweak precision tests and direct searches at the tevatron . finally , we assess the discovery potential of the early lhc .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in @xcite aerts constructed an remarkable macroscopic model in which he could operationally define coincidence experiments violating ( the chsh version of ) bell s inequality ( bi ) , with exactly the same numerical value @xmath0 as the one obtained in typical coincidence experiments with entangled microscopic entities in a singlet state @xcite ( which corresponds to the maximal violation obtainable in quantum mechanics @xcite ) . subsequently , the model was generalized in @xcite , with the introduction of two parameters , @xmath1 $ ] , quantifying the degree of indeterminism and of non - locality present in the model , respectively . more precisely , @xmath2 corresponds to the classical situation of absence of indeterminism , whereas @xmath3 to the situation of maximum indeterminism , typical of pure quantum systems . on the other hand , @xmath4 corresponds to the situation of maximum locality , when the two pairs forming the double - system are totally disconnected , whereas @xmath5 corresponds to the opposite situation of perfect correlation . the authors of @xcite obtain that bi can only be violated if @xmath6 , and that the violation takes its maximal numerical value @xmath7 when @xmath5 ( maximum correlation ) and @xmath2 ( minimum indeterminism ) . also , they find that for any @xmath8 , it is always possible to restore the validity of bi by increasing @xmath9 , whereas this is not any more possible if @xmath10 . to sum up , the study of the model described in @xcite has showed that the source of the violation of bi is the existence of a non zero correlation between the two pairs forming the double - system ( @xmath6 ) , whereas the only effect of increasing the level of indeterminism ( increasing @xmath9 ) is to decrease the value the inequality can take . the purpose of the present paper is to analyze a different macroscopic model in which two parameters @xmath9 and @xmath11 will also be introduced , as in @xcite , to continuously vary the level of indeterminism and non - locality ( correlation ) . this will allow us to confirm that a non zero correlation ( @xmath6 ) is the necessary condition for the violation of the inequality . however , we will also show that by increasing the indeterminism ( increasing @xmath9 ) we can either increase or decrease the value bi can take , depending on the state in which the system is prepared before the coincidence experiments . before describing our macroscopic model , we briefly recall the expression of bell s inequality ( bi ) . @xcite . on a given physical entity , we assume that four different experiments can be performed : @xmath12 , @xmath13 , @xmath14 and @xmath15 . we call @xmath16 , @xmath17 , @xmath18 and @xmath19 the outcomes associated to these four experiments , which can only take the values @xmath20 or @xmath21 . we also assume that experiments @xmath12 and @xmath13 can be performed together with either of experiments @xmath14 and @xmath15 , so defining additional _ coincidence _ experiments : @xmath22 , @xmath23 , @xmath24 and @xmath25 . to every coincidence experiment @xmath26 , @xmath27 , @xmath28 , we can then associate the expectation value @xmath29 of the product of outcomes @xmath30 , by : @xmath31 where @xmath32 is the probability that the coincidence experiment @xmath26 yields the outcomes @xmath33 . under certain hypothesis ( usually referred to as _ bell locality _ , which have to do with the existence of hidden variables independently determining the experiments outcomes ) , one can prove the following ( bell ) inequality @xcite : @xmath34 inequality ( [ bell inequalities ] ) is generally violated by quantum systems in entangled states , like for instance those formed by two spin-@xmath35 entities in a _ singlet ( zero ) spin state _ , for which one can show that @xmath36 . @xcite this means that entangled quantum systems generally violate bell s locality assumption , a fact which remains true even when the two subsystems are separated by a very large spatial distance . in other terms , no local physical theory in the sense specified by bell can agree with all statistical implications of quantum mechanics , and spatial separation is not a sufficient condition for _ experimental separation_. let us consider a solid object , of homogeneous density , shaped as a rectangular @xmath37-prism ( @xmath37 even ) , i.e. , as a polyhedron formed by @xmath37 identical parallelogram - faces , and two lateral regular polygon - faces with @xmath37 sides ( see fig . [ esagono singolo ] for the case @xmath38 , of an hexagonal prism ) . we consider the prism as a special kind of die with @xmath37 faces , which can be rolled on a flat surface ( perpendicular to the gravitational field ) , along a given `` roll - direction , '' parallel to the two short sides of the rectangular upper face ( see fig . [ esagono singolo ] ) . each rectangular face of the prism shows either a symbol `` @xmath39 , '' or a symbol `` @xmath40 . '' the `` + '' symbol is only printed on @xmath41 parallel ( opposed ) rectangular faces of the prism , whereas the symbol `` - '' is printed on all the remaining @xmath42 rectangular faces . we also assume that the `` + '' and `` - '' faces are made of different materials . the two `` + '' faces can slide with an extremely low friction on the flat surface , whereas the @xmath42 `` - '' faces present a very high coefficient of friction with respect to it . considering that the upper and lower faces of the prism always present the same symbol ( and therefore are made of the same material ) , we have that when the prism presents a `` + '' upper face , it can slide with almost no friction , whereas it can not do so if the upper face has a `` - '' symbol . in the following , when the prism presents a `` + '' ( respectively , `` - '' ) upper face , we shall simply say that it is in state ( + ) [ ( respectively , state ( - ) ] . let us now describe what we shall call a _ rolling experiment _ with the prism . it consists in placing a specific shooter ( similar to a `` flipper ball shooter '' ) behind the prism , along the roll - direction , pulling firmly its knob and then releasing it , thus communicating to the prism an a priori unpredictable impulsion ( see fig . [ shooter esagono ] ) . one then waits until the prism stops completely , and read the symbol on its upper face . if it is a `` + , '' the outcome of the experiment is the value @xmath20 , otherwise the value @xmath21 . if the prism is in state @xmath43 , then , because of the very low friction of the face in contact with the surface , it will not roll , but only glide on it , until all translational kinetic energy will be converted into heat . therefore , the rolling experiment will not change the prism s upper face , and the outcome will be @xmath20 , with certainty . on the other hand , if the prism is in state @xmath44 , then the face in contact with the surface will present an extremely high coefficient of friction with it , so that the prism will not anymore slide but roll ( i.e. , rotate around its longitudinal axis ) . typically , most of the energy communicated to it by the shooter will be initially transformed into rotational kinetic energy , then , because of the positive work performed by the friction forces , the rotational energy will be gradually transformed into translational kinetic energy and heat , and of course in the end the prism will stop and show a specific upper face . however , the prism will roll only for as long as the non - elastic effects associated with the _ rolling frictions _ remain lower than the _ sliding frictions _ , since in this case the prism requires less energy to be moved by rolling than by sliding . but since two of the @xmath37 faces involved in the rolling movement ( those with the `` + '' symbol ) present a very low sliding friction , it is highly probable that the prism will conclude its run sliding on one of them , before it will ultimately totally stops . in other terms , apart from exceptional circumstances , which we can simply ignore not to complicate unnecessarily our discussion , we can _ ideally _ assume that , following a rolling experiment , if the initial state is @xmath44 then the final state will be @xmath43 , with certainty . so , our prism is so conceived that , independently of its initial state [ @xmath43 or @xmath44 ] , the outcome of a rolling experiment will always be @xmath20 [ i.e. , its final state will always be @xmath43 ] . we now consider two identical @xmath37-prisms , and use them to construct an entangled system by connecting them through space , by means of a rigid rod ( the length of the rod is arbitrary , but we assume it is made of an extremely light and rigid material ) whose two ends are glued at the center of the two opposed polygon - faces of the two prisms , as indicated in fig . [ esagoni - connected ] . clearly , the connecting rod creates correlations between the @xmath37 different rectangular faces of the two prisms . here we assume that the two prisms have been connected in such a way that the correlations in question are those described in fig . [ n - prism - correspondence ] . we assume that the glue used to connect the rod to the two prisms is strong enough , so that if the two prisms are hit together , simultaneously ( in the same direction ) they will be able to maintain their connection while rolling , i.e. , to remain a whole entity . but we also assume that the glue , although strong , is not as strong as to allow the two prisms to remain connected if only one prism is hit at a time . in other terms , if we hit with the shooter only one of the two prism , then , because of the inertia of the other prism , the impact will cause the rod to suddenly detach and fall , thus disconnecting the two solids ( one of which will then move in the roll - direction , whereas the other one will remain still ) . on the other hand , if two experimenters hit with two shooters both prisms at the same time , then the torque experienced by the rod will be much lower , so that the latter will not detach and the two prisms will be able to roll together on the surface , as a one piece entity . on the double - prism system we have just defined , we perform four different coincidence experiments and show that they produce a violation of bi . for this , we assume that two experimenters ( who we shall call experimenter @xmath47 and experimenter @xmath48 ) are placed close to each one of the two prisms . experimenter @xmath47 can perform on his prism ( say , the left one ) two different experiments , @xmath12 and @xmath13 , which are defined as follow . experiment @xmath12 is the rolling experiment we have defined in sec . [ a rolling prism ] : it consists in hitting the left prism with a shooter along the roll - direction , then reading the symbol marked on the obtained upper face , producing in this way one of the two outcomes : @xmath49 , or @xmath50 . experiment @xmath13 is much simpler : it simply consists in looking at the prism s upper face and check whether it is flat or not . if it is so , then the outcome is @xmath51 , otherwise it is @xmath52 . experimenters @xmath48 can perform on its prism ( the right one ) the same two experiments as experimenters @xmath47 . in other terms , @xmath14 is defined as @xmath12 , and @xmath15 as @xmath13 . clearly , all of the four above mentioned experiments , when singly performed , can only produce the outcome @xmath20 . also , when the coincidence experiments @xmath23 , @xmath24 and @xmath25 are performed by the two experimenters , the only possible outcome for them is @xmath53 , so that according to ( [ expectation value ] ) , @xmath54 . the situation is however more articulate for experiment @xmath22 , which creates correlations between upper faces . indeed , if the two experimenters hit simultaneously the two prisms , then , as we explained , the rod will not separate and they will remain connected as they roll ( the double - prism system can not slide , but only roll , as one of its two lower faces is always a high - friction face ) . therefore , according to fig . [ n - prism - correspondence ] , we have that the probability for the outcome @xmath55 is zero , the probability for each one of the two outcomes @xmath56 and @xmath57 is @xmath58 , and the probability for the outcome @xmath55 is @xmath59 . thus , @xmath60 = 1-(8/n)$ ] . inserting all this in ( [ bell inequalities ] ) , we obtain : @xmath61 eq . ( [ violation - n ] ) clearly violates bi ( [ bell inequalities ] ) . it does so in a maximal way ( @xmath62 ) for the case @xmath63 of two tetragonal prisms , and with a value which is very close to the quantum mechanical maximum of @xmath64 for the case @xmath65 of two decagonal prisms ( @xmath66 ) . before continuing in our analysis , let us mention that it was diederik aerts who was the first , in the early eighties of last century , to conceive an explicit macroscopic model with non - local correlations violating bi in a maximal way , @xcite , as our double - prism system with @xmath63 can do ( see also ref . @xcite and the references cited therein ) . an important difference between our double - prism model and the historical ( connected vessels of water ) model of aerts , is that in the latter it is the fact that the system is broken which is the mechanism responsible for the creation of correlations , whereas in our model it is exactly the contrary : it is only when the double system is _ not _ broken that the coincidence experiment @xmath22 can produce correlations between upper faces ( it is important to distinguish `` faces '' from `` upper faces , '' as only the latter are created in a rolling experiment : if we do nt create upper faces , by rolling the prisms , bi can not be violated @xcite ) . we want now to vary the amount of indeterminism and non - locality in our model , to see how this can affect the degree of violation of bi , thus elucidating their role in the violation . first of all , let us observe that if the number @xmath37 of rectangular faces of the two prisms tends to infinity , the probability @xmath67 tends to @xmath68 , so that in this limit , according to ( [ violation - n ] ) , @xmath69 . one could be tempted then to affirm that the level of indeterminism decreases as @xmath70 , and that this produces a corresponding decrease of the degree of violation of bi , contrary to what was obtained by aerts and collaborators , who showed instead that a decrease of indeterminism produces a stronger violation of bi . @xcite this however would be a wrong conclusion . indeed , if by increasing @xmath37 we can reduce the value taken by bi , this is so because for each different @xmath37 we have a different physical system , and not because we are decreasing the level of indeterminism in a given physical system . what we have to do , instead , is to vary such level within a same double - prism system , characterized by a fixed value of @xmath37 . there are of course different ways to do so . a simple one is to consider that the two experimenters , who have to hit the prisms in the rolling experiments , may not just do it aimlessly , but trying to obtain the specific effect of having the two prisms rolling over a predetermined distance , thus producing a predetermined total angle of rotation . of course , this will make a difference only when the two experimenters are executing the coincidence experiment @xmath22 , as is clear from the fact that when they do nt hit the two prisms together ( in experiments @xmath23 and @xmath24 ) , then because of the extreme low friction of the `` + '' faces ( and the fact that , according to the protocol , they have to pull the knob firmly , i.e. , hit the prism strongly ) they will not succeed in altering the predetermined @xmath71 outcome . now , let us assume that if the two players are successful in producing the chosen rotation in the coincidence experiment @xmath23 , then , considering the state in which the system was prepared , the outcome of the rolling experiment will be @xmath72 ( or @xmath73 ) . in general terms , we can characterize the ability of the two players in obtaining the desired effect by means of a continuous parameter @xmath74 $ ] , such that @xmath75 corresponds to the totally random situation , of maximum indeterminism , when the experimenters hit the two prisms without trying to obtain any specific result , and @xmath2 corresponds to the opposite situation where the two players are able to perfectly control their shot and therefore produce the @xmath72 ( or @xmath73 ) outcome , without fail . in other terms , by varying @xmath9 from @xmath68 to @xmath76 , we can decrease the degree of indeterminism in the measurement processes . following ref . @xcite , we also want to introduce , in addition to the indeterminism parameter @xmath9 , an additional continuous parameter @xmath77 $ ] , characterizing the connectedness of the two prisms ( i.e. , the strength of the correlations between the rectangular faces of the two prisms ) . a simple and natural way to do so is to consider the possibility that , sometimes , the rod can also detach and fall during the execution of the @xmath22 experiment . let us simply assume that @xmath11 corresponds to the probability for the rod of remaining duly glued to the two prisms , when @xmath22 is executed . this means that @xmath5 corresponds to the case of maximum correlation , whereas @xmath4 to the case of absence of correlation . in other terms , by varying @xmath11 from @xmath68 to @xmath76 , we can decrease the degree of connectedness ( non - locality ) of the two entangled prism - entities . clearly , when the rod detaches , the two prism will necessarily end their respective roll by showing an upper face with the `` + '' symbol , and this means that the outcome @xmath53 is now also possible for the experiment @xmath22 , with probability @xmath78 . assuming for simplicity a linear variation of the probabilities as a function of the ability parameter @xmath9 , we can write for the four different outcomes of experiment @xmath22 : @xmath79 then , observing that the probabilities associated with the other three coincidence experiments are not directly affected by the values taken by @xmath9 and @xmath11 ( this because , when the two prisms are rolled independently from one another , they will inevitably end their run by showing a `` + '' upper face ) , we obtain , after a short calculation : @xmath80.\ ] ] visibly , ( [ i - epsilon - rho ] ) generalizes ( [ violation - n ] ) , and in accordance with the analysis of the model in @xcite , we can observe the following . only for the value @xmath4 , corresponding to two totally disconnected prisms , bi is obeyed , and this means that it is the correlation between the two subsystems , i.e. , the non - locality ingredient , which is really responsible for the violation . also , we can see that when we increase @xmath9 , i.e. , the amount of uncertainty in the outcome of the experiments , we clearly also diminish the value of ( [ i - epsilon - rho ] ) , i.e. , the degree of violation of bi , which means that , in accordance with @xcite , not only indeterminism ( associated here to parameter @xmath9 ) does nt contribute to the violation , but actually tends to reduce it . clearly , the maximum violation , @xmath62 , is obtained for @xmath81 and @xmath2 , i.e. , for the situation of perfect correlation and full predictability . it is however important to observe that although inequality ( [ violation - n ] ) is independent of the choice of the initial state of the double - prism system ( which can either be @xmath82 , @xmath83 or @xmath84 ) , this is not anymore the case when the outcomes of the experiments are affected by the ability parameter @xmath9 . indeed , for a given rolling distance ( i.e. , for a given total angle of rotation ) that the two experimenters will try to obtain , if we change the initial state of the system , then we will also change the expected outcome of the experiment , and therefore the value taken by bi . to see this , let us assume this time that the system is prepared in a state such that when the two experimenters can successfully impart to the double - prism the chosen total rotation angle , the final outcome is now @xmath55 , instead of the previous @xmath72 ( or @xmath73 ) . then , we obtain the following probabilities for the four outcomes of experiment @xmath22 : @xmath85.\end{aligned}\ ] ] observing once more that the probabilities associated to the other three coincidence experiments are not affected by the specific values taken by @xmath9 and @xmath11 , we find after a short calculation : @xmath86 as we can see , ( [ i - epsilon - rho - bis ] ) differs sensibly from ( [ i - epsilon - rho ] ) , and the effect produced by a variation of parameter @xmath9 is now exactly opposite : an increase of the level of indeterminism ( i.e. , an increase of @xmath9 ) produces a strengthening of the violation of bi , and not a weakening of it . also , the situation of full predictability ( @xmath2 ) is not anymore associated to a maximal violation of the inequality , but to the non - violation of it ! we conclude with a few comments . first of all , it could be objected that , contrary to the model explored in @xcite , @xmath9 is not the only source of indeterminism in our model , considering that we have defined the parameter @xmath11 as a probability , and that the outcomes of the coincidence experiments @xmath22 clearly depend on whether the rod will detach or not during their execution . in other terms , @xmath11 also contributes to the degree of unpredictability of the outcomes of @xmath22 . this is obviously true , but can not alter our conclusion . indeed , if we keep @xmath11 fixed , then an increase of @xmath9 does actually correspond to a global increase of the level of indeterminism in our model , which , according to ( [ i - epsilon - rho ] ) and ( [ i - epsilon - rho - bis ] ) , can either reinforce or weaken the degree of the violation of bi , depending on the state in which the system was prepared . in fact , our probabilistic description of non - locality in the model highlights an additional mechanism through which a variation of the amount of indeterminism can affect the value taken by bi , in a way that is not unique . indeed , as regards the randomness incorporated in @xmath11 , we can consider that the situation @xmath87 corresponds to the one of maximum uncertainty . then , when from that value of maximum uncertainty we increase @xmath11 , and therefore reduce the uncertainty , according to ( [ i - epsilon - rho ] ) and ( [ i - epsilon - rho - bis ] ) we increase the violation of bi , and this independently of the initial state of the system . but when starting from the same value @xmath87 we decrease @xmath11 , also in this case we reduce the uncertainty , yet this time we decrease the violation of bi ( which for the limit value @xmath4 is obeyed ) , independently of the initial state of the system . having said that , let us now explain why , when the level of indeterminism is increased in our model , by increasing @xmath9 , we can either increase or decrease the violation of bi , depending on the initial state of the system . for simplicity , let us set @xmath5 ( perfect correlation ) . considering that in our model @xmath88 , independently of the value taken by @xmath9 , we can write : @xmath89 . now , having assumed perfect correlation , it is clear that only three outcomes are possible for experiment @xmath22 : @xmath84 , @xmath83 and @xmath82 . according to ( [ expectation value ] ) , outcome @xmath84 contributes positively to @xmath90 , whereas outcomes @xmath83 and @xmath82 contribute negatively to it . therefore , if the outcome of experiment @xmath22 is predetermined , and corresponds to @xmath84 , @xmath91 and bi is obeyed ( @xmath92 ) . on the other hand , if the predetermined outcome is @xmath83 , or @xmath82 , @xmath93 and bi is maximally violated ( @xmath62 ) . assuming that we are in the situation where @xmath91 , for @xmath94 , then by increasing @xmath9 the system will start sometimes to also explore the outcomes @xmath83 , or @xmath82 , and since the latter contribute negatively to @xmath95 , their possible selection will cause its value to diminish , thus producing an increase in the violation of bi . and of course , the situation is reversed when @xmath93 , for @xmath94 . the reason why aerts et al . could nt highlight in @xcite this double role played by indeterminism in coincidence experiments , is because the statistics of outcomes of their model does not depend on the specific state in which the system is prepared , but only on the relative orientation of the measuring apparatus ( as is the case in spin measurements on singlet states , provided the direction of flight of the entangled pair is orthogonal to the directions of orientation of the stern - gerlach filters ) . therefore , although they have studied a model which is more elaborated than ours ( as meant to reproduce the same statistics as spin measurements on singlet states ) , it was actually too specific to fully elucidate the question of the role played by indeterminism in the violation of bi . to conclude , we briefly summarize our results . in agreement with @xcite , we have found that it is the aspect of non - locality , expressed by the connecting rod in our model , which produces the violation of bi , through the _ creation of correlations _ between outcomes of experiments performed in coincidence . in agreement with @xcite , we have also found that by increasing the indeterminism ( increasing @xmath9 ) , we can decrease the value taken by bi . however , we have also shown that this is not a general fact : depending on the state in which the system is prepared , an increase of the level of indeterminism is also capable of increasing the value taken by bi . an additional source of indeterminism can also be envisioned , that was not considered in @xcite , associated to the possibility of actualizing different degrees of non - locality ( connectedness ) in a coincidence experiment . this possibility was described in our model by assuming that not only correlations between `` upper faces '' had to be considered as potential before a coincidence experiment , but also correlations between `` faces . '' when we do so , we find that if the amount of indeterminism associated to this additional level of potentiality is decreased , it can either increase or decrease the value of bi , depending on whether the decrease produces a strengthening or a weakening of the non - local aspect , respectively . as far as this author can judge , it is still an open experimental question to know if a quantum microscopic system in an entangled state , like a singlet state , has also a probability of `` breaking '' during a coincidence experiment ( in the same way the rod in our macroscopic model has a probability of detaching ) , and produce in this way uncorrelated outcomes , instead of correlated ones . @xcite a last remark is in order . the description of the rolling / sliding behavior of the @xmath37-prisms must be understood , as we already said , in an idealized sense . we have never performed real experiments with systems of this sort , and therefore can not guarantee that the way we have theoretically described their behavior , although reasonable , would be perfectly adequate from the perspective of a real experiment . but of course , this is not an essential point in the present analysis : what is important is that the idealized system we have considered behaves in a logical manner , according to coherent mechanisms . the question of how to exactly implement such behavior in real models , which can be subject to real experiments , is a technological issue which goes beyond the purely conceptual scope of the present note . d. aerts , s. aerts , j. broekaert and l. gabora , the violation of bell inequalities in the macroworld , found . phys . , 30 , pp . 1387 - 1414 ( 2000 ) . d. aerts , a mechanistic classical laboratory situation violating the bell inequalities with @xmath0 , exactly ` in the same way ' as its violations by the epr experiments , helv . acta , 64 , 1 - 23 . ( 1991 ) b. s. cirelson , quantum generalizations of bi , lett . phys . , 4 , 93 ( 1980 ) . j.s . bell , in : b. despagnat ( ed . ) , proceedings of the international school of physics `` enrico fermi , '' course xlix ( academic press , new york , 1971 ) 171 . j. s. bell , physics ( long island city , n.y . ) 1 , 195 ( 1964 ) . reproduced as ch . 2 of j. s. bell , speakable and unspeakable in quantum mechanics ( cambridge university press , 1987 ) . a. aspect et al . , experimental realization of einstein - podolsky - rosen - bohm gedankenexperiment : a new violation of bell s inequalities , phys . lett . , 49 , p. 91 a. aspect , bi test : more ideal than ever , nature ( london ) , 398 , p. 189 d. aerts , example of a macroscopical situation that violates bell inequalities , lettere al nuovo cimento , 34 , pp . 107 - 111 ( 1982 ) . d. aerts , the missing element of reality in the description of quantum mechanics of the epr paradox situation , helv . acta , 57 , 421428 ( 1984 ) . m. sassoli de bianchi , using simple elastic bands to explain quantum mechanics : a conceptual review of two of aerts machine - models , to appear in : central european journal of physics , doi : 10.2478/s11534 - 012 - 0164 - 2 . d. aerts , an attempt to imagine parts of the reality of the micro - world , 325 , in : problems in quantum physics ii ; gdansk 89 , edited by : j. mizerski et al . , world scientific publishing company , singapore ( 1990 ) . m. sassoli de bianchi , quantum dice , arxiv:1301.3038 [ quant - ph ] ( 2013 ) .
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some years ago aerts _ et al . _
@xcite presented a macroscopic model in which the amount of non - locality and indeterminism could be continuously varied , and used it to show that by increasing non - locality one increases , as expected , the degree of violation of bell s inequality ( bi ) , whereas , more surprisingly , by increasing indeterminism one decreases the degree of the violation of bi . in this note
we propose a different macroscopic model in which the amount of non - locality and indeterminism can also be parameterized , and therefore varied , and we find that , in accordance with the model of aerts _ et al .
_ , an increase of non - locality produces a stronger violation of bi .
however , differently from their model , we also find that , depending on the initial state in which the system is prepared , an increase of indeterminism can either strengthen or weaken the degree of violation of bi .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
multi - wavelength high spatial resolution observations of ngc 4151 ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) have provided unique information on the continuum and emission line morphology in the central few 100 pc ( @xmath4 mpc , @xmath5 pc , * ? ? ? * ) , contributing to our understanding of the nature of the nuclear emission and its interaction with the host galaxy ( see @xcite for a review ) . in this paper we investigate further these topics , using high quality deep _ acis observations of the nearby prototypical seyfert 1 galaxy ngc 4151 , which allow the morphological investigation of spectral line emission in the inner circum - nuclear region ( @xmath6 pc in projection ) . in particular , we will address the radio jet host galaxy interaction and the nature of the fe k@xmath2 line emission . high resolution radio surveys show broad existence of collimated radio outflows or jets in seyfert galaxies , albeit less prominent than those of radio galaxies ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? ever since the _ hubble space telescope _ emission line imaging resolved the nlr morphology in a number of seyfert galaxies with linear radio structures @xcite , it has been actively debated whether the radio jet plays a competing role against nuclear photoionization in the narrow - line region ( nlr ) ionization structure @xcite . _ chandra _ x - ray observations of the nuclear regions of nearby seyfert galaxies appear to offer powerful diagnostics for discriminating between photoionization by the nuclear radiation or collisional ionization by a radio jet @xcite , although the importance of radio jets in shaping nlr environments seems to differ case - by - case @xcite . the nuclear region of ngc 4151 is known to host a two - sided , @xmath0300 pc - long radio outflow along position angle ( p.a . ) @xmath077@xmath7 ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . within the central 100 pc of the galaxy , @xcite identified a faint , highly collimated jet underlying discrete components that are shock - like features associated with the jet gas clouds interactions . although the distribution of the ionized gas and the kinematics in the nlr imply that shock ionization originated from jet cloud interactions is unlikely to be the dominant source of ionizing photons compared to the nucleus @xcite , @xcite find evidence that some nlr clouds are responsible for bending the jet . near - ir emission - line mapping of the nuclear region by @xcite also reveals enhancement of [ feii ] emission , suggesting the presence of shock heating in addition to nuclear photoionization @xcite . in our previous work using the _ chandra _ hrc image of ngc 4151 , @xcite found an excess of x - ray emission at these jet cloud interaction locations . however , no spectral information could be obtained because of the very limited energy resolution of the hrc . acis spectral imaging data of the ngc 4151 nuclear region @xcite were limited by both heavy pile - up and sensitivity , and did not allow detailed high spatial resolution comparison to the radio jet and emission line gas , nor spectral studies of x - ray emission from spatially resolved features . our 200 ks acis - s observation of ngc 4151 ( pi fabbiano ; see * ? ? ? * and wang et al . 2011 for previous work on this data set ) allows us to pursue these investigations , by comparing the morphology of the neix and ovii with that of the radio jet . moreover , with these data we can address the outstanding controversy on the spatial properties of the 6.4 kev fe k@xmath2 emission of ngc 4151 . in seyfert galaxies , this emission line is expected to originate from cold matter near the nucleus ( @xmath8 pc ) , either the obscuring torus ( e.g. , * ? ? ? * ) or an accretion disk ( e.g. , * ? ? ? * ; * ? ? ? * ) . these model predictions , however , conflicted with the finding of @xcite , who reported spatially resolved narrow iron k@xmath2 line emission in the _ chandra _ hetg observation , concluding that @xmath9 of this emission originates in the enlr at distance up to 6 from the nucleus ( @xmath0400 pc across ) . this conclusion has been more recently contested by the _ xmm_-newton based work of @xcite . the details of our acis observations and data reduction are described in ( * ? ? ? * hereafter paper i ) . briefly , ngc 4151 was observed by _ chandra _ for a total of 180 ks ( after screening for high background intervals ) with the spectroscopic array of the advanced ccd imaging spectrometer ( acis - s ; * ? ? ? * ) in 1/8 sub - array mode during march 27 - 29 , 2008 . the data were reprocessed following standard procedures , using ciao ( version 4.2 ) with the caldb 4.2.1 provided by the _ chandra _ x - ray center ( cxc ) . subpixel event repositioning ( `` static '' method in li et al . 2004 , using the corner split and 2-pixel split events ) and subpixel binning techniques were applied to the acis images to improve the spatial resolution . during our previous work @xcite and paper i , we became well aware of the complexity in data analysis caused by the bright nuclear emission of ngc 4151 . first , photon pile - up is present in the nuclear region even with the reduced frame - time of our observation . we established that for the soft x - ray emission , pile - up is moderate ( @xmath10 ) at @xmath11 . second , although the _ chandra _ s psf is highly centrally peaked , the contamination from the scattered nuclear emission to the extended emission is not negligible due to the brightness of the nucleus and the broader psf in the higher energy range . we performed _ psf simulations that provide an estimate of the expected contamination from the nuclear emission in an extended feature . detailed explanation of the analysis leading to these conclusions is presented in paper i. we have taken into account this information in the following analysis . the 27 kev emission is dominated by the unresolved nucleus @xcite . since we are interested in the extended x - ray emission , we used the energy range below 2 kev where this emission is prominent ( see paper i ) . to investigate the general spectral dependency of the morphology of this soft x - ray emission , images in three spectral bands below 2 kev were extracted from the merged data : 0.30.7 kev ( `` soft band '' ) , 0.71.0 kev ( `` medium band '' ) , and 12 kev ( `` hard band '' ) . following @xcite , exposure maps were created for the individual bands to obtain exposure - corrected flux images . figure [ 3color ] presents the resulting false color composite image of the central @xmath12 ( 450 pc on a side ) region of ngc 4151 , where the soft , medium , and hard band images are shown in red , green , and blue , respectively . the images in the three band have been smoothed with a @xmath13 gaussian kernel . it clearly shows bright structured soft x - ray emission along the northeast ( ne ) southwest ( sw ) direction , which is also the direction of the radio jet . in particular , the medium band image shows a jet - like @xmath02 linear extension ( contours in figure [ 3color ] ) . this medium band is dominated by the neix emission ( see below ) . in the following we concentrate on the analysis of this inner 2-radius circum - nuclear region . high spectral resolution grating spectra of ngc 4151 @xcite have shown that the emission in these soft spectral bands is almost entirely dominated by lines . although the spectral resolution of acis ccd can not provide unique identifications of the strongest soft x - ray emission lines ( @xmath142 kev ) seen in the nuclear spectrum @xcite , we can identify the dominant transitions guided by the hetg observations @xcite . most notably the blended lines appear as three strong lines in the acis spectrum , approximately centered at 0.57 kev , 0.68 kev , 0.91 kev ( c.f . figure 4 in yang et al . 2001 ; figure 3 in @xcite ) . we then extracted the x - ray emission in three narrow energy intervals ( 0.530.63 kev , blended ovii f , i , r ; 0.630.73 kev , blended oviii ly@xmath2 and ovii rrc ; 0.850.95 kev , blended neix f , i , r ) to create line strength images , highlighting regions of these prominent emission lines . this is a reasonable approach since the line emission dominates over the weak underlying continuum @xcite in these narrow bands and reveal substructures that are not obvious in the broadband images ( e.g. , * ? ? ? the resulting images are shown in figures [ lines ] and [ o3 ] , showing the x - ray emission - line structure close to the nucleus ( the central @xmath15 , @xmath0130 pc ) in the context of the radio outflow and the optical emission line clouds . the alignment between these images is done using the peak of x - ray emission , [ oiii ] emission , and the radio core . since the astrometry of our _ chandra _ image is accurate to @xmath16 , our interpretation of these features is not affected by significant alignment uncertainties . the position of the nucleus is indicated with a cross . the ovii , oviii , and neix emission line images all show extended morphology and some structures , closely following the p.a . of the large scale extended nlr traced by the [ oiii ] emission ( e.g. , * ? ? ? * ; * ? ? ? * ) . in particular , the linear feature seen in the medium - band is clearly present in the neix image . there appears to be two x - ray enhancements bracketed by the optical clouds ( indicated by arrows in figure [ o3 ] ) that are close to the radio knot features c2 and c5 @xcite . these `` hot spots '' are better visualized in the neix / ovii ratio image , shown in figure [ jet ] together with the vlba jet @xcite and the nir [ feii ] line emission @xcite . the locations where the jet appears to be intercepted by the optical clouds and where prominent [ feii ] emission arise , are characterised by a neix / ovii ratio of @xmath17 , which is significantly ( @xmath18 ) higher than the ratio in the surrounding regions ( @xmath19 ) . we also examined the optical clouds with high velocity dispersion , possibly associated with jet - cloud impact ( marked as crosses in figure [ jet ] ; see figure 7 in mundell et al . 2003 ) , and find an elevated neix / ovii ratio of @xmath20 ( with marginal significance ) . the eastern neix / ovii hot spot appears to be associated with clear interactions between the jet and the gas cloud , where shock heating becomes important . we note that the observed line flux of ovii is more suppressed than neix in the presence of the galactic absorption column ( @xmath21 @xmath22 ) , and xspec @xcite simulations show the absorption corrected ratio may systematically decrease by 10% . thus the presence of neix / ovii hot spots is not affected , unless there is significant differential absorption in the region . we extracted x - ray spectra from both the high neix / ovii emission line ratio regions and the surrounding low ratio region ( figure [ jet]b ) , and fit them jointly ( with the same model components but normalizations set free ) for comparison using xspec ( version 12.5 ; arnaud 1996 ) . spectra and instrument responses were generated using ciao tool _ specextract_. the background spectrum is taken from a nearby source - free region on the same ccd node . spectra were grouped to have a minimum of 20 counts per energy bin to allow for @xmath23 fitting . the contribution from the bright nuclear emission can not be neglected here ; to have a self - consistent model , we included a fraction of the scattered emission predicted from the psf simulation of the point - like ngc 4151 nucleus . we have made use of the cloudy photoionization modeling code , last described by @xcite , to model the soft x - ray emission . using cloudy version c08.00 , which enables a cloudy / xspec interface @xcite , we attempted to produce the soft part of the x - ray spectrum assuming an open plane - parallel geometry ( `` slab '' ) . the dimensionless ionization parameter @xcite is defined as @xmath24 , where @xmath25 is the hydrogen number density , @xmath26 is the distance to the inner face of the model slab , @xmath27 is the speed of light , and @xmath28 is the emitting rate of hydrogen ionizing photons ( s@xmath29 ) by the ionizing source . it was previously noted that the spectral fit to the soft x - ray continuum of the nucleus is quite uncertain because of heavy absorption @xcite , therefore we adopted the broken power - law form in @xcite for the agn continuum ( @xmath30 for the energy range between 13.6 ev and 0.5 kev , and @xmath31 for @xmath32 kev ) . normalizing to the observed x - ray luminosity of the nucleus @xcite , we obtain @xmath33 ( photons s@xmath29 ) . we varied @xmath34 and the column density @xmath35 of the model slab to create spectral model grids , which were fed to xspec . we quickly find that , using a single photoionized component alone can not reproduce the observed soft x - ray emission in ngc 4151 . with a high photoionization component ( @xmath36 ) , we were able to produce the hydrogen - like neon and oxygen lines emission ( e.g. , oviii , nex ) , which are unambiguously present in all the grating spectra ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? however , the helium - like transitions , in particular the ovii and neix emission line features , are not fitted acceptably . adding a second lower ionization photoionized component ( @xmath37 ) , as detected in @xcite , we were able to obtain a significantly improved fit ( @xmath38 ) , except for residuals in the 0.71.1 kev range that are particularly strong for the hot spot spectrum . adding additional photoionized components to the model does not further improve the fit when an f - test is performed . allowing different absorption columns for the models also does not give a better fit ( reduced @xmath39 ) , indicating that the high neix / ovii ratio is not due to higher obscuration . the absorption column required by the fits is consistent with the low galactic extinction towards the nlr region found in @xcite . instead , these residuals disappear when a thermal emission component ( @xmath40 ; * ? ? ? * ) with a temperature of @xmath41 kev is added to the model , suggesting the presence of collisionally ionized fe l emission . the spectral fitting results are summarized in table 1 and shown in figure [ 2reg ] . the absorption column is consistent with the low galactic extinction towards the nlr region found in @xcite . lastly , we note that fitting the data with only combinations of absorbed optically - thin thermal emission ( @xmath40 ) gave strong residuals of emission line features ( reduced @xmath42 ) , even when the abundance @xmath43 is allowed to vary . the subpixel resolution acis image also allows a similar comparison of the x - ray emission with the optical nlr clouds resolved in the @xmath44 image . taking advantage of the spectral capability of acis , we measured the 0.32 kev counts for the same [ oiii ] clouds identified in @xcite , and derived their 0.52 kev x - ray fluxes using the spectral model above . to subtract the nuclear contribution , we used a local region at the same radii but with different azimuthal angles than the clouds . the results are listed in table 2 . figure [ ratio ] shows these [ oiii ] to soft x - ray ratios , comparing the measurements with hrc results in @xcite and photoionization models in @xcite . the [ oiii]/x - ray ratios at the two x - ray hot spot locations ( corresponding to radio knots c2 and c5 in @xcite ) were also measured , and found to be uniformly low , @xmath03 . this was previously noted in @xcite , and implies enhanced x - ray emission compared to other clouds under nuclear photoionization , which is likely associated with the jet cloud interaction and consistent with their association with high neix / ovii ratio regions . the 6.4 kev fluorescent iron k@xmath2 emission of neutral or mildly ionized cold material is readily visible in the acis spectra of the nucleus @xcite and the extended region ( figure [ fek1]a ) . another fluorescent feature , sii k@xmath2 emission at 1.74 kev , is also clearly present in the extended emission ( figure [ fek1]b ) . both spectra of extended emission were extracted from a @xmath45 annular region centered on the nucleus . is this `` extended '' fluorescent line emission truly originating from outside the nuclear region , or is it the result of the psf wing spreading out point - like nuclear features ? to determine the spatial distribution of the fe k@xmath2 line emitting material , we followed the procedure in @xcite : a continuum image , taken to be the average of the counts in the 5.96.2 kev and 6.56.8 kev bands , was subtracted from the image extracted in the 6.26.5 kev band , resulting in a `` pure '' fek@xmath2 line image , shown in figure [ fek2]a . the fe k@xmath2 emission appears mostly circular in the @xmath46 , but shows faint extended emission towards the north and west . the si k@xmath2 line is blended with the mgxii ly@xmath47 and sixiii @xmath48 emission @xcite , thus the above continuum subtraction is not applied . we extracted emission in the 1.71.8 kev range to obtain the si k@xmath2 emission map ( figure [ fek2]b ) . it is mostly concentrated within a @xmath49 circle , with a slight elongation along the optical bi - cone direction ( northeast southwest ) . however , at such energies the wings of the energy - dependent telescope psf spread a small but non - negligible amount of the nuclear photons to large radii . thus the presence of `` extended '' fe k@xmath2 emission or si k@xmath2 emission in the @xmath50 region may in fact come from the psf - scattered unresolved nuclear emission . to investigate this point , we further compared the observed radial profile of the fe k@xmath2 emission band ( 6.26.5 kev ) and the radial profile in the same band for a point - like nucleus simulated with marx ( see * ? ? ? * ) , shown in figure [ fek]a . a similar comparison for the si k@xmath2 ( 1.71.8 kev ) is shown in figure [ fek]b . the extracted observed profiles shown in figure [ fek ] agree well with the psf profiles , except for the inner @xmath51 where the simulation overpredicted the observed emission due to heavy pile up in the psf core . the simulated psfs indicate that the @xmath46 region contains 89% of the point source emission in the 6.26.5 kev band and 94% of the point source emission in the 1.71.8 kev band . approximately 6% of the nuclear fe k@xmath2 emission is expected to be spread within the @xmath52 extended regions , the extent of fe k@xmath2 seen in figure [ fek1]a . taking into account the contribution from the psf wings , the remaining extended fe k@xmath2 emission is @xmath53 of the total fe k@xmath2 emission . this should be considered an upper limit for the truly extended emission , because the nuclear fe k@xmath2 emission is likely to be underestimated due to pile up . similarly , we estimated that the extended si k@xmath2 emission is @xmath54 of the total si k@xmath2 emission . as pointed out by the referee , the measured extended emission could easily disappear with a slightly broader simulated psf given that the ditherblur parameter in the marx simulation is not known a priori . therefore , we conclude that although there is faint extended emission seen in the fe and si fluorescent line images , at most 5% of the observed fe k@xmath2 emission and 4% of the si k@xmath2 emission can be truly spatially extended ; most of these emissions are consistent with the psf scattering of a strong nuclear component . both our previous work using hrc image of ngc 4151 nuclear region @xcite and this work using acis find that most of the nlr clouds in the central 150 pc radius region of ngc 4151 have a relatively constant [ oiii ] to soft x - ray ratio ( @xmath010 ; figure [ ratio ] and table 2 ) , no matter the distance of the clouds from the nucleus . this ratio indicates a uniform ionization parameter and a density decreasing as @xmath55 . this may be consistent with a photoionized wind scenario ( e.g. , bianchi et al . 2006 ) as the nlr clouds are outflowing ( e.g. , kaiser et al . the [ oiii]/x - ray ratios at the locations of jet - cloud collision are lower ( @xmath03 ) , leading to the suggestion of enhanced x - ray emission in these regions due to shock heating in addition to photoionization . this conclusion is supported by the results of multiwavelength imaging studies . the high spatial resolution emission line images of ovii , oviii , and neix emission show extended structures , in particular two `` hot spots '' in the neix image , which are close to the radio knot features ( figures [ lines ] and [ o3 ] ) . at these locations ( 1 from the nucleus ) , @xcite noted that the morphology of the highly collimated radio outflows appears more disturbed , suggesting interaction with some dense clouds here . there are also clearly optical clouds with high velocity dispersion associated with the jet cloud impact @xcite . the radio knots are near to the location of strong [ feii ] emission @xcite , falling between the peaks of the [ feii ] emission . altogether , the morphology suggests a scenario where the radio ejecta runs into a denser cloud in the inhomogeneous ism and results in locally enhanced shock heating ( e.g. , * ? ? ? cloud interaction has been suggested to explain the enhanced [ feii ] emission , which could be due to the iron unlocked from grains by the shocks @xcite . the neix emission , which traces the higher ionization gas , becomes more prominent with the extra collisional ionization from the jet cloud interaction . @xcite also noted that the poor fit to the ovii line profile in the grating spectra could be due to presence of a non - photoionization component . this possibility is further supported by our spectral fits , which suggest that the neix / ovii enhancements have a thermal origin , requiring the presence of a collisionally ionized plasma ( the @xmath41 kev component ) in addition to a photoionized nuclear outflow @xcite . the emission measure of the @xmath40 component allows us to estimate the electron density @xmath56 ( @xmath57 ) and the thermal pressure @xmath58 ( @xmath59 ) . the emitting volume ( @xmath60 ) of the hot gas , assuming that the depth along the line of sight is comparable to the other dimensions , is @xmath61 @xmath62 for the high neix / ovii ratio hot spots . the filling factor @xmath63 is assumed to be 100% and both of @xmath56 and @xmath58 have a weak dependence on @xmath63 ( @xmath64 ) . there is @xmath65 uncertainty in @xmath66 , which is propagated to the estimated pressure and energy . for the @xmath67 kev emission , we derived an average density @xmath68 @xmath69 , a hot gas pressure @xmath70 dyne @xmath22 , a thermal energy content of @xmath71 ergs , and a cooling time of @xmath72 yr . the estimated internal pressure of the radio jet based on the synchrotron minimum energy assumption ranges between @xmath73 and @xmath74 dyne @xmath22 ( pedlar et al . 1993 ) , to be confined by the hot gas pressure of @xmath74 dyne @xmath22 , which further supports our collisional ionization scenario . assuming @xmath75 k for the photoionized clouds and @xmath76 @xmath69 @xcite , the thermal pressure from the photoionized gas is @xmath77 dyne @xmath22 , which is comparable to the thermal pressure of the hot ism , implying a possible pressure equilibrium between the collisionally ionized hot gas and the photoionized line - emitting cool clouds . we further estimate the age of the interaction from the approximate crossing time of the @xmath78 region ( dimension of the radio knot ; * ? ? ? * ) with a characteristic velocity of @xmath79 km s@xmath29 , the local sound speed . this is approximately the thermal velocity for the 0.58 kev gas ( @xmath80 km s@xmath29 ) , and also at the order of velocity dispersion of emission line gas seen in @xcite . assuming the strong shock jump conditions , a crude estimate of the shock velocity @xmath81 can be obtained using @xmath82 k @xmath83 km s@xmath84 @xcite . for the @xmath85 k x - ray - emitting gas , a @xmath81 of @xmath0700 km s@xmath29 relative to the downstream material is required . this appears consistent with a supersonic but relatively slow radio jet , which is constrained to be sub - relativistic from radio proper motion estimates ( e.g. , @xmath86 km s@xmath29 ; * ? ? ? thus we obtain a characteristic timescale of @xmath87 yr . if the jet cloud interaction converts kinematic energy into heating of the hot gas , a lower limit can be placed on the kinematic luminosity of the jet , @xmath88 erg s@xmath29 . it is interesting to compare this observed @xmath89 to the jet power directly inferred from the synchrotron emission . using the @xmath90@xmath91 scaling relation ( equation 1 ) in @xcite and a total radio luminosity at 1.4 ghz ( @xmath92 erg s@xmath29 ; * ) , we find @xmath93 erg s@xmath29 , suggesting that @xmath1 of the jet power is deposited in the ism . however , recalling that the @xmath90@xmath91 relation was derived from a sample of radio loud galaxies with x - ray cavities ( see @xcite for review on the @xmath90@xmath91 scaling relation ) while here ngc 4151 is of low radio power , we also estimate directly the jet energy flux using the pressure in the knots @xcite based on the minimal energy assumption , giving a similar @xmath94 erg s@xmath29 . we further note that , the crossing time @xmath95 is comparable to the cooling time @xmath96 , implying that the hot gas is prominent only locally ( close to the radio knots ) . indeed , along the path of the radio jet , the clouds that are spatially close to the jet impact spots ( e.g. , clouds # 3,4 bracketing knot c2 ; # 9,10 for knot c5 ) show [ oiii]/x - ray ratios indistinguishable from others ( figure [ ratio ] ) . if these clouds are physically related to the ism interacting with the jet , instead of being projected close to the knots in our line - of - sight , their x - ray emission shows little evidence for excess shock heating in addition to nuclear photoionization . this localized heating is perhaps related to the highly collimated radio ejecta in ngc 4151 @xcite , in contrast to the more expanded lobe - like radio outflow in mrk 3 @xcite , where the nlr gas appears broadly impacted . we plan to investigate in future work whether the [ oiii]/x - ray ratio , such as @xmath974 measured here and seen in mrk 3 and ngc 1386 @xcite , together with other line ratios ( e.g. , [ feii]/[pii ] ; @xcite ) could be useful diagnostics for strong jet ism interaction in seyfert galaxies . in terms of energetics , we summarize our findings in ngc 4151 : ( a ) the jet power estimated from the radio power is @xmath98 erg s@xmath29 , a few percent of the current agn energy output ( @xmath99 erg s@xmath29 ; * ? ? ? * ) , which is close to the canonical @xmath05% of the radiant energy adopted in theoretical models for efficient agn feedback ( e.g. , * ? ? ? * ; * ? ? ? * ) ; ( b ) @xmath1 of the jet power is observed to have been deposited into the host ism in the nuclear region through the interaction between the radio jet and the dense medium ; ( c ) the shock heating due to the jet cloud interaction appears localized to the impact spots , and most clouds are consistent with being photoionized by the nucleus . @xcite claimed that the narrow iron line emission was spatially resolved in the _ chandra _ hetg observation and @xmath100 of the fei k@xmath2 emission comes from the enlr at distance up to 6extent ( @xmath0400 pc across ) . this is intriguing , since the putative pc - scale torus or more compact broad - line region ( * ? ? ? * and references therein ) are expected to be the primary location of narrow fei k@xmath2 emission . @xcite cautioned that due to the sensitivity of the short grating observation in @xcite , the 13 arcsec region off the peak of cross - dispersion profile contains few photons . therefore the detection of spatially extended iron line emission can not be significant . a high signal - to - noise xmm - newton spectrum of the neutral fe k@xmath2 suggests that all line flux originates in a nearly compton - thick torus that is not resolvable at current resolution @xcite . our results show that the extended fe k@xmath2 emission , if present , is only at @xmath101 level of the observed fe k@xmath2 emission , when the contribution from the psf wings is taken into account . this is consistent with our constraint on the extent of the si fluorescent emission and supports the conclusions of @xcite , while it is in strong disagreement with the @xmath102 reported in @xcite . we note that in @xcite the psf is represented as a narrow gaussian with @xmath103 , when the _ chandra _ mission was new . the marx model is now much more advanced so that a more reliable estimate becomes possible . at 6.4 kev , the encircled energy fraction of a point source is only @xmath104 ( see @xmath105 proposers observatory guide chapter 4 table 4.2 ) in such a psf core . hence a significant fraction of the nuclear emission could have been measured in ogle et al . as spatially extended fe k@xmath2 emission . in this paper we present spectral analysis and emission line images from deep _ chandra _ observation of ngc 4151 , aiming to resolve and characterise the x - ray emission in the inner @xmath0130 pc - radius nuclear region . the findings are summarized as follows : 1 . we have obtained high spatial resolution x - ray narrow - band images of ovii , oviii , and neix line emission , which are blended at the acis spectral resolution . the images show extended structures that are spatially correlated with the radio outflow and optical [ oiii ] emission . 2 . we find strong evidence for jet ism interaction , including morphological correspondences with regions of x - ray enhancement , peaks of nir [ feii ] emission , and optical clouds . this is further strengthened by the presence of a @xmath67 kev collisionally ionized component in the spectral fitting of the hot spots , and by the excess of x - ray emission in addition to nuclear photoionization as indicated by a low [ oiii]/x - ray ratio . we find a possible pressure equilibrium between the collisionally ionized hot gas and the photoionized cool clouds . the estimated velocity of the shocks from the jet cloud impact is @xmath0700 km s@xmath29 . we estimate that the jet power in ngc 4151 is close to a few percent of the current agn energy output ( @xmath106 erg s@xmath29 ) . the derived thermal energy in the hot gas suggests that @xmath1 of the jet power is deposited into the host ism in the nuclear region through the interaction between the radio - jet and the dense medium . the [ oiii]/x - ray ratios of nlr clouds bracketing the radio knots show little deviation from other photoionized clouds , indicating a localized impact on the ism by the highly collimated jet . we investigate the spatial extent of the fluorescent features , including the fe k@xmath2 emission and the si k@xmath2 emission . our results show that both are dominated by point - like emission , consistent with an origin of unresolved inner structure such as a torus . the extended fe k@xmath2 emission is @xmath3 of the observed fe k@xmath2 emission , which is in strong disagreement with the @xmath102 reported in @xcite . we thank the anonymous referee for helpful suggestions . this work is supported by nasa grant go8 - 9101x ( pi : fabbiano ) and grant go1 - 12009x ( pi : wang ) . we acknowledge support from the cxc , which is operated by the smithsonian astrophysical observatory ( sao ) for and on behalf of nasa under contract nas8 - 03060 . cgm acknowledges financial support from the royal society and research councils u.k . j. w. thanks g. ferland , t. kallman , s. bianchi , a. marinucci , and s. chakravorty for extensive advices on photoionization modeling , p. nulsen for jet power discussion , t. storchi - bergmann and r. riffel for providing the gemini nifs maps . this research has made use of data obtained from the _ chandra _ data archive , and software provided by the cxc in the application packages ciao and sherpa . , g. p. , bautz , m. w. , ford , p. g. , nousek , j. a. , & ricker , jr . , g. r. 2003 , in society of photo - optical instrumentation engineers ( spie ) conference series , vol . 4851 , society of photo - optical instrumentation engineers ( spie ) conference series , ed . j. e. truemper & h. d. tananbaum , 2844 250 pc of ngc 4151 ( a ) ovii ; ( b ) oviii+ovii rrc ; ( c ) neix . the nucleus position is indicated with a cross . the contours outline the radio outflow in the 1.4 ghz merlin map ( mundell et al . the acis images have been rebinned to @xmath107 per pixel , and smoothed with a @xmath13 gaussian kernel . [ lines ] ] 250 pc of ngc 4151 as in figure [ lines ] , but with contours showing the @xmath44/foc [ oiii]@xmath108 emission line clouds ( winge et al . ( a ) ovii ; ( b ) oviii+ovii rrc ; ( c ) neix . the acis images have been rebinned to @xmath107 per pixel , and smoothed with a @xmath13 gaussian kernel . [ o3 ] ] high - ratio & 4585 & @xmath109 & 20.5 & @xmath110 & @xmath111 & 22.5 & @xmath112 & @xmath113 & @xmath114 & @xmath115 & @xmath116 + low - ratio & 1317 & @xmath109 & 20.5 & @xmath117 & @xmath111 & 22.5 & @xmath118 & @xmath113 & @xmath119 & @xmath120 & @xmath121 + cccccccc 1 & 1.76 & 114 & 1.1 & 369@xmath12224 & 1.5 & 7.4 & 12 + 2 & 1.44 & 93.6 & 1.9 & 688@xmath12229 & 2.7 & 6.9 & 12 + 3 & 1.01 & 65.6 & 2.2 & 581@xmath12238 & 2.3 & 9.4 & 12 + 4 & 0.78 & 50.7 & 1.3 & 470@xmath12239 & 1.9 & 6.9 & 7 + 5 & 1.0 & 65.0 & 1.5 & 168@xmath12229 & 0.7 & 22.2 & 75 + 6 & 0.6 & 39 . & 1.1 & 99@xmath12244 & 0.4 & 27.8 & 110 + 7 & 0.4 & 26 . & 3.4 & 1320@xmath12296 & 5.3 & 6.4 & 5 + 8 & 0.4 & 26 . & 3.3 & 560@xmath122100 & 2.2 & 14.7 & 13 + 9 & 0.85 & 55.2 & 2.4 & 694@xmath12247 & 2.8 & 8.6 & 3 + 10 & 0.89 & 57.8 & 1.6 & 266@xmath12235 & 1.1 & 15.0 & 8 + 11 & 2.15 & 139.7 & 0.6 & 264@xmath12217 & 1.1 & 5.7 & 15 + c2 & 0.9 & 58.5 & 0.8 & 1180@xmath12244 & 4.7 & 1.7 & 2 + c5 & 0.93 & 60.4 & 1.4 & 1052@xmath12243 & 4.2 & 3.3 & 3 +
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we have studied the x - ray emission within the inner @xmath0150 pc radius of ngc 4151 by constructing high spatial resolution emission line images of blended ovii , oviii , and neix .
these maps show extended structures that are spatially correlated with the radio outflow and optical [ oiii ] emission .
we find strong evidence for jet gas cloud interaction , including morphological correspondences with regions of x - ray enhancement , peaks of near - infrared [ feii ] emission , and optical clouds . in these regions , moreover , we find evidence of elevated neix / ovii ratios ; the x - ray emission of these regions also exceeds that expected from nuclear photoionization .
spectral fitting reveals the presence of a collisionally ionized component .
the thermal energy of the hot gas suggests that @xmath1 of the estimated jet power is deposited into the host interstellar medium through interaction between the radio jet and the dense medium of the circum - nuclear region .
we find possible pressure equilibrium between the collisionally ionized hot gas and the photoionized line - emitting cool clouds .
we also obtain constraints on the extended iron and silicon fluorescent emission .
both lines are spatially unresolved .
the upper limit on the contribution of an extended emission region to the fe k@xmath2 emission is @xmath3 of the total , in disagreement with a previous claim that 65% of the fe k@xmath2 emission originates in the extended narrow line region .
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in an attempt to understand the molecular content and physical characteristics of interstellar gas in the low galactic halo , we exploit infrared and ultraviolet data from two nasa satellites : the _ infrared astronomical satellite _ ( _ iras _ ) mission of 1983 and the _ far ultraviolet spectroscopic explorer _ ( _ fuse _ ) satellite of 19992005 . the combination of infrared emission and ultraviolet absorption along sight lines to 45 active galactic nuclei ( agn ) allows us to correlate the infrared cirrus emission intensity with the molecular hydrogen ( h@xmath1 ) absorption column density at select locations at high galactic latitude . _ iras _ mapped the sky in four infrared bands centered on 12 , 25 , 60 , and 100 @xmath0 m . low et al . ( 1984 ) introduced one of the most surprising results from the _ iras _ maps : diffuse filamentary dust clouds that pervade our galaxy at high latitudes , even in the direction of the galactic poles . these infrared cirrus " clouds are most prominent at long wavelengths , 100 @xmath0 m , but some can be seen in the 60 @xmath0 m , 25 @xmath0 m , and 12 @xmath0 m bands . comparisons between _ iras _ dust maps and maps of 21 cm emission reveal a generally good correlation between neutral hydrogen ( hartmann & burton 1997 ) and cirrus dust filaments ( figure 1 ) . because molecular hydrogen ( h@xmath1 ) forms catalytically on the surface of dust grains ( hollenbach , werner , & salpeter 1971 ) , with significant formation rates for grain temperatures @xmath16 k and gas temperatures @xmath17 k ( hollenbach & mckee 1979 ; shull & beckwith 1982 ) , the cold , dusty conditions of the infrared cirrus clouds are favorable for the formation of h@xmath1 . thus , it is plausible that some fraction of the hydrogen atoms in the cirrus clouds are bound into molecules . previously , the presence of h@xmath1 in infrared cirrus was inferred indirectly . first , under the assumption that the infrared emission and total hydrogen column density , n@xmath3 , are proportional , regions of high dust/ ratio , termed infrared excess " , were attributed to the presence of h@xmath1 ( de vries , heithausen , & thaddeus 1987 ; desert , bazell , & boulanger 1988 ; reach , koo , & heiles 1994 ; moritz et al . 1998 ; schlegel , finkbeiner , & davis 1998 , hereafter sfd98 ) . second , the detection of co in dense cirrus clouds suggests that the diffuse cirrus clouds should contain h@xmath1 as well . weiland et al . ( 1986 ) compared co maps from magnani , blitz , & mundy ( 1985 ) to _ iras _ maps of infrared cirrus . each of the 33 co clouds had a cirrus counterpart with similar morphology . this work established that at least some of the infrared cirrus cloud cores contains co gas . unfortunately , there is currently no experiment that can map diffuse h@xmath1 emission , either in the 2.12 @xmath0 m [ ( 10 ) s(1 ) ] vibrational line or in the s(0 ) , s(1 ) , s(2 ) pure rotational lines at 28 @xmath0 m , 17 @xmath0 m , and 12 @xmath0 m , respectively . although h@xmath1 is over @xmath18 times more abundant than co , the ultraviolet and infrared fluorescent emission of h@xmath1 is very weak . ultraviolet absorption - line spectroscopy is therefore the primary means for detecting cold h@xmath1 in diffuse clouds . however , it requires background sources with sufficient uv flux to provide adequate signal - to - noise ratio ( s / n ) to detect the weak h@xmath1 lines . the first major project to conduct such observations was the _ copernicus _ mission of the 1970s ( spitzer & jenkins 1975 ) . however , its sensitivity limited the possible background sources to early - type stars within about 500 pc of the sun . most ob stars that fit this criterion are at low galactic latitude , and they suffer from confusion and dust extinction in the galactic plane . individual features in the infrared cirrus can not be discerned at low galactic latitudes , and these stellar sight lines are not effective probes of the dusty filaments . the _ fuse _ satellite , which has been observing the ultraviolet sky since 1999 , has expanded the opportunities for detecting h@xmath1 . the increased sensitivity of _ fuse _ ( @xmath19mag ) over _ copernicus _ ( @xmath20mag ) allows us to use more distant stars as well as active galactic nuclei ( agn ) as background sources . our _ fuse _ survey of h@xmath1 toward high - latitude agn ( gillmon et al.2005 ) is particularly well suited for probing h@xmath1 in infrared cirrus . the high - latitude sight lines avoid the confusion of the galactic disk , and they provide long path lengths through the galactic halo . in addition , the random distribution of agn on the sky samples a range of infrared cirrus emission intensities . the main limitation of using pencil - beam " ( absorption ) sight lines to detect h@xmath1 in infrared cirrus is the inability to determine whether the gas and dust detected along a given sight line are physically associated . therefore , we must rely on indirect correlations between cirrus and h@xmath1 absorption . in this paper , we compare the h@xmath1 column densities in the survey by gillmon et al . ( 2005 ) with the corresponding infrared cirrus fluxes ( sfd98 ) . by establishing a correlation between the two , we assert that at least some of the detected h@xmath1 resides in the cirrus clouds . in 2 we describe the data acquisition and analysis for both _ iras _ and _ fuse_. in 3 we compare the cirrus emission and h@xmath1 absorption and discuss the correlation of the two . exploiting this correlation and summing over the distribution of h@xmath1 column densities with 100 @xmath0 m cirrus intensity , we estimate the total h@xmath1 mass ( @xmath21 ) in cirrus clouds around the milky way . with molecular fractions ranging from 130% , the total cirrus mass throughout the milky way is @xmath22 . in 4 we summarize our results and the implications of finding this amount of gas in the low halo of the milky way . the h@xmath1 absorption data were taken from _ fuse _ spectra of agn , using standard data analysis techniques ( tumlinson et al . 2002 ; gillmon et al . studies of h@xmath1 have been a major part of the _ fuse _ science plan . the satellite , its mission , and its on - orbit performance are described in moos et al . ( 2000 ) and sahnow et al . scientific results on interstellar h@xmath1 have appeared in a number of papers ( shull et al . 2000 ; snow et al . 2000 ; rachford et al . 2002 ; richter et al . 2001 , 2003 ; tumlinson et al . 2002 ; shull et al . the resolution of _ fuse _ varies from @xmath23 across the far - uv band . all observations were obtained in time - tag ( ttag ) mode , using the @xmath24 lwrs aperture , with resolution @xmath25 km s@xmath8 at 1050 . the s / n of the co - added data ranges from 211 per pixel ; the s / n per resolution element varies with spectral resolution , which is not fixed in our survey . most of the data were binned by 4 pixels before analysis , with the rare case of binning by 2 or 8 pixels . the data in this paper were taken from our high - latitude h@xmath1 survey ( gillmon et al . 2005 ) , which describes our search for h@xmath1 absorption along 45 sight lines to background agn at galactic latitudes @xmath26 . the 45 agn in the survey by gillmon et al . ( 2005 ) were a subset of the 219 _ fuse _ targets selected in wakker et al . ( 2003 ) as candidates for the analysis of galactic ovi . these targets probe diffuse gas in both the local galactic disk and low galactic halo . of the available agn at high latitude , 45 sight lines were chosen , based on an imposed s / n requirement of @xmath27 or @xmath28 with 4-pixel binning . the observed h@xmath1 lines arise from the lyman and werner electronic transitions , from the ground electronic state , x @xmath29 , to the excited states , b @xmath30 ( lyman bands ) and c @xmath31 ( werner bands ) . the rotational - vibrational lines arise from the ground vibrational state and a range of rotational states . our analysis was restricted to ten vibrational - rotational bands , lyman ( 00 ) to ( 80 ) and werner ( 00 ) , which are located between 1000 @xmath32 and 1126 @xmath32 . the vibrational state notation is ( @xmath33 ) . in most sight lines , we observed absorption lines from rotational states @xmath343 , and sometimes up to @xmath35 . the end product of the h@xmath1 absorption - line analysis is the column density , n@xmath11 ( @xmath36 ) , the physical density of h@xmath1 molecules , integrated along the sight line . each absorption line was fitted with a voigt profile in order to determine the equivalent width , a measure of the absorbed light in the line . the equivalent widths were fitted to a curve of growth , to find the column density , n@xmath37 in each rotational state @xmath38 . the points for each @xmath38 were tied together during the fitting to produce a single , consistent column density for each rotational state , n(@xmath38 ) . the sum of the column densities in all rotational states then gives the total column density , n@xmath11 . fuse spectra of 87% ( 39 of 45 ) of the observed agn showed detectable h@xmath1 absorption , with column densities ranging from n@xmath11 @xmath39 @xmath36 . the _ fuse _ survey is sensitive to n@xmath11 @xmath40 @xmath36 , depending on the s / n ( 211 per pixel ) and spectral resolution ( @xmath4120,000 ) . to obtain infrared cirrus emission intensities ( mjy sr@xmath8 ) , we use the 100 @xmath0 m maps presented by sfd98 . these were a composite of data from the _ iras _ mission of 1983 and the _ cobe _ mission of 19891990 , capitalizing on the strengths of each . because the interstellar dust emits like a grey body " , the emission intensity is sensitive to the dust temperature . as a result , two regions with the same dust column density but different dust temperatures will have different infrared intensities . to correct for this effect , sfd98 used the ratio of the 100 @xmath0 m and 240 @xmath0 m _ cobe _ maps to produce a map of dust color temperature . they used this ratio to correct the 100 @xmath0 m _ iras _ map so that it is proportional to dust column density . _ iras _ mapped the sky in four broadband infrared channels , centered at 12 , 25 , 60 , and 100 @xmath0 m with a resolution of @xmath42 , while _ cobe _ mapped the sky in 10 broad photometric bands from 1 to 240 @xmath0 m at a resolution of @xmath43 . before combining the data sets , sfd98 took great care in the difficult removal of the zodiacal foreground emission and _ iras _ striping artifacts that arise from differences in solar elongation between scans . confirmed point sources were also removed . the maps were combined in such a way as to preserve the _ cobe _ calibration and _ iras _ resolution . maps of the temperature - corrected 100 @xmath0 m intensity , @xmath44 , are presented for the northern galactic hemisphere ( figure 2 ) and for the southern galactic hemisphere ( figure 3 ) . we overplot the locations of the 45 sight lines from the _ fuse _ h@xmath1 survey : 28 northern agn and 17 southern agn . before the _ fuse _ mission , the direct detection of h@xmath1 in infrared cirrus by uv absorption - line spectroscopy was prevented mainly by a lack of background sources at high galactic latitude . even though this problem has been alleviated by _ s ability to observe bright agn as background sources , another problem with absorption - line spectroscopy along pencil - beam " sight lines comes to the forefront . absorption studies lack morphological information and can not provide distances to clouds along the line of sight . therefore , it is difficult to identify the gas that gives rise to the detected column density . the infrared cirrus problem is a classic case . for any given sight line , it is possible that the h@xmath1 absorption is not physically associated with the infrared cirrus along the beam . if this is the case , then comparing n@xmath11 with the cirrus dust column density would lead to erroneous results . it would be helpful to determine whether the diffuse h@xmath1 along all sight lines resides in the cirrus . if no other component of the diffuse ism harbored significant amounts of h@xmath1 , then n@xmath11 for a given sight line could safely be associated with other cirrus properties . in this section , we investigate this possibility , based on a property of h@xmath1 called self - shielding " . following line absorption of uv photons from the mean interstellar radiation field , h@xmath1 decays approximately 11% of the time to the dissociative continuum . as a result , the molecular fraction , @xmath46 is generally larger in clouds with a greater total hydrogen column density , n@xmath3 = n@xmath47 + 2n@xmath11 . molecules on the outside of the cloud shield those in the interior from dissociating uv ( hollenbach , werner , & salpeter 1971 ; black & dalgarno 1976 ; browning , tumlinson , & shull 2003 ) . in optically thin clouds , the density of molecules can be approximated by the equilibrium between formation and destruction , @xmath48 in this formula , the numerical value for @xmath2 is scaled to fiducial values of hydrogen density , @xmath49 ( 30 @xmath10 ) , h@xmath1formation rate coefficient , @xmath50 ( @xmath51 @xmath52 s@xmath8 ) , and mean h@xmath1 pumping rate in the fuv lyman and werner bands , @xmath53 s@xmath8 . the h@xmath1 photodissociation rate is written as @xmath54 , where the coefficient @xmath55 is the average fraction of fuv excitations of h@xmath1 that result in decays to the dissociating continuum . the h@xmath1 formation rate per unit volume is written as @xmath56 , where the coefficient @xmath50 depends on the gas temperature , grain surface temperature , and gas metallicity ( @xmath57 ) . the metallicity dependence comes from the assumed scaling of grain - surface catalysis sites with the grain / gas ratio . for sight lines in the local galactic disk , this rate coefficient has been estimated ( jura 1974 ) to range from @xmath58 @xmath52 s@xmath8 at solar metallicity . this standard value for @xmath50 is expected to apply at suitably low temperatures of gas ( @xmath59 k ) and grains ( @xmath60 k ) as discussed by shull & beckwith ( 1982 ) and hollenbach & mckee ( 1979 ) . the effects of self - shielding were observed by plotting @xmath2 versus n@xmath3 , the total column density of hydrogen . as h@xmath1 absorption lines in the lyman and werner bands become optically thick , the rate ( @xmath61 ) of uv pumping and molecular dissociation diminish . in this way , the presence of h@xmath1 screens molecules in the inner portions of the cloud from dissociation . a transition from low ( @xmath62 ) to high ( @xmath63 ) molecular fractions at n@xmath64 @xmath36 was noted in the _ copernicus _ h@xmath1 survey ( savage et al . 1977 ) . a recent _ fuse _ survey of galactic disk stars ( shull et al . 2005 ) provided similar results ( figure 4 ) . in assessing the molecular content of the cirrus , we consider three generic cases . the first possibility ( case i ) is that all the detected h@xmath1 resides in cirrus clouds , with no contribution from foreground clouds . under the assumption that the total hydrogen density , @xmath65 , is proportional to the number density of dust grains in any given interstellar cloud ( hollenbach & mckee 1979 ) , the cirrus dust column density should be proportional to the amount of n@xmath66 associated with h@xmath1 . a plot of n@xmath11 vs. dust column density should then show the familiar self - shielding transition of molecular hydrogen . in case ii , some of the observed h@xmath1 lies in the cirrus , but some resides in another component of the ism not visible in the cirrus maps . a plot of n@xmath11 vs. dust column density would show some points that follow the self - shielding transition ( those corresponding to cirrus ) . however , the transition would be broadened by points that are randomly distributed . if a significant portion of the detected h@xmath1along a sight line is not in cirrus , n@xmath11 will not correlate with dust column density . this effect would produce sight lines with low dust column density and significant amounts of h@xmath1 . the extreme situation is case iii , in which none of the h@xmath1 resides in cirrus clouds . we consider this unlikely , given the number of observed regions with infrared excess " ( high dust / n@xmath47 ratios ) and the detection of co in denser cirrus clouds . once the n@xmath11 in cirrus exceeds @xmath67 @xmath36 , it should be detectable by _ fuse_. for case iii to be consistent with our survey , the covering fraction of cirrus regions with detectable h@xmath1 would need to be quite small , whereas the cirrus covering factor is observed to be @xmath5% at @xmath6 ( see 3.4 ) . the issue then comes down to whether case i or case ii is a better description of the cirrus - h@xmath1 correlation . figure 5 presents a plot of n@xmath11 along sight lines to the 45 agn in the _ fuse _ survey vs. the temperature - corrected ir flux at each location from the sfd98 maps ( blue diamonds ) . if the temperature - corrected flux is proportional to dust column density , this plot can be used to test the three cases mentioned above . the blue diamonds appear to show the self - shielding transition discussed in case i , implying that much of the detected h@xmath1 is in the cirrus . in this section , we describe five additional sight lines shown in red ( figure 5 ) toward the regions of lowest cirrus . these additional sight lines were chosen to explore whether the correlation between n@xmath11and 100 @xmath0 m cirrus is universal . table 5 of sfd98 lists the coordinates of the regions of lowest ( temperature - corrected ) 100 @xmath0 m intensity , @xmath44 , from their maps . we searched the _ fuse _ archive for targets within @xmath68 of the given coordinates and found five targets behind regions of low cirrus ( @xmath69 mjysr@xmath8 ) that also had _ fuse _ data with sufficient ( s / n)@xmath70 to conduct an h@xmath1 analysis . these five targets are in addition to those in the gillmon et al . ( 2005 ) sample of 45 agn . they are listed in table 1 and shown as asterisks in figure 5 . four of these five targets showed no evidence of h@xmath1 , with a typical upper limit of n@xmath11 @xmath71 @xmath36 . this result lends further credence to the theory that most of the observed h@xmath1 is in the cirrus ( case i ) . however , one target , ugc5720 , showed significant h@xmath1 absorption lines ( figure 6 ) . an analysis , as described in 2.1 , yielded a significant column density n@xmath11 @xmath72 @xmath36 . in 3.3 , we explore possible explanations for this anomalous sight line and for the spread in the self - shielding transition . figure 5 plots n@xmath11 vs. temperature - corrected intensity , @xmath44 ( mjysr@xmath8 ) , and exhibits a clear self - shielding transition of h@xmath1 . this indicates that a significant fraction of the detected h@xmath1 resides in the infrared cirrus . however , with small - number statistics on 50 agn sight lines , the transition is not sharp , occurring between log @xmath44 = 0.20.5 . there is one outlying point ( ugc5720 ) with low cirrus intensity , log @xmath73 , but a significant column density , log n@xmath11 @xmath74 . this outlier suggests that not all the detected h@xmath1 resides in cirrus clouds , and that other components of the diffuse gas may harbor detectable amounts of molecules . in some sight lines that intercept non - cirrus clouds with low 100 @xmath0 m intensity , the gas may undergo an early molecular transition and appear with higher n@xmath11 there are other explanations that might produce the same effects , even if all the h@xmath1 resides in cirrus clouds . one consideration is how well the temperature - corrected flux map traces the dust column density . to correct for temperature variations , sfd98 created a temperature map from the ratio of the _ cobe _ 100 @xmath0 m and 240 @xmath0 m maps . thus , low - resolution ( 1.1@xmath75 ) temperature maps were used to correct the @xmath76 resolution _ iras _ map to produce a 100 @xmath0 m map at @xmath77 resolution . it is likely that the dust temperature varies on smaller scales than can be resolved by this method , and a more accurate temperature correction might tighten the self - shielding transition . it is also possible that ugc5720 lies behind a region of cold cirrus dust unresolved by the temperature map . thus , a significant portion of the dust column density might not be indicated by the observed flux . a related consideration is that the _ iras _ map might not resolve significant variations in dust column density . the sight lines observed by _ fuse _ absorption probe gas along a very narrow beam , while the _ iras _ beam is considerably larger . thus , if there were significant structure on smaller scales than the _ iras _ beam , _ fuse _ could observe a denser clump of h@xmath1 while the _ iras _ emission would be beam - diluted " . a higher resolution infrared map might tighten the self - shielding transition and/or bring the outlier ugc5720 onto the correlation . figure 7 shows a @xmath78 section of the sfd98 flux map centered on ugc5720 . large fluctuations in temperature - corrected flux near the position of the agn suggest that the sight line may be picking up a clump of cirrus with higher dust column density unresolved by these maps . finally , there is the possibility that there are multiple cirrus features superimposed along a line of sight . duel & burton ( 1990 ) compared the morphology of cirrus clouds and h i maps in various velocity intervals to show that cirrus features that appear simple are , in some cases , superpositions of kinematically distinct components . this idea of a concatenation of clouds " was proposed for translucent h@xmath1 clouds ( browning et al . 2003 ) to explain the high levels of h@xmath1 rotational excitation in systems with large n@xmath11 . these authors suggested that , if a sight line intercepts multiple , physically distinct cloud components , the h@xmath1 will be exposed to a radiation field enhanced over that expected for a single , contiguous cloud with the same total column density . the enhanced h@xmath1 destruction rate from the stronger uv radiation would reduce the mean molecular fraction and produce a gradual transition to higher n@xmath11 . some sight lines with seemingly high dust column density , but low n@xmath11 , may actually be probing multiple , superimposed filaments with high integrated dust column density . with the empirical correlation ( figure 5 ) between ir cirrus and h@xmath1 column density , we can make a quantitative estimate of the h@xmath1 mass in the diffuse cirrus clouds . we begin with the _ iras _ maps of the northern galactic hemisphere . we assume that these cirrus clouds lie at elevation @xmath79 above the milky way plane , and that clouds of intensity @xmath80 cover a fraction @xmath81 of the planar area at @xmath82 . the total h@xmath1 mass in this planar cloud deck is then , @xmath83 \left [ \frac { \langle f_c \rangle } { 0.5 } \right ] z_{100}^2 \ ; , \ ] ] where n@xmath11@xmath84 is the mean h@xmath1 column density corresponding to cirrus intensity @xmath80 ( figure 5 ) and where @xmath85 is the radius of the cirrus disk , at elevation @xmath86 subtended by the cone at @xmath87 . for this estimate , we have assumed that the cirrus covers a fraction @xmath88 of the sky at @xmath82 , independent of intensity . we now make a more careful calculation , summing over the actual data from _ iras _ and _ fuse_. figure 8 shows the distribution of cirrus covering factors , @xmath81 , averaged over northern hemisphere regions with @xmath89 greater than @xmath90 , @xmath91 , and @xmath92 , respectively . for our fiducial cone at @xmath82 , approximately 50% of the sky is covered by cirrus with intensity log @xmath93 , the value corresponding to the h@xmath1 transition ( figure 5 ) . performing the full summation ( eq . 3 ) over 8 logarithmic intensity bins above the transition , of width @xmath94(log @xmath80 ) = 0.1 , between log @xmath95 and 1.0 , we find a total h@xmath1 mass of @xmath96 contained in the cirrus at @xmath82 . to convert this calculation to the inner milky way , within the solar circle , we multiply by a factor @xmath97 , for the northern and southern galactic hemispheres , and scale by a factor @xmath98 to account for the number of similar conical areas around the galactic disk . note that this total h@xmath1 mass is independent of the assumed cirrus elevation , @xmath86 , since the area - scaling cancels the factor @xmath99 in equation ( 3 ) . we arrive at an extrapolated total molecular mass over the inner milky way , @xmath100 , assuming that the cirrus along the agn sightlines at @xmath6 is typical . the molecular fractions of these cirrus clouds , with n@xmath11 @xmath101 @xmath36 , range from 130% for log @xmath44 = 0.20.5 ( see figure 6 of gillmon et al . 2005 ) , with an average @xmath102 for clouds with log n@xmath11 @xmath103 . therefore , we estimate the total gas mass in the cirrus to be @xmath104 . the characteristics of the cirrus clouds along our agn sight lines can be verified by computing the dust - to - gas " ratio , defined as the ratio of 100 @xmath0 m cirrus intensity , @xmath44 ( mjy sr@xmath8 ) , to h i column density , n@xmath47 ( @xmath36 ) . table 2 gives these values and their ratio for 16 of our agn sight lines observed in 21 cm ( lockman & savage 1995 ) . this ratio ranges from @xmath105 mjy sr@xmath8 @xmath106 toward pg 0953 + 414 to @xmath107 mjy sr@xmath8 @xmath106 toward 3c 273 . the mean value and standard deviation are @xmath108 mjy sr@xmath8 @xmath106 , when we exclude the anomalous sight line to 3c 273 , which lies behind radio loop i and the north polar spur . this mean ratio is in excellent agreement with the mean value , @xmath109 mjy sr@xmath8 @xmath106 , found by boulanger , baud , & van albada ( 1985 ) in a @xmath110 field at high galactic latitude . we have undertaken a comparison between the column density , n@xmath11 , and the 100 @xmath0 m cirrus intensity for a total of 50 sight lines . for the cirrus , we used the temperature - corrected maps of schlegel , finkbeiner , & davis ( 1998 ) , and adopted h@xmath1 column densities from our high - latitude survey ( gillmon et al . the presence of a clear correlation between uv ( h@xmath1 ) absorption and ir ( cirrus ) emission indicates that a significant fraction of the h@xmath1 is physically associated with the cirrus clouds . however , the self - shielding transition of h@xmath1 fraction used to define the correlation is not sharp . the existence of one outlying sight line suggests either that some of the detected h@xmath1 may exist in another component of the diffuse ism , or that the limited resolution of the infrared maps is obscuring the physical conditions . of the three possible cases for h@xmath1cirrus connections laid out in 3.1 , case i or ii best describe the data . put simply , h@xmath1 is contained in most , if not all diffuse cirrus clouds . at galactic latitudes @xmath6 , approximately 50% of the sky is covered with cirrus , at temperature - corrected 100 @xmath0 m intensities @xmath7 mjy sr@xmath8 . with this correlation , we have found a convenient means of identifying the best extragalactic sight lines for h@xmath1-clean " far - uv absorption studies of intergalactic or interstellar matter . conversely , if the goal is to study h@xmath1 at the disk - halo interface , the cirrus maps would be a good guide . we also made a rough estimate of the h@xmath1 mass contained in these cirrus clouds . exploiting the h@xmath1cirrus correlation , we summed the distributions of ir cirrus intensity and h@xmath1 column density to find @xmath21 in cirrus h@xmath1 and @xmath22 in total hydrogen , distributed over the milky way disk - halo interface , within the solar circle . above the self - shielding transition , these diffuse halo clouds have molecular fractions ranging from 130% , for column densities n@xmath111 @xmath36 and n@xmath11 @xmath101 @xmath36 ( gillmon et al . 2005 ) . to support such high molecular fractions by h@xmath1 formation on grain surfaces , the cirrus clouds are probably compressed sheets with densities @xmath9 @xmath10 , in which the gas and grains remain sufficiently cold to form h@xmath1 . on average , the ir cirrus clouds may actually have higher molecular fractions , on average , than diffuse clouds in the galactic disk . this point was also made by reach et al . ( 1994 ) , who estimated that the h@xmath1/h i transition in denser cirrus clouds occurs at n@xmath112 @xmath36 , on the basis of fits to the far - infrared excess . this transition column density is almost twice that found here , probably because the infrared - excess technique requires larger molecular fractions than used for the uv absorbers ( @xmath113 ) . this initial survey opens up considerable opportunities for future studies of the h@xmath1cirrus correlation . higher resolution infrared maps with more accurate temperature corrections , such as those obtainable with the _ spitzer space telescope _ , would greatly improve the effectiveness of this method . a comparison with h i 21-cm maps ( e.g. , lockman & condon 2005 ) could delineate the gaseous structures associated with the cirrus and measure the dust - to - gas ratios in these diffuse clouds . expanding the uv sample to include more sight lines to background agn would improve the statistics of such a small sample . the 45 agn in the survey by gillmon et al . ( 2005 ) were a subset of the 219 _ fuse _ targets selected in wakker et al . ( 2003 ) as candidates for the analysis of galactic ovi . the next 50 brightest targets have an average flux of @xmath114 erg @xmath36 s@xmath8 @xmath8 . to achieve a s / n of 3 per pixel with _ fuse _ would require approximately 20 ksec per agn , for a program total of 1000 ksec . an ultraviolet telescope with sensitivity greater than fuse is probably necessary for feasible exposure times in a @xmath115 target survey . another promising avenue for future exploration is to compare the cirrus maps and h@xmath1 absorption lines with other tracers , such as h i , co and @xmath116-ray emission . on the basis of such comparisons , grenier , casandjian , & terrier ( 2005 ) suggest that many interstellar clouds in the solar neighborhood have extensive dark regions that bridge the dense cloud cores to atomic phases . these details are beyond the scope of our current paper . perhaps the most direct way to investigate the connection between diffuse h@xmath1 and ir cirrus would be to map h@xmath1 in uv or ir fluorescent emission . this method would provide the morphological information lacking in pencil - beam " uv - absorption sight lines . there is currently no high - resolution experiment that can map diffuse h@xmath1 emission , either in the mid - infrared ( 28 and 17 @xmath0 m ) or the far - ultraviolet ( 10001100 ) . intriguing results for h@xmath1 ultraviolet fluorescent emission at @xmath117 resolution may soon be available from the _ spectroscopy of plasma evolution from astrophysical radiation _ ( spear ) mission ( edelstein et al . with appropriate ir and uv telescopes , these methods could help map the gaseous galactic halo . we thank ken sembach and jay lockman for useful discussions . this work was based in part on data obtained for the guaranteed time team team by the nasa - cnes - csa _ fuse _ mission operated by the johns hopkins university . financial support to u.s . participants has been provided by nasa contract nas5 - 32985 . the colorado group also received _ fuse _ support from nasa grant nag5 - 10948 for studies of interstellar h@xmath1 . lccccc ms0354 - 3650 & 238.87 & @xmath118 & 0.37 & 3 & @xmath119 + har03 & 152.41 & 52.87 & 0.38 & 2 & @xmath120 + ugc5720 & 156.20 & 52.80 & 0.41 & 3 & @xmath121 + ngc5447 & 102.82 & 59.83 & 0.44 & 2 & @xmath120 + m101d & 102.67 & 59.75 & 0.46 & 2 & @xmath122 lccc & 1.98 & 20.14 & 1.43 + 3c 273 & 1.13 & 19.42 & 4.29 + h 1821 + 643 & 2.34 & 20.34 & 1.07 + hs 0624 + 6907 & 5.29 & 20.62 & 1.27 + mrc 2251 - 178 & 2.14 & 20.12 & 1.62 + mrk 205 & 2.28 & 20.25 & 1.28 + mrk 421 & 0.83 & 19.73 & 1.54 + pg 0804 + 761 & 1.92 & 20.41 & 0.74 + pg 0844 + 349 & 2.00 & 20.24 & 1.15 + pg 0953 + 414 & 0.66 & 20.00 & 0.66 + pg 1116 + 215 & 1.23 & 19.83 & 1.82 + pg 1211 + 143 & 1.86 & 20.33 & 0.87 + pg 1259 + 593 & 0.44 & 19.67 & 0.95 + pg 1302 - 102 & 2.36 & 20.22 & 1.42 + pks 0405 - 12 & 3.22 & 20.28 & 1.69 + pks 2155 - 304 & 1.19 & 20.06 & 1.03 +
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we combine data from our recent _ fuse _ survey of interstellar molecular hydrogen absorption toward 50 high - latitude agn with cobe - corrected _ iras _ 100 @xmath0 m emission maps to study the correlation of infrared cirrus with h@xmath1 .
a plot of the h@xmath1 column density vs. ir cirrus intensity shows the same transition in molecular fraction , @xmath2 , as seen with total hydrogen column density , n@xmath3 .
this transition is usually attributed to h@xmath1 self - shielding " , and it suggests that many diffuse cirrus clouds contain h@xmath1 in significant fractions , @xmath4 130% .
these clouds cover @xmath5% of the northern sky at @xmath6 , at temperature - corrected 100 @xmath0 m intensities @xmath7 mjy sr@xmath8 .
the sheetlike cirrus clouds , with hydrogen densities @xmath9 @xmath10 , may be compressed by dynamical processes at the disk - halo interface , and they are conducive to h@xmath1 formation on grain surfaces . exploiting the correlation between n@xmath11 and 100 @xmath0 m intensity , we estimate that cirrus clouds at @xmath6 contain @xmath12 in h@xmath1 . extrapolated over the inner milky way , the cirrus may contain @xmath13 of h@xmath1 and @xmath14 in total gas mass . if elevated to 100 pc , their gravitational potential energy is @xmath15 erg .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
weakly interacting massive particles ( wimps ) , and in particular the lightest neutralino in supersymmetric models , are a well motivated dark matter candidate @xcite . wimps can be detected directly , in the lab , via the elastic scattering of wimps on detector nuclei @xcite . experiments now have the sensitivity required to probe the theoretically favoured regions of parameter space @xcite and the cdms experiment has recently observed two events in its wimp signal region @xcite . neutrons , from cosmic - ray induced muons or natural radioactivity , can produce nuclear recoils which ( on an event by event basis ) are indistinguishable from wimp induced recoils . furthermore perfect rejection of other backgrounds is impossible , for instance ` surface events ' ( electron recoils close to the detector surface ) in the case of cdms . as highlighted by the cdms events , as well as the long - standing dama annual modulation signal @xcite , demonstrating the wimp origin of a putative signal is absolutely crucial . the direction dependence of the wimp scattering rate ( due to the earth s motion with respect to the galactic rest frame ) @xcite provides a potential wimp ` smoking gun ' . assuming that the wimp distribution is predominantly smooth , the peak wimp flux comes from the direction of solar motion ( towards the constellation cygnus ) and the recoil rate is then peaked in the opposite direction . the recoil rate , in the galactic rest frame , is highly anisotropic ; the rate in the forward direction is roughly an order of magnitude larger than that in the backward direction . a detector capable of measuring the nuclear recoil vectors ( including the sense + @xmath1 versus -@xmath2 ) in 3-dimensions , with good angular resolution , could reject isotropy of the 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 . confirmation of the wimp origin could be obtained by verifying that the properties of the anisotropy match the expectations for wimp induced recoils . assuming the wimp distribution is dominated by a smooth component , the median inverse recoil direction should be compatible with the direction of solar motion . to summarise , a wimp search strategy with a directional detector could be divided into a sequence of phases : + 1 . search phase ( detection of non - zero recoil signal ) + 2 . detection of anisotropy + 3 . study of properties of anisotropy + which require successively larger numbers of events ( and hence larger exposures ) . the initial simple search phase aims to detect an anomalous recoil signal above that expected from backgrounds . to claim an anomalous signal inconsistent with zero at 95% confidence requires 4 - 5 events . the second step , as discussed above , would be to check whether the galactic recoil directions are anisotropic and would require of order 10 events ( assuming zero background ) . in this paper we focus on the third phase , examining how measuring the median recoil direction could be used to provide confirmation of the wimp origin of an anisotropic recoil signal . in section [ model ] we describe the input to our monte carlo simulations . in section [ res ] we present our results , before concluding with discussion in sec . we use the same statistical techniques and methods for calculating the directional nuclear recoil spectrum as in refs . @xcite . we briefly summarise these procedures here , for further details see these references and ref . @xcite . many of the directional detectors currently under development @xcite are low pressure gas time projection chambers ( tpcs ) , e.g. dmtpc @xcite , drift @xcite and newage @xcite . we therefore simulate a fairly generic tpc based detector . we assume that the the recoil directions , including their senses , are reconstructed perfectly in 3d . these are optimistic assumptions , therefore our results provide a lower limit on the number of events / exposure required by a real tpc based detector . for concreteness we use a @xmath3 target with an energy threshold of 20 kev . finite angular resolution does not significantly affect the number of events required to detect anisotropy @xcite , provided it is not worse than of order @xmath4 . axial and/or 2-d read - out would , however , significantly degrade the detector capability @xcite . the detailed angular dependence of the recoil rate depends on the exact form of the wimp velocity distribution @xcite . however , if the wimp velocity distribution is dominated by a smooth component the main features of the recoil distribution ( rear - front asymmetry , median direction opposite to the direction of solar motion ) are robust ( see e.g. ref . @xcite ) ) . for concreteness we use the standard halo model halo , an isotropic sphere with local density @xmath5 and a maxwellian / gaussian velocity distribution with three dimensional velocity dispersion @xmath6 , and fix the wimp mass at @xmath7 . numerical simulations find velocity distributions with stochastic features at large speeds @xcite . kuhlen et . @xcite find that for high speed wimps ( @xmath8 ) the direction in which the wimp flux is largest can deviate by more than @xmath9 from the direction of solar motion . the effect on the median recoil direction will be substantially smaller than this however . firstly the set - up we are considering ( @xmath3 target with energy threshold of 20 kev and @xmath7 ) is sensitive to wimps with @xmath10 . therefore a high speed feature will only contribute a small fraction of the wimp flux . this will be true in general unless the wimp mass is small and/or the threshold energy is large . secondly , see e.g. fig . 3 of ref . @xcite , due to the elastic scattering the recoil rate is less anisotropic than the wimp flux . the deviation of the inverse median direction from the direction of solar motion will therefore be substantially smaller than the deviation of the peak wimp flux . it will also depend on the ( a priori unknown ) velocity and density of the feature . for instance , for a stream with velocity , in galactic coordinates , @xmath11 the difference between the inverse median direction and the solar direction only exceeds @xmath12 if the fraction of the local density in the stream exceeds @xmath13 @xcite . since the effect of a feature in the speed distribution on the median recoil direction is expected to be small we do not investigate it in this study . if , with a large number of events , a statistically significant deviation of the inverse median direction were found , its origin could be investigated by studying the energy dependence of the deviation . we defer an investigation of this to future work . finally , direct detection experiments probe the ultra - local dark matter distribution on scales many orders of magnitude smaller than those resolved by simulations . it is not ( and may never be ) possible to directly calculate , or otherwise measure , the dark matter distribution on the relevant scales . in this case if wimps are detected , then the directional recoil rate could be used to reconstruct the ultra - local dark matter distribution @xcite . the recoil rate is peaked in the direction opposite to the direction of solar motion . to allow ease of comparison with the direction of solar motion we use the inverse recoil directions ( i.e. the directions from which the recoils originate ) in our analyses . we first examine how the accuracy with which the median galactic recoil direction is determined depends on the number of wimp events . the median direction is defined as the direction @xmath14 which minimises the sum of the arclengths between @xmath15 and the individual inverse recoil directions @xmath16 it is found by minimising @xmath17 where @xmath18 is the number of events . for a given number of wimp events , @xmath19 , we simulate @xmath20 experiments and in each determine the direction , @xmath15 , by minimizing eq . ( [ m ] ) and hence calculate @xmath21 , @xmath22 the angle between the median direction and the direction of solar motion , @xmath23 . in fig . 1 we plot the 50% and 95% percentiles of the distribution of @xmath21 as a function of @xmath24 . we also investigate the effects of non - zero ( isotropic ) backgrounds . we parametrize the background rate in terms of the fraction of events which are signal , @xmath25 where @xmath3 and @xmath26 are the signal and background rates respectively ( c.f . @xcite ) . for comparison we also plot the @xmath27 and @xmath28 percentiles for an purely isotropic distribution . , as a function of the number of wimp events , @xmath19 , for varying signal fractions ( from top to bottom ) @xmath29 and @xmath30 . the solid and dashed lines are the @xmath31 and @xmath28 percentiles . the dotted lines are ( from top to bottom ) the @xmath28 and @xmath27 percentiles for an isotropic recoil distribution.,width=321 ] we now determine the number of events required to confirm the direction of solar motion as the median inverse recoil direction at 95% confidence . we do this using the same methodology as in ref . briefly , we use the distribution of @xmath21 for wimp induced recoils and for the null hypothesis of isotropic recoils , to calculate the rejection and acceptance factors , @xmath32 and @xmath33 . the rejection factor gives the confidence level with which the null hypothesis can be rejected given a particular value of the test statistic , while the acceptance factor is the probability of measuring a larger value of the test statistic if the alternative hypothesis is true , we then find the number of wimp events required to give @xmath34 i.e. to reject the median direction being random at 95@xmath35 confidence in @xmath31 of experiments . this is shown in fig . [ nlambda ] , as a function of the signal fraction @xmath36 . for zero background ( i.e. @xmath37 ) 31 events are required , a factor of @xmath0 more than are required to simply reject isotropy . the number of events required increases as @xmath36 is decreased , initially fairly gradually and then , once the signal becomes subdominant ( @xmath38 ) , more rapidly . billard et al . @xcite have recently investigated using a sky - map based likelihood analysis to probe the correlation of the directional recoil rate with the direction of solar motion ( and hence confirm the wimp origin of a signal ) . their results are qualitatively consistent with ours . for instance they find that , for a signal fraction @xmath39 , the peak signal direction can be confirmed to be within @xmath40 of the direction of solar motion with as few as 25 wimp events . the median direction method is however faster and more robust . it particular it has the advantage of being model independent ( i.e. one does not need to assume an exact form of the wimp velocity distribution ) . , required to reject the median direction being random at 95@xmath35 confidence in @xmath31 of experiments on the signal fraction , @xmath36.,width=321 ] if desired , further confirmation of the wimp origin of an anisotropic recoil signal could be obtained by studying the variation of the median recoil direction in the lab frame @xcite . over the course of a sidereal day the peak in the recoil distribution traces out a small circle on the sky . for a detector located in the northern hemisphere during a sidereal day the peak recoil direction in the lab rotates ( roughly ) from down to south and back . a periodogram analysis could be used to verify that median direction varies over a sidereal , rather than solar , day ( c.f . @xcite for the annual modulation ) . in this paper we have investigated how the median recoil direction in directional detection experiments can be used as a wimp signal . assuming a smooth wimp distribution , the peak wimp flux is from the direction of solar motion and the median recoil direction is in the opposite direction . an ideal detector could reject isotropy of recoils with only of order 10 events @xcite . confirmation of the wimp origin of an anisotropic recoil signal could be obtained by studying the details of the anisotropy and in particularly confirming that the median inverse galactic recoil direction coincides with the direction of solar motion . we find that , with an ideal detector and zero - background , to confirm the direction of solar motion as the median inverse recoil direction at 95% confidence requires 31 events ( see also ref . non - zero isotropic background would increase this number , significantly if the signal is sub - dominant . for concreteness ( and in the absence of a definitive alternative ) we have used the standard halo model halo , which has an isotropic maxwellian speed distribution . while the detailed angular dependence of the recoil rate depends on the exact form of the wimp velocity distribution , if the wimp velocity distribution is dominated by a smooth component the median inverse recoil distribution will be close to the direction of solar motion ( see e.g. ref . recent high resolution simulations have found stochastic , features in the speed distribution at large speeds @xcite . the direction in which the flux of high speed wimps is largest can deviate by more than @xmath9 from the direction of solar motion @xcite . however the deviation of the median inverse direction will be small compared with that expected from an isotropic recoil direction . we therefore conclude that features in the speed distribution at high speed are unlikely to affect the utility of the median recoil direction as a wimp signal . if , with a large number of events , a statistically significant deviation of the inverse median direction were found , its origin could then be investigated by studying the energy dependence of the deviation .
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direct detection experiments have reached the sensitivity to detect dark matter wimps . demonstrating that a putative signal is due to wimps , and not backgrounds , is a major challenge however .
the direction dependence of the wimp scattering rate provides a potential wimp ` smoking gun ' .
if the wimp distribution is predominantly smooth , the galactic recoil distribution is peaked in the direction opposite to the direction of solar motion .
previous studies have found that , for an ideal detector , of order 10 wimp events would be sufficient to reject isotropy , and rule out an isotropic background .
we examine how the median recoil direction could be used to confirm the wimp origin of an anisotropic recoil signal .
specifically we determine the number of events required to confirm the direction of solar motion as the median inverse recoil direction at 95% confidence .
we find that for zero background 31 events are required , a factor of @xmath0 more than are required to simply reject isotropy .
we also investigate the effect of a non - zero isotropic background . as the background rate
is increased the number of events required increases , initially fairly gradually and then more rapidly , once the signal becomes subdominant .
we also discuss the effect of features in the speed distribution at large speeds , as found in recent high resolution simulations , on the median recoil direction .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
with the introduction of efficient multi - object spectrographs on 4m - class telescopes , it has become possible to construct large samples of faint galaxies with measured redshifts . with such a sample , one can compute the luminosity function ( lf ) of galaxies as a function of redshift and thereby directly observe the evolution ( or lack thereof ) of the galaxy population . several groups have now presented the results of deep , faint galaxy redshift surveys ( @xcite , cfrs ; @xcite , autofib ; @xcite ; @xcite , cnoc ) . the conclusions from these surveys are in broad agreement : the population of blue , star - forming galaxies has evolved strongly since @xmath16 while the population of red galaxies shows at most modest signs of evolution ( although , see kauffmann , charlot , & white ( 1996 ) for an alternative analysis of the red galaxies ) . however , there are important differences as well . lin et al . ( 1996a ) demonstrate that the lfs from the various groups are formally inconsistent with each other . since there are many selection effects involved with the construction and analysis of faint redshift surveys , it is difficult to pinpoint the reasons for the disagreement between the various groups . while it is likely that the small numbers of galaxies in each survey and the small areas covered are partly responsible , it is also likely that systematic errors are in important contributor to the differences in detail . quantitative estimates of the evolution are , of course , dependent upon having a reliable measurement of the local lf , and it is , therefore , of concern that there remain considerable uncertainties about the _ local _ lf . the lfs derived from large - area local redshifts survey ( e.g. , the stromlo / apm survey , loveday et al . 1992 ; the cfa survey , marzke , huchra , & geller 1994a ; the las campanas redshift survey , lin et al . 1996b ) all have similar shapes , but there are still substantial differences over the overall normalization , the characteristic luminosity , and the slope at low luminosities . the rapid evolution at @xmath17 required to match steep @xmath18-band counts at intermediate magnitudes @xmath19 ( maddox et al . 1990 ) could be reduced if the normalization or the faint - end slope have been underestimated . the results of the largest of the local surveys , the las campanas redshift survey ( lcrs ) with 18678 galaxies used in the lf analysis and a median redshift of @xmath20 , are seemingly consistent with both a low normalization and a flat faint - end slope . the lcrs is selected from ccd drift scans rather than photographic plates and surveys what should be a fair volume of the universe ( shectman et al . 1996 , davis 1996 ) . it also probes both the southern and northern galactic caps . accordingly , the local luminosity function computed from their data should be free from systematic photometric errors and fluctuations in large - scale structure in the distribution of galaxies . however , both the cfa survey and the autofib survey find a normalization which is a factor of 2 higher than that obtained from the lcrs . while the normalization of the cfa survey can be questioned on the grounds that it does not sample a fair volume , the autofib survey is the concatenation of many fields distributed across the sky . the autofib survey is particularly important because the galaxy sample was selected with a much fainter surface brightness threshold than any of the other local surveys . mcgaugh ( 1994 ) emphasizes that a large population of intrinsically luminous but low surface brightness galaxies may be missed in the shallow photometry on which all the local surveys , except autofib , are based . a steep faint - end slope of the lf , with a power law exponent of @xmath21 , is a natural prediction of galaxy formation theories based on hierarchical structure formation models ( kauffmann , guiderdoni , & white 1994 ) . there is only weak evidence for a steep faint - end slope in the local field galaxy lf . marzke et al . ( 1994b ) report an upturn in the luminosity function of late - type galaxies with @xmath22 , but lcrs , autofib , and cowie et al . ( 1996 ) all derive a flat faint - end slope . there is , however , evidence for a steep faint - end slope in galaxy clusters ( e.g. , de propris et al . 1995 , bernstein et al . environmental influences on galaxy evolution may be reflected in variations of the lf for galaxies in different environments , and it is therefore important to measure the lf in a variety of environments . in this paper , we investigate the evolution and environmental dependence of the galaxy lf based on data obtained during the course of our redshift survey of the corona borealis supercluster . the primary motivation for the survey was to study the dynamics of the supercluster . however , the majority of galaxies for which we measured redshifts actually lie behind the corona borealis supercluster , thus providing a sample suitable for study of the evolution of the lf . the galaxies were originally selected from plates taken as part of the second palomar observatory sky survey ( poss - ii ; @xcite ) and have been calibrated in the gunn @xmath23 and @xmath7 bands , which correspond roughly to the photographic @xmath24 and @xmath25 bands . previous redshift surveys have generally been either selected in bluer bands ( @xmath18 ) , for sensitivity to changes in star - formation rates , or redder bands ( @xmath26 and @xmath27 ) , for sensitivity to old stellar populations which more reliably trace stellar mass . although we had no option but to use the @xmath23 and @xmath7 bands , the two bands turn out fortuitously to have the virtue that corrections to the rest @xmath18 band , where lfs are traditionally computed and compared , are small since the @xmath23 band matches the rest @xmath18 band at @xmath28 and the @xmath7 band matches the rest @xmath18 band at @xmath29 . the cnoc survey used photometry in @xmath23 and @xmath7 as well , and so it is particularly interesting to compare our results to that survey since there should be no systematic effects due to using different passbands for galaxy selection . finally , with over 400 redshifts in the corona borealis supercluster , and roughly 300 in a background supercluster , we can explore the variation of the lf from the field to the supercluster environment . the paper , the second in the series presenting results from the norris survey of the corona borealis supercluster , is organized as follows . in 2 , we summarize our survey , particularly emphasizing those features that are directly relevant to the computation of the lf . we discuss the details of the computation of the lf in 3 . the results are given in 4 for both field galaxies and for the two superclusters individually and are discussed in 5 . finally , we summarize our conclusions in 6 . we use a hubble constant @xmath30 km s@xmath31 mpc@xmath32 and a deceleration parameter @xmath33 . for comparison to the most recent work in the field ( e.g. , cfrs and cnoc ) , we use the ab - normalized @xmath18 band , @xmath6 ( oke 1974 ) . the offsets from @xmath6 to @xmath8 and @xmath18 are @xmath34 and @xmath35 ( fukugita , shimasaku , & ichikawa 1995 ) . the norris survey of the corona borealis supercluster has been described in detail in small et al . ( 1996 ) , paper i of the current series , and will be only briefly reviewed here . the core of the supercluster covers a @xmath36 region of the sky centered at right ascension @xmath37 , declination @xmath38 and consists of 7 rich abell clusters at @xmath11 . since the field - of - view of the 176-fiber norris spectrograph is only 400 arcmin@xmath39 , we planned to observe 36 fields arranged in a rectangular grid with a grid spacing of 1 . as it turned out , we successfully observed 23 of the fields and 9 additional fields along the ridge of galaxies between abell 2061 and abell 2067 , yielding redshifts for 1491 extragalactic objects . we have extended our survey with 163 redshifts from the literature , resulting in 1654 redshifts in the entire survey . 1022 of these galaxies lie beyond the corona borealis supercluster , although of these 1022 , 325 ( 318 with @xmath4 ) galaxies are in a background supercluster ( @xmath13 ) which we have dubbed the `` abell 2069 supercluster . '' the survey fields are distributed across an area of 25 deg@xmath39 . the total area covered by the 32 observed fields , albeit sparsely sampled , is 2.99 deg@xmath39 . as noted above and described in detail in paper i , the objects have been selected from poss - ii photographic plates of poss - ii field 449 , which neatly covers the entire core of the supercluster . we have both a @xmath24 ( kodak iii - aj emulsion with a gg395 filter ) plate and an @xmath25 ( kodak iii - af emulsion with a rg610 filter ) plate . the plates were digitized with 1 arcsec@xmath39 pixels at the space telescope science institute and then processed using the sky image cataloging and analysis tool ( skicat , @xcite ) . the instrumental intensities recorded by skicat were calibrated with ccd sequences in the @xmath23 and @xmath7 bands of galaxies in abell 2069 . the random magnitude errors are 0.25@xmath40 for @xmath23 and @xmath7 brighter than 21@xmath40 and become substantially worse at fainter magnitudes . ( we describe how we correct our computed lfs for these magnitude errors in 3.2 . ) with the skicat system , the star - galaxy separation is 90% accurate to @xmath41 . our lf analysis is limited to galaxies with @xmath42 . since our original motivation for the survey was to study the dynamics of the corona borealis supercluster , we chose a comparatively high spectral resolution for a faint galaxy redshift survey . a third of the objects were observed with @xmath43spectral resolution during the period when the largest ccd available at palomar was a 1024@xmath39 device with 24 pixels ; the rest were observed with @xmath44 resolution with a very efficient 2048@xmath39 ccd ( also with 24 pixels ) . since the operation of retrieving the fibers for one set - up and redeploying the fibers for another takes roughly an hour with the norris spectrograph , we decided to observe only two fields per night in order to minimize the amount of time lost due to changing fields . thus , our exposures were 2 - 4 hours long , and we generally obtained high quality spectra on @xmath45 galaxies . in figure [ figures : success ] , we plot our success rate , the fraction of objects ( i.e. , including stars and quasars ) on which fibers were deployed for which we successfully measured redshifts , as a function of magnitude . figure [ figures : success ] shows that our success rate falls substantially below unity beyond @xmath46 . therefore , we have computed weights for each galaxy to correct for a our incomplete sampling . the weight for a particular object is defined to be simply the ratio of the total number of objects in the photometric catalog to the number of objects with redshifts in an interval of @xmath47 centered on the magnitude of the object . this prescription assumes that the redshift distribution of the objects for which we failed to measure redshifts is identical to the redshift distribution of the objects for which we successfully measured redshifts . the color distribution of the objects that we observed but failed to identify is similar to that of the objects that we successfully observed , leading us to conclude that we do not suffer biases against particular types of galaxies ( paper i ) . moreover , we do not believe that we should have redshift - dependent biases in our success rate . since we limit the computation of the lf to @xmath48 , we are unlikely to be affected by a bias in redshift . the 4000 break and ca h , ca k lines of old stellar populations and the [ ] line of star - forming galaxies are all within our spectral range out to @xmath1 . we plot the calculated weights as a function of @xmath7 magnitude for all galaxies with @xmath42 and which satisfy our surface brightness threshold ( see below ) in figure [ figures : weights ] . the weights are greater than unity even for bright galaxies because of the sparse sampling of our survey area . ( since the very brightest galaxies ( @xmath49 ) produce scattered light contamination of nearby spectra on the ccd , we usually did not place fibers on galaxies with @xmath49 , and thus the weights increase for the very brightest galaxies . ) in paper i , we carefully studied the surface brightness selection effects present in our sample . we found that by restricting our sample to objects with @xmath42 and with core magnitudes @xmath50 ( where the core magnitude is the integrated magnitude within the central 9 arcsec@xmath39 ) , we are free from surface brightness selection effects . for comparison , @xmath51 corresponds to a central surface brightness of @xmath52 @xmath7 mag arcsec@xmath53 for a galaxy with an @xmath54 ( @xmath55 mag in the @xmath7 band ) disk . we compute galaxy lfs in the rest - frame @xmath6 band and , for the local lf , in the rest - frame gunn @xmath7 band as well . rest - frame colors and @xmath56-corrections are computed from the spectral energy distributions compiled by coleman , wu , & weedman ( 1980 , hereafter ccw ) . we assign each galaxy a spectral type based on its @xmath57 color and its redshift . following lilly et al . ( 1995 ) , the spectral type is a real number which takes the values 0 for an elliptical galaxy , 2 for an sbc galaxy , 3 for an scd galaxy , and 4 for an i m galaxy . we then interpolate between the ccw spectral energy distributions to construct the spectral energy distribution appropriate for the given spectral type . galaxies whose colors are redder than a ccw e galaxy or bluer than a ccw i m galaxy are simply assigned the spectral energy distribution of an e galaxy or an i m galaxy , respectively . the number of galaxies with colors outside the limits defined by the ccw e and i m types is known to be small even to large redshifts ( e.g. , crampton et al . the fact that many of our galaxies lie outside the ccw limits ( see figure 17 , paper i ) is due to the large errors in our colors ( @xmath58 ) . we compute the absolute rest - frame @xmath6-band magnitude as follows : @xmath59 where @xmath60 incorporates the corrections based on the spectral energy distribution and @xmath61 is the luminosity distance in mpc . the @xmath62 term represents the change in the bandwidth with redshift and is included in the traditional @xmath56-correction . again following lilly et al . ( 1995 ) , we separate the bandwidth stretching term , which has negligible error since it depends only on the accurately measured redshift , from the terms which depend on the spectral energy distribution and are therefore much more uncertain . we plot @xmath60 for the @xmath23 and @xmath7 bands in figure [ figures : keff ] . by converting from @xmath63 for objects with @xmath64 and from @xmath65 for objects with @xmath66 , @xmath60 may be kept less than @xmath67 for @xmath68 for all spectral types . we use the step - wise maximum - likelihood ( swml ) method of efstathiou , ellis , & peterson ( 1988 ) to estimate the lf . the probability of observing a galaxy of absolute magnitude @xmath69 at redshift @xmath70 in a flux - limited catalog is given by , @xmath71 where @xmath72 is the lf and @xmath73 is the intrinsically faintest galaxy observable at @xmath70 in the flux - limited catalog . the lf is parameterized as a set of @xmath74 numbers @xmath75 such that @xmath76 and then the likelihood , @xmath77 where @xmath78 is the number of galaxies in the sample , is maximized with respect to the @xmath75 . we constrain the values of @xmath75 to satisfy @xmath79 the virtues of the swml method are that it is not biased by the presence of clustering since the normalization of the lf cancels out of the expression for the probability @xmath80 and also that one does not have to assume a particular functional form for the lf . in order to include the weights , we make the substitution @xmath81 where @xmath82 is the weight of galaxy @xmath83 ( zucca , pozzetti , & zamorani 1994 , lin et al . one must then estimate the mean galaxy density separately . we use a standard technique which we describe below . since the weights are greater than one , their use will increase the calculated likelihood for the sample and thus lead to artificially small error estimates . by renormalizing the weights so that @xmath84 , the error estimates are appropriate for the true sample size ( lin et al . 1996b ) . note that the normalization constraint on the @xmath75 reduces the estimated errors . we do not use the traditional @xmath85 method ( @xcite ) employed by ellis et al . ( 1996 ) and lilly et al . ( 1995 ) since the method is sensitive to clustering . we have , however , compared the results of the two techniques for samples with @xmath86 , where the clustering in our survey is not pronounced , and found that they agree satisfactorily . for @xmath87 , we can construct volume - limited sub - samples with @xmath4 in which any galaxy with @xmath88 is visible in the entire volume . of course , the value of the lf in a given magnitude bin for a volume - limited sample is estimated by counting the number galaxies with absolute magnitudes in the bin and then dividing by the volume of the sample and the width of the bin . the swml lfs for @xmath87 agree well with the lfs estimated from the volume - limited samples . we compute the mean density @xmath89 of a magnitude - limited sub - sample using the following estimator : @xmath90 where @xmath91 is the volume of the sample , @xmath78 is the number of objects in the sample , and @xmath92 is the selection function . the selection function , @xmath93 gives the fraction of the lf observable at a given redshift . here , @xmath94 is the lf , @xmath95 is the maximum absolute magnitude that an object can have at redshift @xmath96 and still be included in the sample , and @xmath97 is the absolute magnitude of the most instrinsically faint galaxy in the sample . in practice , one does not begin evaluating the integrals at @xmath98 , but rather at the absolute magnitude of the most instrinsically bright galaxy in the sample . the estimator in equation [ equations : mean_density ] is almost identical to the minimum variance estimator derived by davis & huchra ( 1982 ) for @xmath99 and is unbiased by density inhomogeneities . an additional complication of computing a lf in the @xmath6 band where the objects have been selected in the @xmath7 band is that one must ensure that any object , regardless of its color , would have been detectable in both bands . if one ignores this complication , then the faintest objects at a given redshift will be biased in color . in our survey , since the @xmath7 band is centered at a longer wavelength than the @xmath18 band , the faintest objects would be biased to the red . in order to avoid such a bias , we adjust our absolute @xmath6 magnitude limits as a function of redshift so that the bluest galaxy at any @xmath6 magnitude limit would be observable in the @xmath7 band . for the local lf computed in the @xmath7 band , the bias works in the opposite sense , and so we adjust our rest - frame absolute @xmath7 magnitude limits to ensure that the reddest galaxy a given limit would be detected in the observed @xmath7 band . for each lf , we estimate the parameters of the best - fitting schechter ( 1976 ) function , @xmath100 where @xmath101 is the normalization , @xmath102 determines the location of the bright - end exponential cutoff , and @xmath103 is the faint - end slope . the fitting was performed using a standard @xmath104-minimization algorithm ( press et al . 1992 ) with the schechter function integrated over the width of the adopted magnitude bin . we intend these fits to be useful for comparisons with other work . usually , there are too few points for the fits to be well defined . our random magnitude errors ( @xmath105 ) will artificially brighten the characteristic magnitude @xmath102 of the lf and steepen the faint end . we correct for the magnitude errors by fitting to the data points a schechter function ( equation [ equations : schechter ] ) convolved with a gaussian of dispersion @xmath106 ( efstathiou et al . in fact , however , the corrections to the schechter function parameters are substantially less than the 1@xmath107 statistical errors . the error in the normalization from the @xmath104-fitting only includes the uncertainties due to @xmath102 and @xmath103 . the error due to large - scale structure fluctuations is @xmath108 ( davis & huchra 1982 ) , where @xmath109 is the second moment of the two - point spatial correlation function ( peebles 1980 ) and @xmath91 is the appropriate volume . we use @xmath110 ( @xmath111 mpc)@xmath112 ( tucker et al . 1997 ) and record the error from density fluctuations alongside the uncertainty in @xmath101 due to @xmath102 and @xmath103 . in the following subsections , we report our results for the local lf , the evolution of the lf , and the lfs of the corona borealis and abell 2069 superclusters . all samples are magnitude - limited at @xmath113 . the parameters of the best fitting schechter functions are summarized in table 1 , where the sample is given in the first column , the number of galaxies used to compute the lf in the second column , the absolute range over which the fit is valid in the third column , @xmath101 in the fourth column , @xmath102 in the fifth column , @xmath103 in the sixth column , the reduced @xmath104 in the seventh column , and the estimate of the variance due to density fluctuations in the eighth column . the numbers of galaxies listed in the second column are slightly smaller than the total number of galaxies satisfying the sample listed in the first column because a few galaxies have been trimmed from each sample , as described above , to ensure that there are no color biases in the faintest bins . corrections to @xmath101 to match galaxy counts , as discussed below , have _ not _ been applied to the values listed in table 1 . we wish to emphasize that the fitted schechter functions are intended only to guide the eye and that comparisons of the various lfs in this paper are best done by comparing the individual data points in the plots . the @xmath6-band local galaxy lf is plotted in figure [ figures : local_lf ] . the unfilled circles show the lf for @xmath114 with the superclusters removed . the filled circles show the lf for @xmath114 with the the superclusters included . in order to remove the superclusters , we simply delete all objects with @xmath115 . the median redshift of our local sample with the superclusters removed is @xmath116 . the stromlo / apm lf is plotted with the solid line . we also plot the autofid local lf with the dashed line . in figure [ figures : local_lf ] and all subsequent figures where appropriate , we convolve the schechter function fits to the lfs from other surveys with a gaussian of dispersion @xmath106 in order to facilitate comparisons with our lf data points , which are constructed with galaxies whose photometry suffers from random magnitude errors of @xmath106 . all of the local luminosity functions have similar shapes , but the normalizations and low luminosity ends vary significantly . in order to investigate further the normalization of the local luminosity function , we compute the _ shape _ of local luminosity in the rest - frame @xmath7 band and then normalize this lf to the @xmath7-band counts of weir , djorgovski , & fayyad ( 1995 ) . we plot our @xmath7-band local lf , normalized to the counts of weir et al . ( 1995 ) , in figure [ figures : local_lf_r ] , along with the @xmath7-band local lf from lcrs . to convert from isophotal @xmath117 magnitudes to total gunn @xmath7 magnitudes , we apply a 25% isophotal - to - total light correction and then use @xmath118 ( shectman et al . thus , the corrections compensate for each other and @xmath119 . the counts of weir et al . ( 1995 ) are based on 4 overlapping , high galactic latitude plates taken as part of the poss - ii survey . knowing the shape of the lf , we can estimate the differential number counts as @xmath120 { dv \over dz } dz \nonumber \\ & = & \phi^\ast \int_0^\infty \phi^\prime[m(m , z ) ] { dv \over dz } dz \\ & = & \phi^\ast { di \over dm } \nonumber , \label{equations : diff_counts}\end{aligned}\ ] ] where @xmath121 is @xmath72 with @xmath101 set equal to 1 , @xmath122 is the absolute magnitude of an object at redshift @xmath96 with apparent magnitude @xmath123 , and the @xmath124 are the magnitude intervals . we estimate @xmath101 by minimizing the quantity @xmath125 ^ 2 } \over { \phi^\ast di(m_i)}}\ ] ] with respect to @xmath101 ( efstathiou et al . 1988 ) , which yields @xmath126 ^ 2 / di(m_i ) } \over { \sum_i di(m_i)}}.\ ] ] we find @xmath127 mpc@xmath32 where the errors reflect the changes due to varying jointly @xmath102 and @xmath103 by @xmath128 . this is a 21% reduction from the normalization determined in the norris field itself . although the median redshift of our local sample is @xmath116 , the agreement with lcrs ( @xmath129 ) in the @xmath7 band leads us to believe that we have computed a fair estimate of the local lf . in figure [ figures : r_counts ] , we plot the @xmath7-band differential number counts . the histogram shows the counts from the poss - ii plate from which we selected our objects . the thick solid line is the counts from weir et al . ( 1995 ) , and the triangles are ccd counts from metcalfe et al . the dashed line represents the predicted counts based on our field galaxy lf , including evolution at @xmath86 ( see 4.2 below ) , with the superclusters removed . even with the superclusters removed , our local lf appears to be normalized too high . the solid line , however , gives the predicted counts with the normalization reduced by 21% . with the reduced normalization , the predicted counts agree quite well with the observed counts to @xmath113 . for comparison , we also show , as the dotted line , the predicted counts from the lcrs . these counts fall below the weir et al . ( 1995 ) counts for @xmath130 because they do not include the evolution of the lf beyond @xmath131 . since the volume of the norris region , with the superclusters removed , is @xmath132 mpc@xmath112 , we would expect from equation [ equations : delta_n ] @xmath133 . it is therefore not cause for concern that this region is overdense by 21% . we compute the field galaxy lf in two redshift intervals : @xmath134 and @xmath135 . the results are plotted in figure [ figures : field_lf ] . the normalization of the local lf ( unfilled circles ) has been reduced by 21% to match the @xmath7-band counts . the filled circles are the lf of the high redshift interval . we also plot the lfs of the stromlo / apm survey ( solid line ) , the cnoc survey ( @xmath136 , dashed line ) , and the cfrs survey ( @xmath135 , dotted line ) , all of which have been convolved with a gaussian with @xmath137 to account for random photometry errors . since the lf for the autofib survey was divided into redshift intervals which do not neatly match our redshift intervals and , more importantly , since lin et al . ( 1996a ) have already performed a detailed comparison with the autofib survey , we do not plot the autofib lfs . the @xmath135 lf has clearly evolved with respect to the local lf . we create two sub - samples of galaxies according to the rest - frame equivalent width of [ ] @xmath93727 . the division is made at a rest - frame equivalent width of 10 , which roughly corresponds to dividing the sample into types earlier and later than sbc ( kennicutt 1992 ) and thus allows direct comparison with the results of cnoc and cfrs . for galaxies with @xmath139 , [ ] is not redshifted into our observed wavelength range , and so we use the strength of h@xmath140 to divide our sample . after correcting for stellar absorption , we have 7 galaxies with @xmath139 and @xmath138(h@xmath140 ) @xmath141 5 , which we include in the @xmath138 ( [ ] ) @xmath141 10 sample . we have also investigated separating our sample by color , but we have found that the large errors on our colors tend to dilute trends which are seen clearly in samples defined by @xmath138 ( [ ] ) . the lfs for galaxies with @xmath138 ( [ ] ) @xmath142 10 and @xmath138 ( [ ] ) @xmath141 10 are shown in figure [ figures : no_o2_lf ] and figure [ figures : o2_lf ] , respectively . we also plot in each figure the lfs for the corresponding color - selected samples from cnoc and cfrs . it is important to remember when considering possible detailed discrepancies between our lfs and the cnoc and cfrs lfs that our sample is divided by @xmath138 ( [ ] ) , which , while roughly equivalent to color selection , is not identical . there is no significant indication that the population of early - type galaxies ( i.e. , @xmath138 ( [ ] ) @xmath142 10 ) has evolved since @xmath1 ; this result is not surprising given that the light of early - type galaxies is dominated by red , long - lived stellar populations . in contrast , the lfs of the late - type galaxies ( i.e. , those with @xmath138 ( [ ] ) @xmath141 10 ) show striking evidence for evolution , even though the sample sizes are small and the error bars are large . the lfs of the corona borealis supercluster and the abell 2069 supercluster are given in figure [ figures : clusters_lf ] . we take the redshift range of the corona borealis supercluster to be @xmath143 and that of the abell 2069 supercluster to be @xmath144 . the normalization of the corona borealis supercluster function is a factor of 2 greater than that of the abell 2069 supercluster , but the shapes of the lfs of the two superclusters are similar . note that the volumes used to normalize the superclusters lfs are in redshift space ; the real - space volumes may be quite different ( see 5.3 below for a detailed discussion ) . since the bright ends of the supercluster lfs are clearly not well - described by a schechter function , we restrict our fit to @xmath145 . in addition , the fit to the corona borealis supercluster is limited to @xmath146 since the faintest two data points appear to describe a sharp upturn in the lf . prior evidence for rapid evolution of the galaxy lf to @xmath17 from galaxy counts was based crucially on normalizing the local lf to the bright ( @xmath147 ) galaxy counts from schmidt - telescope photographic surveys ( e.g. , maddox et al . . with this low normalization , predicted counts from the no - evolution model fall well short of the observed counts for @xmath148 . however , the evolution of the lf which we and others observe is not enough to make up for this shortfall . in figure [ figures : b_counts ] , we plot the observed counts from the apm survey ( maddox et al . 1990 ) and from the ccd survey of metcalfe et al . ( 1991 ) , along with various predicted counts . the dotted line shows the expected counts using the loveday et al . ( 1992 ) lf for @xmath134 and the cnoc @xmath6 lf for @xmath86 . despite including the observed evolution of the lf , the predicted counts only begin to agree with the observed counts for @xmath149 , by which point the predicted counts are dominated by galaxies with @xmath86 . in contrast , the predicted counts computed using the evolving lfs measured for our survey and for the autofib survey , while substantially overpredicting the counts for @xmath150 , match the observed counts for @xmath151 . in order to help unravel this confusing situation , we plot in figure [ figures : b_and_r_lf ] various @xmath6- and @xmath7-band local lfs on the same diagram . @xmath6-band lfs are plotted with respect to the bottom axis , while @xmath7-band lfs are plotted with respect to the top axis . the two axes are offset by @xmath152 , which is the median color we measure for field galaxies with @xmath87 . our @xmath6- and @xmath7-band local lfs , both of which have been reduced by 21% following the discussion in 4.1 , are plotted with filled and unfilled circles , respectively . with the color offset , they agree extremely well . the stromlo / apm lf , represented by the solid line , lies consistently below our @xmath6-band lf . the lcrs @xmath7-band lf is consistent with our @xmath7-band lf . unlike lin et al . ( 1996b ) , we do _ not _ conclude that the lcrs @xmath7-band lf matches the stromlo / apm lf . the reason for the disagreement lies in the different measurements of the median color of local galaxies . our median color is that of an sb galaxy , whereas the lcrs median color is that of a much redder e galaxy . the median color of galaxies in the _ third reference catalogue of bright galaxies _ ( de vaucouleurs et al . 1991 ) is @xmath153 ( see table 2 of fukugita et al . 1995 ) , which is roughly that of an sb galaxy . sebok ( 1986 ) also concludes that the typical local galaxy has the color of an sb galaxy . it is thus surprising that the mean color of the lcrs galaxies is so red . since our @xmath7-band lf agrees with the lcrs @xmath7-band lf and since the lcrs computed the colors of their galaxies by matching directly to the apm catalog , the most natural explanation for the anomalous red colors of the lcrs is a systematic error in the bright apm magnitudes . we note that weir et al . ( 1995 ) conclude that magnitudes derived from @xmath24 plates are only reliable for @xmath154 or , equivalently , @xmath155 . galaxies brighter than @xmath156 are saturated on the photographic plates . a systematic error in the bright apm counts would remove the need for rapid , and otherwise unsubstantiated , galaxy evolution at @xmath17 . we note also that there is possible evidence for a rise in the local lf above an @xmath157 for the least luminous galaxies in our survey . such a rise is evident in the data of marzke et al . ( 1994b ) for irregular galaxies , the cfrs , and the low surface brightness galaxy redshift survey of sprayberry et al . similar behavior is also perhaps visible in the local lf of the autofib survey . although ellis et al . ( 1996 ) argue against a rise in the lf for @xmath158 , the three faintest points in the their local lf ( their figure 8) all lie above their preferred schechter function fit . however , since such faint galaxies are only visible in our survey in a quite small volume , we refrain from attempting to make a definitive statement . we asserted in 4.2 that the lf of star - forming galaxies ( @xmath138 ( [ ] ) @xmath141 10 ) evolved from @xmath0 to @xmath1 and that the luminosity function of galaxies with weak [ ] emission ( @xmath142 10 ) did not . a powerful method to verify this result is to compute @xmath159 for appropriate samples ( @xcite ) . if there is no evolution in the number density of objects , @xmath160 ; if the number density declines , @xmath161 ; and if the number density increases , @xmath162 . our @xmath159 analysis is complicated by the need to remove the superclusters and to account for the 21% overdensity of our local field . in order to excise the superclusters , we simply remove all galaxies with @xmath115 . if the maximum redshift @xmath163 at which a galaxy could be observed in our survey lies in the range 0.06 to 0.13 , we set @xmath164 . we correct for the overdensity of our local field by reducing the weights of galaxies with @xmath114 by 21% . the values of @xmath159 for various samples of galaxies are given table 2 , both with and without the 21% correction to the weights of the galaxies with @xmath114 . for samples selected by color , we compute the @xmath159 statistic for galaxies in the redshift range @xmath2 ( with the supercluster region excluded ) . for samples selected by the strength of [ ] , we use the redshift range @xmath165 ( with the supercluster region excluded ) since [ ] from objects with @xmath139 is not redshifted into our observed wavelength range . the differences between our weighted and unweighted statistics are small . the @xmath159 test supports our claim , at the @xmath166 level for the weighted statistic , that the population of star - forming galaxies is evolving . the rate of evolution increases with the strength of [ ] . the population of galaxies with @xmath138 ( [ ] ) @xmath141 20 has @xmath167 . this result is analogous to the results from cfrs in which the rate of evolution is the strongest for the bluest population of galaxies . @xmath159 for the population of red galaxies is consistent with no evolution , in agreement with the lf analysis . broadly speaking , our results are in accord with the results of cfrs , autofib , cowie et al ( 1996 ) , and cnoc . since the cnoc survey , like our survey , uses photometry in the @xmath23 and @xmath7 bands , it is particularly interesting to compare our results in detail to theirs since many of the systematic effects associated with @xmath56- and color - corrections ought to be the same . it is encouraging to see ( figure [ figures : no_o2_lf ] and [ figures : o2_lf ] ) that , given the small samples , our lfs agree well with those of cnoc . the agreement is significantly improved if one reduces the normalization of the cnoc lfs by 20% , as , in fact , is recommended by lin et al . ( 1996a ) . now that we have confirmed that the population of blue galaxies is evolving with redshift , we wish to investigate whether we can detect differences in the colors and the spectral properties of the evolving population with redshift . first , we reiterate that the color distribution of objects with measured redshifts is similar to the color distribution of unidentified objects , leading us to believe that the type distribution of the identified objects is not strongly biased ( paper i ) . although emission lines are generally easier to detect than absorption lines , the difficulty of identifying emission lines at observed wavelengths longer than 5577 , where there are many strong night sky features , combined with the strength of ca h , ca k , and the 4000 break in absorption line objects at @xmath168 mitigate the bias in favor of emission line objects . a sample of the spectra of 8 absorption line objects in this redshift range is shown in figure [ figures : ab_line_objs ] to illustrate the ease of detection of their characteristic absorption features . in figure [ figures : obs_g - r ] , we plot the observed @xmath5 color as a function of redshift of all the objects in our survey along with the tracks of five representative model galaxy spectra . the bluest spectrum is simply a flat - spectrum object , @xmath169 the four other spectra are typical of the hubble types e , sbc , scd , and i m and are taken from ccw . the large , solid diamonds mark the observed median color in the redshift ranges @xmath170 ( arranged to exclude the superclusters ) , @xmath171 , @xmath172 , @xmath173 , and @xmath174 . perhaps counter - intuitively , we see that the observed median color does not become progressively bluer with respect to the model spectra with increasing redshift . as discussed by lilly et al . ( 1995 ) and illustrated in our figure [ figures : rest - frame_g - r ] , the median color does not become bluer because the color - magnitude relation in the local universe ( i.e. , the fact that more luminous galaxies are redder ) breaks down for @xmath131 as the population of blue galaxies brightens while the population of red galaxies does not evolve significantly . we plot in figure [ figures : rest - frame_g - r ] the absolute @xmath6-band magnitudes versus rest - frame @xmath5 for our galaxies divided into four intervals in redshift . in the low - redshift interval , the color - magnitude relation is apparent , but it disappears in the higher redshift intervals . at @xmath175 , we observe far down the lf to absolute magnitudes where blue galaxies are dominant . at @xmath131 , we do not observe as far down the lf , but the population of blue galaxies has brightened so as to be included in the samples . the increase in the luminosity of the population of blue galaxies is presumably associated with a change in the star formation activity at earlier times . in our spectra , there are two convenient star formation indicators , [ ] @xmath93727 and h@xmath10 @xmath94101 . [ ] emission is found in galaxies with ongoing star formation , and its strength is proportional to the strength of h@xmath103 ( @xcite ) . strong h@xmath10 absorption is a signature of the presence of a population of a - stars , which are visible @xmath1761 gyr after a burst of star formation . these lines are reliably measured by automated programs ( see paper i ) since both occur in regions of the spectrum where the continuum is featureless and there is little crowding from other lines . an important virtue of the h@xmath10 line is that , since it appears in absorption , a galaxy with detectable h@xmath10 would have been identified no matter what its spectral characteristics , which implies that there is no bias towards detecting objects with h@xmath10 absorption . a galaxy with [ ] emission is , of course , easier to identify than if it had had only absorption lines . however , as we discussed above , the combination of the difficulty of identifying weak emission lines in the face of strong sky subtraction residuals and of the ease of identifying the strong features characteristic of absorption - line galaxies at moderate redshifts suggests that our survey is not strongly biased towards detecting objects with [ ] . by adding together spectra of late - type galaxies in two redshift intervals in order to obtain two spectra with very high signal - to - noise ratios , heyl et al . ( 1996 ) find that the two star - formation indicators [ ] @xmath93727 and h@xmath10 have both become stronger in the higher redshift mean spectrum . those authors interpreted the increase in the strength of [ ] and h@xmath10 in the @xmath86 spectra to imply that not only were the higher - redshift galaxies forming stars more rapidly , but that also the nature of the star formation was changing with redshift . specifically , they asserted that the strong h@xmath10 absorption in the higher redshift sample is evidence that the star formation in that sample is dominated by bursts . with our high - quality spectra , we can reliably measure [ ] and h@xmath10 without adding together the spectra of many galaxies . in figure [ figures : o2_and_hd ] , we plot as a function of redshift the fraction of galaxies with [ ] emission and the fraction of galaxies with h@xmath10 absorption . the vertical dashed line marks the redshift beyond which h@xmath10 is shifted into the region of the spectrum in which sky subtraction becomes increasingly difficult . while the fraction of galaxies with [ ] emission increases with redshift , the fraction of galaxies with strong h@xmath10 absorption shows no significant variation with redshift , suggesting that the star formation is _ not _ occurring in short - lived bursts ( c.f . , hammer et al . 1997 ) in order to investigate this disagreement more carefully , we repeat the analysis of heyl et al . ( 1996 ) and construct high signal - to - noise ratio composite spectra . the individual spectra have been median - combined after scaling by the median count level . the magnitude weights , which correct for our incompleteness at faint magnitudes , have not been applied since scaling by the median count level in each spectrum effectively incorporates the magnitude weights . in figure [ figures : coadd ] , we plot the composite spectra of galaxies with @xmath138 ( [ ] ) @xmath141 20 , @xmath177 , and either @xmath178 ( thick line , @xmath179 ) or @xmath180 ( thin line , @xmath181 ) . @xmath138 ( [ ] ) @xmath141 20 is typical for the late - type galaxies in which heyl et al . ( 1996 ) observe spectral changes with redshift . while the [ ] line is modestly stronger in the higher redshift composite spectrum , there is no difference in the strength of h@xmath10 ( which is partially filled - in by emission in both spectra ) , which indicates that the nature of the star formation has not changed from @xmath1 to @xmath182 in contrast to heyl et al . ( 1996 ) , we again do not conclude that the spectra of intermediate - redshift blue galaxies show spectral signs of short bursts of star formation . ( we do not believe that the small differences in the rest - equivalent widths of the ca h and k lines are significant since it is difficult to fit a reliable continuum in the vicinity of the 4000 break . ) the modest increase in the strength of the [ ] line with redshift for galaxies with @xmath183 is due , as previously discussed by cowie et al . ( 1996 ) , to the increase in the intrinsic luminosity of star - forming galaxies with redshift . in figure [ figures : o2_ew ] , we plot absolute magnitude versus the rest equivalent width of [ ] for the galaxies in our sample divided into four redshift intervals . as the redshift increases , more and more luminous galaxies exhibit strong [ ] . notice , however , that the range of rest equivalent widths does not vary significantly with redshift . it is difficult to draw firm conclusions about the nature of the evolving galaxy population from the lfs alone . the lfs provide only a statistical view of the entire population and include information about the evolution of individual galaxies only indirectly . furthermore , the luminosity of a galaxy is an unreliable tracer of the physical state of the galaxy . the luminosity of a galaxy , especially in the rest - frame ultraviolet , can change dramatically on short timescales , making the identification of the descendants of distant galaxies in the local population very difficult . in order to make further progress , additional data on the nature of distant galaxies is required . the morphologies of distant galaxies , measured with the _ hubble space telescope _ ( hst ) , are already providing crucial clues ( @xcite , @xcite , @xcite ) . the counts of faint elliptical and early - type spiral galaxies match predictions based on counts in the local universe , provided that the local lf is normalized , as advocated here , a factor of @xmath1761.5 - 2 higher than found by loveday et al . in contrast , the hst number counts of late - type and irregular galaxies are far in excess of the counts expected from observations of nearby galaxies , even with a high normalization of the local lf . an alternative method for selecting distant galaxies is by their gas absorption cross section . steidel , dickinson , & persson ( 1994 ) study the evolution of 58 galaxies selected by @xmath1842796 , 2803 absorption seen in the spectra of high - redshift quasars . these authors found that the galaxies responsible for quasar absorption lines in the range @xmath185 , which typically have luminosities near @xmath54 , do not evolve over the redshift range . however , their sample is small and , when divided by color , is not inconsistent with results from surveys of galaxies selected by apparent magnitude ( lilly et al . in addition , steidel et al . ( 1994 ) note that intrinsically faint blue galaxies do not appear in their sample , and so the population of galaxies which is observed to be evolving most rapidly is not included in their survey . it is now possible , especially with 10m - class telescopes , to measure the masses ( e.g. , vogt et al . 1993 , 1996 ; rix et al . 1996 , guzmn et al . 1996 ) and chemical abundances of distant galaxies . since the masses and metallicities of galaxies evolve more smoothly than the luminosities , measurements of these two quantities will allow more easily interpretable comparisons of distant and local populations ( guzmn et al . 1996 ) . in addition , the mass and metallicity are , compared to morphology and even gas absorption cross section ( churchill , steidel , & vogt 1996 ) , straightforward to define and interpret . a survey to measure the masses and chemical abundances of the faint blue galaxies ought to yield important insights into the nature of this rapidly evolution galaxy population and aid in the identification of their present - day counterparts . the lfs of the two superclusters have quite similar shapes and generally resemble the field galaxy lf . since the normalizations of the superclusters lfs are computed in redshift space , in which distances along the line - of - sight may be substantially altered with respect to real space , it is not straighforward to compare the normalization of the supercluster lfs with that of the local field galaxy lf . for the purpose of the discussion here , we introduce a factor @xmath186 , the ratio of the redshift - space volume to the real - space volume . we expect @xmath186 to be in the range @xmath187 . the lower limit corresponds to assuming that the peculiar velocities in the supercluster regions are small ; the upper limit corresponds to assuming that the depth of the superclusters along the line - of - sight is similar to the linear sizes of the superclusters on the plane of the sky ( @xmath188 mpc ) . the real - space mean density of galaxies in the corona borealis supercluster , obtained simply by integrating the measured lf data points ( @xmath189 ) , is @xmath190 mpc@xmath32 . the mean density of field galaxies in the same range of absolute magnitude is @xmath191 mpc@xmath32 . similarly , the real - space mean density of galaxies in the a2069 supercluster ( @xmath192 ) is @xmath193 mpc@xmath32 , while the mean density of field galaxies in the same range of absolute magnitude is @xmath194 mpc@xmath32 . thus , the overdensities are @xmath195 and @xmath196 for the corona borealis and a2069 superclusters , respectively . it is also important to know whether our sampling of the superclusters is biased towards either the abell clusters within the superclusters or the `` field '' of the superclusters . owing to the difficulties of converting redshift - space volumes to real - space volumes , we assess our sampling of the superclusters using projected surface densities , which are not affected by redshift - space distortions . for the corona borealis supercluster , the project surface density @xmath197 is the mean redshift - space galaxy volume density multiplied by the depth in redshift space of the supercluster along the line - of - sight , @xmath198 mpc . thus , @xmath199 mpc@xmath53 @xmath200 mpc@xmath53 . we compare this value with the median surface density of the regions surrounding successfully observed corona borealis supercluster galaxies . we compute the surface density around a given corona borealis supercluster galaxy by counting the number of supercluster galaxies in a 6 arcmin diameter circle surrounding the chosen galaxy . we either simply count the number of galaxies with measured redshifts within the corona borealis supercluster , or we count the total number of galaxies on the original poss - ii @xmath25 plate , weighted by the empirically - determined fraction of galaxies at a given magnitude which are in the supercluster . the results of these two methods agree well : the median values of the raw and weighted surface densities are @xmath201 mpc@xmath53 and @xmath202 mpc@xmath53 . these two values bracket the projected surface density measured by multiplying the mean galaxy density ( computed from the lf ) by the line - of - sight depth of the supercluster . we conclude , therefore , that we have fairly sampled the corona borealis supercluster . we perform an identical analysis for the background a2069 supercluster . the redshift - space depth of the a2069 supercluster is also @xmath203 mpc . given a mean galaxy volume density of @xmath204 mpc@xmath32 , the projected surface density is @xmath205 mpc@xmath53 . the median surface density of the regions surrounding the successfully observed a2069 supercluster galaxies , computed in the same fashion as for the corona borealis supercluster , is @xmath206 mpc@xmath53 ( unweighted ) or @xmath207 mpc@xmath53 ( weighted ) . thus , we are slightly biased to the denser regions of the a2069 supercluster . the overall resemblance between the supercluster lfs and the field galaxy lf suggests that the fundamental physical processes which drive galaxy formation and evolution must not depend strongly on environment . there are , however , important differences between the lfs in the field and in the superclusters that must ultimately be due to environmental effects . the most striking difference between the field and supercluster lfs is that the supercluster lfs to do not continue the exponential decline for galaxies brighter than @xmath208 . both superclusters evidently contain a population of very luminous galaxies . despite the fact that there are only 6 galaxies with @xmath209 in the two superclusters combined , it is clear that these galaxies are giant ellipticals found in the densest regions of the superclusters . the 4 of the 6 for which we have spectra ( the other 2 were taken from the literature ) are dominated by the light of an old , red stellar population . all of the galaxies are found in regions in the upper 27th percentile of local surace density , with half of them found in regions in the upper 10th percentile . in fact , the four galaxies in the corona borealis supercluster are all found in the dense ridge of galaxies between the abell clusters a2061 and a2067 . for the corona borealis supercluster , the characteristic magnitude @xmath102 is @xmath210 brighter than in the field and is quite close to the value measured by colless ( 1989 ) for rich clusters . there is no significant difference between @xmath102 for the abell 2069 supercluster and that of the field . for the corona borealis supercluster lf , the data points for the two faintest magnitude bins hint that the lf may steepen significantly for galaxies fainter than @xmath211 . since the hint is based on only two data points , which are themselves based on only 29 galaxies , we must be cautious in our interpretation . however , a steepening of the supercluster lf fainter than @xmath211 would be in accord with observations of the faint end of the lf in galaxy clusters and groups , in which a number of workers report steep ( @xmath212 ) lfs ( impey , bothun & malin 1988 ; ferguson & sandage 1991 ; biviano et al . 1995 ; de propris et al . 1995 ; driver & phillipps 1996 ) . with the exception of the study of the coma cluster by biviano et al . ( 1995 ) , the observations of steep faint ends in cluster and group lfs have depended , since redshifts were not available , on the subtraction of a background component from the foreground cluster , a procedure which is prone to systematic errors . although it would be unwise to draw strong conclusions from our two data points , they do have the virtue of being based on galaxies with measured redshifts . we have presented an analysis of the lf of galaxies in the norris survey of the corona borealis supercluster . our @xmath7-band lf of local field galaxies , when normalized to counts in high galactic latitude fields , agrees well with the lcrs . however , the normalization of our @xmath6 local lf is roughly a factor of 1.6 higher than that of the stromlo / apm survey . since lin et al . ( 1996b ) claim that the lcrs local lf agrees well with the stromlo / apm survey , the difference must lie in a systematic photometry error in one ( or more ) of the three surveys . a clue to the nature of this error is provided by examining the mean colors of norris and lcrs galaxies . the mean color of local norris galaxies is that of an sb galaxy , whereas the mean color of lcrs galaxies , computed by matching lcrs galaxies directly with galaxies in the apm catalog , is that of an e galaxy . given the agreement of the norris and lcrs @xmath7-band lfs , we therefore believe that the error is most likely in the apm catalog . indeed , brightening the magnitudes of apm galaxies with @xmath213 by @xmath1760.25@xmath40 would bring all of the local lfs into agreement . a ccd - based local redshift survey ( e.g. , the sloan survey , gunn & weinberg 1995 ) will certainly resolve any remaining questions about the local lf . we have observed evolution of the field galaxy lf within our sample , thereby confirming the conclusions drawn from several previous redshift surveys . the evolution is limited to the population of blue , star - forming galaxies . the population of blue galaxies becomes more luminous with increasing redshift , and thus the median color of the field galaxy population does not change . the evolution of the population of blue galaxies is reflected in the larger fraction of galaxies at higher redshift exhibiting spectral signatures of ongoing star formation . in contrast to the results of heyl et al . ( 1996 ) , but in agreement with hammer et al . ( 1996 ) , we find that the star formation is long - lived . we do not see evidence for short - term bursts of star formation . we are unable to detect any evolution of the population of galaxies with @xmath138 ( [ ] ) @xmath142 10 . the fact that the evolution which we observe in our @xmath23- and @xmath7-band selected survey is consistent with the results of the surveys of lilly et al . ( 1995 ) , ellis et al . ( 1996 ) , cowie et al . ( 1996 ) , and lin et al . ( 1996 ) adds to the already strong evidence that a consistent picture of the evolution of the galaxy lf is emerging . in particular , it is quite reassuring that our lfs agree well with those of lin et al . ( 1996a ) since both used the @xmath23 and @xmath7 bands and should therefore have very similar systematic effects . the lfs of the two superclusters show significant differences from the field galaxy lf , despite considerable overall similarity . since the superclusters are @xmath214 denser than the field , we are likely to be observing the influence of the environment on galaxy formation and evolution . the most prominent difference is an excess of very bright galaxies ( @xmath208 ) relative to the best - fitting schechter function , which accurately describes the field lf over the observed absolute magnitude range . these very bright galaxies are found in very dense regions of the superclusters and have spectra dominated by an old , red stellar population . in the corona borealis supercluster , the characteristic magnitude @xmath102 is @xmath210 brighter than in the field . @xmath102 for the abell 2069 supercluster is , however , very close to the value in the field . we have also presented suggestive evidence that there is a sharp upturn in the supercluster lf for @xmath215 . while there is also a suggestion of an upturn in the local field galaxy lf for the least luminous galaxies in our survey , it does not appear as dramatic as the upturn seen in the supercluster lf , but more data are needed before the possible difference can be quantified . we are grateful to the kenneth t. and eileen l. norris foundation for their generous grant for construction of the norris spectrograph . we thank the staff of the palomar observatory for the expert assistance we have received during the course of the survey , david hogg for many enlightening discussions , and the referee for a careful reading of this paper and helpful suggestions . this work has been supported by an nsf graduate fellowship ( tas ) and nsf grant ast-92213165 ( wlws ) .
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we measure the field galaxy luminosity function ( lf ) as a function of color and redshift from @xmath0 to @xmath1 using galaxies from the norris survey of the corona borealis supercluster .
the data set consists of 603 field galaxies with @xmath2 and spans a wide range in apparent magnitude ( @xmath3 ) , although our field galaxy lf analysis is limited to 493 galaxies with @xmath4 .
we use the observed @xmath5 colors of the galaxies to compute accurate corrections to the rest @xmath6 and @xmath7 bands .
we find that our local @xmath7-band lf , when normalized to counts in high galactic latitude fields , agrees well with the local lf measured in the las campanas redshift survey .
our @xmath6-band local lf , however , does not match the @xmath8-band lf from the stromlo / apm survey , having a normalization 1.6 times higher .
we see compelling evidence that the @xmath6-band field galaxy lf evolves with redshift .
the evolution is strongest for the population of star - forming galaxies with [ ] @xmath93727 rest - frame equivalent widths greater than 10 .
the population of red , quiescent galaxies shows no sign of evolution to @xmath1 .
the evolution of the lf which we observe is consistent with the findings of other faint galaxy redshift surveys .
the fraction of galaxies with [ ] emission increases rapidly with redshift , but the fraction of galaxies with strong h@xmath10 absorption , a signature of a burst of star - formation , does not .
we thus conclude that the star formation in distant galaxies is primarily long - lived .
we also compute the lfs of the corona borealis supercluster ( @xmath11 , 419 galaxies with @xmath12 ) and the abell 2069 supercluster ( @xmath13 , 318 galaxies with @xmath14 ) .
the shapes of the two supercluster luminosity functions are broadly similar to the shape of the local luminosity function .
however , there are important differences .
both supercluster lfs have an excess of very bright galaxies .
in addition , the characteristic magnitude of the corona borealis supercluster lf is roughly half a magnitude brighter than that of the local field galaxy lf , and there is a suggestion of an upturn in the lf for galaxies fainter than @xmath15 .
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in a universe dominated by collisionless , cold dark matter ( cdm ) , the central densities of galaxy halos are high , and rise to form cusps with @xmath1 , as @xmath2 @xcite . in contrast , a large class of alternatives to cdm , here generically called warm dark matter ( wdm ) , are expected to have lower central densities and constant density cores at small radii @xcite . in the majority of wdm models , phase space arguments predict that density cores will be more prominent in lower mass halos . current observations of the smallest rotationally - supported galaxies are somewhat ambiguous . some galaxies prefer cored halos , some prefer cusps , and others are well fit by either @xcite . moreover , the effects of non - circular motions and other systematic issues are yet to be sorted out for these systems . the dwarf spheroidal ( dsph ) satellite galaxies of the milky way provide a potentially superior laboratory for studying the nature of dark matter . observationally , their proximity ( @xmath3 kpc ) allows kinematic studies of individual stars . theoretically , their small masses make them ideal candidates for prominent wdm cores . moreover , the cdm prediction of cuspy central density profiles is robust for these satellite systems because the cusps are stable to any tidal interactions that may have occurred @xcite . unfortunately , obtaining dsph density profile slopes has been difficult . the best current constraints on their mass profiles come from analyses of @xmath4 line - of - sight ( los ) velocities ( e.g. * ? ? ? * ; * ? ? ? the stellar velocity anisotropy is a major source of degeneracy in los velocity dispersion modeling , and , as we show below , the log - slope of the underlying density profile remains practically unconstrained even if the number of observed stars is increased by a factor of @xmath5 . in this _ , we examine the prospects of constraining the dark matter density profiles of dsphs by combining stellar los velocities with proper motion measurements . we show that @xmath4 proper motions at @xmath6 km s@xmath7 accuracy can break the relevant degeneracies and determine the dark matter distribution in the vicinity of the dsph stellar core radius , @xmath8 . specifically both the dark halo density and the local log - slope of the dark halo density profile at @xmath9 may be determined with @xmath10 times higher precision than from los velocity dispersion data alone . using two well - motivated examples of a cusp and core halo , we show that the local log - slope measurement can distinguish between them at greater than the 3-@xmath11 level . we discuss our results in the context of nasa s _ sim planetquest _ ( space interferometry mission ) which will have the sensitivity to measure proper motions of stars at this accuracy in multiple dsphs . astrometry as a means to constrain the central density profiles of dsphs was previously considered by @xcite . using a two - parameter family of models for the dsphs , these authors show that proper motions will be enough to completely reconstruct the profile if the underlying shape follows their adopted form . however , cdm halos , and likely their wdm counterparts , are described by a minimum of four unknown parameters : a scale density , a scale radius , an _ asymptotic _ inner slope , and an _ asymptotic _ outer slope . by marginalizing over all of these parameters as well as the velocity anisotropy of the stars , we demonstrate that the asymptotic slopes are never well - determined , even with proper motions . we show that what is best constrained is the log - slope , @xmath12 , at a radius @xmath13 comparable to the king core radius , specifically @xmath14 . we note that unlike the asymptotic @xmath15 slope , the log - slope at @xmath16 kpc is known from cdm simulations . this radius corresponds to @xmath17 of the relevant halo virial radius , and is well - resolved in current simulations . we define the three - dimensional ( spatial , not projected ) components of a star s velocity to be @xmath18 , @xmath19 , and @xmath20 . the velocity component along the line of sight is then @xmath21 , where @xmath22 and @xmath23 is the line - of - sight direction . the components parallel and tangential to the radius vector @xmath24 in the plane of the sky are @xmath25 and @xmath26 , respectively . for each component , the velocity dispersion is defined as @xmath27 . we will assume @xmath28 . the velocity dispersion for each observed component can be constructed by solving the jeans equation for the three - dimensional stellar radial velocity dispersion profile @xmath29 and integrating along the line of sight . we note that even in the case of tidally disturbed dwarfs , @xcite have shown that dsph velocity dispersions are well modeled by the jeans equation , as long as unbound , interloper stars are removed with standard procedures . we derive the three resulting observable velocity dispersions : @xmath30 here @xmath31 is the stellar velocity anisotropy , @xmath32 is the surface density of stars , and @xmath33 is the three - dimensional density of stars . it is clear from inspection that each component depends on @xmath34 in a different fashion , and therefore can be used together to constrain its value . for @xmath32 and @xmath35 we use a king profile @xcite , which is characterized by a core radius , @xmath36 , and tidal radius , @xmath37 . we adopt values that describe the surface density of draco : @xmath38 kpc and @xmath39 kpc . note that the remaining dsphs have similar king concentrations , @xmath40 , with sextans having the largest ratio @xmath41 . our results do not change significantly as we vary @xmath42 ( equivalent to looking at different dsphs ) . in eqs . ( [ eq : losdispersion ] ) , ( [ eq : rdispersion ] ) and ( [ eq : phidispersion ] ) , the radial stellar velocity dispersion , @xmath43 , depends on the total mass distribution , and thus the parameters describing the dark matter density profile . we will consider the following general parameterization of the dark matter density profile , @xmath44^{(c - a)/b}}. \label{eq : densityprofile}\ ] ] here , the value of @xmath45 sets the _ asymptotic _ inner slope , and different combinations of @xmath46 and @xmath47 set the transition to the asymptotic outer slope . for the specific choice @xmath48 , we have an nfw profile @xcite . we denote this as our _ cusp _ case below . for our _ core _ case we use @xmath49 , corresponding to a burkert profile @xcite . we take these two models to be representative of the predictions of cdm and wdm models . the burkert profile for the core case is motivated by the expectation that wdm halos will mimic cdm halos at large radius . this was seen in the wdm simulations of @xcite . the burkert choice is also conservative compared to the often - used isothermal core with @xmath50 , which is more divergent in shape from an nfw and would be easier to distinguish observationally . regardless , our methods are robust to changes in the underlying form of the density profile . ( [ eq : densityprofile ] ) allows considerable flexibility in overall form , and the five _ shape _ parameters ( @xmath51 ) are in many cases degenerate . however , there are a number of physically relevant quantities that may be derived for any set of the five shape parameters . the first is the log - slope of the dark matter density profile , defined as @xmath52 . for the density profile in eq . ( [ eq : densityprofile ] ) this is given by @xmath53 $ ] . other quantities of physical interest are the integrated mass within a given radius , @xmath54 , and the physical density at a given radius , @xmath55 , which are clearly obtained for a degenerate set of shape parameters . below , we show that while the _ shape _ parameters are not well constrained by dsph velocity data , the _ physical _ quantities of interest at the scale of the stellar core radius , @xmath56 , may be constrained to high precision . our goal is to estimate the accuracy with which the velocity components of stars in dsphs can be used to probe the underlying dark matter distribution . we will consider a model with six independent parameters : @xmath45 , @xmath46 , @xmath47 , @xmath57 , @xmath58 , and @xmath59 constant . we will consider generalized @xmath60 forms below . in order to keep the profile shape relatively smooth ( as is expected for dark matter halo profiles ) we restrict the range of @xmath46 and @xmath47 by adding gaussian priors of @xmath61 . the errors attainable on these parameters will depend on the covariance matrix , which we will approximate by the @xmath62 fisher information matrix @xmath63 @xcite . the inverse of the fisher matrix , @xmath64 , provides an estimate of the covariance between the parameters , and @xmath65 approximates the error in the estimate on the parameter @xmath66 . the cramer - rao inequality guarantees that @xmath65 is the minimum possible variance on the @xmath67th parameter for an unbiased estimator . using @xmath64 in place of the true covariance matrix involves approximating the likelihood function of the parameters as gaussian near its peak , so @xmath64 will be a good approximation to the errors on parameters that are well - constrained . the fisher matrix also provides information about degeneracies between parameters but obviously should not be trusted for estimates of the error along these degeneracy directions . lll|ccccc @xmath68 & 1.3 ( 0.32 ) & @xmath69 & 2.0 ( 2.2 ) & 0.91 ( 1.0 ) & 0.28 ( 0.24 ) & 0.21 ( 0.23 ) & 0.16 ( 0.15 ) + @xmath70 [ @xmath71 & 1.3 ( 5.3 ) & @xmath72 & 0.27 ( 0.38 ) & 0.12 ( 0.17 ) & 0.11 ( 0.11 ) & 0.07 ( 0.08 ) & 0.06 ( 0.05 ) + @xmath73 [ @xmath74 kpc@xmath75 & 4.0 ( 2.5 ) & @xmath76 & 1.6 ( 0.73 ) & 0.73 ( 0.33 ) & 0.16 ( 0.16 ) & 0.13 ( 0.12 ) & 0.09 ( 0.09 ) + @xmath77 [ @xmath78 & 22 ( 28 ) & @xmath79 & 1.6 ( 1.7 ) & 0.71 ( 0.76 ) & 0.68 ( 0.93 ) & 0.45 ( 0.62 ) & 0.38 ( 0.52 ) + @xmath34 & 0 ( 0 ) & @xmath80 & 2.1 ( 1.4 ) & 0.97 ( 0.62 ) & 0.16 ( 0.16 ) & 0.15 ( 0.15 ) & 0.10 ( 0.10 ) + @xmath45 & 1 ( 0 ) & @xmath81 & 4.4 ( 4.6 ) & 2.0 ( 2.1 ) & 0.78 ( 0.87 ) & 0.55 ( 0.66 ) & 0.44 ( 0.51 ) + @xmath57 [ @xmath74 kpc@xmath75 & 1.0 ( 3.2 ) & @xmath82 & 20 ( 9.2 ) & 9.1 ( 4.2 ) & 5.9 ( 2.0 ) & 4.0 ( 1.5 ) & 3.3 ( 1.1 ) + @xmath83 [ kpc ] & 2.0 ( 1.5 ) & @xmath84 & 8.3 ( 5.1 ) & 3.8 ( 2.3 ) & 3.3 ( 1.8 ) & 2.2 ( 1.2 ) & 1.8 ( 1.0 ) + we pick large radial bins to compute the velocity dispersions and check that this uncorrelates the different bins . then the elements of @xmath85 are given by @xmath86 the sum is over @xmath87 radial bins and @xmath88 refers to the three velocity `` methods '' one line - of - sight and two components in the plane of the sky . the errors on the velocity dispersion are represented by @xmath89 . we choose bins of equal width in distance , so that there are approximately an equal number of stars in each radial bin . as long as we distribute equal number of stars in each bin , the results we present below are insensitive to the binning scheme , except in the limit of very few bins , or in the limit of small numbers of stars per bin . to model the errors on the velocity dispersions , we define @xmath90 ^ 2 \rangle$ ] . we assume that the errors on the velocity of each star are gaussian and that the theory error ( from the distribution function ) and experimental error are summed in quadrature , @xmath91 ^ 2 \ , , \label{eq : totalerror}\ ] ] where @xmath92 is the number of stars in each bin ( taken here to be constant from bin to bin ) . here @xmath93 is the dispersion in each bin determined from eqs . ( [ eq : losdispersion])-([eq : phidispersion ] ) , and @xmath94 represents the measurement error in the velocities of stars in that bin . for los velocities , the spectroscopic resolution implies errors @xmath95 , which are negligible compared to the underlying dispersion , @xmath96 . sim can achieve @xmath97as yr@xmath7 proper motions for faint stars . at 100 kpc , this translates to an accuracy of @xmath6 km s@xmath7 . we will take this as a typical measurement error for velocities in the plane of the sky . the derivatives in eq . ( [ eq : fishergaussian ] ) , and thus the errors attainable on any of the parameters , depend on the location in the true set of parameter space . to examine how the errors vary as a function of the true set of parameters , we choose two fiducial models . both models produce a `` typical '' dsph velocity dispersion profile , which is roughly flat at @xmath98 out to @xmath37 . this requirement does not completely fix the profile . for our _ cusp _ model ( nfw ) we add the further restriction that @xmath83 and @xmath57 fall within the expected cdm range @xcite . for our _ core _ model ( burkert ) we set @xmath83 and @xmath57 to give a central phase space density @xmath99 , which can be produced in some wdm models @xcite . the values of @xmath83 and @xmath57 for both models are listed in table [ tab : parameterstable ] as are the implied log - slopes @xmath100 ( cusp ) and @xmath101 ( core ) at the characteristic radius @xmath102 , which is set below at the value @xmath103 kpc . the log - slope of the density profile varies with radius , and so does the minimal error attainable on it . we are interested in searching for the radius , @xmath102 , where the log - slope is best constrained . [ fig : gammastar ] shows the @xmath104 error on @xmath105 computed as a function of radius using 1000 line - of - sight velocities and 200 proper motions for the cusp ( left ) and core ( right ) models . in both cases , the error reaches a minimum of @xmath106 at @xmath107 and we explicitly adopt @xmath108 for the rest of the discussion . we find that the value of @xmath102 and the minimum error depend somewhat on the fiducial model . as shown in fig . [ fig : gammastar ] , profiles with cusps are slightly better constrained than profiles with cores , and @xmath102 occurs at slightly smaller radii for cusped models . we also find that as @xmath42 increases , the minimum error decreases and @xmath102 increases . we have chosen a conservatively small ratio here , @xmath109 . table [ tab : parameterstable ] summarizes our results . we list 1-@xmath11 errors attainable on several quantities determined by marginalizing over the 6 fit parameters discussed above . we present errors for five different combinations for the number of los stars and proper motion stars . in all cases , the errors on the `` physical '' parameters , @xmath110 , @xmath111 , and @xmath112 , decrease significantly when proper motions are included . quantities listed without ( with ) parentheses correspond to the cusp ( core ) halo case . note that the errors on the halo shape parameters @xmath46 and @xmath47 are essentially the same as the priors we set on them of @xmath61 and hence we do not list them in the table . a quantity that is well constrained without the addition of proper motion information is the mass @xmath113 , which can be determined to @xmath114 ( @xmath115 ) for cusped ( cored ) models with current data . the mass can be determined to a remarkable @xmath116 accuracy when 200 sim stars are added . the maximum circular velocity is relatively unconstrained by los data alone , but becomes determined to within @xmath117 for cusped ( cored ) models with @xmath118 proper motions . more interesting for the nature of dark matter is the log - slope of the density profile at @xmath102 . even with 1000 los velocities , the log - slope in the cusp case is virtually unconstrained , but with the addition of @xmath118 sim stars the log - slope is well - determined , e.g. @xmath119 for the case of an underlying cusp . the power of velocity dispersions to constrain the density and mass at @xmath120 is relatively simple to understand by examining the observable velocity dispersion components ( eqs . [ eq : losdispersion ] , [ eq : rdispersion ] and [ eq : phidispersion ] ) . these observable quantities depend on the three - dimensional stellar velocity dispersion , which scales as @xmath121 for a power - law stellar distribution @xmath35 and constant @xmath34 . the majority of stars reside at projected radii @xmath122 , where the stellar distribution is falling rapidly @xmath123 . in this case , for @xmath124 , the los component scales as @xmath125 and is dominated by the mass and density profile at the smallest relevant radii , @xmath126 . for @xmath127 , @xmath128 and @xmath129 is similarly dominated by @xmath130 contributions . therefore we expect the strongest constraints on the dark matter profile at @xmath120 . for @xmath131 similar arguments hold but there is a manifest degeneracy between @xmath34 and the log - slope @xmath132 . as discussed above and illustrated in table 1 , this degeneracy can be broken by including both los information ( eq . [ eq : losdispersion ] ) with tangential information ( eqs . [ eq : rdispersion ] , [ eq : phidispersion ] ) . with the constraints on @xmath132 , are we able to distinguish cored from cusped models ? to answer this question , in figure [ fig : contours ] we present @xmath133 and @xmath134 error contours in the two - dimensional plane of @xmath132-@xmath34 . this figure clearly shows the utility of combining both los and proper motions . due to the degeneracy with @xmath34 , a sample of @xmath135 line - of - sight stars is unable to distinguish between our two fiducial models , though the models can be clearly distinguished with the addition of proper motions . in the @xmath34-@xmath132 plane , the principal constraining power comes from combining the los and @xmath136 velocity dispersions because the @xmath34 dependence in these two quantities comes with opposite signs ( eqs . [ eq : losdispersion ] , [ eq : rdispersion ] ) . finally , we address the fact that @xmath34 could in principle vary significantly with radius . we have repeated our analysis assuming that @xmath137 , and marginalize over @xmath138 , @xmath139 and @xmath140 . we find very small changes to the @xmath141 error . for reasons similar to those given above , the three different velocity dispersion profiles constrain @xmath142 to high accuracy , breaking the @xmath34-@xmath143 degeneracy . we have shown that the measurement of proper motions for @xmath4 stars in a typical dsph can be combined with current line - of - sight velocity measurements to constrain the dark halo log - slope to @xmath0 and normalization to @xmath144 at about twice the king radius . this is a factor of @xmath6 better than currently possible with los velocity dispersion data alone . the results from such observations will provide a very sensitive test of the cdm paradigm and an incisive tool for investigating the microscopic nature of dark matter . we estimate that with @xmath3 days of observing time over the 5 year lifetime of sim , it will be possible to obtain the proper motions of @xmath4 stars in multiple dsphs @xcite . we thank s. kazantzidis , s. kulkarni , s. majewski , and r. munoz for useful discussions . les is supported in part by a gary mccue postdoctoral fellowship through the center for cosmology at the university of california , irvine . jsb , les , and mk are supported in part by nsf grant ast-0607746 .
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we show that measurements of stellar proper motions in dwarf spheroidal galaxies provide a powerful probe of the nature of dark matter . allowing for general dark matter density profiles and stellar velocity anisotropy profiles ,
we show that the log - slope of the dark matter profile at about twice the stellar core ( king ) radius can be measured to within @xmath0 when the proper motions of 200 stars are added to standard line - of - sight velocity dispersion data .
this measurement of the log - slope provides a test of cold and warm dark matter theories at a sensitivity not possible with line - of - sight velocity dispersion measurements alone .
the upcoming sim planetquest will have the sensitivity to obtain the required number of proper motions in milky way dwarf spheroidal galaxies .
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according to dijkgraaf and vafa @xcite , the effective glueball superpotential of @xmath0 supersymmetric @xmath1 gauge theory has an asymptotic expansion given by the planar part of the topological expansion of a matrix model . to give the effective potential for various vacua of the gauge theory , dijkgraaf and vafa propose a formula obtained by saddle point expansion of the matrix integral around different critical points . the filling fractions , i.e. , the fraction of the eigenvalues sitting close to each of the critical points of the potential are the parameters selecting the vacua . in this paper we give a way to define non - perturbatively ( i.e. , beyond the saddle point expansion ) the matrix integrals . different integrals are integrals over different cycles in the space of normal matrices . in the case of generic polynomial potentials it is possible to construct a cycle for each of the critical points , so that all effective potentials considered by dijkgraaf and vafa arise in the asymptotic expansion of our integrals . we note that the idea of integrating over eigenvalues along suitable contours in the complex plane appears in special situations in @xcite . david s contours were also more recently considered in @xcite , where a ( non - holomorphic ) modification of the integral was proposed to obtain arbitrary filling factors . a second result of this paper is the description of the asymptotic distribution of eigenvalues for each critical point . the eigenvalues , as predicted by dijkgraaf and vafa , lie asymptotically along arcs connecting branch points of a two - fold cover of the complex plane . we give a reality condition which specifies the shape of the arcs and the density of eigenvalues . before considering the case of general @xmath2 matrices , it is instructive to consider the case @xmath3 of ordinary integrals . consider the airy integral @xmath4 this integral gives a formal solution of the airy differential equation @xmath5 . indeed the saddle point expansion around the two critical points of the integrand give the two linear independent formal power series solutions . the non - perturbative solutions can be obtained by integrating along paths in the complex plane connecting any two of the three directions at infinity where the integrand goes to zero . more generally , the integrals of the form @xmath6 for a generic polynomial @xmath7 of degree @xmath8 are solutions of a differential equation of order @xmath9 . we show that a system of @xmath9 linearly independent solutions are obtained by choosing @xmath8 integration contours in the complex plane going to infinity in directions where the integrand decays exponentially . moreover the contours can be chosen so that each of them passes through exactly one of the @xmath8 critical points . it follows that the @xmath9 saddle point expansions at each of the critical points are the asymptotic expansions of true solutions given by convergent integrals . in the case of matrices we consider integrals of the form @xmath10 over real @xmath11-dimensional cycles @xmath12 in the space of complex @xmath2 matrices . the potential @xmath13 is a polynomial with complex coefficients , @xmath14 and the normalization factor @xmath15 is such that @xmath16 . the observables @xmath17 are holomorphic functions invariant under conjugation . in the well - studied case of integrals over hermitian matrices , with a polynomial @xmath13 with real coefficients bounded below , the saddle point approximation becomes exact in the large @xmath18 limit with fixed @xmath19 and the relevant critical point is described by an asymptotic _ density of eigenvalues _ which has support on a union of intervals on the real axis . if @xmath19 is small , these intervals are small neighborhoods of the minima of the potential @xmath13 . to obtain the densities of eigenvalues needed to make contact to gauge theory one considers the variational problem for critical points subject to the side condition that the fraction of eigenvalues in the vicinity of each of the critical points ( not just minima ) are given numbers . we consider here the case of a generic polynomial @xmath13 of degree @xmath8 with complex coefficients , with distinct critical points @xmath20 , and propose to consider integrals over cycles in the space of normal matrices ( a matrix is normal if it commutes with its adjoint or , equivalently , if it is conjugated to a diagonal matrix by a unitary matrix ) . the cycles we consider are parametrized by integers @xmath21 whose sum is @xmath18 and are characterized by the condition that @xmath22 eigenvalues belong to a path @xmath23 in the complex plane going through @xmath24 and going to infinity in a direction where @xmath25 . in the limit @xmath26 with @xmath27 fixed , the eigenvalues in the saddle point approximation ( supposed to be exact in the limit ) are distributed along @xmath9 arcs in the complex plane . for small @xmath19 the arcs are close to the critical points and the @xmath28th arcs contains the fraction @xmath29 of the eigenvalues . let @xmath30 be a polynomial of degree @xmath31 with complex coefficients . we want to consider integrals of the form @xmath32 for polynomials @xmath33 . before considering the question of integration cycles we may evaluate such integrals as asymptotic series as @xmath34 by formal application of the saddle point method at each of the critical points @xmath20 , which we assume to be distinct . in this way we get @xmath9 asymptotic series of the form @xmath35 and the question is whether these are asymptotic expansions of our integral for suitable cycles @xmath23 . the cycles which we should consider here are ( linear combinations of ) paths for which the integral converges . as the integrand is holomorphic , homotopic paths will give the same answer and what matters is the behavior at infinity . as @xmath36 , @xmath37 , so there are @xmath8 directions in the complex plane for which @xmath38 as @xmath39 tends to infinity in these directions . let us call these asymptotic directions valleys as in these directions the integrand decays exponentially . neighboring valleys are separated by hills , which are directions of exponential increase of the integrand . so the cycles one needs to consider are linear combinations of infinite paths connecting pairs of distinct valleys . as paths connecting two valleys can be deformed into sums of paths connecting the two valleys with any third one , there are only @xmath9 linearly independent cycles . let us assume for simplicity that the critical values @xmath40 have distinct imaginary parts . then there is a canonical way to associate to each critical point @xmath24 a path @xmath23 in such a way that the asymptotic expansion of the integral over @xmath23 as @xmath34 is obtained by the saddle point expansion at @xmath24 . namely , we take the _ steepest descent _ paths(see , e.g. , @xcite and @xcite ) emerging from @xmath24 , defined by the condition that the tangent vector at each point points in the direction of the gradient of @xmath41 . as @xmath42 is holomorphic , the gradient of @xmath43 is orthogonal to the gradient of @xmath44 by the cauchy riemann equations . thus steepest descent paths are level lines for the imaginary part of @xmath42 . each non - degenerate critical point @xmath24 is at the intersection of two such level lines . one of these two lines , the one along which @xmath41 takes its minimum at @xmath24 , is the steepest descent path @xmath23 . along the other line @xmath45 , the real part takes its maximum at @xmath24 . we claim that @xmath23 is a smooth path going to infinity in both directions and connecting two valleys indeed , @xmath23 is ( in suitable parametrization ) given by a solution of the differential equation @xmath46 . it follows that latexmath:[$\frac d{dt}\mathrm{re}(p(z(t)))= imaginary parts , so the steepest descent path passing through @xmath24 may not come close to any other critical point . thus @xmath48 is bounded below so that , as we go away from @xmath24 , @xmath49 must go to infinity and the path @xmath23 connects two valleys . similarly @xmath50 connects two hills . as @xmath23 and @xmath50 cross at @xmath24 , the two valleys connected by @xmath23 are separated by hills and are thus different . we have thus shown that for each critical point @xmath24 there is a steepest descent path @xmath23 going through @xmath24 and connecting pairs of different valleys . on @xmath23 the real part of @xmath42 is minimal at @xmath24 so that the saddle point expansion at @xmath24 indeed gives the asymptotic expansion of the integral over @xmath23 . we consider matrix integrals of the form for @xmath42 a polynomial of degree @xmath8 . they are integrals of holomorphic differential forms over @xmath11 dimensional cycles . for each set of natural numbers @xmath51 summing up to @xmath18 , we have a cycle in the normal matrices , characterized by the condition that @xmath22 eigenvalues run over the path @xmath23 of the previous section . more precisely , the cycle @xmath12 is parametrized by @xmath52 : @xmath53 the first @xmath54 diagonal elements @xmath55 are parametrizations of @xmath56 , the next @xmath57 are parametrizations of @xmath58 and so on . the usual argument to reduce the integral to an integral over the eigenvalues ( see @xcite ) gives @xmath59 the integral is over @xmath60 and the function @xmath61 , a function on matrices invariant under conjugation , is regarded as a symmetric function of the eigenvalues . to study the large @xmath18 limit , it is useful to introduce the trace of the resolvent : @xmath62 as products of such traces are generating functions of polynomial functions invariant under conjugation . the `` loop equation '' @xcite for this quantity is @xmath63 where @xmath64 is a polynomial of degree @xmath65 with leading coefficient @xmath66 . this equation can be derived from the identity @xmath67 in the limit @xmath68 , the matrix integral is supposed to be dominated by an integral over a region where the eigenvalues are close ( for small @xmath19 ) to critical points of @xmath13 . with our choice of integration cycles and keeping @xmath69 fixed as @xmath68 , there will be a fraction @xmath70 of the eigenvalues close to @xmath71 . in the limit @xmath68 one expects that the saddle point approximation becomes exact and thus @xmath72 converges to @xmath73 for some probability measure @xmath74 with support in regions around the critical points of @xmath13 and so that @xmath29 is the measure of the region around @xmath24 . technically one assumes that the limit @xmath68 exists and that @xmath75 . the function @xmath76 is defined and holomorphic for @xmath39 outside the support of the measure . setting @xmath77 we finally obtain @xmath78 for some polynomial @xmath79 of degree @xmath65 with leading coefficient @xmath66 . thus the function @xmath80 , which is a priori defined on the complement of the support of the measure , has an analytic continuation to a two - fold covering of the complex plane . the original function @xmath81 is the branch of the function defined by which behaves at infinity as @xmath82 . with the analogy with the case of hermitian matrices in mind , it is reasonable to assume that the measure @xmath74 has support on a collection of arcs @xmath83 which for small @xmath19 are close to the critical points of @xmath13 , and that @xmath84 for some _ density of eigenvalues _ @xmath85 defined on the arcs : namely , the measure @xmath86 of a set @xmath87 intersecting one of the arcs , say @xmath88 , in a piece @xmath89 is @xmath90 note that for the right - hand side to be defined we need to fix an orientation on @xmath88 . then @xmath80 is a holomorphic function outside on the complement of the arcs and the density on @xmath88 is related to the discontinuity of @xmath81 : @xmath91 here @xmath92 ( @xmath93 ) denotes the limit of @xmath80 as @xmath39 tends to @xmath94 from the left ( from the right ) of the oriented curve @xmath88 . from this information we deduce that the arcs @xmath88 connect pairs of zeros of @xmath81 , the branch points of the hyperelliptic curve @xmath95 of genus @xmath65 . the measure of the @xmath96th arc is then the period @xmath97 over a cycle @xmath98 enclosing the pair of branch points in counterclockwise direction . it follows from the condition on the leading coefficient of @xmath99 that @xmath100 . we also note that since @xmath101 , we have the formula @xmath102 . there remains to determine the precise form of the arcs and the coefficients in @xmath99 as functions of the filling fractions @xmath70 . first of all , the relation between the @xmath65 free complex coefficients @xmath103 of @xmath79 ( recall that the leading coefficient is fixed to be @xmath104 ) and the periods @xmath70 ( @xmath105 ) subject to @xmath100 is , locally around any point where the branch points are distinct , a holomorphic diffeomorphism , since the jacobian matrix @xmath106 is non - degenerate , being the matrix of @xmath107-periods of a basis of abelian differentials on a smooth curve . it follows that locally there exists a real @xmath108-dimensional submanifold in the complex space @xmath109 of coefficients @xmath110 which maps to real positive @xmath70 . the condition that fixes the shape of the arcs is the reality and positivity condition for the density : if @xmath111 is a parametrization of the arc @xmath88 respecting its orientation , the condition is @xmath112 using @xmath113 , we may parametrize the arcs ( away from the endpoints ) to be solutions of the differential equation @xmath114 connecting branch points . alternatively , arcs may be described as level lines of a function : introduce the hyperelliptic integral @xmath115 it is a holomorphic many - valued function on the complement of the support of the measure . as we go around an arc @xmath88 , @xmath116 increases by @xmath70 so @xmath117 is single valued . the measure of a piece between two points @xmath118 , @xmath119 on a curve @xmath88 is @xmath120 which is real . thus the arcs @xmath88 are level lines of the imaginary part of @xmath116 . around a branch point @xmath121 which is a simple zero of @xmath122 , @xmath123 . therefore there are three smooth level lines of @xmath61 emerging from every simple branch points . in the most general situation the support of the measure may then be a graph consisting of level lines of @xmath124 joining branch points . a more precise description is possible in the case of small @xmath19 to which we turn . for small @xmath125 we claim that the arcs and the density of eigenvalues are determined completely by the filling fractions @xmath70 through and . to show this , notice first that as @xmath34 , pairs of branch points @xmath126 converge to the critical points @xmath71 and the periods @xmath70 ( eq . ) , regarded as functions of @xmath19 and the coefficients @xmath110 of @xmath99 are holomorphic at @xmath127 . we have @xmath128 since there is a bijective holomorphic correspondence between values @xmath129 at the distinct points @xmath71 and coefficients @xmath110 of @xmath99 , we have at @xmath127 , and by analyticity also for small @xmath19 , a biholomorphic map @xmath130 , @xmath100 . in particular , we can invert this map and find a unique @xmath99 for each set of @xmath131 , such that @xmath100 . it remains to show that for all small @xmath125 there is a level line @xmath88 of @xmath117 connecting @xmath132 to @xmath133 . as @xmath34 , @xmath134 the level lines of @xmath135 at @xmath127 are thus the level lines of @xmath43 . in the neighborhood of a non - degenerate critical point @xmath71 they look like the left picture in fig . [ f-1 ] : on any small circle around @xmath71 each value is taken on at most four times . ( 150,130)(60,0 ) for positive small @xmath19 the critical point @xmath71 splits into two branch points @xmath132 , @xmath133 from each of which three level lines emerge . the condition that the period @xmath70 is real implies that @xmath136 . the function @xmath137 is defined up to a sign in a neighborhood of @xmath126 , so that its zero level line is uniquely defined . it follows that there is a level line @xmath88 of @xmath124 , namely the zero set of @xmath138 , joining @xmath132 to @xmath133 as on the right picture in fig . [ f-1 ] : if none of the level lines emerging from @xmath132 and @xmath133 were to join , @xmath139 would take the value zero six times on any small circle encircling @xmath126 , which can not be , as this does not happen at @xmath127 . also , a level line can not go from a point @xmath132 or @xmath133 to itself as the real part of @xmath61 is monotonic along level lines . we conclude that for any small @xmath125 , and any given filling fractions @xmath140 summing up to 1 , there is a unique polynomial @xmath141 , so that the curve @xmath142 has @xmath107-periods @xmath70 . the zeros of @xmath81 are connected in pairs by arcs @xmath88 obeying the reality condition . according to the discussion above , these arcs are are the support of the measure and the density of eigenvalue is @xmath143 , @xmath144 . we have given a non - perturbative definition of the matrix integrals that in the large @xmath18 limit give the superpotentials considered in @xcite . we considered the case of a generic polynomial potential with complex coefficients . for small t hooft coupling @xmath19 , the density of eigenvalues was shown to be given by arcs connecting pairs of branch points of a hyperelliptic curve . the shape of the arcs is uniquely determined by a reality condition . for larger @xmath19 or for potentials with degenerate critical points , one expect the arcs to combine into graphs in the complex plane . it would be interesting to understand what kind of graphs can arise in this way . r. dijkgraaf , c. vafa , `` matrix models , topological strings , and supersymmetric gauge theories '' ( arxiv : hep - th//0206255 ) , nucl.phys . b644 ( 2002 ) 320 r. dijkgraaf , c. vafa , `` a perturbative window into non - perturbative physics '' ( arxiv : hep - th/0208048 ) , 2002 f. david , `` non - perturbative effects in matrix models and vacua of two dimensional gravity '' phys.lett . b302 ( 1993 ) 403410 c. i. lazaroiu `` holomorphic matrix models , '' hep - th/0303008 , jhep 0305 ( 2003 ) 044 n.g . de bruijn , `` asymptotic methods in analysis , '' north - holland publishing , amsterdam , 1958 a. erd ' elyi , `` asymptotic expansions , '' dover publications , new york , 1956 s. wadia , `` dyson - schwinger equation approach to the large - n limit : model systems and string representation of yang mills theory '' phys . d 24 ( 1981 ) 970978 a. a. migdal , `` loop equations and 1/n expansion '' phys . 102 ( 1983 ) 199290 d. bessis , c. itzykson , j. b. zuber , `` quantum field theory techniques in graphical enumeration , '' advances in applied mathematics , 1:109157 , 1980 .
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we study a class of holomorphic matrix models .
the integrals are taken over middle dimensional cycles in the space of complex square matrices . as the size of the matrices tends to infinity
, the distribution of eigenvalues is given by a measure with support on a collection of arcs in the complex planes .
we show that the arcs are level sets of the imaginary part of a hyperelliptic integral connecting branch points .
_ department of mathematics , eth zurich _ _ 8092 zurich , switzerland _
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the _ fermi _ gamma - ray space telescope hosts two instruments , the large area telescope @xcite and the gamma - ray burst monitor @xcite , which together are capable of measuring the spectral parameters of gamma - ray bursts ( grbs ) across seven decades in energy . since the start of gbm and lat science operations in early august 2008 , emission at energies @xmath4100mev has been detected from ten grbs . these detections were made possible by the lat s greater sensitivity and shorter deadtime ( 26@xmath5s ) compared to previous instruments . prior to _ fermi _ , high - energy gamma - rays from grbs with energies up to 18 gev were observed by the egret instrument on - board the _ compton gamma - ray observatory_. the egret observations suggested three types of high - energy emission : an extrapolation of the low energy spectra to the @xmath4100 mev band ( e.g. , * ? ? ? * ) , an additional spectral component during the prompt emission @xcite and in the case of grb 940217 , a gev afterglow which was detectable for 90 minutes after the trigger @xcite . the redshifts of these events were not determined . recently , reported that grb 080514b which triggered agile at lower energies , was detected by the grid instrument up to 300 mev . a photometric redshift of @xmath6 was reported for this event . in the _ fermi _ era , due to the advanced localization capabilities of the lat and the rapid follow - up by the _ swift _ narrow field instruments @xcite and the ground - based follow - up community , redshifts for five of the ten lat bursts have been measured . these include grb 080916c with @xmath7 , a long burst that has the highest inferred isotropic energy , @xmath8ergs ( 10 kev10 gev ) @xcite , and grb 090510 with @xmath9 @xcite , the second short burst seen by the lat and the first short burst to show definitively an additional hard power - law component in the gev band during the prompt phase @xcite . grb 090902b is a long , fairly intense burst with a redshift of @xmath10 @xcite and fluence of @xmath11erg@xmath12 ( 10 kev10 gev ) over the first 25 seconds of the prompt emission . these data give an isotropic energy @xmath13ergs , comparable to that of grb 080916c . similar to grb 090510 , grb 090902b has a significant additional , hard power - law component that appears during the prompt phase . furthermore , a spectral feature at energies @xmath14kev is evident in the gbm spectrum of grb 090902b that is consistent with an extrapolation of the @xmath4100mev power - law emission down to those energies . in previous analyses , @xcite reported evidence for an additional low - energy spectral component below 20kev for @xmath15% of batse bursts . we report on the observations and analysis of gamma - ray emission from grb 090902b measured by the gbm and lat instruments . in section 2 , we present details of the detections by both instruments and summarize the follow - up observations . in section 3 , we show the light curves of the prompt emission as seen by the various detectors and describe the extended emission found in the lat data out to 1 ks after the trigger . in section 4 , we present the time - resolved spectral analysis of the burst emission during the prompt phase . finally , in section 5 , we discuss the physical interpretation of the gbm and lat data , focusing on the implications of the power - law component for models of grb physics . on 2009 september 2 at 11:05:08.31 ut , the _ fermi _ gamma - ray burst monitor triggered on and localized the bright burst grb 090202b ( trigger 273582310 / 090902462 , * ? ? ? * ) . the burst was within the lat field of view initially at an angle of 51@xmath16 from the boresight . this event was sufficiently bright in the gbm that an autonomous repoint request was made , and the spacecraft began slewing within 10 seconds towards the burst . after @xmath1200 seconds , it had pointed the lat boresight to within a few degrees of the final burst localization . it maintained that pointing until @xmath11 ks post - trigger , when the earth s limb began to enter the lat field - of - view ( fov ) . this burst was detected up to @xmath15 mev by gbm , and emission was significantly detected by the lat , with 39 photons above 1 gev . the highest energy photon had @xmath17 gev and arrived 82 seconds after the gbm trigger ; and the initial analyses detected photons as late as 300 seconds after the trigger @xcite . from the lat data , the burst was localized to r.a.(j2000 ) , dec(j2000 ) = 265.00 , 27.33 with a statistical uncertainty of 0.04@xmath16 ( + @xmath180.1@xmath16 systematic ) , enabling target of opportunity observations to begin @xmath112.5 hours after the trigger with the narrow field instruments on _ swift_. a candidate x - ray afterglow within the lat error circle was detected by the x - ray telescope ( xrt , * ? ? ? this source was confirmed to be fading @xcite , and uvot observations revealed the optical afterglow @xcite . the earliest ground - based optical observations were obtained by rotse - iiia @xmath11.4 hours post trigger @xcite . other detections were reported in the optical @xcite , in the near infrared by grond @xcite and in the radio @xcite . the location of the fading source detected by grond was r.a.(j2000 ) , dec(j2000 ) = @xmath19 , + 271927.1 , 3.3 arcminutes from the lat location @xcite . the afterglow redshift of @xmath10 was measured by @xcite using the gmos spectrograph mounted on the gemini - north telescope . in figure [ fig : prompt light curves ] , we show the gbm and lat light curves in several energy bands . the top three panels show data from the most brightly illuminated nai and bgo detectors of the gbm , and the bottom three panels show the lat data with various event selections . in the bottom panel , the measured photon energies are plotted as a function of time , including the highest energy event ( @xmath20gev ) that arrived 82 seconds after the gbm trigger time , @xmath21 . from the gbm light curves , we see that at energies @xmath221mev the prompt phase ends approximately 25 seconds after @xmath21 . detailed analysis of the gbm data for energies 50300 kev yields a formal t90 duration of 21.9 seconds starting at @xmath23s . by contrast , the lat emission @xmath4100mev clearly continues well after this time range . on time scales longer than the prompt phase , the lat detects emission from grb 090902b as late as 1 ks after the gbm trigger . the spectrum of this emission is consistent with a power - law with photon index @xmath24 , and its flux ( @xmath4100mev ) declines as @xmath25 over the interval ( @xmath26s ) . as we note above , the lat observations are interrupted by entry of the earth s limb into the fov , but analysis of data after @xmath27s , when the source location is again unocculted , shows that any later emission lies below the lat sensitivity ( figure [ fig : gev afterglow ] ) . the upper limit we obtain for data after @xmath27 is consistent with an extrapolation of the @xmath2 decay . similar late time emission for energies @xmath4100mev that extends well beyond the prompt phase has been seen for five earlier bursts by _ fermi _ : grb 080916c @xcite ; grb 090323 @xcite ; grb 090328 @xcite ; grb 090510 , independently seen by agile @xcite and by _ fermi _ @xcite ; and grb 090626 @xcite . spectral analysis was performed using the data from both the gbm and the lat . these analyses include data from the nai detectors 0,1,2,9,10 and bgo detectors , and lat `` transient '' class data , with front- and back - converting events considered separately . the nai data are fit from 8 kev to 1 mev and the bgo from 250 kev to 40 mev using the time tagged event ( tte ) data , which are high time resolution data that allow us to define the time intervals based on the structure of the gbm and lat light curves . the lat data are fit from 100 mev to 200 gev . an effective area correction of 0.9 has been fit to the bgo data to match the model normalizations given by the nai data ; this correction is consistent with the uncertainties in the gbm detector responses . the fits were performed with the spectral analysis software package rmfit ( version 3.1 ) . for further details on the data extraction and spectral analysis procedures see @xcite and @xcite . the time - integrated spectrum of grb 090902b is best modeled by a band function @xcite and a power - law component ( table [ tab : spectral params ] ) . the power - law component significantly improves the fit between 8kev and 200gev both in the time - integrated spectrum and in the individual time intervals where there are sufficient statistics . it is also required when considering only the gbm data ( 8kev40mev ) for the time - integrated spectrum , as its inclusion causes an improvement of @xmath282000 in the cstat statistic over the band function alone . when data below @xmath150 kev are excluded , a power - law component can be neglected in the gbm - only fits . we conclude that this power - law component contributes a significant part of the emission both at low ( @xmath1850kev ) and high ( @xmath4100mev ) energies . figure [ fig : prompt spectrum ] shows the counts and unfolded @xmath29 spectra for a band function with a power - law component fit to the data for interval * b * ( when the low energy excess is most significant ) using the parameters given in table [ tab : spectral params ] . spectral evolution is apparent in the band function component from the changing @xmath30 values throughout the burst , while @xmath31 remains soft until interval * e * when it hardens significantly . @xmath31 is similarly hard in interval * f * , after which the band function component is no longer detected . the hardening of @xmath31 is accompanied by an apparent hardening of the power - law index , @xmath32 , which until interval * e * does not exhibit much variation . however , this is not definitive since the flux is too low to constrain @xmath32 in intervals * e * and * f * separately . a spectral fit of the sum of these two intervals confirms the presence of both a harder @xmath31 and a harder @xmath32 , with a clear statistical preference for the inclusion of the power - law component . an equally good fit is obtained in the combined * e * + * f * interval if this power - law has an exponential cut - off at high energies , with the preferred cut - off energy lying above 2gev . finally , we note that in interval * b * , a marginally better fit is achieved using a model with the additional power - law component having an exponential cut - off at high energies . the improvement is at the @xmath33 level and indicates weak evidence for a cutoff in the second component , placing a lower limit on the cutoff energy in this interval of about 1gev . the _ fermi _ data for grb 090902b show for the first time clear evidence of excess emission both at low energies ( @xmath34kev ) and at high energies ( @xmath4100mev ) , while the band function alone fits data at intermediate energies adequately . these excesses are well - fit by a single power - law component suggesting a common origin . this power - law component accounts for @xmath35 of the total fluence in the 10 kev10 gev range , and its photon index is hard , with a value @xmath36 throughout most of the prompt phase . such a hard component producing the observed excess at low energies is difficult to explain in the context of leptonic models by the usual synchrotron self - compton ( ssc ) mechanisms . in the simplest versions of these models , the peak of the ssc emission is expected to have a much higher energy than the synchrotron peak at mev energies , and the ssc component has a soft tail that is well below the synchrotron flux at lower energies and so would not produce excess emission below @xmath37 kev . hadronic models , either in the form of proton synchrotron radiation @xcite or photohadronic interactions @xcite , can produce a hard component with a similar low energy excess via direct and cascade radiation ( e.g. , synchrotron emission by secondary pairs at low energies ) . however , the total energy release in hadronic models would exceed the observed gamma - ray energy of @xmath38ergs significantly and may pose a challenge for the total energy budget . collimation into a narrow jet may alleviate the energy requirements , since the actual energy release from grb 090902b can be smaller by a jet beaming factor @xmath39 from the apparent isotropic value , where @xmath40 is the bulk lorentz factor of the fireball . from the observation of a @xmath41 gev photon in interval * c * , the highest energy during the prompt phase and thus the most constraining , we derive a minimum value of the bulk lorentz factor @xmath42 using the flux variability time scale of @xmath43ms found in the bgo data . this limit follows from the constraint that the opacity for @xmath44 pair production with target photons fitted by the band+pl model in interval * c * is less than unity for the 11.16gev photon . this high @xmath45 value is of the same order as the values derived for grb 080916c @xcite and grb 090510 @xcite , both of which have been detected at @xmath46gev with the lat . the delayed onset of the @xmath47100 mev emission from the gbm trigger has been modeled for grb 080916c as arising from proton synchrotron radiation in the prompt phase @xcite and for grb 090510 as arising from electron synchrotron radiation in the early afterglow phase @xcite . in order to produce the peak of the lat emission at @xmath48s in the early afterglow scenario for grb 090902b from deceleration of the grb fireball , a value of @xmath49 is required . this is similar to @xmath45 that we calculate , but the observed large amplitude variability on short time scales ( @xmath50 ms ) in the lat data , which is usually attributed to prompt emission , argues against such models . also , the appearance of the power - law component extending down to @xmath51 kev within only a few seconds of the grb trigger disfavors an afterglow interpretation . the proton synchrotron model , on the other hand , requires a rather large total energy budget , as mentioned previously . yet another interpretation of the observed excess in the high and low energies may be provided by two non - thermal power - law components along with a thermal component from the jet photosphere @xcite . the thermal component , broadened by temperature variations , then accounts for the @xmath52kev few mev emission with @xmath53 @xcite , although fits of such a model to our data do not improve over the band+pl model . furthermore , it is difficult for the photospheric model to explain the delayed onset of the @xmath54100mev emission . the detection of the @xmath55gev photon , @xmath56seconds after the grb trigger and well after the soft gamma - ray emission subsided , may help constrain the origin of the late - time decay of the power - law component , which goes as @xmath2 . a synchrotron origin of the 33.4 gev photon would be difficult since it would require significant energy gain by electrons over a gyroradius and a bulk lorentz factor @xmath41500 . in the case of diffusive shock - acceleration , the energy losses in the upstream region of the shock may dominate ( see , e.g. , * ? ? ? * ) and prevent acceleration of electrons to an energy high enough to radiate a 33.4 gev photon . an interpretation by afterglow ssc emission is still possible , however . the constraints on the quantum gravity mass scale from grb 090902b using the time - of - flight test @xcite are much weaker than those from grb 090510 @xcite due to the larger interval , 82 seconds , between @xmath21 and the arrival time of the 33.4gev photon . however , the moderately high redshift ( @xmath0 ) of grb 090902b allows us to use this photon to probe and constrain models of the extragalactic background light . the 33.4 gev photon would not be absorbed by the ebl in any models except for the `` fast evolution '' and the `` baseline '' models by @xcite , which give optical depths of @xmath57 and 5.8 , respectively . we have performed spectral fits of the lat data with and without the predicted ebl absorption from stecker s models assuming a simple power - law as the intrinsic emission model . based on monte - carlo simulations , we found that stecker s fast evolution and baseline models are disfavored at a @xmath58 level . in summary , grb 090902b is one of the brightest bursts detected by the gbm and lat instruments on _ fermi_. it clearly shows excess emission at high and low energies during the prompt phase , requiring a hard power - law component in addition to the usual band function in order to fit the data . the origin of this component is not understood , and its presence in this burst poses genuine challenges for the theoretical models . like the other two bright _ fermi _ bursts detected by the lat , grb 080916c and grb 090510 , grb 090902b appears to possess a very high lorentz factor for the bulk outflow , @xmath59 , and has some suggestion of a delayed onset of the emission above @xmath1100 mev . finally , the 33.4gev photon , the highest energy yet detected from a grb , and the @xmath0 redshift of this burst have allowed us to place significant constraints on some models of the extragalactic background light . the _ fermi _ lat collaboration acknowledges support from a number of agencies and institutes for both development and the operation of the lat as well as scientific data analysis . these include nasa and doe in the united states , cea / irfu and in2p3/cnrs in france , asi and infn in italy , mext , kek , and jaxa in japan , and the k. a. wallenberg foundation , the swedish research council and the national space board in sweden . additional support from inaf in italy and cnes in france for science analysis during the operations phase is also gratefully acknowledged . , b. l. , catelli , j. r. , & schneid , e. j. 1998 , in american institute of physics conference series , vol . 428 , gamma - ray bursts , 4th hunstville symposium , ed . c. a. meegan , r. d. preece , & t. m. koshut , 349353 100mev and @xmath41gev , respectively . the vertical lines indicate the boundaries of the intervals used for the time - resolved spectral analysis . those time boundaries are at @xmath60 seconds . the insets show the counts for the corresponding dataset binned using these intervals in order to illustrate the relative numbers of counts considered in each spectral fit . ] + s ) . top : counts spectrum ; separate model components are plotted , band ( dashed ) , power - law ( solid ) . bottom : unfolded @xmath61 spectrum . the extension of the @xmath62mev power - law component to the lowest energies ( @xmath63kev ) is shown.,title="fig : " ] cccccccccccc * @xmath64 & 0.030.0 & 726 ( @xmath658 ) & -0.61 ( @xmath650.01 ) & -3.8 ( @xmath66 ) & -1.93(@xmath67 ) & 2562/963 & 2005 & ( 4.59 @xmath65 0.05)@xmath6810@xmath69 + * a. & 0.04.6 & 526 ( @xmath6512 ) & -0.09 ( @xmath650.04 ) & -3.7 ( @xmath70 ) & -1.87(@xmath71 ) & 901/963 & 43 & ( 3.72 @xmath65 0.13)@xmath6810@xmath72 + * b. & 4.6 9.6 & 908 ( @xmath73 ) & 0.07 ( @xmath650.03 ) & -3.9 ( @xmath66 ) & -1.94 ( @xmath650.02 ) & 1250/963 & 3165 & ( 1.44 @xmath65 0.03)@xmath74 + * c. & 9.613.0 & 821 ( @xmath6516 ) & -0.26 ( @xmath650.03 ) & -5.0(@xmath75 ) & -1.98 ( @xmath650.02 ) & 1310/963 & 2109 & ( 9.42 @xmath65 0.24)@xmath76 + * d. & 13.019.2 & 529 ( @xmath659 ) & -0.65 ( @xmath65 - 0.02 ) & -3.2 ( @xmath77 ) & -1.86 ( @xmath65 0.02 ) & 1418/963 & 199 & ( 1.29 @xmath650.03)@xmath74 + * e. & 19.222.7 & 317 ( @xmath658 ) & -0.78 ( @xmath65 - 0.02 ) & -2.4 ( @xmath650.1 ) & @xmath64 & 1117/965 & @xmath64 & ( 4.8 @xmath65 0.2 ) @xmath76 + * f. & 22.725.0 & 236 ( @xmath78 ) & -1.30 ( @xmath79 ) & -2.2 ( @xmath650.1 ) & @xmath64 & 1077/965 & @xmath64 & ( 1.0 @xmath65 0.1)@xmath80 + * e.+f . & 19.225.0 & 327 ( @xmath81 ) & -0.91 ( @xmath650.02 ) & -2.6 ( @xmath82 ) & -1.59 ( @xmath83 ) & 1219/963 & 16 & ( 6.1 @xmath650.4)@xmath80 + * g. & 25.030.0 & @xmath64 & @xmath64 & @xmath64 & -1.93 ( @xmath84 ) & 1209/967 & @xmath64 & ( 6.8 @xmath65 0.8)@xmath85 + * * * * * * * * *
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we report on the observation of the bright , long gamma - ray burst , grb 090902b , by the gamma - ray burst monitor ( gbm ) and large area telescope ( lat ) instruments on - board the _ fermi _ observatory .
this was one of the brightest grbs to have been observed by the lat , which detected several hundred photons during the prompt phase . with a redshift of @xmath0 ,
this burst is among the most luminous detected by _
fermi_. time - resolved spectral analysis reveals a significant power - law component in the lat data that is distinct from the usual band model emission that is seen in the sub - mev energy range .
this power - law component appears to extrapolate from the gev range to the lowest energies and is more intense than the band component both below @xmath150kev and above 100mev .
the band component undergoes substantial spectral evolution over the entire course of the burst , while the photon index of the power - law component remains constant for most of the prompt phase , then hardens significantly towards the end .
after the prompt phase , power - law emission persists in the lat data as late as 1 ks post - trigger , with its flux declining as @xmath2 .
the lat detected a photon with the highest energy so far measured from a grb , @xmath3 gev .
this event arrived 82 seconds after the gbm trigger and @xmath150seconds after the prompt phase emission had ended in the gbm band .
we discuss the implications of these results for models of grb emission and for constraints on models of the extragalactic background light .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the hyperbolic distance has turned out to be a useful tool in geometric function theory . the basic models for the hyperbolic distance are the unit ball model and the upper half space model . using these models in the plane case @xmath0 , we can find the hyperbolic distance in any domain with at least 2 boundary points via the riemann mapping theorem . in higher dimensions @xmath1 , there are no such results we could use to consider the hyperbolic distance in general domains . a solution to this is to use other distance functions , which approximate the hyperbolic distance and are easier to evaluate . we call this kind of distance functions hyperbolic type distances . the study of hyperbolic distances was initiated four decades ago by gehring , palka , martin and osgood @xcite . thereafter many researchers have studied hyperbolic type metrics or used them as a tool in their work , see for example @xcite . in this article we are interested in the growth of hyperbolic type distances in proper subdomains @xmath2 . we consider the growth along a euclidean line segment from @xmath3 to @xmath4 . by a linear transformation we may assume that @xmath5 . to ensure that the line segment @xmath6 is in @xmath7 we restrict our study to starlike domains , which means that for every @xmath8 the euclidean line segment @xmath9 $ ] is contained in @xmath7 . let @xmath10 be a starlike domain and @xmath4 . let @xmath11 and denote @xmath12 . we study the behaviour of the function @xmath13 and the function @xmath14 . ] where @xmath15 is a hyperbolic type distance , see figure [ fig : function f ] . note that now @xmath16 is a continuous mapping from @xmath17 to @xmath18 . to study this function we start of with an easier one @xmath19 all the results of our study are true in a more general setting as long as the line segment @xmath20 is contained in the domain . however , we consider starlike domains as then this condition is clearly fulfilled . our main result is the following schwarz lemma type theorem . for definition of the distances see the section named after the distance . [ thm : main theorem ] let @xmath2 be a domain with @xmath21 and @xmath22 be the function defined in for any @xmath4 . for @xmath23 @xmath24 and for @xmath25 @xmath26 where the upper and lower bounds are best possible , and @xmath27 in this section we introduce notation and consider examples of the hyperbolic distance in the upper half space and a ball . for @xmath29 we denote the closed euclidean line segment between the points by @xmath30 = \ { c \in { { \mathbb{r}^n}}\colon c = x+t(y - x ) , \ , 0 \le t \le 1 \}$ ] . we also use notation @xmath31 , @xmath32 and @xmath33 $ ] for open and half - closed line segments in @xmath34 . we denote the smallest angle between line segments @xmath30 $ ] and @xmath35 $ ] by @xmath36 . we denote euclidean balls and spheres with centre @xmath37 and radius @xmath38 , respectively , by @xmath39 and @xmath40 . a domain @xmath2 , @xmath21 , is said to be starlike , if it is strictly starlike with respect to @xmath41 : for every @xmath4 the line segment @xmath20 is contained in @xmath7 . we say that distance function @xmath42 in @xmath43 is hyperbolic type , if @xmath44 , where @xmath45 is the hyperbolic distance in the unit ball defined in . let @xmath46 . the hyperbolic distance for all @xmath47 is defined as @xmath48 note that we use shifted version of the upper half space , as the usual upper half space @xmath49 does not contain the origin . [ exa : rho in hnb ] let @xmath50 and @xmath11 with @xmath51 . we show that in this case @xmath52 . now for @xmath53 we have @xmath54 and by differentiation we obtain @xmath55 we consider @xmath56 by taking the limit @xmath57 as @xmath58 . @xmath59 let @xmath60 be a ball with @xmath37 and @xmath61 . for @xmath62 the hyperbolic distance is defined by @xmath63 [ exa : rho in b ] let @xmath64 and @xmath65 . we show that in this case @xmath66 . now for @xmath67 @xmath68 and @xmath69 . taking the limit we obtain @xmath70 as @xmath58 . @xmath59 examples [ exa : rho in hnb ] and [ exa : rho in b ] suggest that for hyperbolic type distances @xmath71 could hold for @xmath72 or perhaps even for @xmath73 . it turns out that this conjecture is not true in general . however , based on our study in this article , it seems that the following is true for a hyperbolic type distance @xmath15 there exists constants @xmath72 and @xmath74 such that @xmath75 in this section we study the problem for the distance to the boundary function @xmath77 formulated in . let us begin our study with few simple planar examples , which are easy to reconstruct in higher dimensions . [ exm : simple domain ] we consider starlike domain @xmath78 where @xmath79 with @xmath80 . for a given @xmath81 we derive a formula for @xmath77 and show that @xmath82 and thus linear for large values of @xmath83 . we choose @xmath84 for @xmath85 and note that @xmath4 . of examples [ exm : simple domain ] ( on left ) and [ exm : circular domain ] ( on right).,title="fig : " ] of examples [ exm : simple domain ] ( on left ) and [ exm : circular domain ] ( on right).,title="fig : " ] when @xmath83 is small , then @xmath86 is close to @xmath41 and @xmath87 . to be more explicit we can write @xmath88 where @xmath89 is the point in @xmath90 $ ] , with @xmath91 , see figure [ fig : simple example ] . if @xmath83 is faraway from @xmath41 , the we have @xmath92 where @xmath93 and @xmath94 . next we want to express our function @xmath77 in terms of @xmath83 and points @xmath95 and @xmath96 . we easily obtain @xmath97 , @xmath98 , @xmath99 , @xmath100 and @xmath101 putting all together gives @xmath102 and here @xmath98 . @xmath59 [ exm : circular domain ] let us consider starlike domain @xmath103 and @xmath104 , see figure [ fig : simple example ] . for @xmath11 and @xmath105 we have @xmath106 now @xmath107 implying @xmath108 as @xmath109 . @xmath59 [ exm : polynomial domain ] in example [ exm : circular domain ] we could place the circular arc with any decreasing function : first define a decreasing function @xmath110 \to [ 0,1]$ ] with @xmath111 and @xmath112 . then reflect the function across the real axis and then reflect both function across the imaginary axis . for @xmath113 we can define polynomial function @xmath114 to obtain a domain @xmath7 . now for @xmath104 we have @xmath115 and since @xmath116 we obtain @xmath117 as @xmath109 . @xmath59 let us then consider general case @xmath2 . how quickly and how slowly @xmath77 can decrease for large values of @xmath83 ? how quickly and how slowly @xmath77 can increase for @xmath83 close to @xmath41 ? before considering the bounds for @xmath77 , we introduce angular domain . for @xmath118 and @xmath119 we define @xmath120 if @xmath83 is close to @xmath41 , then the slowest growth for @xmath77 occurs in the following case : @xmath121 for some @xmath122 and @xmath123 , see figure [ fig : domain g2 ] . now for @xmath124 $ ] , we have @xmath125 defined in for the extremal case , when @xmath83 is close to @xmath41 . ] if @xmath83 is faraway from the boundary , then the slowest growth occurs in the case @xmath126 for any @xmath122 and @xmath127 . now @xmath128 for @xmath129 . as we will later see , it turns out that the domains @xmath130 and @xmath131 defined respectively in and can be used as extremal domains for many hyperbolic type distances . note that if @xmath7 was not starlike , we could use @xmath132 and @xmath133 instead of @xmath130 and @xmath131 . next we consider how quickly @xmath77 can decrease . close to the origin the fastest decrement occurs in the domain @xmath134 and it is @xmath135 for @xmath136 . faraway from the origin the decrement can made arbitrarily slow as can be observed from examples [ exm : simple domain ] , [ exm : circular domain ] and [ exm : polynomial domain ] . we have arrived to the following theorem : [ thm : distance to the boundary estimates ] for any starlike domain @xmath7 and point @xmath4 we have @xmath137 moreover , @xmath138 in this section we estimate @xmath140 for the distance ratio distance @xmath139 , which is defined in any open subset @xmath2 for points @xmath141 by @xmath142 the distance ratio distance was introduced by vuorinen in the 1980 s @xcite and in a slightly different form by gehring and osgood @xcite . for @xmath140 we can use the same domains as for @xmath77 to consider the extremal cases . note that the following result is true also in non starlike domains . [ thm : distance ratio estimates ] let @xmath2 be a starlike domain and @xmath4 . for the distance ratio distance @xmath139 we have @xmath143 , @xmath144 and @xmath145 let us first consider lower bound for @xmath146 . we denote a closest boundary @xmath147 with @xmath148 and @xmath4 . now @xmath149 with @xmath51 we have @xmath150 this is obtained for example as in in the domain @xmath151 for some @xmath122 and @xmath123 . let us then consider upper bound for @xmath146 . now as @xmath152 is a constant we may assume @xmath153 and obtain by @xmath154 this situation is obtained in the domain @xmath155 , because for any @xmath7 we have @xmath156 . the upper and lower bounds of @xmath140 give us @xmath157 which yields @xmath143 . note that the lower bound for @xmath140 in theorem [ thm : distance ratio estimates ] works for all @xmath83 whereas the upper bound holds only close to the origin . as a matter of fact , by theorem [ thm : distance to the boundary estimates ] , the function @xmath158 can be arbitrarily close to zero near the boundary and therefore we can not find an upper bound for @xmath140 in this case . next we consider the quasihyperbolic distance . let @xmath2 be a domain . we define the quasihyperbolic length of a curve @xmath160 by @xmath161 for @xmath141 the quasihyperbolic distance between @xmath162 and @xmath163 is define by @xmath164 where the infimum is taken over all rectifiable curves @xmath165 joining @xmath162 and @xmath163 in @xmath7 . the quasihyperbolic distance was introduced in the 1970 s by gehring and palka @xcite . [ thm : quasihyperbolic estimates ] let @xmath2 be a starlike domain and @xmath4 . for the quasihyperbolic distance @xmath159 we have @xmath166 , @xmath167 and @xmath168 we start by finding a lower bound for @xmath169 . let @xmath4 and @xmath11 . for @xmath170 we estimate @xmath171 } \frac{|du|}{d_g(0)+u } = \int_0^t \frac{du}{d_g(0)+u}\\ & = & \log(d_g(0)+t)-\log d_g(0 ) = \log \left ( 1+\frac{t}{d_g(0 ) } \right ) . \end{aligned}\ ] ] for @xmath172 and @xmath173 clearly @xmath174 and we obtain @xmath175 the lower and the upper bounds of @xmath169 are equal to the corresponding bounds for @xmath140 and the expression for @xmath176 follows as in the proof of theorem [ thm : distance ratio estimates ] . exactly same observation for @xmath169 near the boundary can be made that we made for the distance ratio distance . theorem [ thm : quasihyperbolic estimates ] gives a lower bound and an upper bound can not be obtained . for a domain @xmath2 and points @xmath141 we define the triangular ratio distance by @xmath179 geometrically the supremum is attained at a point @xmath180 such that it is either on line segment @xmath181 $ ] or if this is not possible , then @xmath180 is on the largest ellipsoid with focii @xmath162 and @xmath163 contained in @xmath7 . the triangular ratio distance was introduced by hst in the 2000 s @xcite . since @xmath182 $ ] we observe that @xmath177 is not hyperbolic type . for @xmath141 we define @xmath183 to obtain a hyperbolic type distance . as in the case of the distance to the boundary function and the distance ratio distance we can use the same extremal domains . the domain @xmath151 for some @xmath122 and @xmath123 gives lower bound @xmath184 for @xmath129 . the domain @xmath185 for any @xmath122 and @xmath127 gives upper bound @xmath186 for @xmath187 . combining these estimates we obtain : [ thm : sigma distance ] let @xmath2 be a starlike domain and @xmath4 . then @xmath188 for @xmath129 and @xmath189 for @xmath187 . moreover , @xmath190 . the bounds for @xmath191 follow from and . by differentiation we obtain @xmath192 and thus @xmath193 . [ rem : sigma remark ] theorem [ thm : sigma distance ] suggests that an alternative way to define a hyperbolic type distance by using the triangular ratio distance , could be @xmath194 . for this distance function @xmath195 . for a domain @xmath2 and points @xmath141 we define the cassinian distance by @xmath197 the cassinian distance was introduced by ibragimov in the 2000 s @xcite . in the plane case the supremum is attained at a point @xmath180 that is on the largest cassinian oval with focii @xmath162 and @xmath163 contained in @xmath7 . [ thm : cassinian distance ] let @xmath2 be a starlike domain and @xmath4 . then @xmath198 for @xmath129 and @xmath199 for @xmath187 . moreover , @xmath200 . the domain @xmath151 for some @xmath122 and @xmath123 gives lower bound @xmath198 for @xmath129 . the domain @xmath185 for any @xmath122 and @xmath127 gives upper bound @xmath201 for @xmath187 . differentiation gives @xmath202 and clearly @xmath200 . let @xmath204 be a domain such that @xmath205 is not contained in a sphere in @xmath206 . then for @xmath141 we define the apollonian distance ( * ? ? ? * theorem 1.1 ) by @xmath207 the apollonian distance was introduced by barbilian in the 1930 s @xcite and reintroduced by beardon in the 1990 s in connection with the hyperbolic metric @xcite . it is worth pointing out that we can write @xmath208 and in the case @xmath0 each quotient defines an apollonian circle . taking the supremum means that we take the largest possible apollonian circles in @xmath209 . [ lem : apollonian estimate ] let @xmath210 , @xmath38 and @xmath211 . denote @xmath212 . then @xmath213 and for @xmath214 and @xmath215 @xmath216 to simplify notation we may assume that @xmath217 and @xmath218 for @xmath219 . now @xmath220 } \left\ { \frac{|w - re^{i \zeta}|}{r } \right\ } = \frac{|w - re^{i \pi}|}{r } = \frac{r+|w|}{r},\ ] ] because @xmath221 and the function @xmath222 is increasing for @xmath223 $ ] . similarly we obtain @xmath224 } \left\ { \frac{r}{|w - re^{i \zeta}| } \right\ } = \frac{r}{|w - re^{i 0}| } = \frac{r}{r-|w|}.\ ] ] next we note that the function @xmath225 , @xmath226 , is decreasing on @xmath227 , because @xmath228 . the function @xmath229 , @xmath226 , is decreasing on @xmath230 , because @xmath231 . combining monotonicity of @xmath232 with gives @xmath233 and similarly monotonicity of @xmath234 with gives @xmath235 [ thm : f for alpha ] let @xmath236 be a starlike domain and @xmath4 . then @xmath237 for @xmath129 and @xmath238 for @xmath136 . moreover , we have @xmath239 and the upper bound for @xmath240 and the lower bound for @xmath241 are best possible . by we can estimate @xmath242 by estimating @xmath243 and @xmath244 separately . by lemma [ lem : apollonian estimate ] we obtain upper bounds @xmath245 and @xmath246 these inequalities give @xmath247 we can also use lemma [ lem : apollonian estimate ] for lower bound @xmath248 and we can trivially estimate @xmath249 together these two inequalities give us @xmath250 differentiation of gives @xmath251 and differentiation of gives @xmath252 . thus @xmath239 . finally , we give two example domains , which show that the upper bound for @xmath240 and the lower bound for @xmath241 are best possible . the domain @xmath253 for some @xmath122 and @xmath127 shows that the upper bound is best possible . now the suprema in the definition of the apollonian distance are obtained at @xmath254 and @xmath95 , and thus @xmath255 the domain @xmath131 defined in shows that the lower bound for @xmath241 is best possible . we recall that @xmath256 for any @xmath122 and @xmath127 . for @xmath257 with @xmath258 we have @xmath259 and differentiation together with taking the limit gives @xmath260 as @xmath58 . note that in the theorem [ thm : f for alpha ] also the upper bound for @xmath241 is best possible , because the upper bound for @xmath240 is best possible and both upper and lower bounds of @xmath240 tend to @xmath41 as @xmath58 . let @xmath2 be a domain . for @xmath141 we define the seittenranta distance by @xmath262 the seittenranta distance was introduced by seittenranta in the 1990 s ( * ? ? ? * theorem 3.3 ) . before estimating @xmath263 we find general upper and lower bound for @xmath264 . [ lem : upper bound for delta ] let @xmath2 be a domain . then for all @xmath141 we have @xmath265 by the euclidean triangle inequality we obtain @xmath266 and the assertion follows . the following lower bound for @xmath264 is from ( * ? ? ? * theorem 3.11 ) . [ pro : lower bound for delta ] let @xmath10 be a domain and @xmath205 is not contained in a sphere in @xmath206 . then for all @xmath267 we have @xmath268 we also need exact formulas for the seittenranta distance in two starlike domains . [ lem : formula for delta in special domains ] let @xmath131 be the domain defined in for some @xmath122 and define a starlike domain @xmath269 \(1 ) for @xmath270 we have @xmath271 \(2 ) for @xmath270 we have @xmath272 in both cases we have @xmath273 where @xmath274 . the idea of our proof is to first show that for any @xmath275 the supremum over @xmath276 is attained at @xmath277 . using this property we can find supremum over @xmath278 . \(1 ) we denote @xmath279 and @xmath280 . for @xmath281 the angle @xmath282 and by the law of cosines @xmath283 we fix @xmath278 and find the point @xmath89 , @xmath284 , which gives the minimum value for @xmath285 since @xmath286 we have @xmath287 and @xmath288 the denominator of @xmath289 is positive , because @xmath290 , @xmath291 and @xmath292 is equivalent to @xmath293 . the numerator of @xmath289 equals zero , when @xmath294 and the function @xmath295 obtains its minimum either as @xmath296 or as @xmath297 . we have @xmath298 and @xmath299 let us now consider @xmath300 . if @xmath301 , then @xmath302 and the suprema in the definition of the seittenranta metric are obtained at @xmath303 , @xmath304 implying @xmath305 if @xmath306 , then @xmath307 and the suprema in the definition of the seittenranta metric are obtained at @xmath308 , @xmath309 implying @xmath305 \(2 ) it is easy to see that @xmath310 consists of @xmath311 and a line @xmath312 . if both @xmath313 , then the assertion follows from ( 1 ) . we assume @xmath314 and denote @xmath315 . for any @xmath316 with @xmath284 we may assume that @xmath317 is maximal . this immediately implies that @xmath318 . we denote @xmath319 . now for @xmath320 we have by the pythagorean theorem @xmath321 and we want to find maximum of @xmath322 we show that @xmath323 is a decreasing function . differentiation gives @xmath324 and the numerator of @xmath325 equals zero whenever @xmath326 for @xmath327 if we choose minus in @xmath328 then clearly @xmath329 . if we choose plus in @xmath328 , then the equation @xmath330 has solution @xmath331 and @xmath332 . now @xmath333 is either positive or negative for @xmath334 . we estimate @xmath335 we have obtained @xmath329 and since @xmath336 we have @xmath337 for all @xmath338 . now we are ready to consider @xmath339 . for any fixed @xmath340 , the largest value for @xmath341 is obtained for @xmath303 . thus the suprema in the definition of the seittenranta metric are obtained at @xmath303 , @xmath342 implying @xmath343 where the second equality is obtained by the pythagorean theorem as @xmath344 implying @xmath345 and @xmath346 . now we are ready to find bounds for @xmath263 and @xmath347 . [ thm : seittenranta distance ] let @xmath2 be a starlike domain . then @xmath348 for @xmath129 and @xmath349 for @xmath136 . moreover , @xmath350 and the bounds for @xmath347 are best possible . lower bound for @xmath263 and @xmath347 follow from proposition [ pro : lower bound for delta ] and theorem [ thm : f for alpha ] . for the upper bound we use lemma [ lem : upper bound for delta ] to obtain @xmath351 for @xmath136 . differentiation and taking the limit gives @xmath352 as @xmath58 , which proves the upper bound for @xmath347 . finally , we show that the bounds for @xmath347 are best possible . sharpness of the lower bound occurs in the domain @xmath131 defined in . by lemma [ lem : formula for delta in special domains ] ( 1 ) @xmath353 and by differentiation we obtain @xmath252 . now as @xmath58 we get the @xmath354 . sharpness of the upper bound occurs in the domain @xmath355 defined in lemma [ lem : formula for delta in special domains ] . now by lemma [ lem : formula for delta in special domains ] ( 2 ) @xmath356 by differentiation and taking the limit we obtain @xmath357 as @xmath58 . let @xmath2 be a domain . for @xmath141 we define the visual angle distance by @xmath360 the visual angle distance was introduced in the 2010 s in @xcite . clearly @xmath361 $ ] and therefore we define for @xmath141 a hyperbolic type distance @xmath362 [ pro : increasing visual angle distance ] let @xmath363 be a line and @xmath364 . then the function @xmath365 is strictly increasing for @xmath366 . now @xmath367 and the point @xmath368 moves along the line @xmath369 . it is easy to see that @xmath370 , @xmath371 and @xmath372 is strictly increasing as a function of @xmath373 . by the law of sines @xmath374 and since @xmath375 is constant we obtain for @xmath376 @xmath377 now @xmath378 is a constant and @xmath372 is strictly increasing . thus we need to show that for @xmath379 the function @xmath380 is strictly increasing in @xmath381 . by differentiation we obtain @xmath382 and the assertion follows . in proposition [ pro : increasing visual angle distance ] the line @xmath369 did not contain 0 . if @xmath383 with @xmath384 , then for any @xmath385 we have @xmath386,\\ \pi , & \textrm{if } 0 \in [ u , w].\\ \end{array } \right.\ ] ] [ thm : tau distance ] let @xmath2 be a starlike domain . then @xmath387 for all @xmath129 and @xmath388 for all @xmath136 . moreover , the bounds for @xmath389 are best possible and @xmath390 . by we note that @xmath391 for all @xmath129 and by proposition [ pro : increasing visual angle distance ] we obtain @xmath392 for all @xmath129 . the lower bound @xmath41 is obtained in the domain @xmath131 introduced in . the upper bound for @xmath389 is attained in @xmath393 . by * ( 3.3 ) ) , @xmath394 and thus @xmath395 by differentiation and taking the limit we obtain @xmath396 as @xmath58 . [ rem : tau remark ] as in remark [ rem : sigma remark ] , theorem [ thm : tau distance ] suggests that we could define @xmath397 . for this distance function @xmath398 . in this final section we prove our main result and consider the estimates in domains other than starlike . until now we have considered starlike domains and found estimates for the function @xmath22 . moreover , the upper bound for @xmath22 is obtained only when @xmath83 is close to the origin ( @xmath136 ) , whereas the lower bound is valid also for large values ( @xmath129 ) . let now @xmath2 be any domain with @xmath21 and @xmath4 . our results hold also in @xmath7 , when @xmath399 . in other words , the lower and upper bounds for @xmath22 are true for @xmath136 and the results for @xmath400 are also true . we initially choose @xmath41 to simplify notation . it could be any point in @xmath7 . the result for @xmath139 follows from theorem [ thm : distance ratio estimates ] , for @xmath159 from theorem [ thm : quasihyperbolic estimates ] , for @xmath401 form theorem [ thm : sigma distance ] and remark [ rem : sigma remark ] , and for @xmath196 from [ thm : cassinian distance ] . in theorem [ thm : main theorem ] we assumed @xmath4 . this is not needed , if we only consider @xmath295 close to the origin . we can define for any @xmath403 @xmath404 as an application of theorem [ thm : main theorem ] we obtain if we restrict to starlike john domains , then we can easily get lower bound for @xmath22 . a domain @xmath7 is @xmath408-john domain , @xmath409 , if there is a distinguished point @xmath410 such that any @xmath411 can be connected to @xmath412 by a rectifiable curve @xmath413 \to g$ ] , which is parametrised by arclength and with @xmath414 , @xmath415 and @xmath416 for every @xmath417 $ ] . we can choose @xmath418 and it is easy to see that for example @xmath419 . we consider next monotonicity of the function @xmath22 . by * theorem 4.8 ) the metric balls @xmath420 are starlike whenever @xmath7 is starlike . this implies that in starlike domains @xmath140 is an increasing function . same is also true for the quasihyperbolic distance ( * ? ? ? * theorem 2.10 ) , the apollonian distance ( * ? ? ? * theorem 3.5 ) , the triangular ratio distance ( * ? ? ? * theorem 1.2 ) and the visual angle distance ( by definition ) . for the seittenranta distance this is not know ( * ? ? ? * open problem 4.10 ( 1 ) ) . monotonicity is not true for the function @xmath77 . however , in convex domains @xmath77 behaves well : there exists a point @xmath421 such that @xmath77 is increasing on @xmath422 $ ] and decreasing on @xmath423 . the next example shows that @xmath77 has not this property in starlike domains . [ exm : comb ] we show that @xmath77 misbehaves in the domain @xmath424 \right ) , \quad a_l = ( 1 - 2^{-l})e_1 + 2^{-(l+1 ) } e_2,\ ] ] for @xmath104 . @xmath7 is not strictly starlike , but by replacing line segments @xmath425 $ ] with angular domains @xmath426 , where @xmath427 is small enough , we could construct a strictly starlike domain with the same effect . we stick to the line segment version to make the computation easier to follow . we demonstrate that on each interval @xmath428 $ ] the function @xmath77 obtains its maximum in @xmath429 and @xmath430 . this means that @xmath77 has infinitely many local maxima and minima . of example [ exm : comb ] . right : function @xmath77 in example [ exm : comb].,title="fig:",scaledwidth=30.0% ] of example [ exm : comb ] . right : function @xmath77 in example [ exm : comb].,title="fig:",scaledwidth=50.0% ] let us fix @xmath431 . now @xmath432 and thus @xmath433 . we show that for @xmath434 , @xmath435 by construction of @xmath7 it is clear that @xmath436),|b_le_1-a_{l+1}| \}.\ ] ] by geometry we have @xmath437 ) \ge 2^{-(l+1 ) } = g(1 - 2^{-l}).\ ] ] we calculate @xmath438 and @xmath439 implying @xmath440 inequalities and imply and thus @xmath77 misbehaves ( it has infinitely many local maxima and minima ) . @xmath59
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we study the growth of hyperbolic type distances in starlike domains .
we derive estimates for various hyperbolic type distances and consider the asymptotic sharpness of the estimates .
@equation=@theorem keywords : hyperbolic type distance , starlike domain msc2010 : 30f45 , 51m10 , 30c65
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the goal of the development of the code was to have a simple and efficient tool for the computation of adiabatic oscillation frequencies and eigenfunctions for general stellar models , emphasizing also the accuracy of the results . not surprisingly , given the long development period , the simplicity is now less evident . however , the code offers considerable flexibility in the choice of integration method as well as ability to determine all frequencies of a given model , in a given range of degree and frequency . the choice of variables describing the equilibrium model and oscillations was to a large extent inspired by @xcite . as discussed in section [ sec : eqmodel ] the equilibrium model is defined in terms of a minimal set of dimensionless variables , as well as by mass and radius of the model . fairly extensive documentation of the code , on which the present paper in part is based , is provided with the distribution packagejcd / adipack.n ] . @xcite provided an extensive review of adiabatic stellar oscillations , emphasizing applications to helioseismology , and discussed many aspects and tests of the aarhus package , whereas @xcite carried out careful tests and comparisons of results on polytropic models ; this includes extensive tables of frequencies which can be used for comparison with other codes . the equilibrium model is defined in terms of the following dimensionless variables : @xmath0 here @xmath1 is distance to the centre , @xmath2 is the mass interior to @xmath1 , @xmath3 is the photospheric radius of the model and @xmath4 is its mass ; also , @xmath5 is the gravitational constant , @xmath6 is pressure , @xmath7 is density , and @xmath8 , the derivative being at constant specific entropy . in addition , the model file defines @xmath4 and @xmath3 , as well as central pressure and density , in dimensional units , and scaled second derivatives of @xmath6 and @xmath7 at the centre ( required from the expansions in the central boundary condition ) ; finally , for models with vanishing surface pressure , assuming a polytropic relation between @xmath6 and @xmath7 in the near - surface region , the polytropic index is specified . the following relations between the variables defined here and more `` physical '' variables are often useful : @xmath9 we may also express the characteristic frequencies for adiabatic oscillations in terms of these variables . thus if @xmath10 is the buoyancy frequency , @xmath11 is the lamb frequency at degree @xmath12 and @xmath13 is the acoustical cut - off frequency for an isothermal atmosphere , we have @xmath14 where @xmath15 is the adiabatic sound speed , and @xmath16 is the pressure scale height , @xmath17 being the gravitational acceleration . finally it may be noted that the squared sound speed is given by @xmath18 these equations also define the dimensionless characteristic frequencies @xmath19 , @xmath20 and @xmath21 as well as the dimensionless sound speed @xmath22 , which are often useful . as is well known the displacement vector of nonradial ( spheroidal ) modes can be written in terms of polar coordinates @xmath23 as @xmath24 \exp ( - { { \rm i}}\omega t ) \right\ } \ ; . \nonumber\end{aligned}\ ] ] here @xmath25 is a spherical harmonic of degree @xmath12 and azimuthal order @xmath2 , @xmath26 being co - latitude and @xmath27 longitude ; @xmath28 is an associated legendre function , and @xmath29 is a suitable normalization constant . also , @xmath30 , @xmath31 , and @xmath32 are unit vectors in the @xmath1 , @xmath26 , and @xmath27 directions . finally , @xmath33 is time and @xmath34 is the angular frequency of the mode . similarly , e.g. , the eulerian perturbation to pressure may be written @xmath35 \ ; . \label{eq : e2.2}\ ] ] as the oscillations are adiabatic ( and only conservative boundary conditions are considered ) @xmath34 is real , and the amplitude functions @xmath36 , @xmath37 , @xmath38 , etc . can be chosen to be real . the equations of adiabatic stellar oscillations , in the nonradial case , are expressed in terms of the following variables : , @xmath39 results from the earlier use of an unconventional sign convention for @xmath40 ; now , as usual , @xmath40 is defined such that the perturbed poisson equation has the form @xmath41 , where @xmath42 is the eulerian density perturbation . ] @xmath43 here @xmath40 is the perturbation to the gravitational potential . also , we introduce the dimensionless frequency @xmath44 by @xmath45 corresponding to eqs [ eq : buoy ] [ eq : cutoff ] . these quantities satisfy the following equations : @xmath46 y_1 + ( a - 1 ) y_2 + \eta a y_3 \ ; , \\ \label{eq : ea.3 } x { { { \rm d}}y_3 \over { { \rm d}}x } & = & y_3 + y_4 \ ; , \\ \label{eq : ea.4 } x { { { \rm d}}y_4 \over { { \rm d}}x } & = & - a u y_1 - u { v_g \over \eta } y_2 \\ & & + [ l ( l + 1 ) + u(a - 2 ) + u v_g ] y_3 + 2(1 - u ) y_4 \ ; . \nonumber\end{aligned}\ ] ] here @xmath47 , and the notation is otherwise as defined in eq . [ eq : fivea ] . in the @xcite approximation , where the perturbation to the gravitational potential is neglected , the terms in @xmath48 are neglected in eqs [ eq : ea.1 ] and [ eq : ea.2 ] and eqs [ eq : ea.3 ] and [ eq : ea.4 ] are not used . the dependent variables @xmath49 in the nonradial case have been chosen in such a way that for @xmath50 they all vary as @xmath51 for @xmath52 . for large @xmath12 a considerable ( and fundamentally unnecessary ) computational effort would be needed to represent this variation sufficiently accurately with , e.g. , a finite difference technique , if these variables were to be used in the numerical integration . instead i introduce a new set of dependent variables by @xmath53 these variables are then @xmath54 in @xmath55 near the centre . they are used in the region where the variation in the @xmath56 is dominated by the @xmath51 behaviour , for @xmath57 , say , where @xmath58 is determined on the basis of the asymptotic properties of the solution . this transformation permits calculating modes of arbitrarily high degree in a complete model . for radial oscillations only @xmath59 and @xmath60 are used , where @xmath59 is defined as above , and @xmath61 here the equations become @xmath62 y_1 + a y_2 \ ; . \label{eq : ea.6}\end{aligned}\ ] ] the equations are solved on the interval @xmath63 $ ] in @xmath55 . here , in the most common case involving a complete stellar model @xmath64 , where @xmath65 is a suitably small number such that the series expansion around @xmath66 is sufficiently accurate ; however , the code can also deal with envelope models with arbitrary @xmath67 , typically imposing @xmath68 at the bottom of the envelope . the outermost point is defined by @xmath69 where @xmath70 is the surface radius , including the atmosphere ; thus , typically , @xmath71 . the centre of the star , @xmath72 , is obviously a singular point of the equations . as discussed , e.g. , by @xcite boundary conditions at this point are obtained from a series expansion , in the present code to second significant order . in the general case this defines two conditions at the innermost non - zero point in the model . for radial oscillations , or nonradial oscillations in the cowling approximation , one condition is obtained . the surface in a realistic model is typically defined at a suitable point in the stellar atmosphere , with non - zero pressure and density . here the simple condition of vanishing lagrangian pressure perturbation is implemented and sometimes used . however , more commonly a condition between pressure perturbation and displacement is established by matching continuously to the solution in an isothermal atmosphere extending continuously from the uppermost point in the model . a very similar condition was presented by @xcite . in addition , in the full nonradial case a condition is obtained from the continuous match of @xmath40 and its derivative to the vacuum solution outside the star . in full polytropic models , or other models with vanishing surface pressure , the surface is also a singular point . in this case a boundary condition at the outermost non - singular point is obtained from a series expansion , assuming a near - surface polytropic behaviour ( see * ? ? ? * for details ) . the code also has the option of considering truncated ( e.g. , envelope ) models although at the moment only in the cowling approximation or for radial oscillations . in this case the innermost boundary condition is typically the vanishing of the radial displacement @xmath73 although other options are available . the numerical problem can be formulated generally as that of solving @xmath74 with the boundary conditions @xmath75 @xmath76 here the order @xmath77 of the system is 4 for the full nonradial case , and 2 for radial oscillations or nonradial oscillations in the cowling approximation . this system only allows non - trivial solutions for selected values of @xmath78 which is thus an eigenvalue of the problem . the programme permits solving these equations with two basically different techniques , each with some variants . the first is a shooting method , where solutions satisfying the boundary conditions are integrated separately from the inner and outer boundary , and the eigenvalue is found by matching these solutions at a suitable inner fitting point @xmath79 . the second technique is to solve the equations together with a normalization condition and all boundary conditions using a relaxation technique ; the eigenvalue is then found by requiring continuity of one of the eigenfunctions at an interior matching point . for simplicity i do not distinguish between @xmath80 and @xmath56 ( cf . section [ sec : eq ] ) in this section . it is implicitly understood that the dependent variable ( which is denoted @xmath56 ) is @xmath80 for @xmath57 and @xmath56 for @xmath81 . the numerical treatment of the transition between @xmath80 and @xmath56 has required a little care in the coding . it is convenient here to distinguish between @xmath77 = 2 and @xmath77 = 4 . for @xmath77 = 2 the differential eqs [ eq : e3.1 ] have a unique ( apart from normalization ) solution @xmath82 satisfying the inner boundary conditions [ eq : e3.2 ] , and a unique solution @xmath83 satisfying the outer boundary conditions [ eq : e3.3 ] . these are obtained by numerical integration of the equations . the final solution can then be represented as @xmath84 . the eigenvalue is obtained by requiring that the solutions agree at a suitable matching point @xmath85 , say . thus @xmath86 these equations clearly have a non - trivial solution @xmath87 only when their determinant vanishes , i.e. , when @xmath88 equation [ eq : e3.5 ] is therefore the eigenvalue equation . for @xmath77 = 4 there are two linearly independent solutions satisfying the inner boundary conditions , and two linearly independent solutions satisfying the outer boundary conditions . the former set @xmath89 is chosen by setting @xmath90 and the latter set @xmath91 is chosen by setting @xmath92 the inner and outer boundary conditions are such that , given @xmath59 and @xmath48 , @xmath60 and @xmath39 may be calculated from them ; thus eqs [ eq : e3.6 ] and [ eq : e3.7 ] completely specify the solutions , which are obtained by integrating from the inner or outer boundary . the final solution can then be represented as @xmath93 at the fitting point @xmath79 continuity of the solution requires that @xmath94 this set of equations only has a non - trivial solution if @xmath95 where , e.g. , @xmath96 . thus eq . [ eq : e3.9 ] is the eigenvalue equation in this case . clearly @xmath97 as defined in either eq . [ eq : e3.5 ] or eq . [ eq : e3.9 ] is a smooth function of @xmath78 , and the eigenfrequencies are found as the zeros of this function . this is done in the programme using a standard secant technique . however , the programme also has the option for scanning through a given interval in @xmath78 to look for changes of sign of @xmath97 , possibly iterating for the eigenfrequency at each change of sign . thus it is possible to search a given region of the spectrum completely automatically . the programme allows the use of two different techniques for solving the differential equations . one is the standard second - order centred difference technique , where the differential equations are replaced by the difference equations @xmath98 , \quad i = 1 , \ldots , i \ ; . \label{eq : e3.11}\ ] ] here i have introduced a mesh @xmath99 in @xmath55 , where @xmath10 is the total number of mesh points ; @xmath100 , and @xmath101 . these equations allow the solution at @xmath102 to be determined from the solution at @xmath103 . the second technique was proposed by @xcite ; here on each mesh interval @xmath104 we consider the equations @xmath105 with constant coefficients , where @xmath106 . these equations may be solved analytically on the mesh intervals , and the complete solution is obtained by continuous matching at the mesh points . this technique clearly permits the computation of solutions varying arbitrarily rapidly , i.e. , the calculation of modes of arbitrarily high order . on the other hand solving eqs [ eq : e3.12 ] involves finding the eigenvalues and eigenvectors of the coefficient matrix , and therefore becomes very complex and time consuming for higher - order systems . thus in practice it has only been implemented for systems of order 2 , i.e. , radial oscillations or nonradial oscillations in the cowling approximation . if one of the boundary conditions is dropped , the difference equations , with the remaining boundary condition and a normalization condition , constitute a set of linear equations for the @xmath107 which can be solved for any value of @xmath44 ; this set may be solved efficiently by forward elimination and backsubstitution ( e.g. , * ? ? ? * ) , with a procedure very similar to the so - called henyey technique ( e.g. , * ? ? * see also christensen - dalsgaard 2007 ) used in stellar modelling . the eigenvalue is then found by requiring that the remaining boundary condition , which effectively takes the role of @xmath108 , be satisfied . however , as both boundaries , at least in a complete model , are either singular or very nearly singular , the removal of one of the boundary conditions tends to produce solutions that are somewhat ill - behaved , in particular for modes of high degree . this in turn is reflected in the behaviour of @xmath97 as a function of @xmath44 . this problem is avoided in a variant of the relaxation technique where the difference equations are solved separately for @xmath109 and @xmath110 , by introducing a double point @xmath111 in the mesh . the solution is furthermore required to satisfy the boundary conditions [ eq : e3.2 ] and [ eq : e3.3 ] , a suitable normalization condition ( e.g. @xmath112 ) , and continuity of all but one of the variables at @xmath85 , e.g. , @xmath113 ( when @xmath77 = 2 clearly only the first continuity condition is used ) we then set @xmath114 and the eigenvalues are found as the zeros of @xmath97 , regarded as a function of @xmath78 . with this definition , @xmath97 may have singularities with discontinuous sign changes that are not associated with an eigenvalue , and hence a little care is required in the search for eigenvalues . however , close to an eigenvalue @xmath97 is generally well - behaved , and the secant iteration may be used without problems . as implemented here the shooting technique is considerably faster than the relaxation technique , and so should be used whenever possible ( notice that both techniques may use the difference eqs [ eq : e3.11 ] and so they are numerically equivalent , in regions of the spectrum where they both work ) . for _ second - order systems _ the shooting technique can probably always be used ; the integrations of the inner and outer solutions should cause no problems , and the matching determinant in eq . [ eq : e3.5 ] is well - behaved . for _ fourth - order systems _ , however , this needs not be the case . for modes where the perturbation to the gravitational potential has little effect on the solution , the two solutions @xmath115 and @xmath116 , and similarly the two solutions @xmath117 and @xmath118 , are almost linearly dependent , and so the matching determinant nearly vanishes for any value of @xmath78 . this is therefore the situation where the relaxation technique may be used with advantage . this applies , in particular , to the calculation of modes of moderate and high degree which are essential to helioseismology . to make full use of the increasingly accurate observed frequencies the computed frequencies should clearly at the very least match the observational accuracy , for a given model . only in this way do the frequencies provide a faithful representation of the properties of the model , in comparisons with the observations . however , since the numerical errors in the computed frequencies are typically highly systematic , they may affect the asteroseismic inferences even if they are smaller than the random errors in the observations , and hence more stringent requirements should be imposed on the computations . also , the fact that solar - like oscillations , and several other types of asteroseismically interesting modes , tend to be of high radial order complicates reaching the required precision . the numerical techniques discussed so far are generally of second order . this yields insufficient precision in the evaluation of the eigenfrequencies , unless a very dense mesh is used in the computation ( see also * ? ? ? the code may apply two techniques to improve the precision . one technique ( cf . * ) uses the fact that the frequency approximately satisfies a variational principle @xcite . the variational expression may formally be written as @xmath119 where @xmath120 and @xmath121 are integrals over the equilibrium model depending on the eigenfunction , here represented by @xmath122 . the variational property implies that any error @xmath123 in @xmath122 induces an error in @xmath124 that is @xmath125 . thus by substituting the computed eigenfunction into the variational expression a more precise determination of @xmath78 should result . this has indeed been confirmed @xcite . the second technique uses explicitly that the difference scheme [ eq : e3.11 ] , which is used by one version of the shooting technique , and the relaxation technique , is of second order . consequently the truncation errors in the eigenfrequency and eigenfunction scale as @xmath126 . if @xmath127 and @xmath128 are the eigenfrequencies obtained from solutions with @xmath129 and @xmath10 meshpoints , the leading - order error term therefore cancels in @xmath130 \ ; . \label{eq : e3.18}\ ] ] this procedure , known as _ richardson extrapolation _ , was used by @xcite . it provides an estimate of the eigenfrequency that is substantially more accurate than @xmath131 , although of course at some added computational expense . indeed , since the error in the representation [ eq : e3.11 ] depends only on even powers of @xmath132 , the leading term of the error in @xmath133 is @xmath134 . even with these techniques the precision of the computed frequencies may be inadequate if the mesh used in stellar - evolution calculations is used also for the computation of the oscillations . the number of meshpoints is typically relatively modest and the distribution may not reflect the requirement to resolve properly the eigenfunctions of the modes . @xcite discussed techniques to redistribute the mesh in a way that takes into account the asymptotic behaviour of the eigenfunctions ; a code to do so , based on four - point lagrangian interpolation , is included in the adipls distribution package . on the other hand , for computing low - order modes ( as are typically relevant for , say , @xmath135 scuti or @xmath136 cephei stars ) , the original mesh of the evolution calculation may be adequate . it is difficult to provide general recommendations concerning the required number of points or the need for redistribution , since this depends strongly on the types of modes and the properties of the stellar model . it is recommended to carry out experiments varying the number and distribution of points to obtain estimates of the intrinsic precision of the computation ( e.g. , * ? ? ? * ; * ? ? ? in the latter case , considering simple polytropic models , it was found that 4801 points yielded a relative precision substantially better than @xmath137 for high - order p modes , when richardson extrapolation was used . in the discussion of the frequency calculation it is important to distinguish between _ precision _ and _ accuracy _ , the latter obviously referring to the extent to which the computed frequencies represent what might be considered the ` true ' frequencies of the model . in particular , the manipulations required to derive eq . [ eq : varprinc ] and to demonstrate its variational property depend on the equation of hydrostatic support being satisfied . if this is not the case , as might well happen in an insufficiently careful stellar model calculation , the value determined from the variational principle may be quite precise , in the sense of numerically stable , but still unacceptably far from the correct value . indeed , a comparison between @xmath138 and @xmath133 provides some measure of the reliability of the computed frequencies ( e.g. * ? ? ? the programme finds the order of the mode according to the definition developed by @xcite and @xcite , based on earlier work by @xcite . specifically , the order is defined by @xmath139 here the sum is over the zeros @xmath140 in @xmath59 ( excluding the centre ) , and @xmath141 is the sign function , @xmath142 if @xmath143 and @xmath144 if @xmath145 . for a complete model that includes the centre @xmath146 for radial oscillations and @xmath147 for nonradial oscillations . thus the lowest - order radial oscillation has order @xmath148 . although this is contrary to the commonly used convention of assigning order 0 to the fundamental radial oscillation , the convention used here is in fact the more reasonable , in the sense that it ensures that @xmath149 is invariant under a continuous variation of @xmath12 from 0 to 1 . with this definition @xmath150 for p modes , @xmath151 for f modes , and @xmath152 for g modes , at least in simple models . it has been found that this procedure has serious problems for dipolar modes in centrally condensed models ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? the eigenfunctions @xmath59 are shifted such that nodes disappear or otherwise provide spurious results when eq . [ eq : e4.1 ] is used to determine the mode order . a procedure that does not suffer from this difficulty has recently been developed by @xcite ; i discuss it further in section [ sec : develop ] . a powerful measure of the characteristics of a mode is provided by the _ normalized inertia_. the code calculates this as @xmath153 \rho r^2 { { \rm d}}r \over m [ \xi_r ( r_{\rm phot } ) ^2 + l(l+1 ) { \xi_{\rm h}}(r_{\rm phot } ) ^2 ] } \nonumber \\ & = & { \int_{x_1}^{{x_{\rm s } } } \left [ y_1 ^ 2 + y_2 ^ 2 / l ( l + 1 ) \right ] q u { { \rm d}}x / x \over 4 \pi [ y_1 ( x_{\rm phot } ) ^2 + y_2 ( x_{\rm phot } ) ^2/l(l+1 ) ] } \ ; .\end{aligned}\ ] ] ( for radial modes the terms in @xmath60 are not included . ) here @xmath154 and @xmath155 are the distance of the innermost mesh point from the centre and the surface radius , respectively , and @xmath156 is the fractional photospheric radius . the normalization at the photosphere is to some extent arbitrary , of course , but reflects the fact that many radial - velocity observations use lines formed relatively deep in the atmosphere . a more common definition of the inertia is @xmath157 where @xmath158 is the so - called _ mode mass_. the code has the option to output the eigenfunctions , in the form of @xmath159 . in addition ( or instead ) the displacement eigenfunctions can be output in a form indicating the region where the mode predominantly resides , in an energetical sense , as @xmath160 ( for radial modes only @xmath161 is found ) . these are defined in such a way that @xmath162 { { \rm d}}x / x \over 4 \pi [ y_1 ( x_{\rm phot } ) ^2 + y_2 ( x_{\rm phot } ) ^2/l(l+1 ) ] } \ ; . \label{eq : e4.4}\ ] ] the form provided by the @xmath163 is also convenient , e.g. , for computing rotational splittings @xmath164 ( e.g. , * ? ? ? * ) , where @xmath165 is the frequency of a mode of radial order @xmath149 , degree @xmath12 and azimuthal order @xmath2 . for slow rotation the splittings are obtained from first - order perturbation analysis as @xmath166 characterized by _ kernels _ @xmath167 , where in general the angular velocity @xmath168 depends on both @xmath1 and @xmath26 . the code has built in the option to compute kernels for first - order rotational splitting in the special case where @xmath168 depends only on @xmath1 . several revisions of the code have been implemented in preliminary form or are under development . a substantial improvement in the numerical solution of the oscillation equations , particularly for high - order modes , is the installation of a fourth - order integration scheme , based on the algorithm of @xcite . this is essentially operational but has so far not been carefully tested . comparisons with the results of the variational expression and the use of richardson extrapolation , of the same formal order , will be particularly interesting . as discussed by @xcite the use of @xmath169 ( or , as here , @xmath170 ) as one of the integration variables has the disadvantage that the quantity @xmath171 enters into the oscillation equations . in models with a density discontinuity , such as results if the model has a growing convective core and diffusion is neglected , @xmath171 has a delta - function singularity at the point of the discontinuity . in the adipls calculations this is dealt with by replacing the discontinuity by a very steep and well - resolved slope . however , it would obviously be an advantage to avoid this problem altogether . this can be achieved by using instead the lagrangian pressure perturbation @xmath172 as one of the variables . implementing this option would be a relatively straightforward modification to the code and is under consideration . the proper classification of dipolar modes of low order in centrally condensed models has been a long - standing problem in the theory of stellar pulsations , as discussed in section [ sec : results ] . such a scheme must provide a unique order for each mode , such that the order is invariant under continuous changes of the equilibrium model , e.g. , as a result of stellar evolution . as a major breakthrough , takata in a series of papers has elucidated important properties of these modes and defined a new classification scheme satisfying this requirement @xcite . a preliminary version of this scheme has been implemented and tested ; however , the latest and most convenient form of the takata classification still needs to be installed . a version of the code has been established which computes the first - order rotational splitting for a given rotation profile @xmath173 , in addition to setting up the corresponding kernels . this is being extended by k. burke , sheffield , to cover also second - order effects of rotation , based on the formalism of @xcite . an important motivation for this is the integration , discussed by @xcite , of the pulsation calculation with the astec evolution code to allow full calculation of oscillation frequencies for a model of specified parameters ( mass , age , initial rotation rate , etc . ) as the result of a single subroutine call . i am very grateful to w. dziembowski and d. o. gough for illuminating discussions of the properties of stellar oscillations , and to a. moya and m. j. p. f. g. monteiro for organizing the comparisons of stellar oscillation and model calculations within the esta collaboration . i thank the referee for useful comments which , i hope , have helped improving the presentation . this project is being supported by the danish natural science research council and by the european helio- and asteroseismology network ( helas ) , a major international collaboration funded by the european commission s sixth framework programme . christensen - dalsgaard , j. , berthomieu , g. : theory of solar oscillations . in : cox , a. n. , livingston , w. c. , matthews , m. ( eds ) , _ solar interior and atmosphere _ , p. 401 space science series , university of arizona press ( 1991 ) moya , a. , christensen - dalsgaard , j. , charpinet , s. , lebreton , y. , miglio , a. , montalbn , j. , monteiro , m. j. p. f. g. , provost , j. , roxburgh , i. , scuflaire , r. , surez , j. c. , suran , m. : inter - comparison of the g- , f- and p - modes calculated using different oscillation codes for a given stellar model . apss , this volume ( 2007 )
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development of the aarhus adiabatic pulsation code started around 1978 .
although the main features have been stable for more than a decade , development of the code is continuing , concerning numerical properties and output .
the code has been provided as a generally available package and has seen substantial use at a number of installations .
further development of the package , including bringing the documentation closer to being up to date , is planned as part of the helas coordination action .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
in 1893 and 1897 lodge argued that a device using electromagnetic fields should in principle be able to detect angular acceleration by the interference of light . sixteen years later sagnac @xcite observed such an effect and realised the potential for light interferometers as accelerometers . the continued development of such instruments and their careful exploitation by michelson and others was pivotal in the establishment of our current world view of classical physics . sagnac recognised that his observations could open up new avenues in the development of accelerometers , an insight that has been dramatically verified . today the ring - laser is a cornerstone of most inertial guidance systems and optoelectronics is the backbone of the communications industry ( see @xcite for historical accounts of ring - laser development ) . the basic principles leading to the sagnac effect are well understood in broad outline . as with all interferometers , a guided wave phenomenon ( such as that experienced by electromagnetic fields in a wave guide ) is used to distinguish different propagation paths in spacetime . the traditional electromagnetic `` passive '' type of sagnac interferometer involves splitting a light beam into two coherent components , forcing them to travel different paths in spacetime to an event where they combine to produce an interference pattern . the nature of the interference fringes can then be correlated with the difference in the optical paths . in such a device the frequency of the light employed is controlled externally and the circuits in spacetime are often determined by mirrors or continuous dielectric fibres . with the discovery of the laser it was found that the electromagnetic modes of a closed tubular cavity containing an active lasing medium ( excited initially by an external rf field ) could be made to produce an interference pattern that varied with the state of rotation of the whole apparatus . such an instrument has been labelled an `` active '' device to distinguish it from the traditional passive sagnac interferometer . certain modes of a non - rotating cavity can be made to form standing waves . however , the co- and counter - propagating modes in a rotating lasing cavity acquire different resonant frequencies leading to a beat - frequency as observed by an on - board probe . in 1963 sperry - rand recognised that the active ring - laser could be used as an inertial - guidance device . today such miniturised interferometers are used routinely in civil and military applications . considerable increase in sensitivity can be achieved using larger ring - lasers . insulated from thermal and seismic noise , sensitivities of @xmath0 are routinely obtained in the ug1 sagnac interferometer housed in the cashmere cavern in christchurch , new zealand . with an effective area of @xmath1 and perimeter @xmath2 it operates with a he - ne laser at @xmath3 and an intracavity power of @xmath4 at each mirror . high resolution can be achieved by pattern matching to periodic variation of the sagnac frequency induced by rotation of the earth . using matching based on geophysical modelling , detection of lunar perturbations of the earth s rotation and the amplitudes of the oppolzer modes have recently been reported in @xcite . the instantaneous direction of the earth s rotation axis has been measured to a precision of @xmath5 part in @xmath6 when averaged over a time interval of several hours . the enhanced sensitivity of such large ring - lasers owes much to the significant improvement of mirror design in recent years . it is imperative for the maintenance of a sagnac beat signal that pollution of the resonant state by competing modes is inhibited . this requires considerable engineering skill , the use of thermally inert materials and mirrors that can maintain the dominant cavity resonant mode over long periods of time . with such advances in technology it is natural to seek the sensitivities that are necessary to discriminate the effects of non - newtonian gravitation on the behaviour of light @xcite . the most important of these is the gravito - magnetic field of the rotating earth , predicted by einstein s description of gravitation . however , even crude estimates of the effects of terrestrial gravito - magnetism show that a number of competing effects that may modify the classical sagnac effect ( due to the rotation of the apparatus ) need careful consideration . such effects are often dependent on the structure of the interferometer and so invite one to contemplate alternative designs . although , in general , one expects that the sensitivity of an active ring - laser will increase with its spatial dimensions one may also enquire about the dependence of sensitivity on its geometry and spatial shape . in order to discriminate gravito - magnetic effects it then becomes necessary to understand in some detail how the geometry and shape of the interferometer affect the resonant mode structure of the cavity . in this way one can hope to discriminate between a number of features that compete to modify the classical sagnac frequency . the effect of gravitation on the sagnac beat frequency is usually discussed in terms of the high frequency ray - optic description of light . in this way one can simply relate the interference of co- and counter - propagating electromagnetic modes to the proper time difference between events that terminate their ray histories in spacetime @xcite . while such an analysis can give valuable insights into the leading terms in an asymptotic expansion in the lasing frequency it can not address the effects due to the vector nature of the electromagnetic modes and the effects of the infinity of competing modes possessed by the lasing cavity . an exact analysis of the wave nature of light in an active sagnac cavity is a non - trivial mathematical problem that has to our knowledge never been fully considered . the problem is complicated by the fact that the harmonic electromagnetic modes rarely split into simpler transverse electric ( te ) and transverse magnetic ( tm ) modes that greatly simplify the analysis in most simply - connected axially symmetric cavity geometries . such a separation relies on a number of factors such as the shape of the entire cavity in space , the homogeneity of the lasing medium , the nature of the acceleration of the cavity and the presence of even a weak non - newtonian gravitational field . in order to accommodate these complexities a viable approximation is mandatory . here the problem is approached in terms of an expansion scheme determined by scales related to the geometry of the cavity . thus slender cavities are considered where the ratio of the transverse dimension to the cavity length is small . such cavities will be called `` wavetubes '' . in order to confidently predict near - earth general relativistic contributions to the active sagnac beat frequency a careful electromagnetic mode analysis of the cavity must be considered . this can be done in terms of the covariant maxwell equations @xcite for moving media @xcite on a background spacetime determined by einstein s equations for gravitation . the procedure will exploit a local system of coordinates adapted to the geometry of a wavetube . such a wavetube can be characterised in terms of the locus in space of the centre of each of its transverse cross - sections . this locus is regarded as a closed space curve with possible non - zero frenet torsion and curvature . the approximation scheme will be developed for a wavetube with central locus having local curvature departing slightly from that of a fixed plane circle . in sections 2 and 3 it is shown how to construct te and tm type solutions to the covariant maxwell equations for a particular class of coframes . these include the fields in rotating cavities containing a homogenous , isotropic and dispersionless medium in the presence of terrestrial gravito - magnetism . the language of exterior differential forms is used throughout since it offers the most succinct formulation for the field equations and makes clear the role of the spacetime metric and local coframe that feature in the subsequent computational scheme . the relation to the inertial 3-d formulation can be found in @xcite and references therein . in section 4 a particular coordinate system is introduced that , together with a particular frenet frame , encodes the geometry of the wavetube and permits one to introduce dimensionless parameters that control the nature of the approximation scheme . in these coordinates differences between non - inertial motion and gravito - magnetic effects are made explicit in sections 5.1 and 5.2 . a complete mode analysis of the rotating wavetube is given in section 6 and the effects of multi - mode excitation and the description of sagnac beats is explored in section 7 . in particular attention is drawn to the effects of non - planarity of the wavetube on the beat frequency by reference to a particular geometry generated by the structure of a torus - knot . to accommodate the ideas above it is first shown how a class of spacetime metrics that admit local orthonormal coframes with particular properties are sufficient to construct te and tm type electromagnetic modes relative to such a coframe . the language of exterior forms is used throughout range over the integers @xmath7 , indices in the middle of the latin alphabet @xmath8 range over the integers @xmath9 and greek indices @xmath10 range over @xmath11 . expressions involving repeated indices imply use of the einstein summation convention . ] since this greatly facilitates the discussion and leads in a direct manner to an explicit construction of the fields below . the hollow wavetube is supposed to be constructed of a perfectly conducting material which in this section contains no lasing medium . let @xmath12 be a spacetime and let @xmath13 be a local orthonormal coframe with respect to the metric tensor @xmath14 with dual frame @xmath15 , @xmath16 and orientation ( volume form ) defined by the hodge map @xmath17 with @xmath18 the inverse metric tensor @xmath19 is @xmath20 define the split of the cotangent space @xmath21 with @xmath22 spanned by @xmath23 and @xmath24 spanned by @xmath25 at @xmath26 . these subspaces inherit hodge maps with @xmath27 and @xmath28 so @xmath29 now restrict to spacetime metrics @xmath30 that admit local orthonormal coframes satisfying the conditions @xmath31 these conditions permit a viable approximation scheme leading to a tractable decomposition of electromagnetic modes in a wide context to be discussed below . a differential form @xmath32 ( vector field @xmath33 ) on @xmath34 that satisfies @xmath35 ( @xmath36 ) for all @xmath37 will be called _ _ longitudinal _ _ is the interior derivative on forms with respect to the vector field @xmath33 . ] . a _ transverse _ differential form @xmath38 ( vector field @xmath39 ) will satisfy @xmath40 ( @xmath41 ) for all @xmath42 . let the @xmath43-form @xmath44 be a solution to the vacuum maxwell equations @xmath45 in a spacetime region bounded by a perfectly conducting wavetube hypersurface @xmath46 and subject to the boundary condition @xmath47 the wavetube interior on spacetime is topologically @xmath48 where @xmath49 is a @xmath43-disc . solutions to maxwell s equations are sought by adapting the above coframe to the wavetube geometry . furthermore , we shall suppose that ( [ fresnel_bc ] ) must be satisfied with @xmath50 , for all @xmath51 , a transverse @xmath5-form . since @xmath52 the pair @xmath23 are , by frobenius theorem , normal @xmath5-forms to a local foliation @xmath53 of @xmath34 . the leaves of @xmath53 contain the wavetube cross - sections , each of whose intersection with @xmath54 is the image of a closed curve . locally , we seek separable propagating solutions of the form @xmath55 for some @xmath43-form @xmath56 where @xmath57 for constant components @xmath58 . we decompose @xmath56 into longitudinal and transverse parts @xmath59 thus @xmath60 are transverse @xmath5-forms and @xmath61 are transverse @xmath43-forms . using the identity @xmath62 where @xmath63 for any vector field @xmath33 on @xmath34 and @xmath42 is any @xmath64-form on @xmath34 it can be shown that @xmath65 where @xmath32 is a longitudinal @xmath64-form , @xmath38 is a transverse @xmath66-form and @xmath67 inserting the expression for @xmath44 into maxwell s equations ( [ maxwell_df_dstarf ] ) gives @xmath68 where ( [ hodge_as_parhodge_perphodge ] ) has been used . the form structure indicates that the above pair splits into @xmath69 acting with @xmath17 on ( [ split_maxwell_dphi ] ) and ( [ split_maxwell_dpsi ] ) and using ( [ hodge_as_parhodge_perphodge ] ) yields @xmath70 where @xmath71 and @xmath72 . acting with @xmath73 , where @xmath74 , on equation ( [ split_maxwell_ealpha ] ) yields @xmath75 where @xmath76 . similarly , ( [ split_maxwell_parhodgeealpha ] ) leads to @xmath77 thus , inserting ( [ phi_kkappa_related ] ) in ( [ dkkappa_phi_related ] ) and ( [ psi_kkappa_related ] ) in ( [ dkkappa_psi_related ] ) yields @xmath78 the helmholtz equations ( [ equation_for_phi ] ) and ( [ equation_for_psi ] ) determine @xmath79 and @xmath80 subject to ( [ fresnel_bc ] ) . the boundary condition ( [ fresnel_bc ] ) may be expanded @xmath81 implying @xmath82 in general there are three possibilities for the metric norm @xmath83 of @xmath84 : @xmath85 @xmath86 . * triviality of @xmath79 * + let @xmath87 be a @xmath43-chain for which the image of @xmath88 is in @xmath54 and @xmath89 and note @xmath90 using ( [ equation_for_phi ] ) and ( [ phi_boundary_condition ] ) where @xmath91 is the complex - conjugate of @xmath79 . since @xmath92 where @xmath93 is any transverse @xmath64-form and , by hypothesis , @xmath86 it follows from ( [ integration_phi ] ) that @xmath94 and so , using ( [ phi_boundary_condition ] ) , @xmath95 . * triviality of @xmath80 and @xmath44 * + since @xmath84 is null @xmath96 is proportional to @xmath84 and so ( [ phi_kkappa_related ] ) and ( [ psi_kkappa_related ] ) with @xmath95 imply that @xmath97 vanishes and @xmath80 is constant . let @xmath98 be the null longitudinal @xmath5-form with constant components @xmath99 that satisfies @xmath100 . acting with @xmath101 on ( [ split_maxwell_parhodgeealpha ] ) and ( [ split_maxwell_ealpha ] ) and noting @xmath102 where @xmath103 it is found that @xmath104 since @xmath97 vanishes @xmath105 where @xmath106 and @xmath107 . the differential form @xmath108 is transverse and so @xmath109 follows from ( [ kkappa_boundary_condition ] ) . by integrating ( [ dkkappa_l_psi_related ] ) over the @xmath43-chain @xmath87 introduced earlier , noting that ( [ kkappa_l_boundary_condition ] ) implies @xmath110 and recalling that @xmath80 is constant , it follows that @xmath111 and @xmath112 cross - sections of the wavetube interior are simply - connected and so by the poincar lemma @xcite @xmath113 it follows from ( [ div_kkappa_l ] ) and ( [ kkappa_l_boundary_condition ] ) that @xmath114 satisfies @xmath115 subject to the dirichlet boundary condition @xmath116 where @xmath117 is a complex constant . the solution to ( [ laplace_in_null_case ] ) and ( [ laplace_in_null_case_boundary_cond ] ) is @xmath118 and so , using ( [ exactness_of_kkappa_zeta ] ) , @xmath119 and @xmath120 @xmath121 . * triviality of @xmath79 * + using ( [ integration_phi ] ) and ( [ positivity_parhodge ] ) it can be seen that @xmath79 vanishes since , by hypothesis , @xmath121 . * triviality of @xmath80 and @xmath44 * + introduce the spacelike normalized @xmath5-form @xmath122 and write @xmath123 where @xmath124 and @xmath125 and @xmath126 vanishes by ( [ phi_kkappa_related ] ) because @xmath95 . thus ( [ kkappa_boundary_condition ] ) yields @xmath127 since @xmath84 is longitudinal and @xmath128 is transverse ( [ psi_kkappa_related ] ) gives the neumann boundary condition @xmath129 the simple - connectedness of the wavetube cross - sections implies that the eigenvalues @xmath130 in ( [ equation_for_psi ] ) associated with the neumann boundary condition ( [ psi_boundary_condition ] ) are negative . to prove this an argument similar to that used in the dirichlet case is employed . let @xmath87 be the @xmath43-chain introduced earlier and note that ( [ psi_boundary_condition ] ) implies @xmath131 , where @xmath132 is a @xmath133-form , and @xmath134 since @xmath135 . thus @xmath136 using ( [ equation_for_psi ] ) and ( [ psi_boundary_condition_s ] ) where @xmath137 is the complex - conjugate of @xmath80 . by hypothesis @xmath121 and using ( [ positivity_parhodge ] ) it follows that @xmath111 . therefore @xmath120 @xmath138 . this is the only situation that leads to a non - trivial expression for @xmath44 . introduce the timelike normalized @xmath5-form @xmath139 and write @xmath140 where @xmath141 and ( [ phi_kkappa_related ] ) and ( [ psi_kkappa_related ] ) have been used . inserting the above equation for @xmath142 into the boundary condition ( [ kkappa_boundary_condition ] ) yields @xmath143(p ) = 0 { \quad\forall\ , p\in{\ensuremath{\mathcal{b}}}}.\ ] ] however , ( [ phi_boundary_condition ] ) means that @xmath144 on @xmath54 is proportional to @xmath145 and so again ( [ psi_boundary_condition ] ) is obtained . expressed entirely in terms of @xmath146 the maxwell 2-form @xmath44 can be written @xmath147 where @xmath148\exp(iw_{tm}),\\ \label{te_f_in_terms_of_psi } f_{te } = \biggl [ \psi{\ensuremath{\#_\perp}}1 - i\frac{dw_{te}}{|dw_{te}|^2 } \wedge { \ensuremath{\#_\perp}}d\psi\biggr]\exp(iw_{te}).\end{gathered}\ ] ] the helmholtz equations for @xmath79 and @xmath80 are @xmath149 solved subject to the boundary conditions @xmath150 where @xmath151 and @xmath152 are timelike . the constants @xmath153 are determined by the appropriate boundary conditions and , in general , @xmath154 . to excite electromagnetic modes in a wavetube one may fill it with a gas such as a helium - neon mixture and induce it to lase with an external rf field . the presence of a lasing medium requires that one take into account its electrical and magnetic susceptibility . the electromagnetic field in a material medium is described by a pair of 2-forms @xmath155 on spacetime that satisfy the equations : @xmath156 where @xmath37 is a source 3-form . for a given @xmath37 , the 2-form @xmath157 must be related to @xmath44 in order to have a closed system . this is usually done by relating their components relative to decompositions of the form : @xmath158 where the timelike vector @xmath159 is normalised with @xmath160 and the forms @xmath161 are all annihilated by @xmath162 . with a spacetime metric tensor @xmath30 , having physical dimensions of length squared , a coherent dimensional scheme arises with the aid of the permittivity constant @xmath163 and permeability @xmath164 of free space satisfying @xmath165 where @xmath166 is the speed of light in vacuum . in terms of conventional mks components of electric and magnetic fields one has in any @xmath30-orthonormal coframe @xmath167 adapted to @xmath159 : @xmath168 for @xmath157 one has @xmath169 and @xmath170 . for a linear isotropic homogeneous dispersionless non - conducting medium with relative permittivity @xmath171 and relative permeability @xmath172 one has the simple constitutive relations @xmath173 and @xmath174 or @xmath175 such a constitutive relation assumes that the dimensionless quantities @xmath171 and @xmath172 are frame - independent constant scalars on spacetime . the refractive index @xmath176 of the medium is @xmath177 propagating solutions to the source - free maxwell equations @xmath178 that satisfy the boundary condition @xmath179 at the wavetube surface @xmath54 , @xmath180 have a similar structure to ( [ tm_f_in_terms_of_phi ] ) and ( [ te_f_in_terms_of_psi ] ) when @xmath159 has the form @xmath181 where @xmath182 are constant and @xmath183 belong to the frame @xmath184 dual to the coframe @xmath185 satisfying the conditions ( [ coframe_assumptions ] ) . it can be shown that @xmath186 and @xmath187 are solutions to ( [ source_free_maxwell_medium ] ) where @xmath188\exp(iw_{tm}),\\ f_{te } = \biggl [ \psi{\ensuremath{\#_\perp}}1 - i\frac{dw_{te}}{|\alpha_{te}|^2 } \wedge { \ensuremath{\#_\perp}}d\psi\biggr]\exp(iw_{te } ) \label{te_f_medium_in_terms_of_psi}\end{gathered}\ ] ] with @xmath189 the helmholtz equations for @xmath79 and @xmath80 are @xmath190 and are solved subject to the boundary conditions @xmath191 as before @xmath151 and @xmath152 where @xmath192 and @xmath193 are constant . a moving space curve in euclidean @xmath194 may be represented by a @xmath194-valued mapping @xmath195 . suppose that @xmath196 parameterises arc length and @xmath197 is a time variable . the instantaneous geometry of the space curve may be described by three mutually orthogonal unit vectors @xmath198 on the image of @xmath199 at each time @xmath200 . for a frenet frame with @xmath201 tangent to @xmath202 one has : @xmath203 where @xmath204 denotes the derivative of @xmath205 with respect to @xmath196 and @xmath206 and @xmath207 denote the curvature and torsion of @xmath202 respectively . we shall restrict our discussion to space curves that admit continuous frenet frames along their entire length . since the triad @xmath208 forms an orthonormal basis on the curve the time derivative @xmath209 is related to @xmath208 by an anti - symmetric matrix for each @xmath210 . we are interested in space curves that rotate rigidly with uniform angular velocity @xmath211 without deformation . thus @xmath212 , @xmath213 and @xmath214 . these imply @xmath215 in a fixed global cartesian triad @xmath216 one may write in cylindrical coordinates : @xmath217 if this triad is oriented so that @xmath218 then @xmath219 and @xmath220 are independent of @xmath197 and @xmath221 . let the vector @xmath222 with cartesian components @xmath223 in the frame @xmath216 , @xmath224 locate a point at the instant @xmath200 . if this point lies in a tubular neighbourhood about the space curve @xmath202 it may also be labelled by the frenet coordinates @xmath225 such that : @xmath226 thus for points in this region one has a transformation between the cartesian coordinates @xmath223 and the frenet coordinates @xmath227 . when the curve moves @xmath228 and so for rigid motion the above gives : @xmath229dt\\ & + [ 1 -\kappa(s ) x_1]ds\,{{\mathbf t}}+[d x_1 - x_2 \tau(s ) ds]{{\mathbf n}}+[d x_2 + x_1 \tau(s ) ds]{{\mathbf b}}\\ = & dy_1{{\mathbf i}}+ dy_2{{\mathbf j}}+ dy_3{{\mathbf k}}\end{split}\ ] ] where @xmath230 , @xmath231 and @xmath232 . these equations imply a @xmath197 dependent coordinate transformation between @xmath223 and @xmath225 . the following is concerned with cavities that have small circular cross - section compared with their length and are represented by a space curve that approximates a plane circle rotating about its axis . to this end it is convenient to introduce a number of dimensionless parameters and new coordinates . first introduce polar variables @xmath233 and @xmath234 by @xmath235 where @xmath236 , @xmath237 and @xmath238 is the radius of the circular cross - section . it is supposed that the absolute values of the curvature and torsion of the closed curve @xmath199 are bounded from above by @xmath239 and @xmath240 respectively and that the scale of @xmath199 is given by that of a circle of radius @xmath241 . introduce the dimensionless parameters @xmath242 the dimensionless parameter @xmath243 indicates the order of small quantities that will be used below to approximate the metric appropriate for a cavity with small cross section . alternative assumptions about the assignment of @xmath243 to the quantities @xmath244 will give different approximation schemes . the above choice will be shown to give the classical sagnac beat frequency to first order in a wavetube based on a planar space curve . the calculation will proceed for @xmath245 to lowest order in @xmath243 for a curve that approximates a circle of radius @xmath241 rotating with uniform angular speed @xmath246 about its axis . introduce the scaled polar coordinates @xmath247 and @xmath248 where @xmath249 and the scaled quantities @xmath250 and consider the class of @xmath199 for which @xmath251(s , t ) = a_0\bigl[1 + \epsilon\gamma_1({\hat{s}},{\hat{\lambda}})\bigr],\\ \notag [ { { \mathbf n}}\cdot({{\mathbf k}}{{\mathbf \times}}{{\mathbf c}})](s , t ) = a_0\epsilon\gamma_2({\hat{s}},{\hat{\lambda}}),\\ \label{approximate_curve_conditions } [ { { \mathbf b}}\cdot({{\mathbf k}}{{\mathbf \times}}{{\mathbf c}})](s , t ) = a_0\epsilon\gamma_3({\hat{s}},{\hat{\lambda}}).\end{gathered}\ ] ] thus , to lowest ( zeroth ) order in @xmath243 @xmath252 + { { \mathbf b}}[d{\hat{x}}_2 + \mu_4\hat\tau({\hat{s } } ) { \hat{x}}_1 d{\hat{s } } ] + { { \mathbf t}}\mu_2 d{\hat{\lambda}}.\ ] ] in terms of @xmath247 and @xmath248 @xmath253{\ensuremath{\otimes}}[d\phi+\mu_4{\hat{\tau}}({\hat{s } } ) d{\hat{s } } ] + d{\hat{\rho}}{\ensuremath{\otimes}}d{\hat{\rho}}\end{split}\ ] ] and the zeroth order space curve is a circle of radius @xmath241 orthogonal to the rotation axis @xmath254 . we shall consider only stationary axisymmetric spacetimes . thus @xmath12 admits a commuting pair of killing vector fields @xmath255 and @xmath256 : @xmath257 = 0\end{gathered}\ ] ] where @xmath258 is the lie derivative with respect to @xmath33 . the vector field @xmath255 is timelike and future - pointing and the vector field @xmath256 is spacelike and has closed orbits . here by definition , material points that are `` rotating '' and @xmath256 above . ] follow timelike worldlines whose tangents at any event are a linear combination @xmath159 of @xmath255 and @xmath256 at that event . if the spacetime is asymptotically flat and the normalised @xmath159 is timelike asymtotically it also offers a field of stationary observers . before constructing a local coframe adapted to a twisted wavetube rotating in a spacetime with gravitation we consider the simpler case of minkowski spacetime . the metric tensor @xmath30 on minkowski spacetime is @xmath259 in terms of the inertial coordinates @xmath260 . it possesses the timelike future - pointing killing vector field @xmath261 and the spacelike killing vectors @xmath262 and @xmath263 where @xmath264 on the wavetube spacetime domain @xmath260 is related to the spacetime frenet coordinates @xmath265 by @xmath266,\\ y_2 = { { \mathbf j}}\cdot[{{\mathbf c}}(s , t)+x_1{{\mathbf n}}(s , t)+x_2{{\mathbf b}}(s , t)],\\ y_3 = { { \mathbf k}}\cdot[{{\mathbf c}}(s , t)+x_1{{\mathbf n}}(s , t)+x_2{{\mathbf b}}(s , t)].\end{gathered}\ ] ] the wavetube apparatus follows integral curves of @xmath267 in spacetime . by expressing the vector field @xmath267 with respect to @xmath260 : @xmath268 it can be seen that @xmath267 is a linear combination of the timelike future - pointing killing vector field @xmath269 and the spacelike killing vector field @xmath256 : @xmath270 where @xmath271=0 $ ] . at the instant @xmath272 write @xmath199 in the form @xmath273 where @xmath274 is a plane circle in @xmath194 rotating with angular speed @xmath275 about its axis @xmath254 . the curve image @xmath276 has tangent @xmath277 , normal @xmath278 , constant binormal @xmath279 and radius @xmath241 . note that @xmath196 is _ not _ necessarily the arc parameter of @xmath274 . the vector @xmath276 can be expressed @xmath280 and it is supposed that the tangent @xmath201 , normal @xmath281 and binormal @xmath282 of @xmath199 satisfy @xmath283{{\mathbf n}}_t(s ) + \sin[\beta(s , t)]{{\mathbf b}}_t(s ) + o(\epsilon),\\ \label{moving_curve_requirements } { { \mathbf b}}(s , t ) = -\sin[\beta(s , t)]{{\mathbf n}}_t(s ) + \cos[\beta(s , t)]{{\mathbf b}}_t(s ) + o(\epsilon)\end{gathered}\ ] ] for some angle @xmath284 . using @xmath285 it follows that @xmath286 which is consistent with ( [ approximate_curve_conditions ] ) . thus , in terms of @xmath265 the metric tensor @xmath30 is @xmath287 = \rho [ d\phi + \tau(s)d s].\end{gathered}\ ] ] where @xmath13 is orthonormal to lowest order in @xmath243 . note that the approximate minkowski metric is insensitive to @xmath288 and the curvature of @xmath199 . the metric tensor at large distances from an isolated compact rotating body in an asymptotically flat spacetime , with newtonian gravitational mass @xmath289 and angular momentum @xmath290 , may be approximated ( for @xmath291 where @xmath292 is newton s gravitational constant ) by @xmath293 in spherical polar coordinates @xmath294 . the use of the metric ( [ kerrm ] ) is of course an idealisation for physical applications such as terrestrial ring lasers , where one does not have two killing vectors . in particular , due to interactions with other celestial bodies the positions of both the rotation and angular momentum axes will not quite coincide with each other and their positions will vary with time . as reported in ref . @xcite for the case of ring lasers at the earth s surface such effects are three to four orders of magnitude larger than the lense - thirring effect . nonetheless the calculation presented here serves a useful purpose in isolating effects due to gravito - magnetism . the implications of time variation of the rotation and angular momentum axes of the earth on terrestrial lense - thirring measurements will be discussed elsewhere @xcite . the vector fields @xmath261 and @xmath295 are a commuting pair of killing vectors . furthermore , @xmath269 is timelike and future - pointing and @xmath296 is spacelike and has closed orbits . consider the acceleration of matter on one of the integral curves of the 4-velocity field @xmath297 with respect to a stationary observer belonging to the 4-velocity field @xmath298 . at events where the integral curves of @xmath159 and @xmath299 coincide @xmath159 has spatial acceleration @xmath300 where @xmath301 projects the 4-acceleration to the instantaneous 3-space of @xmath299 . in the non - relativistic weak gravitational field limit one finds that the g - orthonormal components of @xmath302 yield a centripetal acceleration with dominant magnitude @xmath303 at an event with coordinates @xmath304 . thus each observer within @xmath299 would interpret such events within @xmath159 to have instantaneous 3-acceleration produced by rotation with angular speed @xmath275 . the asymptotically minkowski coordinate chart @xmath260 is given in terms of @xmath294 as @xmath305 the dot @xmath306 and cross @xmath307 products of the pair of vectors @xmath308 are defined with respect to the tensor @xmath132 : @xmath309 and the @xmath132-orthonormal frame @xmath216 : @xmath310 and , as before , the vector @xmath311 , @xmath312 locates the point @xmath260 . the metric tensor @xmath30 has the form @xmath313 \end{split}\ ] ] where @xmath260 and the spacetime coordinates @xmath314 satisfy @xmath315,\\ y_2 = { { \mathbf j}}\cdot[{{\mathbf c}}({{\underline{s}}},t)+{{\underline{x}}}_1{{\mathbf n}}({{\underline{s}}},t)+{{\underline{x}}}_2{{\mathbf b}}({{\underline{s}}},t)],\\ y_3 = { { \mathbf k}}\cdot[{{\mathbf c}}({{\underline{s}}},t)+{{\underline{x}}}_1{{\mathbf n}}({{\underline{s}}},t)+{{\underline{x}}}_2{{\mathbf b}}({{\underline{s}}},t)]\end{gathered}\ ] ] and the @xmath132-orthonormal frame @xmath208 solves the frenet - serret equations @xmath316 the functions @xmath317 are underlined as a reminder that they are normalized with respect to the tensor @xmath132 rather than the spacetime metric @xmath30 . as in the previous section consider curves of the form @xmath318 subject to ( [ moving_curve_requirements ] ) , with @xmath196 replaced by @xmath319 , where @xmath320 is the rotating plane circle of @xmath132-radius @xmath241 centred at the point @xmath321 with the frenet frame @xmath322 and @xmath218 . here one is considering @xmath323 perturbations about a `` circular '' space curve parallel to the `` equatorial plane '' of the kerr geometry where elements of the wavetube apparatus follow integral curves of @xmath324 in this spacetime . as before , introduce the dimensionless functions @xmath325 since @xmath326 and the position vector @xmath327 has the form @xmath328\ ] ] and using @xmath329 + { { \mathbf b}}[d{\hat{\underline{x}}}_2 + \mu_4\hat{{\underline{\tau}}}({\hat{\underline{s } } } ) { \hat{\underline{x}}}_1 d{\hat{\underline{s } } } ] + { { \mathbf t}}\mu_2 d{\hat{\lambda}},\\ { { \mathbf b}}= { { \mathbf k}}\end{gathered}\ ] ] with ( [ moving_curve_requirements ] ) it follows that @xmath330.\ ] ] the metric tensor @xmath30 adapted to @xmath199 is @xmath331\\ & - \mu_6 \bigl[(d{\hat{\underline{s}}}+ \mu_2 d{\hat{\lambda}}){\ensuremath{\otimes}}d{\hat{\lambda}}+ d{\hat{\lambda}}{\ensuremath{\otimes}}(d{\hat{\underline{s}}}+ \mu_2 d{\hat{\lambda}})\bigr ] \end{split}\ ] ] to lowest order in @xmath243 where @xmath332 and @xmath333 a @xmath30-orthonormal coframe to lowest order in @xmath243 is @xmath334^{1/2}c dt,\\ \notag e^1 = \bigl[(1+\mu_5)^{1/2}\mu_2-\mu_6/(1+\mu_5)^{1/2}\bigr ] cdt + ds,\\ \notag e^2 = d\rho,\\ \label{lowest_order_kerr_coframe } e^3 = \rho [ d\phi + \tau(s)ds]\end{gathered}\ ] ] where @xmath335 both coframes ( [ lowest_order_minkowski_coframe ] ) and ( [ lowest_order_kerr_coframe ] ) satisfy ( [ coframe_assumptions ] ) and so single - valued propagating electromagnetic @xmath44 modes inside a wavetube with circular cross - section of radius @xmath238 can be generated from particular solutions of the helmholtz equations discussed earlier . solutions regular in the wavetube follow from the forms @xmath336 $ ] where @xmath337 , @xmath338 and @xmath339 or @xmath340 ( see equations ( [ alpha_t_beta_t_explicit ] ) ) . the role of the frenet torsion in solutions of the scalar helmholtz equation in coordinates adapted to non - planar curves has been noted before in a number of different contexts @xcite . the @xmath186 and @xmath187 modes follow from @xmath341\ ] ] with @xmath342 satisfying @xmath343 and @xmath344\ ] ] with @xmath345 satisfying @xmath346 where @xmath347 is the @xmath348th regular bessel function of the first kind and @xmath349 . for both te and tm modes @xmath350 and , using @xmath351 , @xmath352 is the frequency ( hz ) , measured by the wavetube apparatus , of the field mode associated with @xmath353 or @xmath354 . positive wavenumbers @xmath355 correspond to propagation in the direction of the tangent @xmath201 and negative wavenumbers to propagation in the opposite direction . since the wavetube is closed , i.e. the wavetube interior on spacetime is topologically @xmath48 , the maxwell @xmath43-form satisfies @xmath356 where @xmath357 is the length of the wavetube and so using ( [ xi_in_terms_omega_k ] ) and either ( [ tm_f_medium_in_terms_of_phi ] ) or ( [ te_f_medium_in_terms_of_psi ] ) @xmath358 \equiv k_\xi = \frac{2\pi n}{l } - n{\bar{\tau}},\quad\,n\in{{\mathbf z}}\ ] ] where @xmath359 is the average frenet torsion . the mode spectrum is classified using the triple of integers @xmath360 , where the positive integer @xmath64 labels a solution of @xmath361 in the tm case or @xmath362 in the te case . the longitudinal members of ( [ lowest_order_minkowski_coframe ] ) and ( [ lowest_order_kerr_coframe ] ) have the form @xmath363 where @xmath364 and @xmath365 are constants with the dimensions of @xmath166 . for each @xmath366 , where @xmath339 or @xmath340 ( see equations ( [ alpha_t_beta_t_explicit ] ) ) and @xmath367 is determined by the boundary conditions , the dispersion relation @xmath368 with @xmath369 yields @xmath370\ ] ] where @xmath371 it is assumed that , for terrestrial applications , @xmath372 and so the sign of the square root of the discriminant in ( [ wavetube_omega_spectrum ] ) is chosen to be positive . in addition to @xmath373 $ ] introduce the scalars @xmath374 dependent on the mode indices @xmath360 and the type @xmath375 as follows : @xmath376,\\ \zeta_t \equiv \zeta_t[p , n]\end{gathered}\ ] ] where @xmath377)=0 $ ] and @xmath378)=0 $ ] . the wavetube apparatus measures the frequency @xmath379 = \frac{c}{2\pi}dw_t\bigl(v)\ ] ] and so here @xmath380=\frac{\omega_t[n , p , n]c}{2\pi\sqrt{a^2-b^2}}.\ ] ] we may approximate this expression further by taking into account the relative magnitudes of @xmath381 , @xmath348 , @xmath382 , @xmath383 and @xmath384 . for a lasing medium excited by a rf field @xmath385 is expected and it is assumed that @xmath386 and @xmath387 . then ( [ wavetube_omega_spectrum ] ) has the form @xmath388 \simeq \frac{|k[n , n]|(a^2-b^2)}{a{{\ensuremath{\mathcal{n}}}}+{\text{sgn}}(k[n , n])b}\ ] ] where @xmath389 using ( [ k_in_terms_of_n_tau ] ) and @xmath390 hence @xmath391 and @xmath392 are independent of the mode type @xmath393 and the bessel root index @xmath64 in this approximation . henceforth we write @xmath394 and @xmath395 and drop their @xmath64 dependencies . for terrestrial applications we expect that @xmath396 ( as well as @xmath397 ) . in this case ( [ wavetube_frequency_spectrum ] ) becomes @xmath398 \simeq \frac{c|n|}{l{{\ensuremath{\mathcal{n } } } } } - { \text{sgn}}(n)\biggl[\frac{c|n|}{l}\frac{b}{a{{\ensuremath{\mathcal{n}}}}^2 } + \frac{cn{\bar{\tau}}}{2\pi{{\ensuremath{\mathcal{n}}}}}\biggr]\ ] ] where ( [ mod_k_in_terms_of_n_tau ] ) and ( [ high_freq_wavetube_omega_spectrum ] ) have been used . if a number of electromagnetic modes of similar frequency are active in the wavetube then the time histories of the electromagnetic fields measured by the wavetube apparatus will exhibit beating . the classical @xcite sagnac beat frequency @xmath399 induced by rotation on flat spacetime follows from ( [ approx_high_frequency_wavetube_frequency_spectrum ] ) with @xmath400 and by choosing a pair of rotationally symmetric ( @xmath401 ) modes that differ only in their directions of propagation and have @xmath402 : @xmath403 - \nu[-n,0]\\ & \simeq -\frac{2\omega n a_0}{l } \end{split}\ ] ] where @xmath364 and @xmath365 have been obtained from ( [ lowest_order_minkowski_coframe ] ) . to reveal the classical expression for the sagnac beat frequency the parameters @xmath241 and @xmath357 and the mode index @xmath381 are eliminated by introducing the area @xmath404 and perimeter @xmath405 of the plane circle approximated by the wavetube locus and the lasing wavelength @xmath406 . it follows that @xmath407 the most general frequency difference is constructed from a pair of modes of type @xmath408 with indices @xmath409 and @xmath410 . in the approximation introduced earlier we only need consider @xmath381 and @xmath348 dependences : @xmath411 \equiv \nu[n_1,n_1 ] - \nu[n_2,n_2].\ ] ] thus , with @xmath402 , ( [ standard_sagnac_beat_frequency ] ) has the generalization @xmath412 \simeq -\frac{2n\omega a_0}{l{{\ensuremath{\mathcal{n}}}}^2 } - \frac{cn{\bar{\tau}}}{\pi{{\ensuremath{\mathcal{n}}}}}.\ ] ] for terrestrial applications the external rf field generates counter- and co - propagating modes with indices @xmath413 and @xmath414 whose magnitudes are so large it is possible that @xmath415 . more generally , there is no reason to assume that @xmath416 and @xmath417 are identical . with @xmath418 and @xmath419 and using ( [ approx_high_frequency_wavetube_frequency_spectrum ] ) it follows that @xmath420 \simeq & \frac{c}{l{{\ensuremath{\mathcal{n}}}}}(n_1+n_2 ) - \frac{a_0\omega}{l{{\ensuremath{\mathcal{n}}}}^2}(n_1-n_2)\\ & - \frac{c{\bar{\tau}}}{2\pi{{\ensuremath{\mathcal{n}}}}}(n_1+n_2 ) . \end{split}\ ] ] the dimensionless parameters @xmath421 , @xmath422 contain the _ euclidean _ radius @xmath241 of the multiply - wound plane circle @xmath423 that approximates the wavetube locus on kerr spacetime @xmath12 . one may eliminate the parameter @xmath241 in favour of a length determined by the spacetime metric @xmath30 rather than the euclidean metric @xmath424 . for each constant @xmath425 the multiply - wound planar circle may be described as the image of the map @xmath426 & \rightarrow { \ensuremath{\mathcal{m}}}\\ u & \mapsto ( t = t_0,\,r = r_0,\,\theta=\theta_0,\,\varphi = u ) \end{split}\ ] ] where @xmath294 are the coordinates used in ( [ kerrm ] ) , @xmath427 and the positive integer @xmath428 is a winding number . the arclength @xmath429 of @xmath423 is @xmath430 where @xmath431 is tangent to @xmath423 . using ( [ kerrm ] ) , ( [ spacetime_gamma_map ] ) and ( [ g_length_of_circle ] ) it follows that @xmath432 where @xmath433 introduce dimensionless parameters @xmath434 @xmath435 so that , by using ( [ a0_in_terms_of_l0 ] ) to eliminate @xmath241 : @xmath436 typically , for terrestrial applications , the magnitudes of the dimensionless parameters @xmath437 are much less than unity . thus , it is reasonable to approximate @xmath438 by a polynomial obtained from the multivariate taylor expansion of @xmath438 with respect @xmath439 about @xmath440 . however , when truncating the taylor expansion the numerical values of the terms in the series must be carefully scrutinized . for example , the parameters associated with a wavetube whose locus approximates @xmath423 with `` radius '' @xmath441 fixed on the earth and centred on the poles have magnitudes @xmath442 , @xmath443 and @xmath444 . for terrestrial conditions @xmath445 is generally true . it follows that @xmath446 is a good approximation ( the other terms in the taylor series are about 10 orders of magnitude smaller ) . using ( [ approx_high_frequency_wavetube_frequency_spectrum ] ) and ( [ approx_kerr_b_over_a ] ) it follows that @xmath447 \simeq \frac{c|n|}{l{{\ensuremath{\mathcal{n } } } } } -{\text{sgn}}(n)\biggl[\frac{l_0}{2\pi m}\frac{\omega|n|}{l{{\ensuremath{\mathcal{n}}}}^2 } \biggl(1+\frac{{g_n}m}{r_0 c^2 } -\frac{2{g_n}j}{\omega r_0 ^ 3 c^2 } \biggl ) + \frac{cn{\bar{\tau}}}{2\pi{{\ensuremath{\mathcal{n}}}}}\biggr]\ ] ] and , more generally , for a pair of modes with indices @xmath448 and @xmath449 , where @xmath418 and @xmath419 , @xmath450 \simeq & \frac{c}{l{{\ensuremath{\mathcal{n}}}}}(n_1+n_2)\\ & -\frac{l_0}{2\pi m}\frac{\omega}{l{{\ensuremath{\mathcal{n}}}}^2 } \left[1+{{g_n}m\over r_0 c^2}-{2{g_n}j\over\omega r_0 ^ 3 c^2}\right ] ( n_1-n_2)\\ & - \frac{c{\bar{\tau}}}{2\pi{{\ensuremath{\mathcal{n}}}}}(n_1+n_2 ) . \end{split}\ ] ] by choosing a pair of rotationally symmetric ( @xmath401 ) modes that differ only in their directions of propagation and have @xmath451 we find the classical sagnac beat frequency ( [ standard_sagnac_beat_frequency ] ) is modified in this approximation to @xmath452 - \nu[-n,0]\\ & \simeq -\frac{1}{{{\ensuremath{\mathcal{n}}}}^2}\frac{4\omega{{\ensuremath{\mathcal{a}}}}}{\lambda{{\ensuremath{\mathcal{p } } } } } \left[1+{{g_n}m\over r_0 c^2}-{2g_nj\over\omega r_0 ^ 3c^2}\right ] \end{split}\ ] ] where the `` area '' @xmath453 and `` perimeter '' @xmath454 of @xmath423 are defined as @xmath455 in terms of total arclength @xmath429 and winding number @xmath428 of @xmath423 and @xmath456 . using ( [ scales_and_barred_scales ] ) it can be seen that the average torsion @xmath457 is related to the euclidean torsion @xmath458 of the wavetube locus by @xmath459 where @xmath460 . to illustrate the significance of the frenet torsion contribution to ( [ generalized_sagnac_beat_frequency ] ) we consider a wavetube based on a particular torus knot . a type @xmath461 torus knot is a curve @xmath462 specified by a pair of integers @xmath463 and a triple of real numbers @xmath464 : @xmath465\\ { { \bm{\gamma}}}_1(\sigma ) = { { \mathbf i}}\cdot{{\bm{\gamma}}}(\sigma ) = [ v_1 + v_2\cos(u_2\sigma)]\cos(u_1\sigma),\\ { { \bm{\gamma}}}_2(\sigma ) = { { \mathbf j}}\cdot{{\bm{\gamma}}}(\sigma ) = [ v_1 + v_2\cos(u_2\sigma)]\sin(u_1\sigma),\\ { { \bm{\gamma}}}_3(\sigma ) = { { \mathbf k}}\cdot{{\bm{\gamma}}}(\sigma ) = v_3\sin(u_2\sigma),\\ { { { \mathbf c}}}({{\underline{s } } } ) = { { \bm{\gamma}}}\bigl(\sigma({{\underline{s}}})\bigr)\end{gathered}\ ] ] where the euclidean arc parameter @xmath319 satisfies @xmath466 the torus knot @xmath462 is approximated by a multiply - wound circle of euclidean radius @xmath467 . the type @xmath468 torus knot is shown in figure [ figure : knot ] and its frenet curvature and torsion are shown in figures [ figure : curvature ] and [ figure : torsion ] . choosing mks units , so @xmath469 , this particular knot has @xmath470 and so @xmath471 . the classical sagnac beat frequency ( [ standard_sagnac_beat_frequency ] ) of a circular ne wavetube laser ( @xmath472 ) and `` radius '' @xmath473 fixed on the earth ( @xmath474 ) and centred on the poles is @xmath475 . clearly , for @xmath476 and @xmath400 the standard sagnac and torsion contributions to the beat frequency are quite similar and , moreover , in order to determine @xmath275 from the beat frequency both contributions would have to be taken into account . a particular approximation scheme has been developed that permits one to determine the electromagnetic mode structure for a rotating slender wavetube containing a non - conducting isotropic homogeneous dispersionless medium . the possible effects of a weak non - newtonian gravito - magnetic field on these modes has been included . the approximation enables one to identify te and tm type field configurations with respect to the longitudinal axis of the wavetube . although this need not be fixed in space , provided the dimensionless parameters @xmath477 and @xmath288 are smaller than @xmath478 the decomposition remains valid to the order prescribed . in this regime one finds that such a wavetube with non - zero integrated frenet torsion can produce a modification to the classical sagnac beat frequency due to any rotation of the interferometer . to detect effects of gravito - magnetism in this scenario one must adjust the parameters @xmath479 so that the effects due to the dimensionless parameters @xmath480 can be distinguished experimentally from those produced by @xmath481 . there are of course practical limitations that bound @xmath429 and @xmath238 and the possibility of increasing @xmath275 by rotating the interferometer . furthermore the example given in section 7.1 shows that there exist slender geometrical structures based on space curves having small frenet curvature but integrated frenet torsion giving rise to interference effects commensurate with those produced by the rotation of the earth . it is difficult to construct geometrical configurations that meet all the above competing constraints needed to reveal terrestrial gravito - magnetic effects without releasing the condition of relatively constant local frenet curvature . to overcome this limitation it is clear that a more complete analysis of the mode spectra should take into account perturbations induced by variations of frenet curvature of the wavetube . this is a non - trivial modification since it implies that the maxwell equations no longer separate into twisted te and tm modes . however , for appropriate local curvature variations one may extend the approximation scheme using a perturbation approach and the basis of twisted modes found in this analysis . this would yield a clear separation of the effects of frenet torsion and frenet curvature on the sagnac beat frequency . such developments would provide a more complete picture of the delicate interplay between acceleration , geometry , electromagnetism and gravity that arises in any attempt to design an effective ring - laser capable of detecting terrestrial gravito - magnetism . dab , an and rwt are most grateful for the hospitality provided by the department of physics and astronomy , university of canterbury , christchurch , new zealand and for valuable discussions with g stedman , r hurst and r reeves . they are also grateful to m hamilton , university of adelaide , for information on optical polarising devices . dab acknowledges financial support from the royal society , dlw from the marsden fund of the royal society of new zealand and rwt is grateful to bae systems for their support and interest in this research .
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a new approximation scheme , designed to solve the covariant maxwell equations inside a rotating hollow slender conducting cavity ( modelling a ring - laser ) , is constructed .
it is shown that for well - defined conditions there exist te and tm modes with respect to the longitudinal axis of the cavity .
a twisted mode spectrum is found to depend on the integrated frenet torsion of the cavity and this in turn may affect the sagnac beat frequency induced by a non - zero rotation of the cavity .
the analysis is motivated by attempts to use ring - lasers to measure terrestrial gravito - magnetism or the lense - thirring effect produced by the rotation of the earth .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
observations of a sample of three late - type galaxies with low surface - brightness and the radio - weak edge - on galaxy ngc 5907 ( all with a low sfr ) revealed that they all have an unusually high thermal fraction and weak total and regular magnetic fields ( chyy et al . 2007 , dumke et al . however , these objects still follow the total radio - fir correlation , extending it to the lowest values measured so far . hence , these galaxies have a lower fraction of synchrotron emission than galaxies with higher sfr . it is known that the thermal intensity is proportional to the sfr . our findings fit to the equipartition model for the radio - fir correlation ( niklas & beck 1997 ) , according to which the nonthermal emission increases @xmath0 and the _ total _ magnetic field strength @xmath1 increases @xmath2 . + no similar simple relation exists for the _ regular _ magnetic field strength . we integrated the polarization properties in 41 nearby spiral galaxies and found that ( independently of inclination effects ) the degree of polarization is lower ( @xmath3 ) for more luminous galaxies , in particular those for @xmath4 ( stil et al . the radio - brightest galaxies are those with the highest sfr . though a dynamo action needs star formation and supernova remnants as the driving force for velocities in vertical direction , we conclude from our observations that stronger star formation seems to reduce the magnetic field regularity . on kpc - scales , chyy ( 2008 ) analyzed the correlation between magnetic field regularity and sfr locally within one galaxy , ngc 4254 . while he found that the total and random field strength increase locally with sfr , the regular field strength is locally uncorrelated with sfr . we determined the exponential scale heights of the total power emission at @xmath5 cm for four edge - on galaxies ( ngc 253 , ngc 891 , ngc 3628 , ngc 4565 ) for which we have combined interferometer and single - dish data ( vla and the 100-m effelsberg ) . in spite of their different intensities and extents of the radio emission , the vertical _ scale heights _ of the thin disk and the thick disk / halo are similar in this sample ( 300 pc and 1.8 kpc ) ( dumke & krause 1998 , heesen et al . we stress that our sample includes the brightest halo observed so far , ngc 253 , with a very high sfr , as well as one of the weakest halos , ngc 4565 , with a small sfr . for ngc 253 heesen et al . ( this volume ) argued that the synchrotron lifetime ( which is @xmath6 ) mainly determines the vertical scale height of the synchrotron emission and estimated the cosmic ray bulk velocity to @xmath7 km / s . as this is similar to the escape velocity , it shows the presence of a galactic wind in this galaxy . the fact that we observe similar averaged scaleheights at @xmath5 cm for the four galaxies mentioned above imply that the galactic wind velocity is proportional to @xmath8 , and hence proportional to @xmath9 . in a larger sample of 11 edge - on galaxies we found in all of them ( except the inner part of ngc 4631 , see krause 2009 ) mainly a disk - parallel magnetic field along the galactic midplane together with an x - shaped poloidal field in the halo . our sample includes spiral galaxies of different hubble types and sfr , ranging from @xmath10 . the disk - parallel magnetic field is the expected edge - on projection of the spiral magnetic field within the disk as observed in face - on galaxies . it is generally thought to be generated by a mean - field @xmath11-dynamo for which the most easily excited field pattern is the axismmetric spiral ( ass ) field ( e.g. beck et al . the poloidal part of the ass dynamo field alone , however , can not explain the observed x - shaped structures in edge - on galaxies as the field strength there seems to be comparable to that of the large - scale disk field . model calculations of the mean - field @xmath12-dynamo for a disk surrounded by a spherical halo including a _ galactic wind _ ( brandenburg et al . 1993 ) simulated similar field configurations as the observed ones . new mhd simulations are in progress ( see e.g. gressel et al . this volume , hanasz et al . this volume ) which include a galactic wind implicitely . a galactic wind can also solve the helicity problem of dynamo action ( e.g. sur et al . 2007 ) . hence , a galactic wind may be essential for an effective dynamo action , and to explain the observed similar vertical scale heights and x - shaped magnetic field structure in edge - on galaxies .
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from our radio observations of the magnetic field strength and large - scale pattern of spiral galaxies of different hubble types and star formation rates ( sfr ) we conclude that though a high sfr in the disk increases the total magnetic field strength in the disk and the halo the sfr does not change the global field configuration nor influence the global scale heights of the radio emission .
the similar scale heights indicate that the total magnetic field regulates the galactic wind velocities
. the galactic wind itself may be essential for an effective dynamo action .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the first planetary nebula ( pn ) belonging to a globular cluster ( gc ) was discovered more than 85 years ago , in m15 ( pease 1928 ) . it was another six decades before a second gcpn was found , this time in m22 ( gillett et al . 1989 ) . jacoby et al . ( 1997 , hereafter jmf97 ) then carried out a systematic ground - based ccd survey of 133 milky way gcs , using a narrow - band [ ] 5007 filter along with a filter in the neighboring continuum . they discovered two more pne , in the galactic clusters ngc 6441 and pal 6 . the number of pne known in gcs in the local group was raised to five by the serendipitous discovery of a pn in the cluster h5 belonging to the fornax dwarf spheroidal galaxy ( larsen 2008 ) . outside the local group , [ ] emission has been detected in the integrated light of a handful of gcs during spectroscopic investigations , as summarized by minniti & rejkuba ( 2002 ) , zepf et al . ( 2008 ) , chomiuk , strader , & brodie ( 2008 ) , and peacock , zepf , & maccarone ( 2012 ) . ( however , as discussed in 5 , not all of these distant emission sources are actually pne . ) pne in gcs raise two issues related to stellar evolution . the first is _ why are there so few pne in gcs ? _ jmf97 posed this question because one would expect to find @xmath016 pne in the milky way gcs on the basis of the total luminosity of the galactic gc system , a pn lifetime of @xmath02@xmath1 yr , and the assumption that every star produces a visible pn near the end of its life . ( the prediction comes basically from an application of the `` fuel - consumption theorem , '' as defined by renzini & buzzoni 1986 . ) in order to explain the smaller number actually observed , jmf97 suggested that the assumption that every star produces a pn may be incorrect in gcs . in fact , single stars in very old populations , having started their lives at about @xmath2 , leave the asymptotic giant branch ( agb ) with masses reduced to as low as @xmath0@xmath3 ( alves , bond , & livio 2000 ; hereafter abl00 ) to @xmath0@xmath4 ( kalirai et al . 2009 ) . the theoretical post - agb evolutionary timescales of such low - mass remnants are so long ( e.g. , schoenberner 1983 ) that any nebular material ejected at the end of the agb phase has time to disperse before the central star becomes hot enough to ionize it . thus , the single stars now evolving in gcs would not be expected to produce any visible ionized pne . now the question becomes _ why are there any pne in gcs at all ? _ the answer probably lies in the evolution of binary stars . there are ( at least ) two ways that binaries can produce pne in populations in which single stars can not . ( 1 ) coalescence of two stars in a binary near the main sequence could produce first a blue straggler , and eventually a higher - mass post - agb remnant that _ would _ evolve rapidly enough to ionize a pn . abl00 detected no photometric variations for k648 , the central star of the pn ps 1 in m15 , consistent with it being a merger remnant . ( 2 ) or a red giant or agb star may undergo a common - envelope ( ce ) interaction with a companion , rapidly exposing the giant s hot core , and thus promptly subjecting the ejecta to ionizing radiation . these and other scenarios to account for the presence of pne in gcs have been discussed by abl00 , ciardullo et al . ( 2005 ) , buell ( 2012 ) , jacoby et al . ( 2013 ; hereafter jcd13 ) , and others . they are part of a larger conceptual framework in which it has been increasingly recognized that binary interactions are likely to be a major , if not dominant , formation channel for pne in all populations ( e.g. , bond & livio 1990 ; bond 2000 ; de marco 2009 ; and references therein ) . the binary - merger hypothesis can be tested by determining the luminosities of central stars of pne in gcs , and then inferring their masses from theoretical core - mass / luminosity relations . abl00 used the wide field planetary camera 2 ( wfpc2 ) on the _ hubble space telescope _ ( _ hst _ ) to carry out photometry of k648 . the absolute luminosity of the star implied a mass of @xmath5 . this is significantly higher than the masses of remnants of single stars in gcs ( see above ) , giving the star a fast enough post - agb evolution for it to have ionized the ejecta before they had time to dissipate . abl00 concluded that the central star must have achieved its high mass as a result of a merger . _ hst _ imaging of all four pne in galactic gcs , and photometry of their central stars , have been collected and discussed by jacoby et al . ( 2014 , 2015 ) . apart from k648 , the evidence for high stellar masses resulting from binary mergers has remained less compelling . in fact , if the pn were ejected as a consequence of a ce interaction , the mass of its central star would be unlikely to differ much from those of remnants of single - star evolution , or could even be lower . jacoby et al . do , however , argue that the morphologies of these gcpne are at least suggestive of ejection from a binary interaction . a potential test is to search for x - ray emission , arising from a synchronously rotating , active , late - type companion star to the pn nucleus . variable x - ray emission from k648 has in fact been detected by hannikainen et al . ( 2005)which , if due to the central star rather than the surrounding pn , would argue against the merger scenario i discussed above . further progress could be made with a larger sample than the five known pne in the milky way and fornax gc populations . there are many hundreds of gcs known in the andromeda galaxy , m31 ; and smaller numbers are known in m33 , the magellanic clouds , and other members of the local group . as noted by many authors , galaxies like m31 have experienced different evolutionary histories than our galaxy , which might be reflected in systematic differences in their gc systems ( e.g. , van den bergh 2010 and references therein ) . conceivably , for example , there could be a population of gcs with intermediate rather than extremely old ages , which would be richer in pne than the milky way gcs . jcd13 obtained integrated spectra of 274 old gcs in m31 , searching for emission at [ ] 5007 that would indicate the presence of a pn in the cluster . three candidate pne were found among this sample of old gcs . in the complementary survey discussed here , i have used _ hst _ to search for pne in gcs belonging to local group galaxies , especially m31 , by obtaining direct narrow - band [ ] images . in my search for pne in local group gcs , i used _ hst _ in its `` snapshot '' mode , in which short exposures are inserted into gaps in the telescope schedule that remain once the primary observations have been sequenced as optimally as possible . my observations were made during two _ hst _ proposal cycles . in cycle 16 ( program i d snap-11218 ) i submitted a list of 125 target gcs , of which 41 were successfully observed . in cycle 17 ( snap-11714 ) i submitted 100 targets , of which 23 were observed . the cycle 16 observations were made with wfpc2 , from 2007 july 10 to 2008 november 23 . in cycle 17 , i used the uvis channel of the newly installed wide field camera 3 ( wfc3 ) , and these observations were made between 2009 september 1 and 2011 september 29 . nine of the cycle 17 observations were repeats of wfpc2 targets , using the more sensitive wfc3 . eleven additional gcs were serendipitously present in the frames along with the primary targets , for a total of 66 unique clusters , belonging to eight different galaxies . the list of candidate targets for both cycles was developed as follows . the overall aim was primarily to maximize the total integrated luminosity of the cluster sample , in order to maximize the probability of discovering pne ; and secondarily to provide targets widely distributed across the sky . ( 1 ) for the magellanic clouds , i chose the 8 `` population ii '' clusters with integrated visual absolute magnitudes of @xmath6 , as listed by olszewski , suntzeff , & mateo ( 1996 , their table 1 ) . ( 2 ) in m33 , i selected the 5 brightest red ( i.e. , old ) gcs listed by christian & schommer ( 1988 , their table vii ) ; all of them are brighter than @xmath7 ( @xmath8 ) . ( 3 ) in the dwarf irregular galaxies ngc 6822 and wlm , i chose hubble vii in the former , and wlm-1 in the latter . hubble vii , with an integrated @xmath9 ( @xmath10 ) is an old , metal - poor gc ( e.g. , wyder , hodge , & zucker 2000 ) . wlm-1 is the lone gc known in its galaxy , and is also very old ( e.g. , hodge et al . 1999 , who give @xmath11 and @xmath12 ) . ( 4 ) for the fornax dwarf spheroidal , out of its five known gcs i included the two bright clusters ngc 1049 and h2 , for which strader et al.(2003 ) give @xmath13 magnitudes of 12.6 and 13.5 ( @xmath14 and @xmath15 ) . ( 5 ) for the dwarf elliptical ngc 147 , i selected hodge ii , with a @xmath13 magnitude of 16.5 ( sharina et al . 2006 ) , corresponding to @xmath16 . ( 6 ) i then selected gcs in the m31 system in order of increasing integrated @xmath13 magnitudes from the catalog of galleti et al . ( 2004 ) , which is maintained online , until the target lists were filled to the allocated numbers . these lists contained clusters with integrated @xmath13 magnitudes ranging from 13.8 to 16.8 ( @xmath17 to @xmath18 ) . with both wfpc2 and wfc3 , i observed each cluster in the narrow - band f502n [ ] 5007 filter , and the broad - band f555w @xmath13 " filter . exposure times for most of the gcs were @xmath19 s and @xmath20 s in f502n and f555w , respectively , for wfpc2 , or @xmath21 s and @xmath22 s in f502n and f555w for wfc3 . for the nearby magellanic cloud gcs , the wfpc2 exposures were shortened to @xmath23 s and @xmath24 s. the field of view of wfpc2 is @xmath25 for the pc chip ( i placed the target gcs near the corner of the pc field least affected by charge - transfer inefficiency [ cti ] ) , plus @xmath26 for each of the three neighboring wide - field ( wf ) chips . the plate scale is @xmath27 for pc , and @xmath28 for the wf . for the wfc3 imaging i centered each target in a @xmath29 pixel subarray , and the plate scale is @xmath30 , giving a field of view of @xmath31 . table 1 lists the 66 gcs that were imaged . the nomenclature is as given in 2.2 ; for m31 i have used the designations of the galleti et al . ( 2004 ) and peacock et al . ( 2010 ) catalogs . column 2 gives their integrated visual absolute magnitudes , calculated using integrated @xmath13 magnitudes and distances and reddenings from the sources cited in the previous subsection . serendipitously , nine of the wfpc2 frames contained a second , usually fainter gc in one of the wf chips , as indicated in the footnotes to the table . in one case , the smaller wfc3 field contained a second cluster : ngc 147 hodge iii ( @xmath32 , @xmath33 ) was in the field of view when hodge ii was observed . all exposures were dithered for cosmic - ray removal by taking two exposures in each filter and moving the telescope pointing by a few pixels between exposures . for the wfpc2 frames , i used iraf routines to align and combine them , while for wfc3 i used the pipeline drizzle - combined images from the _ hst _ archive . a pn will be bright in the narrow - band f502n filter centered near the emission line of [ ] at 5007 . the broad - band f555w filter also transmits at 5007 ( where the system throughput is actually higher than for f502n ) , as well as at other emission lines in pne , the strongest of which are [ ] 4959 , h@xmath34 and h@xmath35 , and 5876 . to estimate the expected relative count rates for a typical pn in these two filters , i used the line intensities for the halo pn bobn 1 given by otsuka et al . i convolved them with the system throughputs for both cameras in the two filters , as given in the respective _ instrument handbooks _ ( wfpc2 : mcmaster & biretta 2008 ; wfc3 : bond & quijano 2007 ) . the resulting predicted count - rate ratios for the pn are @xmath36 for wfpc2 , and 2.1 for wfc3 . drizzled wfc3 images from the _ hst _ archive are in units of counts @xmath37 , so a typical pn would be approximately twice as bright in f555w as in f502n . however , wfpc2 images from the archive pipeline give total counts accumulated during the exposure . since the ratio of exposure times was @xmath38 ( in most cases ) for wfpc2 , the ratio of expected integrated counts for a pn in the two filters is thus @xmath39 . by contrast , a star will appear much brighter in the broad - band f555w filter than in narrow - band f502n . for a typical gc red giant , the count - rate ratios for a star will be @xmath40 and 30 for wfpc2 and wfc3 , respectively ( the wfc3 ratio being smaller because its f502n bandpass is wider than the one used in wfpc2 ) . at the distance of m31 , the fwhm of stellar images in _ hst _ frames of @xmath0@xmath41 corresponds to a linear diameter of @xmath00.2 pc . all but the largest pne will appear essentially stellar at this resolution . thus i searched the images for point sources of comparable brightness ( within factors of @xmath02 or 0.5 , depending on the camera ) in both filters , and rejected sources that were considerably brighter in f555w . i did this using two different methods : ( 1 ) first i simply visually blinked the f502n and f555w images . this technique works very well , because there are relatively few point - like sources detected in the f502n frames , especially with wfpc2 , and those that are present are therefore conspicuous . apart from the rare true pne , nearly all of the apparent sources seen at f502n fell into two categories : ( a ) bright stars ( mostly gc red giants ) , easily recognized because they are much brighter in the f555w frames , and ( b ) spurious artifacts , particularly cases where cosmic rays struck the same pixels in both of the f502n frames that were combined , sometimes producing artifacts that resemble real sources ; however , these are also very easy to recognize and reject because they lack counterparts in the f555w images . ( this valuable check would not have been possible if a broad - band filter had been chosen that rejected the 5007 line . ) ( 2 ) it was , however , difficult to blink the images near the centers of the gcs , because of the high surface brightness in the f555w frames . thus i also created an image of each gc in which i calculated the ratio of the two frames , and searched them ( again visually ) with a display windowing that emphasized sources with similar count rates in both filters . this method worked well in the centers of the clusters . as noted above , the target clusters were placed in the pc chip of wfpc2 ; however , i also searched the three neighboring wf chips , and found several new pne in the fields surrounding the gcs , as well as recovering many previously cataloged pne . i also searched the entire frames taken with wfc3 , likewise finding a few field pne . figure 1 illustrates the two search techniques . in the example shown , i independently recovered a known field pn in m31 , no . 285 in the catalog of ford & jacoby ( 1978 ) . the left frame shows the f502n [ ] image , with the pn marked in the center , and also showing a bright star to the lower left . the middle frame shows the @xmath13-band f555w image ; now numerous stars are also detected , and the field star is much brighter , but the pn is nearly unchanged . the right frame shows the ratio image , f502n / f555w , with the display stretched to show objects that have similar brightnesses in both frames and reject sources with low f502n / f555w ratios . now the stars have disappeared , leaving only the pn , and verifying that it is a source emitting at 5007 . i determined a nominal limiting magnitude for the wfpc2 f502n frames as follows . i marked apparent point - like stellar sources on a frame of the well - resolved fornax h2 cluster over a range of brightnesses from obvious detections down to sources so faint that they were likely to be spurious . ( by 20078 , the wfpc2 , which had been onboard _ hst _ since 1993 , was suffering badly from cti and hot pixels in addition to the usual cosmic - ray hits , leading to the presence of many low - level streaks and other artifacts in these frames . ) i then compared the f502n frame with the f555w frame to determine which of the marked sources had brightened ( indicating that they were real stars ) or disappeared ( indicating that they were artifacts ) . i found a strong cutoff at a flux level such that 100% of the sources brighter than this level were real , but below this level the fraction of spurious detections rose rapidly . i then calibrated this flux level , using aperture photometry and the reduction procedure described in the _ wfpc2 data handbook _ ( gonzaga & biretta 2010 , 5.2.5 ) . i included cti corrections as formulated by dolphin ( 2009 ) . the resulting limiting line flux for the wfpc2 frames is @xmath42 . it is customary in work on extragalactic pne ( e.g. , ciardullo et al . 1989 ) to use `` [ ] @xmath435007 magnitudes , '' defined by @xmath44 . on this scale , the source detections in my wfpc2 f502n frames are complete to @xmath45 . at the distance of m31 , this limit reaches more than 3.5 mag into the pn luminosity function in the andromeda bulge ( e.g. , fig . 9 in jcd13 ) . this magnitude limit applies to isolated sources . the limit is brighter in the central regions of gcs , due to the bright backgrounds and source confusion . using artificial - star experiments ( i.e. , adding an artificial pn image to the center of a gc ) , i find that the limiting magnitude is about 0.9 mag brighter , or about @xmath46 , at the center of a m31 gc of typical central surface brightness . for wfc3 , the limiting magnitudes are considerably fainter , since wfc3 is a factor of @xmath03.2 more sensitive , has lower readout noise , and was very considerably less affected by cti than was wfpc2 at the late stage of its lifetime . the wfc3 frames are cosmetically quite superior to the wfpc2 images , exhibiting very few low - level artifacts apart from occasional instances of a cosmic ray striking the same pixels in both f502n images . using the same techniques , i estimated that the limiting detection magnitude for the wfc3 exposures was @xmath47 for isolated sources , and again about 0.9 mag brighter for pne at the centers of typical m31 gcs . i found only one out of the 66 gcs that were imaged in the two snapshot programs to have nearby candidate [ ] @xmath435007 emission sources . i identified two sources in the vicinity of the m31 cluster b086 , as shown in figure 2 . details of these sources are given in table 2 . the brighter of them , here designated m31 b086 - 2 , has been cataloged previously by meyssonnier , lequeux , & azzopardi ( 1993 ; no . 481 in their list of unresolved emission - line objects in m31 ) , and by azimlu , marciniak , & barmby ( 2011 ; no . 1596 in their catalog of objects considered to be regions in m31 ) . the fainter source , m31 b086 - 1 , has to my knowledge not been cataloged previously . both objects appear to be unresolved at wfc3 resolution . i converted the measured count rates for both of them to @xmath48 magnitudes , using the wfc3 photometric zero - points given online . results are in table 2 . since both sources are unresolved , and have values of @xmath48 consistent with those of pne in m31 , it appears plausible that they are pne ( although it can not be ruled out entirely that either one is a symbiotic binary , nova at the nebular stage , a very compact region , or other 5007 - emitting source ) . however , b086 lies in a very rich field only @xmath49 from the nucleus of m31 , and fairly near a spiral arm and dust lane . moreover , both pn candidates , at separations of @xmath50 and @xmath51 from the center of b086 , are well outside the half - light radius of b086 , measured on my images to be @xmath52 . thus , in the absence of a spectroscopic followup and radial - velocity confirmation , it is questionable whether either object is physically associated with b086 . if the brighter source had been centered in the gc , it would have added only 3.5% to the flux of the cluster within a diameter of @xmath53 , in a bandpass similar to that of the f502n filter . the fainter source would constitute only 0.6 % of the cluster flux if centered in the cluster . thus , ground - based direct imaging would be unlikely to have detected either pn candidate if located near the cluster center . b086 was observed by jcd13 , but since my pn candidates would have been well outside their spectroscopic fiber aperture , it is not surprising that they were not detected . in the course of examining the _ hst _ frames , i noted a number of candidate @xmath54 sources in the fields surrounding the target gcs . these field sources are listed in table 3 . although they are too distant from the gcs to be considered cluster members , i give them designations based on the name of the target cluster . as an index of my survey completeness , i include previously cataloged objects that i independently recovered , as well as new discoveries . the table includes @xmath48 values measured from my material ; i adjusted the zero - point for these magnitudes so as to match the absolute line fluxes listed by ciardullo et al . ( 1989 ) for the sources in common . the rms scatter was 0.2 mag , consistent with the moderately low snr of the wfpc2 [ ] snapshot images . the final column in table 3 gives cross references for the previously known sources . nearly all of the sources brighter than @xmath55 have been cataloged previously . all of the listed objects appear stellar at _ hst _ resolution , making it likely that most of them are pne , even though some were included in earlier compilations of regions . however , one object appears to coincide with a classical nova . i augmented my snapshot observations by searching the _ hst _ archive for frames taken by other investigators with any of the _ hst _ cameras , using a [ ] f502n filter , in the vicinity of m31 , and then checking whether any gcs were within the fields of view . surprisingly , only two _ hst _ programs , other than my own , have ever used this filter at pointings lying within @xmath56 of the center of m31 . fortuitously , 12 cataloged gcs lie within the fields observed in these two programs . images were obtained with the advanced camera for surveys ( acs ) or wfc3 , as listed in table 4 . three of these clusters had also been imaged in my snapshot survey , so the grand total number of local group gcs outside the milky way imaged by _ hst _ in a f502n filter is 75 . the f502n images in the two archival programs were accompanied either by frames in f547 m or f550 m ( which exclude the 5007 emission line ) or in f555w ( which includes 5007 as described above ) . out of the 12 clusters , one very convincing pn candidate was found within the m31 cluster b477 , as illustrated in figure 3 . details are given in table 4 . this emission - line object had been listed by walterbos & braun ( 1992 ; no . 683 , which they described as being unresolved , in their catalog of h@xmath34/ [ ] sources in m31 ) , and by merrett et al . ( 2006 , hereafter mmd06 ; no . 446 in their catalog of pne in m31 ) . the @xmath48 value in table 4 is quoted from mmd06 . m31-b477 was not one of the clusters observed spectroscopically by jcd13 , and the association with the star cluster has apparently not been recognized until now . b477 is , however , not considered to be an old gc of the type targeted in my snapshot survey : kang et al . ( 2012 ) give an age of only 325 myr , based on _ galaxy evolution explorer _ uv photometry . morphologically , b477 also does not resemble a gc , as can be seen in figure 3 . the present imaging survey is complementary in several ways to the ground - based spectroscopic search for pne in the m31 gcs carried out by jcd13 . my survey concentrated on the brightest gcs in m31 ( and the rest of the local group ) , while their large - scale spectroscopic program targeted a wide range of cluster luminosities in m31 . their method is particularly favorable for pne in fainter gcs , in which there is less dilution of the pn 5007 emission by cluster starlight in the integrated spectrum . their spectroscopic apertures had diameters of @xmath57 , which captures most of the cluster light but would still miss more widely separated pne , whereas i covered the entire cluster plus a large surrounding field . jcd13 observed 274 old gcs in m31 , as compared to only 75 imaged in my survey ( plus two archival programs ) throughout the local group . of the 56 gcs in m31 that i observed or had archival f502n images , jcd13 obtained spectra of 39 of them . jcd13 were able to reach fainter limiting magnitudes than my wfpc2 frames , down to about @xmath58 vs. my 24.1 , because they used a larger telescope ( 3.5 m vs. 2.4 m ) , and much longer integration times ( 210 minutes vs. 6001000 s ) . the limiting magnitudes for my wfc3 frames are , however , similar to those of jcd13 . the spectroscopic observations are susceptible to spurious detections of ambient [ ] emission from diffuse nebulae superposed on the gc , whereas direct high - resolution imaging of point sources does not suffer from this problem . one inadequacy of direct imaging in [ ] is that it does not provide a line profile , line ratios such as [ ] /h@xmath35 , or other spectroscopic information , all of which help discriminate pne from other emission - line sources . for example , zepf et al . ( 2008 ) found very broad [ ] emission from a source within a gc in the virgo galaxy m49 , attributed to an accreting black hole ( bh ) rather than a pn . irwin et al . ( 2010 ) found [ ] emission , unaccompanied by balmer emission , from a gc in the fornax galaxy ngc1399 ; they discussed this source in terms of a white - dwarf being tidally disrupted by an intermediate - mass bh , and alternative non - pn scenarios for this object have been proposed by maccarone & warner ( 2011 ) and clausen et al.(2012 ) . jcd13 found candidate pne in three old gcs in m31 : nb89 , b115 , and bh16 . of these , only nb89 was observed in my snapshot programs , and only because it happens to lie a few arcseconds away from the bright primary target gc b127 . much deeper _ hst _ [ ] imaging of nb89two exposures of 2700 s each was obtained in the archival wfc3 program go-12174 , as summarized in table 4 . the pn candidate in nb89 has @xmath59 according to jcd13 . an isolated point source this bright would have been detected easily in the deep wfc3 frames , yet no 5007 point source is seen in the images . it is conceivable that the pn coincides exactly with a red giant in the cluster . however , a more likely possibility is that the emission arises from ambient nebular emission that filled jcd13 s aperture , rather than from a point source indeed , jcd13 raised this as a strong possibility . in fact , nb89 does lie exactly superposed on a filament of the spiral - shaped emission region discovered by ciardullo et al . ( 1988 ) in their deep ground - based h@xmath34 imaging of the m31 nucleus . although it is unclear whether this low - excitation region has sufficiently strong 5007 emission for this explanation , the lack of a point - like source in the _ hst _ [ ] images supports this view . there is no available narrow - band 5007 _ hst _ imaging for the other two clusters , b115 and bh16 , in which jcd13 detected [ ] emission , with @xmath48 values of 22.6 and 25.3 , respectively . however , there are archival broad - band optical and near - uv images for both clusters , obtained in the andromeda treasury program go-12058 ( pi j. dalcanton ) . veyette et al . ( 2014 ) have shown that pne have distinctive optical and nuv broad - band colors . this is especially true shortward of the balmer jump , where there are significant contributions to the flux from the nebular continuum and numerous emission lines , and from the underlying photosphere of the central star . in figure 4 i show _ hst _ images of b115 and bh16 , taken with wfc3 in the near uv ( f275w ) and optical uv ( f336w ) , and with acs at approximately the @xmath60 band ( f475w ) . the sizes of the frames are similar to those of the spectroscopic apertures used by jcd13 . in both cases , there is a conspicuous , very blue star - like object within the field . it is probable that these are the pne detected by jcd13 , although high - resolution narrow - band imaging would be needed to absolutely confirm the identifications . similar sets of broad - band wfc3 and acs images exist for nb89 , but in this case there is no obvious blue star in or near the cluster . this is consistent with the above suggestion that the [ ] emission in nb89 is from ambient diffuse nebulosity rather than a pn . the current online version of the galleti et al . ( 2004 ) m31 catalog ( version 5 , 2012 august ) has 2060 entries , which include confirmed gcs , candidate gcs , and candidate objects subsequently shown not to be genuine gcs . the mmd06 catalog of pne in m31 contains 3300 confirmed pne and pn candidates . i intercompared these two catalogs , finding 44 cases where the coordinates agreed within @xmath61 . a majority of these objects are actually candidate gcs that have been reclassified as regions by caldwell et al . ( 2009 ) , and are thus neither gcs nor pne . sixteen of the gcs have not been classified ; most of them are from the catalog of kim et al . ( 2007 ) , and i suspect that they are also misclassified regions . of these 44 clusters , only seven were included in the spectroscopic observations of jcd13 ( b075 , b105 , bh16 , b530 , sk052a , sk070a , and sk104a ) . jcd13 reported 5007 emission only in the old cluster bh16 ( see 5 above ) . for the other six , the nominal separations of the emission - line objects from the clusters are larger than the jcd13 aperture radius . there are five cases where an m31 cluster classified as an `` old '' gc by caldwell et al . ( 2009 , 2011 ) , peacock et al . ( 2010 ) , strader et al . ( 2011 ) , and/or jcd13 has an mmd06 source lying within @xmath61 . these are listed in table 6 . of these five , there are archival _ hst _ images available for only two . ( 1 ) bh16 was discussed in the previous section , in which i suggested that a blue point - like source to the se of the cluster center , which would have been within jcd13 s spectroscopic aperture , is the pn . curiously , however , the pn detected by jcd13 does not appear to be the object cataloged as mmd06 1360 , which differs substantially in both radial velocity and @xmath48 from the jcd13 pn . the _ hst _ images show a bright blue star lying @xmath62 wsw of the cluster , which may be the mmd06 source . ( 2 ) sk104a also has a neighboring blue star in broad - band _ hst _ images , lying @xmath63 nne from the cluster center . this is possibly the pn candidate , but narrow - band imaging and/or spectroscopy is needed to confirm this . i used _ hst _ snapshot observing to obtain [ ] 5007 images of 55 of the brightest `` population ii '' gcs in local group galaxies outside the milky way . an additional 11 fainter clusters serendipitously fell within these frames . a search of the _ hst _ archive found [ ] images of nine more gcs in m31 , for a grand total of 75 local - group gcs . i searched all 75 of these clusters for point - like 5007 sources , which would be strong pn candidates . among these old gcs of the local group , i found only the m31 cluster b086 to have two pn candidates in its vicinity . however , they are at such large angular separations that their cluster membership is doubtful . one very convincing pn candidate was found in images of the young m31 cluster b477 . i also investigated _ hst _ images of the three candidate pne found in old clusters by jcd13 in their ground - based spectroscopic survey of m31 gcs . one of them appears to be a case where ambient diffuse nebulosity produced the 5007 emission , rather than a true pn . for the other two , there are only broad - band _ hst _ images available , but both clusters contain a bright blue point source which is plausibly the pn . a comparison of two extensive catalogs of gcs and pne in m31 revealed several additional candidate pne within a few arcseconds of old gcs , but spectroscopy and narrow - band _ hst _ images will be needed to verify the cluster membership . the total luminosity of the 157 known gcs in the milky way system , determined from the data in the harris ( 1996 ) compliation ( 2010 editionharris / mwgc.dat ] ) , corresponds to a visual absolute magnitude of @xmath64 . for my sample of 75 local - group gcs , although only numbering about half of the milky way clusters , the total luminosity is actually slightly higher , at @xmath65 . in the milky way gc system , there are four known pne ( jmf97 ; table 4 of jcd13 ) . if they were at the distance of m31 , they would have @xmath48 magnitudes of 21.3 , 25.3 , 27.4 , and 29.7 . given the limiting magnitudes of my surveys ( 3.2 ) , only one of them would be bright enough to be detected in my wfpc2 snapshots , and one more in my deeper wfc3 frames . in the actual old - population m31 clusters , we have two very probable pne those in b115 and bh16along with several more candidates ( 4.1,4.3 ) , which are as yet unconfirmed . as noted in 1 , there is also a pn in one of the gcs of the fornax dwarf spheroidal ( larsen 2008 ) . thus , within the small - number statistics , the incidences of pne in the old gcs of the milky way and in the rest of the local group are very similar . according to our understanding of stellar evolution , we do not expect that pne can be formed by single stars in the ancient populations of gcs2357 in m22 ( gillett et al . 1989 ) and the pn in the fornax gc h5 ( larsen 2008 ) . this may raise the possibility for these two objects of a late thermal pulse ( in a single star ) , similar to that invoked to explain the compact h - deficient [ ] -emitting nebula surrounding v605 aquilae ( clayton et al . 2013 and references therein ) . detailed discussion is given by jacoby et al . ( 2015 ) ] . binary - star mergers and ce events provide a plausible explanation for the fact that a few pne nevertheless exist in these clusters . one observational test would be to search for evidence of binarity in the central stars of pne in milky way gcs : those whose masses appear to be too low to have resulted from mergers are likely to have gone through a ce interaction , leaving them as close but still un - merged binaries . i also found some 60 point - like [ ] 5007 sources in the fields surrounding the gcs ( all but one of them in m31 ) ; many of them were already cataloged , but the stellar appearances in the _ hst _ images verify that they are highly probable field pne . support for program numbers snap-11218 and snap-11714 was provided by nasa through grants from the space telescope science institute , which is operated by the association of universities for research in astronomy , incorporated , under nasa contract nas5 - 26555 . helpful assistance and advice on the _ hst _ scheduling was given by alison vick , denise taylor , sylvia baggett , ronald gilliland , and larry petro . this research has made extensive use of the simbad database , operated at cds , strasbourg , france ; and of the catalogs of galleti et al.(2004 ) , merrett et al . ( 2006 ) , and harris ( 1996 ) . lcll fornax : h2 & @xmath15 & 2008 - 11 - 23 & 2010 - 01 - 10 + fornax : ngc 1049 & @xmath66 & 2008 - 09 - 19 & 2010 - 02 - 01 + lmc : ngc 1466 & @xmath67 & & 2009 - 09 - 01 + lmc : ngc 1786 & @xmath67 & 2008 - 09 - 19 & 2010 - 02 - 06 + lmc : ngc 1835 & @xmath68 & 2007 - 08 - 28 & + lmc : ngc 1916 & @xmath69 & 2008 - 06 - 01 & + lmc : ngc 2005 & @xmath70 & 2008 - 11 - 22 & 2009 - 12 - 05 + lmc : ngc 2019 & @xmath67 & 2008 - 08 - 26 & 2009 - 10 - 28 + lmc : ngc 2210 & @xmath66 & 2007 - 12 - 26 & + m31-b005 & @xmath69 & 2008 - 11 - 15 & + m31-b008 & @xmath71 & & 2010 - 01 - 02 + m31-b019 & @xmath72 & 2008 - 08 - 23 & + m31-b023 & @xmath73 & 2008 - 08 - 05 & 2011 - 09 - 05 + m31-b027 & @xmath69 & 2007 - 08 - 28 & + m31-b037 & @xmath71 & 2007 - 09 - 13 & + m31-b041 & @xmath74 & 2007 - 09 - 13 & + m31-b058 & @xmath72 & 2008 - 07 - 22 & + m31-b061 & @xmath75 & 2007 - 07 - 17 & + m31-b063 & @xmath76 & 2007 - 07 - 17 & + m31-b082 & @xmath69 & 2008 - 07 - 21 & + m31-b086 & @xmath77 & & 2011 - 01 - 26 + m31-b097d & @xmath78 & 2008 - 07 - 28 & + m31-b103d & @xmath79 & 2008 - 09 - 09 & + m31-b110 & @xmath80 & 2008 - 11 - 15 & + m31-b124 & @xmath81 & 2008 - 11 - 16 & + m31-b127 & @xmath82 & 2008 - 11 - 16 & + m31-b158 & @xmath83 & 2008 - 07 - 28 & + m31-b171 & @xmath77 & & 2011 - 09 - 29 + m31-b174 & @xmath84 & 2008 - 11 - 15 & + m31-b179 & @xmath68 & 2008 - 08 - 10 & + m31-b192 & @xmath85 & 2008 - 11 - 15 & + m31-b193 & @xmath68 & 2008 - 11 - 15 & + m31-b225 & @xmath73 & 2008 - 06 - 16 & + m31-b293 & @xmath86 & 2007 - 09 - 24 & + m31-b311 & @xmath84 & 2008 - 06 - 19 & + m31-b312 & @xmath69 & 2008 - 11 - 17 & 2010 - 09 - 03 + m31-b327 & @xmath66 & 2007 - 09 - 12 & + m31-b338 & @xmath87 & 2008 - 06 - 19 & + m31-b343 & @xmath86 & & 2011 - 08 - 14 + m31-b366 & @xmath88 & & 2010 - 09 - 04 + m31-b373 & @xmath76 & 2008 - 07 - 24 & + m31-b379 & @xmath89 & 2007 - 09 - 30 & + m31-b384 & @xmath90 & & 2009 - 10 - 09 + m31-b386 & @xmath69 & 2008 - 07 - 28 & + m31-b405 & @xmath77 & 2008 - 06 - 14 & + m31-b407 & @xmath91 & 2008 - 06 - 18 & + m31-b472 & @xmath77 & 2008 - 09 - 09 & + m31-b480 & @xmath92 & 2008 - 07 - 24 & + m31-b514 & @xmath90 & 2007 - 07 - 19 & + m31-g001 & @xmath93 & 2007 - 07 - 16 & + m31-g002 & @xmath90 & 2008 - 09 - 24 & + m31-mcgc1 & @xmath91 & & 2009 - 09 - 30 + m31-mcgc3 & @xmath86 & & 2009 - 11 - 09 + m31-mcgc5 & @xmath89 & & 2009 - 12 - 02 + m31-nb89 & @xmath94 & 2008 - 11 - 16 & + m31-vdb0 & @xmath81 & 2007 - 09 - 12 & + m33-mkk33 & @xmath66 & & 2011 - 06 - 16 + m33-r12 & @xmath66 & & 2009 - 12 - 02 + m33-r14 & @xmath75 & & 2010 - 02 - 26 + m33-sm310 & @xmath95 & & 2009 - 12 - 02 + m33-u49 & @xmath96 & & 2009 - 11 - 26 + ngc 147 : hodge ii & @xmath97 & 2008 - 11 - 21 & 2010 - 06 - 17 + ngc 147 : hodge iii & @xmath79 & 2008 - 11 - 21 & + ngc 6822 : h vii & @xmath98 & 2007 - 07 - 10 & + smc : ngc 121 & @xmath71 & 2007 - 11 - 19 & + wlm : wlm-1 & @xmath90 & 2008 - 09 - 08 & 2009 - 10 - 03 + lllll ngc 147 hii-1 & 00:33:12.62 & 48:28:17.5 & 22.6 & fjj ngc 147 - 1 + m31 b005 - 1 & 00:40:16.29 & 40:43:23.2 & 24.0 & amb 551 ; mmd 1944 + m31 b195d-1 & 00:40:25.75 & 40:37:06.0 & 23.2 & mmd 2052 + m31 b195d-2 & 00:40:30.82 & 40:36:53.8 & 20.7 & mla 47 ; hkpn 47 ; mmd 2049 + m31 b023 - 1 & 00:40:58.27 & 41:14:16.8 & 23.5 & + m31 b037 - 1 & 00:41:37.51 & 41:15:10.6 & 22.2 & mla 378 ; mmd 1052 + m31 b063 - 1 & 00:42:06.30 & 41:29:29.5 & 24.0 & + m31 b086 - 3 & 00:42:16.99 & 41:14:02.3 & 24.6 & + m31 b082 - 1 & 00:42:17.84 & 41:01:15.6 & 22.4 & mla 485 + m31 b127 - 1 & 00:42:37.07 & 41:14:34.8 & 20.6 & fj 54 ; mmd 2811 + m31 b127 - 2 & 00:42:37.47 & 41:14:34.3 & 21.9 & fj 71 ; mmd 2826 + m31 b127 - 3 & 00:42:38.26 & 41:15:33.4 & 21.4 & fj 35 ; mmd 2815 + m31 b127 - 4 & 00:42:38.36 & 41:14:26.9 & 22.6 & fj 143 ; mmd 2827 + m31 b127 - 5 & 00:42:38.38 & 41:14:33.9 & 21.3 & fj 55 ; mmd 2810 + m31 b127 - 6 & 00:42:38.95 & 41:14:56.0 & 21.9 & fj 70 ; mmd 2812 + m31 b127 - 7 & 00:42:39.39 & 41:14:17.1 & 22.1 & fj 65 ; mmd 2809 + m31 b127 - 8 & 00:42:39.72 & 41:15:36.3 & 21.0 & fj 18 ; mmd 2817 + m31 b127 - 9 & 00:42:39.75 & 41:15:49.3 & 20.9 & fj 17 ; mmd 2825 + m31 b127 - 10 & 00:42:40.08 & 41:14:37.8 & 22.0 & fj 66 ; mmd 2828 + m31 b127 - 11 & 00:42:40.12 & 41:13:23.9 & 22.8 & fj 159 ; mmd 2822 + m31 b127 - 12 & 00:42:40.18 & 41:15:32.7 & 23.1 & fj 334 + m31 b127 - 13 & 00:42:40.20 & 41:13:50.3 & 23.2 & fj 160 ; mmd 2824 + m31 b127 - 14 & 00:42:40.28 & 41:15:44.2 & 23.3 & fj 333 + m31 b127 - 15 & 00:42:40.34 & 41:14:09.5 & 22.0 & fj 181 ; mmd 2823 + m31 b127 - 16 & 00:42:40.69 & 41:14:09.4 & 20.9 & fj 56 ; mmd 1250 + m31 b127 - 17 & 00:42:41.34 & 41:14:06.9 & 22.1 & fj 141 ; mmd 1251 + m31 b127 - 18 & 00:42:42.10 & 41:14:09.0 & 21.5 & fj 57 ; mmd 1249 + m31 b127 - 19 & 00:42:42.15 & 41:14:22.3 & 22.6 & fj 174 ; mmd 1255 + m31 b127 - 20 & 00:42:42.33 & 41:15:53.3 & 21.1 & fj 4 ; mmd 1339 + m31 b127 - 21 & 00:42:42.43 & 41:13:55.9 & 22.1 & fj 99 ; mmd 1246 + m31 b127 - 22 & 00:42:42.62 & 41:13:33.8 & 23.9 & + m31 b127 - 23 & 00:42:42.91 & 41:15:19.8 & 22.6 & fj 73 ; mmd 1303 + m31 b127 - 24 & 00:42:43.02 & 41:15:31.8 & 22.2 & fj 19 ; mmd 1306 + m31 b127 - 25 & 00:42:43.03 & 41:15:36.5 & 21.6 & fj 20 ; mmd 1290 + m31 b127 - 26 & 00:42:43.07 & 41:14:08.8 & 22.5 & fj 139 ; mmd 1252 + m31 b127 - 27 & 00:42:43.27 & 41:15:56.1 & 21.8 & fj 7 ; mmd 1323 + m31 b127 - 28 & 00:42:43.79 & 41:16:01.0 & 21.5 & fj 8 ; mmd 1324 + m31 b127 - 29 & 00:42:43.81 & 41:15:38.9 & 22.7 & fj 319 + m31 b127 - 30 & 00:42:44.03 & 41:15:01.7 & 23.5 & m31 nova 2006 - 11b + m31 b127 - 31 & 00:42:44.39 & 41:15:04.9 & 22.5 & mmd 1349 + m31 b127 - 32 & 00:42:44.90 & 41:15:20.6 & 21.8 & fj 74 ; mmd 1347 + m31 b127 - 33 & 00:42:44.99 & 41:16:03.9 & 22.3 & fj 324 + m31 b127 - 34 & 00:42:45.16 & 41:15:23.2 & 21.1 & fj 21 ; mmd 1266 + m31 b127 - 35 & 00:42:45.21 & 41:16:05.4 & 21.4 & fj 9 + m31 b127 - 36 & 00:42:45.25 & 41:15:29.2 & 22.3 & fj 323 ; mmd 1295 + m31 b127 - 37 & 00:42:45.76 & 41:16:01.2 & 23.5 & fj 454 ? + m31 b127 - 38 & 00:42:45.85 & 41:16:01.3 & 23.3 & + m31 b127 - 39 & 00:42:46.05 & 41:15:16.1 & 21.6 & fj 59 ; mmd 1287 + m31 b127 - 40 & 00:42:46.28 & 41:15:22.5 & 22.3 & fj 22 ; mmd 1296 + m31 b127 - 41 & 00:42:46.36 & 41:15:46.0 & 21.5 & fj 72 ; mmd 1299 + m31 b127 - 42 & 00:42:46.69 & 41:16:09.6 & 21.2 & fj 10 ; mmd 1309 + m31 b179 - 1 & 00:43:32.62 & 41:17:20.9 & 23.2 & amb 272 ; mmd 1391 + m31 b179 - 2 & 00:43:32.71 & 41:18:32.8 & 21.4 & mla 796 ; amb 2249 ; mmd 2767 + m31 b193 - 1 & 00:43:40.11 & 41:36:01.0 & 23.1 & amb 2331 ; mmd 647 + m31 b472 - 1 & 00:43:41.55 & 41:28:02.1 & 22.4 & fj 274 ; mla 836 ; mmd 3022 + m31 b472 - 2 & 00:43:43.26 & 41:27:30.2 & 22.8 & fj 285 ; mmd 2706 + m31 b193 - 2 & 00:43:49.91 & 41:37:34.3 & 24.0 & + m31 b225 - 1 & 00:44:27.52 & 41:22:29.0 & 22.6 & mla 998 ; amb 2930 ; mmd 979 + m31 b225 - 2 & 00:44:31.02 & 41:22:25.7 & 22.9 & wb 445 ; amb 334 ; mmd 978 + m31 b386 - 1 & 00:46:30.45 & 42:01:09.6 & 25.3 & + lcll m31-b080 & @xmath99 & 2010 - 12 - 21 & + m31-b084 & @xmath85 & 2010 - 12 - 21 & + m31-b096 & @xmath75 & 2010 - 12 - 21 & + m31-b124 & @xmath81 & & 2010 - 12 - 23 + m31-b127 & @xmath82 & & 2010 - 12 - 23 + m31-b132 & @xmath100 & & 2010 - 12 - 23 + m31-b264 & @xmath101 & & 2010 - 12 - 23 + m31-b477 & @xmath95 & 2003 - 09 - 27 & + m31-nb21 & @xmath92 & & 2010 - 12 - 23 + m31-nb39 & @xmath94 & & 2010 - 12 - 23 + m31-nb41 & @xmath102 & & 2010 - 12 - 23 + m31-nb89 & @xmath94 & & 2010 - 12 - 23 + lllc b075 & 00:42:08.824 & 41:20:21.25 & + 759 & 00:42:09.100 & 41:20:22.80 & 3.3 + b105 & 00:42:30.747 & 41:30:27.34 & + 534 & 00:42:30.700 & 41:30:24.60 & 2.8 + bh16 & 00:42:46.090 & 41:17:35.99 & + 1360 & 00:42:45.900 & 41:17:36.50 & 2.3 + sk052a & 00:42:37.315 & 41:50:53.58 & + 2598 & 00:42:37.500 & 41:50:56.00 & 3.0 + sk104a & 00:45:44.311 & 41:57:27.80 & + 139 & 00:45:44.600 & 41:57:27.00 & 3.4 +
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single stars in ancient globular clusters ( gcs ) are believed incapable of producing planetary nebulae ( pne ) , because their post - asymptotic - giant - branch evolutionary timescales are slower than the dissipation timescales for pne .
nevertheless , four pne are known in galactic gcs .
their existence likely requires more exotic evolutionary channels , including stellar mergers and common - envelope binary interactions .
i carried out a snapshot imaging search with the _ hubble space telescope _ ( _ hst _ ) for pne in bright local group gcs outside the milky way .
i used a filter covering the 5007 nebular emission line of [ ] , and another one in the nearby continuum , to image 66 gcs .
inclusion of archival _ hst _ frames brought the total number of extragalactic gcs imaged at 5007 to 75 , whose total luminosity slightly exceeds that of the entire galactic gc system .
i found no convincing pne in these clusters , aside from one pn in a young m31 cluster misclassified as a gc , and two pne at such large angular separations from an m31 gc that membership is doubtful . in a ground - based spectroscopic survey of 274 old gcs in m31 , jacoby et al .
( 2013 ) found three candidate pne .
my _ hst _ images of one of them suggest that the [ ] emission actually arises from ambient interstellar medium rather than a pn ; for the other two candidates , there are broad - band archival uv _ hst _ images that show bright , blue point sources that are probably the pne . in a literature search ,
i also identified five further pn candidates lying near old gcs in m31 , for which follow - up observations are necessary to confirm their membership .
the rates of incidence of pne are similar , and small but non - zero , throughout the gcs of the local group .
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pairs of closely stacked quantum dots ( qds ) coupled via inter - band dipole moments @xcite ( double quantum dots , dqds ) or by tunneling resulting from carrier wave function overlap and coulomb correlations @xcite ( quantum dot molecules , qdms ) attract much attention due to the richness of their physical properties which show huge technological promise for nanoelectronics , spintronics and quantum information processing applications . the unique features of these systems , as compared to individual qds , can be used as the basis for long - time storage of quantum information @xcite and conditional optical control of carrier states @xcite which pave the way to an implementation of a two - qubit quantum gate @xcite . double dot structures enable also coherent optical spin control and entangling @xcite or may act as sources of entangled photons @xcite . it has been shown that the exciton spectrum of a qdm can be used to define an excitonic qubit with an extended life time @xcite , that information can be written on the spin state of the dopant mn ion located in one of the dots forming a dqd @xcite , and that a photon emitted by a nearby quantum point contact may induce carrier transfer in a dqd system @xcite . the richness and complexity of the physical properties of these systems have been manifested in many optical experiments @xcite . the spacing between the dots forming intentionally manufactured dqds and qdms is typically on the order of nanometers which is two to three orders of magnitude smaller than the wavelength of a resonantly coupled photon . under such conditions , in atomic samples the effect of superradiant emission is observed @xcite . a similar effect of coupling to common radiative modes appears also in the emission from qd systems . here also superradiance - like phenomena occur in the evolution of the exciton occupation and polarization . apart from the modified evolution for spontaneous emission , collective coupling to the radiation field can lead to other physical effects , one of which is the vacuum induced coherence ( vic ) @xcite . this effect consists in spontaneous , partial coherent excitation transfer from an initially occupied qd to initially empty one which results in exciton occupation trapping in a decoherence - resistant state . like the superradiance phenomena , the vic effect can be expected to occur also in qd systems . on the other hand , although natural atoms and qds have many properties in common , the characteristic feature of the latter is the inhomogeneity of transition energy which , for technologically realistic structures , is on the order of milli - electron - volts . moreover , small spatial separation of the dots leads to coupling between them @xcite . in our previous works we studied the impact of the transition energy mismatch and the coupling between the two emitters on the stability of collective effects @xcite and their role in the linear @xcite and nonlinear optical response of the dqd systems @xcite . we showed that , in the absence of coupling the between the qds , the appearance of an optically inactive subradiant state and a rapidly decaying superradiant one are extremely sensitive to the fundamental transition energy mismatch and already for a mismatch on the order of micro - electron - volts the collective character of the evolution is replaced by oscillation around the average exponential decay . the destructive effect of the system inhomogeneity may be , to some extent , overcome by sufficiently strong coupling between the dots which rebuilds collective behavior even in structures with technologically realistic values of energy mismatch @xcite . we have also pointed out that phonon - induced dynamics can slow down the decay of a superradiant state or speed up the emission from the superradiant one @xcite . based on these previous investigations of superradiance phenomena in qds , one can expect that the vic effect should , in principle , also be observable in these systems , at least in the presence of sufficiently strong coupling @xcite . however , to our knowledge , its stability against various inhomogeneities and perturbations typical for the solid state environmnet ( in particular non - parallel orientation of the interband dipoles in the two dots as well as phonon perturbations ) have not been studied . in this paper , we study the necessary conditions for the vic effect to appear in a system of two vertically stacked semiconductor qds . we investigate the role of the fundamental transition energy mismatch , coupling between the dots , and phonon - induced kinetics in this process . we also pay particular attention to the difference of the magnitude of the interband dipole moments as well as to their non - parallel alignment ( due to subband mixing ) . we show that in spite of all these inhomogeneities that are typical to qd structures ( as opposed to natural atoms ) the vacuum - induced coherence can be almost fully stabilized in realistic pairs of non - identical qds in a certain range of parameters . in particular , different interband dipole moments for the two dots lead to the appearance of a state which is perfectly immune to radiative decay for a particular choice of the system parameters . the paper is organized as follows . in sec . [ sec : system ] , we describe the system under investigation and define its model . in sec . [ sec : evolution ] , the method for describing the evolution is described . section [ sec : results ] contains the discussion of the results . concluding remarks are contained in section [ sec : concl ] . the investigated system is composed of two vertically stacked semiconductor qds interacting with quantum electromagnetic field and lattice vibrations . we restrict the discussion to the ground - level transitions with fixed polarization and spin orientations . we take into consideration only ` spatially direct ' states in which electron - hole pairs reside in the same qd . due to the strong coulomb coupling these states have a much lower energy than the ` dissociated ' states ( the external electric fields which would change this picture @xcite are not considered in our discussion ) . in this manner , the dqd or qdm may be described as a four - level system , with the state @xmath0 denoting empty dots , states @xmath1 and @xmath2 representing single - exciton states with electron - hole pairs residing in the lower or higher qd , and the @xmath3 corresponding to the biexciton state , that is , to both qds occupied with an exciton . as it is currently impossible to produce on demand pairs of qds with identical fundamental transition energies , we assume that the exciton transition energies for the two dots are different , @xmath4 where @xmath5 is the average transition energy and @xmath6 is the energy mismatch . as in ref . , we describe the evolution in the ` rotating basis ' defined by the unitary transformation @xmath7t/\hbar},\ ] ] where @xmath8 and @xmath9 are the standard free photon and phonon hamiltonians , respectively . the hamiltonian of the system is then @xmath10 the first term describes exciton states in a dqd structure . @xmath11 where @xmath12 is a biexciton shift due to the interaction of static dipole moments and @xmath13 is the amplitude of the coupling between the single - exciton states of the dots . the qds are separated by a distance on the order of a few nm which is much smaller than the relevant wave length of the electromagnetic field with which the dots interact . this allows us to neglect the space dependence of the electromagnetic field and describe the coupling of excitons to the photon modes in the dicke limit @xcite . the relevant hamiltonian in the dipole and rotating wave approximations is then @xmath14 where @xmath15 are the creation and annihilation operators for the exciton in the @xmath16th qd , @xmath17 is the creation operator of the photon mode with the wave vector @xmath18 , and @xmath19 is a coupling constant for the @xmath16th qd , where @xmath20 is the inter - band dipole moment for the @xmath16th qd , @xmath21 is the unit polarization vector of the photon mode with polarization @xmath22 , @xmath23 is the corresponding frequency , @xmath24 is the vacuum dielectric constant , @xmath25 is the relative dielectric constant of the semiconductor and @xmath26 is the normalization volume . we investigate only wide - gap semiconductors with fundamental transition energies on the order of 1 ev for which zero - temperature approximation for the electromagnetic modes may be used at any reasonable temperature . interaction of the carriers confined in the two dots with phonon modes is modeled by the hamiltonian @xmath27 where @xmath28 and @xmath29 are creation and annihilation operators of the phonon mode with the wave vector @xmath30 and @xmath31 are the system reservoir coupling constants for the first and second qd , respectively . we model the electron and hole wave functions by identical gaussians with extensions @xmath32 in the @xmath33 plane and @xmath34 along the growth direction , @xmath35 for such wave functions and for the deformation potential couplings between the confined carriers and longitudinal phonon modes , the coupling constants have the form@xcite @xmath36,\ ] ] where @xmath37 is the distance between the dots and @xmath38.\ ] ] here @xmath39 are deformation potential constants for electrons / holes , @xmath40 is the crystal density , @xmath41 is the speed of longitudinal sound ( linear phonon dispersion is assumed ) and @xmath42 are momentum components in the @xmath33 plane and along the @xmath43 axis . analytical formulas describing the evolution of a pair of qds are available for uncoupled systems ( @xmath44 ) interacting only with phonon modes @xcite and , in the markov limit , if only radiative decay is included @xcite . in this paper we use the previously proposed method @xcite that allows us to deal with the simultaneous action of both the phonon and photon surroundings . our approach is based on the equation of motion for the reduced density matrix of the carrier subsystem in the interaction picture , @xmath45+\mathcal{l}_{\mathrm{ph}}[\rho].\ ] ] here the first term accounts for the effects induced by the radiative environment , which is described in the markov limit by the lindblad dissipator @xmath46= } \nonumber \\ & & \sum_{\alpha\beta=1}^{2}\gamma_{\alpha\beta } \left[\sigma^{\alpha}_{-}(t)\rho\sigma^{\beta}_{+}(t ) -\frac{1}{2}\left\{\sigma^{\beta}_{+}(t)\sigma^{\alpha}_{-}(t),\rho\right\}\right ] , \label{superlindblad}\end{aligned}\ ] ] where @xmath47^{\dagger}= \exp\left[\frac{ih_{\mathrm{dqd}}t}{\hbar}\right]\sigma^{\alpha}_{- } \exp\left[-\frac{ih_{\mathrm{dqd}}t}{\hbar}\right]\ ] ] and @xmath48 if we assume that the spontaneous decay rates for the two qds are @xmath49 and @xmath50 then it follows directly from eq . ( [ gamma ] ) that @xmath51 where @xmath52 . by redefining the relative phase of the exciton states in the two dots one can assume without any loss of generality that @xmath53 and @xmath54 are real . the second term in eq . ( [ evolution ] ) accounts for the effects due to interaction with phonon surrounding , allowing for non - markovian dynamics . to describe these effects we use the time - convolutionless equation @xmath55=}\\ & & -\int\limits^{t}_{0}d\tau\mathrm{tr_{ph}}\bigl[h_{\mathrm{dqd - ph}}(t ) , \left[h_{\mathrm{dqd - ph}}(\tau),\rho(t)\otimes\rho_{\mathrm{ph}}\right]\bigr],\end{aligned}\ ] ] where @xmath56h_{\mathrm{dqd - ph}}\exp\left[-\frac{ih_{\mathrm{dqd}}t}{\hbar}\right],\ ] ] @xmath57 is the phonon density matrix at thermal equilibrium , and @xmath58 denotes partial trace with respect to the phonon degrees of freedom . .parameters used in numerical simulations . the values correspond to a self - assembled inas / gaas system . [ cols="<,^,^",options="header " , ] below , we present our results of simulations of the vic process in pairs of qds . in all the analyzed cases , we assume that the system is prepared initially in a localized state @xmath1 . in sec . refsubsec : identical we explain the effect in a system of identical , uncoupled dots . then , in sec . [ subsec : dipole ] , we analyze the role of the relative magnitude and orientation dipole moments and in sec . [ subsec : dv ] the effect of the energy mismatch and coupling between the dots . in sec . [ subsec : all ] we analyze the interplay of all the parameters that distiguish qds from natural atoms in the evolution of qds interacting only with radiation reservoir . the phonon impact on the vic is discussed in sec . [ subsec : fonony ] . , @xmath59 ns@xmath60 ) two - level systems . ( a ) exciton occupation of the system and occupations of the states @xmath1 ( @xmath61 ) and @xmath2 ( @xmath62 ) . ( b ) the off - diagonal density matrix element @xmath63 . ] collective coupling of two identical qds ( systems with identical fundamental transition energies and parallel dipole moments of equal magnitudes ) to the quantum electromagnetic vacuum leads to the appearance of a short - living ( bright ) superradiant state , @xmath64 , and an optically inactive ( dark ) subradiant state @xmath65 . the initial state of the analyzed system is a localized single - exciton state ( @xmath1 or @xmath2 ) which can appear naturally , e.g. , as an effect of incoherent trapping or controlled tunnel injection of carriers in an injection structure similar to that studied in ref . . such a state of a system of two identical and uncoupled qds may be expressed as an equal combination of the sub- and superradiant states , @xmath66 and @xmath67 , @xmath68a ) . while this process is taking place , coherence builds up spontaneously in the system ( fig . [ fig : vic_delta=0]b ) . as a consequence , the pair of identical qds is trapped in a delocalized and decoherence - resistant state with the exciton occupation number equal @xmath69 and real off - diagonal matrix element equal @xmath70 . this effect is referred to as vic . ) qds with identical transition energies ( @xmath71 ) but with non - identical values of the spontaneous recombination rates ( @xmath72 ) . ( b - d ) the total exciton occupation ( red solid line ) and the occupations of the individual dots ( green and blue dashed lines ) for the three cases shown in ( a ) . the value of @xmath73 ns@xmath60 is the same for all the graphs in this figure . ] vertically stacked semiconductor qds differ slightly in size and shape . if the system was formed in a self - assembled two - layer process then the upper qd is usually bigger than the lower one @xcite . the geometry and the structure of the qds is reflected in the carrier wave functions and thus in the interband dipole moments . according to eq . ( [ gamma ] ) , this leads to different spontaneous decay rates for the two qds forming the investigated system . the dipoles corresponding to the upper and lower qd may differ in amplitude and , if the hole states in the two structures have different light - hole admixtures , also in orientation ( see appendix ) . the equation of motion describing the evolution of a system of two energetically identical ( @xmath71 ) and uncoupled ( @xmath44 ) qds interacting only with the radiation reservoir is described by eq . with @xmath74 and @xmath75 given by eq . ( [ superlindblad ] ) with @xmath76 . it is known that in three - level open systems of this kind , non - radiating superposition states occur under certain conditions @xcite . in our case , a non - trivial ( different than the ground state @xmath77 ) stationary solution to the open system evolution equation , corresponding to the spontaneously formed , stable , delocalized dark state discussed above , exists only for @xmath78 , i.e. , for parallel dipole moments [ eq . ( [ gamma ] ) ] ( such that @xmath79 up to a phase factor ) . for dipoles that are parallel but have different amplitudes , the evolution is similar to the case of identical systems , i.e. , coupling to photon surrounding leads to excitation transfer and occupation trapping . as can be seen in fig . [ fig : delta=0_rozne_dipole ] , the fraction of trapped exciton occupation depends on the values of the single dot decay rates and stabilizes at the level @xmath80 . as in the case of identical qds , the suppression of the exciton decay is due to the existence of a dark state which , for @xmath72 , is not strictly anti - symmetric , @xmath81 due to unequal contribution from the single - exciton states to the dark state ( [ darkg11g22 ] ) the final occupation of the dots is non - symmetric , either , and equals @xmath82 respectively . different dipole moments allow one to achieve different final situations . in the trapped state , the occupation of the initially excited dot may be lower [ figs . [ fig : delta=0_rozne_dipole](b ) and [ fig : delta=0_rozne_dipole](c ) ] or higher [ fig . [ fig : delta=0_rozne_dipole](d ) ] than that of the initially empty dot . ) qds with identical transition energies ( @xmath71 ) and different spontaneous recombination rates ( @xmath72 ) for a few values of the angle @xmath83 between the dipole moments . the values of @xmath49 and @xmath50 shown in fig . ( a ) are valid for both figures . ] another factor that influences the interband dipole moments is the light - hole admixture . if this admixture is different for the two dots then the dipole moments corresponding to the qds forming the qdm or dqd become non - parallel ( see appendix ) this means that the off - diagonal decay rate @xmath84 , where @xmath83 is the angle between the interband dipole moments of the two dots [ see eq . ] . consequently , the stationary solution to eq . does not exist and quenching of the final exciton occupation is observed . the values of @xmath83 are determined by the light hole admixture and typically are on the order of @xmath85 @xcite . however , as can be seen in fig . [ fig : teta_rozne_dipole ] , even for much larger values of the angle between the dipoles , quenching of the occupation is weak . although the system looses its coherence at long times , the character of the evolution remains the same as in the case of parallel dipoles on time scales much longer than the nominal exciton lifetime , i.e. , initially excitation transfer between the dots takes place until the occupation rate @xmath86 close to that defined by the decay rates @xmath49 and @xmath50 is reached and , then the impact of of hole subband mixing is manifested as a slow and equal decay of occupations [ fig . [ fig : teta_rozne_dipole](b ) ] . in spite of rapid technological progress , manufacturing of dqds or qdms with identical fundamental transition energies is still not feasible . as can be seen in fig . [ fig : vic_d_v](a ) , for the realistic case of non - zero energy mismatch , the effect of trapping the system in an occupied and optically inactive state is destroyed . already for the energy splitting on the order of tens of @xmath87ev quenching of the final occupation is observed ( the relevant energy scale is the transition energy line width @xmath88ev ) . qds constituting such an inhomogeneous double dot systems interact with disjoint energy ranges of the electromagnetic field which destroys the collective character of coupling to photon modes . the energy mismatch of the dots slows the decay of the superradiant state and induces emission from the subradiant one @xcite . due to the lack of a stable exciton state in which the system might be trapped the vic effect in inhomogeneous qdms is destroyed . ns@xmath60 ) . ] initially , the character of the evolution of an exciton occupation of the inhomogeneous pairs of qds does not differ considerably from the corresponding case of a pair of identical dots . until @xmath89 , the coupling to photon modes maintains its collective character and excitation transfer from the initially occupied qd to the initially empty one takes place . later , due to the emission from the subradiant as well as from the superradiant state , occupations of both dots decay @xcite . if the distance between the qds is sufficiently small coupling between the systems ( frster or tunneling ) becomes effective and affects the evolution of carriers confined in the structure . since sub- and superradiant states are eigenstates of the coupling part of the hamiltonian [ eg.([hamiltoniandqd ] ) ] separated by the energy @xmath90 , sufficiently strong interaction between the dots rebuilds the collective character of the interaction even in structures with technologically realistic energy mismatches on the order of 1 mev @xcite . this also enables the vic effect in inhomogeneous dqds to be rebuilt . as can be seen in fig . [ fig : vic_d_v](b ) , for @xmath91 , a transition from a localized initial single - exciton state to a nearly stable state is possible for a system with a technologically achievable energy mismatch . although full stabilization of the vic effect is impossible for non - identical dots and quenching of the exciton occupation always takes place sufficiently strong coupling between the dots considerably reduces the decay of the trapped state . for a weaker coupling between the dots , the exciton occupation decay is faster but still reduced compared to the system of uncoupled qds . in contrast to identical dots , the two localized states have different contributions from the from sub- and superradiant states . this results in a different degree of trapping depending on the choice of the initially occupied state @xcite . mev , @xmath92 mev , @xmath73 ns@xmath60 , and for a few values of @xmath50 . ( b ) the total occupation and the occupations of the individual dots for the last case shown in ( a ) . ( c ) exciton occupation for three sets of parameters stabilizing the occupation trapping . red solid line : @xmath93 mev , @xmath94 mev , @xmath95 ns@xmath60 , @xmath96 ns@xmath60 . green dashed line : @xmath93 mev , @xmath97 mev , @xmath98 ns@xmath60 , @xmath99 ns@xmath60 . blue dotted line : @xmath100 mev , @xmath92 mev , @xmath73 ns@xmath60 , @xmath101 ns@xmath60 . ( d ) the total occupation and the occupations of the individual dots corresponding to the red line in ( c ) . ] for technologically realistic dqds and qdms , the fundamental energy mismatch of the two qds forming the system is on the order of milli - electron - volts , the dipole moments differ slightly between the dots and the systems are coupled with one another via frster or tunneling coupling . the evolution of such systems , for parallel dipole moments , is shown in fig . [ fig : realistic ] . in fig . [ fig : realistic](a ) , the dynamics of a system strongly stabilized by a coupling between the dots is presented for a few values of the decay rates . as can be seen , the rate of occupation decay depends on the choice of the two single - dot decay rate parameters . in such a system , the coherent excitation transfer between the dots takes place on time scales similar to the previously discussed cases . however , the initial state is now a superposition of non - degenerate system eigenstates , which leads to fast oscillations of the exciton occupation between the two dots [ fig . [ fig : realistic](b ) ] . in the present general case , the spontaneously formed coherence and the trapped occupation can again be fully stabilized by an appropriate choice of parameters in a system with parallel dipoles . the single - exciton eigenstates of the system hamiltonian ( [ hamiltonianh ] ) are @xmath102 where @xmath103 and @xmath104 . the corresponding eigenenergies will be denoted by @xmath105 . if the coupling between the single exciton states satisfies the relation @xmath106 ) and ( [ wlasne2 ] ) ] corresponds to the dark state ( [ darkg11g22 ] ) . in such case , the evolution of a realistic system also leads to occupation trapping in a decoherence - resistant state [ fig . [ fig : realistic](c ) ] at the occupation @xmath107 . as can be seen from eq . ( [ vvvv ] ) , non - equal values of the spontaneous decay rate for the two dots open a possibility to stabilize the evolution of the system for wide range of parameters . as for a system of identical dots , the occupations of single dots also stabilizes , but again initial fast oscillations due to contribution from two energy eigenstates are observed [ fig . [ fig : realistic](d ) ] . mev , @xmath72 ) and coupled ( @xmath92 mev ) qds for a few values of @xmath83 ( a ) and for two values of @xmath50 ( b ) . ] for the non - parallel dipole moments , i.e. , for @xmath108 , the effect of occupation trapping is in principle destroyed but for strongly coupled dots , the decay is very weak . for the interesting time scale on the order of ns the impact of non - parallel dipoles becomes visible only for the values of @xmath83 that exceed the realistic ones by an order of magnitude @xcite ( fig . [ fig : realisticqds_teta ] ) . ( color online ) the phonon spectral density for @xmath109 nm.,width=226 ] spontaneous emission from a system of qds is affected by phonon dynamics @xcite . coupling between the qds and lattice vibrations induces excitation transfer between the two single exciton eigenstates [ eq . and ] . for a pair of identical qds ( @xmath110 ) , these eigenstates exactly coincide with the bright ( superradiant ) and dark ( subradiant ) states @xmath67 and @xmath66 defined in sec . [ subsec : identical ] . if the dots are non - identical , there is no perfect correspondence between these two pairs of states but , for non - zero coupling , still one of them is brighter and the other is darker . as we have shown previously @xcite , the phonon - induced redistribution of single exciton occupations among these two states strongly affects spontaneous emission in a temperature - dependent way . for the system of vertically stacked dots analyzed in this paper , the amplitudes of both tunneling and frster couplings are negative , so that @xmath111 . in this case , the darker eigenstate has a higher energy and will be affected by relaxation even at low temperatures . as we show below , phonon dynamics indeed breaks the relative stability of the darker state and leads to considerably accelerated decay of the excitonic occupation , except for special parameter choices . while , in general , non - markovian effects may be important in the carrier - phonon dynamics that affects spontaneous emission @xcite , the phonon - related effects to be discussed below can be understood within a markovian picture of transitions between energy eigenstates . the corresponding rate for a transition between the eigenstates @xmath112 and @xmath113 is @xmath114 , where @xmath115 , @xmath116 and the spectral density is defined as @xmath117 the spectral density for the inter - dot spacing @xmath109 nm is plotted in fig . [ fig : spdens ] . apart from the usual cutoff at frequencies larger than @xmath118 due to the restriction on the momentum non - conservation @xmath119 , where @xmath120 is the average dot size , it is characterized by the oscillations with the period @xmath121 which result from the double dot structure of the system @xcite . ratio ( corresponding to @xmath122 ) : the comparison of the total exciton occupation in a system coupled to phonons and without phonons . ( a ) @xmath123 mev ; ( b ) short - time section of ( a ) with solid ( red ) line representing the total exciton occupation , dotted ( magenta ) line corresponding to the total occupation without phonons and the dashed ( green ) and dash - dotted ( blue ) lines showing the occupations of the two dots ; ( c ) @xmath124 mev ( corresponding to the 3rd minimum of the spectral density ) ; ( d ) @xmath125 mev ( 5th minimum of the spectral density ) . ] the phonon - related effects on the spontaneous buildup of coherence is shown in fig . [ fig : vic_ph ] . here we keep the mixing angle @xmath126 constant ( there is , a constant ratio @xmath127 ) and use the energy splitting @xmath128 as a parameter . in fig . [ fig : vic_ph](a ) we plot the evolution of the total exciton occupation for @xmath129 mev , where the spectral density is relatively large , corresponding to phonon transition rates on the order of 1 ps@xmath60 . one can see here that indeed the phonon - induced relaxation suppresses the vacuum - induced coherence , which is manifested by the rapid decay of the excitonic occupation . the role of phonons in this strong change of the occupation dynamics is clear in fig . [ fig : vic_ph](b ) , where we show the first 100 ps of the same evolution . a very fast phonon - induced redistribution of occupations of the two dots ( shown in green dashed and blue dash - dotted lines ) takes place on a picosecond time scale after which the radiative effects are completely dominated by the phonon - induced thermalization which , at low temperatures , forces the system to stay in the bright state @xmath130 . the detrimental effect of carrier - phonon coupling can only be avoided if the phonon - induced occupation dynamics is made slow compared to the spontaneous emission . this can be achieved by using the oscillating form of the phonon spectral density ( see fig . [ fig : spdens ] ) . as can be seen in fig . [ fig : vic_ph](c , d ) , if choosing the energy splitting @xmath131 such that it corresponds to the 3rd minimum of the spectral density has little positive effect but for @xmath131 in the 5th minimum , the phonon effects become very weak and the long living tail of the excitonic occupation is restored . an interesting additional effect can be seen if one compares figs . [ fig : vic_ph](a ) and [ fig : vic_ph](d ) : if the phonon effects are strong then increasing the temperature slows down the occupation decay . however , for weak phonon influence the temperature dependence is opposite . this can be explained by noting that in the presence of fast phonon - induced redistribution of single exciton occupations , the occupation of these two states remains in quasi - equilibrium , which means that the occupation of the higher energy , darker state increases as the temperature grows . on the contrary , if the phonon - induced dynamics is slow the system state is almost unperturbed and close to the spontaneously formed dark superposition . phonons lead to transitions out of this stable state with intensity growing with temperature . we have studied the formation of vacuum - induced coherence and the associated long - living trapped excitonic population in a pair of vertically stacked semiconductor qds . we focused on the features that distinguish qd systems from natural atoms : the mismatch of transition energies , coupling , possibly non - identical dipole moments for the optical transition , and strong interactions with the phonon environment . we have shown that the vic effect is very sensitive to the inhomogeneity of the qds . already for the fundamental transition energy mismatch on the order of the emission line width , the exciton occupation is quenched . however , the destructive effect of the energy inhomogeneity can be strongly reduced by coupling between the dots . while for dots with identical magnitudes of the interband dipole moments full stabilization of the spontaneously formed coherence can only be achieved in the limit of infinite coupling , in pairs of qds with different interband dipoles it is possible to adjust the energy mismatch , the coupling between the qds , and the dipole moments in such a way that a perfectly stable state forms in the spontaneous emission process and a fraction of the initial excitonic population attains a formally infinite life time . the non - parallel orientation of the interband dipoles , that may be caused by heavy - light hole mixing leads to negligibly weak effects in realistic structures . while carrier - phonon coupling typically destroys the vacuum induced coherence on picosecond time scales , it can be overcome by appropriately selecting the energy splitting between the single exciton states . these results show that the vic effect can be observed in realistic systems with energy splitting on the order of milli - electron - volts provided that the system parameters ( interband dipoles , coupling and energy mismatch ) can be controlled with sufficient flexibility . this seems to be possible by appropriately designing the system on the manufacturing stage and then employing the dependence of various parameters on external fields ( e.g , via stark effect or modification of electron - hole wave function overlap ) . let us note , finally , that the major experimentally detectable consequence of the appearance of vacuum - induced coherence in the double dot system is the long - living tail in the exciton occupation ( or in luminescence intensity ) . since this effect is of considerable amplitude and evolves on long , nanosecond time scales it should be relatively easily detectable with time - resolved luminescence or pump probe spectroscopy . a. s. acknowledges support within a `` moda kadra 2015 plus '' project , co - financed by the polish ministry of science and higher education ( mnisw ) and the european union within the european social fund , and within a scholarship for outstanding young scientists granted by the polish mnisw . in this appendix , we discuss the effect of light hole admixture on the non - parallel alignment of the interband dipoles for ( nominally ) heavy hole transitions . within the 4-band luttinger model @xcite , the electron and hole wave functions are @xmath132 where @xmath133 refers to a higher or lower qd , respectively , @xmath134 is a bloch function for the valence subband @xmath22 ( heavy holes : @xmath135 , light holes : @xmath136 ) , @xmath137 is the space coordinate , @xmath138 denotes spin , @xmath139 is an electron bloch function with a fixed spin @xmath140 ( for definiteness ) and @xmath141 refer to electron and hole envelope functions , respectively . the matrix element of the interband dipole moment for the @xmath16th qd is @xmath142 to calculate the above integral we sum over unit cells ( labeled by @xmath143 ) and integrate over one unit cell ( @xmath144 labels the position within the cell ) . the envelope functions vary slowly which allows us to assume that they are constant over one unit cell , @xmath145 . the bloch functions are periodic , @xmath146 , and orthogonal for different bands . as a result , we can write @xcite @xmath147 here @xmath148 is the bulk interband dipole moment , @xmath149 @xmath26 is the unit cell volume , @xmath150 is the envelop function overlap integral , and we have replaced the summation over unit cells with integration over @xmath143 . we are investigating bright heavy hole excitons , hence @xmath151 and the other coefficients @xmath152 are much smaller . the non - vanishing bulk dipole moment matrix elements involving the spin-@xmath153 electron state are @xcite @xmath154 because the wave functions differ for the two qds , the values of the heavy hole overlap integrals @xmath155 may vary . since , to the leading order , @xmath156 the difference of the overlap integrals leads to different magnitudes of the dipole moments @xmath157 , @xmath158 . moreover , one has @xmath159 hence , @xmath160 thus , if the light hole admixture is different for the two dots , @xmath161 then @xmath162 , that is , the dipoles are non - parallel . as the subband mixing is typically small in self - assembled structures the angle @xmath83 between the dipole moments is small . therefore , one can write @xmath163 where @xmath164 is an irrelevant phase . comparing eqs . ( [ dipole1 ] ) and ( [ dipole2 ] ) one gets @xmath165
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we present a theoretical study of vacuum - induced coherence in a pair of vertically stacked semiconductor quantum dots .
the process consists in a coherent excitation transfer from a single - exciton state localized in one dot to a delocalized state in which the exciton occupation gets trapped .
we study the influence of the factors characteristic of quantum dot systems ( as opposed to natural atoms ) : energy mismatch , coupling between the single exciton states localized in different dots , different and non - parallel dipoles due to subband mixing , as well as coupling to phonons .
we show that the destructive effect of the energy mismatch can be overcome by an appropriate interplay of the dipole moments and coupling between the dots which allows one to observe the trapping effect even in a structure with technologically realistic energy splitting on the order of milli - electron - volts .
we also analyze the impact of phonon dynamics on the occupation trapping and show that phonon effects are suppressed in a certain range of system parameters .
this analysis shows that the vacuum induced coherence effect and the associated long - living trapped excitonic population can be achieved in quantum dots .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
polyelectrolyte ( pe ) solutions are systems widely studied since they show properties that are of fundamental interest for applications in health science , food industry , water treatment , surface coatings , oil industry , among other fields . in fact , one of the problems found in genetic engineering in the appearance of conformational changes of the adn molecule , which is a charged polyelectrolyte.@xcite . + here we study an infinite dilution polyelectrolyte solution , so that , the interaction among polyelectrolyte macromolecules are negligible . we model the polyelectrolyte as having dissociable functional groups that give rise to charged sites and counter - ions in aqueous solution . the long range interactions arising from these multiple charges are responsible for their macroscopic complex properties , which can not be explained by regular polymer theories . the spatial structures of these materials in solution have been studied extensively , particularly with a scaling theory@xcite that are not appropriate for highly charged pe . the first simulations carried out for a single chain predicted the formation of groups of monomers , as the fraction of charged monomers increased . such structures are known as pearl necklaces . the size of such pearls and the distance between then is determined by the balance between the electrostatic repulsion and steric effects . + these pearl necklace structures have also been found in molecular dynamics ( md ) simulations@xcite . in this paper we are interested in the application of the much simpler cellular automata simulation to characterize the main features of a polyelectrolyte that could be responsible for such conformations . the complete simulation of this complex system requires the description of a model in terms of potential or forces . in the md simulations of limbach and holm @xcite , the monomers are connected along the chain by the finite extendible nonlinear elastic ( fene ) bond represented by the potential energy . @xmath0 where @xmath1 is the distance between two bonded monomers , @xmath2 , is the elastic bond constant , @xmath3 is the monomer diameter , @xmath4 is boltzmann s constant , @xmath5 is the absolute temperature and the parameter @xmath6 represents the maximum extension of the bond between two neighbor monomers . two charged sites i and j , with charges @xmath7 and @xmath8 , a distance @xmath9 apart , interact with the electrostatic coulomb potential @xmath10 this potential is weighed by the bjerrum length @xmath11 $ ] , where @xmath12 and @xmath13 are the solvent permitivity and the vacuum permitivity respectively , @xmath14 is the electric charge unit . the parameter @xmath15 is a measure of the strength of the electrostatic force as compared to the kinetic energy . the length ratio @xmath16 is a measure of the reduced temperature @xmath17 $ ] . the short range and van der waals interaction between any two particles or monomers is represented in the md simulation by a typical truncated lennard - jones potential @xmath18 + \epsilon_{cut } & r_{ij } < r_{c},\\ 0 & r_{ij } > r_{c } \end{array}\right.\ ] ] where @xmath19 is the potential energy well depth and @xmath20 is the cut off energy . this potential prevents the superposition of the bonding monomers . counter - ions interact via a purely repulsive lj interaction with @xmath21 . even though in the cellular automata simulation we do not use any form of potential energies or forces in an explicit manner , the _ rules _ for the movement of the different particles must be inspired on a model defined in terms of such potentials . we therefore establish our rules based on the essence of the previous three potentials . + the polymer is constructed by placing the monomers in a three dimensional cubic network of side @xmath22 and volume @xmath23 . each cell then has @xmath24 neighbors and represents a monomer with a monovalent charge @xmath25 , @xmath26 , or , for a neutral monomer , @xmath27 , as depicted in fig.(1 ) in two dimensions . out of a total of @xmath28 monomers in the chain , it is assumed that a given fraction @xmath29 is charged . the polymer is then constructed by randomly binding consecutive sites in the network . each monomer could be charged or uncharged , with a distribution chosen randomly . a key step on the construction of the polyion is the spatial location of the dissociated counter - ions . we place the counter - ions also randomly in free cells in the volume around the charged monomer within a distance @xmath15 , that is , in a volume @xmath30 centered on the charged site . the use of the bjerrum parameter , which is related to the quality of the solvent , ensures the conservation of the total electroneutrality but gives a spatial distribution of counter - ions around the charged sites . + so , each monomer @xmath31 of the system is represented in a @xmath32 matrix where each element @xmath33 indicates the @xmath34 polymer with charge @xmath35 , that could be @xmath36 or @xmath27 , at the positions @xmath37 given by the cell label @xmath38 $ ] . the counter - ions with opposite charge are represented by a similar @xmath39 matrix @xmath40 . + for simplicity we chose monomers with dissociable groups that give a site with a positive charge . we then set the following displacement rules for the different particles : + * _ neutral monomer particle _ * 1 . locate the unoccupied nearest neighbor sites . the new position where a move could be acceptable are those where it does not superimposed with any other particle in the system and where no bond is broken . 2 . count the amount of monomer particles around the current position and around every unoccupied neighbor , within a cube of volume @xmath41 centered on it . 3 . move the test neutral monomer to the position that has the higher amount of monomers around it , including the current one if it were the case . * _ positively charged monomer particle _ * 1 . as before , locate the unoccupied acceptable nearest neighbor sites . 2 . count the amount of charge around the current position and around every unoccupied neighbor , within a cube of volume @xmath30 centered on it . 3 . move the test positive charged monomer to the position that has the lowest positive charge around it , including the current one if it were the case . * _ negatively charged counter - ion _ * 1 . move randomly to an unoccupied site within a cube of volume @xmath30 centered on the accepted new position of the corresponding positive monomer in the polymeric chain . we denote the position of monomer i with @xmath42 and the distance between two particles i and j with @xmath43 . the center of mass for the chain is then @xmath44 . and the center of mass coordinates are @xmath45 . a parameter that is useful in the study of the spatial conformations of a polymer is the radius of gyration @xmath46 , defined as @xmath47 according with our construction and movement rules , we can vary the length of the chain @xmath22 , or the number of monomers @xmath28 , satisfying @xmath48 and the number of charged monomers , @xmath49 . we also take as an independent variable the parameter @xmath15 , which determines the number of cells , of size @xmath3 , where the range of the electrostatic attraction between a charged monomer and contra - ion extends . the charge distribution of the sequence of charged and uncharged monomers is determined randomly depending on the initial random seed . + in fig.(2 ) we show some of the equilibrium conformations obtained for a polyelectrolyte with @xmath50 monomers , with @xmath51 , for several charge fractions . the line represents the polyelectrolyte , the filled black bonded circles represent the charged monomers . the counter - ions in solution are represented by open non - bonded circles . for clarity , the neutral monomers are not shown . for a low charge fraction of @xmath52 , the polyion presents an elongated string appearance . as the charge fraction is increased the polyion contracts and some groups of monomers tend to form clusters , so that , already for @xmath53 , it shows the locally collapsed structures known as pearl necklace . these results are very similar to those obtained by the molecular dynamics simulations of limbach and holm @xcite + it is important to notice that for a given total charge , determined by the fraction @xmath29 , different distributions of the charged monomers give different conformations . to study this behavior , we have carried out simulations with several initial seeds for @xmath50 monomers , with a fixed fraction @xmath53 and a range parameter of @xmath54 . in fig.(3 ) , we show the temporal evolution of the radius of gyration @xmath55 . in all cases the conformations change from the initial given @xmath55 value to a plateau value that correspond to the equilibrium structures . fig.(3 ) clearly show that the plateau @xmath55 values , and thence , the final conformations depend on the charge distribution . in fig.(4 ) we show snapshots of final structures corresponding with the different seeds of fig.(3 ) . + as we can see from fig.(4 ) , the formation of the pearl necklace structures is independent of the charge sites distribution , for a given @xmath29 and @xmath15 . these clusters seem to be stabilized by the counter - ions as a consequence of the electroneutrality condition that we force to be satisfied . this is so because in our model the degree of freedom of the mobile counter - ions is much higher that that of the monomers tied to the chain . the strong repulsions that originate by the formation of clusters of neutral and positively charged monomers is compensated by the counter - ion cloud that forms around it . in order to study the reproducibility of the configurations found , we carried out a large number of simulation runs , for fixed values of @xmath56 , @xmath53 , @xmath57 and @xmath58 and for the same initial charge distribution . in fig.(5 ) we show the histograms for the frequency with which a given value of @xmath55 appears . we can see that the distribution of the @xmath55 is very closely a gaussian with a reasonably low dispersion of less than a @xmath59 about the mean value . + in fig.(5b ) we can see that similar distribution is obtained when the range parameter @xmath60 increases to a value of @xmath58 . we have further tested the effect of the parameter @xmath15 by generating structures for different values of it . in fig.(6 ) we show some equilibrium conformations for @xmath60 equal to a ) 3,b ) 6,c ) 9 and d ) 12 . here we use a large charge fraction @xmath53 and @xmath50 and we used the same initial charge distributions in all cases . as we can observe as bjerrum length increases the final polyelectrolyte structures become more compact . the number or pearls or conglomerates is higher for the lower values of @xmath15 , a result similar to that obtained by md simulations of limbach y holm @xcite-@xcite with the simple technique described here , we were able to reproduce the complex structure of model polyelectrolytes that fare very well with those predicted by the more sophisticated molecular dynamic and monte carlo simulations@xcite . we even predict situations with single conglomerates and with pearl necklace type conglomerates . we thus show the potentiality of the celular automata in the simulation of the trends in the formation of the various types of spatial conformations of polyelectrolytes . we remark on the importance of the charge distribution once the fractional charge is fixed . this work was supported in part by the grant 04 - 005 - 01 from the decanato de investigacin of universidad nacional experimental del tchira , in part by the grant g-9700741 of fonacyt , and in part by the grant c-1279 - 0402-b from consejo de desarrollo cientco humanstico y tecnolgico of universidad de los andes . + numerical calculations were carried out at the computer center cecalcula . + 25 a.v . dobrynin , r.h . colby and m. rubinstein,_macromolecules _ , * 28*,1859,(1995 ) . a. v. dobrynin ; m. rubinstein ; s. p. obukhok,_macromolecules_,*29*,2974,(1996 ) . y. yasmasaki , y. teramoto , and k. yoshikawa,_biophys . j._,*80*,2974,(2001 ) . h. j. limbach y c. holm,_j . b_,*107*,8041 - 8055,(2003 ) . u. micka ; c. holm ; k. kremer,_langmuir_,*15*,4033,(1999 ) . u. micka ; k. kremer,_europhys . lett._,*49*,189 - 195,(2000 ) . h. j. limbach y c. holm , _ j. chem . phys._,*114*,9674 - 9682,(2002 ) . h. j. limbach y c. holm,_comput . commun._,*147*,321 - 324,(2002 ) . h. j. limbach y c. holm ; k. kremer , _ j. phys : condens . matter_,*15*,s205-s211,(2003 ) . h. j. limbach y c. holm ; k. kremer , _ europhys . lett._,*60*,566 - 572,(2002 ) .
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resumen + + abstract + . +
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You are an expert at summarizing long articles. Proceed to summarize the following text:
globular clusters have dynamical properties that distinguish them from other stellar systems and allow one to test dynamical models . they can be very long - lived , with typical ages for galactic globular clusters of 7 - 13 gyr @xcite , yet they are dynamically active in that their dynamical relaxation times are significantly less than a hubble time @xcite . for the well - observed galactic globular clusters , the ratio of the age to the relaxation time ( at the half - light radius ) lies in the range 2 - 40 ( a median of 7 - 8 ) , representing clusters that display modest dynamical evolution to those that have undergone core collapse . binary stars are expected to play a central role in the dynamical evolution of globular clusters through the process of binary burning . in this process , the dynamical interactions of binaries with other stars or binaries both hardens the binary and adds kinetic energy to the interacting star ( other binary ) , thereby slowing the contraction of the cluster , especially in the core region . therefore , there are predicted relationships between the globular cluster binary properties and its dynamical state . some of these predictions can be tested by measuring the binary fraction of globular cluster stars between systems in different dynamical states and as a function of radius within individual clusters . the process of measuring binary fractions with globular cluster color - magnitude diagrams relies on accurate magnitude determinations of @xmath1 main sequence stars in crowded environments . this approach is very challenging for ground - based observers , but it became feasible with the high angular resolution capabilities of the hubble space telescope @xcite . hst has been used to observe globular clusters with the wfpc2 and more recently with the wfc / acs as part of a treasury project . the results of these efforts on the topic of binary fractions were recently completed using these samples @xcite , with a variety of interesting results . the methods used by @xcite are published in @xcite ( henceforth , paper i ) . the present paper analyzes the trends of binary fraction with cluster properties and with radius . both methods and analysis are contained within the thorough analysis of @xcite , and while there are a many similarities in the approaches of both efforts , there are a few differences , both technically and in sample selection . several of the results by @xcite and @xcite are confirmed by us , but there are some important differences that we highlight in this paper . the sample , which was frozen in 2010 , derives from the galactic globular cluster list @xcite for which there were sufficiently long wfpc2 or acs observations ( e.g. , snapshot observations were excluded ) . we avoided clusters with high extinctions ( e(b - v ) @xmath2 0.4 ) to avoid the challenges associated with the variations in the extinction across the globular cluster ( paper i ) . @xcite included more globular clusters with high extinction , for which they developed an approach to correct for small - scale extinction issues . they also included other data sets , some taken after our cutoff date , so our sample contains 35 clusters @xcite while theirs contains 59 . the earlier survey by @xcite examines 13 lower surface brightness globular clusters . briefly , the color magnitude method of determining the main sequence binary fraction relies on the fact that when a second star is within the resolution element , the combination departs from the main sequence along a well - defined locus in color and magnitude . this approach necessitates that the measurement error is small compared to the deviation from the main sequence of a typical binary , such as one with a mass ratio of 1:2 . practically , this is best accomplished with a magnitude error near 0.01 mag , which can be challenging in crowded fields , but this challenge has been met through the development of point source extraction software that includes detailed treatments of the point spread function ( psf ) in such situations @xcite . we calculate three measures of the binary fraction , two of which are non - parametric . one approach is only to count stars with a mass fraction greater than q = 0.5 and form a binary fraction , f@xmath3(high q ) . another method is to fit a gaussian distribution of the star density perpendicular to the straightened main sequence ( as a function of color ) , mostly using the blue side . the fit is extended to the red side and subtracted from the total distribution , having made corrections for contamination by false binaries and field stars . the remaining stars are assumed to be binaries whose mass ratios extend below q = 0.5 , typically reaching 0.3 ; the binary fraction is given by f@xmath3(count ) . the final method is parametric in that we produce a best - fit using a binary fraction with a power - law dependence on mass ( for q @xmath4 0.3 ) , along with the gaussian described above and the contamination corrections . this resulting binary fraction is denoted as f@xmath3(fit ) . contamination is a critical issue because the chance superposition of two unrelated stars is indistinguishable from a true binary . this problem is solved by simulations that lead to estimates for the number of chance superpositions as a function of magnitude and position in the cluster . another type of contamination comes from foreground and background stars . this is not a serious problem except at low latitudes , where we correct for it through galactic models that predict the contributions from such stars @xcite . however , when the field is dense with foreground and background stars , these corrections are not always of sufficient precision . a correction that is too large will produce a negative star density in the region that we are counting binaries , leading to a non - physical negative binary fraction . this contamination most affects the @xmath5 method and we have two globular clusters where this occurred . multiple stellar populations have been discovered in an increasing number of globular clusters @xcite and it is suspected that two generations of stars exist in globular clusters , formed typically within @xmath6 years of each other , a period of time far less than the age of these systems . the evidence for the two ( or more ) populations comes from color magnitude diagrams that use ultraviolet and optical colors , and with some spectral data @xcite . for example , in ngc 6352 @xcite , the width in color of the main sequence below the turnoff is 0.116 mag in @xmath7 , which is wider than the color range used for binary fitting . if this were the case for the cmd constructed with f606w and f814w data , determining binary fractions would be particularly challenging , if not impossible . however , the spread of the main sequence is significantly less when using @xmath8 , because metallicity variations shift the relevant part of the main sequence in a vector nearly aligned with the main sequence ( paper i ) . therefore , multiple populations can have a nearly degenerate color sequence on the lower main sequence . for ngc 6352 , the distribution of the stars about the modal color value is largely fit by us with a gaussian with @xmath9 mag , well less than the @xmath10 mag range used for binary determinations . for ngc 6753 @xcite , populations a and c , which differ mainly in their @xmath11 enhancement and less so in the fe abundance , are distinct in uv colors and are successfully modeled with stellar atmosphere and isochrone codes . however , when using @xmath8 , the isochrones of the two populations are degenerate , based upon the stellar atmosphere models ( their figure 11 ) . we can quantify the spread a bit better for the @xmath8 colors in the lower main sequence ( @xmath12 in their figure 6 ) , based on the models used for ngc 7089 @xcite ; this is an unusual globular cluster not included in our study . for fixed helium , age , and iron abundance ( y = 0.248 , 12 gyr , [ fe / h ] = -1.0 ) , a change in the alpha abundances from [ @xmath11/fe ] = 0.2 to 0.4 leads to a color shift in the main sequence of 0.008 mag . the large difference in the metallicity from [ fe / h ] = -1.7 ( y = 0.246 , 13 gyr , [ @xmath11/fe ] = 0.4 ) to a slightly younger but more metal rich population with [ fe / h ] = -1.0 ( y = 0.248 , 12 gyr , [ @xmath11/fe ] = 0.4 ) leads to a shift of 0.029 mag . such large metallicity differences are uncommon and more typical case is ngc 6656 @xcite , where the authors consider two populations with [ fe / h ] = -1.82 and -1.68 . this difference leads to a measurable broadening near the turnoff , but further down the main sequence , we calculate that the broadening is 0.008 mag for the lower main sequence in the @xmath8 colors . populations with enhanced he abundances are considered for ngc 7089 by @xcite , which lead to similar shifts , although these populations do not seem to account for much of the population on their lower main sequence or they fit poorly near the turnoff region ( their figure 6 ) . also , there is no direct ( spectroscopic ) proof that the he abundance is enhanced by the large amounts they considered ( y = 0.33 ) . multiple stellar populations certainly can add to the dispersion of the lower main sequence , but not necessarily to a degree that invalidates the determination of binary fractions . a measure of the contamination is the breadth of the gaussian used to fit the central part of the aligned cmd in color space . for cases where the gaussian fit has @xmath13 , the main sequence is sufficiently wide to invalidate the cmd approach to determining the binary fraction , as in the case of ngc 2808 @xcite ; other objects were excluded from this study at the outset . however , when the main sequence is narrow , multiple main sequences are sufficiently degenerate to allow an estimation of the binary fraction . the determination of @xmath14 is probably the most reliable , as it does not depend on the main sequence having a symmetric structure . to estimate the binary fraction to smaller values of @xmath15 ( to about 0.3 ) requires a fitting procedure that makes use of the entire distribution , but depends upon the assumption that most of an individual color distribution can be represented by gaussians . while such fits lead to acceptable values of @xmath16 , small intrinsic asymmetries 2 - 3@xmath17 from the center of the distribution can lead to overestimates of the binary fraction . many of the fitted binary fractions , @xmath18 are consistent with an extension of @xmath19 to @xmath20 , yet it is difficult to predict a - priori which are reliable and which are not . therefore , we advise investigators to use @xmath18 with caution , with @xmath19 being more reliable . addition discussion on these topics is found in paper i. analysis of the whole field of view statistically gives the most reliable results , as it includes the most number of stars . this allows greater precision when comparing the binary fractions from different techniques . however , there are shortcomings to this approach . due to the variation in distances , the sampling of each cluster extends to different radii . for example , the half light radius of ngc 104 is not contained within the acs field . this issue would not be relevant except that there is a gradient in the binary fraction as a function of radius . we do not include the region near the ccd edge and between the gap of the two chips because the photometry is not sufficiently accurate . in table [ tab : whole ] , we list the analysis results using different methods for the whole field of view . in this table , we give the region size in terms of their half mass radii in column 2 . in column 3 , 4 , and 5 , we list the binary fractions with 1 @xmath17 errors for the high mass - ratio ( @xmath5 ) method , the star counting method , and the fitting method , respectively . column 6 and 7 are the fitted parameters for the third method , where power @xmath21 is the power index of the power - law function for the binary mass - ratio distribution , and @xmath22 is the minimum binary mass ratio we can get . column 8 gives the @xmath23 and degrees of freedom for the fitting method . the last column gives the binary fraction quality flags . they indicate the main error sources . only the results including one quality flag can be used , such as g ( good estimate ) , d ( dense core ) , and n ( small number of stars ) . results including flag f ( field stars ) or e ( large photometric errors ) should be used with caution , as the contaminations are asymmetric and with large uncertainties ( some fractions become negative values due to the asymmetric distribution of field stars ) . from the table , we can calculate the mean binary fractions including 25 reliable binary fractions from the three methods : 5.2% ( high q ) , 6.3% ( counting ) , and 7.3% ( fitting ) , respectively . in figure [ fig : compall ] , we show the comparisons of binary fractions obtained through different methods . blue filled circles are for binary fractions obtained by high q method compared to the counting method . red open circles are for binary fractions obtained by the fitting method compared to the counting method . the black solid line shows where the two methods give the same results . from the figure , we can see that the binary fractions obtained through the counting and fitting method are consistent with each other , while they are usually larger than the values obtained through the high mass - ratio method . this is because the latter method does not include binaries with small mass ratio that are closer to the main sequence , while the former methods can statistically recover part of the signals from the small mass - ratio binaries . for the fitting method in table [ tab : whole ] , the fitted power @xmath21 has a mean value of @xmath24 ( from the 25 reliable clusters ) , suggesting that most binaries tend to have small mass ratios . the fitted minimum binary fraction @xmath22 has a mean value of @xmath25 , which is smaller than the cut - off ratio in the high mass - ratio method ( @xmath26 ) , indicating that the fitting method can recover part of mass - ratio binaries that are hidden in the main sequence . the comparison between the minimum whole - field binary fractions with other efforts is not as good as one would expect , given that they are using the same data sets . the most direct comparison uses the binary fraction above a certain value of @xmath15 , such as 0.5 for @xcite and this work , which was chosen because the distance of these binaries from the main sequence ridge line is generally above the 3@xmath17 uncertainties of a star . @xcite also presents a binary fraction above some minimum binary ration , their @xmath22 , but they do not give values for @xmath22 , which may differ between clusters ( their table 3 ) . for the 13 globular clusters considered by sollima , terzan 7 and palomar 12 have binary fractions that are significantly higher than the other clusters , a result confirmed by milone and this work , although the values differ between the groups . for the remaining 11 clusters , we do not find a meaningful relationship between the binary fractions from sollima and milone ( figure 2 ) . the values from sollima are systematically larger than those of milone , but that may be due to a difference in how their binary fractions are calculated . there is the same offset and lack of a correlation when comparing the sollima values to our values ( figure 2 ) . when comparing our values with those of milone , there is no systematic offset and most of the values are consistent with each other , but there are a few significant discrepancies , which we discuss further below ( figure 3 ) . although these various binary fractions are determined from the same data sets , the number of stars used is somewhat different and the photometric software is different . however , each of the photometric methods has a heritage and is well - tested , although the method used by milone @xcite appears to be more efficient and more magnitudes are extracted for the same fields in approximately the same magnitude ranges . one might expect that the different photometric methods would lead to different numbers of stars extracted but would not lead to systematically incorrect magnitudes on a scale greater than 3@xmath17 . furthermore , the various groups performed simulations to demonstrate the reliability of recovering artificial stars that were placed into the very fields being analyzed . for most of these clusters , the other corrections are too small to account for the differences , such as overlapping stars and field star contamination . the reasons for these differences remain unclear and a cross - method investigation is beyond the scope of this effort . the analysis for the binary fractions within the half - mass radius has physical importance in that it provides a uniform basis to make comparisons between clusters and with theoretical models . one difference with the fitting method is that we fixed the values of the power index @xmath21 and the minimum mass - ratio @xmath22 to the ones obtained by whole field analysis , because there are more stars to constrain the parameters using the whole chips . the binary fractions are listed in table [ tab : rh ] , which is similar to table [ tab : whole ] , except that we omit the size column . we also exclude the columns of power index @xmath21 and the minimum mass - ratio @xmath22 , as they are all fixed to the values in table [ tab : whole ] . the average binary fractions within the half - mass radius from this table is 5.6% ( for the @xmath5 binary fractions ) , which includes 27 reliable clusters with flags of d , n , or g. there is a very tight relationship between the fitting and the counting method , so either are equivalently good for further analysis ) . however , there is a poorer relationship between the high q binary fraction and the one from the counting method ( or from the fitting method ) . this is also a poorer relationship than when using the whole fields . based on the tests conducted in paper i , the weaker relationship can not be attributed to uncertainties in the methods unless there are mitigating factors , such as an incorrect estimate for the foreground / background stellar contamination , significant differential reddening , or multiple populations . if these mitigating factors can be ignored , we would attribute the binary fraction difference to an increasing binary fraction in the range @xmath27 , which are included in the counting method but not in the high - q method . from the binary fractions that we calculated , we examine whether they are correlated with the physical properties of age , dynamical age ( age divided by the relaxation time at r@xmath28 ) , metallicity ( [ @xmath29 ) , and absolute magnitude m@xmath30 . to make the comparison , we assume that the binary fraction and above quantities are linearly related , as @xmath31 where y is the binary fraction and x is the different properties of globular clusters . figure [ fig : rh_fb ] shows the relationships between the half - mass binary fractions and the ages , dynamical ages ( absolute ages divided by the relaxation time at half - mass radius ) , metallicity [ fe / h ] , and the absolute v magnitudes . the fitted line parameters and the non - parametric spearman rank coefficients are shown in table [ tab : fb_rh ] . the strongest correlation is between the half - mass binary fractions and the cluster absolute v magnitudes , meaning that less luminous ( massive ) clusters have higher binary fractions . this correlation is also confirmed by the spearman rank coefficient , with reliable significance . this relationship was also discovered by @xcite . the lowest luminosity globular cluster , e3 , is not included in our work , but as evident in @xcite , it supports the increase of binary fraction with decreasing luminosity . the lower density end of this trend would correspond to the open clusters , where they usually have higher binary frequencies ( @xmath32 for hyades by @xcite , @xmath33 % for praesepe by @xcite , and 50% for m67 by @xcite ) . a more recent study of 9 open clusters show 15%-54% of binary fraction @xcite . for field stars near the solar neighborhood , the binary frequency can be as high as 50% @xcite . from this table , we can see that the half - mass binary fractions appears to be anti - correlated with the cluster ages at above the 95% confidence level . however , two clusters have a significant effect on this result , terzan 7 and palomar 12 , which are both relatively young and have low absolute luminosities . we return to this relationship when considering the core binary fractions , below . we do not see a correlation of the half - mass radius binary fractions with either the dynamical time or the metallicity [ fe / h ] , which is the same result as that of @xcite , who used a larger sample but did the analysis with binary fractions within the core radius as well as between the core and the half light radius . the binary fractions within the core regions are expected to respond most rapidly to dynamical interactions between stars . in our sample , not all the clusters are suitable for the core binary fraction analysis , because the core regions for some dense clusters are too crowded to recover stars or the core regions for some core - collapsed clusters are too small to include enough stars from which to measure a binary fraction . with such limitations ( and selection effects ) , we analyzed 25 clusters in our sample with the high mass - ratio method . table [ tab : corefb ] shows the core binary fractions for 25 clusters with the high mass - ratio ( @xmath5 ) method . the mean binary fractions within the core regions is about 7.0% ( excluding 8 clusters with field star contamination or due to large photometric errors and low star numbers ) . the correlation of these binary fractions with those of milone is a good but not perfect correlation , with a few clusters in which our binary fractions are sytematically higher . the relationship between the binary fraction and the physical properties of the clusters is similar to the results using the binary fractions within r@xmath28 , with one exception . there is a strengthening of the inverse correlation between the binary fraction and the cluster age ( figure [ fig : corefb ] ) , and this relationship remains significant even when the two youngest clusters are excluded . our ages @xmath34 , as reported in paper i , are mainly from @xcite , although a few additional ages were used from @xcite , @xcite , and @xcite . the correlation coefficient of the correlation is -0.595 , a significance of 99.8% , and with the values @xmath35 and @xmath36 . we can examine this relationship using a more uniform set of ages , where @xcite determined the absolute magnitudes of the main sequence turnoff , and using isochrones , obtained a relative age scale . the relative age scale is normalized at the mean , so one can multiply it by the appropriate age ( e.g. , 12 gyr ) to obtain an absolute age . the results are not strongly dependent on the isochrone library , so we use their d07@xmath37 values and find a similar anticorrelation ( figure [ fig : corefb2 ] ) with slope of @xmath38 ( for a mean age of 12 gyr , this would correspond to @xmath39 ) and intercept @xmath40 . the spearman correlation indicates a significance at the 99.4% level , although if the points are normally distributed , we could use the pearson correlation test , which yields a higher levels of significance , @xmath41 . in this case , we included clusters for which the formal binary fraction is negative ( mainly systems with large numbers of foreground / background stars ) , except ngc 6656 , where the field stars are so dominant that the binary fraction is unreliable . however , the result does not change significantly if those points are excluded ( or if ngc 6656 were included ) . this result has implications that we comment upon below . another important prediction of the models is that there will be a significant radial dependence on the binary fraction as well as on the stellar mass function . this was explored by both sollima and milone , who both found radial gradients in the binary fraction . we study the radial distribution of the binary fraction and also find evidence for a radial gradient . in addition we investigate whether this is an intrinsic property or one that developed due to the dynamical interactions in the cluster . in deriving the radial distribution of the binary fractions , we divide the cluster up into three radial bins of approximately equal numbers of stars , so that the uncertainties are comparable in the three bins . the uncertainties are greater when compared to the whole field values , due to the fewer number of stars , and for the inner bin in clusters with dense central regions , the results have greater uncertainty . the methods that involve fitting , @xmath42 and @xmath18 have larger uncertainties than @xmath43 due to the additional components that must be constrained . the results are given in table 4 . due to the smaller uncertainties , we use @xmath43 to examine the radial binary dependence in figure [ fig : fb_r ] , we plot all the high mass - ratio ( @xmath5 ) binary fractions at different annular bins from all the clusters in our sample as a function of their radial distances from the cluster center . the left panel is the high mass - ratio binary fractions normalized to their core fractions , and the distances are normalized to their core radii . there are only 17 clusters with reliable core binary fractions measured in our sample . the straight line fitted results are shown in table [ tab : fb_r ] , first row , which shows a moderate correlation ( slope @xmath44 ) . the spearman rank correlation coefficient , however , shows a significant correlation ( coefficient of @xmath45 , with a highly reliable significance ) . the right panel in figure [ fig : fb_r ] and the second row in table [ tab : fb_r ] show the high mass - ratio binary fractions normalized to their half - mass fractions as a function of the distances normalized to their half - mass radii . the straight line fit shows a strong correlation ( slope @xmath46 ) , with a spearman rank correlation coefficient of -0.61 , designating a highly significant correlation . even though there can be significant differences between investigators for the value of a binary fraction for the same cluster , the radial distributions are very similar between investigators . when a result is found , such as the radial decrease in the binary fraction , it could be due to evolutionary forces , but it could also be due to initial conditions the cluster was simply born with that property . to examine this issue , we compare the six clusters where the ratio of the age to dynamical relaxation time ( at @xmath47 ) is less than four , ngc 104 , ngc 5053 , ngc 5272 ( m3 ) , ngc 5466 , ngc 5897 , and arp 2 . if the radial decrease steepens with the number of elapsed relaxation times , these six clusters should show shallower radial binary fraction slopes . of these six , only two show a radial decrease where the binary fraction drops by at least 20% . for the remaining 25 clusters for which we have good radial information , only four fail to show a radial decrease of 20% or greater . clusters with dynamical ages above four relaxation times show a radial decrease in 84% of the cases while those with younger ages show the decrease 33% of the time . this implies that the decrease in the binary fraction with radius is due to evolution rather than birth . the astrophysical question we sought to address is whether globular clusters follow the predictions of the dynamical models . as a cluster progresses through several relaxation times , mass segregation should occur and soft binaries are rendered unbound . mass segregation leads to binaries sinking deeper into the cluster , increasing the binary fraction , but strong dynamical interactions can destroy binaries , having the opposite effect . to understand which effect prevails requires both theoretical and observational efforts , with the theoretical work indicating that the core binary fraction will rise with time prior to the core collapse period ( fregau 2009 ) . another consideration in comparing theory to observation is that the initial properties of a cluster can mimic that expected from dynamical evolution . for example , frank ( 2013 ) studied the globular cluster palomar 14 , which has a long half - mass relaxation time ( 20 gyr ) and lies in the outer - halo , where it does not feel strong tidal forces from the galaxy . they find that palomar 14 exhibits significant mass segregation despite having an age less than even a single relaxation time . the most likely conclusion is that it was born as a mass- segregated cluster , and if this formation history is common , it complicates the ability to compare data to models . a final consideration is that the cluster - to - cluster dispersion may be larger than the effect one is trying to measure , so it is helpful to understand the range of initial conditions in a globular cluster . a significant result is that the core binary fraction decreases inversely with the age of the globular clusters . this was first found by @xcite also at a confidence level exceeding 99% ; both we and sollima used ages from @xcite . however , when we used ages from @xcite , we obtain the same result , with only a slight decrease in significance . when @xcite used the ages from @xcite , or from @xcite , they did not find a significant anticorrelation when using their entire sample . restricted to just lower density clusters , they do not find an anticorrelation when using the relative ages from @xcite , but they appear to find an anticorrelation ( their figure b,10 ) when using the absolute ages from @xcite and @xcite . since the anticorrelation is with age and not dynamical age , it suggests that the younger clusters were born with a higher binary fraction . evidently , the conditions from which younger globular clusters formed were more conducive to binary formation . a examination of the binary fraction as a function of dynamical time can be used to obtain insight into the likely range of initial conditions . to examine this issue , we use the core binary fraction from milone for @xmath48 in figure [ fig : binary_dyn_t ] , which leads to a more statistically significant result than using the binary fractions presented in this paper . at the shortest dynamical times , @xmath49 , the range in the binary fraction is about an order of magnitude . this is probably representative of the range of initial binary fractions in clusters , although the absolute value of the binary fraction must have been larger for a few reasons . binaries where one or both stars are remnants ( white dwarfs or neutron stars ) are recorded as single stars in the cmd method , while many of the soft binaries are destroyed even by the first relaxation time . another impression one has from this figure is that the binary fraction lies within an envelope that is slowly decreasing with dynamical age , by about a factor of two over the range of the figure . that is , above 10 relaxation times , there are no clusters with binary fractions above 6% and nearly half have binary fractions below 2% . this is in contrast to the earlier times , where larger binary fractions are relatively more common . it will be difficult to improve on this data set as the hst images are of excellent quality and the best globular clusters have been observed . the radial decrease in the binary fraction is a robust result and one can account for an intrinsic dispersion in the binary fraction by dividing by the binary fraction either in the core or within half light radius . this radial decrease is nearly always seen in clusters older than four dynamical times and seen less commonly for shorter dynamical times . this implies that the radial binary fraction distribution is a result of relaxation and tidal effects rather than due to initial conditions . the mass function for the binaries is still not very well known . for @xmath48 , one can simply count the stars in the color - magnitude diagram to obtain a binary fraction , but to extend this to lower mass ratios , a parametric fitting method is needed because the magnitude of the departure from the main sequence from photometry errors becomes comparable to that due to the pairing of two stars . our fitting procedure finds that the number of binaries increases with decreasing mass ratio ( down to @xmath50 ) , although there is significant uncertainty in the value of the exponent . this would be consistent with models in which binary stars are random pairs of stars , yet the work by milone indicates that the mass function is flat above @xmath51 . we caution the reader that our fitting procedure depends upon an assumed gaussian symmetry in the width of the main sequence . if this assumption is not generally valid , then our result of a rising binary mass distribution is thrown into question . there are other avenues for studying binary populations in star clusters and one of the most effective is through spectroscopic programs . a number of these clusters have stars bright enough for ground - based studies on large telescopes with multi - object capabilities . sufficient spectroscopic monitoring can determine the period and velocity amplitude of the binary star system , which determines the mass function . with the accurate photometric measurements from the hst data sets and a mapping between the absolute magnitude and stellar mass , the orbital parameters can be deduced . such data provide information on the hardness of the binaries , essential data not provided by the cmd method used here . spectroscopy will help in another important area in which a star has a degenerate partner , which will lead to a periodic shift in the stellar absorption lines . this will help to give a fuller picture for the total number of stars in binaries , and these are probably binaries that formed relatively early . to help to facilitate such spectroscopic programs , we compiled a list of 6421 binaries with @xmath48 , with ra , dec , magnitudes , and distance from the center of the globular cluster ( ji 2011 ; also available from the authors upon request ) . the photometry for all stars in the fields is available from the acs globular cluster treasury program , with links from the mast website ( their magnitudes are slightly different from ours ) . the authors would like to thank a.e . dolphin for answering our many questions that arose when using the photometry package dolphot . for their many questions and suggestions , we would like to thank from mario mateo , jon miller , eric bell , sally oey , patrick seitzer , and a very helpful referee . we gratefully acknowledge financial support through hst programs 10939 and 11125 from nasa . fan , x. , burstein , d. , chen , j. s. , zhu , j. , jiang , z. , wu , h. , yan , h. , zheng , z. , zhou , x. , fang , l. z. , chen , f. , deng , z. , chu , y. , hester , j. j. , windhorst , r. a. , li , y. , lu , p. , sun , w. h. , chen , w. p. , tsay , w. s. , chiueh , t. h. , chou , c. k. , ko , c. m. , lin , t. c. , guo , h. j. , & byun , y. i. 1996 , , 112 , 628 piotto , g. , milone , a. p. , anderson , j. , et al . 2012 , , 760 , 39 piotto , g. , bedin , l. r. , anderson , j. , king , i. r. , cassisi , s. , milone , a. p. , villanova , s. , pietrinferni , a. , & renzini , a. 2007 , apjl , 661 , 53 piotto , g. , milone , a. p. , bedin , l. r. , et al . 2015 , , 149 , 91 ccrrrrcrc ngc104 & 0.75 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 204.1/162 & d + ngc288 & 1.06 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 120.1/106 & g + ngc362 & 2.89 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 176.9/142 & d + ngc1851 & 4.64 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 126.4/136 & d + ngc2808 & 2.96 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 299.5/164 & d , p , e + ngc4590 & 1.57 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 141.6/ 96 & g + ngc5053 & 0.91 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 43.9/ 70 & g + m3 & 1.02 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 200.3/161 & d + ngc5466 & 1.03 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 34.6/ 63 & g + ngc5897 & 1.15 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 100.4/ 98 & g + ngc5904 & 1.34 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 113.6/122 & d + ngc5927 & 2.15 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 157.3/148 & f + ngc6093 & 3.88 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 207.6/125 & d + ngc6121 & 0.55 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 66.0/ 62 & f + ngc6101 & 2.25 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 105.8/100 & g + m13 & 1.40 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 147.1/145 & d + ngc6218 & 1.34 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 92.6/ 91 & g + ngc6341 & 2.32 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 247.0/146 & d + ngc6352 & 1.15 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 110.4/ 99 & f + ngc6362 & 1.15 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 108.2/ 95 & g + ngc6397 & 0.82 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 103.1/ 79 & g + ngc6541 & 2.23 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 227.7/140 & d , f + ngc6624 & 2.89 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 200.2/144 & d , f + ngc6637 & 2.82 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 142.3/135 & d , f + ngc6652 & 4.93 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 92.7/ 85 & d , f + ngc6656 & 0.70 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 488.0/124 & f + ngc6723 & 1.55 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 113.2/118 & g + ngc6752 & 1.24 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 124.2/ 98 & g + terzan7 & 3.07 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 40.5/ 63 & g + arp2 & 1.34 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 57.9/ 74 & g + ngc6809 & 0.84 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 64.8/ 77 & g + ngc6981 & 2.54 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 85.3/102 & d + ngc7078 & 2.37 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 229.2/160 & d , e + ngc7099 & 2.30 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 85.7/ 68 & g + palomar12 & 1.38 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 13.1/ 33 & n + crrrcc ngc104 & 3.03@xmath52 0.13 & [email protected] & [email protected] & 205.6/162 & d + ngc288 & 6.29@xmath52 0.32 & [email protected] & [email protected] & 96.9/ 99 & g + ngc362 & 5.69@xmath52 0.36 & [email protected] & [email protected] & 122.8/118 & d + ngc1851 & 3.89@xmath52 0.85 & [email protected] & [email protected] & 40.3/ 73 & d , e + ngc2808 & -12.09@xmath52 0.96 & [email protected] & [email protected] & 71.9/ 90 & d , e , p + ngc4590 & 6.24@xmath52 0.30 & [email protected] & [email protected] & 112.3/ 87 & g + ngc5053 & 5.57@xmath52 0.40 & [email protected] & [email protected] & 42.5/ 70 & g + m3 & 5.03@xmath52 0.17 & [email protected] & [email protected] & 197.4/161 & d + ngc5466 & 5.23@xmath52 0.35 & [email protected] & [email protected] & 33.4/ 63 & g + ngc5897 & 5.63@xmath52 0.28 & [email protected] & [email protected] & 82.5/ 96 & g + ngc5904 & 3.15@xmath52 0.15 & [email protected] & [email protected] & 111.6/122 & d + ngc5927 & 3.87@xmath52 0.28 & [email protected] & [email protected] & 107.8/127 & f + ngc6093 & 7.59@xmath52 0.58 & [email protected] & [email protected] & 80.8/ 98 & d + ngc6121 & 4.79@xmath52 0.48 & [email protected] & [email protected] & 46.7/ 40 & g + ngc6101 & 5.50@xmath52 0.38 & [email protected] & [email protected] & 105.7/100 & g + m13 & 3.46@xmath52 0.16 & [email protected] & [email protected] & 149.2/144 & d + ngc6218 & 3.21@xmath52 0.24 & [email protected] & [email protected] & 93.3/ 88 & g + ngc6341 & 5.31@xmath52 0.29 & [email protected] & [email protected] & 151.0/136 & d + ngc6352 & -0.57@xmath52 0.88 & [email protected] & [email protected] & 104.7/ 98 & f + ngc6362 & 4.31@xmath52 0.29 & [email protected] & [email protected] & 99.9/ 94 & g + ngc6397 & 4.47@xmath52 0.49 & [email protected] & [email protected] & 100.7/ 79 & g + ngc6541 & 3.94@xmath52 0.28 & [email protected] & [email protected] & 184.6/123 & d + ngc6624 & 1.31@xmath52 0.57 & [email protected] & [email protected] & 63.7/ 93 & f , d + ngc6637 & 4.16@xmath52 0.40 & [email protected] & [email protected] & 102.5/103 & f , d + ngc6652 & 3.17@xmath52 0.78 & [email protected] & [email protected] & 42.6/ 41 & f , d + ngc6656 & -4.94@xmath52 0.28 & [email protected] & [email protected] & 486.9/124 & f + ngc6723 & 4.66@xmath52 0.21 & [email protected] & [email protected] & 101.3/115 & g + ngc6752 & 0.74@xmath52 0.16 & [email protected] & [email protected] & 118.3/ 96 & g + terzan7 & 17.15@xmath52 1.40 & [email protected] & [email protected] & 43.9/ 62 & g + arp2 & 8.05@xmath52 0.55 & [email protected] & [email protected] & 30.9/ 50 & g + ngc6809 & 3.31@xmath52 0.22 & [email protected] & [email protected] & 33.4/ 43 & g + ngc6981 & 7.18@xmath52 0.38 & [email protected] & [email protected] & 85.7/102 & d + ngc7078 & 5.85@xmath52 0.32 & [email protected] & [email protected] & 231.0/160 & d + ngc7099 & 3.78@xmath52 0.44 & [email protected] & [email protected] & 103.6/ 62 & d + palomar12 & 13.36@xmath52 1.30 & [email protected] & [email protected] & 7.2/20.0 & g + [ tab : rh ] cccccc age & [email protected]&[email protected] & 1055.2/ 25&-0.392&0.043 + dynamical age & [email protected] & [email protected] & 1042.0/ 25&-0.068&0.737 + @xmath53$]&[email protected] & [email protected] & 1054.4/ 25&-0.016&0.936 + mv & [email protected] & [email protected] & 1017.0/ 25 & 0.490&0.010 + [ tab : fb_rh ] crrrrrc ngc104 & 0.00 - 0.29 & [email protected] & [email protected] & [email protected] & 144.7/130 & d + & 0.29 - 0.42 & [email protected] & [email protected] & [email protected] & 75.7/110 & d + & 0.42 - 0.56 & [email protected] & [email protected] & [email protected] & 104.7/100 & d + ngc288 & 0.00 - 0.43 & [email protected] & [email protected] & [email protected] & 64.0/ 63 & g + & 0.43 - 0.70 & [email protected] & [email protected] & [email protected] & 77.0/ 71 & g + & 0.70 - 1.05 & [email protected] & [email protected] & [email protected] & 38.8/ 67 & g + ngc362 & 0.00 - 0.97 & [email protected] & [email protected] & [email protected] & 127.7/118 & d + & 0.97 - 1.48 & [email protected] & [email protected] & [email protected] & 91.9/101 & d + & 1.48 - 2.16 & [email protected] & [email protected] & [email protected] & 89.0/ 87 & g + ngc1851 & 0.00 - 1.55 & [email protected] & [email protected] & [email protected] & 91.5/113 & d + & 1.55 - 2.34 & [email protected] & [email protected] & [email protected] & 76.0/ 87 & d + & 2.34 - 3.55 & [email protected] & [email protected] & [email protected] & 79.1/ 76 & d + ngc2808 & 0.00 - 1.35 & [email protected] & [email protected] & [email protected] & 186.9/141 & d , e , p + & 1.35 - 1.73 & [email protected] & [email protected] & [email protected] & 180.3/132 & d , e , p + & 1.73 - 2.26 & [email protected] & [email protected] & [email protected] & 204.4/130 & d , e , p + ngc4590 & 0.00 - 0.44 & [email protected] & [email protected] & [email protected] & 47.3/ 64 & g + & 0.44 - 0.74 & [email protected] & [email protected] & [email protected] & 38.4/ 55 & g + & 0.74 - 1.14 & [email protected] & [email protected] & [email protected] & 74.5/ 60 & g + ngc5053 & 0.00 - 0.36 & [email protected] & [email protected] & [email protected] & 22.2/ 43 & g + & 0.36 - 0.52 & [email protected] & [email protected] & [email protected] & 26.6/ 40 & g + & 0.52 - 0.68 & [email protected] & [email protected] & [email protected] & 25.1/ 41 & g + m3 & 0.00 - 0.35 & [email protected] & [email protected] & [email protected] & 112.2/135 & d + & 0.35 - 0.54 & [email protected] & [email protected] & [email protected] & 82.8/111 & d + & 0.54 - 0.76 & [email protected] & [email protected] & [email protected] & 128.6/102 & d + ngc5466 & 0.00 - 0.39 & [email protected] & [email protected] & [email protected] & 31.8/ 39 & g + & 0.39 - 0.58 & [email protected] & [email protected] & [email protected] & 24.6/ 38 & g + & 0.58 - 0.79 & [email protected] & [email protected] & [email protected] & 34.4/ 39 & g + ngc5897 & 0.00 - 0.44 & [email protected] & [email protected] & [email protected] & 41.0/ 63 & g + & 0.44 - 0.64 & [email protected] & [email protected] & [email protected] & 29.6/ 59 & g + & 0.64 - 0.85 & [email protected] & [email protected] & [email protected] & 32.6/ 61 & g + ngc5904 & 0.00 - 0.43 & [email protected] & [email protected] & [email protected] & 75.4/103 & d + & 0.43 - 0.68 & [email protected] & [email protected] & [email protected] & 75.9/ 78 & g + & 0.68 - 0.99 & [email protected] & [email protected] & [email protected] & 52.3/ 67 & g + ngc5927 & 0.00 - 0.75 & [email protected] & [email protected] & [email protected] & 117.0/112 & df + & 0.75 - 1.15 & [email protected] & [email protected] & [email protected] & 87.3/ 96 & f + & 1.15 - 1.64 & [email protected] & [email protected] & [email protected] & 122.6/ 94 & f + ngc6093 & 0.00 - 1.17 & [email protected] & [email protected] & [email protected] & 100.7/108 & d + & 1.17 - 1.82 & [email protected] & [email protected] & [email protected] & 89.0/ 85 & d + & 1.82 - 2.96 & [email protected] & [email protected] & [email protected] & 47.2/ 73 & g + ngc6101 & 0.00 - 0.84 & [email protected] & [email protected] & [email protected] & 73.0/ 69 & g + & 0.84 - 1.25 & [email protected] & [email protected] & [email protected] & 46.1/ 64 & g + & 1.25 - 1.72 & [email protected] & [email protected] & [email protected] & 46.2/ 61 & g + ngc6121 & 0.00 - 0.20 & [email protected] & [email protected] & [email protected] & 53.9/ 33 & n + & 0.20 - 0.30 & [email protected] & [email protected] & [email protected] & 29.0/ 35 & n + & 0.30 - 0.41 & [email protected] & [email protected] & [email protected] & 16.4/ 38 & n + m13 & 0.00 - 0.50 & [email protected] & [email protected] & [email protected] & 113.3/120 & d + & 0.50 - 0.75 & [email protected] & [email protected] & [email protected] & 84.0/102 & d + & 0.75 - 1.04 & [email protected] & [email protected] & [email protected] & 86.8/ 90 & g + ngc6218 & 0.00 - 0.47 & [email protected] & [email protected] & [email protected] & 51.6/ 59 & g + & 0.47 - 0.72 & [email protected] & [email protected] & [email protected] & 39.7/ 55 & g + & 0.72 - 1.00 & [email protected] & [email protected] & [email protected] & 54.4/ 57 & g + ngc6341 & 0.00 - 0.75 & [email protected] & [email protected] & [email protected] & 113.5/121 & d + & 0.75 - 1.18 & [email protected] & [email protected] & [email protected] & 76.3/102 & d + & 1.18 - 1.74 & [email protected] & [email protected] & [email protected] & 104.0/ 87 & g + ngc6352 & 0.00 - 0.43 & [email protected] & [email protected] & [email protected] & 36.8/ 57 & f + & 0.43 - 0.64 & [email protected] & [email protected] & [email protected] & 36.9/ 54 & f + & 0.64 - 0.86 & [email protected] & [email protected] & [email protected] & 34.6/ 55 & f + ngc6362 & 0.00 - 0.44 & [email protected] & [email protected] & [email protected] & 58.4/ 60 & g + & 0.44 - 0.65 & [email protected] & [email protected] & [email protected] & 28.4/ 56 & g + & 0.65 - 0.85 & [email protected] & [email protected] & [email protected] & 61.7/ 56 & g + ngc6397 & 0.00 - 0.28 & [email protected] & [email protected] & [email protected] & 36.4/ 46 & g + & 0.28 - 0.45 & [email protected] & [email protected] & [email protected] & 33.6/ 45 & g + & 0.45 - 0.61 & [email protected] & [email protected] & [email protected] & 43.3/ 49 & g + ngc6541 & 0.00 - 0.71 & [email protected] & [email protected] & [email protected] & 97.4/111 & d + & 0.71 - 1.14 & [email protected] & [email protected] & [email protected] & 96.3/ 93 & d + & 1.14 - 1.66 & [email protected] & [email protected] & [email protected] & 177.0/ 87 & g + ngc6624 & 0.00 - 0.91 & [email protected] & [email protected] & [email protected] & 64.2/ 86 & f + & 0.91 - 1.45 & [email protected] & [email protected] & [email protected] & 61.0/ 84 & f + & 1.45 - 2.13 & [email protected] & [email protected] & [email protected] & 73.2/ 90 & f + ngc6637 & 0.00 - 0.87 & [email protected] & [email protected] & [email protected] & 102.9/ 96 & f + & 0.87 - 1.38 & [email protected] & [email protected] & [email protected] & 67.2/ 82 & f + & 1.38 - 2.15 & [email protected] & [email protected] & [email protected] & 56.1/ 76 & f + ngc6652 & 0.00 - 1.34 & [email protected] & [email protected] & [email protected] & 34.5/ 50 & f + & 1.34 - 2.34 & [email protected] & [email protected] & [email protected] & 28.7/ 48 & f + & 2.34 - 3.77 & [email protected] & [email protected] & [email protected] & 36.2/ 47 & f + ngc6656 & 0.00 - 0.26 & [email protected] & [email protected] & [email protected] & 86.4/ 84 & f + & 0.26 - 0.39 & [email protected] & [email protected] & [email protected] & 126.0/ 82 & f + & 0.39 - 0.52 & [email protected] & [email protected] & [email protected] & 168.6/ 80 & f + ngc6723 & 0.00 - 0.50 & [email protected] & [email protected] & [email protected] & 93.7/ 92 & d + & 0.50 - 0.80 & [email protected] & [email protected] & [email protected] & 115.5/ 79 & g + & 0.80 - 1.16 & [email protected] & [email protected] & [email protected] & 132.6/ 72 & g + ngc6752 & 0.00 - 0.41 & [email protected] & [email protected] & [email protected] & 77.4/ 75 & g + & 0.41 - 0.66 & [email protected] & [email protected] & [email protected] & 70.2/ 65 & g + & 0.66 - 0.93 & [email protected] & [email protected] & [email protected] & 64.5/ 64 & g + ngc6809 & 0.00 - 0.32 & [email protected] & [email protected] & [email protected] & 39.3/ 49 & g + & 0.32 - 0.47 & [email protected] & [email protected] & [email protected] & 35.7/ 48 & g + & 0.47 - 0.60 & [email protected] & [email protected] & [email protected] & 44.3/ 48 & g + ngc6981 & 0.00 - 0.67 & [email protected] & [email protected] & [email protected] & 54.4/ 75 & d + & 0.67 - 1.12 & [email protected] & [email protected] & [email protected] & 41.2/ 66 & g + & 1.12 - 1.89 & [email protected] & [email protected] & [email protected] & 53.7/ 58 & g + ngc7078 & 0.00 - 0.85 & [email protected] & [email protected] & [email protected] & 163.3/140 & d , e + & 0.85 - 1.25 & [email protected] & [email protected] & [email protected] & 136.9/122 & d , e + & 1.25 - 1.78 & [email protected] & [email protected] & [email protected] & 252.2/107 & d , e + ngc7099 & 0.00 - 0.64 & [email protected] & [email protected] & [email protected] & 74.6/ 52 & g + & 0.64 - 1.11 & [email protected] & [email protected] & [email protected] & 43.0/ 42 & g + & 1.11 - 1.74 & [email protected] & [email protected] & [email protected] & 36.0/ 37 & g + arp2 & 0.00 - 0.49 & [email protected] & [email protected] & [email protected] & 13.0/ 28 & g + & 0.49 - 0.72 & [email protected] & [email protected] & [email protected] & 11.6/ 29 & g + & 0.72 - 1.02 & [email protected] & [email protected] & [email protected] & 9.1/ 27 & g + palomar12 & 0.00 - 0.40 & [email protected] & [email protected] & [email protected] & 2.2/ 13 & n + & 0.40 - 0.71 & [email protected] & [email protected] & [email protected] & 1.5/ 13 & n + & 0.71 - 1.05 & [email protected] & [email protected] & [email protected] & 3.6/ 13 & n + terzan7 & 0.00 - 0.86 & [email protected] & [email protected] & [email protected] & 13.0/ 28 & n + & 0.86 - 1.48 & [email protected] & [email protected] & [email protected] & 11.6/ 29 & n + & 1.48 - 2.35 & [email protected] & [email protected] & [email protected] & 9.1/ 27 & n + [ tab : radial2 ] ccc ngc104 & 4.26 @xmath52 2.68 & d , n + ngc288 & 5.64 @xmath52 0.41 & g + ngc4590 & 9.43 @xmath52 0.69 & g + ngc5053 & 5.67 @xmath52 0.40 & g + m3 & 5.42 @xmath52 1.07 & d + ngc5466 & 5.58 @xmath52 0.43 & g + ngc5897 & 5.08 @xmath52 0.34 & g + ngc5904 & 7.73 @xmath52 0.72 & d + ngc5927 & 5.41 @xmath52 0.82 & d , f + ngc6101 & 6.13 @xmath52 0.40 & g + ngc6121 & 4.94 @xmath52 0.63 & g + m13 & 6.17 @xmath52 0.47 & d + ngc6218 & 3.75 @xmath52 0.47 & g + ngc6341 & 1.20 @xmath52 2.47 & d , n + ngc6352 & 1.76 @xmath52 1.30 & f + ngc6362 & 4.81 @xmath52 0.44 & g + ngc6541 & -0.12 @xmath52 5.47 & d , n + ngc6637 & 8.51 @xmath52 1.27 & d , f + ngc6656 & -2.39 @xmath52 0.33&f + ngc6723 & 6.89 @xmath52 0.38 & g + ngc6752 & 5.27 @xmath52 2.47&d , n + ngc6809 & 3.43 @xmath52 0.24&g + ngc6981 & 11.56 @xmath52 0.81&g + arp2 & 7.42 @xmath52 0.69 & n + terzan7 & 20.02 @xmath52 2.10 & n + [ tab : corefb ]
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binary stars are predicted to have an important role in the evolution of globular clusters , so we obtained binary fractions for 35 globular clusters that were imaged in the f606w and f814w with the acs on the hubble space telescope . when compared to the values of prior efforts @xcite , we find significant discrepancies , despite each group correcting for contamination effects and having performed the appropriate reliability tests .
the most reliable binary fractions are obtained when restricting the binary fraction to @xmath0 .
our analysis indicates that the range of the binary fractions is nearly an order of magnitude for the lowest dynamical ages , suggesting that there is a broad distribution in the binary fraction at globular cluster formation .
dynamical effects also appears to decrease the core binary fractions by a factor of two over a hubble time , but this is a weak relationship .
we confirm a correlation from previous work that the binary fraction within the core radius decreases with cluster age , indicating that younger clusters formed with higher binary fractions .
the strong radial gradient in the binary fraction with cluster radius appears to be a consequence of dynamical interactions .
it is often not present in dynamically young clusters but nearly always present in dynamically old clusters .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
soft x ray transients ( sxrts ) , when in outburst , show properties similar to those of persistent low mass x ray binaries containing a neutron star ( lmxrbs ; white et al . 1984 ; tanaka & shibazaki 1996 ; campana et al . 1998 ) . the large variations in the accretion rate that are characteristic of sxrts allow the investigation of a variety of regimes for the neutron stars in these systems which are inaccessible to persistent lmxrbs . while it is clear that , when in outbursts , sxrts are powered by accretion , the origin of the low luminosity x ray emission that has been detected in the quiescent state of several sxrts is still unclear . an interesting possibility is that a millisecond radio pulsar ( msp ) turns on in the quiescent state of sxrts ( stella et al . this would provide a missing link " between persistent lmxrbs and recycled msps . aql x-1 is the most active sxrt known : more than 30 x ray and/or optical outbursts have been detected so far . the companion star has been identified with the k1v variable star v1333 aql and an orbital period of 19 hr has been measured ( chevalier and ilovaisky 1991 ) . the outbursts of aql x-1 are generally characterised by a fast rise ( 510 d ) followed by a slow decay , with an @xmath4folding time of 3070 d ( see tanaka & shibazaki 1996 and campana et al . 1998 and references therein ) . type i x ray bursts were discovered during the declining phase of an outburst ( koyama et al . 1981 ) , testifying to the presence of a neutron star . ray luminosities are in the @xmath5 range ( for the @xmath6 kpc distance inferred from its optical counterpart ; thorstensen et al . 1978 ) . close to the outburst maximum the x ray spectrum is soft with an equivalent bremsstrahlung temperature of @xmath7 kev . sporadic observations of aql x-1 during the early part of the outburst decay ( czerny et al . 1987 ; tanaka & shibazaki 1996 ; verbunt et al . 1994 ) showed that when the source luminosity drops below @xmath8 the spectrum changes to a power law with a photon index of @xmath2 , extending up to energies of @xmath9 kev ( harmon et al . 1996 ) . rosat pspc observations revealed aql x-1 in quiescence on three occasions at a level of @xmath1 ( 0.42.4 kev ; verbunt et al . 1994 ) . in this lower energy band the spectrum is considerably softer and consistent with a black body temperature of @xmath10 kev . an outburst from aql x-1 reaching a peak luminosity of @xmath11 ( 210 kev ) was discovered and monitored starting from mid - february , 1997 with the rossixte all sky monitor ( asm ; levine et al . six observations were carried out with the bepposax narrow field instruments ( nfis ) starting from march 8@xmath12 , 1997 ( see table 1 ) , with the aim of studying the final stages of the outburst decay . 1a shows the light curve of the aql x-1 outburst as observed by the rossixte asm and bepposax mecs . the first part of the outburst can be fit by a gaussian with sigma @xmath13 d. this is not uncommon in sxrts ( e.g. in the case of 4u 160852 ; lochner & roussel - dupr 1994 ) . the flux decay rate changed dramatically around mjd 50512 ( march 5@xmath12 , 1997 ) . at the time of the first bepposax observation ( which started on march 8@xmath12 , 1997 ) the source luminosity was decreasing very rapidly , fading by about 30% in 11 hr , from a maximum level of @xmath14 . the second observation took place on march 12@xmath12 , 1997 when the source , a factor of @xmath15 fainter on average , reduced its flux by about 25% in 12 hr . in the subsequent four observations the source luminosity attained a constant level of @xmath16 , consistent with previous measurements of the quiescent luminosity of aql x-1 ( verbunt et al . 1994 ) . the sharp decrease after mjd 50512 is well described by an exponential decay with an @xmath4folding time @xmath17 d. the bepposax lecs , mecs and pds spectra during the fast decay phase , as well as those obtained by summing up all the observations pertaining to quiescence , can be fit with a model consisting of a black body plus a power law ( see table 2 and fig . the soft black body component remained nearly constant in temperature ( @xmath18 kev ) , but its radius decreased by a factor of @xmath19 from the decay phase to quiescence . the equivalent radius in quiescence ( @xmath20 km ) is consistent with the rosat results ( verbunt et al . the power law component changed substantially from the decay phase to quiescence : during the decay the photon index was @xmath21 , while in quiescence it hardened to @xmath22 . the two values are different with @xmath23 confidence ( see table 1 ) . the ratio of the 0.510 kev luminosities in the power law and black body components decreased by a factor of five between the first bepposax observation and quiescence . [ cols="^,^,^,^,^,^ " , ] @xmath25 spectra from the lecs and mecs ( and pds for the first observation ) detectors have been considered . the spectra corresponding to the quiescent state have been summed up , in order to increase the statistics and an upper limit from the summed pds data was also used to better constrain the power law slope . @xmath26 unabsorbed x ray luminosity in the energy range 0.510 kev . in the case of the first observation the pds data were included in the fit ( the unabsorbed 0.5100 kev luminosity amounts to @xmath27 ) . the bepposax observations enabled us to follow for the first time the evolution of a sxrt outburst down to quiescence . the sharp flux decay leading to the quiescent state of aql x-1 is reminiscent of the final evolution of dwarf novae outbursts ( e.g. ponman et al . 1995 ; osaki 1996 ) , although there are obvious differences with respect to the x ray luminosities and spectra involved in the two cases , likely resulting from the different efficiencies in the gravitational energy release between white dwarfs and neutron stars . models of low mass x ray transient outbursts hosting an old neutron star or a black hole are largely built in analogy with dwarf novae outbursts . in particular , van paradijs ( 1996 ) showed that the different range of time - averaged mass accretion rates over which the dwarf nova and low mass x ray transient outbursts were observed to take place is well explained by the higher level of disk irradiation caused by the higher accretion efficiency of neutron stars and black holes . however , the outburst evolution of low mass x ray transients presents important differences . in particular , the steepening in the x ray flux decrease of aql x-1 has no clear parallel in low mass x ray transients containing black hole candidates ( bhcs ) . the best sampled light curves of these sources show an exponential - like decay ( sometimes with a superposed secondary outburst ) with an @xmath4folding time of @xmath29 d and extending up to four decades in flux , with no indication of a sudden steepening ( chen et al . in addition , bhc transients display a larger luminosity range between outburst peak and quiescence than neutron star sxrts ( garcia et al . 1998 and references therein ) . being the mass donor stars and the binary parameters quite similar in the two cases , it appears natural to attribute these differences to the different nature of the underlying object : neutron stars possess a surface and , likely , a magnetosphere , while bhcs do not . when in outburst accretion down to the neutron star surface takes place in sxrts , as testified by the similarity of their properties with those of persistent lmxrbs , especially the occurrence of type i bursts and the x ray spectra and luminosities . the mass inflow rate during the outburst decay decreases , causing the expansion of the magnetospheric radius , @xmath30 . thus , accretion onto the neutron star surface can continue as long as the centrifugal drag exerted by the corotating magnetosphere on the accreting material is weaker than gravity ( illarionov & sunyaev 1975 ; stella et al . this occurs above a limiting luminosity @xmath31 , where @xmath32 is the gravitational constant ; @xmath33 , @xmath34 , @xmath35 g and @xmath36 ms are the neutron star mass , radius , magnetic field and spin period , respectively ( here and in the following we assume @xmath37 and @xmath38 cm ) . as @xmath30 reaches the corotation radius , @xmath39 , accretion onto the surface is inhibited and a lower accretion luminosity ( @xmath40 ) of @xmath41 is released . after this luminosity gap the source enters the propeller regime . if the mass inflow rate decreases further , the expansion of @xmath30 can continue up to the light cylinder radius , @xmath42 , providing a lower limit to the accretion luminosity that can be emitted in the propeller regime @xmath43 . below @xmath44 the radio pulsar mechanism may turn on and the pulsar relativistic wind interacts with the incoming matter pushing it outwards . matter inflowing through the roche lobe is stopped by the radio pulsar radiation pressure , giving rise to a shock front ( illarionov & sunyaev 1975 ; shaham & tavani 1991 ) . clearly these regimes have no equivalent in the case of black hole accretion . during the february - march 1997 outburst of aql x-1 , rossixte observations led to the discovery of a nearly coherent modulation at @xmath45 hz ( @xmath46 ms ) during a type i x ray burst . a single qpo peak , with a centroid frequency ranging from @xmath47 to 830 hz , was also observed at two different flux levels , when the persistent luminosity was @xmath48 and @xmath49 ( zhang et al . 1998 ) . in the presence of a single qpo peak , the magnetospheric and sonic point beat frequency models ( alpar & shaham 1985 ; miller et al . 1997 ) interpretation is ambiguous in that the qpo peak could represent either the keplerian frequency at the inner disk boundary or the beat frequency . moreover , the burst periodicity at @xmath45 hz may represent the neutron star spin frequency , @xmath50 , or half its value ( zhang et al . 1998 ) . in either cases , the possibility that accretion onto the neutron star surface takes place even in the quiescent state of aql x-1 faces serious difficulties : for a quiescent luminosity of order @xmath51 a magnetic field of only @xmath52 g would be required , in order to fulfill the condition @xmath53 . for such a low magnetic field , aql x-1 and , by inference , lmxrbs with khz qpos can hardly be the progenitors of recycled msps . more crucially , the marked steepening in the outburst decay that takes place below @xmath54 , is accompanied by a marked spectral hardening , resulting from a sudden decrease of the flux in the black body spectral component . this is clearly suggestive of a transition taking place deep in the gravitational well of the neutron star , where most of the x rays are produced . the most appealing mechanism is a transition to the propeller regime , where most of the inflowing matter is stopped at the magnetospheric boundary ( zhang , yu & zhang 1998 ) . in fig . 1a , the luminosity at mjd 50512 is identified with @xmath55 and the lower horizontal lines indicate the luminosity interval during which aql x-1 is likely in the propeller regime . additional information on the neutron star magnetic field ( and spin ) can be inferred as follows . the observed ratio of the luminosity , @xmath56 , when the qpo at @xmath57 hz were detected and the luminosity @xmath58 when the centrifugal barrier closes is @xmath59 . at @xmath58 the keplerian frequency of matter at the magnetospheric boundary is , by definition , equal to the spin frequency , i.e. @xmath60 or @xmath61 hz for aql x-1 . based on beat - frequency models , at @xmath56 the keplerian frequency at the inner disk boundary can be either @xmath62 hz or @xmath63 hz , depending on whether the single khz qpos observed corresponds to the keplerian or the beat frequency . in the magnetospheric beat - frequency models , simple theory predicts that the keplerian frequency at the magnetospheric boundary is @xmath64 ; in the radiation pressure - dominated regime relevant to the case at hand , the ghosh and lamb ( 1992 ) model predicts instead @xmath65 . therefore we expect @xmath66 and @xmath67 , in the two models , respectively . such a low ratio clearly favors the interpretation in which @xmath62 hz and @xmath68 hz . in the sonic point beat - frequency model ( miller et al . 1997 ) , the innermost disk radius is well within the magnetosphere , implying that the keplerian frequency at the magnetospheric boundary is @xmath69 . in this case all possible combinations of @xmath50 and @xmath70 are allowed . by using the observed @xmath58 , a neutron star magnetic field of @xmath71 g ( depending on the adopted model of the disk - magnetosphere interaction ) is obtained in the case @xmath68 hz and @xmath72 g in the case @xmath73 hz . once in the propeller regime , the accretion efficiency decreases further as the magnetosphere expands for decreasing mass inflow rates ( @xmath74 ) . the @xmath75 d exponential - like luminosity decline observed with bepposax is considerably faster than the propeller accretion luminosity extrapolated from the first part of the outburst ( e.g. the gaussian profile shown by the dashed line in fig . we note here that the spectral transition accompanying the onset of the centrifugal barrier may also modify the irradiation properties of the accretion disk , contributing to x ray luminosity turn off . alternatively , an active contribution of the `` propeller '' mechanism or the neutron star spin - down energy dissipated into the inflowing matter can not be excluded . it is unlikely that the quiescent luminosity of aql x-1 is powered by magnetospheric accretion in the propeller regime . as shown in fig . 1a , the quiescent x ray luminosity is probably lower than the minimum magnetospheric accretion luminosity @xmath76 allowed in the propeller phase ( this remains true for @xmath77 g if @xmath68 hz , and for @xmath78 g if @xmath73 hz ) . moreover the bepposax x ray spectrum shows a pronounced decrease in the power law to black body flux ratio together with a flattening of the power law component between the fast decay phase and quiescence , suggesting that a transition to shock emission from the interaction of a radio pulsar wind with the matter outflowing from the companion star has taken place . note that an x ray spectrum with a slope of @xmath79 has been observed from the radio pulsar psr b125963 immersed in the wind of its be star companion . models of this interaction predict that a power law x ray spectrum with a slope around @xmath79 should be produced for a wide range of parameters ( tavani & arons 1997 ) . the additional soft x ray component observed during the outburst decay ( see table 2 ) might be emitted at the polar caps as a result of the residual neutron star accretion in the propeller phase . note that the equivalent black body radius decreases for decreasing x ray luminosities , just as it would be expected if the magnetospheric boundary expanded . alternatively , the black body - like spectral component observed in quiescence could be due to the streaming of energetic particles that hit the polar caps , in close analogy to the soft x ray component observed , in msps , at the weaker level of @xmath80 ( becker & trmper 1997 ) . assuming a magnetic field in the range derived in section 3.1 ( i.e. @xmath81 g for @xmath82 hz and @xmath83 g for @xmath84 hz ) , we can consistently explain the @xmath1 quiescent x ray luminosity as powered by a radio pulsar enshrouded by matter outflowing from the companion star , if the conversion efficiency of spin - down luminosity to x ray is @xmath85% . this is consistent with modeling and observations of enshrouded pulsars ( tavani 1991 ; verbunt et al . 1996 ) . there are chances of observing a msp ( a simple scaling from msps implies a signal at 400 mhz of @xmath86 mjy ; see kulkarni et al . 1990 ) , even though the emission would probably be sporadic , like in the case of the pulsar psr 174424a due to the large amount of circumstellar matter ( see lyne et al . 1991 ; shaham & tavani 1991 ) . in summary aql x-1 appears to provide the first example of an old fast rotating neutron stars undergoing a transition to the propeller regime at first , followed by a transition to the radio pulsar regime , as the transient x ray emission approaches its quiescent level . therefore , aql x-1 ( and possibly sxrts in general ) likely represents the long - sought `` missing link '' between lmxrbs and recycled msps .
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we report on the march - april 1997 bepposax observations of aql x-1 , the first to monitor the evolution of the spectral and time variability properties of a neutron star soft x
ray transient from the outburst decay to quiescence .
we observed a fast x ray flux decay , which brought the source luminosity from @xmath0 to @xmath1 in less than 10 days .
the x ray spectrum showed a power law high energy tail with photon index @xmath2 which hardened to @xmath3 as the source reached quiescence .
these observations , together with the detection by rossixte of a periodicity of a few milliseconds during an x ray burst , likely indicate that the rapid flux decay is caused by the onset of the propeller effect arising from the very fast rotation of the neutron star magnetosphere . the x ray luminosity and hard spectrum that characterise the quiescent emission can be consistently interpreted as shock emission by a turned - on rotation - powered pulsar .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the identification of the higgs boson(s ) is one of the main goals of the large hadron collider ( lhc ) being built at cern . there are expectations that there exists a ` light ' higgs boson with mass @xmath3 gev . in this mass range , its detection at the lhc will be challenging . there is no obvious perfect detection process , but rather a range of possibilities , none of which is compelling on its own . some of the processes are listed in table 1 , together with the predicted event rates for the integrated luminosity of 30 fb@xmath4 expected over the first two or three year period of lhc running . we see that , _ either _ large signals are accompanied by a huge background , _ or _ the processes have comparable signal and background rates for which the number of higgs events is rather small . here we wish to draw particular attention to process ( c ) , which is often disregarded ; that is the exclusive signal @xmath5 , where the + sign indicates the presence of a rapidity gap . it is possible to install proton taggers so that the ` missing mass ' can be measured to an accuracy @xmath6 gev @xcite . then the exclusive process will allow the mass of the higgs to be measured in two independent ways . first the tagged protons give @xmath7 and second , via the @xmath1 decay , we have @xmath8 , although now the resolution is much poorer with @xmath9 gev . the existence of matching peaks , centered about @xmath10 , is a unique feature of the exclusive diffractive higgs signal . besides its obvious value in identifying the higgs , the mass equality also plays a key role in reducing background contributions . another advantage of the exclusive process @xmath11 , with @xmath1 , is that the leading order @xmath12 background subprocess is suppressed by a @xmath13 selection rule @xcite . [ cols="<,^,^,^,^ " , ] the radiation associated with the @xmath14 hard subprocess is not the only way to populate and to destroy the rapidity gaps . there is also the possibility of soft rescattering in which particles from the underlying event populate the gaps . the probability , @xmath15 , that the gaps survive the soft rescattering was calculated using a two - channel eikonal model , which incorporates high mass diffraction @xcite . including this factor , and the nlo @xmath16 factor , the cross section is predicted to be @xcite @xmath17 for the production of a standard model higgs boson of mass 120 gev at the lhc ) at the tevatron , 0.2 fb , is too low to provide a viable signal . ] . it is estimated that there may be a factor two uncertainty in this prediction @xcite . the event rate in entry ( c ) of table 1 includes a factor 0.6 for the efficiency associated with proton tagging , 0.6 for @xmath18 and @xmath19 tagging , 0.5 for the @xmath20 jet polar angle cut , @xmath21 , ( necessary to reduce the @xmath2 qcd background ) and 0.67 for the @xmath1 branching fraction @xcite . hence the original @xmath22 events is reduced to an observable signal of 11 events , as shown in table 1 . the advantage of the @xmath23 signal is that there exists a @xmath13 selection rule , which requires the leading order @xmath24 background subprocess to vanish in the limit of massless quarks and forward outgoing protons limit , the two born - level diagrams ( figs . 2(a , b ) _ without _ the emission of the gluon ) cancel each other . ] . however , in practice , lo background contributions remain . the prolific @xmath25 subprocess may mimic @xmath2 production since we may misidentify the outgoing gluons as @xmath18 and @xmath19 jets . assuming the expected 1% probability of misidentification , and applying @xmath21 jet cut , gives a background - to - signal ratio @xmath26 . secondly , there is an admixture of @xmath27 production , arising from non - forward going protons which gives @xmath28 . thirdly , for a massive quark there is a contribution to the @xmath13 cross section of order @xmath29 , leading to @xmath26 , where @xmath30 is the transverse energy of the @xmath18 and @xmath19 jets . next , we have the possibility of nlo @xmath31 background contributions . of course , the extra gluon may be observed experimentally and these background events eliminated . however , there are exceptions . the extra gluon may go unobserved in the direction of a forward proton . this background may be effectively eliminated by requiring the equality @xmath32 . then we may have soft gluon emission . first , we note that emission from an outgoing @xmath18 or @xmath19 is not a problem , since we retain the cancellation between the crossed and uncrossed graphs . emission from the virtual @xmath18 line is suppressed by at least a factor of @xmath33 ( in the amplitude ) , where @xmath34 and @xmath35 are the energies of the outgoing soft gluon and an outgoing @xmath18 quark in the @xmath24 centre - of - mass frame . the potential danger is gluon emission from an incoming gluon , see fig . 2 . the first two diagrams no longer cancel , as the @xmath2 system is in a colour - octet state . however , the third diagram has precisely the colour and spin structure to restore the cancellation . thus soft gluon emissions from the initial colour - singlet @xmath36 state factorize and , due to the overriding @xmath13 selection rule , qcd @xmath2 production is still suppressed . the remaining danger is large angle hard gluon emission which is collinear with either the @xmath18 or @xmath19 jet , and therefore unobservable . if the cone angle needed to separate the @xmath37 jet from the @xmath18 ( or @xmath19 ) jet is @xmath38 then the expected background from unresolved three jet events leads to @xmath39 . the nnlo @xmath40 background contributions are found to be negligible ( after requiring @xmath41 ) , as are soft pomeron - pomeron fusion contributions to the background ( and to the signal ) @xcite . so , in total , double - diffractive higgs production has a signal - to - background ratio of about three , after including the @xmath16 factors . identifying a ` light ' higgs will be a considerable experimental challenge . all detection processes should be considered . from table 1 we see that valuable information can be obtained from weak boson fusion , where the higgs and the accompanying jets are produced at high @xmath42 . for example , process ( d ) is based on the @xmath43 decay for which the background is small @xcite , whereas process ( f ) exploits rapidity gaps so that the larger @xmath1 signal may be isolated @xcite , provided the pile - up problems can be overcome @xcite . here we have drawn attention to the exclusive @xmath11 signal , process ( c ) . the process has the advantage that the signal exceeds the background . the favourable signal - to - background ratio is offset by a low event rate , caused by the necessity to preserve the rapidity gaps so as to ensure an exclusive signal . nevertheless , entry ( c ) of table 1 shows that the signal has reasonable significance in comparison to the standard @xmath44 and @xmath45 search modes . moreover , the advantage of the matching higgs peaks , @xmath46 , can not be overemphasized . spectrum , see process ( a ) of table 1 . ] we thank albert de roeck , risto orava and andrei shuvaev for valuable discussions , and the eu , pparc and the leverhulme trust for support . xx d. zeppenfeld et al . , phys . rev . * d62 * ( 2000 ) 013009 . v. drollinger , t. mller and d. denegri , cms note , hep - ph/0111312 ; + j. goldstein et al . * 86 * ( 2001 ) 1694 . a. de roeck , v.a . khoze , a.d . martin , r. orava and m.g . ryskin , durham report , hep - ph/0207042 . khoze , a.d . martin and m.g . ryskin , eur . j. * c23 * ( 2002 ) 311 . d. zeppenfeld , hep - ph/0203123 ; n.kauer , t. plehn , d. rainwater and d. zeppenfeld , phys . * b503 * ( 2001 ) 113 . khoze , a.d . martin and m.g . ryskin , eur . j. * c21 * ( 2001 ) 99 . z. bern , l. dixon and c. schmidt , hep - ph/0206194 . khoze , a.d . martin and m.g . ryskin , eur . j. * c14 * ( 2000 ) 525 . khoze , a.d . martin and m.g . ryskin , eur . j. * c19 * ( 2001 ) 477 . khoze , a.d . martin and m.g . ryskin , eur . j. * c18 * ( 2000 ) 167 .
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we show that exclusive double - diffractive higgs production , @xmath0 , followed by the @xmath1 decay , could play an important role in identifying a ` light ' higgs boson at the lhc , provided that the forward outgoing protons are tagged .
we predict the cross sections for the signal and for all possible @xmath2 backgrounds .
ippp/02/41 + dcpt/02/82 + 3 july 2002 * forward proton tagging as a way to identify a light higgs boson at the lhc * a.d .
martin , v.a .
khoze and m.g .
ryskin institute for particle physics phenomenology , + university of durham , dh1 3le , uk
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the striking feature of the lightcurves of known gamma - ray pulsars are relatively long duty cycles as well as phase shifts in comparison to the radio pulses ( thompson et al . 1999 , thompson 2001 , kanbach 2002 ) . the lightcurve shapes fall into three categories . the three brightest gamma - ray pulsars - the crab pulsar , vela and geminga , along with b1951 + 32 exhibit two well defined , sharp peaks separated in phase by @xmath0 , and connected by the interpeak bridge of a considerable level . b1706 - 44 and b1055 - 52 show two peaks separated by about 0.2 in phase , whereas b1509 - 58 exhibits a broad single pulse . similar properties are present in the x - ray domain of the high - energy emission . moreover , in the case of the crab pulsar its optical pulsed emission is usually considered jointly with the high - energy emission : the phase - averaged spectra in gamma and x - rays connect smoothly to optical , and the pulses have similar shapes and phases . this suggests that gammas , x - rays and optical light may come from the same regions . these properties , along with the spectral properties which are not the subject of this paper , prompted substantial refinements of the physical models of pulsar high - energy activity , both outer - gap models ( cheng et al . 1986 , romani & yadigaroglu 1995 , zhang & cheng 1997 ) as well as polar - cap models ( sturrock 1971 ; ruderman & sutherland 1975 ; daugherty & harding 1982 , sturner et al . 1995 ) , ( see rudak , dyks & bulik 2002 for recent critical review ) . despite reiterated arguments in favour of polar - gap models ( e.g. baring 2001 ) and outer - gap models ( e.g. yadigaroglu & romani 1995 ) , both classes of models still suffer from serious problems . according to the polar cap model , the characteristic double - peak lightcurve forms when the line of sight intersects the polar - cap beam where the highest - energy emission originates : upon entering the beam a leading peak is produced , followed by a bridge emission due to the inner parts of the beam , when the line of sight exits the beam the trailing peak forms . because of long duty cycles in the high - energy domain on one hand , and narrow opening angles for gamma - ray emission on the other hand , polar cap models have to rely on nearly aligned rotators , where inclination ( the angle @xmath1 ) of the magnetic axis ( the magnetic moment @xmath2 ) to the spin axis ( the angular velocity @xmath3 ) is comparable to the angular extent of the polar cap ( daugherty & harding 1994 ) . the outer - gap model , on the contrary , prefers highly inclined rotators ( chiang & romani 1992 ; romani & yadigaroglu 1995 ; cheng et al . however , the model is unable ( without additional postulates ) to account for the presence of outer wings in the double - peak lightcurves . more importantly , the model in its present form ( e.g. zhang & cheng 2002 ) is unable to account for a substantial level of the off - pulse emission in the crab pulsar . also on theoretical grounds the existence of outer gaps in the present - day ` vacuum ' approach ( i.e. with the gap extending between the null surface and the light cylinder ) has been questioned . the outer gap remains anchored to the conventional null surface , provided that no current is injected at the boundaries of the accelerator ( no pairs are created ) ; otherwise the gap extention becomes very sensitive to the details of pair creation ( hirotani & shibata 2001 , hirotani et al . 2003 ) . these problems motivated us to propose a new picture of the origin of high - energy radiation within the pulsar magnetospheres - in regions confined to the surface of last open magnetic field lines ( similarly to thin outer - gap accelerators ) but extending between the polar cap and the light cylinder . the most important consequence of such extended accelerators , as far as the lightcurves are concerned , is a caustic nature of the high - energy peaks : special relativity effects ( aberration of photon emission directions and time of flight delays due to the finite speed of light @xmath4 ) cause that photons emitted at different altitudes within some regions of the magnetosphere are piled up at the same phase of a pulse ( morini 1983 ; romani & yadigaroglu 1995 ) . the term ` high - energy radiation ' used throughout the paper refers to nonthermal radiation in the energy domain of gamma - rays and hard x - rays ( i.e. above several kev ) . the soft x - ray radiation ( @xmath5 kev ) is not included into our considerations because it is heavily affected by thermal emission from the neutron star surface in some objects , like geminga which exhibits a complex pattern of soft x - ray pulses ( e.g. fig.4 in jackson et al . 2002 ) . for reasons mentioned at the beginning of this section , in the case of the crab pulsar our definition includes also the optical band . the paper is arranged in the following way : in section 2 we introduce the two - pole caustic model for the high - energy lightcurves of pulsars and present the results of numerical calculations . section 3 contains discussion of the generic features of the model as well as comparison with the properties of other models . special relativity ( sr ) effects which affect pulsar lightcurves include the aberration of photon emission directions and the time of flight delays caused by the finite speed of light @xmath4 . morini ( 1983 ) was the first to prove that these effects were able to produce prominent peaks in pulsar lightcurves . he obtained the peaks of caustic origin in his version of the polar cap model , which included photon emission from high altitudes , where the sr effects are important . unlike in the case of the morini s model , in the standard polar cap model the strongest gamma - ray emission takes place very close to the star surface , where the caustic effects are not important and , therefore , pulsar lightcurves are mostly determined by the altitudinal extent of the accelerator ( polar gap ) and by the emissivity profile along the magnetic field lines ( eg . daugherty & harding 1996 ; dyks & rudak 2000 ) . in the recent version of the outer gap model ( chiang & romani 1992 ; romani & yadigaroglu 1995 ; yadigaroglu 1997 ; cheng et al . 2000 ) the peaks in the lightcurves are purely due to the caustic effects the fact first emphasized by romani & yadigaroglu ( 1995 ) . however , the altitudinal extent of accelerator ( limited by the position of the inner boundary of the outer gap ) is crucial in limiting the possible shapes of lightcurves within this model . interestingly , yadigaroglu ( 1997 ) also considered photon emission along the entire length of all last open magnetic field lines , relaxing thus the outer - gap extent limit ( see the bottom panel in fig . 3.1 of his thesis ) . however , he did nt pursue this possibility in any further details . hereby we propose that the observed high - energy emission from known gamma - ray pulsars originates from the regions considered already by yadigaroglu ( 1997 ) . let us assume that the gap region ( the region where particle acceleration is taking place as well as high - energy photons originate ) posesses the following properties : + - the gap region extends from the polar cap to the light cylinder , + - the gap is thin and confined to the surface of last open magnetic - field lines , + - photon emissivity is uniform within the gap region . + fig.1 shows schematically the location and the extension of the proposed gap ( for the sake of comparison the location of the outer gap is shown as well ) . the resulting lightcurves are dominated by strong peaks ( either double or single ) of caustic origin . in the outer - gap model , the inner boundary of the gap is located at an intersection of the null - charge surface with the last closed field lines . its radial distance @xmath6 is , therefore , a function of azimuthal angle @xmath7 , with a minimum value @xmath8 in the @xmath9-plane . for highly inclined rotators @xmath8 becomes a small fraction of the light cylinder radius @xmath10 : @xmath11 ^ 2 $ ] ( halpern & ruderman 1993 ) ; for the inclination angle @xmath12 the ratio is @xmath13 . however , for azimuthal directions departing the @xmath9-plane , the inner radius @xmath6 tends to @xmath14 . the radiation region extends out to the light cylinder and the radiation escaping the magnetospheric region comes from particles moving outwards along the magnetic lines . + in our model we assume , however , that the actual gap extends to the polar cap , i.e. @xmath15 for all azimuthal angles @xmath7 . this assumption is essential for the expected performance of the model since it implies that the observer can detect radiation originating from _ both _ magnetic hemispheres . as in the outer - gap model of chiang & romani ( 1992 ) , and cheng et al . ( 2000 ) we assumed that the photon emissivity is uniform everywhere in the proposed gap region . for simplicity , we consider a rigidly rotating static - like magnetic dipole . departures of the retarded field lines from the static case are of the order of @xmath16 and they are insignificant since prominent features in modelled lightcurves ( peaks ) which we discuss below arise due to radiation at radial distances @xmath17 . also , rotationally - driven currents can be neglected : longitudinal currents suspected to flow within the open field line region can not modify @xmath18 by a factor exceeding @xmath19 whereas toroidal currents due to plasma corotation change @xmath18 barely by @xmath16 ( @xmath20 is the local corotation velocity in the speed of light units ; @xmath21 ) . we considered radiating particles moving from the outer rim of the polar cap toward the light cylinder . the photons were emitted tangentially to local magnetic field lines in the corotation frame , and then they were followed crossing the magnetosphere with no magnetic attenuation . our treatment of rotational effects in the numerical code used for the calculations is the same as in yadigaroglu ( 1997 ) and in dyks & rudak ( 2002 ) . specifically , we used the following standard aberration formula to transform the unit vector of photon propagation direction @xmath22 from the corotating frame to its value @xmath23 in the inertial observer frame : @xmath24 where @xmath25 . the above formula results directly from the general lorentz transformation . by replacing @xmath26 and @xmath27 in the formula with @xmath28 and @xmath29 ( respectively ) , one obtains a general lorentz transformation for a velocity vector @xmath30 . + photon travel delays were taken into account by adding @xmath31 to the azimuth @xmath32 of photon emission direction @xmath23 ( @xmath33 is the radial position of emission point , and @xmath23 is the unit vector of photon propagation direction after aberration effect is included ) . the result , taken with a minus sign , is a phase of detection @xmath34 : @xmath35 our numerical code performs a raytracing followed by a numerical integration of the observed pulse profile . all results described below are just due to the dipolar shape of the magnetic field , the aberration effect ( eq . 1 ) , as well as the light travel time delays ( eq . 2 ) . the results are presented in fig.[fig2 ] in the form of photon mapping onto the @xmath36-plane , with accompanying lightcurves for five viewing angles @xmath37 ( the angle between the spin axis and the line of sight ) . inclination angle @xmath12 , and spin period @xmath38 s were assumed for the rotator . for each magnetic pole two caustics form in the @xmath36-plane : 1 ) the dominant caustic , and 2 ) the subdominant caustic . + the dominant caustic ( easy to identify in the photon mapping ) is associated with the trailing part of the emission region with respect to its magnetic pole . the subdominant caustic ( much weaker ) is associated with the leading part of the emission region . characteristic features in the lightcurves due to caustic crossing for large values of @xmath37 are marked with capital letters from a to d. the dominant - caustics crossing yields two prominent peaks d and b , in the lightcurve . the peaks consist of photons emitted over a very wide range of altitudes ( e.g. for @xmath39 almost all altitudes between the star surface and the light cylinder contribute to the peaks ; see also fig . 2 in morini 1983 ) . the subdominant - caustics crossing yields the features a and c , which actually contribute to the trailing wings of the peaks d and b , respectively . features a and c consist of photons emitted very close to the light cylinder ( cf . fig . 9 in cheng this result is in clear contrast with the results obtained for the conventional outer - gap model ( see fig.[fig3 ] ) , where the leading peak ( a ) is due to the subdominant caustic , and the trailing peak ( b ) is due to the dominant caustic . unlike in the outer gap model , in our model each of the two prominent peaks in the resulting lightcurve is associated with a different magnetic pole . therefore , we propose the name two - pole caustic model " . the position of the last closed magnetic field lines and the magnitude of special relativity effects are governed by the proximity of the light cylinder and , therefore , the radiation pattern as well as the lightcurves shown in fig . 2 do not depend on the rotation period @xmath40 but solely on the inclination angle @xmath1 ( with one exception : the size of blank spots corresponding to polar caps does depend on @xmath40 ) . therefore , the two - pole caustic model is relevant for pulsars with any rotation period . we find that noticeable peaks of caustic origin appear in the lightcurves practically for any inclination of magnetic dipole @xmath41 and for any viewing angle @xmath42 . for small inclinations ( @xmath43 ) the peaks are broader and it is more probable to observe single - peaked lightcurves ; such lightcurves form for a wide range of viewing angles for which the coventional outer gap model predicts no emission . in the glast era , the detectability of moderately inclined pulsars " viewed far from the equatorial plane will serve as a clear discriminator between the two - pole caustic model and the traditional outer gap model . [ fig4 ] presents a comparison of a lightcurve calculated within the two - pole caustic model for the vela pulsar with a lightcurve obtained for this pulsar with egret ( kanbach 1999 ) . the model lightcurve was calculated for electrons distributed evenly along the polar cap rim ; the density profile across the rim was assumed to be the gaussian function @xmath44 , symmetrical about @xmath45 , with @xmath46 ( @xmath47 is the magnetic colatitude of magnetic field lines footprints at the star surface , and @xmath48 is the magnetic colatitude of the rim ) . photon emission was followed up to @xmath49 . the shape of the egret lightcurve is very well reproduced by the two - pole caustic model : the leading peak is narrower than the trailing peak , which connects smoothly with the bridge emission ; the leading peak seems to present a separate entity it does not connect smoothly with the bridge , and is followed by a characteristic ` interpulse bump ' . these features result generically from the two - pole caustic model , since it predicts that the trailing peak , the bridge emission , and the ` interpulse bump ' arise from sampling a single , continuous radiation pattern from one magnetic pole ( cf . [ fig2]a ) . the leading peak and the offpulse emission are produced by sampling an emission pattern from the opposite pole . the ` interpulse bump ' predicted by the two - pole caustic model is not observed in the crab pulsar . this might be caused by a decline in the photon emissivity above @xmath50 . single peaked gamma - ray lightcurves as the one observed for b1509@xmath5158 ( kuiper et al . 1999 ) can be reproduced in the two - pole caustic model for small viewing angles ( fig . [ fig2]b ) , however , the predicted phase lag between the gamma - ray and the radio peak ( @xmath52 ) is smaller than the one observed for b1509@xmath5158 ( @xmath53 ) . moreover , double - peak lightcurves with small separation between the peaks in b1706@xmath5144 ( thompson et al . 1996 ) , and b1055@xmath5152 ( thompson et al . 1999 ) can not be interpreted in the same way as the lightcurves with wide separation of the peaks . these difficulties forced chiang & romani ( 1992 ) ( and probably yadigaroglu 1997 ) to abandon the geometry of the two - pole caustic model ( r. w. romani , private communication ) . we agree with r. w. romani that these particular lightcurves should be interpreted in terms of the outer gap caustics a and b ( figs . [ fig2 ] and [ fig3 ] ) . this interpretation can be accommodated by the two - pole caustic model only when the outer gap part of the lightcurves ( between a and b in fig . [ fig2 ] ) dominates over the leading peak formed by the trailing caustic d. as can be noticed in fig . [ fig2 ] , the intensity of the outer gap part of lightcurves ( between a and b ) increases with respect to the intensity of the peak d , as the viewing angle @xmath37 approaches the value for which the line of sight barely skims the outer gap part of the radiation pattern . for some viewing gometries the intensity of the outer gap part ( a b ) may exceed by a few times the intensity of the peak d. fig . [ fig5 ] shows an example of such a lightcurve . given the low intensity and large width of the peak d , it may stay unresolved in the low statistics data of b1509@xmath5158 , b1706@xmath5144 , and b1055@xmath5152 . table 1 summarizes major similarities and differences between our model and other models . for these purposes we choose the model of morini 1983 ( it was the first model where caustic effects were noticed ) and the model of smith et al . 1988 , along with the polar - cap model and the outer - gap model . we introduced a two - pole caustic model for the high - energy lightcurves of pulsars ( for the crab pulsar in particular ) . the effects of aberration and light travel delays , as well as the geometry of the last closed magnetic field lines are essential for forming the lightcurves of a caustic nature . the generic features in the lightcurves provided by the two - pole caustic model are consistent with the observed characteristics of pulsar lightcurves : + 1 . the most natural ( in terms of probability ) shape consists of two peaks ( separated by 0.4 to 0.5 in phase for large viewing angles ) , + 2 . the peaks posess well developed wings , + 3 . there is a bridge ( inter - peak ) emission component , + 4 . there is a non - vanishing off - pulse emission level , + 5 . the radio pulse ( or pulse precursor , in the case of crab ) comes ahead of the leading peak ( by @xmath52 in phase for large viewing angles ) . features 1 . , 3 . , and 5 . are not a unique property of the two - pole caustic model , they can be easily obtained within the outer - gap model ( compare the lightcurves in figs.[fig2 ] and [ fig3 ] ) . feature 2 . may in principle be obtained within the outer - gap model , but no consensus exists among the proponents of that model on the actual physical reason behind this feature ( yadigaroglu 1997 , cheng , ruderman & zhang 2000 ) . in our model the trailing wings are formed by the subdominant caustics ( a ) and ( c ) which often blend with peaks ( d ) and ( b ) , respectively ( the effect of blending is not shown in fig . 2 since the calculation were truncated at @xmath54 . ) feature 4 . , however , may play a decisive role in showing the advantage of the two - pole caustic model over the outer - gap model : this particular feature of our model may explain the presence of the significant x - ray flux from the crab pulsar at pulse minimum discovered by tennant et al . it has not been demonstrated so far how such a feature could be obtained within the outer - gap model ; in particular , it is absent in the x - ray pulse profile calculated for the crab pulsar in the recent model of zhang & cheng 2002 . an interesting property of the double - peak lightcurve , inherent to the two - pole caustic model , emerges for viewing angles @xmath37 close to @xmath55 : the trailing peak ( together with its wings ) assumes the shape which is roughly similar ( in the sense of translations in the rotation - phase @xmath34 space ) to the shape of the leading peak and its wings ( cf . fig.[fig2 ] ) . such effect is not possible in the case of the outer - gap model ( cf . fig.[fig3 ] ) , nor it is possible in the polar cap model ( see wona et al . 2002 ) ; these two models lead to approximate ` mirror'-symmetry in the double - peak lightcurves . in principle then , this property might also be used to discriminate between the two - pole caustic model and other models . an important testing - ground for any models will be polarization properties of the high - energy radiation . for the time being , good - quality polarization information is available only for optical light from crab ( smith et al . 1988 ) . an argument in favor of the caustic origin of the optical peaks of the crab pulsar is that the degree of polarization drops to minimum values at the phases of both peaks ( cf . 4c of smith et al . such a drop is justified by virtue of the caustic nature of the peaks : it results from a pile - up of polarized radiation with different polarization angles . smith et al . @xcite emphasize , that the behaviour of the polarization as a function of rotation - phase for crab is strikingly similar for both peaks , i.e. the polarization behaviour at the phase of the leading peak repeats at the trailing peak . the outer gap model is able to reproduce this feature , even though each of the two peaks arises from a very different type of caustic in this model : the leading peak is due to the caustic formed close to the light cylinder at the _ leading _ part of the emission region , whereas the trailing peak is due to the caustic formed within the _ trailing _ part . romani & yadigaroglu ( 1995 ) consider the ability to reproduce the double sweep in the polarization position angle to be one of major successes of the outer - gap model . in the two - pole caustic model , the two peaks arise due to crossing the same type of caustic - the dominant caustic associated with the trailing part of the emission region ( cf . section 2 ) . we suspect , therefore , that such a double sweep should even more naturally be produced by the two - pole caustic model . a comprehensive treatment of the polarization properties of high - energy radiation in the two - pole caustic model will be the subject in our future work . we emphasize that the characteristic form of the high - energy lightcurves of pulsars ( the double - peak structure with bridge emission in high - energy , preceded by the peak in radio ) is an inherent property of a rotating source with a magnetic dipole , with roughly uniform high - energy emissivity along the last open field lines . two recently proposed models may provide physical grounds for the geometry of the two - pole caustic model : the slot - gap model of arons & scharlemann ( 1979 ) , in the modern version of muslimov & harding ( 2003 ) ; and the model of hirotani et al . ( 2003 ) of an outer gap extended on either side of the null surface due to the currents . when a realistic physical model for the extended gaps becomes available , calculations of spectral characteristics within our model will be possible . we acknowledge fruitful discussions with a. harding , k. hirotani and a. muslimov . we thank j. gil for bringing our attention to the problem of missing wings in the double - peak lightcurves in the outer - gap model and r. w. romani for pointing out some shortcomings of the two - pole caustic model . comments by the anonymous referee helped us to clarify the paper significantly . this work was supported by the grant pbz - kbn-054/p03/2001 . part of the work was performed while jd held a national research council research associateship award at nasa / gsfc . arons , j. & scharlemann , e. t. , 1979 , , 231 , 854 baring , m. g. 2001 , proc . tonantzintla workshop , eds . a. carrami~ nana et al . , astrophysics and space science library , volume 267 ( kluwer academic publishers dordrecht ) , p.167 cheng , k. s. , ho , c. , & ruderman , m. 1986 , , 300 , 500 cheng , k. s. , ruderman , m. a. , & zhang , l. 2000 , , 537 , 964 chiang , j. , & romani , r. w. 1992 , , 400 , 629 kuiper , l. , hermsen , w. , krijger , j. m. , bennett , k. , carramiana , a. , et al . 1999 , a&a , 351 , 119 daugherty , j.k . , & harding , a.k . 1982 , apj , 252 , 337 daugherty , j.k . , & harding , a.k . 1994 , apj , 42 , 325 daugherty , j.k . , & harding , a.k . 1996 , apj , 458 , 278 dyks , j. , & rudak , b. 2000 , , 319 , 477 dyks , j. , & rudak , b. 2002 , a&a , 393 , 511 halpern , j. p. , & ruderman , m. a. 1993 , , 415 , 286 hirotani , k. , & shibata , s. , 2001 , , 325 , 1228 hirotani , k. , harding , a.k . , & shibata , s. , 2003 , , 591 , 334 jackson , m. s. , halpern , j. p. , gotthelf , e. v. , & mattox , j. r. 2002 , , 578 , 935 kanbach , g. , 1999 , in proc . of the third integral workshop , eds . a. bazzano , g. g. c. palumbo , & c. winkler , astrophys . comm . , 38 , 17 kanbach , g. , 2002 , in proc . of the 270 . we - heraeus seminar on neutron stars , pulsars , and supernova remnants , eds . w. becker , h. lesch , and j. trmper , mpe report 278 , 91 ( astro - ph/0209021 ) morini , m. 1983 , , 202 , 495 muslimov , a. , & harding , a.k . , 2003 , , 588 , 430 romani , r. w. & yadigaroglu , i .- a . , 1995 , , 438 , 314 rudak , b. , dyks , j. , & bulik , t. , 2002 , in proc . of the 270 . we - heraeus seminar on neutron stars , pulsars , and supernova remnants , eds . w. becker , h. lesch , and j. trmper , mpe report 278 , 142 ( astro - ph/0206101 ) ruderman , m. a. , & sutherland , p. g. 1975 , , 196 , 51 smith , f. g. , et al . 1988 , , 233 , 305 sturner , s. j. , dermer , c. d. , & michel , f. c. 1995 , , 445 , 736 sturrock , p. a. 1971 , , 164 , 529 tennant , a. p. , et al . 2001 , , 554 , l173 thompson , d. j. , bailes , m. , bertsch , d. l. , esposito , j. a. , fichtel , c. e. , et al . 1996 , , 465 , 385 thompson , d. j. , bailes , m. , bertsch , d. l. , cordes , j. , damico , n. , et al . , 1999 , , 516 , 297 thompson , d. j. , 2001 , gamma - ray pulsars : observations , in high energy gamma - ray astronomy , aip proceedings , eds . f. a. aharonian and h. j. v " olk , 558 , 103 yadigaroglu , i .- a . 1997 , ph.d . thesis , stanford university yadigaroglu , i .- a . , & romani , r. w. 1995 , , 449 , 211 wona , a. , dyks , j. , rudak , b. , & bulik , t. 2002 , proc . of the xxviii moriond meeting the gamma - ray universe , eds . a. goldwurm et al . , 539 ( astro - ph/0205224 ) zhang , l. , & cheng , k. s. 1997 , , 487 , 370 zhang , l. , & cheng , k. s. 2002 , , 569 , 872 & @xmath59 & @xmath60 + caustic origin of the peaks & @xmath51 & @xmath61 @xmath62 & @xmath51 & + & + + -1.7mm@xmath63 & @xmath51 & @xmath51 & + & @xmath51 & + + -1.7mm@xmath64 & + & + & @xmath51 & @xmath51 & + + ( the acceleration region is extended ) & @xmath51 & @xmath51 & @xmath51 & + & + +
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we present a new model of high - energy lightcurves from rotation powered pulsars .
the key ingredient of the model is the gap region ( i.e. the region where particle acceleration is taking place and high - energy photons originate ) which satisfies the following assumptions : i ) the gap region extends from each polar cap to the light cylinder ; ii ) the gap is thin and confined to the surface of last open magnetic - field lines ; iii ) photon emissivity is uniform within the gap region .
the model lightcurves are dominated by strong peaks ( either double or single ) of caustic origin . unlike in other pulsar models with caustic effects ,
the double peaks arise due to crossing two caustics , each of which is associated with a different magnetic pole .
the generic features of the lightcurves are consistent with the observed characteristics of pulsar lightcurves : 1 ) the most natural ( in terms of probability ) shape consists of two peaks ( separated by 0.4 to 0.5 in phase for large viewing angles ) ; 2 ) the peaks posess well developed wings ; 3 ) there is a bridge ( inter - peak ) emission component ; 4 ) there is a non - vanishing off - pulse emission level ; 5 ) the radio pulse occurs before the leading high - energy peak .
the model is well suited for four gamma - ray pulsars - crab , vela , geminga and b1951 + 32 - with double - peak lightcurves exhibiting the peak separation of 0.4 to 0.5 in phase .
hereby , we apply the model to the vela pulsar .
moreover , we indicate the limitation of the model in accurate reproducing of the lightcurves with single pulses and narrowly separated ( about 0.2 in phase ) pulse peaks .
we also discuss the optical polarization properties for the crab pulsar in the context of the two - pole caustic model .
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big bang cosmology predicts the existence of a background gas of free photons and neutrinos . the measured cosmic microwave background ( cmb ) radiation supports the applicability of standard cosmology back to photon decoupling which occured approximately one hundred thousand years after the big bang . the relic neutrinos , on the other hand , have decoupled when the universe had a temperature of one mev and an age of just one second . thus , a measurement of the relic neutrinos , with a predicted average number density of @xmath3 per light ( @xmath4 mev ) neutrino species @xmath5 , would provide a new window to the early universe . their predicted number density is comparable to the one of the microwave photons . however , since neutrinos interact only weakly , the relic neutrinos have not yet been detected directly in laboratory experiments @xcite . illustration of a z - burst resulting from the resonant annihilation of an ultrahigh energy cosmic neutrino on a relic ( anti-)neutrino . , width=188 ] recently , an indirect detection possibility for relic neutrinos has been discussed @xcite . it is based on so - called z - bursts resulting from the resonant annihilation of ultrahigh energy cosmic neutrinos ( uhec@xmath6s ) with relic neutrinos into @xmath7 bosons @xcite ( cf . [ illu ] ) . on resonance , the corresponding cross section is enhanced by several orders of magnitudes . if neutrinos have non - vanishing masses @xmath8 for which there is rather convincing evidence in view of the apparent observation of neutrino oscillations @xcite the respective resonance energies , in the rest system of the relic neutrinos , correspond to @xmath9 with @xmath10 denoting the mass of the z boson . these resonance energies are , for neutrino masses of @xmath11 ev , remarkably close to the energies of the highest energy cosmic rays observed at earth by collaborations such as agasa @xcite , fly s eye @xcite , haverah park @xcite , hires @xcite , and yakutsk @xcite ( for a review , see ref . indeed , it was argued @xcite that the ultrahigh energy cosmic rays ( uhecrs ) above the predicted greisen - zatsepin - kuzmin ( gzk ) cutoff @xcite around @xmath12 ev are mainly protons ( and , maybe , photons ) from z decay . in this way , one possibly also solves one of the outstanding problems of ultrahigh energy cosmic ray physics @xcite , namely the apparent observation of cosmic rays with energies above the gzk cutoff , in an elegant and economical way without invoking new physics beyond the standard model , except for neutrino masses . the gzk puzzle hinges on the fact that nucleons with super - gzk energies have a short attenuation length of about @xmath13 mpc , due to inelastic interactions with the cosmic microwave background , while plausible astrophysical sources for those energetic particles are much farther away @xcite . ultrahigh energy neutrinos produced at cosmological distances , on the other hand , can reach the gzk zone unattenuated and their resonant annihilation on the relic neutrinos could just result in the observed cosmic rays of the highest energies . moreover , at present this annihilation process may be the only way to detect the relic neutrino background , a basic ingredient of our cosmological picture . the z - burst hypothesis for the ultrahigh energy cosmic rays was discussed in many papers @xcite . the required uhec@xmath6 fluxes were estimated in ref . @xcite for different spectral indices . in ref . @xcite , particle spectra were determined numerically for case studies which supported the z - burst scenario . the effect of possible lepton asymmetries was studied in ref . @xcite . in ref . @xcite , the analysis of the z - burst mechanism was advocated as one of the few possibilities for an absolute neutrino mass determination . in the present paper , we present the details of ( and extend ) our recent quantitative investigation of the z - burst scenario @xcite ( see also refs . @xcite ) , where we have determined the required mass of the heaviest relic neutrino as well as the necessary ultrahigh energy cosmic neutrino flux via a maximum likelihood analysis . but before we start this enterprise , we shall briefly review the current knowledge of the absolute scale of neutrino masses . neutrinos almost certainly have non - vanishing masses and mix . this follows from the apparent observation of neutrino oscillations whose evidence is compelling for atmospheric neutrinos @xcite , strong for solar neutrinos @xcite , and so - far unconfirmed for neutrinos produced in the laboratory and studied by the lsnd collaboration and others @xcite . however , neutrino oscillations are sensitive only to the mass ( squared ) splittings @xmath14 , not to the individual masses , @xmath15 , themselves . only a lower bound on the mass of the heaviest neutrino can be derived from these observations , e.g. @xmath16 from the atmospheric mass splitting in a three neutrino flavour scenario and @xmath17 from the still allowed value of the lsnd mass splitting in a four flavour scenario , respectively ( for a recent compilation of global analyses on mass splitting and mixing parameters , see for example refs . @xcite and references therein ) . for an investigation of the absolute scale of neutrino masses one has to exploit different types of experiments , such as the search for mass imprints in the endpoint spectrum of tritium beta ( @xmath18 ) decay or the search for neutrinoless double beta ( @xmath19 ) decay . at present , these direct kinematical measurements of neutrino masses provide only upper limits , e.g. @xmath20 with @xmath21 being the leptonic mixing matrix , for the effective mass measured in tritium @xmath18 decay @xcite , and @xmath22 for the majorana neutrino mass parameter @xcite appearing in @xmath19 decay . the recently reported evidence for neutrinoless double beta decay and correspondingly deduced parameter range @xcite @xmath23 has been seriously challenged by refs . combining the experimental constraints from oscillations and from tritium @xmath18 decay , one infers upper bounds on the mass of the heaviest neutrino @xcite , @xmath24 in a three flavour , and @xmath25 in a four flavour scenario , respectively . for further recent investigations of the relationship between oscillation phenomena , @xmath18 decay and @xmath19 decay , in particular with respect to the neutrino mass spectrum , one may also consult refs . @xcite and references cited therein . further information on the absolute scale of neutrino masses can be obtained through cosmological and astrophysical considerations . neutrinos in the @xmath26 ev mass range have cosmological implications since they represent a non - negligible part of dark matter . this gives the opportunity to put upper limits on neutrino masses from cosmology @xcite . analyses of galaxy clustering , including recent cmb measurements and other cosmological constraints , give an upper bound @xmath27 on the sum of the neutrino masses @xcite . from the spread of arrival times of neutrinos from supernova sn 1987a , coupled with the measured neutrino energies , a time - of - flight limit of @xmath28 ev can be derived @xcite , which , however , is not competitive with the direct limit ( [ lim_beta ] ) . it is extremely welcome that the z - burst scenario opens a further and timely window to the absolute neutrino mass scale , since the opportunities to determine this crucial quantity are rare at present . in addition , its verification would give us an indirect detection of the so elusive relic neutrinos and , finally , offer an explanation of the origin of the highest energy cosmic rays . the organization of the present paper is as follows . in section [ sect : spectra ] we describe our determination of the proton and photon spectra at earth , which originate from z - bursts taking place mainly at extragalactic distances . we discuss in detail the main ingredients in the predictions of the spectra , such as the details of z production and hadronic decay , the propagation of nucleons and photons through the diffuse extragalactic photon background , the diffuse flux of ultrahigh energy cosmic neutrinos , and the relic neutrino number density , as well as their anticipated uncertainties . the comparison of the z - burst spectra with the observed ultrahigh energy cosmic ray spectrum is presented in section [ sect : determination ] . the required absolute neutrino mass , as well as the necessary ultrahigh energy cosmic neutrino flux , are determined via a maximum likelihood analysis for various assumptions about the nature of the background of ordinary cosmic rays from unresolved astrophysical sources and variations of the diffuse extragalactic photon background notably in the radio band . on the basis of these studies we find that the z - burst determinations of the mass of the heaviest neutrino as well as of the neutrino flux are fairly robust . we find a required neutrino mass range of @xmath0 ev @xmath1 ev at the 68% confidence level , if the background of ordinary cosmic rays is of extragalactic origin . this range narrows down considerably if a particular universal radio background is assumed , e.g. to @xmath0 ev @xmath2 ev for a large one . in section [ sect : discussion ] we discuss the implications of our findings for future laboratory studies such as the tritium beta decay and neutrinoless double beta decay , as well as for astrophysical and cosmological neutrino investigations . this section contains also our conclusions . our comparison of the z - burst scenario with the observed uhecr spectrum proceeds as follows . first , we determine the probability of z production as a function of the distance from earth . secondly , we exploit collider experiments to derive the energy distribution of the produced protons and photons in the laboratory ( lab ) system . thirdly , we consider the propagation of the protons and photons , i.e. we determine their energy losses due to pion and/or @xmath29 production through scattering on the diffuse extragalactic background photons and due to their redshift . the last step is the comparison of the predicted and the observed spectra and the extraction of the required mass of the heaviest relic neutrino and of the necessary uhec@xmath6 flux . our prediction of the differential proton flux , i.e. the number of protons arriving at earth with energy @xmath30 per units of energy ( @xmath30 ) , of area ( @xmath31 ) , of time ( @xmath32 ) and of solid angle ( @xmath33 ) , @xmath34 from z - bursts can be summarized as @xmath35 \nonumber & & \times \left [ \int\limits_0^\infty { \rm d } e_{\nu_i}\,f_{\nu_i}(e_{\nu_i},r)\ , n_{\bar\nu_i}(r ) \right . \\[1ex]\nonumber & & \left . \hspace{6ex } + \int\limits_0^\infty { \rm d } e_{\bar\nu_i}\ , f_{\bar\nu_i}(e_{\bar\nu_i},r)\ , n_{\nu_i}(r ) \right ] \\[1ex ] \nonumber & & \times \sigma_{\nu_i\bar\nu_i}(s)\ , { \rm br}(z\to { \rm hadrons})\ , \frac{{\rm d } n_{p+n}}{{\rm d } e_p } \\[1ex ] \nonumber & & \times ( -)\frac{\partial}{\partial e } p_p(r , e_p;e ) \,,\end{aligned}\ ] ] with the following important building blocks : the uhec@xmath6 fluxes @xmath36 at the energies @xmath37 ( @xmath38 ) and at the distance @xmath39 of z production to earth , the number density @xmath40 of the relic neutrinos , the z production cross section @xmath41 at centre - of - mass ( cm ) energy squared @xmath42 , the branching ratio @xmath43 , the energy distribution @xmath44 of the produced protons ( and neutrons ) with energy @xmath45 , and the probability @xmath46 that a proton created at a distance @xmath39 with energy @xmath45 arrives at earth above the threshold energy @xmath30 . a similar expression as eq . ( [ p - flux ] ) holds for the differential photon flux from z - bursts , @xmath47 \nonumber & & \times \left [ \int\limits_0^\infty { \rm d } e_{\nu_i}\,f_{\nu_i}(e_{\nu_i},r)\ , n_{\bar\nu_i}(r ) \right . \\[1ex ] \nonumber & & \left . \hspace{6ex } + \int\limits_0^\infty { \rm d } e_{\bar\nu_i}\,f_{\bar\nu_i}(e_{\bar\nu_i},r)\ , n_{\nu_i}(r ) \right ] \\[1ex ] \nonumber & & \times \sigma_{\nu_i\bar\nu_i } ( s)\ , { \rm br}(z\to { \rm hadrons})\ , \frac{{\rm d } n_\gamma}{{\rm d } e_\gamma } \\[1ex ] \nonumber & & \times ( -)\frac{\partial}{\partial e } p_\gamma ( r , e_\gamma ; e ) \,.\end{aligned}\ ] ] here , the photon propagation function @xmath48 gives the expected number of detected photons above the threshold energy @xmath30 if one photon started from a distance of @xmath39 with energy @xmath49 . note , that the @xmath50 function has a different interpretation than the @xmath51-function of protons . this arises from the fact that the number of photons in distinction to the number of protons is not conserved during their propagation . the building blocks related to z - production and decay @xmath52 , the hadronic branching ratio , and the momentum distributions @xmath53 are very well , and the propagation functions @xmath54 are fairly well determined , whereas the first two ingredients , the flux of uhec@xmath6s , @xmath36 , and the radial distribution of the relic neutrino number density @xmath40 , are much less accurately known . in the following we shall discuss all these ingredients in detail . at lep and slc millions of z bosons were produced and their decays analyzed with extreme high accuracy . due to the large statistics , the uncertainties of our analysis related to z decay turned out to be negligible . let us start with a review of the standard model neutrino annihilation cross section . the s - channel z - exchange annihilation cross section into any fermion anti - fermion ( @xmath55 ) pair is given by ( see e.g. refs . @xcite ) @xmath56 where @xmath57 is the cm energy squared , @xmath58 is the total width of the z boson , and @xmath59 is the effective number of annihilation channels , @xmath60 \nonumber & & \times \ , \frac{2}{3}\,n_f \left ( 1 - 8\ , t_{3\,f}\ , q_f\ , \sin^2\theta_w + 8\,q_f^2\sin^4\theta_w\right ) \,.\end{aligned}\ ] ] here the sum is over all fermions with @xmath61 , with charge @xmath62 ( in units of the proton charge ) , isospin @xmath63 ( @xmath64 for @xmath65 and neutrinos ; @xmath66 for @xmath67 and negatively charged leptons ) , and @xmath68 for leptons ( quarks ( @xmath69 ) ) . with @xmath70 for the @xmath71 of the effective weinberg angle and @xmath72 gev@xmath73 for the fermi coupling constant @xcite , formula ( [ z - res - cs - roulet ] ) gives , at the @xmath7-mass , @xmath74 \label{cs - tot - zpeak } \sigma ( \nu_i \bar{\nu}_i \to z^\ast \to { \rm all}\ f\bar f)\mid_{s = m_z^2 } & = & 455.6\ { \rm nb}\,,\end{aligned}\ ] ] with a branching ratio @xmath75 , in good agreement with the experimental result @xcite , @xmath76 later we shall exploit the following simplification which arises due to the fact that the cross section ( [ z - res - cs - roulet ] ) is sharply peaked at the resonance cm energy squared @xmath77 . correspondingly , it acts essentially like a @xmath78-function in the integration over the energies @xmath37 of the incident neutrinos in eqs . ( [ p - flux ] ) and ( [ ph - flux ] ) , and we can assume that the uhec@xmath6 fluxes are constant in the relevant energy region . thus , introducing the energy - averaged annihilation cross section @xcite , @xmath79 which is the effective cross section for all neutrinos within @xmath80 of their peak annihilation energy , we can write @xmath81\nonumber & & \hspace{8ex } \simeq e_{\nu_i}^{\rm res}\,f_{\nu_i}(e_{\nu_i}^{\rm res})\,\langle \sigma_{\rm ann}\rangle \,.\end{aligned}\ ] ] in view of the expected rapid decrease of the uhec@xmath6 flux at increasing energies ( cf . section [ fluxes ] ) , we neglect @xmath32-channel w- and z - exchange annihilation processes . on resonance , the @xmath57-channel z - exchange processes completely overwhelm them . fairly above the resonant energies , on the other hand , they eventually dominate but are probably unobservable due to the lack of an appreciable uhec@xmath6 flux . next , we turn to the energy distribution of the protons and photons in z decay . we combined existing published and some improved unpublished data on the momentum distribution , @xmath82 of protons ( @xmath83 ) ( plus antiprotons ( @xmath84 ) ) in z decays @xcite , see fig . [ prot - mom - dist ] ( top ) . the experimental data , ranging down to @xmath85 , were combined with the predictions from the modified leading logarithmic approximation ( mlla ) @xcite at low @xmath86 . the @xmath87 multiplicity is @xmath88 in the hadronic channel @xcite . momentum distributions in hadronic z decays . _ top : _ combined data from refs . @xcite on proton ( plus antiproton ) momentum distribution ( solid ) , normalized to @xmath89 , and charged pion momentum distributions ( dotted ) , normalized to @xmath90 . _ bottom : _ distribution of protons ( `` p '' ; solid ) , photons ( `` @xmath91 '' ; dotted ) and electrons ( `` e '' ; dashed ) in the lab system , in which the target relic neutrino is at rest . , title="fig:",width=326 ] momentum distributions in hadronic z decays . _ top : _ combined data from refs . @xcite on proton ( plus antiproton ) momentum distribution ( solid ) , normalized to @xmath89 , and charged pion momentum distributions ( dotted ) , normalized to @xmath90 . _ bottom : _ distribution of protons ( `` p '' ; solid ) , photons ( `` @xmath91 '' ; dotted ) and electrons ( `` e '' ; dashed ) in the lab system , in which the target relic neutrino is at rest . , title="fig:",width=326 ] in the cm system of the z production the angular distribution of the hadrons is determined by the spin @xmath64 of the primary quarks and thus proportional to @xmath92 ( here @xmath93 is the angle between the incoming neutrinos and the outgoing hadrons ( cf . @xcite ) ) . the energy distribution of the produced protons with energy @xmath45 entering the z - burst spectrum ( [ p - flux ] ) , @xmath94 with @xmath95 , is finally obtained after a lorentz transformation from the cm system to the lab system , @xmath96 where @xmath97 is the proton mass . the first line comes from the angular distribution , the second line is the jacobian and the third one is the momentum distribution at the inverted momentum . the scaled energy distribution @xmath98 , as defined in eq . ( [ q - def ] ) and given by eq . ( [ q - dist ] ) , is displayed in fig . [ prot - mom - dist ] ( bottom ) . neutrons produced in z decays will decay and end up as uhecr protons . they are taken into account according to @xmath99 where the neutron ( @xmath100 ) ( plus antineutron ( @xmath101 ) ) multiplicity , @xmath102 , is @xmath103 smaller than the proton s @xcite . photons are produced in hadronic z decays via fragmentation into neutral pions , @xmath104 ( cf . [ illu ] ) . the corresponding scaled energy distribution in the lab system , defined analogously to eq . ( [ q - def ] ) , reads @xmath105 where the scaled energy distribution @xmath106 , with @xmath107 , is given by eq . ( [ q - dist ] ) , with @xmath108 and @xmath109 , the momentum distribution of pions in hadronic z decay . for the latter distribution , we take the measured one of charged pions @xmath110 from hadronic z decay @xcite ( cf . [ prot - mom - dist ] ( top ) ) , normalized such that @xmath111 @xcite . note , that the distributions @xmath112 , @xmath113 , presented in fig . [ prot - mom - dist ] ( bottom ) , compare favorably with the ones presented in ref . @xcite based on the event generator pythia @xcite . electrons ( and positrons ) from hadronic z decay are also relevant for the development of electromagnetic cascades . they stem from decays of secondary charged pions , @xmath114 ( cf . fig . [ illu ] ) , and their scaled energy distribution in terms of @xmath115 reads @xmath116 \nonumber & & \times { \mathcal q}_{\pi^\pm } \left ( \frac{2\ , y}{xw+\sqrt{x^2+(2\,m_e / m_\pi)^2 } } \right ) { \mathcal p}_e ( x ) \,,\end{aligned}\ ] ] where @xmath117 is the momentum distribution of the electrons in the rest system of the charged pion . the energy distribution @xmath118 is also displayed in fig . [ prot - mom - dist ] ( bottom ) . the cosmic microwave background is known to a high accuracy . it plays the key role in the determination of the probability @xmath46 that a proton created at a distance @xmath39 with energy @xmath45 arrives at earth above the threshold energy @xmath30 , suggested in ref . @xcite and determined for a wide range of parameters in ref . the propagation function @xmath119 takes into account the fact that protons of extragalactic ( eg ) origin and energies above @xmath120 ev lose a large fraction of their energy due to pion and @xmath29 production through scattering on the cmb and due to their redshift @xcite . in the present analysis we have included new results for @xmath46 which include now variations in the cosmological parameters , in extension to the already published form @xcite exploited in ref . @xcite . in our analysis we go , according to @xmath121 out to distances @xmath122 ( cf . ( [ p - flux ] ) ) corresponding to redshift @xmath123 ( cf . we use the expression @xmath124\ ] ] for the relation of the hubble expansion rate at redshift @xmath125 to the present one . uncertainties of the latter , @xmath126 100 km / s / mpc , with @xmath127 @xcite , are included . in eq . ( [ h - omega ] ) , @xmath128 and @xmath129 , with @xmath130 , are the present matter and vacuum energy densities in terms of the critical density . as default values we choose @xmath131 and @xmath132 , as favored today . our results turn out to be pretty insensitive to the precise values of the cosmological parameters . _ top : _ the intensity spectrum of the diffuse extragalactic photon background at redshift @xmath133 from ref . @xcite ( solid ) . different estimates from ref . @xcite of the universal radio background ( urb ) are also indicated : high urb ( long - dashed ) and moderate urb ( short - dashed ) . _ bottom : _ photon energy attenuation length @xmath134 at @xmath133 corresponding to the photon background shown above @xcite ( solid ) . variations of @xmath134 arising from different assumptions about the urb ( cf . ref . @xcite and top ) are also indicated : high urb ( long - dashed ) and moderate urb ( short - dashed ) . shown is furthermore the energy attenuation length for electrons due to synchrotron radiation ( long - dashed - short - dashed ) , for different magnitudes of the extragalactic magnetic fields . , title="fig:",width=326 ] _ top : _ the intensity spectrum of the diffuse extragalactic photon background at redshift @xmath133 from ref . @xcite ( solid ) . different estimates from ref . @xcite of the universal radio background ( urb ) are also indicated : high urb ( long - dashed ) and moderate urb ( short - dashed ) . _ bottom : _ photon energy attenuation length @xmath134 at @xmath133 corresponding to the photon background shown above @xcite ( solid ) . variations of @xmath134 arising from different assumptions about the urb ( cf . ref . @xcite and top ) are also indicated : high urb ( long - dashed ) and moderate urb ( short - dashed ) . shown is furthermore the energy attenuation length for electrons due to synchrotron radiation ( long - dashed - short - dashed ) , for different magnitudes of the extragalactic magnetic fields . , title="fig:",width=326 ] the simulations needed for the computation of @xmath46 require large computer power . we have used a farm of personal computers ( pcs ) consisting of 128 parallel @xmath135 ghz pentium 4 processors normally devoted to qcd lattice calculations @xcite and have exploited about 3 hours cpu time to determine @xmath46 in a range of @xmath136 mpc , @xmath137 ev @xmath138 ev , and @xmath137 ev @xmath139 ev , for each fixed value of the cosmological parameters . the simulation was carried out in small ( @xmath140 kpc ) steps in @xmath39 . for each step , the statistical energy losses due to pion/@xmath29 production and redshift are taken into account @xcite . in this connection , the advantage of our formulation of the z - burst spectrum ( cf . ( [ p - flux ] ) ) in terms of the probability @xmath46 becomes evident . we have to determine the latter only once and for all . without the use of @xmath46 , we would have to perform a simulation for any variation of the input spectrum , notably for any change in the neutrino mass . our maximum likelihood analysis , involving the neutrino mass as a free parameter , would thus require excessive computer power on the order of 300 hours on the above pc farm for fixed cosmological parameters . since @xmath46 is of universal usage , we have decided to make the corresponding numerical data for the probability distribution @xmath141 available for the public via the world - wide - web url _ http://www.desy.de/~uhecr _ . the determination of the photon propagation function @xmath48 , entering the photon flux prediction ( [ ph - flux ] ) , was done as follows . in distinction to the case of the proton propagation function , we used here the continuous energy loss ( cel ) approximation which largely simplifies the work and reduces the required computer resources we have estimated the necessary cpu time for a full simulation on the pc farm mentioned above to 1000 hours per fixed set of the cosmological parameters . in the cel approximation , the energy ( and number ) of the detected photons is a unique function of the initial energy and distance , and statistical fluctuations are neglected . a full simulation of the photon propagation function will be the subject of a later work . the processes that are taken into account are pair production on the diffuse extragalactic photon background ( cf . [ ph_mfp ] ( top ) ) , double pair production and inverse compton scattering of the produced pairs . we comment also on synchrotron radiation in a possible extragalactic magnetic field ( egmf ) . for the energy attenuation length of the photons due to these processes , we exploited the values quoted in ref . @xcite ( see also ref . @xcite ) and the further ones presented in fig . [ ph_mfp ] ( bottom ) which incorporate various assumptions about the poorely known universal radio ( urb ) ( from ref . @xcite ) and infrared ( irb ) backgrounds . we shall analyse later the dependence of the neutrino mass and other fit parameters on these variations . note , that , in view of the recent urb estimates in ref . @xcite , the ones presented in fig . [ ph_mfp ] ( top ) ) , which are based on ref . @xcite , can be referred to as `` minimal '' urb . the computation of the photon propagation function @xmath48 was carried out in the following way . the energy attenuation of photons in the cel approximation was calculated according to @xmath142 where @xmath143 is the energy attenuation length at redshift @xmath125 . the number of photons was assumed to be constant at ultrahigh energies @xmath144 ev , due to the small inelasticities in this energy range . below , it was increased in a way to maintain energy conservation ( except for the redshift contribution ) : @xmath145 the @xmath48 function was then obtained by integration of these equations . in the ultrahigh energy region which is most relevant for us since we perform our fit to the cosmic ray data there the approximation described above gives the photon flux quite reliable , while at lower energies it yields an upper bound . presently unknown ingredients in the evaluation of the z - burst spectra ( [ p - flux ] ) and ( [ ph - flux ] ) are the differential fluxes @xmath146 of ultrahigh energy cosmic neutrinos ( see e.g. refs . @xcite for recent reviews ) . present experimental upper limits on these fluxes are rather poor ( cf . fig . [ flux_upp_lim ] and refs . @xcite ) . upper limits on differential neutrino fluxes in the ultrahigh energy regime . shown are experimental upper limits on @xmath147 from fly s eye @xcite and on @xmath148 from the goldstone lunar ultrahigh energy neutrino experiment glue @xcite , as well as theoretical upper limits on @xmath149 from `` visible '' ( `` wb '' @xcite ) and `` hidden '' ( `` mpr '' @xcite ) hadronic astrophysical sources . , width=302 ] what are the theoretical expectations for diffuse uhec@xmath6 fluxes ? more or less guaranteed are the so - called cosmogenic neutrinos which are produced when ultrahigh energy cosmic protons scatter inelastically off the cosmic microwave background radiation @xcite in processes such as @xmath150 , where the produced pions subsequently decay @xcite . these fluxes ( for recent estimates , see refs . @xcite ) represent reasonable lower limits , but turn out to be insufficient for the z - burst scenario . recently , theoretical upper limits on the ultrahigh energy cosmic neutrino flux have been given in refs . @xcite . per construction , the upper limit from `` visible '' hadronic astrophysical sources , i.e. from those sources which are transparent to ultrahigh energy cosmic protons and neutrons , is of the order of the cosmogenic neutrino flux and shown in fig . [ flux_upp_lim ] ( `` wb '' ; cf . refs . @xcite ) . also shown in this figure ( `` mpr '' ) is the much larger upper limit from `` hidden '' hadronic astrophysical sources , i.e. from those sources from which only photons and neutrinos can escape @xcite . even larger fluxes at ultrahigh energies may arise if the hadronic astrophysical sources emit photons only in the sub - mev region thus evading the `` mpr '' bound in fig . [ flux_upp_lim ] or if the neutrinos are produced via the decay of superheavy relic particles @xcite , for which also the fragmentation function of the decay is of major interest @xcite , or topological defects @xcite . in this situation of insufficient knowledge , we take the following approach concerning the flux of ultrahigh energy cosmic neutrinos , @xmath36 . it is assumed to have the form @xmath151 where @xmath125 is the redshift and where @xmath152 characterizes the cosmological source evolution ( see also refs . the flux at zero redshift , @xmath153 , is left open . for hadronic astrophysical sources it is expected to fall off power - like , @xmath154 , @xmath155 , at high energies . due to this fact and because of the strong resonance peaks in the @xmath156 annihilation cross section ( [ z - res - cs - roulet ] ) at the resonance energies ( [ eres ] ) , the z - burst rate will be only sensitive to the flux at the resonant energy of the heaviest neutrino . of course , the latter may be nearly degenerate with the other neutrino mass eigenstates , @xmath157 , as it is the case for @xmath158 ev in a three flavour scenario . correspondingly , our later fit to the uhecr data will be sensitive only to @xmath159 \,,\ ] ] where the sum extends over the number of mass eigenstates which are quasi - degenerate with the heaviest neutrino . note , finally , that , independently of the production mechanism , neutrino oscillations result in a uniform @xmath146 mixture for the different mass eigenstates @xmath5 . _ top : _ mass density fluctuation field @xmath78 along the supergalactic plane as obtained from peculiar velocity measurements @xcite . shown are contours in intervals of @xmath160 , surface maps on a grid of spacing @xmath161 kms@xmath162 , corresponding to @xmath163 mpc , with the height proportional to @xmath78 , and contrast maps . one recognizes some well - known structures in the nearby volume such as the great attractor at supergalactic coordinates ( sgx @xmath164 kms@xmath162 , sgy @xmath165 kms@xmath162 ) , the perseus - pisces complex ( sgx @xmath166 kms@xmath162 , sgy @xmath167 kms@xmath162 ) , and the large void ( sgx @xmath168 kms@xmath162 , sgy @xmath169 kms@xmath162 ) in between . _ bottom : _ mass density fluctuation field obtained from above data , averaged over all directions , for @xmath170 . the overdensities at around 20 and 80 mpc reflect the great attractor and the perseus - pisces complex , respectively . , title="fig:",width=453 ] _ top : _ mass density fluctuation field @xmath78 along the supergalactic plane as obtained from peculiar velocity measurements @xcite . shown are contours in intervals of @xmath160 , surface maps on a grid of spacing @xmath161 kms@xmath162 , corresponding to @xmath163 mpc , with the height proportional to @xmath78 , and contrast maps . one recognizes some well - known structures in the nearby volume such as the great attractor at supergalactic coordinates ( sgx @xmath164 kms@xmath162 , sgy @xmath165 kms@xmath162 ) , the perseus - pisces complex ( sgx @xmath166 kms@xmath162 , sgy @xmath167 kms@xmath162 ) , and the large void ( sgx @xmath168 kms@xmath162 , sgy @xmath169 kms@xmath162 ) in between . _ bottom : _ mass density fluctuation field obtained from above data , averaged over all directions , for @xmath170 . the overdensities at around 20 and 80 mpc reflect the great attractor and the perseus - pisces complex , respectively . , title="fig:",width=326 ] the dependence of the relic neutrino number density @xmath171 on the distance @xmath39 is treated in the following way . the question is whether there is remarkable clustering of the relic neutrinos within the local gzk zone of about 50 mpc . it is known that the density distribution of relic neutrinos as hot dark matter follows the total mass distribution ; however , with less clustering @xcite . in fact , for @xmath172 ev , one expects pretty much that the neutrino number density equals the big bang prediction ( [ standard_number_dens ] ) @xcite . to take above facts into account , the shape of the @xmath40 distribution is varied , for distances below 100 mpc , between the standard cosmological homogeneous case ( [ standard_number_dens ] ) and that of the total mass distribution obtained from peculiar velocity measurements @xcite ( cf . [ dens - prof ] ( top ) ) . these peculiar measurements suggest relative overdensities of at most a factor @xmath173 , depending on the grid spacing ( cf . [ dens - prof ] ( bottom ) ) . a relative overdensity @xmath174 in our neighbourhood , as it was assumed in earlier investigations of the z - burst hypothesis @xcite , seems unlikely in view of these data . our quantitative results turned out to be rather insensitive to the variations of the overdensities within the considered range . for scales larger than 100 mpc the relic neutrino density is taken according to the big bang cosmology prediction , @xmath175 @xmath176 . possible uniform neutrino density enhancements due to eventual lepton asymmetries , as advocated in ref . @xcite , are negligible in view of the recent , very stringent bounds on the neutrino degeneracies @xcite . in any case , such uniform enhancements change only the normalization of the z - burst spectra and have no effect on their shape . therefore , they could possibly milder the required uhec@xmath6 flux , but the value of the neutrino mass inferred from the z - burst scenario is not affected by them . the predicted spectra of protons and photons from z - bursts , ( [ p - flux ] ) and ( [ ph - flux ] ) , can now be compared with the observed uhecr spectrum ( cf . [ fit_normal ] ) . our analysis includes published uhecr data of agasa @xcite , fly s eye @xcite , haverah park @xcite , and hires @xcite , as well as unpublished one from the world wide web pages of the experiments on 17/03/01 ( for a review , see ref . @xcite ) . due to normalization difficulties we did not use the yakutsk @xcite results . we shall take into account the fact that above @xmath12 ev less than @xmath177 of the cosmic rays can be photons at the @xmath178 confidence level ( c.l . ) @xcite ( see also refs . @xcite ) . as usual , each logarithmic unit between @xmath179 and @xmath180 is divided into ten bins . the predicted number of uhecr events in a bin is taken as @xmath181,\ ] ] where @xmath182 m@xmath183s@xmath184sr is the total exposure ( estimated from the highest energy events and the corresponding fluxes ) and where @xmath185 ev is the lower bound of the @xmath186 energy bin . the first term in eq . ( [ flux ] ) , @xmath187 , corresponds to the diffuse background of ordinary cosmic rays from unresolved astrophysical sources . below the gzk cutoff , it should have the usual and experimentally observed power - law form @xcite . the second term represents the sum of the proton and photon spectra , @xmath188 , eqs . ( [ p - flux ] ) and ( [ ph - flux ] ) , from z - bursts . the separation of the flux into two terms ( one from the power - law background and one from the z - burst ) is physically well motivated . the power - law part below the gzk cutoff is confirmed experimentally and , for extragalactic sources , it should suffer from the gzk effect . in the z - burst scenario , cosmic rays are coming from another independent source ( z - bursts ) , too . what we observe is the sum of the two . as the detailed fits in the next section will show , the flux from z - bursts is much smaller in the low energy region than the flux of the power - law background . correspondingly , the low energy part of the spectrum ( between @xmath189 and @xmath190 ev ) has very little influence on the z - burst fit parameters , notably on the neutrino mass . the available uhecr data with their error bars and the best fits ( long - dashed ) from ordinary cosmic ray protons originating from our local neighbourhood ( `` halo background '' ; _ top _ ) and originating from diffuse extragalactic sources ( `` eg background '' ; _ bottom _ ) , respectively . for the latter case , the bump at @xmath12 ev represents protons injected at high energies and accumulated just above the gzk cutoff due to their energy losses . the predicted fall - off for energies above @xmath12 ev can be observed . , title="fig:",width=298 ] the available uhecr data with their error bars and the best fits ( long - dashed ) from ordinary cosmic ray protons originating from our local neighbourhood ( `` halo background '' ; _ top _ ) and originating from diffuse extragalactic sources ( `` eg background '' ; _ bottom _ ) , respectively . for the latter case , the bump at @xmath12 ev represents protons injected at high energies and accumulated just above the gzk cutoff due to their energy losses . the predicted fall - off for energies above @xmath12 ev can be observed . , title="fig:",width=298 ] we shall study several possibilities for the background term @xmath187 . the first one is based on the assumption that the diffuse background of ordinary cosmic rays even above the gzk cutoff at @xmath12 ev consists of protons which are produced in our neighborhood within our galactic halo or at least within the gzk zone of about 50 mpc . we shall refer to this possible cosmic ray background in the following as the `` halo background '' . we should note , however , that there are no apparent suitable astrophysical sources within the gzk distance to which the highest energy events point @xcite a fact which is difficult to reconcile with the halo background model . for the halo background , no gzk attenuation is included . the spectrum is assumed to have the usual power - law behavior which describes the data well for smaller energies @xcite ( cf . [ fit_normal ] ( top ) ) , @xmath191\nonumber & & \hspace{6ex } \frac{a}{1\,{\rm ev}}\ , \left ( \frac{e}{1\,{\rm ev}}\right)^{-\beta } \hspace{3ex } ( { \rm halo\ bk'd } ) \,.\end{aligned}\ ] ] a similar background model , however with a fixed power - law index @xmath192 , as found by agasa to fit the data from @xmath193 ev ( cf . table v in ref . @xcite ) , has been exploited in a recent z - burst simulation @xcite . in the second case , we assume that the diffuse background of ordinary cosmic rays comes from protons which originate from uniformly distributed , extragalactic sources hence we name it `` extragalactic background '' . in view of the observed distribution of arrival directions of uhecrs , this assumption seems to be phenomenologically more realistic than the halo background model . though there are some peculiarly clustered events , the overall distribution at present statistics seems to be practically uniform @xcite . the extragalactic background suffers of course from gzk attenuation . correspondingly , we take the above power - law ( [ pow - law - halo ] ) , @xmath194 , as an injection spectrum and take its modification due to the interactions with the cmb photons into account with the help of the proton propagation function @xmath46 , @xmath195\nonumber & & \hspace{3ex } \int_0^\infty { \rm d}e_p \int_0^{r_{\rm max } } { \rm d}r \,\left(1+z(r)\right)^3 \\[1ex ] \label{pow - law - eg } & & \hspace{5ex } \times\ , \frac{a}{1\,{\rm ev}}\ , \left ( \frac{e_p}{1\,{\rm ev}}\right)^{-\beta}\ , \\[1ex]\nonumber & & \hspace{5ex } \times\ , ( -)\frac{\partial p_p(r , e_p;e)}{\partial e } \hspace{3ex } ( { \rm eg\ bk'd})\,.\end{aligned}\ ] ] the predicted spectrum of the extragactic background protons shows an accumulation at around the gzk scale @xmath12 ev and a sharp drop beyond ( see fig . [ fit_normal ] ( bottom ) ) . as far as the z - burst spectra entering our prediction ( [ flux ] ) are concerned , we proceed as follows . we shall mainly concentrate on the case where there is only one relevant neutrino mass scale @xmath196 , either because there are three neutrino types with nearly degenerate neutrino masses , @xmath157 , or there is one neutrino which is much heavier than the other ones such that the contribution of the latter to the cosmic ray spectrum can be neglected since the corresponding resonance energies are much larger and the uhec@xmath6 fluxes are expected to fall with increasing energy . therefore , we fit only one neutrino mass parameter @xmath196 . in this case , the spectra ( [ p - flux ] ) and ( [ ph - flux ] ) for protons and photons from z - bursts can be written as @xmath197\nonumber & & b\ , \int_0^\infty { \rm d}e_i \int_0^{r_{\rm max } } { \rm d}r \,\left(1+z(r)\right)^{3+\alpha}\,\delta_n ( r ) \\[1ex]\label{fit - zburst - spectra } & & \frac{4\,m_\nu}{m_z^2}\ , { \mathcal q_i}\left ( y = \frac{4\,m_\nu\,e_i}{m_z^2 } \right ) \\[1ex]\nonumber & & \times\ , ( -)\frac{\partial p_i(r , e_i;e)}{\partial e } , \hspace{6ex } i = p,\gamma\,,\end{aligned}\ ] ] where @xmath152 is the cosmological evolution parameter , @xmath198 is the mass density fluctuation field ( cf . fig . [ dens - prof ] ( bottom ) ) , normalized to one , and @xmath199 are the boosted momentum distributions from hadronic z decay , normalized to @xmath200 , for @xmath201 , and to @xmath202 , for @xmath203 . we are left here with two fit parameters , the mass @xmath196 of the heaviest neutrino and the overall normalization @xmath204 , which may be expressed , on account of eqs . ( [ eresfres ] ) and ( [ fnures ] ) , in terms of the original quantities entering eqs . ( [ p - flux ] ) and ( [ ph - flux ] ) , as @xmath205 note , that the neglection of finite width effects , @xmath80 , in our implementation ( [ fit - zburst - spectra ] ) of the z - burst spectra is perfectly adequate in view of the relative errors @xmath206% which we will find later from our fits . the expectation value for the number of events in a bin is given by eq . ( [ flux ] ) . to determine the most probable value for @xmath207 we use the maximum likelihood method and minimize @xcite the @xmath208 , @xmath209 \nonumber & & \sum_{\log\left(\frac{e_i}{{\rm ev}}\right)=18.5}^{\log\left(\frac{e_i}{\rm ev}\right)=26.0 } 2\left [ n(i)-n_{\rm o}(i)+n_{\rm o}(i ) \ln\left ( n_{\rm o}(i)/n(i)\right ) \right],\end{aligned}\ ] ] where @xmath210 is the total number of observed events in the @xmath186 bin . since the z - burst scenario results in a quite small flux for lower energies , we take the lower bound just below the `` ankle '' : @xmath211 ev . our results are insensitive to the definition of the upper end ( the flux is extremely small there ) for which we choose @xmath212 . the uncertainties of the measured energies are about 30% which is one bin . by means of a monte carlo analysis , we take these uncertainties into account and include the corresponding variations in our final error estimates . for comparison with recent work on the z - burst scenario @xcite , we have done also fits with no background component , @xmath213 in eq . ( [ flux ] ) , and a larger lower end , @xmath214 . such a scenario , with @xmath215 ev , was advocated in ref . @xcite as appropriate to explain all uhecr data above the gzk cutoff by the z - burst model and to attribute all uhecr data below the cutoff to ordinary astrophysical extragalactic sources . qualitatively , our analysis can be understood as follows . in the z - burst scenario a small relic neutrino mass needs a large incident neutrino energy @xmath216 ( [ eres ] ) in order to produce a z. large @xmath216 results in a large lorentz boost , thus large @xmath45 resp . @xmath49 . in this way the _ shape _ of the detected energy ( @xmath30 ) spectrum determines the mass of the relic neutrino . the sum of the necessary uhec@xmath6 fluxes @xmath217 , on the other hand , is determined by the over - all _ normalization _ @xmath204 . our fitting procedure involves four parameters : @xmath218 and @xmath196 . the minimum of the @xmath219 function is @xmath220 at @xmath221 which is the most probable value for the mass , whereas the 1@xmath222 ( 68% ) confidence interval for @xmath196 is determined by @xmath223 here @xmath224 , @xmath225 , @xmath226 are defined in such a way that the @xmath227 function is minimized in @xmath228 and @xmath204 , at fixed @xmath196 . the available uhecr data with their error bars and the best fits from z - bursts , for a strong uhe@xmath91 attenuation such that the z - burst photons can be neglected ( @xmath229 ) . _ top : _ best fit for the case of a halo background ( solid line ) . the bump around @xmath12 ev is mainly due to the z - burst protons ( dash - dotted ) , whereas the almost horizontal contribution ( long - dashed ) is the first , power - law - like term of eq . ( [ flux ] ) . _ middle : _ the case of an `` extragalactic '' uhecr background . the first bump at @xmath12 ev represents protons produced at high energies and accumulated just above the gzk cutoff due to their energy losses . the bump at @xmath230 ev is a remnant of the z - burst energy . the long - dashed line shows the contribution of the power - law - like spectrum with the gzk effect included . _ bottom : _ the case of no uhecr background above @xmath231 . , title="fig:",width=298 ] the available uhecr data with their error bars and the best fits from z - bursts , for a strong uhe@xmath91 attenuation such that the z - burst photons can be neglected ( @xmath229 ) . _ top : _ best fit for the case of a halo background ( solid line ) . the bump around @xmath12 ev is mainly due to the z - burst protons ( dash - dotted ) , whereas the almost horizontal contribution ( long - dashed ) is the first , power - law - like term of eq . ( [ flux ] ) . _ middle : _ the case of an `` extragalactic '' uhecr background . the first bump at @xmath12 ev represents protons produced at high energies and accumulated just above the gzk cutoff due to their energy losses . the bump at @xmath230 ev is a remnant of the z - burst energy . the long - dashed line shows the contribution of the power - law - like spectrum with the gzk effect included . _ bottom : _ the case of no uhecr background above @xmath231 . , title="fig:",width=298 ] the available uhecr data with their error bars and the best fits from z - bursts , for a strong uhe@xmath91 attenuation such that the z - burst photons can be neglected ( @xmath229 ) . _ top : _ best fit for the case of a halo background ( solid line ) . the bump around @xmath12 ev is mainly due to the z - burst protons ( dash - dotted ) , whereas the almost horizontal contribution ( long - dashed ) is the first , power - law - like term of eq . ( [ flux ] ) . _ middle : _ the case of an `` extragalactic '' uhecr background . the first bump at @xmath12 ev represents protons produced at high energies and accumulated just above the gzk cutoff due to their energy losses . the bump at @xmath230 ev is a remnant of the z - burst energy . the long - dashed line shows the contribution of the power - law - like spectrum with the gzk effect included . _ bottom : _ the case of no uhecr background above @xmath231 . , title="fig:",width=298 ] @xmath152 & @xmath196 [ ev ] & @xmath220 & @xmath31 & @xmath204 & @xmath18 + @xmath232 & @xmath233 & @xmath234 & @xmath235 & @xmath236 & @xmath237 + @xmath238 & @xmath239 & @xmath240 & @xmath241 & @xmath242 & @xmath243 + @xmath244 & @xmath245 & @xmath246 & @xmath247 & @xmath248 & @xmath243 + + @xmath152 & @xmath196 [ ev ] & @xmath220 & @xmath31 & @xmath204 & @xmath18 + @xmath232 & @xmath249 & @xmath250 & @xmath251 & @xmath252 & @xmath253 + @xmath238 & @xmath254 & @xmath255 & @xmath256 & @xmath257 & @xmath258 + @xmath244 & @xmath259 & @xmath260 & @xmath261 & @xmath262 & @xmath263 + + @xmath264 & @xmath196 [ ev ] & @xmath220 & @xmath31 & @xmath204 & @xmath18 + @xmath265 & @xmath266 & @xmath267 & @xmath268 & @xmath269 & @xmath268 + @xmath270 & @xmath271 & @xmath272 & @xmath268 & @xmath273 & @xmath268 + @xmath274 & @xmath275 & @xmath276 & @xmath268 & @xmath277 & @xmath268 + @xmath278 & @xmath279 & @xmath280 & @xmath268 & @xmath281 & @xmath268 + @xmath282 & @xmath283 & @xmath284 & @xmath268 & @xmath285 & @xmath268 + @xmath286 & @xmath287 & @xmath288 & @xmath268 & @xmath289 & @xmath268 + @xmath290 & @xmath291 & @xmath292 & @xmath268 & @xmath293 & @xmath268 + as already mentioned , presently there is no evidence that the observed highest energy cosmic rays are photons . let us start therefore with the assumption ( cf . ref . @xcite ) that the ultrahigh energy photons from z - bursts can be neglected in the fit in comparison to the protons . this is certainly true for a sufficiently large universal radio background , e.g. on the level of the maximal one estimated in ref . @xcite and/or for a sufficiently strong extragalactic magnetic field @xmath294 g. we shall refer to this scenario in the following as `` strong '' uhe@xmath91 attenuation . our best fits to the observed data from this scenario ( cf . table [ fit_noph ] ) can be seen in fig . [ fit_nu ] , for evolution parameter @xmath295 . we find a neutrino mass of @xmath296 ev for the case that the uhecr background protons are of halo type ( [ pow - law - halo ] ) , @xmath297 ev , if they are of extragalactic type ( [ pow - law - eg ] ) , and @xmath298 ev , if there are are no background protons above @xmath299 ev , respectively . the first numbers are the 1@xmath222 , the numbers in the brackets are the 2@xmath222 errors . this gives an absolute lower bound on the mass of the heaviest neutrino of @xmath300 ev at the 95% c.l . , which is comparable to the one obtained from the atmospheric mass splitting in a three flavour scenario , eq . ( [ lim_low_atm ] ) . the available uhecr data with their error bars and the @xmath301 fits ( solid lines ) from z - bursts , for a halo ( _ top _ ) , an extragalactic ( _ middle _ ) , and no ( _ bottom _ ) uhecr background ( long - dashed lines ) , and a strong uhe@xmath91 attenuation such that the z - burst photons can be neglected and only the z - burst protons ( dash - dotted lines ) have to be taken into account in the fit ( @xmath229 ) . , title="fig:",width=325 ] the available uhecr data with their error bars and the @xmath301 fits ( solid lines ) from z - bursts , for a halo ( _ top _ ) , an extragalactic ( _ middle _ ) , and no ( _ bottom _ ) uhecr background ( long - dashed lines ) , and a strong uhe@xmath91 attenuation such that the z - burst photons can be neglected and only the z - burst protons ( dash - dotted lines ) have to be taken into account in the fit ( @xmath229 ) . , title="fig:",width=325 ] the available uhecr data with their error bars and the @xmath301 fits ( solid lines ) from z - bursts , for a halo ( _ top _ ) , an extragalactic ( _ middle _ ) , and no ( _ bottom _ ) uhecr background ( long - dashed lines ) , and a strong uhe@xmath91 attenuation such that the z - burst photons can be neglected and only the z - burst protons ( dash - dotted lines ) have to be taken into account in the fit ( @xmath229 ) . , title="fig:",width=325 ] the surprisingly small uncertainties are based on the @xmath227 analysis described in section [ gen ] . the inclusion of the already mentioned 30% uncertainties in the observed energies by a monte carlo analysis increases the error bars by about 10% . note , that the relative errors in the extragalactic and in the no background cases are of the same order . this shows that the smallness of these errors does not originate from the low energy part of the background component . the fits are rather good : for 21 non - vanishing bins and 4 fitted parameters they can be as low as @xmath302 and @xmath303 in the halo and the extralactic background case , respectively , whereas in the no background case , for @xmath304 ev , we have 9 bins with 2 fitted parameters and a @xmath305 , see table [ fit_noph ] . in the latter case , however , which was advocated strongly in ref . @xcite , a remarkable dependence of the fitted mass on the value of @xmath306 is observed . the @xmath307 fits are shown in fig . [ fit_errors ] . as it should be , the spread is small in the region where there are data , whereas it can be quite large in the presently unexplored ultrahigh energy regime . finally , let us mention that , both in the `` halo '' as well as in the extragalactic background cases , the @xmath227 fits _ without _ a z - burst component ( cf . [ fit_normal ] ) are far worse : we find @xmath308 in the former , and @xmath309 in the latter , for 21 non - vanishing bins and 2 fitted parameters . this finding suggests that the power - law background terms alone can not describe the data . in addition to the case of strong uhe@xmath91 attenuation , corresponding to a large universal radio background or a large extragalactic magnetic field , let us consider now the case of a `` minimal '' urb . here , we assume a vanishing egmf and exploit , as in ref . @xcite , a universal radio background on the level of the one from ref . @xcite ( cf . section [ prop ] and fig . [ ph_mfp ] ( top ) ) . the corresponding fits , for our three background scenarios , can be seen in fig . [ fit_nu_minurb ] and table [ fit_minurb ] . note , that our best fits are still compatible with the already mentioned upper limits on the photon fraction of the observed ultrahigh energy cosmic rays @xcite . we observe that the value of the neutrino mass found in both the halo background scenario , with @xmath310 ev , as well as in the extragalactic background scenario , with @xmath311 ev , are compatible , at about the 2 @xmath222 level , with the corresponding values found in the case of strong uhe@xmath91 attenuation . as before , in the `` no '' background scenario , we find a strong dependence of the fitted mass on the value of @xmath306 . the available uhecr data with their error bars and the best fits from z - bursts , for an energy attenuation of photons as in ref . @xcite and a vanishing egmf ( @xmath312 ) . _ top : _ best fit for the case of a halo uhecr background ( solid line ) , corresponding to the sum of the background protons ( long - dashed ) , the z - burst protons ( dash - dotted ) and the z - burst photons ( short - dashed ) . _ middle : _ the case of an extragalactic uhecr background . _ bottom : _ the case of no uhecr background above @xmath231 . , title="fig:",width=325 ] the available uhecr data with their error bars and the best fits from z - bursts , for an energy attenuation of photons as in ref . @xcite and a vanishing egmf ( @xmath312 ) . _ top : _ best fit for the case of a halo uhecr background ( solid line ) , corresponding to the sum of the background protons ( long - dashed ) , the z - burst protons ( dash - dotted ) and the z - burst photons ( short - dashed ) . _ middle : _ the case of an extragalactic uhecr background . _ bottom : _ the case of no uhecr background above @xmath231 . , title="fig:",width=325 ] the available uhecr data with their error bars and the best fits from z - bursts , for an energy attenuation of photons as in ref . @xcite and a vanishing egmf ( @xmath312 ) . _ top : _ best fit for the case of a halo uhecr background ( solid line ) , corresponding to the sum of the background protons ( long - dashed ) , the z - burst protons ( dash - dotted ) and the z - burst photons ( short - dashed ) . _ middle : _ the case of an extragalactic uhecr background . _ bottom : _ the case of no uhecr background above @xmath231 . , title="fig:",width=325 ] @xmath152 & @xmath196 [ ev ] & @xmath220 & @xmath31 & @xmath204 & @xmath18 + @xmath232 & @xmath313 & @xmath314 & @xmath315 & @xmath316 & @xmath317 + @xmath238 & @xmath318 & @xmath319 & @xmath320 & @xmath321 & @xmath322 + @xmath244 & @xmath323 & @xmath324 & @xmath325 & @xmath326 & @xmath327 + + @xmath152 & @xmath196 [ ev ] & @xmath220 & @xmath31 & @xmath204 & @xmath18 + @xmath232 & @xmath328 & @xmath329 & @xmath330 & @xmath331 & @xmath258 + @xmath238 & @xmath332 & @xmath333 & @xmath334 & @xmath335 & @xmath263 + @xmath244 & @xmath336 & @xmath337 & @xmath338 & @xmath339 & @xmath340 + + @xmath264 & @xmath196 [ ev ] & @xmath220 & @xmath31 & @xmath204 & @xmath18 + @xmath265 & @xmath341 & @xmath342 & @xmath268 & @xmath343 & @xmath268 + @xmath270 & @xmath344 & @xmath345 & @xmath268 & @xmath346 & @xmath268 + @xmath274 & @xmath347 & @xmath348 & @xmath268 & @xmath349 & @xmath268 + @xmath278 & @xmath350 & @xmath351 & @xmath268 & @xmath352 & @xmath268 + @xmath282 & @xmath353 & @xmath354 & @xmath268 & @xmath355 & @xmath268 + @xmath286 & @xmath356 & @xmath357 & @xmath268 & @xmath358 & @xmath268 + @xmath290 & @xmath359 & @xmath360 & @xmath268 & @xmath361 & @xmath268 + besides the no uhecr background scenario , the z - burst determination of the neutrino mass seems reasonably robust . its robustness with regard to changes of presently unknown quantities within their anticipated variations is further illustrated in tables [ fit_modurb ] and [ fit_hubble ] and in fig . [ mass_res ] . for a wide range of cosmological source evolution ( @xmath362 ) , hubble parameters @xmath363 , @xmath364 , @xmath365 , @xmath366 , for variations of the possible relic neutrino overdensity in our gzk zone within the limits discussed in section [ nunumb ] , and for different assumptions about the diffuse extragalactic photon background , the results remain within the above error bars . the main uncertainties concerning the central values originate from the different assumptions about the background of ordinary cosmic rays . in the case that the ordinary cosmic rays above @xmath189 ev are protons and originate from a region within the gzk zone of about 50 mpc ( `` halo '' ) , the required mass of the heaviest neutrino seems to lie between @xmath367 ev @xmath368 ev at the 68% c.l . ( @xmath369 , cf . below ) , if we take into account the variations between the minimal and moderate urb cases and the strong uhe@xmath91 attenuation case ( cf . [ mass_res ] ) . note , that a value of @xmath370 ev , as studied recently in ref . @xcite in a simulation of the z - burst scenario in the context of a halo background model , seems to lie more than 3@xmath222 away from our best fit values . the much more plausible assumption that the ordinary cosmic rays above @xmath189 are protons of extragalactic origin leads to a required neutrino mass of @xmath0 ev @xmath1 ev at the 68% c.l . ( @xmath369 ) ( cf . [ mass_res ] ) . @xmath152 & @xmath196 [ ev ] & @xmath220 & @xmath31 & @xmath204 & @xmath18 + @xmath232 & @xmath371 & @xmath314 & @xmath372 & @xmath373 & @xmath374 + @xmath238 & @xmath375 & @xmath376 & @xmath377 & @xmath378 & @xmath379 + @xmath244 & @xmath380 & @xmath381 & @xmath382 & @xmath383 & @xmath384 + + @xmath152 & @xmath196 [ ev ] & @xmath220 & @xmath31 & @xmath204 & @xmath18 + @xmath232 & @xmath385 & @xmath386 & @xmath387 & @xmath388 & @xmath253 + @xmath238 & @xmath389 & @xmath390 & @xmath391 & @xmath392 & @xmath258 + @xmath244 & @xmath393 & @xmath394 & @xmath395 & @xmath396 & @xmath258 + + @xmath264 & @xmath196 [ ev ] & @xmath220 & @xmath31 & @xmath204 & @xmath18 + @xmath265 & @xmath397 & @xmath398 & @xmath268 & @xmath399 & @xmath268 + @xmath270 & @xmath400 & @xmath401 & @xmath268 & @xmath402 & @xmath268 + @xmath274 & @xmath403 & @xmath404 & @xmath268 & @xmath405 & @xmath268 + @xmath278 & @xmath406 & @xmath407 & @xmath268 & @xmath408 & @xmath268 + @xmath282 & @xmath409 & @xmath410 & @xmath268 & @xmath411 & @xmath268 + @xmath286 & @xmath412 & @xmath413 & @xmath268 & @xmath414 & @xmath268 + @xmath290 & @xmath415 & @xmath416 & @xmath268 & @xmath417 & @xmath268 + @xmath418 & @xmath196 [ ev ] & @xmath220 & @xmath31 & @xmath204 & @xmath18 + @xmath419 & @xmath420 & @xmath421 & @xmath422 & @xmath423 & @xmath424 + @xmath425 & @xmath426 & @xmath427 & @xmath428 & @xmath429 & @xmath430 + + @xmath418 & @xmath196 [ ev ] & @xmath220 & @xmath31 & @xmath204 & @xmath18 + @xmath419 & @xmath431 & @xmath432 & @xmath433 & @xmath434 & @xmath424 + @xmath425 & @xmath435 & @xmath436 & @xmath437 & @xmath438 & @xmath439 + + @xmath418 & @xmath196 [ ev ] & @xmath220 & @xmath31 & @xmath204 & @xmath18 + @xmath419 & @xmath440 & @xmath441 & @xmath442 & @xmath443 & @xmath444 + @xmath425 & @xmath445 & @xmath446 & @xmath447 & @xmath448 & @xmath430 + summary of the masses of the heaviest neutrino required in the z - burst scenario , with their 1@xmath222 ( solid ) and 2@xmath222 ( dotted ) error bars , for the case of an extragalactic and a halo background of ordinary cosmic rays and for various assumptions about the diffuse extragalactic photon background in the radio band ( @xmath229 ) . from left : strong @xmath91 attenuation , moderate and minimal urb . , width=326 ] we performed a monte carlo analysis studying higher statistics . in the near future , the pierre auger observatory @xcite will provide a ten times higher statistics , which reduces the error bars in the neutrino mass to about one third of their present values . let us consider next in more detail the @xmath91 ray spectra from z - bursts , notably in the @xmath449 gev region . as illustrated in fig . [ fit_egret ] , the egret measurements of the diffuse @xmath91 background in the energy range between 30 mev and 100 gev @xcite put non - trivial constraints on the evolution parameter @xmath152 . whereas different values of @xmath152 lead to very similar spectra in the ultrahigh energy region which determines the neutrino mass they are easily discriminated in @xmath450 tev photons . only evolution parameters @xmath451 seem to be compatible with the egret measurements ( cf . [ fit_egret ] ) , quite independently of different assumptions about the urb . these numerical findings are in fairly good agreement with other recent simulations @xcite . the necessary uhec@xmath6 flux at the resonant energy @xmath216 is obtained via eq . ( [ b - uhecnu - fluxes ] ) from the fitted overall normalization @xmath204 , @xmath452 where @xmath453 mpc is the present size of the universe and where , on account of eqs . ( [ z - h ] ) and ( [ h - omega ] ) , the ratio of the maximally considered distance of z production , @xmath122 , and the size of the universe is given by @xmath454 this ratio is of order one ; it equals @xmath455 for @xmath456 , @xmath457 and @xmath458 . the required fluxes are summarized in fig . [ eflux ] , together with some existing upper limits and projected sensitivities of present , near future and future observational projects . they appear to be well below present upper limits and are within the expected sensitivity of amanda @xcite , rice @xcite , and auger @xcite . clearly , these fluxes are higher than the ones advocated in ref . @xcite based on local neutrino overdensities @xmath459 on scales of about @xmath460 mpc . however , since we also took into account a background from ordinary cosmic rays , our normalization of the z - burst component is different and correspondingly our fluxes are somewhat less than a factor of @xmath461 higher . they are consistent with the ones found in refs . @xcite . as far as power - law extrapolations of the fluxes , @xmath462 , below the resonance energy are concerned , we find , in agreement with refs . @xcite , that indices @xmath463 are excluded by the fly s eye limits @xcite ( cf . [ eflux ] ) . the available uhecr data with their error bars and the best fit from z - bursts , for various cosmological evolution parameters @xmath152 and an energy attenuation of photons as in fig . [ ph_mfp ] ( bottom ) exploiting a `` minimal '' urb ( @xmath464 ) . also shown is the diffuse @xmath91 background in the energy range between 30 mev and 100 gev as measured by egret @xcite ( solid ) . , width=326 ] the required neutrino flux , for power - law extrapolations with index @xmath465 to lower energies , is larger than the theoretical upper limit from `` hidden '' hadronic astrophysical sources , from which only photons and neutrinos can escape ( cf . fig . [ flux_upp_lim ] ( `` mpr '' ) ) . from an analysis of the assumptions behind this limit @xcite one finds that one has to invoke hidden astrophysical sources whose photons somehow do nt show up in the diffuse @xmath91 ray background measured by egret @xcite . in general , astrophysical sources for the required uhe neutrinos should be distributed with @xmath466 , accelerate protons up to energies @xmath467 ev , be opaque to primary nucleons and emit secondary photons only in the sub - mev region . it is an interesting question whether such challenging conditions can be realized in bl lac objects , a class of active galactic nuclei for which some evidence of zero or negative cosmological evolution has been found ( see ref . @xcite and references therein ) and which were recently discussed as possible sources of the highest energy cosmic rays @xcite . alternatively , one may invoke top - down scenarios @xcite for the sources of the highest energy cosmic neutrinos such as unstable superheavy relic particle decays @xcite or topological defect decays @xcite . it should be stressed that , besides the neutrino mass , the uhec@xmath6 flux at the resonance energy is one of the most robust predictions of the z - burst scenario which can be verified or falsified in the near future . in this connection it is worthwhile to recall ( cf . section [ nunumb ] ) that the current limits on neutrino degeneracies @xcite do not allow a remarkable uniform enhancement due to lepton asymmetries which otherwise would be welcome for a relaxation of the huge flux requirement @xcite . neutrino fluxes , @xmath468 , required by the z - burst hypothesis for the case of a halo and an extragalactic background of ordinary cosmic rays , respectively ( @xmath229 ) . shown are the necessary fluxes obtained from the fit results of table [ fit_noph ] for the case of a strong uhe@xmath91 attenuation . the horizontal errors indicate the 1@xmath222 ( solid ) and 2@xmath222 ( dotted ) uncertainties of the mass determination and the vertical errors include also the uncertainty of the hubble expansion rate . also shown are upper limits from fly s eye @xcite on @xmath147 and the goldstone lunar ultrahigh energy neutrino experiment glue @xcite on @xmath469 , as well as projected sensitivities of amanda @xcite on @xmath149 and auger @xcite on @xmath147 . the sensitiviy of rice is comparable to the one of auger @xcite.,width=326 ] we have presented a detailed comparison of the predicted spectra from z - bursts resulting from the resonant annihilation of ultrahigh energy cosmic neutrinos with relic neutrinos with the observed ultrahigh energy cosmic ray spectrum , extending our earlier study @xcite . the mass of the heaviest relic neutrino turned out to have to lie in the range @xmath367 ev @xmath368 ev on the 1 @xmath222 level , if the background of ordinary cosmic rays above @xmath189 ev consists of protons and originates from a region within a distance of about 50 mpc . in the phenomenologically most plausible case that the ordinary cosmic rays above @xmath189 are protons of extragalactic origin one is lead to a required neutrino mass in the range @xmath0 ev @xmath1 ev at the 68% c.l .. we have also analysed a case where there is no background of ordinary cosmic rays above , say , @xmath299 ev . here , we find @xmath470 ev @xmath471 ev . the above neutrino mass ranges include variations in presently unknown quantities , like the amount of neutrino clustering , the universal radio background , and the extragalactic magnetic field , within their anticipated uncertainties . if it turns out that the highest energy cosmic rays are extragalactic protons rather than photons one has to invoke a large universal radio background and/or large extragalactic magnetic field to suppress the photons from z - bursts . in this case and for our extragalactic background scenario , the neutrino mass range narrows down to @xmath0 ev @xmath2 ev , with a best fit value of @xmath472 ev . it is remarkable , that the mass ranges required in the z - burst scenario coincide nearly perfectly with the present knowledge about the mass of the heaviest neutrino from oscillations and tritium @xmath18 decay from eqs . ( [ lim_low_atm ] ) and ( [ lim_comb_osc_beta ] ) , @xmath473 ev , in a three flavour or , from eqs . ( [ lim_low_lsnd ] ) and ( [ lim_comb_osc_beta_lsnd ] ) , @xmath474 ev , in a four flavour scenario . our values , in the extragalactic background scenario , are still compatible with a hierarchical neutrino mass spectrum , with the largest mass suggested by the atmospheric neutrino oscillation , @xmath475 ev . however , our best fit values point more into the direction of a quasi - degenerate scenario , @xmath476 . it is amusing to observe that the recently reported evidence for neutrinoless double beta decay , with @xmath477 ev , if true , would point also into the same direction @xcite . the above neutrino masses are in a range which can be explored by near - future laboratory experiments , like the katrin tritium @xmath18 decay experiment @xcite , with a projected sensitivity of @xmath478 ev within @xmath479 years , or the @xmath19 decay experiments nemo-3 @xcite and cuore @xcite , which aim at @xmath480 ev , and their next - generation follow - up projects genius @xcite and exo @xcite , with @xmath481 ev . our mass range also compares favourably with the expected sensitivity @xmath482 ev of near - future global cosmological analyses involving also new cmb measurements @xcite , as well as @xmath483 ev expected from neutrino observations from supernovae explosions @xcite . from our fits , we could also determine the necessary ultrahigh energy neutrino flux at the resonance energy . its prediction , in the context of the z - burst scenario , turned out to be of similar robustness as the prediction of the mass of the heaviest neutrino . it was found to be consistent with present upper limits and detectable in the near future by the already operating neutrino telescopes amanda and rice , and by the pierre auger observatory , presently under construction . a search at these facilities is the most promising and timely step in testing the z - burst hypothesis . one does not have to wait for future projects such as euso @xcite for this critical investigation . the required neutrino fluxes are enormous . if such tremendous fluxes of ultrahigh energy neutrinos are indeed found , one has to deal with the challenge to explain their origin . it is fair to say , that at the moment no convincing astrophysical sources are known which meet the requirements for the z - burst hypothesis , i.e. which have no or a negative cosmological evolution , accelerate protons at least up to @xmath484 ev , are opaque to primary nucleons and emit secondary photons only in the sub - mev region . ideas to create the required neutrino fluxes from the decays of superheavy particles or topological defects do not seem to be too appealing since they invoke further new physics beyond neutrino masses . we thank s. barwick , o. biebel , s. bludman , w. buchmller , p. di bari , m. kachelriess , k. mannheim , h. meyer , w. ochs , k. petrovay , s. sarkar , f. schrempp , d. semikoz , g. sigl , and e. waxman for useful discussions . we thank the opal collaboration for their unpublished results on hadronic z decays . this work was partially supported by hungarian science foundation grants no . otka - t34980/t29803/t22929/m28413/omfb-1548/om - mu-708 . l. stodolsky , phys . * 34 * , 110 ( 1975 ) [ erratum - ibid . * 34 * , 508 ( 1975 ) ] . f. smith and j. d. lewin , phys . b * 127 * , 185 ( 1983 ) . b. f. shvartsman , v. b. braginsky , s. s. gershtein , y. b. zeldovich and m. y. khlopov , jetp lett . * 36 * , 277 ( 1982 ) [ pisma zh . eksp . teor . fiz . * 36 * , 224 ( 1982 ) ] . p. langacker , j. p. leveille and j. sheiman , phys . d * 27 * , 1228 ( 1983 ) . i. ferreras and i. wasserman , phys . d * 52 * , 5459 ( 1995 ) . c. hagmann , astro - ph/9905258 . g. duda , g. gelmini and s. nussinov , phys . d * 64 * , 122001 ( 2001 ) . d. fargion , b. mele and a. salis , astrophys . j. * 517 * , 725 ( 1999 ) . t. j. weiler , astropart . phys . * 11 * , 303 ( 1999 ) . t. weiler , phys . . lett . * 49 * , 234 ( 1982 ) . e. roulet , phys . d * 47 * , 5247 ( 1993 ) . s. yoshida , h. y. dai , c. c. jui and p. sommers , astrophys . j. * 479 * , 547 ( 1997 ) . y. fukuda _ et al . _ [ super - kamiokande collaboration ] , phys . rev . lett . * 81 * , 1562 ( 1998 ) . y. fukuda _ et al . _ [ super - kamiokande collaboration ] , phys . b * 433 * , 9 ( 1998 ) . y. fukuda _ et al . _ [ super - kamiokande collaboration ] , phys . b * 436 * , 33 ( 1998 ) . s. fukuda _ et al . _ [ super - kamiokande collaboration ] , phys . rev . lett . * 85 * , 3999 ( 2000 ) . m. ambrosio _ et al . _ [ macro collaboration ] , phys . b * 517 * , 59 ( 2001 ) . b. t. cleveland _ et al . _ , astrophys . j. * 496 * , 505 ( 1998 ) . w. hampel _ et al . _ [ gallex collaboration ] , phys . b * 447 * , 127 ( 1999 ) . j. n. abdurashitov _ et al . _ [ sage collaboration ] , phys . c * 60 * , 055801 ( 1999 ) . s. fukuda _ et al . _ [ superkamiokande collaboration ] , phys . rev . * 86 * , 5651 ( 2001 ) . s. fukuda _ et al . _ [ super - kamiokande collaboration ] , phys . lett . * 86 * , 5656 ( 2001 ) . m. altmann _ et al . _ [ gno collaboration ] , phys . b * 490 * , 16 ( 2000 ) . q. r. ahmad _ et al . _ [ sno collaboration ] , phys . rev . * 87 * , 071301 ( 2001 ) . c. athanassopoulos [ lsnd collaboration ] , phys . lett . * 75 * , 2650 ( 1995 ) . c. athanassopoulos _ et al . _ [ lsnd collaboration ] , phys . c * 54 * , 2685 ( 1996 ) . c. athanassopoulos _ et al . _ [ lsnd collaboration ] , phys . lett . * 77 * , 3082 ( 1996 ) . c. athanassopoulos _ [ lsnd collaboration ] , phys . * 81 * , 1774 ( 1998 ) . c. athanassopoulos _ et al . _ [ lsnd collaboration ] , phys . c * 58 * , 2489 ( 1998 ) . a. aguilar _ et al . _ [ lsnd collaboration ] , phys . d * 64 * , 112007 ( 2001 ) . y. declais _ et al . _ , nucl . b * 434 * , 503 ( 1995 ) . m. apollonio _ et al . _ [ chooz collaboration ] , phys . b * 466 * , 415 ( 1999 ) . f. boehm _ et al . _ , phys . * 84 * , 3764 ( 2000 ) . k. eitel [ karmen collaboration ] , nucl . suppl . * 91 * , 191 ( 2000 ) . b. armbruster _ et al . _ [ karmen collaboration ] , hep - ex/0203021 . s. avvakumov _ et al . _ , hep - ex/0203018 . m. takeda _ et al . _ , phys . lett . * 81 * , 1163 ( 1998 ) . d. j. bird _ [ hires collaboration ] , phys . * 71 * , 3401 ( 1993 ) . d. j. bird _ [ hires collaboration ] , astrophys . j. * 424 * , 491 ( 1994 ) . d. j. bird _ et al . _ , astrophys . j. * 441 * , 144 ( 1995 ) . m. a. lawrence , r. j. reid and a. a. watson , j. phys . g * 17 * , 733 ( 1991 ) . m. ave , j. a. hinton , r. a. vazquez , a. a. watson and e. zas , phys . lett . * 85 * , 2244 ( 2000 ) . kieda00 d. kieda _ et al . _ , in _ proceedings of the 26th international cosmic ray conference _ , salt lake , 1999 . efimov91 n.n . et al . _ , in _ proc . of the astrophysical aspects of the most energetic cosmic rays _ ( world scientific , singapore , 1991 ) . m. nagano and a. a. watson , rev . phys . * 72 * , 689 ( 2000 ) . k. greisen , phys . * 16 * , 748 ( 1966 ) . g. t. zatsepin and v. a. kuzmin , jetp lett . * 4 * , 78 ( 1966 ) [ pisma zh . fiz . * 4 * , 114 ( 1966 ) ] . e. waxman , astro - ph/9804023 . s. yoshida , g. sigl and s. j. lee , phys . * 81 * , 5505 ( 1998 ) . j. j. blanco - pillado , r. a. vazquez and e. zas , phys . d * 61 * , 123003 ( 2000 ) . g. gelmini and a. kusenko , phys . lett . * 82 * , 5202 ( 1999 ) . g. gelmini and a. kusenko , phys . * 84 * , 1378 ( 2000 ) . t. j. weiler , hep - ph/9910316 . j. l. crooks , j. o. dunn and p. h. frampton , astrophys . j. * 546 * , l1 ( 2001 ) . g. b. gelmini , hep - ph/0005263 . h. ps and t. j. weiler , phys . d * 63 * , 113015 ( 2001 ) . d. fargion , m. grossi , p. g. de sanctis lucentini , c. di troia and r. v. konoplich , astro - ph/0102426 . z. fodor , s. d. katz and a. ringwald , hep - ph/0105064 . z. fodor , s. d. katz and a. ringwald , hep - ph/0105336 . b. h. mckellar , m. garbutt , g. j. stephenson and t. goldman , hep - ph/0106123 . a. ringwald , hep - ph/0111112 . o. e. kalashev , v. a. kuzmin , d. v. semikoz and g. sigl , hep - ph/0112351 . g. gelmini and g. varieschi , hep - ph/0201273 . g. l. fogli , e. lisi , d. montanino and a. palazzo , phys . d * 64 * , 093007 ( 2001 ) . j. n. bahcall , m. c. gonzalez - garcia and c. pena - garay , jhep * 0108 * , 014 ( 2001 ) . m. c. gonzalez - garcia and y. nir , hep - ph/0202058 . c. weinheimer _ et al . _ , b * 460 * , 219 ( 1999 ) . v. m. lobashev _ et al . _ , phys . b * 460 * , 227 ( 1999 ) . j. bonn _ et al . _ , nucl . suppl . * 91 * , 273 ( 2001 ) . v. m. lobashev _ et al . _ , nucl . suppl . * 91 * , 280 ( 2001 ) . c. e. aalseth _ [ igex collaboration ] , phys . c * 59 * , 2108 ( 1999 ) . l. baudis _ et al . _ , phys . * 83 * , 41 ( 1999 ) . c. e. aalseth _ et al . _ , phys . atom . nucl . * 63 * , 1225 ( 2000 ) [ yad . fiz . * 63 * , 1299 ( 2000 ) ] . h. v. klapdor - kleingrothaus _ et al . . phys . j. a * 12 * , 147 ( 2001 ) . c. e. aalseth _ et al . _ , hep - ex/0202026 . h. v. klapdor - kleingrothaus , a. dietz , h. l. harney and i. v. krivosheina , mod . a * 16 * , 2409 ( 2002 ) . f. feruglio , a. strumia and f. vissani , hep - ph/0201291 . c. e. aalseth _ et al . _ , hep - ex/0202018 . v. d. barger , t. j. weiler and k. whisnant , phys . lett . b * 442 * , 255 ( 1998 ) . h. v. klapdor - kleingrothaus , h. pas and a. y. smirnov , phys . d * 63 * , 073005 ( 2001 ) . f. vissani , nucl . suppl . * 100 * , 273 ( 2001 ) . s. m. bilenky , s. pascoli and s. t. petcov , phys . d * 64 * , 053010 ( 2001 ) . s. m. bilenky , s. pascoli and s. t. petcov , phys . d * 64 * , 113003 ( 2001 ) . y. farzan , o. l. peres and a. y. smirnov , nucl . b * 612 * , 59 ( 2001 ) . p. osland and g. vigdel , phys . b * 520 * , 143 ( 2001 ) . m. czakon , j. gluza , j. studnik and m. zralek , phys . d * 65 * , 053008 ( 2002 ) . r. a. croft , w. hu and r. dave , phys . lett . * 83 * , 1092 ( 1999 ) . m. fukugita , g. c. liu and n. sugiyama , phys . * 84 * , 1082 ( 2000 ) . e. gawiser , astro - ph/0005475 . x. wang , m. tegmark and m. zaldarriaga , astro - ph/0105091 . t. j. loredo and d. q. lamb , in _ dallas 1988 , proceedings , relativistic astrophysics _ , pp . 601 - 630 . p. j. kernan and l. m. krauss , nucl . b * 437 * , 243 ( 1995 ) . p. gondolo , g. gelmini , and s. sarkar , nucl . b * 392 * , 111 ( 1993 ) . d. e. groom _ et al . _ [ particle data group collaboration ] , eur . phys . j. c * 15 * , 1 ( 2000 ) . d. buskulic _ et al . _ [ aleph collaboration ] , z. phys . c * 66 * , 355 ( 1995 ) . k. abe _ et al . _ [ sld collaboration ] , in _ proc . of the 19th intl . on photon and lepton interactions at high energy lp99 _ , ed . jaros and m.e . peskin , hep - ex/9908033 . opal pn299 , unpublished . y. i. azimov , y. l. dokshitzer , v. a. khoze and s. i. troian , z. phys . c * 27 * , 65 ( 1985 ) . y. i. azimov , y. l. dokshitzer , v. a. khoze and s. i. troian , z. phys . c * 31 * , 213 ( 1986 ) . c. p. fong and b. r. webber , nucl . b * 355 * , 54 ( 1991 ) . t. sjostrand , comput . commun . * 82 * , 74 ( 1994 ) . j. n. bahcall and e. waxman , astrophys . j. * 542 * , 543 ( 2000 ) . z. fodor and s. d. katz , phys . d * 63 * , 023002 ( 2001 ) . e. waxman , astrophys . j. * 452 * , l1 ( 1995 ) . z. fodor , s. d. katz and g. papp , hep - lat/0202030 . s. lee , phys . d * 58 * , 043004 ( 1998 ) . r. j. protheroe and p. a. johnson , astropart . * 4 * , 253 ( 1996 ) . r. j. protheroe and p. l. biermann , astropart . * 6 * , 45 ( 1996 ) [ erratum - ibid . * 7 * , 181 ( 1996 ) ] . t. a. clark , l. w. brown and j. k. alexander , nature * 228 * , 847 ( 1970 ) . r. j. protheroe , nucl . suppl . * 77 * , 465 ( 1999 ) . r. gandhi , nucl . suppl . * 91 * , 453 ( 2000 ) . j. g. learned and k. mannheim , ann . nucl . part . sci . * 50 * , 679 ( 2000 ) . r. m. baltrusaitis _ et al . _ , phys . d * 31 * , 2192 ( 1985 ) . p. w. gorham , k. m. liewer , c. j. naudet , d. p. saltzberg and d. r. williams , astro - ph/0102435 . e. waxman and j. n. bahcall , phys . d * 59 * , 023002 ( 1999 ) . j. n. bahcall and e. waxman , phys . d * 64 * , 023002 ( 2001 ) . k. mannheim , r. j. protheroe and j. p. rachen , phys . d * 63 * , 023003 ( 2001 ) . d. seckel _ et al . _ [ rice collaboration ] , in _ proc . 27th international cosmic ray conference _ , hamburg , germany , 2001 , pp . 1137 - 1140 . v. s. beresinsky and g. t. zatsepin , phys . b * 28 * , 423 ( 1969 ) . berezinsky , v.s . and zatsepin , g.t . , j. nucl . phys . * 11 * , 111 ( 1970 ) [ yad . fiz . * 11 * , 200 ( 1970 ) ] . s. yoshida and m. teshima , prog . phys . * 89 * , 833 ( 1993 ) . r. engel and t. stanev , phys . d * 64 * , 093010 ( 2001 ) . berezinsky , v.s . , in : _ proc . dumand summer workshops _ , learned , j.g . ( ed . ) , khabarovsk and lake baikal , 22 - 31 aug 1979 , hawaii dumand center , university of hawaii , 1980 , pp . 245 - 261 . v. berezinsky , m. kachelriess and a. vilenkin , phys . lett . * 79 * , 4302 ( 1997 ) . v. a. kuzmin and v. a. rubakov , phys . atom . nucl . * 61 * , 1028 ( 1998 ) [ yad . fiz . * 61 * , 1122 ( 1998 ) ] . m. birkel and s. sarkar , astropart . * 9 * , 297 ( 1998 ) . s. sarkar and r. toldra , nucl . b * 621 * , 495 ( 2002 ) . v. berezinsky and m. kachelriess , phys . b * 434 * , 61 ( 1998 ) . v. berezinsky and m. kachelriess , phys . d * 63 * , 034007 ( 2001 ) . r. toldra , comput . commun . * 143 * , 287 ( 2002 ) . r. toldra , astro - ph/0201151 . c. barbot and m. drees , hep - ph/0202072 . a. ibarra and r. toldra , hep - ph/0202111 . v. s. berezinsky and a. vilenkin , phys . d * 62 * , 083512 ( 2000 ) . c. p. ma , astro - ph/9904001 . j. r. primack and m. a. gross , astro - ph/0007165 . j. r. primack , private communication . l. n. da costa , w. freudling , g. wegner , r. giovanelli , m. p. haynes and j. j. salzer , astrophys . j. * 468 * , l5 ( 1996 ) . a. dekel _ et al . _ , astrophys . j. * 522 * , 1 ( 1999 ) . j. p. kneller , r. j. scherrer , g. steigman and t. p. walker , phys . d * 64 * , 123506 ( 2001 ) . a. d. dolgov , s. h. hansen , s. pastor , s. t. petcov , g. g. raffelt and d. v. semikoz , hep - ph/0201287 . m. ave , j. a. hinton , r. a. vazquez , a. a. watson and e. zas , astro - ph/0110613 . k. shinozaki _ et al . _ [ agasa collaboration ] , in _ proc . 27th international cosmic ray conference _ , hamburg , 2001 , pp . 346 - 349 . s. l. dubovsky and p. g. tinyakov , jetp lett . * 68 * , 107 ( 1998 ) . v. berezinsky and a. a. mikhailov , phys . b * 449 * , 237 ( 1999 ) . g. a. medina tanco and a. a. watson , astropart . * 12 * , 25 ( 1999 ) . c. k. guerard , nucl . * 75a * , 380 ( 1999 ) . x. bertou , m. boratav and a. letessier - selvon , int . j. mod . a * 15 * , 2181 ( 2000 ) . p. sreekumar _ et al . _ , astrophys . j. * 494 * , 523 ( 1998 ) . s. barwick , talk at radhep 2000 , ucla workshop , www.ps.uci.edu/~amanda a. osipowicz _ et al . _ [ katrin collaboration ] , hep - ex/0109033 . c. marquet [ nemo collaboration ] , nucl . suppl . * 87 * , 298 ( 2000 ) . a. alessandrello _ et al . _ [ cuore collaboration ] , hep - ex/0201038 . h. v. klapdor - kleingrothaus , j. hellmig and m. hirsch , j. phys . g * 24 * , 483 ( 1998 ) . m. danilov _ et al . _ , phys . b * 480 * , 12 ( 2000 ) . j. f. beacom , r. n. boyd and a. mezzacappa , phys . lett . * 85 * , 3568 ( 2000 ) . n. arnaud , m. barsuglia , m. a. bizouard , f. cavalier , m. davier , p. hello and t. pradier , phys . d * 65 * , 033010 ( 2002 ) . o. catalano , nuovo cim . * 24c * , 445 ( 2001 ) .
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we consider the possibility that a large fraction of the ultrahigh energy cosmic rays are decay products of z bosons which were produced in the scattering of ultrahigh energy cosmic neutrinos on cosmological relic neutrinos .
we compare the observed ultrahigh energy cosmic ray spectrum with the one predicted in the above z - burst scenario and determine the required mass of the heaviest relic neutrino as well as the necessary ultrahigh energy cosmic neutrino flux via a maximum likelihood analysis .
we show that the value of the neutrino mass obtained in this way is fairly robust against variations in presently unknown quantities , like the amount of neutrino clustering , the universal radio background , and the extragalactic magnetic field , within their anticipated uncertainties .
much stronger systematics arises from different possible assumptions about the diffuse background of ordinary cosmic rays from unresolved astrophysical sources . in the most plausible case that these ordinary cosmic rays are protons of extragalactic origin ,
one is lead to a required neutrino mass in the range @xmath0 ev @xmath1 ev at the 68% confidence level .
this range narrows down considerably if a particular universal radio background is assumed , e.g. to @xmath0 ev @xmath2 ev for a large one .
the required flux of ultrahigh energy cosmic neutrinos near the resonant energy should be detected in the near future by amanda , rice , and the pierre auger observatory , otherwise the z - burst scenario will be ruled out .
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in unified models for radio sources , bl lac objects are the low - power radio galaxies whose jets are the most highly beamed along our line of sight ( e.g. browne 1983 ; antonucci & ulvestad 1985 ; ulrich 1989 ; urry & padovani 1995 ) . their rapid variability , apparent superluminal motion , strong point - like emission in radio , optical and x - ray , and the detection of some sources in @xmath2-rays , are all explained if we are seeing emission from a relativistic jet closely aligned with our line of sight . low - power radio galaxies then represent the ` parent population ' of unaligned objects whose jets are less favourably aligned . these low - power radio galaxies are likely mostly to be fanaroff & riley ( 1974 ) class i ( fri ) objects , but the parent population may also include some transitional objects and low - excitation frii radio galaxies . an important test of this unified model is the degree to which the isotropic ( alignment - independent ) properties of bl lacs are similar to those of the parent population of radio galaxies . such tests have been made , on the whole successfully , by looking at the extended radio emission ( e.g. antonucci & ulvestad 1985 ; kollgaard et al . 1992 ; perlman & stocke 1993 , 1994 ) and properties of the host galaxies ( e.g. ulrich 1989 ; abraham , mchardy & crawford 1991 ; wurtz , stocke & yee 1996 ; falomo 1996 ) although there is some evidence that there are too few bl lacs associated with the most luminous host galaxies ( wurtz et al . 1996 ) . another isotropic indicator is the clustering environment . using two - point correlation analysis of optical fields , it has been shown that fri radio galaxies are normally found in groups or clusters of galaxies ( longair & seldner 1979 ; prestage & peacock 1988 ) and bl lacs seem also to inhabit groups or poor clusters ( pesce , falomo & treves 1995 ; smith , odea & baum 1995 ; wurtz , stocke & ellingson 1997 ) though it appears that , at least at low redshift , bl lacs are not often found in the dominant galaxies of rich clusters ( prestage & peacock 1988 ; owen , ledlow & keel 1996 ; wurtz et al.1997 ) ; for example , owen et al . ( 1996 ) find no bl lacs at the centres of abell clusters , a result inconsistent at the 95% confidence level with the numbers expected from the unified models of urry & padovani . clustering environment may also be investigated by x - ray observations . it has long been known that many objects close to the fri - frii luminosity boundary are associated with rich clusters having luminous x - ray haloes . recent observations with _ rosat _ have shown that more typical fri radio galaxies have extended thermal x - ray emission on scales characteristic of groups or poor clusters ( worrall & birkinshaw 1994 ) . this offers a new way to test the unification hypothesis ; such emission should be isotropic , and so we expect all bl lacs to have x - ray haloes comparable to those of fris . this test is difficult because it requires us to separate any extended x - ray emission from the bright unresolved emission of the bl lac nucleus . in this paper we describe such an analysis of _ rosat _ observations of the bl lac pks 0521@xmath1365 . pks 0521@xmath1365 is a well - studied bl lac with a redshift of 0.055 , comparable to the redshifts of the radio galaxies studied by worrall & birkinshaw ( 1994 ) . it is variously described in the literature as a blazar , a bl lac object , or an n - galaxy , and on multifrequency spectral index plots like those of sambruna , maraschi & urry ( 1996 ) is placed among radio - selected bl lacs . its host galaxy is easily detectable in the optical [ contributing @xmath3 per cent of the source luminosity at 5500 in an @xmath4 arcsec effective aperture ; falomo , scarpa & bersanelli ( 1994 ) ] and it exhibits strong , variable broad emission lines ( scarpa , falomo & pian 1995 ) . the host galaxy is a giant elliptical ( wurtz et al . pesce et al . ( 1995 ) suggest that the excess galaxy count around the object corresponds to a cluster of abell richness 0 or more ; they identify at least one , and up to four associated galaxies within 100 kpc . however , the cross - correlation analysis of wurtz et al . ( 1997 ) suggests a poorer cluster , with richness class @xmath5 . [ unbeamed ] in the radio , the source has a 408-mhz flux of 36.1 jy ( wright & otrupcek 1990 ) , corresponding to a power at that frequency of @xmath6 w hz@xmath7 sr@xmath7 ; this places it slightly above the nominal fri - frii luminosity boundary ( @xmath8 w hz@xmath7 sr@xmath7 at this frequency ) , though of course some of the 408-mhz emission is contributed by the core , presumed to be beamed . it exhibits a core - halo - hotspot morphology on arcsecond scales ( wardle , moore & angel 1984 ; ekers et al . 1989 ; see also section [ discuss ] ) , which , together with its comparatively high radio power , may suggest that it is an aligned version of a transitional fri - frii object . the prominent radio jet is also seen in optical synchrotron emission , extending about 6 arcsec from the nucleus ( e.g.keel 1986 , macchetto et al . no motion of the core components was detected in vlbi observations ( tingay et al . 1996 ) and this , together with the comparatively low ratios of nuclear to host - galaxy optical emission ( falomo et al . ) and radio core to extended radio flux ( antonucci & ulvestad 1985 ) , suggests a source that is only moderately relativistically boosted along the line of sight compared to the most extreme bl lacs . it was for this reason that we selected it as a suitable candidate for an x - ray search for extended emission with the _ rosat _ hri . pks 0521@xmath1365 has already been extensively observed at x - ray wavebands , with _ einstein _ ( worrall & wilkes 1990 ) , _ exosat _ ( sambruna et al . 1994 ) and the _ rosat _ pspc ( pian et al . 1996 ) , and was detected in @xmath2-rays by _ egret _ ( lin et al . 1996 ) , but none of these observations provides the resolution necessary to separate compact from extended structure . the source appears to have a concave spectrum in the x - ray waveband ; the energy index is @xmath9 in the soft band , flattening to @xmath10 at higher energies . throughout the paper we use a cosmology in which @xmath11 km s@xmath7 mpc@xmath7 , @xmath12 . at the distance of pks 0521@xmath1365 1 arcsec corresponds to 1.475 kpc . we observed pks 0521@xmath1365 with the _ rosat _ hri for 46.6 ks between 1995 feb 09 and 1995 feb 11 . the data were analysed using the iraf post - reduction off - line software ( pros ) . pks 0521@xmath1365 is detected at @xmath13 net counts in a 2.5-arcmin radius circle with background taken from an annulus between 2.5 and 3.5 arcmin . the source circle was chosen by expanding the radius until the background - subtracted counts enclosed stayed constant within the errors . a 25-arcsecond radius circle around a nearby point source ( 1 arcmin to the north ) was excluded from the analysis . the detected counts correspond to a rate of @xmath14 counts s@xmath7 . the source centroid is consistent to within a few arcseconds with the best available optical and radio positions . [ var ] a simple analysis , binning the data into 1-ks intervals , suggests that the source was not strongly variable during the 110 ks that contain our observations . to compare the source flux with the earlier _ rosat _ pspc observations , we re - analysed the 4.8 ks of pspc data from the _ rosat _ public archive , taking account of the nearby background sources . we detect the source at @xmath15 pspc counts in the same source region , or @xmath16 counts s@xmath7 . a spectral fit to a power - law model between 0.1 and 2.4 kev using the galactic @xmath17 ( @xmath18 @xmath19 ) of elvis , lockman & wilkes ( 1989 ) gives an energy spectral index of @xmath20 ( @xmath21 with 32 d.o.f . ; errors are @xmath22 for one interesting parameter ) , consistent with the results of pian et al.(1996 ) and sambruna ( 1997 ) ; the unabsorbed 1-kev flux density from the pspc data is then @xmath23 @xmath24jy , while the flux density from the hri data on the same assumptions is @xmath25 @xmath24jy . if @xmath17 is allowed to vary the best - fit spectral index is @xmath26 with @xmath27 @xmath19 ( @xmath28 with 31 d.o.f . ; errors @xmath22 for two interesting parameters ) . incorporating the spectral uncertainties , the flux in the 0.12.4 kev _ rosat _ band seems to be @xmath29 per cent lower in the hri data than it was at the epoch of the pspc observations ( 1992 aug 29 ) . we return to this point below . fig . [ contour ] ( left ) shows that the source is approximately symmetrical ; radial profiling is therefore appropriate . to search for extension in pks 0521@xmath1365 we first compared its radial profile with the standard empirical expression ( david et al . 1997 ) for the hri point response function ( prf ) . it will be seen from fig . [ radial ] ( left ) that the radial profile is not well fitted by a standard _ prf ( normalised to the central regions , which we expect to be dominated by the point source ) ; there is a deficit of counts at small radii and an excess of counts at large ones . it is well known that images from the hri can be badly affected by errors in the _ rosat _ aspect correction ; the effect is to produce spurious elongation ( ` smearing ' ) in the images of point sources , thus broadening the effective , azimuthally - averaged prf . such smearing is likely to be significant compared to the detector and instrument point response on scales of @xmath30 arcsec . as shown in fig . [ contour ] , the source appears elongated , in its central regions , in position angle roughly @xmath31 ; aspect smearing could therefore be responsible for some of the deviations from the nominal prf shown in fig . [ radial ] . the aspect errors are thought to arise because the star tracking system does not always calculate positions correctly , due to variations in the gains of the pixels of the star tracker ccd . the spacecraft wobble exacerbates this problem because it causes the guide stars to move across the tracking ccd on short timescales . methods for correcting aspect smearing ( e.g. morse 1994 ) therefore rely on binning the data according to the phase of the wobble ; so long as the satellite roll angle and the properties of the ccd are constant , the aspect error should be a function of the wobble phase only . to do this , we applied iraf / pros scripts provided by j. silverman and d. harris , based on suggestions by g. hasinger ( for more details see harris et al . we first established that all our data were taken at the same roll angle ; this , along with the relatively small number of closely spaced observation intervals , allowed us to analyse all the data together . the scripts bin the data as a function of wobble phase , utilising the fact that the relative wobble phase at any time is given simply by the spacecraft clock ( modulo clock resets , none of which occured during our observation ) . we chose to divide the data into 20 phase bins , giving approximately 500 counts per binned observation ; with 20 phase bins there are enough counts for centroiding in each bin , and examination of images derived from each bin showed them to be circular . a centroid was then found , using the _ detect _ suite of tasks within pros , for each individual bin , and the images were restacked so that the centroids aligned . the resulting image ( fig . [ contour ] , right ) appears less elliptical than the uncorrected image ( fig . [ contour ] , left ) and is considerably better fit , in its inner regions , by the nominal hri prf ( figure [ radial ] , right ) ; it appears that the stacking procedure has made the core slightly narrower than the nominal prf , which is of course derived from observations that may themselves have been affected by aspect uncertainties . we conclude that the inner regions of pks 0521@xmath1365 are well described as a point source . more importantly , even after dewobbling we still see an excess of counts over the prf at radii of 10 - 40 arcsec . there is no reason to doubt that this corresponds to real extended emission . to characterise the scale size of this extended emission we fit models consisting of a @xmath32-model ( sarazin 1986 ) and a point - source component , both convolved with the nominal prf , to the background - subtracted radial profile of the dewobbled image . to avoid large contributions to the fitting statistic due to the fact that the restacked data are narrower than the nominal prf , we sampled the radial profile more coarsely in its central regions for these fits . we performed fits for various different values of the parameter @xmath32 and the core radius . figure [ chi2 ] shows a plot of @xmath33 as a function of @xmath32 and core radius ; the best fit ( @xmath34 with 15 degrees of freedom ) has @xmath35 and a core radius of 8 arcsec , with a central normalisation of the @xmath32-model of @xmath36 counts arcsec@xmath37 . figure [ radialmod ] shows the best - fit model plotted with the data . fitting a point source alone to the data gives unacceptably high values of the fitting statistic ( @xmath38 ) . for comparison , we fitted a similar range of @xmath32-models to the archival pspc dataset . because of the broader intrinsic prf of the pspc , smearing is not considered to be a problem with this instrument . the best - fitting point source and @xmath32-model had a core radius of 35 arcsec with @xmath35 , as shown in fig . the data are better fit with this model than with a point - source model alone ( best - fit @xmath39 with 9 degrees of freedom ) , although the point - source fit is formally acceptable ( @xmath40 with 10 degrees of freedom ) . a large range of possible core radii are acceptable , and the hri best - fitting @xmath32 model is among those consistent with the pspc data within the 90 per cent confidence contour ; the normalisations of the models are also consistent . if an appropriately normalised @xmath32-model with core radius 8 arcsec and @xmath41 is subtracted from the pspc data the remaining emission is adequately modelled as a point source with no significant support for an additional extended component . we therefore conclude that the extended emission is well represented by the hri best - fit model . the best - fit point - source count rates in this model in the pspc and hri data are consistent to within a few per cent , suggesting that the inconsistency between the total pspc and hri count rates ( section [ var ] ) is a result of our not having taken into account the two spectral components corresponding to compact and extended emission . the @xmath32-model fits to the hri data described above imply that the extended x - ray emission has a core radius of 12 kpc and contributes @xmath42 counts to the total in a 2.5-arcmin radius circle . the corresponding 0.2 - 1.9 kev luminosity , assuming a raymond - smith model with @xmath43 kev and using galactic @xmath17 , is approximately @xmath44 w. this is considerably more luminous than the extended emission seen in most of the radio galaxies of worrall & birkinshaw ( 1994 ) , although the fwhm of 14 kpc is comparable to some of the smaller objects . the central cooling time , on the same temperature assumption , is @xmath45 yr , which , if the emission were thermal , would imply rapid cooling , with expected mass deposition rates of @xmath46 @xmath47 yr@xmath7 . the radius at which the cooling time is comparable to the hubble time would be @xmath48 kpc , larger than the core radius , which means that a simple single - temperature model of the extended emission is not self - consistent . cooling takes place for any reasonable choice of temperature of the x - ray gas . the extended x - ray component detected by the hri is comparable in size to the extended radio emission in pks 0521@xmath1365 which might be thought to suggest an explanation in terms of inverse - compton ( ic ) scattering by the energetic electrons in the radio lobes of photons from the cosmic microwave background radiation ( cmb ) and from the bl lac nucleus itself ( cf . brunetti , setti & comastri 1997 ) . [ overlay ] shows the x - ray image superposed on a 1.4-ghz vla radio map made from data supplied by g.g . since the flux on the shortest baselines of this radio observation ( made at bna configuration ) is equal to the single - dish parkes radio flux ( wright & otrupcek 1990 ) of 16.3 jy , we can be confident that the extended structure is well represented . the largest angular size ( las ) of the radio emission is about 50 arcsec , whereas the fwhm of the extended x - ray component is 10 arcsec , and fig . [ overlay ] shows that the radio emission is quite asymmetrically distributed around the nucleus , in contrast to the symmetry of the x - ray emission . however , we note that inverse - compton emission would come preferentially from the parts of the radio lobes closest to the active nucleus , so this in itself does not make an inverse - compton model implausible . since we do not know the unprojected geometry of the source , we model the extended radio - emitting component crudely as a uniform sphere centred around the nucleus . the flux density from the core is 2.7 jy [ consistent with the value of antonucci & ulvestad ( 1985 ) ] leaving 13.6 jy in extended emission . projection effects make it difficult to assess the true size ( and therefore volume ) of the lobes , but the flux expected from ic scattering from an object of a given synchrotron ( radio ) flux is only weakly dependent on the object s size , and so we assume a radius for the sphere of half the las . because the frequency of a scattered photon is raised by approximately the square of the lorentz factor ( @xmath2 ) of the scattering electron , and because the electron energy spectrum is dominated by electrons with @xmath49 [ i.e. electrons near the assumed low - energy cutoff in the energy distribution carilli et al . ( 1991 ) ] the most significant contribution in the x - ray band from the ic process is due to far - ir photons . we accordingly model the spectrum of the bl lac s nucleus in this region as a power law with spectral index 0.78 normalised to the measured 100-@xmath24 m flux of impey & negebauer ( 1988 ) . pian et al.(1996 ) argue that the doppler factor in the nucleus is of order unity , based on the relatively low radio core prominence discussed in section [ unbeamed ] and on models for the nuclear x - ray and @xmath2-ray emission , and we adopt this value in order to infer a luminosity from the observed flux ; we neglect the effects of possible circumnuclear obscuration , so that our source model is effectively an isotropically radiating central source surrounded by a uniform , spherically symmetrical electron distribution . using a modified version of the code of hardcastle , birkinshaw & worrall ( 1998a ) we find that the ic process in this model can not produce enough x - rays to account for the observed flux unless the lobes have magnetic field strength approximately an order of magnitude weaker than the value implied by equipartition of energy between magnetic field and relativistic electrons ( for electron filling factor unity ) ; this is relatively much weaker than fields that have been inferred in other sources from observations of x - ray emission attributed to inverse - compton or synchrotron - self - compton emission ( harris et al . 1994 ; feigelson et al . 1995 ; hardcastle et al . 1998a ; brunetti et al . 1998 ) . at equipartition , the x - rays from the ic process contribute at most 2 per cent of the observed flux , or less if the filling factor is lower or there is a significant contribution from relativistic protons . as a result of this simple analysis , we believe that it is unlikely that the extended x - rays seen around the source are due in large part to inverse - compton emission . instead , it seems more likely that they are indeed thermal emission from a rapidly cooling central region , and that pks 0521@xmath1365 inhabits a cooling flow ; this means that its environment is significantly different from those of the fris observed by worrall & birkinshaw ( 1994 ) , which tended to lie in less dense environments with much longer cooling times . this model is qualitatively consistent with the bright extended emission line region seen in pks 0521@xmath1365(boisson , cayatte & sol 1989 ) and with the strong polarization asymmetry seen in the radio observations . we attempt to fit the radial profile of the source with a cooling flow model based on that described by hardcastle , lawrence & worrall ( 1998b ) . in this version of the model , the outer regions of the source are fit with an isothermal @xmath32 model while the temperature and density inside the cooling radius ( @xmath50 ) are power law functions of radius . the electron density follows @xmath51 while the temperature is given by @xmath52 here @xmath53 is the core radius of the @xmath32 model , and @xmath54 is the inner limiting radius of the cooling flow , allowing us to avoid infinities at @xmath55 . @xmath56 , @xmath57 and @xmath58 are scale electron densities and @xmath59 , @xmath60 and @xmath61 scale temperatures ; @xmath61 corresponds to the temperature of the non - cooling gas . it is clear that matching temperatures and densities allows us to write all the scale factors in terms of @xmath58 and @xmath61 . the results are only weakly dependent on @xmath54 so long as it is small and we fix it at a value corresponding to 0.01 arcsec in what follows . the parameters @xmath62 and @xmath63 set the slope of the density and temperature distributions ; for a realistic cooling model we expect @xmath64 and @xmath65 , so that density increases and temperature decreases with decreasing radius . the ideal gas law implies that pressure goes as @xmath66 . if the atmosphere is required to be close to hydrostatic equilibrium , then matching mass as a function of radius inside and outside the cooling zone leads to a constraint on @xmath67 as a function of @xmath62 and @xmath63 : @xmath68 and if we assume some law for the radial dependence of pressure ( @xmath69 is consistent with observations of other cooling regions ) then the model has only five free parameters : @xmath32 , @xmath53 , @xmath62 , @xmath70 and @xmath58 , the last being a normalisation parameter that can be determined in the fit . in addition , such a model can only be physically realistic if the cooling time at @xmath50 is comparable to the system lifetime , or the hubble time : @xmath71 ( sarazin 1986 ) where @xmath72 is in years , @xmath73 in kev and @xmath74 in @xmath75 . we fit a range of representative cooling models to the data . as before , @xmath32 was chosen from a small number of possible values ( 0.5 , 0.667 , 0.9 ) while @xmath53 ranged from 11000 arcsec . we tried values of 1 , 1.5 , 2 , 5 and 10 kev for @xmath76 . @xmath62 was allowed to take the values 1.5 , 1.75 or 2.0 . no model consisting of a cooling flow alone was a good fit to the data . a number of models consisting of a cooling flow and a central point source were comparable in goodness of fit to the combination of @xmath32-model and point source discussed above , but most of these were ruled out by the constraint on cooling time . those that were not have small core radii , low external temperatures and steep temperature and density power laws : for example , the model with @xmath77 kev , @xmath78 , @xmath79 , @xmath80 arcsec is an acceptable fit to the hri radial profile ( @xmath81 ) . in this model just under 10 per cent of the total counts ( @xmath82 ) are assigned to the cooling flow component . the nominal cooling time at the core radius is @xmath83 years ; the mass deposition rate is approximately @xmath84 yr@xmath7 and the implied densities at 20 kpc correspond roughly with those required for pressure equilibrium with the cold gas inferred from observations of the extended emission - line region ( boisson et al . though the details of the model may not be correct , this shows that the extended emission of pks 0521@xmath1365 can plausibly be modelled as a cooling flow of this type . a strong nuclear source is still necessary ; the flux of the point - like component in the cooling - flow model is reduced by only @xmath85 per cent compared to that derived from the simple @xmath32-model fits of section [ analysis ] . we have carried out the first separation of nuclear and galaxy - scale extended x - ray emission in a bl lac object and found evidence that pks 0521@xmath1365 inhabits a dense and , presumably , rapidly cooling region of x - ray emission , a much more extreme environment than those found for ` typical ' fri radio galaxies ; we have shown that it can be modelled as a cooling flow in low - temperature cluster gas . if this result were extended to other bl lac objects , it would cause difficulties for models that seek to unify fris and bl lacs , and might imply some causal relationship between a dense and rapidly cooling atmosphere and the bl lac phenomenon . however , it may be that pks 0521@xmath1365 , with its high power and intermediate radio structure , is not representative of the bl lac class . further observations are planned both to verify the thermal nature of the galaxy - scale halo and to see whether pks 0521@xmath1365 is unusual among bl lacs in this respect . we are grateful to guy pooley for allowing us to use his vla observations of pks 0521@xmath1365 , and to the vla analysts for help in recovering the vla data from the archive . this research has made use of the nasa / ipac extragalactic database ( ned ) which is operated by the jet propulsion laboratory , california institute of technology , under contract with nasa . the digitized sky surveys were produced at the space telescope science institute under u.s . government grant nag w-2166 . the national radio astronomy observatory is operated by associated universities inc . , under co - operative agreement with the national science foundation . this work was supported by nasa grant nag 5 - 2312 and pparc grant gr / k98582 . david l.p . , harnden f.r . , kearns k.e . , zombeck m.v . , harris d.e . , prestwich a. , primini f.a . , silverman j.d . , snowden s.l . , 1997 , rosat science data center report , available at url : @xmath86http://hea - www.harvard.edu / rosat / rsdc_www / hricalrep.html@xmath87 lin y.c . , bertsch d.l . , dingus b.l . , esposito j.a . , fichtle c.e . , hartman r.c . , hunter s.d . , kanback g. , kniffen d.a . , mayer - hasselwander h.a . , michelson p.f . , von montigny c. , mukherjee r. , nolan p.l . , scheid e. , sreekumar p. , thompson d.j . , 1995 , apj , 442 , 96 macchetto f. , albrecht r. , barbieri c. , blades j.c . , boksenberg a. , crane p. , deharveng j.m . , disney m.j . , jakobsen p. , kamperman t.m . , king i.r . , mackay c.d . , paresce f. , weigelt g. , baxter b. , greenfield p. , jedrzejewski r. , nota a. , sparks w.b . , 1991 , apj , 369 , l55 tingay s.j . , edwards p.g . , costa m.e . , lovell j.e.j . , mcculloch p.m. , jauncey d.l . , reynolds j.e . , tzioumis a.k . , migenes v. , gough r. , king e.a . , jones d.l . , preston r.a . , murphy d.w . , meier d.l . , van ommen t.d . , st . john m. , hoard d.w . , nicolson g.d . , wan t.-.s . , shen z.-.q . , 1996 , apj , 464 , 170
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models that seek to unify bl lacs and low - power radio galaxies predict that the two types of object should show similar isotropically emitted x - ray emission .
testing this is usually limited by difficulties in separating strong x - ray emission from a bl lac nucleus and surrounding low - surface brightness emission . in this paper
we report rosat hri observations of the @xmath0 bl lac object pks 0521@xmath1365 .
we are able to separate a luminous extended x - ray component from the bright nucleus . using a new radio map ,
we show that it is unlikely that the extended emission is due to inverse - compton scattering of photons from the active nucleus , and instead interpret it as thermal emission from dense , rapidly cooling gas .
this is a more extreme environment than is found in typical fri radio galaxies , and may pose a problem for unified models .
x - rays : galaxies galaxies : individual : pks 0521@xmath1365 galaxies : active
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You are an expert at summarizing long articles. Proceed to summarize the following text:
after the discovery of half - metallic ferromagnetism in half - heusler alloy nimnsb by de groot et al.@xcite many of the compounds were found to be half - metallic in different experimental @xcite and theoretical studies@xcite . the half - metallic ferromagnets ( hmfs ) have band gap at the fermi level ( _ e@xmath3 _ ) in one spin channel while the other spin channel is strongly metallic . these materials show a complete spin polarization of the conduction electrons at the _ _ e@xmath3__. because of their exceptional band structures at the _ e@xmath3 _ , these materials are of great interest from theoretical and applications point of view . co containing full - heusler alloys were firstly proposed by ishida et al.@xcite and were synthesised by webster@xcite . fe@xmath0mnz type heusler alloys have also been proposed to show half - metallic ferromagnetism by fujii et al.@xcite . in the family of fe based heusler alloys only fe@xmath0mnsi@xcite and fe@xmath0crz ( z = si , ge , sn ) @xcite were predicted to be half - metallic ferromagnts theoretically . fe@xmath0mnsi is a ferromagnetic material with a curie temperature of 214 k and transforms at 69 k to the low temperature phase with smaller spontaneous magnetization @xcite . the total magnetic moment of fe@xmath0mnsi obtained by magnetization measurements is 2.1 @xmath4/f.u . ( f.u.@xmath5 formula unit ) at 4.2 k @xcite which is smaller than the calculated moment of 3.0 @xmath4/f.u . fe@xmath0mnal on the other hand is not a half - metallic as there exist slight density of states at the _ e@xmath3 _ for minority spin channel @xcite . it is well known that the local spin density approximation ( lsda ) and the generalized gradient approximation ( gga ) schemes for the exchange - correlation potential are not sufficient to describe the electronic structure and magnetism of some heusler alloys like co@xmath0fesi@xcite . such strongly correlated systems which contain atoms with open _ d _ or _ f _ shells , can be treated by adding on - site coulomb interaction ( _ u _ ) term as modification to lsda i.e. by using lsda + _ u _ approach @xcite . the lsda + _ u _ method accounts an orbital - dependent on - site electron - electron coulomb interaction which is not included in the pure lda or gga . rai et al.@xcite have studied some co based heusler alloys and reported the increase in band gap , hybridization of _ d_-_d _ orbitals as well as _ d_-_p _ orbitals when on - site coulomb interaction is added to lsda approach . they have also concluded that some co - based heusler alloys show half - metallic character when are treated with lsda + _ u_. it is well known that the systems in which there exist the density of states ( dos ) at the fermi energy for one spin , the on - site coulomb interaction ( _ u _ ) may bring a drastic change into their electronic and magnetic properties @xcite . as the electron - electron correlation plays an important role in the heusler compounds so it can be expected that it will also affect the fe - based heusler alloys . in the present work we have employed the full - potential linearized augmented - plane wave methods to compute the electronic and magnetic properties of three full - heusler alloys viz . fe@xmath0mnsi , fe@xmath0mnal , and co@xmath0mnge . the electronic structures in the fm solutions without _ u _ show that fe@xmath0mnsi is half - metallic compound whereas fe@xmath0mnal is metallic . the large effect of _ u _ is observed on the electronic structures and magnetic moments of fe - based compounds whereas on co@xmath0mnge compound its effect is negligibly small . the magnetic moment of fe atom increases with _ u _ and in the mn atom it decreases with _ u_. the magnetic moments of fe and mn atoms are found to be coupled ferromagnetically in fe@xmath0mnsi for all values of _ u _ , whereas in fe@xmath0mnal they coupled antiferromagnetically below _ u _ = 2 ev and ferromagnetically above it . the ground state of fe - based compounds remains half - metallic for all values of _ u _ , whereas it changes from metallic to semiconductor in fe@xmath0mnal after _ u _ = 2 ev . also the ferromagnetic coupling between fe and mn atom is found to be responsible for the presence of semiconducting ground state in this compound . the electronic and magnetic properties of fe@xmath1mnsi , fe@xmath1mnal and co@xmath1mnge were calculated by using the full - potential linearized augmented plane - wave ( fp - lapw ) method within the density functional theory ( dft ) implemented in wien2k code@xcite . the local spin density approximation ( lsda ) of perdew and wang @xcite was employed for exchange - correlation energy of electrons . the effect of on - site coulomb interaction ( _ u _ ) under lsda + _ _ u _ _ @xcite formulation of the dft was also considered in the calculations . the around - the - mean - field ( amf ) version of the lsda+__u _ _ @xcite method was employed to account for the `` double - counting '' correction terms in the energy functional . the effective coulomb - exchange interaction _ _ u__@xmath6 is given by ( _ u _ - _ j _ ) , where _ u _ and _ j _ are the coulomb and exchange parameter . the value of _ u _ was varied from 1 to 5 ev and _ j _ was kept fixed to 0 ev , therefor _ _ u__@xmath6 was equal to _ u _ in our calculations . the values of muffin - tin radii used in the calculations were 2.2 bohr for fe , mn , si and al atoms . the maximum _ l _ value ( _ l@xmath7 _ ) for the expansion of wave function in spherical harmonics inside the atomic spheres was equal to 10 . for convergence of energy eigenvalues the wave function in the interstitial regions were expanded in plane waves with cutoff @xmath8 = 8 , where @xmath9 is the smallest atomic sphere radius and @xmath10 is largest k vector in the plane wave expansion . the electronic and magnetic properties of these compounds were calculated by using the experimental lattice parameters . the self consistent iteration was repeated until calculated total energy / cell and charge / cell of the systems converge to less than 10@xmath11 ry and 0.001e , respectively . all these compounds belong to the family of full heusler alloys and crystallizes in @xmath12 crystal structure with space group @xmath13-@xmath14 . these compounds have composition x@xmath0yz , where x and y are transition metals and z is the main groups element . x atoms ( fe and co ) are placed at wyckoff position 8c ( 1/4 , 1/4 , 1/4 ) . y atoms ( mn ) and z atoms ( si , al and ge ) are located at wyckoff position 4a ( 0 , 0 , 0 ) and 4b ( 1/2 , 1/2 , 1/2 ) , respectively @xcite . the dispersion curves of fe@xmath1mnsi and fe@xmath1mnal along the high symmetry directions of the first brillouin zone are presented in fig . 1(a and b ) . in the dispersion curve of fe@xmath1mnsi , the bands labeled by 1 , 2 and 3 are lying above the _ e@xmath3 _ , bands 4 , 5 , 6 and 7 are crossing it at 14 different k - points and band 8 is lying below it . around w - point , bands 4 - 7 are concentrated in the energy range of about -0.2 to 0.2 ev . the total energy of the system can be minimized if there will be shifting of these bands and this shifting may lead to the fm ground state for this compound . from fig . 1(b ) it is clear that in fe@xmath1mnal the bands labeled by 1 - 5 are lying above the _ e@xmath3 _ , bands 6 and 7 which are crossing the _ e@xmath3 _ at 10 different k - points and band 8 is lying just below it . bands 4 - 7 are concentrated around the w - point in the energy range of about -0.2 to 0.2 ev . therefore here also one can expect that shifting of these bands will minimize the total energy of the system which may lead to the ferromagnetic ground state in the compounds . these results are consistent with our earlier reported results for the paramagnetic phase of co@xmath1mnge@xcite . the total and partial density of states plots of fe@xmath1mnsi and fe@xmath1mnal are presented in fig . 2 and fig . 3 , respectively . from total density of states plots ( tdos ) ( fig . 2(a ) and 3(a ) ) it is clear that there is very large density of states of about 6 states / ev / f.u . ( f.u.@xmath5 formula unit ) at _ e@xmath3 _ for both the spins . according to the stoner theory the large value of tdos may be considered as the indication of the ferromagnetic ground states in the compounds@xcite . the antibonding bands are extended upto 0.4 ev and 0.3 ev below the _ e@xmath3 _ for fe@xmath1mnsi and fe@xmath1mnal , respectively . as per stoner theory the total energy of the systems will be minimized if there is a shifting in spin - up and spin - down bands by @xmath20.4 ev and 0.3 ev below and above the _ e@xmath3 _ for fe@xmath1mnsi and fe@xmath1mnal , respectively . this may be responsible for the half - metallic fm ground state in the compounds as observed in the case of co@xmath1mnge@xcite . also the total energies of fm phase of fe@xmath1mnsi and fe@xmath1mnal are about 0.77 ev and 0.54 ev less than pm phase , which further confirm that both compounds should have fm ground states . the partial density of states ( pdos ) plots for fe , mn and si atoms of fe@xmath1mnsi are presented in fig . 2(b - d ) . from fig . 2(b ) it is evident that the pdos of fe atom at _ e@xmath3 _ is mainly contributed by _ _ t@xmath15 _ _ and _ e@xmath16 _ states with contribution of about 0.3 states / ev / atom and about 2.0 states / ev / atom , respectively for both spin channels . the pdos of mn atom is shown in fig . 2(c ) and in both spin channels the contribution from _ t@xmath15 _ and _ e@xmath16 _ is @xmath21 states / ev / atom and @xmath21.6 states / ev / atom , respectively . the pdos of si atom ( fig . 2(d ) ) show that the occupancy of 3__s _ _ and 3__p _ _ orbitals at _ e@xmath3 _ is very small , which can be neglected . the pdos plots for fe , mn and al atoms of fe@xmath1mnal are shown in fig . 3(b - d ) . it is clear from fig . 3(b ) that the contribution of _ t@xmath15 _ is @xmath20.3 states / ev / atom and _ e@xmath16 _ is @xmath21 states / ev / atom at _ e@xmath3 _ for both spin channels . in the pdos of mn ( fig . 3(c ) ) the occupancy of _ t@xmath15 _ at _ e@xmath3 _ is @xmath20.7 states / ev / atom and _ _ e@xmath16 _ _ at _ e@xmath3 _ is @xmath22.0 states / ev / atom . from pdos plot of al atom ( fig . 3(d ) ) it is evident that there is negligibly small contribution from 3__s _ _ and 3__p _ _ orbitals at _ e@xmath3_. it is also clear from these figures that _ t@xmath15 _ and _ e@xmath16 _ have main contribution to the total dos at _ e@xmath3 _ for fe@xmath1mnsi and fe@xmath1mnal . the _ e@xmath16 _ state has the largest contribution to the tdos of both compounds at _ e@xmath3_. the spin resolved dispersion curves of fe@xmath1mnsi and fe@xmath1mnal along the high symmetry directions of the first brillouin zone are shown in fig . 4 . from the dispersion curve shown in fig . 4(a ) it is clear that fe@xmath1mnsi is metallic for spin - up channel . bands labeled by 1 and 2 are lying just above the _ e@xmath3 _ , bands 3 and 4 are crossing the _ e@xmath3 _ at 7 different k - points and bands 5 - 7 are lying just below it . 4(b ) shows that this compound behave as semiconductor for down spin channel and there exist an indirect gap of about 0.44 ev from @xmath17 to x - direction . the computed value of indirect band gap using lsda is less than the reported value @xcite . this compound is found to be half - metallic similar to the previous studies @xcite . bands labeled by 1 - 6 have shifted into the conduction band while bands 7 and 8 have shifted into the valence band and thus there is presence of ferromagnetic ground state in this compound as stated earlier . from the dispersion curves shown in fig . 4(c and d ) , it is very clear that fe@xmath1mnal is metallic in nature . in the spin - up channels bands 1 - 4 are lying above the _ e@xmath3 _ and bands 5 - 8 are crossing it at 8 different k - points . in down spin channel bands 1 - 7 are lying above _ e@xmath3 _ and only one band , labelled by 8 is crossing it at two different k - points . thus there exist a slight density of states at _ e@xmath3 _ in the spin - dn channel and making this compound metallic in nature . there is almost flat conduction band along @xmath17 to x direction in the down spin channel of both compounds and this can be responsible for large value of effective masses of these compounds . total and partial dos plots of fe@xmath1mnsi for fm solution are shown in fig . 5 . from fig . 5(a ) , it is clear that spin - up channel is occupied at _ e@xmath3 _ with occupancy of about 5 states / ev / f.u . whereas spin - down channel is unoccupied . thus this compound behave as metal for majority spin states and semiconductor for minority spin states . after comparing fig . 2(a ) and 5(a ) , one can conclude that tdos shifts towards lower energy in spin - up channel whereas , in spin - down channel it shifts towards higher energy . because of this shift there is creation of band gap in the minority spin channel as is observed in co@xmath1mnge . this band shift appears to be responsible for the presence of fm ground state in this compound . however , in this compound the shift in tdos is very small in comparison to rigid shift observed in spin - up channel of co@xmath1mnge@xcite . also the value of tdos at _ e@xmath3 _ is very large in comparison to tdos of co@xmath1mnge studied earlier and one can expect very large effect of on - site coulomb interaction ( _ u _ ) on such systems . from the pdos of fe atom ( fig . 5(b ) ) it is clear that _ t@xmath15 _ ( @xmath20.5 states / ev / atom ) and _ e@xmath16 _ ( @xmath21.4 states / ev / atom ) states have main contribution at _ e@xmath3 _ for spin - up channel while the minority spin channel is empty . near the _ e@xmath3 _ in spin - dn channel , the valence band maximum has main contribution from _ t@xmath15 _ and conduction band minimum is contributed by _ e@xmath16 _ states . the pdos of mn atom is presented in fig . 5(c ) and it is clear from this plot that _ t@xmath15 _ bands have occupation of @xmath21.0 states / ev / atom with negligibly small contribution from _ e@xmath16 _ band at _ e@xmath3 _ for spin - up channel . the _ e@xmath16 _ band of mn atom has shifted towards lower energy with no contribution at the _ e@xmath3_. the pdos of si atom ( fig . 5(d ) ) shows negligibly small contribution from 3__s _ _ and 3__p _ _ orbitals . the total and partial density of states plots of spin - up and spin - down channels for fe@xmath1mnal are shown in fig . the spin - up channel of fig . 6(a ) shows that total density of states at _ e@xmath3 _ is @xmath21 states / ev / f.u . and spin - down channel show very small ( @xmath20.1 states / ev / f.u . ) density of states at _ e@xmath3_. on comparing fig . 3(a ) and 6(a ) one can find that there is no rigid shifting of bands rather there is splitting of bands at _ e@xmath3 _ , which we have not observed in co@xmath1mnge and fe@xmath1mnsi . the pdos of fe , mn and al atoms for both spin channels are shown in fig . 6(b - d ) . from figs . 3(b ) and 6(b ) it is clear that there is no rigid shift in band as per stoner theory as is observed in co@xmath1mnge@xcite . e@xmath16 _ states split in such a way that there exists a minimum at _ e@xmath3 _ with contribution of @xmath20.3 states / ev / atom and 0.1 states / ev / atom , from _ t@xmath15 _ and _ e@xmath16 _ bands , respectively . this minimum is responsible for the existence of the pseudo gap . in the minority spin channel the valence band maximum has mainly _ t@xmath15 _ character and conduction band minimum has _ e@xmath16 _ character . this compound show metallic nature for down spin also because of very small contribution from _ t@xmath15 _ bands at _ e@xmath3_. in fig . 6(c ) the pdos for mn _ t@xmath15 _ and _ e@xmath16 _ bands are shown . the spin - up channel of pdos of mn atom is contributed by _ t@xmath15 _ state ( @xmath20.2 states / ev / atom ) with negligibly small contribution from _ e@xmath16 _ state at _ e@xmath3_. it is evident from fig . 6(d ) that in pdos of al atom there is negligibly small contribution of 3__s _ _ and 3__p _ _ orbitals . the total magnetic moment per formula unit for fe@xmath1mnsi is 3.0 @xmath4 with contribution from fe , mn , si and interstitial region is 0.22 , 2.52 , -0.01 and 0.042 @xmath4 , respectively . the similar value of total magnetic moment is also predicted theoretically in @xcite , whereas the experimental results @xcite gave a saturation magnetic moment less than this value . this may be due to the reason that it is difficult to obtain the pure phases experimentally . the mn atom coupled ferromagnetically with fe atom as is found by galanakis et al.@xcite . the magnetic moment of si is very small and it is coupled antiferromagnetically with fe and mn atoms . the total magnetic moment per formula unit for fe@xmath1mnal is 2.0 @xmath4 and contribution from fe , mn , al and interstitial region is -0.23 , 2.44 , -0.008 and 0.05 @xmath4 , respectively . the calculated value of the total magnetic moment matches with earlier reported value @xcite . the mn atom coupled antiferromagnetically with fe atom in this compound . by using the full - potential screened korringa - kohn - rostoker ( fskkr ) green s function method in conjunction with the local spin density approximation galanakis et al.@xcite have also found similar results . but fujii et al . @xcite have observed ferromagnetic coupling between fe and mn atoms which is contradictory to our results . the magnetic moment of al is very small but it is coupled ferromagnetically with fe atom and antiferromagnetically with mn atom . the mn atom carries the largest magnetic moment in all these compounds , while the fe atom of fe@xmath1mnsi and fe@xmath1mnal carry the modest magnetic moment which is in agreement with the earlier reported results @xcite . from the study of electronic structure of fe@xmath1mnsi it is clear that for spin - up channel there is very large density of states at _ e@xmath3 _ and spin - down channel is empty . the electronic structure of fe@xmath1mnal show comparatively small value of tdos from 3__d _ _ bands at _ e@xmath3 _ for both spin channels . it is well known that 3__d _ _ bands are less dispersive therefore on - site coulomb interaction can have very large effect on the electronic and magnetic properties of such compounds . the lsda+__u _ _ method may bring a drastic change on the magnetic properties and electronic properties of these compounds . so we have study the effect of _ u _ varying from 1 to 5 ev on the magnetic moment and electronic structure of these compounds along with previously studied co@xmath0mnge compound @xcite . firstly , we discuss the effect of _ u _ on the magnetic moments of all these compounds . the total number of 3__d _ _ electrons in both spin channels and local magnetic moments in the presence of _ u _ are presented in table 1 for fe@xmath1mnsi , fe@xmath1mnal and co@xmath1mnge . it is clear from table that in fe@xmath1mnsi , the number of 3__d _ _ electrons of fe atom in up spin channel increases while that of down spin channel decreases with _ u_. however the total number of electrons remains fixed to the value @xmath26.1 . the local magnetic moment of fe atom is found to increase with increasing the value of _ u_. the magnetic moment increases from 0.29 @xmath4 at _ u _ = 1 ev to 0.87 @xmath4 at _ u _ = 5 ev . in mn atom , the total number of 3__d _ _ electrons in spin - up channel decreases and spin - dn channel increases with _ u _ in such a way that total number of 3__d _ _ electrons remains fixed to the value @xmath24.9 . also the magnetic moment of mn atom decreases from 2.41 @xmath4 at _ u _ = 1 ev to 1.29 @xmath4 at _ u _ = 5 ev . the above results suggest that the hund s like exchange interactions between fe 3__d _ _ electrons are increasing and that of mn 3__d _ _ electrons decreasing with increase in _ u_. in fe@xmath1mnal , also the number of electrons of fe ( mn ) atom in up spin channel increases ( decreases ) while that of down spin channel decrease ( increases ) with _ u_. the value of local magnetic moment of fe atom also goes on increasing with increasing the value of _ u_. the value of magnetic moment increases from -0.14 @xmath4 at _ u _ = 1 ev to 0.55 @xmath4 at _ u _ = 5 ev . there is also a very anomalous effect of _ u _ on the magnetic moment of fe atom , as the magnetic moment changes its sign after _ u _ = 2 ev and coupled ferromagnetically with mn atom . on the other hand the magnetic moment of mn decreases from 2.24 @xmath4 at _ u _ = 1 ev to 0.77 @xmath4 at _ u _ = 5 ev . at _ u _ = 2 ev , the magnetic moment of fe atom become almost zero and the total magnetic moment is contributed only by the mn atom . in the fe atom the hund s like exchange interactions are also found to increase with _ u _ while in mn atom decrease with _ u_. in comparison to fe@xmath1mnsi the effect of _ u _ on the magnetic properties of fe@xmath1mnal compound is found to be more . the on - site coulomb interactions are affecting above two compounds drastically but we have observed no drastic effect on the magnetic moment of co@xmath1mnge . in this compounds the number of electrons in spin - up and spin - dn channels are no changing significantly with _ u_. the magnetic moment of co atom increases slightly from 1.07 @xmath4 at _ u _ = 1 ev to 1.15 @xmath4 at _ u _ = 5 ev . the magnetic moment of mn atom decreases from 2.80 @xmath4 at _ u _ = 1 ev to 2.61 @xmath4 at _ u _ = 5 ev . the value of _ u _ is affecting the magnetic properties of these compounds although it has very less effect on co@xmath1mnge . therefore we have also studied the electronic structures of all these compounds in the presence of _ u_. the total and partial dos of fe@xmath1mnsi , fe@xmath1mnal and co@xmath1mnge are presented in fig . 7 - 9 only for two selected values of _ u _ i.e. _ u _ = 2 and 4 ev . the spin - up channel of fig . 7(a ) shows that the tdos at _ e@xmath3 _ decrease very slowly with _ u _ and there exists a gap at _ e@xmath3 _ in down spin channel . the value of band gap in spin - up channel increases from @xmath20.8 ev at _ u _ = 2 ev to @xmath20.9 ev at _ u _ = 4 ev . tdos in the spin - up and spin - dn channels shifts towards the lower energy as the value of _ u _ is increased from 2 ev to 4 ev . the pdos of fe atom is presented in fig . 7(b ) and it is evident from figure that for these two values _ u _ , _ e@xmath16 _ and _ t@xmath15 _ states are contributing to the tdos at _ e@xmath3 _ with more contribution from _ e@xmath16 _ states . also the contribution from _ t@xmath15 _ and _ e@xmath16 _ states at _ e@xmath3 _ decreases from @xmath20.45 states / ev / atom at _ u _ = 2 ev to @xmath20.27 states / ev / atom at _ u _ = 4 ev and @xmath21.7 states / ev / atom at _ u _ = 2 ev to @xmath21.0 states / ev / atom at _ u _ = 4 ev , respectively . the contribution from both these states decrease with _ u _ and pdos appears to shift towards lower energy in the both spin channels . the pdos of mn ( fig . 7(c ) ) at _ e@xmath3 _ is occupied by _ t@xmath15 _ states with negligibly small contribution from _ e@xmath16 _ states . also the occupancy of _ t@xmath15 _ states decreases very slowly with _ u _ and _ t@xmath15 _ and _ e@xmath16 _ states shift towards lower energy in spin - up channel . in the down spin channel the _ t@xmath15 _ states also shift towards lower energy while , _ e@xmath16 _ states shift towards higher energy . the pdos of si atom has negligibly small contribution from 3__s _ _ and 3__p _ _ orbitals for these values of _ u_. the tdos of fe@xmath1mnal for _ u _ = 2 ev and 4 ev are shown in fig . 8(a ) . for _ u _ = 2 ev , the soft gap is appearing in the spin - up channel and when value of _ u _ is increased this compound become semiconductor with band gap of @xmath20.7 ev . in the spin - dn channel the band gap does not change significantly with _ u_. this is also observed in magnetic moment calculations , where at _ u _ = 2 ev , the magnetic moment of fe atom become almost zero and for _ u _ = 4 ev the value of magnetic moment is positive . the tdos for both the spin channels shift towards higher energy at _ u _ = 4 ev . the pdos of fe , mn and al atoms shown in fig . 8(b - d ) for _ u _ = 2 ev and _ u _ = 4 ev . from pdos of fe atom it is clear that _ t@xmath15 _ and _ e@xmath16 _ states shifts towards higher energy for both the spin channels . the pdos of mn atom show that in spin - up channel shifting of _ t@xmath15 _ states is dominating and in spin - dn channel shifting of _ e@xmath16 _ states is more dominating . also shifting is taking place at higher rate in the conduction band . the pdos of al atom has also negligibly small contribution from 3__s _ _ and 3__p _ _ orbitals for these values of _ u_. the tdos of co@xmath1mnge in the presence of _ u _ are shown in fig . this figure shows that the spin - up channel of co@xmath1mnge is metallic for all values of _ u _ and down spin channel is semiconducting . as the value of _ u _ is increased from 2 to 4 ev , there is broadening of the band gap in the spin - dn channel . the band gap is found to increase from @xmath21.0 ev at _ u _ = 2 ev to @xmath21.4 ev at _ u _ = 4 ev . we have not observed any significant shifting in the tdos for both spin channels . thus the on - site coulomb interactions are not playing a significant effect on this compound , which is also evident from the calculations of magnetic moments . it is clear from pdos of co atom shown in fig . 9(b ) that at _ e@xmath3 _ , the _ t@xmath15 _ and _ e@xmath16 _ states contribute same for both values of _ u_. increasing value of _ u _ is not going to affect the pdos of co atom . the pdos of mn atom ( fig . 9(c ) ) also shows no difference in the dos at _ e@xmath3 _ for these two values of _ 9(d ) shows that contribution from 3__s _ _ and 3__p _ _ states to the tdos is negligibly small and also not affected by the on - site coulomb interactions . from the above results it is clear that the metallic ground state of fe@xmath1mnal compound changes directly to the semiconducting ground state when the value of _ u _ is increased , whereas ground state of other two heusler alloys remain half - metallic ferromagnetic . very few experimental work related to fe@xmath1mnal compound is found in the literature . in order to verify our predicted result it is necessary to perform electrical conductivity and neutron diffraction experiments , which directly probe the electronic transport behaviour and the nature of magnetic coupling between the fe and mn moments . to the best of our knowledge there are no experimental data on electrical conductivity in the @xmath12 phase . however liu et al . @xcite have reported resistivity data of the compound when it is in the b@xmath1 phase , which show insulating behaviour at the low temperature . similarly neutron diffraction experiments on fe@xmath1mnsi compound are also desirable to know the magnetic moments of the fe and mn atoms . these experiments will help in understanding the role of on - site coulomb interaction among the 3__d _ _ electrons of fe and mn by studying the magnitude and directions of the magnetic moments of fe and mn atoms in the fe@xmath1mnsi and fe@xmath1mnal compounds . the full - potential linearized augmented - plane wave methods have been employed to study the electronic and magnetic properties of fe@xmath0mnsi , fe@xmath0mnal and co@xmath0mnge . the ferromagnetic ( fm ) solutions without using on - site coulomb interaction _ u _ show the presence of half - metallic fm ground state in fe@xmath0mnsi however , in fe@xmath0mnal the ground state is found to be metallic . the total magnetic moment is contributed by mn atom with small contribution from fe atom in both cases . the electronic and magnetic properties of fe@xmath0mnsi and fe@xmath0mnal are affected significantly by _ u _ , whereas the almost negligible effect of _ u _ is found in co@xmath1mnge . the magnetic moment of fe atom in fe - based compounds is found to increase with _ u _ and for mn atom its value decreases . in fe@xmath1mnsi the fe and mn moments are coupled ferromagnetically for all values of u , whereas in fe@xmath1mnal they coupled antiferromagnetically below u = 2 ev and ferromagnetically above it . the study of electronic structures show that in fe@xmath0mnsi and co@xmath1mnge the ground state remains half - metallic ferromagnetic for all values of _ u _ , whereas in fe@xmath0mnal compound the ground state become semiconducting after _ u _ = 2 ev . the ferromagnetic coupling between fe and mn moments appears to be responsible for this in fe@xmath0mnal compound . r. a. degroot , f. m. mueller p. g. van engen and k. h. j. buschow , phys . lett . * 50 * , 2024 ( 1983 ) . k. e. h. m. hanseen and p. e. mijnarends , phys . b * 34 * , 5009 ( 1986 ) . m. m. kirillova , a. a. makhnev , e. i. shreder , v. p. dyakina and n. b. gorina , phys . status solidi b * 187 * , 231 ( 1995 ) . i. galanakis , p. h. dederichs and n. papanikolaou , phys . b * 66 * , 134428 ( 2002 ) . j. k@xmath18bler , a. r. williams and c. b. sommers , phys . b * 28 * , 1745 ( 1983 ) . s. ishida , s. fujii , s. kashiwagi and s. asano , j. phys . soc . jpn . * 64 * , 2152 ( 1995 ) . s. fujii , s. ishida and s. asano , j. phys . * 64 * , 185 ( 1995 ) . s. picozzi , a. continenza and a. j. freeman , phys . b * 66 * , 094421 ( 2002 ) . j. barth , g. h. fecher , b. balke , s. ouardi , t. graf , c. felser , a. shkabko , a. weidenkaff , p. klaer , h. j. elmers , h. yoshikawa , s. ueda and k. kobayashi , phys . b * 81 * , 064404 ( 2010 ) . a. candan , g. u@xmath19ur , z. charifi , h. baaziz and m. r. ellialt@xmath20o@xmath19lu , j. alloys compd . * 560 * , 215 ( 2013 ) . p. j. webster , j. phys . solids * 32 * , 1221 ( 1971 ) . k. r. a. ziebeck and p. j. webster , philos . mag . * 34 * , 973 ( 1976 ) . m. kawakami , physica b * 186 - 188 * , 1037 ( 1993 ) . s. fujii , s. ishida and s. asano , j. phys . jpn . * 63 * , 1881 ( 1994 ) . b. hamad and q. m. hu , phys . status solidi b * 248 * , 2893 ( 2011 ) . h. c. kandpal , g. h. fecher and c. felser , phys . b * 73 * , 094422 ( 2006 ) . s. wurmehl , g. h. fecher , h. c. kandpal , v. ksenofontov , c. felser , h. -j . lin and j. morais phys . b * 72 * , 184434 ( 2005 ) . m.t . czy@xmath21yk and g. a. sawatzky phys . b * 49 * , 14211 ( 1994 ) . d. p. rai , sandeep , m. p. ghimire and r .k . thapa , bull.mat.sc . * 34 * , 1219 ( 2011 ) . d. p. rai and r .k . thapa , j. alloys and comp . * 542 * , 257 ( 2012 ) . s. ishida , d. nagatomo , s. fujii and s. asano , mater . 49 * , 114 ( 2008 ) . p. blaha , k. schwarz , g. k. h. madsen , d. kvasnicka , and j. luitz , wien2k , an augmented plane wave plus local orbitals program for calculating crystal properties ( vienna university of technology , vienna , 2001 ) . j. p. perdew and y. wang , phys . b * 45 * , 13244 ( 1992 ) . j. brown , k. u. neumann , p. j. webster and k. r. a. ziebeck , j. phys . : condens . matter * 12 * , 1827 ( 2000 ) . h. c. kandpal , g. h. fecher and c. felser j. phys . * 40 * , 1507 ( 2007 ) . b. hamad , j. khalifeh , i. a. aljarayesh , c. demangeat , h. -b . luo and q. -m . hu , j. appl . phys . * 107 * , 093911 ( 2010 ) . i. galanakis , p. h. dederichs and n. papanikolaou , phys . b * 66 * , 174429 ( 2002 ) . k. ueda , k. hamaya , k. yamamoto , y. ando , t. sadoh , y. maeda and m. miyao , appl . . lett . * 93 * , 112108 ( 2008 ) . l. hongzhi , z. zhiyong , m. li , x. shifeng , l. heyan , q. jingping , l. yangxian and w. guangheng , j. phys . d * 40 * , 7121 ( 2007 ) . s. plogmann , t. schlath@xmath22lter , j. braun , m. neumann , yu . m. yarmoshenko , m. v. yablonskikh , e. i. shreder , e. z. kurmaev , a. wrona and a. @xmath23 lebarski , phys . b * 60 * , 6428 ( 1990 ) . z. liu , x. ma , f. meng and g. wu , j. alloys compd . * 509 * , 3219 ( 2011 ) . + & & & + [ 0.5ex ] compound&_u _ ( ev ) & 3__d__@xmath24 & 3__d__@xmath25 & m(@xmath26 ) & 3__d__@xmath24 & 3__d__@xmath25 & m(@xmath26 ) + + & 1&3.21&2.92&0.29&3.67&1.26&2.41 + & 2&3.25&2.89&0.36&3.62&1.32&2.30 + fe@xmath1mnsi&3&3.32&2.82&0.50&3.49&1.44&2.05 + & 4&3.39&2.74&0.65&3.34&1.59&1.75 + & 5&3.50&2.63&0.87&3.11&1.82&1.29 + [ 1 ex ] & 1&2.97&3.10&-0.14&3.55&1.31&2.24 + & 2&3.03&3.03&0.00&3.41&1.44&1.97 + fe@xmath1mnal&3&3.11&2.94&0.17&3.24&1.60&1.64 + & 4&3.20&2.84&0.36&3.03&1.81&1.22 + & 5&3.29&2.74&0.55&2.80&2.03&0.77 + [ 1 ex ] & 1&4.16&3.09&1.07&3.91&1.11&2.80 + & 2&4.16&3.08&1.08&3.89&1.13&2.76 + co@xmath1mnge&3&4.17&3.06&1.11&3.88&1.15&2.73 + & 4&4.17&3.05&1.12&3.85&1.17&2.68 + & 5&4.18&3.03&1.15&3.82&1.21&2.61 + [ 1 ex ]
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the electronic band structures , density of states plots and magnetic moments of fe@xmath0mnsi , fe@xmath0mnal , and co@xmath0mnge are studied by using the first principles calculations .
the fm solutions using lsda without _ u _ show the presence of half - metallic ferromagnetic ( hfm ) ground state in fe@xmath0mnsi , whereas the ground state of fe@xmath0mnal is found to be metallic . in both compounds
the maximum contribution to the total magnetic moment is from the mn atom , while the fe atom contributes very less .
the electronic structures and magnetic moments of fe - based compounds affected significantly by _ u _ , whereas its effect is very less on co@xmath1mnge .
the magnetic moment of fe atom in fe@xmath0mnsi ( fe@xmath0mnal ) , increased by @xmath2 70 % ( @xmath2 75 % ) and in mn atom it decreases by @xmath2 50 % ( @xmath2 70 % ) when the value of _ u _ is increased from 1 to 5 ev .
the hund s like exchange interactions are increasing in fe atom while decreasing in mn atom with increase in _
u_. the fe and mn moments are ferromagnetically coupled in fe@xmath0mnsi for all values of _ u _ , whereas in fe@xmath0mnal they coupled antiferromagnetically below _
u _ = 2 ev and ferromagnetically above it . above _ u _ = 2 ev the metallic ground state of fe@xmath0mnal changes to semiconducting ground state and the ferromagnetic coupling between fe and mn atoms appears to be responsible for this .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
recently , babichev and fabbri @xcite have shown that the massive linearized equation around the schwarzschild black hole in both de rham , gabadadze , and tolley ( drgt ) theory @xcite and its bigravity extension @xcite gives rise to an instability of @xmath0-mode ( spherically symmetric mode ) with @xmath1 the spheroidal harmonic index . this was done by comparing it with the four - dimensional linearized equation around the five - dimensional black string where the gregory - laflamme ( gl ) instability was found @xcite . it turned out that the bimetric black hole is unstable provided a mass of @xmath2 satisfies a bound of @xmath3 with @xmath4 the horizon radius in the metric function @xmath5 . we note that the limit of @xmath6 recovers the black hole in the drgt theory . the black hole in the drgt theory is also unstable because @xmath7 satisfies a bound of @xmath8 . in addition , the authors @xcite have confirmed this result by considering the schwarzschild - de sitter black hole and extending the @xmath9 mode to generic modes of @xmath10 . these results may indicate an important fact that the static black holes do not exist in massive gravity theory . on the other hand , whitt @xcite has insisted thirty years ago that provided both massive spin-0 and spin-2 gravitons are non - tachyonic , the schwarzschild black hole is classically stable in a massive theory of fourth - order gravity when he uses the linearized - ricci tensor equation . in this case , one does not worry about the ghost instability arising from the fourth - order gravity theory because the linearized - ricci tensor satisfies a second - order tensor equation . recently , the author has revisited this stability issue . as expected , it was shown that the black hole in fourth - order gravity with @xmath11 is unstable provided the graviton mass of @xmath12 satisfies a bound of @xmath13 @xcite . this was performed by comparing the linearized - ricci tensor equation with the four - dimensional metric perturbation equation around the five - dimensional black string @xcite . in this work , we wish to reexamine the classical stability of schwarzschild - ads ( sads ) black hole in einstein - weyl gravity which was known to be stable against the metric perturbation @xcite . by contrast , it is shown that solving both the linearized - einstein tensor equation and the metric perturbation equation exhibit unstable modes featuring the gl instability of five - dimensional ads black string @xcite . it confirms that the gl instability of the black hole in the einstein - weyl gravity is due to the massiveness but not a feature of fourth - order gravity giving ghost states . taking into account the number of degrees of freedom ( dof ) , it is helpful to show why the sads black hole is physically stable in the einstein gravity @xcite , whereas the sads black hole is unstable in the einstein - weyl gravity . the number of dof of the metric perturbation is 2 in the einstein gravity , while the number of dof is 5 in the einstein - weyl gravity . the @xmath14-mode analysis of the massive graviton with @xmath15 dof shows the gl instability . the @xmath14-mode analysis is relevant to the massive graviton in the einstein - weyl gravity but not to the massless graviton in the einstein gravity . we start with the fourth - order gravity in ads@xmath16 spacetimes @xmath17 \label{action}\end{aligned}\ ] ] with two arbitrary parameters @xmath18 and @xmath19 . although this theory is renormalizable in minkowski spacetimes @xcite , the massive spin-2 graviton suffers from having ghosts . a massive spin-0 graviton is decoupled for the choice of @xmath11 , which leads to a critical gravity with @xmath20 @xcite . for a higher dimensional critical gravity , see a reference of @xcite . the einstein - weyl gravity is defined under the condition of @xmath11 as @xmath21 \label{ewaction}\end{aligned}\ ] ] with @xmath22 here the last of gauss - bonnet term could be neglected because it does not contribute to equation of motion . from ( [ action ] ) , the einstein equation is derived to be @xmath23 where the einstein tensor is given by @xmath24 and @xmath25 takes the form @xmath26 it is wellknown that eq.([equa1 ] ) provides the sads black hole solution @xcite @xmath27 with the metric function @xmath28 here @xmath29 denotes the curvature radius of ads@xmath16 spacetimes . we note that a mass parameter of @xmath30 is not the horizon radius @xmath31 which is obtained as a solution to @xmath32 . hereafter we denote the background quantities with the `` overbar '' . in this case , the background ricci tensor is given by @xmath33 it is easy to show that the sads black hole solution ( [ sch ] ) to the einstein equation of @xmath34 is also the solution to the einstein - weyl gravity when one substitutes ( [ beeq ] ) together with @xmath35 into ( [ equa2 ] ) . to perform the stability analysis , we usually introduce the metric perturbation around the sads black hole @xmath36 then , the linearized einstein equation takes the form @xmath37\delta g _ { \mu\nu}&+&\alpha\big[\bar{\nabla}^2\delta g_{\mu\nu}+2\bar{r}_{\rho\mu\sigma\nu}\delta g^{\rho\sigma}-\frac{2\lambda}{3 } \delta r \bar{g}_{\mu\nu}\big ] \nonumber \\ & + & ( \alpha+2\beta)\big[-\bar{\nabla}_\mu\bar{\nabla}_\nu+\bar{g}_{\mu\nu}\bar{\nabla}^2 + \lambda \bar{g}_{\mu\nu}\big ] \delta r=0,\end{aligned}\ ] ] where the linearized einstein tensor , ricci tensor , and ricci scalar are given by @xmath38 with @xmath39 . it is very difficult to solve the linearized equation ( [ lin - eq ] ) directly because it is a coupled second - order equation for @xmath40 and @xmath41 . thus , we attempt to decouple @xmath41 from ( [ lin - eq ] ) . for this purpose , we take the trace of ( [ lin - eq ] ) which leads to @xmath42\delta r=0.\ ] ] it implies that the dalembertian operator could be removed if one chooses @xmath43 in this case , the linearized ricci scalar is constrained to vanish @xmath44 plugging @xmath45 into eq . ( [ lin - eq ] ) leads to the equation for the linearized einstein tensor solely @xmath46 this shows clearly why we consider the einstein - weyl gravity ( [ ewaction ] ) with @xmath11 instead of the fourth - order gravity action ( [ action ] ) with arbitrary @xmath18 and @xmath19 . before we proceed , we wish to mention that the metric perturbation is not suitable for analyzing the sads black hole stability in the einstein - weyl gravity . for simplicity , we consider the ads@xmath16 spacetimes background whose curvature tensor takes a simple form @xmath47 after choosing the transverse - traceless gauge ( ttg ) @xmath48 eq . ( [ slin - eq ] ) leads to a fourth - order differential equation @xcite @xmath49 which may imply a massless spin-2 graviton equation @xmath50 and a massive spin-2 graviton equation @xmath51 here the mass squared is given by @xmath52 in ads@xmath16 spacetimes , the stability condition is given by the absence of tachyonic instability ( @xmath53 ) @xcite , which implies that @xmath19 must satisfy @xmath54 in the massless case of @xmath55 , eq.([four - eq ] ) leads to that for a critical gravity @xmath56 however , it was shown that a general mode of @xmath57 suffers from negative norm states unless one truncates out the log - mode by imposing appropriate ads@xmath16 boundary conditions . up to now , there is no consistent truncation mechanism to eliminate the log - mode . we recall that this problem arises because we work with the fourth - order derivative equation ( [ four - eq ] ) for the metric perturbation . in this work , we do not consider a new unitary gravity for @xmath58 @xcite because it has still a non - unitarity problem like the critical gravity . going back to the sads black hole ( [ sch ] ) , we rewrite eq . ( [ slin - eq ] ) as a second - order equation for the linearized einstein tensor @xmath59 if one introduces the lichnerowicz operator @xmath60 the corresponding equation could be rewritten as @xmath61 taking into account the ttg ( [ ttg ] ) , the linearized einstein tensor reduces to @xmath62 then , one can rewrite ( [ g - eq ] ) as a fourth - order equation for @xmath63 @xmath64 which is similar to ( [ four - eq ] ) in ads@xmath16 spacetimes . ( [ four - eqq ] ) may imply a linearized massless equation around the sads black hole @xcite @xmath65 and a linearized massive equation for @xmath66 @xmath67 at this stage , we wish to point out the difference between ( [ se1m - eq ] ) and ( [ lem2-eq ] ) . the former equation is a second - order equation for the linearized einstein tensor , whereas the latter is a suggesting second - order equation from the fourth - order equation ( [ four - eqq ] ) for the metric perturbation . it is known that the introduction of fourth - order derivative terms gives rise to ghost - like massive graviton @xcite , which may imply an instability of a black hole even if a black hole solution exists . hence , even though ( [ se1-eq])[([se2-eq ] ) ] were frequently used as a linearized massless [ massive ] equation around the ads@xmath16 spacetimes @xcite , their validity is not yet proved because they are free from ghost states . in order to check whether ( [ se2-eq])[([lem2-eq ] ) ] are reliable or not , we note that our action ( [ ewaction ] ) reveals ghosts when we perform the metric perturbation @xmath63 around the minkowski spacetimes with @xmath68 @xcite . ( [ four - eq])[([four - eqq ] ) ] take the form in the minkowski background @xcite @xmath69 with an external source @xmath70 . replacing @xmath71 by @xmath72 , the metric perturbation is given by @xmath73 which the last term spoils the unitarity . hence , splitting ( [ four - eq])[([four - eqq ] ) ] into two second - order equations ( [ se1-eq])[([lem1-eq ] ) ] and ( [ se2-eq])[([lem2-eq ] ) ] is dangerous because the ` @xmath74 ' sign in the front of ( [ se2-eq])[([lem2-eq ] ) ] is missed . as is shown in ( [ h - s ] ) , the ghost arises from this sign when one performs the partial fraction . to this end , the authors in @xcite have found the two on - shell energies on the ads@xmath16 spacetime background @xmath75 when they compute each hamiltonian which satisfies ( [ se1-eq ] ) and ( [ se2-eq ] ) , respectively . thus , for @xmath76 , ghost - like massive excitation is not avoidable . in order for the theory to be free from ghosts , one needs to choose @xmath77 ) which corresponds to the critical gravity where a massive graviton becomes a massless graviton . because of a missing of ` @xmath74 ' sign , we may insist that eq.([se2-eq])[([lem2-eq ] ) ] by itself do not represent a correct linearized equation for studying the stability of the sads black hole in the einstein - weyl gravity . however , the overall ` @xmath74 ' sign in ( [ se2-eq])[([lem2-eq ] ) ] does not make any difference unless an external source is introduced in the right - hand side as eq.([source ] ) does indicate . therefore , the fourth - order gravity does not automatically imply the instability of the black hole even if one uses ( [ lem2-eq ] ) . hopefully , if one uses ( [ se1m - eq ] ) instead of ( [ lem2-eq ] ) , one is free from the ghost issue because ( [ se1m - eq ] ) is a genuine second - order equation . in einstein gravity , the linearized equation around the schwarzschild black hole is given by @xmath78 with @xmath79 ( [ ricc - t ] ) . then , the metric perturbation @xmath63 is classified depending on the transformation properties under parity , namely odd and even . using the regge - wheeler @xcite and zerilli gauge @xcite , one obtains two distinct perturbations : odd with 2 dof and even with 4 dof . this implies that one starts with 6 dof after choosing the regge - wheleer gauge , leading to 2 dof ( 1 for odd and 1 for even ) for a massless spin-2 graviton propagation . the schwarzschild black hole is stable against the metric perturbation @xcite . performing the stability analysis of the sads black hole in einstein gravity , one has to use the linearized equation @xmath80 which was tuned out to be stable by following the regge - wheeler prescription @xcite . in these cases , the @xmath0-mode analysis is not necessary to show the stability of the schwarzschild and sads black holes because the massless spin-2 graviton requires modes with @xmath81 . however , the @xmath14-mode analysis is responsible for detecting an instability of a massive graviton propagating on the sads black hole in einstein - weyl gravity . the even - parity metric perturbation is designed for a single @xmath14-mode analysis in the massive gravity and whose form is given by @xmath82 and @xmath83 as @xmath84 even though one starts with 4 dof , they are related to each other when one uses the ttg ( [ ttg ] ) . hence , we expect to have one decoupled equation for @xmath85 . for a massive gravity theory in the minkowski background , there is correspondence between linearized ricci tensor @xmath86 and ricci spinor @xmath87 when using the newman - penrose formalism @xcite . here the null real tetrad is necessary to specify polarization modes of a massive graviton , as the massive gravity requires null complex tetrad to specify six polarization modes @xcite . this implies that in fourth - order gravity theory , one may take the linearized ricci tensor @xmath86 ( [ ricc - t ] ) with 6 dof as physical observables @xcite . requiring @xmath88 further , the dof of @xmath86 is five which is the same dof for the metric perturbation @xmath63 in massive gravity theory . at this stage , we stress again that ( [ se1m - eq ] ) is considered as the second - order equation with respect to @xmath40 , but not the fourth - order equation ( [ four - eq ] ) for @xmath63 . hence , we propose @xmath40 as physical observables propagating on the sads black hole background instead of @xmath89 on the schwarzschild black hole background . also , we have the tracelessness of @xmath90 and the transversality of @xmath91 from the contracted bianchi identity . then , @xmath40 describe exactly five dof propagating on the sads black hole background without ghosts . since eq.([lem2-eq ] ) is the same linearized equation for four - dimensional metric perturbation around five - dimensional black string , we follow the gl instability analysis in ads@xmath16 spacetimes @xcite . eliminating all but @xmath85 , eq.([lem2-eq ] ) reduces to a second - order equation for @xmath85 @xmath92 where @xmath93 and @xmath94 were given by ( 20 ) in @xcite . we stress again that the @xmath14-mode perturbation is described by single dof but not 5 dof . the authors in @xcite have solved ( [ second - eq ] ) numerically and found unstable modes for @xmath95 . 1 that is generated from the numerical analysis . we note that @xmath96 correspond to @xmath97 , respectively . from the observation of fig . 1 with @xmath98 , we find unstable modes for @xmath99 with the mass @xmath100 similarly , we find eq.([lem2-eq ] ) when we replace @xmath101 by @xmath63 in ( [ se1m - eq ] ) . hence , a relevant equation for @xmath102 takes the same form @xmath103 which shows the same unstable modes appeared in fig . this implies that even if one uses ( [ lem2-eq ] ) as a linearized massive equation @xcite , our conclusion remains unchanged because ( [ lem2-eq ] ) and ( [ se1m - eq ] ) are the same equation for different tensors . consequently , the instability arises from the massiveness ( @xmath104 ) but not from a feature of the fourth - order equation which gives the ` @xmath74 ' sign ( ghost = negative norm state ) when one splits it into two second - order equations . this implies that static black holes in massive gravity theory do not exist and/or they do not form in the gravitational collapse . if a black hole was formed in the massive gravity theory , one may ask what is the end - state of such instability . for unstable black strings of sads@xmath105r with translational symmetry , there are some evidences that break - up occurs @xcite . however , we consider a spherically symmetric black hole in four dimensions . a possible end - state may be a spherically symmetric black hole endowed with a graviton cloud @xcite finally , in the case of @xmath106 , the theory becomes massless and is stable against the einstein tensor perturbation . however , this corresponds precisely to the critical gravity when one uses the metric perturbation . here we have a non - unitarity issue due to the log - mode like ( [ crit - eq ] ) . also , one finds that @xmath107(adt mass)=0 and @xmath108(wald s entropy)=0 at the critical point , leading to a vacuum but not a black hole @xcite . e. babichev and a. fabbri , class . grav . * 30 * , 152001 ( 2013 ) [ arxiv:1304.5992 [ gr - qc ] ] . c. de rham and g. gabadadze , phys . d * 82 * , 044020 ( 2010 ) [ arxiv:1007.0443 [ hep - th ] ] . s. f. hassan and r. a. rosen , jhep * 1202 * , 126 ( 2012 ) [ arxiv:1109.3515 [ hep - th ] ] . v. cardoso and j. p. s. lemos , phys . d * 64 * , 084017 ( 2001 ) [ gr - qc/0105103 ] . a. ishibashi and h. kodama , prog . phys . * 110 * , 901 ( 2003 ) [ hep - th/0305185 ] . k. s. stelle , phys . d * 16 * , 953 ( 1977 ) . s. deser , h. liu , h. lu , c. n. pope , t. c. sisman and b. tekin , phys . d * 83 * , 061502 ( 2011 ) [ arxiv:1101.4009 [ hep - th ] ] . s. -j . hyun , w. -j . jang , j. -h . jeong and s. -h . yi , jhep * 1201 * , 054 ( 2012 ) [ arxiv:1111.1175 [ hep - th ] ] .
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we investigate the classical stability of schwarzschild - ads black hole in a massive gravity theory of the einstein - weyl gravity .
it turns out that the linearized einstein tensor perturbations exhibit unstable modes featuring the gregory - laflamme instability of five - dimensional ads black string , in contrast to the stable schwarzschild - ads black hole in einstein gravity .
we point out that the instability of the black hole in the einstein - weyl gravity arises from the massiveness but not a feature of fourth - order derivative theory giving ghost states .
+ yun soo myung + institute of basic sciences and department of computer simulation , inje university gimhae 621 - 749 , korea + pacs numbers : 04.70.bw , 04.50.kd
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You are an expert at summarizing long articles. Proceed to summarize the following text:
a great deal of attention has been devoted to non - contact friction between nanostructures , including , for example , the frictional drag force between two - dimensional quantum wells @xcite , and the friction force between an atomic force microscope tip and a substrate @xcite . in non - contact friction the bodies are separated by a potential barrier thick enough to prevent electrons or other particles with a finite rest mass from tunneling across it , but allowing interaction via the long - range electromagnetic field , which is always present in the gap between bodies . the presence of inhomogeneous tip - sample electric fields is difficult to avoid , even under the best experimental conditions @xcite . for example , even if both the tip and the sample were metallic single crystals , the tip would still have corners present and more than one crystallographic plane exposed . the presence of atomic steps , adsorbates , and other defects will also contribute to the inhomogeneous electric field . the electric field can be easily changed by applying a voltage between the tip and the sample . the electromagnetic field can also be created by the fluctuating current density , due to thermal and quantum fluctuations inside the solids . this fluctuating electromagnetic field is always present close to the surface of any body , and consist partly of traveling waves and partly of evanescent waves which decay exponentially with the distance away from the surface of the body . the fluctuating electromagnetic field originating from the fluctuating current density inside the bodies gives rise to the well - known long - range attractive van der waals interaction between two bodies @xcite . if the bodies are in relative motion , the same fluctuating electromagnetic field will give rise to a friction which is frequently named as the van der waals friction . van der waals friction can be considered as mediated by photon exchange between the bodies : one body emit a photon , and the other absorbs it , thus transferring momentum between the bodies , resulting in a friction force . at large distances between the bodies , the main contribution to friction comes from photon exchange , corresponding to the propagating electromagnetic waves . however this contribution is very small because the photons corresponding to propagating waves carry very small momentum , no larger than @xmath0 . the photons , corresponding to the evanescent electromagnetic waves , carry the momentum @xmath1 . thus for distances @xmath2 between two bodies smaller characteristic distance @xmath3 , which depends on temperature ( at room temperature @xmath4 ) , the main contribution to friction comes from the evanescent electromagnetic field . in analogy with electron tunneling , this mechanism of momentum transfer can be considered as associated with the photon tunneling . although the dissipation of energy connected with the non - contact friction always is of electromagnetic origin , the detailed mechanism is not totally clear , since there are several different mechanisms of energy dissipation connected with the electromagnetic interaction between bodies . first , the electromagnetic field from one body will penetrate into the other body , and induce an electric current . in this case friction is due to ohmic losses inside the bodies . the fluctuating electromagnetic field can also excite the vibrations of the adsorbates or other surface localized modes , e.g. surface plasmons and polaritons . in this case friction is due to energy relaxation of the surface modes . another contribution to friction from the electromagnetic field is associated with the time - dependent stress acting on the surface of the bodies . this stress can excite acoustic waves , or induce time - dependent deformations which may result in a temperature gradient . it can also induce motion of defects either in the bulk , or on the surface of the bodies . the contribution to friction due to non - adiabatic heat flow , or motion of defects , is usually denoted as internal friction . it is very worthwhile to get a better understanding of different mechanisms of non - contact friction because of it practical importance for ultrasensitive force detection experiments . this is because the ability to detect small forces is inextricably linked to friction via the fluctuation - dissipation theorem . for example , the detection of single spins by magnetic resonance force microscopy , which has been proposed for three - dimensional atomic imaging @xcite and quantum computation @xcite , will require force fluctuations to be reduced to unprecedented levels . in addition , the search for quantum gravitation effects at short length scale @xcite and future measurements of the dynamical casimir forces @xcite may eventually be limited by non - contact friction effects . recently gotsmann and fuchs @xcite observed long - range non - contact friction between an aluminum tip and a gold ( 111 ) surface . the friction force @xmath5 acting on the tip is proportional to the velocity @xmath6 , @xmath7 . for motion of the tip normal to the surface the friction coefficient @xmath8 , where @xmath2 is the tip - sample spacing and @xmath9 @xcite . later stipe _ _ et.al.__@xcite observed non - contact friction effect between a gold surface and a gold - coated cantilever as a function of tip - sample spacing @xmath2 , the temperature @xmath10 , and the bias voltage @xmath11 . for vibration of the tip parallel to the surface they found @xmath12 , where @xmath13 and @xmath14 at 295for the spacing @xmath15 100they found @xmath16 which is @xmath17500 times smaller that reported in ref . @xcite at the same distance using a parallel cantilever configuration . in a recent letter , dorofeev _ et.al . _ @xcite claim that a the non - contact friction effect observed in @xcite is due to ohmic losses mediated by the fluctuating electromagnetic field . this result is controversial , however , since the van der waals friction has been shown @xcite to be many orders of magnitude smaller than the friction observed by dorofeev _ et.al . _ presently , the origin of the difference in magnitude and distance dependence of the long - range non - contact friction effect observed in @xcite and @xcite is not well understood . in order to improve the basic understanding of non - contact friction , we present new results for van der waals friction . in @xcite we developed a theory of van der waals friction for surfaces in parallel relative motion . here we generalize the theory to include also the case when the surfaces are in normal relative motion , and we show that there is drastic difference between these two cases . thus , for normal relative motion of clean good conductor surfaces , the friction is many orders of magnitude larger than for parallel relative motion , but still smaller than observed experimentally . another enhancement mechanism of the non - contact friction can be connected with resonant photon tunneling between states localized on the different surfaces . recently it was discovered that resonant photon tunneling between surface plasmon modes give rise to extraordinary enhancement of the optical transmission through sub - wavelength hole arrays @xcite . the same surface modes enhancement can be expected for van der waals friction if the frequency of these modes is sufficiently low to be excited by thermal radiation . at room temperature only the modes with frequencies below @xmath18 can be excited . for normal metals surface plasmons have much too high frequencies ; at thermal frequencies the dielectric function of normal metals becomes nearly purely imaginary , which exclude surface plasmon enhancement of the van der waals friction for good conductors . however surface plasmons for semiconductors are characterized by much smaller frequencies and damping constants , and they can give an important contribution to van der waals friction . other surface modes which can be excited by thermal radiation are adsorbate vibrational modes . especially for parallel vibrations these modes may have very low frequencies . all information about the long - range electromagnetic interaction between two non - contacting bodies is , in principle , contained in the reflection factors of the electromagnetic field . at present time very little is known about the reflection factors for large wave vectors and for extremely small frequencies . in the calculations of the reflection factors one must take into account the non - local response of the electron gas on the external electromagnetic field . there are two correlation length which determine this nonlocal response . first is the skin depth , which determines the long - range length scale of the nonlocality in the volume , and the second is the screening length . the latter length scale determines the short range correlation length for non - local response in the surface region . in our previous calculations of the van der waals friction @xcite we mostly considered good conductors . in this case it was shown that the important contribution comes from the non - local optic effects in the surface region . however it was shown that the van der waals friction becomes much larger for high resistivity material , for which the volume contribution from non - local effects is also important . it is easy to see that within local optic approximation the van der waals friction diverge when the conductivity of materials tend to zero . this means that the local optic approximation breaks down for high - resistivity materials . this situation is completely different from the heat transfer between bodies via photon tunneling @xcite , where the heat flux is maximal at conductivities corresponding to semi - metals . in order to clarify the situation we study the dependence of the van der waals friction on the dielectric properties of the materials within the non - local dielectric approach , which was proposed some years ago for the investigation of the anomalous skin effects @xcite . we consider two semi - infinite metals * 1 * and * 2 * having parallel flat surfaces . we introduce a coordinate system with @xmath19 plane in the surface of body * 1 * , and the @xmath20 axis along the upward normal . the surface of body * 2 * is located at @xmath21 , performing small amplitude vibrations along the @xmath22 axes with displacement coordinate @xmath23 since the system is translation invariant in the @xmath24 plane , the electromagnetic field can be represented by the fourier integral @xmath25 where @xmath26 and @xmath27 are the electric and magnetic induction field , and @xmath28 is the two - dimensional wave vector in @xmath29 plane . after the fourier transformation it is convenient to choose the coordinate axis in the ( x , y ) plane along the vectors @xmath28 and @xmath30 $ ] . in the vacuum gap between the bodies the electric field @xmath31 , and the magnetic induction field @xmath32 , can , to the linear order in the vibrational coordinate , be written in the form @xmath33 @xmath34e^{-ipz}+ [ \mathbf{k}^{+}\times\mathbf{w}_{0}]e^{ipz}\right ) \right . \nonumber \\ \left . + \frac{1}{\omega + \omega _ 0}\left ( [ \mathbf{k}^{-}_+\times\mathbf{v}_{1}]e^{-ip^+z}+ [ \mathbf{k}^{+}_+ \times\mathbf{w}_{1}e^{ip^+z}]\right)e^{-i\omega _ 0 t } \right ] e^{-i\omega t } \label{two}\end{aligned}\ ] ] where @xmath35 at the surfaces of the bodies the amplitude of the outgoing electromagnetic wave must be equal to the amplitude of the reflected wave plus the amplitude of the radiated wave . it is convenient to decompose the electromagnetic field into the @xmath36- and @xmath37 - polarized electromagnetic waves . for @xmath36-polarized electromagnetic waves the electric field is in the incident plane determined by the vectors @xmath28 and @xmath38 , and for @xmath37- polarized electromagnetic waves the electric field is normal to the incident thus the boundary conditions for the electromagnetic field at @xmath39 can be written in the form @xmath40 @xmath41 where @xmath42 is the reflection factor for surface * 1 * for @xmath43 - polarized electromagnetic field , and where @xmath44 are the components of the fluctuating electric field outside the surface * 1 * in the absence of the body * 2*. the boundary condition at the surface of the body * 2 * must be written in the reference frame where the body * 2 * is at rest . the electric field in this reference frame is determined by a lorentz transformation . performing a lorentz transformation of the electric field to linear order in @xmath45 gives @xmath46 } c \label{five}\ ] ] for the @xmath36-polarized electromagnetic waves the second term in ( [ five ] ) is of the order of magnitude @xmath47 relative to the first one and can be neglected for the most practical cases . however , for the @xmath37-polarized electromagnetic waves the second term is of the order of magnitude @xmath48 and can be of the same order of magnitude as the first term . in the rest frame of body * 2 * there is also mixture of @xmath37- and @xmath49 polarized electromagnetic waves . in @xcite it was was shown that this gives contribution of the order @xmath50 and thus can be neglected . after performing lorentz transformation to linear order in @xmath51 and @xmath52 we get @xmath53 @xmath54 @xmath55 the boundary conditions for the electromagnetic field at @xmath56 in the rest frame of body * 2 * can be written in the form @xmath57 @xmath58 @xmath59 where @xmath60 is the reflection factor for surface * 2 * for @xmath43 - polarized electromagnetic field , and where @xmath61 are the components of the fluctuating electric field outside the surface * 1 * in the absence of the body * 1*. from ( [ three],[four ] ) and ( [ six]-[eight ] ) we get @xmath62 @xmath63 @xmath64 @xmath65 @xmath66 where @xmath67 and @xmath68 . other components of the fluctuating electromagnetic field can be found from the transversality conditions @xmath69 the fundamental characteristic of the fluctuating electromagnetic field is the correlation function , determining the average product of components @xmath70 . accordingly to the general theory of the fluctuating electromagnetic field ( see for a example @xcite ) these correlation function are given by @xmath71 \label{corfs}\ ] ] @xmath72 \label{corfp}\ ] ] where @xmath73 denote statistical average over the random field , and where the bose - einstein factor @xmath74 we note that @xmath36 is real for @xmath75 ( propagating waves ) , and purely imaginary for @xmath76 ( evanescent waves ) . thus for @xmath75 and @xmath76 the correlation functions are determined by the first and the second terms in eqs . ( [ corfs])and ( [ corfp ] ) , respectively . the frictional stress @xmath77 which act on the surfaces of the two bodies can be obtained from @xmath78 component of the maxwell stress tensor @xmath79 , evaluated at @xmath39 : @xmath80 @xmath81 @xmath82 _ { z=0 } \label{3thirteen}\ ] ] to linear order in the vibrational coordinate @xmath83 and the frequency @xmath84 the stress acting on the surface * 1 * can be written in the form @xmath85 here the first term determines the conservative van der waals stress and the second term is the adiabatic change of the conservative van der waals stress during vibration . the last term determines the frictional stress with friction coefficient @xmath86 . for normal relative motion ( see appendix a ) we obtain the friction coefficient @xmath87 , where the contribution to the friction coefficient from the propagating electromagnetic waves is given by @xmath88 @xmath89 @xmath90\frac 1{\left|1-e^{2ipd}r_{1p}r_{2p}\right|^4 } + [ p\rightarrow s ] , \label{3seventeen}\ ] ] and where the contribution to the friction from the evanescent electromagnetic waves is given by @xmath91 @xmath92 @xmath93 \frac 1{\left|1-e^{-2kd}r_{1p}r_{2p}\right|^4 } + [ p\rightarrow s ] , \label{3eighteen}\ ] ] where @xmath94 . the symbol @xmath95 $ ] in eqs . ( [ 3seventeen ] ) and ( [ 3eighteen ] ) denotes the term which is obtained from the first one by replacement of the reflection factors @xmath96 for @xmath49 polarized waves , by the reflection factors @xmath97 for @xmath98 polarized waves . the friction coefficient for two flat surfaces in parallel relative motion was obtained by us before , @xcite and can be written as @xmath99 , where the contribution to the friction coefficient from the propagating electromagnetic waves is given by @xmath100 @xmath101 , \label{3nineteen}\ ] ] and where the contribution to the friction from the evanescent electromagnetic waves is given by @xmath102 @xmath103 . \label{3twenty}\ ] ] there is a principal difference between the friction coefficient for normal and parallel relative motion , related to the denominator in the formulas for the friction coefficient . the resonant condition corresponds to the case when the denominator of the integrand in eqs . ( [ 3seventeen]-[3twenty ] ) , which is due to multiple scattering of the evanescent electromagnetic waves from the opposite surfaces , is small . for two identical surfaces and @xmath104 , where @xmath105 and @xmath106 are the imaginary and real part , respectively , this corresponds to the resonant condition @xmath107 . at resonance the integrand in eqs . ( [ 3nineteen ] ) and ( [ 3twenty ] ) has a large factor @xmath108 , in sharp contrast to the case of parallel relative motion , where there is no such enhancement factor . the resonance condition can be fullfiled even for the case when @xmath109 because for evanescent electromagnetic waves there is no restriction on the magnitude of the real part or the modulus of @xmath110 . this open up the possibility of resonant denominators for @xmath111 . to estimate the friction coefficient @xmath112 for an atomic force microscope tip we can use an approximate formula @xcite @xmath113 where it is assumed that the tip has cylinder symmetry . here @xmath114 denotes the tip - surface distance as a function of the distance @xmath115 from the tip symmetry axis , and the friction coefficient @xmath116 is determined by the expressions for the flat surfaces . this scheme was proposed in @xcite for the calculation of the conservative van der waals interaction . the error of these scheme is not larger than 5 - 10% in practice in an atomic force microscopy experiment , and 25% in a worst case satiation @xcite . although this scheme was proposed for the conservative van der waals interaction , we assume that the same scheme is also valid for the calculation of the van der waals friction . we assume that the tip has a paraboloid shape given [ in cylindrical coordinates ( @xmath117 ) ] by the formula:@xmath118 , where @xmath2 is the distance between the tip and the flat surface , and where @xmath110 is the radius of curvature of the tip . in the case of the power dependence @xmath119 we get @xmath120 in a more general case one must use numerical integration . in the local optic approximation , where the dielectric function is assumed to depend only on the frequency @xmath121 , the reflection factors @xmath122 and @xmath123 for flat surfaces , covered by an adsorbate layer , are given by @xcite : @xmath124 } { p+s/\epsilon-4\pi in_aq[s\alpha_{\parallel}/\epsilon+q \alpha_{\perp } ] } , \label{3twentyone}\ ] ] @xmath125 where @xmath126 and where @xmath127 and @xmath128 are the polarizabilities of adsorbates in a direction parallel and normal to the surface , respectively . here @xmath129 is the bulk dielectric function and @xmath130 is the concentration of adsorbates . for clean surfaces @xmath131 , and in this case formulas ( [ 3twentytwo,3twentythree ] ) reduce to the well - known fresnel formula . at @xmath132 and @xmath133 , where @xmath134 is the electron mean free path , and where @xmath135 and @xmath136 are the fermi velocity and fermi wave number , respectively , the system will be characterized by non - local dielectric function @xmath137 . in this paper we use the non - local optic dielectric approach , proposed some years ago for the investigations of the optical properties of a semi - infinite electron gas @xcite . accordingly to @xcite , the reflection factor for @xmath36 - polarized electromagnetic field , incident on the flat surface , is determined by @xcite @xmath138 where the surface impedance @xmath139 is given by @xmath140 where @xmath141 is the finite- life- time generalization of the longitudinal lindhard dielectric function which accordingly to @xcite can be written as : @xmath142 @xmath143 @xmath144\ln \frac{z - u+1}{z - u-1}% + [ 1-(z+u)^2]\ln \frac{z+u+1}{z+u-1}\right ) , \label{rfive}\ ] ] where @xmath145 , @xmath146 is the plasma frequency , @xmath147 is the drude relaxation time , and where @xmath135 and @xmath136 are the fermi velocity and wave vector , respectively . for @xmath98 polarization the reflection factor is determined by @xmath148 where @xmath149 ^ 2\ln \frac{z - u^{\prime } + 1}{z - u^{\prime } -1}\right . \nonumber \\ & & \left . + [ 1-(z+u^{\prime } ) ^2]^2\ln \frac{z+u^{\prime } + 1}{z+u^{\prime } -1}% \right ) \label{rnine}\end{aligned}\ ] ] with @xmath150 . we show in sec.4 that the maximum of the electromagnetic friction is reached for small electron densities , where the electron gas becomes non - degenerate ( the electro gas is degenerate for @xmath151 and non - degenerate for @xmath152 , where @xmath153 is the fermi energy ) . for non - degenerate electron gas we use the following classical expressions for dielectric functions @xcite @xmath154 \label{rten } \\ \epsilon _ t(\omega , q ) & = & 1+\frac{\omega _ p^2}{\omega ( \omega + i\gamma ) } % f\left ( \frac{\omega + i\gamma } { \sqrt{2}qv_t}\right ) \label{releven}\end{aligned}\ ] ] where the function @xmath155 is defined by the integral @xmath156 and @xmath157 , where @xmath158 is the electron mass . by a well - conducting metal we mean one whose dielectric function @xmath159 ( @xmath77 is the conductivity ) has an absolute value much larger than unity . for good conductors at thermal frequencies @xmath160 and @xmath161 . thus an enhancement in friction is possible only for very small @xmath162 . it is convenient to write the friction coefficient for the two flat surfaces in the form @xmath163 taking into account that @xmath164 , from eq . ( [ 3eighteen ] ) for normal relative motion of clean surfaces within local optic approximation we get contribution to friction from evanescent @xmath36-and @xmath37-polarized electromagnetic waves @xmath165 ^ 2 \big[\big[(k^2+|s/\epsilon|^2)\mathrm{cosh}kd + 2k\big[\mathrm{im } ( s/\epsilon)\big]\mathrm{sinh}kd\big]^2\ ] ] @xmath166\frac{1}{\left|\big((s/\epsilon)^2-k^2\big ) \mathrm{sinh}kd + 2ik(s/\epsilon)\mathrm{cosh}kd\right|^4 } \label{4one}\ ] ] @xmath167 ^ 2 \big[\big[(k^2+|s|^2)\mathrm{cosh}kd + 2k\mathrm{im } s\mathrm{sinh}kd\big]^2\ ] ] @xmath168\frac{1}{\left|\big(s^2-k^2\big ) \mathrm{sinh}kd + 2iks\mathrm{cosh } kd\right|^4 } \label{4two}\ ] ] as @xmath169 , there is no singularity in @xmath170 , and for @xmath171 , for @xmath172 , than taking into account that in this limit @xmath173 and @xmath174 , we get @xmath175 @xmath176 where the surface impedance @xmath177 . for @xmath178 , @xmath179 becomes slowly dependent on @xmath2 : @xmath180 for @xmath181 we get @xmath182 for the propagating electromagnetic waves , taking into account that @xmath183 , we get @xmath184 @xmath185 for @xmath178 the contribution to friction from propagating wave is negligibly small in the comparison with the contribution from the evanescent waves . for @xmath181 as @xmath186 , the integral @xmath187 has no singularity , and we get for @xmath188 @xmath189 in addition , @xmath190 has singularities at the other zeroes of @xmath191 , i.e. , near the values @xmath192 ( @xmath193 is an integer ) . in the vicinity of @xmath194 , putting @xmath195 , we have @xmath196 and @xmath197 , @xmath198 @xmath199 the number @xmath158 of such contribution is obviously equal to the integer part of the quantity @xmath200 ( @xmath201 $ ] ) , so that all @xmath194 ( with the exception of @xmath202 ) make a summary contribution @xmath203 @xmath204 @xmath205 \label{4ten}\ ] ] in the integral @xmath187 , the contribution from the vicinity of the point @xmath194 is @xmath206 @xmath207 and consequently @xmath208 @xmath209 at @xmath210 , when we can assume @xmath211 , the @xmath98 and @xmath49 wave contribution are approximately equal , and for the total contribution from propagating electromagnetic waves in this limit we get @xmath212 the above formulas in this section were obtained from the general eqs . ( [ 3seventeen]-[3twenty ] ) assuming absence of spatial dispersion of the dielectric function . but these formulas contain only surface impedance @xmath213 that describe the ratio of the tangential components of the electric and magnetic fields on the boundary of the body . thus , the results in this section remain in force also in the presence of spatial dispersion , provided only that the surface impedance of the medium is small enough . we would have arrived at the same formulas also if we were to assume from the very beginning that the leontovich boundary condition @xmath214 is satisfied on the surface of the metal . at not too low temperatures , the impedances of metals are given by the formula for normal skin effect @xmath215 where @xmath77 is the conductivity . in the local optic approximation it is assuming that there is no dependence of @xmath77 on @xmath28 . in the wien region of frequencies it is also good approximation to neglect by the frequency dependence of @xmath77 . in this approximation using ( [ 4three ] ) for @xmath216 ( @xmath217 ) we get @xmath218 for the comparison the @xmath36-wave contribution for parallel relative motion for @xmath219 is given by ( @xcite ) @xmath220 it is interesting to note that for normal relative motion in contrast to the parallel relative motion practically for all @xmath221 the main contribution to friction comes from retardation effects because eqs . ( [ 4fourteen ] ) in contrast to eq . ( [ 4fifteen ] ) contains the light velocity . from eq . ( [ 4four ] ) we get @xmath37-wave contribution to friction for @xmath222 @xmath223 for parallel relative motion the @xmath37-wave contribution is in two times smaller . for @xmath224 , taking into account that eq . ( [ 4five ] ) is valid only for @xmath225 , we get @xmath226 from eq . ( [ 4twelve ] ) we get distance independent contribution to friction from propagating electromagnetic waves for @xmath227 @xmath228 figures 1 - 2 show the calculated contribution to the friction coefficient @xmath229 from evanescent electromagnetic waves for two semi - infinite solids , with parameters chosen to correspond to copper ( @xmath230 , @xmath231 at @xmath232 , for parallel ( fig.1 ) and normal ( fig.2 ) relative motion . results are shown separately for both the @xmath37- and @xmath36- wave contribution . the dashed line show the result when the local ( long - wavelength ) dielectric function @xmath233 is used , where @xmath234 in this case the integration in ( [ rtwo ] ) and ( [ rseven ] ) can be performed analytically and we get fresnel formulas . fig . 1 shows that the non - local optic effects become important for parallel relative motion for the @xmath49 wave contribution for sufficiently small separations ( @xmath235 ) . however , for the @xmath98 wave contribution for both parallel and normal motion the non - local optic effects are negligibly small for practically all separations . for normal relative motion for @xmath36-wave contribution the non - local optic effects are less important than for the parallel relative motion . in the present calculations we use the non - local dielectric approach which take into account the non - local optic effects on the length scale of the skin - depth . there are also the short range non - local optic effects coming from the non - local nature of the screening response near the surface . this gives the surface non - local contribution which we investigated in our previous publications @xcite . comparing our previous calculations with the present one , we find that for @xmath236 the volume contribution from the non - local effects is of the same importance as the surface contribution . for @xmath237 the main contribution to the friction coefficient @xmath229 comes from @xmath37-polarized waves . in particularly , at @xmath238 the @xmath37-wave contribution @xmath239 , so that with the surface area @xmath240m@xmath241 ( typical for probe scanning microscopy ) , the friction coefficient is @xmath242 . the @xmath98wave contribution is characterized by weak distance dependence for @xmath243 , and @xmath244 for @xmath245 . for good conductors like copper , even for very short distances , the main contribution to the friction coefficient comes from the @xmath98 polarized electromagnetic waves . this difference between @xmath49 and @xmath98 polarized waves results from screening effects : good conductors are good reflectors for @xmath49 polarized field , which implies that they are ineffective in the emission and absorption of evanescent @xmath49 polarized waves . however these screening effects are less important for @xmath98 polarized waves . as pointed out in @xcite , the @xmath36 -wave contribution increase and the @xmath98wave contribution decrease when the free electron density decrease . within the local optic approximation the force of friction diverges when one go to the limit of zero conductivity . this situation is different from the radiative heat transfer , where , even in the case of local optics , a maximum in the heat transfer occurs for conductivities corresponding to semi - metals . fig.3 shows the dependence of coefficient of friction on the electron density . when the electron density decreases there is transition from a degenerate electron gas to a non - degenerate electron gas at the density @xmath246 . at @xmath232 the transition density @xmath247 . for @xmath248 we use the ( non - local ) dielectric function appropriate for a degenerate electron gas , while for @xmath249 we use an expression corresponding to a non- degenerate electron gas . in the calculations we used the electron mean free path @xmath250 . at @xmath251 the maximum value @xmath252 is obtained for @xmath253 , corresponding to the dc conductivity @xmath254 we rewrite the denominator in eq . ( [ 3eighteen ] ) in the form @xmath255 ^ 2\ ] ] @xmath256 ^ 2 \label{5one}\ ] ] where @xmath106 and @xmath105 are real and imaginary part of @xmath110 , respectively ( @xmath257 ) . let us suppose that @xmath258 . in this case resonant conditions are determined by equation @xmath259 close to resonance we can write @xmath260\ ] ] @xmath261 \label{5three}\ ] ] where @xmath262 which leads to the following contribution to the friction coefficient : @xmath263 } \label{5four}\ ] ] the parameter @xmath264 in this expression defines the value of @xmath265 where the two poles approximation is valid . to proceed further let us make the following simplifications . close to the poles we can use approximation @xmath266 where @xmath267 is a constant . then from resonant condition ( [ 5two ] ) we get @xmath268 for the two poles approximation to be valid the difference @xmath269 . from this condition we get @xmath270 . for the short distances the parameter @xmath264 defines the value of @xmath265 where the solution of eq . ( [ 5two ] ) ceases to exit . for @xmath271 and @xmath272 from eq . ( [ 5four ] ) we get @xmath273 for the parallel relative motion using the same approximation we get @xmath274 interesting , the explicit @xmath2 dependence has dropped out of eq . ( [ 5eight ] ) . however , @xmath275 is still @xmath2 dependent , through the @xmath2 dependence of @xmath264 . for the small distances one can expect that @xmath264 is determined by the dielectric properties of the material and does not depend on @xmath2 . in this case the friction will be also distance independent . probably , the weak distance dependence observed in @xcite can be explained by the resonant photon tunneling . resonant photon tunneling enhancement of the van der waals friction is possible for two semiconductor surfaces which can support low - frequency surface plasmon modes . as an example we consider two clean surfaces of silicon carbide ( sic ) . the optical properties of this material can be described using an oscillator model @xcite @xmath276 with @xmath277 and @xmath278 the frequency of surface plasmons is determined by condition @xmath279 and from ( [ three ] ) we get @xmath280 . in fig.2 we plot the friction coefficient @xmath281 : note that the friction between the two semiconductor surfaces is several order of magnitude larger than between two clean good conductor surfaces . another enhancement mechanism is connected with resonant photon tunneling between adsorbate vibrational modes localized on different surfaces . as an example , let us consider ions with charge @xmath282 adsorbed on metal surfaces . the polarizability for ion vibration normal to the surface is given by @xmath283 where @xmath284 is the frequency of the normal adsorbate vibration , and @xmath285 is the damping constant . in eq . ( [ 3twentyone ] ) the contribution from parallel vibrations is reduced by the small factor @xmath286 . however , the contribution of parallel vibrations to the van der waals friction can nevertheless be important due to the indirect interaction of parallel adsorbate vibration with the electric field , via the metal conduction electron @xcite . thus , the small parallel component of the electric field will induce a strong electric current in the metal . the drag force between the electron flow and adsorbates can induce adsorbate vibrations parallel to the surface . this gives the polarizability : @xmath287 where @xmath193 is the conduction electron concentration . as an illustration , in fig.3 we show coefficient of friction for the two cu(001 ) surfaces covered by a low concentration of potassium atoms ( @xmath288 ) . in the @xmath289 integral in eqs.([3eighteen],[3twenty ] ) we used the cut off @xmath290 ( where @xmath291 is the inter - adsorbate distance ) because our microscopic approach is applicable only when the wave length of the electromagnetic field is larger than double average distance between the adsorbates . in comparison , the friction between two clean surface at separation @xmath292 is seven order of magnitude smaller . at @xmath292 the friction coefficient @xmath112 for an atomic force microscope tip with @xmath293 is @xmath294 ( @xmath295 , see fig.2 ) ; this is of the same order of magnitude as the observed friction @xcite . we have calculated the van der waals friction between two flat surfaces for normal relative motion and found a drastic difference in the comparison with parallel relative motion . this difference is connected with resonance produced by the multiple scattering of the electromagnetic waves from the opposite surfaces . in the case of sharp resonance it gives much larger contribution to friction in the case of normal relative motion than for parallel relative motion . we have studied in the detail the friction between two good conductors and have found that for normal relative motion even for very small distances the main contribution to friction comes from the retardation effects . the non - local optic effects are very important for @xmath36-wave contribution to friction for parallel relative and much less important for normal relative motion . for @xmath37-wave contribution the non - local optic effects are unimportant for both direction of relative motion . in the case of van der waals friction we have found that for distances between bodies @xmath296 , for good conductors with a high free electrons concentration , the main contribution to friction is associated with the @xmath37-polarized electromagnetic waves . for @xmath243 this mechanism gives a friction coefficient per unit area @xmath297kgs@xmath298m@xmath299 nearly independent of the distance @xmath300 while for @xmath245 the friction coefficient @xmath229 depends on distance as @xmath301 for an atomic force microscope tip with the near substrate area @xmath302m@xmath303 we got the friction coefficient @xmath304kgs @xmath298 for @xmath243 . when the concentration of electrons decreases , the @xmath98contribution to the friction decreases while the @xmath49 contribution increases . at @xmath251 and with the electron lifetime @xmath305s , the @xmath49 contribution reaches maximum @xmath306kgs@xmath298m@xmath307 at the electron concentration @xmath308m@xmath309 which corresponds to the conductivity @xmath310(@xmath311m)@xmath312 we have shown that the van der waals friction can be enhanced by several orders of magnitude in the case of resonant photon tunneling between low - frequency surface plasmon modes and adsorbate vibrational modes . in the case of friction for two cu(100 ) surfaces covered by a low concentration of potassium atoms at @xmath313 we have found friction of the same order of magnitude as it was observed in experiment @xcite . however , the distance dependence in this case is more stronger than it was observed in @xcite . further experiments with simple and well defined composition of the tip and sample must be performed to elucidate different energy dissipation mechanisms in the non - contact friction . the obtained results should have broad application in non - contact friction microscopy , and in design of new tools for studying adsorbate vibrational dynamics and optical properties of the surface plasmons . a.i.v acknowledges financial support from dfg and the russian foundation for basic research ( project no . 01 - 02 - 16202 ) b.n.j . p acknowledges financial support from bmbf . we thank r.o . jones for help in the numerical calculations . after substituting ( [ one ] ) and ( [ two ] ) into formula ( [ 3thirteen ] ) to linear order in vibrational coordinate @xmath52 and frequency @xmath51 we get @xmath314 @xmath315 + \left(\frac{c}{\omega}\right)^2p\big[(p+p^{*})(<\left|w_{0y}\right|^2 > + < \left|v_{0y}\right|^2>)\ ] ] @xmath316 + \big(\frac{p^+}{q^2}\big[(p+p^{*})(<w_{1z}w_{0z}^{*}>+<v_{1z}v_{0z}^{*}>+c.c.)\ ] ] @xmath317+\frac{c^2}{\omega(\omega + \omega_0)}p^{+}\big[(p+p^{*})(<w_{1y}w_{0y}^{*}>\ ] ] @xmath318 \big)e^{-i\omega_0t}\big ) \label{aone}\ ] ] from eqs . ( [ 3fifteen ] ) and ( [ aone ] ) it follow that the friction coefficient is determined by the formula @xmath319 @xmath320+\frac{c^2}{\omega(\omega + \omega_0)}p^{+}\big[(p+p^{*})(<w_{1y}w_{0y}^{*}>\ ] ] @xmath321 \big)\big)_{\omega_0=0 } \label{atwo}\ ] ] using eqs . ( [ nine]-[thirteen],[corfs],[corfp ] ) we get @xmath322 @xmath323 \label{athree}\ ] ] @xmath324 @xmath325 @xmath326e^{-2|p|d } \label{afour}\ ] ] other similar expressions for the @xmath37-wave contribution can be obtained from eqs . ( [ athree ] ) and ( [ afour ] ) by replacement of the reflection factors @xmath327 for @xmath36-polarized wave by the reflection factors @xmath328 for @xmath37-polarized wave . after substituting eqs . ( [ athree ] ) and ( [ afour ] ) , and similar expression for @xmath37-polarized waves in eq . ( [ atwo ] ) we get the friction coefficient for normal relative motion which is determined by formulas ( [ 3seventeen]- [ 3eighteen ] ) . 1 . the friction coefficient for two flat surfaces in parallel relative motion as a function of separation @xmath2 at @xmath329k with parameter chosen to correspond to copper ( @xmath330 ) . the contributions from the @xmath98 and @xmath49polarized electromagnetic field are shown separately . the full curves represent the results obtained within the non - local optic dielectric formalism , and the dashed curves represent the result obtained within local optic approximation . ( the log - function is with basis 10 ) fig . 2 . the friction coefficient for two flat surfaces in normal relative motion as a function of separation @xmath2 at @xmath329k with parameter chosen to correspond to copper ( @xmath330 ) . the contributions from the @xmath98 and @xmath49polarized electromagnetic field are shown separately . the full curves represent the results obtained within the non - local optic dielectric formalism , and the dashed curves represent the result obtained within local optic approximation . ( the log - function is with basis 10 ) fig . 3 . the friction coefficient for two flat surface as a function of the free electron density @xmath193 at @xmath329k . the full curve was obtained by interpolation between the result ( dashed lines ) obtained within the non - local optic dielectric approach , with dielectric functions corresponding to a degenerate electron gas for @xmath331m@xmath332 , and to a non - degenerate electron gas for @xmath333 the calculation were performed with the damping constant @xmath230 , separation @xmath251 and @xmath334 ( the log - function is with basis 10 ) fig.4 . the friction coefficient for two clean semiconductor surfaces in ( a ) normal and ( b ) parallel relative motion , as a function of the separation @xmath2 . @xmath335k and with parameters chosen to correspond to a surfaces of silicon carbide ( sic ) ( see text for explanation ) ( the log - function is with basis 10 ) fig . 5 . the friction coefficient for two surface covered by adsorbates in ( a ) normal and ( b ) parallel relative motion , as a function of the separation @xmath2 . @xmath336k and with parameters chosen to correspond to k / cu(001 ) @xcite ( @xmath337 ) ( the log - function is with basis 10 )
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we calculate the van der waals friction between two semi - infinite solids in normal relative motion and find a drastic difference in comparison with the parallel relative motion .
the case of the good conductors is investigated in details both within the local optic approximation , and using a non - local optic dielectric approach .
we show that the friction may increase by many order of magnitude when the surfaces are covered by adsorbates , or can support low - frequency surface plasmons . in this case
the friction is determined by resonant photon tunneling between adsorbate vibrational modes , or surface plasmon modes .
the theory is compared to atomic force microscope experimental data .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the properties of the relativistic fermi gas ( rfg ) model of the nucleus @xcite have inspired the idea of superscaling . in the rfg model , the responses of the system to an external perturbation are related to a universal function of a properly defined scaling variable which depends upon the energy and the momentum transferred to the system . the adjective universal means that the scaling function is independent on the momentum transfer , this is called scaling of first kind , and it is also independent on the number of nucleons , and this is indicated as scaling of second kind . the scaling function can be defined in such a way to result independent also on the specific type of external one - body operator . this feature is usually called scaling of zeroth - kind @xcite . one has superscaling when the three kinds of scaling are verified . this happens in the rfg model . the theoretical hypothesis of superscaling can be empirically tested by extracting response functions from the experimental cross sections and by studying their scaling behaviors . inclusive electron scattering data in the quasi - elastic region have been analyzed in this way @xcite . the main result of these studies is that the longitudinal responses show superscaling behavior . the situation for the transverse responses is much more complicated . the presence of superscaling features in the data is relevant not only by itself , but also because this property can be used to make predictions . in effect , from a specific set of longitudinal response data @xcite , an empirical scaling function has been extracted @xcite , and has been used to obtain neutrino - nucleus cross sections in the quasi - elastic region @xcite . we observe that the empirical scaling function is quite different from that predicted by the rfg model . this indicates the presence of physics effects not included in the rfg model , but still conserving the scaling properties . we have investigated the superscaling behavior of some of these effects . they are : the finite size of the system , its collective excitations , the meson exchange currents ( mec ) and the final state interactions ( fsi ) . the inclusion of these effects produce scaling functions rather similar to the empirical one . our theoretical universal scaling functions , @xmath3 , and the empirical one @xmath4 , have been used to predict electron and neutrino cross sections . the definitions of the scaling variables and functions , have been presented in a number of papers @xcite therefore we do not repeat them here . the basic quantities calculated in our work are the electromagnetic , and the weak , nuclear response functions . we have studied their scaling properties by direct numerical comparison ( for a detailed analysis see ref . @xcite ) . we present in fig . [ fig : fexp ] the experimental longitudinal and transverse scaling function data for the @xmath0c , @xmath2ca and @xmath5fe nuclei given in ref . @xcite for three values of the momentum transfer . we observe that the @xmath6 functions scale better than the @xmath7 ones . the @xmath7 scaling functions of @xmath0c , especially for the lower @xmath8 values , are remarkably different from those of @xmath2ca and @xmath5fe . the observation of the figure , indicates that the scaling of first kind , independence on the momentum transfer , and of zeroth kind , independence on the external probe , are not so well fulfilled by the experimental functions . these observations are in agreement with those of refs . @xcite . , and transverse , @xmath7 , scaling functions obtained from the experimental electromagnetic responses of ref . @xcite . the numbers in the panels indicate the values of the momentum transfer in mev / c . the full circles refer to @xmath0c , the white squares to @xmath2ca , and the white triangles to @xmath5fe . the thin black line in the @xmath6 panel at 570 mev / c , is the empirical scaling function obtained from a fit to the data . the thick lines show the results of our calculations when all the effects beyond the rfg model have been considered . the full lines have been calculated for @xmath0c , the dotted lines for @xmath1o , and the dashed lines for @xmath2ca . the dashed thin lines show the rfg scaling functions.,height=604 ] to quantify the quality of the scaling between a set of @xmath9 scaling functions , each of them known on a grid of @xmath10 values of the scaling variable @xmath11 , we define the two indexes : @xmath12 \ , - \ , \min_{\alpha=1,\ldots , m } \left [ f_\alpha(\psi_i ) \right ] \right\ } \ , , \label{eq : delta}\ ] ] and @xmath13 \ , - \ , \min_{\alpha=1,\ldots , m } \left [ f_\alpha(\psi_i ) \right ] \right\ } \label{eq : erre}\ ] ] where @xmath14 is the largest value of the @xmath15 . the two indexes give complementary information . the @xmath16 index is related to a local property of the functions : the maximum distance between the various curves . since the value of this index could be misleading if the responses have sharp resonances , we have also used the @xmath17 index which is instead sensitive to global properties of the differences between the functions . since we know that the functions we want to compare are roughly bell shaped , we have inserted the factor @xmath18 to weight more the region of the maxima of the functions than that of the tails . .[tab : rdelta]values of the @xmath16 and @xmath17 indexes , for the experimental scaling functions of fig . [ fig : fexp ] . [ cols="^,^,^ " , ] in tab . [ tab : rdelta ] we give the values of the indexes calculated by comparing the experimental scaling functions of the various nuclei at fixed value of the momentum transfer . we consider that the scaling between a set of functions is fulfilled when @xmath19 0.096 and @xmath20 0.11 . these values have been obtained by adding the uncertainty to the values of @xmath17 and @xmath16 for @xmath6 at 570 mev / c . from a best fit of this last set of data we extracted an empirical universal scaling function @xcite represented by the thin full line in the lowest left panel of fig . [ fig : fexp ] . this curve is rather similar to the universal empirical function given in ref . @xcite . let s consider now the scaling of the theoretical functions . the thin dashed lines of fig . [ fig : fexp ] show the rfg scaling functions . the thick lines show the results of our calculations when various effects beyond the rfg are introduced , _ i.e. _ : nuclear finite size , collective excitations , final state interactions , and , in the case of the @xmath7 functions , meson - exchange currents . we have studied the effects of the nuclear finite size , by calculating scaling functions within a continuum shell model . at q=700 mev / c , these scaling functions are very similar to those of the rfg model . at lower values of the momentum transfer , the shell model scaling functions show sharp peaks , produced by the shell structure , not present in the rfg model . we found that shell model scaling functions fulfill the scaling of first kind , the most likely violated , down to 400 mev / c . we have estimated the effects of the collective excitations by doing continuum rpa calculations with two different residual interactions@xcite . the rpa effects become smaller the larger is the value of the momentum transfer . at @xmath21 600 mev / c , the rpa effects are negligible if calculated with a finite - range interaction . collective excitations breaks scaling properties , but we found that scaling of first kind is satisfied down to about 500 mev / c . the presence of the mec violates the scaling of the transverse responses . we included the mec by using the model of ref . @xcite . in our calculations only one - pion exchange diagrams are considered , including those with virtual excitation of the @xmath22 . in our model mec effects start to be relevant for @xmath23 600 mev / c . we found that mec do not destroy scaling in the kinematic range of our interest . the main modification of the shell model scaling functions , are produced by the fsi , we have considered by using the model developed in ref . we obtained scaling functions very different from those predicted by the rfg model , and rather similar to the empirical ones . in any case , the fsi do not heavily break the scaling properties . we found that the scaling of first kind is conserved down to @xmath8=450 mev / c . the same type of scaling analysis applied to @xmath24 reaction leads to very similar results @xcite . to investigate the prediction power of the superscaling hypothesis , we compared responses , and cross sections , calculated by using rpa , fsi and eventually mec , with those obtained by using @xmath3 and @xmath25 . we show in fig . [ fig : ee_xsect ] double differential electron scattering cross sections calculated with complete model ( full ) and those obtained with @xmath3 ( dashed lines ) and @xmath26 ( dotted lines ) . these results are compared with the data of refs . @xcite . c data @xcite have been measured at a scattering angle of @xmath27=37.5@xmath28 , the @xmath1o data @xcite at @xmath27=32.0@xmath28 and the @xmath2ca data @xcite at @xmath27=45.5@xmath28 . the full lines show the results of our complete calculations . the cross sections obtained by using @xmath3 are shown by the dashed lines , and those obtained with @xmath4 by the dotted lines.,height=566 ] the excellent agreement between the results of the full calculations and those obtained by using @xmath3 , indicates the validity of the scaling approach in this kinematic region where the @xmath8 values are larger than 500 mev / c . the differences with the cross sections obtained by using the empirical scaling functions , reflect the differences between the various scaling functions shown in fig . [ fig : fexp ] . the disagreement with the experimental data is probably due to the fact that our models do not consider the excitation of the real @xmath22 resonance , and the pion production mechanism . o . in all the panels the full lines show the result of our complete calculation , the dashed ( dotted ) lines the result obtained with our universal ( empirical ) scaling function . the results shown in panels ( a ) , ( b ) and ( c ) have been obtained for neutrino energy of 300 mev . panel ( a ) : double differential cross sections calculated for the scattering angle of 30@xmath28 as a function of the nuclear excitation energy . panel ( b ) : cross sections integrated on the scattering angle , always as a function of the nuclear excitation energy . panel ( c ) : cross sections integrated on the nuclear excitation energy , as a function of the scattering angle . panel ( d ) : total cross sections , as a function of the neutrino energy.,height=566 ] the situation for the double differential cross sections is well controlled , since all the kinematic variables , beam energy , scattering angle , energy of the detected lepton , are precisely defined , and consequently also energy and momentum transferred to the target nucleus . this situation changes for the total cross sections which are of major interest for the neutrino physics . the total cross sections are only function of the energy of the incoming lepton , therefore they consider all the scattering angles and of the possible values of the energy and momentum transferred to the nucleus , with the only limitation of the global energy , and momentum , conservations . this means that , in the total cross sections , kinematic situations where the scaling is valid and also where it is not valid are both present . we show in the first three panels of fig . [ fig : nue ] various differential charge - exchange cross sections obtained for 300 mev neutrinos on @xmath1o target . in the panel ( a ) we show the double differential cross sections calculated for a scattering angle of 30@xmath28 , as a function of the nuclear excitation energy . the values of the momentum transfer vary from about 150 to 200 mev / c . this is not the quasi - elastic regime where the scaling is supposed to hold , and this explains the large differences between the various cross sections . the cross sections integrated on the scattering angle are shown as a function of the nuclear excitation energy in the panel ( b ) of the figure , while the cross sections integrated on the excitation energy as a function of the scattering angle are shown in the panel ( c ) . the first three panels of the figure illustrate in different manner the same physics issue . the calculation with the scaling functions fails in reproducing the results of the full calculation in the region of low energy and momentum transfer , where surface and collective effects are important . this is shown in panel ( b ) by the bad agreement between the three curves in the lower energy region , and in panel ( c ) at low values of the scattering angle , where the @xmath8 valued are minimal . total charge - exchange neutrino cross sections are shown in panel ( d ) as a function of the neutrino energy @xmath29 . the scaling predictions for neutrino energies up to 200 mev are unreliable . these total cross sections are dominated by the giant resonances , and more generally by collective nuclear excitation . we have seen that these effects strongly violate the scaling . at @xmath29 = 200 mev the cross section obtained with our universal function is still about 20% larger than those obtained with the full calculation . this difference becomes smaller with increasing energy and is about the 7% at @xmath29 = 300 mev . this is an indication that the relative weight of the non scaling kinematic regions becomes smaller with the increasing neutrino energy .
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superscaling analysis of electroweak nuclear response functions is done for momentum transfer values from 300 to 700 mev / c .
some effects , absent in the relativistic fermi gas model , where the superscaling holds by construction , are considered . from the responses calculated for the @xmath0c , @xmath1o and @xmath2ca nuclei ,
we have extracted a theoretical universal superscaling function similar to that obtained from the experimental responses .
theoretical and empirical universal scaling functions have been used to calculate electron and neutrino cross sections .
these cross sections have been compared with those obtained with a complete calculation and , for the electron scattering case , with the experimental data .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the diffusion process of a particle in a potential with the presence of a sink studied through solution of the smoluchowski equation@xcite is of keen interest to many scientists in chemical dynamics as it serves a reference model for a wide variety of dynamical processes . various attempts has been made to study the diffusion processes with a suitable position of the sink@xcite which can find application in diffusion controlled reactions@xcite . such a model is used by wilemski and fixman@xcite to calculate the rate of diffusion controlled reactions as well as cyclization of polymer chain in solutions . ovchinnikova@xcite , zusman@xcite , marucs and nadler@xcite have used such a model for electron transfer reactions in polar solvents . marcus@xcite recently used a diffusive equation with a sink term to develop a theory of uni - molecular reactions in clusters . pressure influence on isomerization reactions is also explained by sumi@xcite by using such type of model.bagchi , fleming@xcite uses a model of this kind to analyze barrier less electron relaxation in solution . exact analytical results for diffusion problems helps in understanding the different parameters like friction and provide an insight to different approximations . as per the authors knowledge , there are no cases where the smoluchowski equation with a time dependent sink is solved by analytical methods . even there is no analytical solution available for the simplest possible case _ i.e. , _ for free particle . there are large number of works on diffusion under time independent sink . in recent work from our group we have given an analytical model in which the problem of particle undergoing diffusive motion under the influence of two potentials where the coupling is time independent@xcite . most of that has focussed on the time evolution / propagator derivation of the case of one or more dirac delta function sinks with constant strength in time . in contrast , this paper presents an work that deals mainly with a dirac delta sink whose strength varies with time and we are the first one to consider this effect explicitly . there are two common methods that one may think of for solving this problem . one method is using a path integral method of the feynman type and the other is using laplace transforms . we use the latter method though the two methods are closely related . our method closely follows the method used for solving schrodinger wave equation . we would like to solve the smoluchowski equation with a time - dependent sink as shown in figure 1 . @xmath2 in the above equation @xmath3 is the diffusion coefficient . it is related to the friction coefficient @xmath4 by @xmath5 . by integrating eq . ( 1 ) from @xmath6 to @xmath7 , we find that the eq.(1 ) reduces to . @xmath8 this is just the case of zero potential with an additional time - dependent boundary condition . if we take the laplace transform @xcite of eq . ( 2 ) , we get @xmath9.\ ] ] for solving eq . ( 3 ) we first consider the homogeneous solution in order to satisfy the boundary condition at the origin . the solution is given below @xmath10 where @xmath11 is the integration constant . by applying the boundary condition at the origin to eq . ( 4 ) we find . @xmath12\\ \nonumber & \rightarrow a(s ) = \frac{l\left[k(t)p(0,t)\right]}{\sqrt { s / a}}.\end{aligned}\ ] ] we now set @xmath13 in eq . ( 4 ) to get @xmath14}{\sqrt { s / a}}+\frac{1}{2 a \sqrt{s / a}}\int dx'(\sqrt{s / a}|x'|)p(x',0),\ ] ] using the convolution theorem for laplace transforms @xcite , we invert eq . ( 6 ) to get @xmath15 the eq.(7 ) may be used to determine the probability distribution at the origin ( _ i.e. , _ @xmath0 ) . once @xmath0 is known , one may find @xmath1 by inverting eq.(4 ) which has the following result : + @xmath16 eq . ( 7 ) and eq.(8 ) together completely determine the problem . as we can see knowing the probability distribution at the origin _ @xmath17 is essentially equivalent to knowing the probability distribution everywhere ( @xmath1 ) . in some cases it may be advantageous to write down the solution in terms of propagator as follows @xmath19 where @xmath20 is propagator . to find the propagator , we use eq.(5 ) and eq.(6 ) to get @xmath21 using eq . ( 4 ) and eq . ( 10 ) , we get @xmath22 so that @xmath23 we now invert eq . ( 11 ) to get @xmath24 \times erfc\left(k_0 \sqrt { t / a}+\frac{|x|+|x'|}{2\sqrt { t / a } } \right)\nonumber \\ + \frac{1}{2 a^2 \sqrt{\pi t / a}}exp \left ( a\frac{(x - x')^2}{4t}\right).\end{aligned}\ ] ] we can see easily that the same propagator can be found by using feynman approach . let us assume that @xmath25 , so that @xmath26 and hence the detailed expression for @xmath1 can be @xmath27 \times erfc\left(k_0 \sqrt { t / a}+\frac{|x|+|a|}{2\sqrt { t / a } } \right)\nonumber \\ + \frac{1}{2 a^2 \sqrt{\pi t / a}}exp \left(a\frac{(x+a)^2}{4t}\right).\end{aligned}\ ] ] we now consider the case , where @xmath28 has the following form : @xmath29 here @xmath30 is a real number . we take the initial condition as @xmath31 where @xmath32 and @xmath30 are real positive numbers . using eq.(6)and the identity @xmath33 = - d l[t\times f(t)]/ds$ ] , we get @xmath34 we can solve this equation under the condition that the laplace transform vanishes at s=@xmath35 and that it must also be bounded at @xmath36 .the last condition is required for the inversion integral to exist . so that we get @xmath37 this laplace domain expression can not be converted to time domain in terms of elementary or special functions . however , it can be done using the integral @xmath38 in the above expression @xmath39 is a positive number taken to be larger than the right - most pole in the complex plane . so we are unable to do the last step analytically , although numerically it is not a problem . we choose @xmath40 , where @xmath30 is a real positive constant . the laplace transform of @xmath28 does not exist . interestingly , the laplace transform of @xmath41 does exist if we choose an initial condition that vanishes at the origin ! our derivation in this subsection is valid under this retriction . we first use the fact , @xmath42=\int^\infty_s ds ' l[f(t),s'].\ ] ] now by using eq . ( 21 ) and eq . ( 6 ) we get @xmath43 now we define a new function @xmath44 as given below @xmath45 now eq.(22 ) can be written as @xmath46 on solving eq . ( 24 ) we get , @xmath47 by equation 29 we find , + @xmath48 we may now invert equation 30 to find , + @xmath49 we now seek @xmath50.to find this quantity , we first write , + @xmath51 to find g , we use results of equations ( 30 ) , ( 27 ) , ( 5 ) and ( 4 ) to find , + @xmath52\ ] ] we may now invert this expression to find , + @xmath53 @xmath54= l[f\;(t),s\;-a].\ ] ] by equation ( 31 ) and ( 6 ) we find @xmath55 to obtain a solution , we could iterate this expression repeatedly to obtain the series solution : + @xmath56 but since we already know the form of the serise , we may substitute equation(37)directly into equation(36 ) . after doing this nad solving for the a s by equating terms with like powers of @xmath39 , we find that @xmath57 ( s ) is given by + @xmath58 @xmath59 in the current work the authors had attempted to find the exact solution for the smoluchowski equation with a time dependent delta function sink in many special cases . these special cases include linear , inversely and exponential dependence . the case of sinusoidal time dependent sink can be treated in a similar manner as the exponential time dependent sink by representing the sine function as a sum of complex exponentials . th authors want to thank iit mandi for professional development fund as well as htra scholarship . h. risken , the fokker planck equation ( springern , berlin 1984 ) . a. samanta and s. k. ghosh , j. chem . 97(12 ) ( 1992 ) 9321 . b. bagchi j. chem . 87 , 5393 ( 1987 ) s. a. rice , diffusion - limited reactions , edited by c. h. bamford , c. f. h. tipper and r. o. compton ( elseiver , amsterdam , 1985 ) , vol 25 . g. wilemsky and m. fixman , j. chem . phys . 84 ( 1986)4894 . g. wilemsky and m. fixman , j. chem . phys . 60 ( 1974)878 . m. ya . ovchinnikova , teor . khim 17 ( 1981)651 . l. d. zusman , chem . 80 ( 1983)29 . h. sumi and r. a. marcus , j. chem . 84 ( 1986)4894 r. a. marcus , j. chem . ( 1996)5446 . h. sumi , j. mol . 65/66 ( 1995)65 . b. bagchi , g. r. fleming , j. chem . 78 ( 1983)7375 . b. bagchi , g. r. fleming , j. phys . 94 ( 1984)9 . a. chakraborty . phys . 139 ( 2013)094101 . j. campbell , j. phys . a : math . theor . 42 ( 2009)365212 . m. abramowitz and i. stegun , _ handbook of mathematical functions _ ( dover , new york , 1970 ) .
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the smoluchowski equation for a free particle with a time dependent sink is solved exactly for many special cases . in this method by knowing the probability distribution at the origin @xmath0 , one may derive the probability distribution at all positions _
i.e. , _ @xmath1 .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
the origin of binary stars is still a puzzle in our understanding of star formation . numerous theoretical simulations support the hypothesis that the fragmentation of collapsing molecular cloud cores , promoted by either rotation or turbulence , is the main mechanism for the formation of binary / multiple stellar systems ( see reviews by bodenheimer et al . 2000 , tohline 2002 , and goodwin et al . 2007 ) . nevertheless , many key questions concerning this fragmentation process , e.g. , when exactly does the fragmentation start , are still under debate ( see tohline 2002 ) . although it is generally assumed that cloud cores do not fragment during the free - fall collapse phase , several groups ( e.g. , bate & burkert 1997 ; tsuribe & inutsuka 1999 ; boss et al . 2000 ) found that fragmentation can occur at the end of the isothermal phase ( generally referred to as prompt fragmentation ) , while others ( e.g. , truelove et al . 1997 , 1998 ; machida et al . 2005 , 2008 ) argue that the isothermal gas is stable against fragmentation and found that fragmentation only occurs during / after the adiabatic phase ( see figure 9 in andr et al . 2009 for a discussion of the different evolutionary phases of core collapse ) . on the observational side , a handful of young protostellar ( i.e. , class0 ) binary systems have been found ( e.g. , looney et al . 2000 ; launhardt 2004 ) , and there are increasing kinematics studies of binarity in the protostellar phase ( e.g. , chen et al . 2007 , 2008 , 2009 ; volgenau et al . 2006 ; maury et al . 2010 ) . however , the number of observed and well - studied protostellar binary ( protobinary ) systems is still small , and these systems have already gone through the fragmentation process . the observational link between the initial conditions in a fragmenting cloud core and the final stellar systems formed therein is still missing . it is therefore critical to observe , at high angular resolution , more dense cores in nearby molecular clouds to search for the earliest phase of fragmentation and study in detail their properties , in order to put direct constraints on fragmentation models . in this letter , we present submillimeter array ( sma ; ho et al . 2004 ) dust continuum observations toward the r corona australis ( rcra ) region . at a distance of @xmath5170pc ( knude & hg 1998 ) , the cra dark cloud is one of the nearest star - forming regions . as the most active star formation site in this cloud , the rcra region has been extensively observed in the past two decades ( see review by neuh@xmath6user & forbrich 2008 , and references therein ) . using scuba on the james clerk maxwell telescope ( jcmt ) , nutter et al . ( 2005 ) found a prestellar core candidate termed smm1a in the rcra region , which is elongated in the east - west direction and has a gas mass of @xmath510@xmath7 , and an effective radius of @xmath53000au . with a maximum velocity dispersion of about 0.8kms@xmath8 ( harju et al . 1993 ) , the smm1a core is gravitationally bound . infall motions in this region of the cloud further confirm that this is a prestellar core ( groppi et al . 2004 ) . in this letter , we report the discovery of a multiple system within the smm1a core , based on high angular resolution millimeter observations . this may represent the earliest phase of core fragmentation observed thus far . the rcra region was observed with the sma on 2006 august 20 in the compact configuration . six antennas were used in the array , providing baselines from 5k@xmath9 to 52k@xmath9 at 220ghz . the sma primary beam is about 55@xmath10 at this frequency . two positions next to each other at ( r.a . , decl.)@xmath11=(19:01:53.3 , @xmath1236:57:21.0 ) and ( r.a . , decl.)@xmath11=(19:01:56.4 , @xmath1236:57:27.0 ) were observed . the digital correlator was set up to cover the frequency ranging [email protected] and [email protected] in the lower and upper sidebands , respectively . the 1.3 mm dust continuum emission was recorded with a total bandwidth of @xmath53.3ghz ( @xmath51.8ghz usb and @xmath51.5ghz lsb ) . system temperatures for rcra ranged from 100 to 280k ( depending on elevation ) , with a typical value of @xmath5200k . the visibility data were calibrated with the mir package ( qi 2005 ) , with quasars 3c454.3 and 1924 - 292 as the bandpass and gain calibrators , respectively . uranus was used for absolute flux calibration , from which we estimate a flux accuracy of @xmath520% , by comparing the measured quasar fluxes with those listed in the sma calibration database . the calibrated visibility data were imaged using the miriad toolbox ( sault et al . the sma synthesized beam size at 1.3 mm dust continuum , with robust _ uv _ weighting 0 , is 5.7@[email protected]@xmath10 . figure 1a shows the sma 1.3 mm dust continuum image of rcra , overlaid with the scuba 850@xmath14 m dust continuum contours ( from groppi et al . the northern part of this image shows clear detection of the dust continuum emission associated with the herbig - ae star rcra and protostellar cores smm2 , smm1b and smm1c these last two are also referred to as sma1 and sma2 in groppi et al . ( 2007 ; see also choi et al . 2008 ) . to the south , the scuba 850@xmath14 m image shows the smm1a core , which is elongated in the east - west direction ( see fig.1a ) . at higher angular resolution , the scuba 450@xmath14 m image in figure 1b shows that the smm1a core is clearly resolved into two sub - cores ( see also van den ancker 1999 ) . at even higher resolution , the sma 1.3 mm continuum observations reveal that the smm1a core is divided into three condensations , aligning roughly from east to west , which we refer to here as smm1a - a , smm1a - b , and smm1a - c ( fig.1b ) . all are detected with signal - to - noise ratio of 4 or more . source smm1a - a coincides with the eastern sub - core observed in the scuba 450@xmath14 m image , while smm1a - b and smm1a - c are coincident with the western sub - core ( fig.1b ) . the single - dish and interferometer observations are in general agreement with each other , which indicates that the three continuum sources detected with the sma are neither artifacts nor are due to noise in the interferometer image46@xmath10 at 270ghz ) ; ( 2 ) more flux was resolved out in their extended configuration observations ; and ( 3 ) the 1.1 mm observations lack the sensitivity to detect these weak sources . ] . also note that the sio(54 ) line and several other molecular lines , e.g. , dcn(32 ) , were included in this sma correlator setup , but no line emission was detected from the three condensations . figure 2 shows the mid - infrared images of the rcra region , taken by the _ spitzer space telescope_. the mips1 band image at 24@xmath14 m ( fig.2a ) clearly shows infrared emission at the position of cores smm1b ( associated with the infrared source irs7b , see wilking et al . 1997 ) , smm2 , and the herbig stars rcra and tcra . there is also 24@xmath14 m emission from the smm1c core , which may be associated with the radio source b9 centered on the core ( see choi et al . 2008 ) , and the infrared source irs7a located @xmath55@xmath10 to the south . the mips2 band image at 70@xmath14 m ( fig.2b ) has a lower angular resolution and is dominated by the emission from the smm1c / irs7a region . the mips3 ( 160@xmath14 m ) image of rcra is extremely saturated , and is thus not shown here . none of the @xmath15 images at bands from 3.6 to 70@xmath14 m shows compact infrared emission in the smm1a core , which suggests that the core is extremely cold . from the sma 1.3 mm images , we derived the positions , fluxes , and ( deconvolved ) fwhm sizes of the three sources in rcra smm1a using a multi - component gaussian fitting routine ( see table 1 ) . angular separations between sources smm1a - a and -b and smm1a - b and -c are 122@xmath1603 and 58@xmath1603 , corresponding to projected separations of @xmath52100au and @xmath51000au ( at a distance of 170pc ) , respectively . assuming that the 1.3 mm dust continuum emission is optically thin , the total gas mass ( @xmath17 ) of the three sources is calculated with the same method as described in launhardt & henning ( 1997 ) . in the calculations , we adopt a dust opacity of @xmath18 ( ossenkopf & henning 1994 ) , a typical value for dense and cold molecular cloud cores , and a mass - averaged dust temperature of @xmath518k ( see below ) . the gas masses of the three sources , derived from the sma 1.3 mm dust continuum observations , range from @xmath50.10 to @xmath50.23@xmath7 ( see table 1 ) . with these mass and the fwhm size of the condensations , we estimate their average densities to be in the order of @xmath19 to @xmath20@xmath21 . note that these are average densities and the local peak densities in these sources may be orders of magnitude higher . these high densities , as well as the infall motions detected in this region ( see groppi et al . 2004 ) , suggest that the three condensations in the smm1a are on their way to form low - mass stars . considering that the smm1a core has a mass of approximately 10@xmath22 , and assuming a core - to - star efficiency of @xmath23% ( evans et al . 2009 ) , it is probable that a multiple stellar system with a total mass of 3 @xmath22 will eventually form in smm1a . table 1 lists the 450@xmath14 m and 850@xmath14 m fluxes estimated from the scuba images , and the 800@xmath14 m fluxes adopted from van den ancker ( 1999 ) . since sources smm1a - b and smm1a - c are not resolved in the jcmt observations , we treat them as one sub - core here . the scuba 450@xmath14 m fluxes are derived by fitting two - dimensional gaussians toward the two sub - cores ( smm1a - a and smm1a - b+c ) , while the 850@xmath14 m fluxes are estimated from the fluxes enclosed within roughly one beam around each sub - core . based on the sma 1.3 mm and jcmt submm data points , as well as the 3@xmath24 upper limits in the @xmath15 mips images , we construct the spectral energy distributions ( seds ) of the two sub - cores ( see figure 3 ) . in order to derive luminosities and temperatures , we first interpolate and then integrate the seds , always assuming spherical symmetry . interpolation between the flux densities is done by a @xmath25@xmath26 single - temperature grey - body fitting to all points ( including the upper limits ) . the resulting bolometric luminosities are @xmath270.08@xmath28 for smm1a - a and @xmath270.09@xmath28 for smm1a - b+c . the dust and bolometric temperatures are between 17 and 19k for the sources . the low luminosities and temperatures , as well as the fact that no compact infrared emission is detected from these sources in the @xmath15 images , resemble the typical properties of prestellar cores ( see review by andr et al . 2009 ) . in the vla survey of the rcra region , no radio source is detected at the positions of the three condensations presented here ( choi et al . 2008 ) . in contrast , the protostars smm1b ( class0/i ) and smm1c ( class0 ) to the north of smm1a are bright and centrally peaked at millimeter wavelengths ( fig.1a ) , and are associated with hard x - ray ( forbrich & preibisch 2007 ) , infrared ( fig.2a ) , and centimeter radio counterparts ( choi et al . 2008 ) . therefore , although the chemical and kinematic properties of smm1a are still poorly known , the observations thus far suggest that the three sources in smm1a are in an earlier evolutionary stage than that of class 0 protostars . as predicted by theoretical studies , there are different phases in the evolution from an initial isothermal ( 10k ) prestellar core to a class 0 protostar ( andr et al . the low luminosities ( @xmath270.1@xmath2 ) and temperatures ( close to 20k ) of the three sources in smm1a are similar to the predicted properties of so - called first hydrostatic ( adiabatic ) cores , which are formed after the isothermal collapse of prestellar cores ( e.g. , masunaga et al . 1998 ; chen et al . 2010 ) . however , it must be noted that large uncertainties remain in our estimates due to the limited observations available . further high angular resolution observations at different wavelengths are needed to constrain the seds of the three sources in order to address more precisely their evolutionary statuses . regardless of this , it is fair to say that _ the rcra smm1a multiple system represents the earliest phase of core fragmentation observed thus far_. prompt fragmentation of rotationally flattened cloud cores with initially flat density profiles , immediately after a phase of free - fall collapse , has long been suggested as one of the most efficient mechanisms for binary star formation ( tohline 2002 ) . since this fragmentation is expected to occur at the end of the isothermal collapse phase , it is also referred to as isothermal fragmentation . the separation of the fragments formed via prompt fragmentation has a scale of 10@xmath26@xmath1210@xmath29au , which corresponds to the jeans length in this phase . in contrast , other groups found that isothermal gas is stable against fragmentation and that core fragmentation can only occur during ( and after ) the adiabatic phase ( e.g. , truelove et al . 1998 ; machida et al . 2008 ) . because this fragmentation occurs after the formation of the first adiabatic core , it is also called the first core fragmentation ( machida et al . 2005 ) . here , the separation of the fragments formed generally has a scale of 3@xmath12300au ( machida et al . clearly , observing the earliest phase of core fragmentation is the most effective way to discriminate between prompt ( isothermal ) and adiabatic fragmentation scenarios . if prompt fragmentation does occur in the isothermal phase , we may expect to find a binary / multiple system consisting of adiabatic cores with the separation of @xmath51000au . as shown in the scuba images ( see figure 1b ) , the rcra smm1a core is highly elongated with an aspect ration of roughly 3 , showing a morphology similar to the bars or filaments seen in the simulations of prompt ( isothermal ) fragmentation ( see , e.g. , bate & burkert 1997 ; tsuribe & inutsuka 1999 ; boss et al . 2000 ) . from the mass and radius of the smm1a core ( see nutter et al . 2005 ) , the average volume density is calculated to be 1@xmath122@xmath1310@xmath30@xmath21 . the corresponding jeans length is @xmath51000au for a temperature of @xmath510k . the separations among the three sources found in the smm1a core are also consistent with this estimated jeans length . moreover , our results from the sed fitting ( see above ) suggest that the three sources are at an evolutionary stage earlier than that of class0 protostars , perhaps in the first ( adiabatic ) core stage . all these results appear to support the view that the three sources are formed through the prompt fragmentation of an elongated collapsing core during the isothermal phase . in this scenario , we would expect the rotational axis of smm1a to be perpendicular to the direction of the core s major axis , which would be roughly perpendicular to the large scale rotation observed in the rcra region by groppi et al . this is not necessarily inconsistent with our picture as other cores show drastic differences in the rotation axes measured at different scales ( e.g. , ohashi et al . we also note that so far most observed protostellar binary / multiple systems ( e.g. , l723 , launhardt 2004 ; or ngc2264 d - mm1 , teixeira et al . 2007 ) have separations in the scale of 1000au , which also supports the scenario of prompt fragmentation , although these protostellar systems have had at least a few 10@xmath29 yrs to evolve from their prestellar stage and thus their separations may not represent real initial conditions . on the other hand , as introduced above , prompt ( isothermal ) fragmentation has trouble accounting for the close binary systems with separations @xmath4100au . the fragmentation of cores at the adiabatic phase is still an attractive mechanism for close binary star formation . considering that the binary separation distribution in ( pre-)main sequence stars peaks at @xmath530au , we actually would expect to find frequently close protobinary systems with separations of @xmath4100au to support the adiabatic fragmentation scenario . in practice , however , the number of this kind of systems is very small . this observational result can be understood by the fact that current large millimeter interferometers ( e.g. , sma and iram - pdbi ) normally reach 12@xmath10 angular resolution under general conditions , and thus mostly resolve wide protobinary systems with separations of 200au or more in nearby star - forming clouds . we believe that routine observations at angular resolution better than 0.5@xmath10 will reveal more close protobinary systems , which will then allow us to study in detail a statistically significant number of protostellar binary / multipe systems with a wide range of separations ( from @xmath510au to 10@xmath29au ) . we present sma 1.3 mm dust continuum observations toward the rcra region . the 1.3 mm dust continuum emission is detected from dense cores smm1b , smm1c , and smm2 , and the herbig - ae star rcra in this region . we discover within the prestellar core candidate smm1a a multiple system made up of three condensations with masses in the range of 0.1 to 0.2@xmath0 . the angular separations between the three new sources are @xmath512.2@xmath10 and @xmath55.8@xmath10 , corresponding to the projected separations of @xmath52100au and @xmath51000au , respectively . based on sma observations and complementary data from the jcmt and @xmath15 telescopes , we construct their spectral energy distributions ( seds ) and find that all three sources have extremely low bolometric luminosities ( @xmath270.1@xmath2 ) and temperatures ( @xmath2720k ) . we suggest that the three condensations in smm1a resulted from the fragmentation of an elongated core in the isothermal phase . the rcra smm1a system appear to be the earliest phase of low - mass core fragmentation observed thus far . smm1a - a & 19:01:55.86 & @xmath1236:57:48.8 & 46@xmath169 & 56@xmath1352 & [email protected] & [email protected] & 1.5 & [email protected] + smm1a - b & 19:01:54.87 & @xmath1236:57:45.2 & 64@xmath1613 & 68@xmath1363 & [email protected] & [email protected]@xmath31 & 0.8@xmath31 & [email protected]@xmath31 + smm1a - c & 19:01:54.56 & @xmath1236:57:40.6 & 110@xmath1620 & 78@xmath1331 & [email protected] & [email protected]@xmath31 & 0.8@xmath31 & [email protected]@xmath31 +
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we report the discovery of multiple condensations in the prestellar core candidate smm1a in the rcra cloud , which may represent the earliest phase of core fragmentation observed thus far .
the separation between the condensations is between 1000 and 2100au , and their masses range from about 0.1 to 0.2@xmath0 .
we find that the three condensations have extremely low bolometric luminosities ( @xmath1@xmath2 ) and temperatures ( @xmath3k ) , indicating that these are young sources that have yet to form protostars .
we suggest that these sources were formed through the fragmentation of an elongated prestellar core .
our results , in concert with other observed protostellar binary systems with separations in the scale of 1000au , support the scenario that prompt fragmentation in the isothermal collapse phase is an efficient mechanism for wide binary star formation , while the fragmentation in the subsequent adiabatic phase may be an additional mechanism for close ( @xmath4100au ) binary star formation .
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You are an expert at summarizing long articles. Proceed to summarize the following text:
specific - heat studies down to very low temperatures provide insight into low - energy excitations of the electronic , phononic , or magnetic subsystems of solids @xcite . being directly linked to the entropy changes , i.e. @xmath9 , the specific heat @xmath10 is a valuable tool to study such excitations as well as phase transitions . one example is the onset of superconductivity in bcs superconductors where a specific heat jump @xmath11@xmath12 appears at @xmath12 ( @xmath13 is the sommerfeld coefficient ) . well below @xmath12 , the specific heat gives direct access to the entropy of cooper - pair breaking and in bcs superconductors it exponentially depends on the isotropic gap @xmath14 . to be more general , the specific heat measures the gap magnitude and structure and provides information on the pairing mechanism . noteworthy , being a thermodynamic quantity the specific heat is sensitive to bulk properties which is in contrast to rather surface - sensitive methods such as arpes or stm . in this work , a commercially available device is applied for specific heat measurements of the unconventional superconductor k@xmath0na@xmath1fe@xmath2as@xmath2 @xcite . the data imply a large @xmath4-contribution to the specific heat well below @xmath12 thereby evidencing @xmath5-wave superconductivity in this material . the calibration of the calorimeter by measurements of high purity ag however indicates a strong schottky - like increase of the addenda heat capacity which increases systematic errors below @xmath15 . in order to study materials with small heat capacity , i.e. with very small sample mass and/or low specific heat , the design of a novel calorimeter is presented . the proposed calorimeter with paramagnetic temperature sensor and a squid - based readout is expected to have a much smaller addenda heat capacity of less than @xmath7 for @xmath16 and promises a temperature resolution of @xmath6 . k@xmath0na@xmath1fe@xmath2as@xmath2 single crystals were grown using kas - flux described in ref . a single crystal with mass @xmath17 was placed on a commercial calorimeter ( heat capacity puck qd - p107h ) from quantum design @xcite . apiezon n grease @xcite ( typically less than @xmath18 ) served as an adhesive between the sample and the measuring platform . this platform consists of a @xmath19 sapphire single crystal borne by kapton strips . two ruthenium oxide thick film resistors ( ruox resistors ) attached to the sapphire platform serve as heater ( with resistance @xmath20 ) and thermometer ( @xmath21 ) . henceforth this setup is referred to as addenda . both resistors are electrically contacted via pt92w8-wires , which also define the thermal link between addenda and thermal bath . the calorimeter is mounted to the mixing chamber of a dilution refrigerator . for calibration purposes , the ruox resistors temperature dependencies @xmath20 and @xmath21 were measured by a standard 4-wire sensing method using an avs-47 ac resistance bridge . for the measurement of the heat capacity we applied a standard pulse - fitting method as described in literature @xcite . heating power for the pulses is supplied using the analog voltage output of the data acquisition hardware ni - usb 6251 box from national instruments @xcite connected in series with a @xmath22-resistance . a lock - in amplifier ( signal recovery 7265 dsp ) performing ac 4-wire sensing in the low @xmath23-range is used to measure the temperature response with desired resolution both in time and amplitude . to suppress parasitic heating , the lock - in amplifier is galvanically detached from the platform and all wires attached to the platform are low - pass - filtered at room temperature with a cut - off frequency of @xmath24 . the lock - in amplifier signal was calibrated against a carbon resistance thermometer placed at the mixing chamber ; this thermometer had been calibrated against a fixed - point thermometer ( srd1000 from hdl @xcite ) and a noise thermometer ( see ref . @xcite ) . a typical heat pulse and the corresponding temperature response is shown in fig . [ singlepulse ] . after the heat pulse , the temperature dependence @xmath25 shows two different relaxation processes associated with the thermal links between addenda and sample and between addenda and thermal bath , respectively . the temperature relaxation is described by two exponentials with relaxation times @xmath26 and @xmath27 . the black curve represents the model fit applied to describe the pulse @xmath25 from which fitting parameters are extracted to determine the total heat capacity of sample and addenda . for calibration , a silver sample with nominal purity of @xmath28 and mass @xmath29 is used . magnetic susceptibility measurements in a commercial squid magnetometer ( quantumdesign mpms - xl5 @xcite ) revealed no detectable magnetic impurities in the temperature range @xmath30 that might contribute to the specific heat of this sample . from theoretical calculations one derives the electronic specific heat of silver @xcite . the debye coefficient is derived from the debye temperature from ref . the silver specific heat is considered to be @xmath31 with @xmath32 and @xmath33 @xcite . an additional correction due to the apiezon n grease specific heat is taken into account as well @xcite . the resulting addenda heat capacity obtained after subtracting the silver heat capacity is shown in fig . [ addenda ] . the error bars shown indicate the statistical error of typically ten individual pulses measured . a continuous @xmath34-curve is obtained by approximating the experimental data by means of an appropriate arbitrary empirical function @xmath35 , with arbitrary parameters @xmath36 , @xmath37 , @xmath38 , @xmath39 , @xmath40 and @xmath41 . the result of this procedure is shown by the black line in fig . [ addenda ] . the data show a nearly linearly increasing addenda heat capacity above a minimum at around @xmath42 and a schottky - like increase below this minimum . while no further details of the thermometer composition have been communicated to the authors , one may however assume that ru nuclear moments with @xmath43 present in the thermometer or the pt92w8-wires cause the observed schottky - like behaviour @xcite . although @xmath44-pairing has been suggested for the entire class of fe - based superconductors ( e.g. , ref . @xcite , and references therein ) , the nature of superconductivity in these materials is still under debate and specific heat studies are one of the major experimental tools to address this issue @xcite . here , single - crystalline k@xmath0na@xmath1fe@xmath2as@xmath2 was studied by means of the device described above in the temperature regime between @xmath45 and @xmath46 ( fig . [ kna122 ] ) . the presence of superconductivity in the crystal was confirmed by measurements of the volume ac susceptibilities ( @xmath47 and @xmath48 ) which yields the superconducting transition temperature @xmath49 , and @xmath50 @xcite . in the inset of fig . [ kna122 ] , where the specific heat at higher temperatures up to @xmath51 obtained by a quantum design ppms system is shown , the associated specific heat jump @xmath52 at @xmath12 is clearly visible @xcite . the specific heat jump at @xmath12 amounts to @xmath53 . at low temperatures the data show a linear - in-@xmath54 decrease of @xmath55 , i.e. @xmath56 , which is superimposed by a schottky - like contribution below @xmath57mk . note , that at @xmath58 the sample heat capacity is larger than the addenda contribution by a factor of two ( see fig . [ addenda ] ) . this ratio strongly increases upon heating to , e.g. , 10 at @xmath59 . the experimentally observed @xmath4-dependence of the specific heat well below @xmath12 evidences quasi - particle excitations near line node(s ) , i.e. the superconducting gap in k@xmath0na@xmath1fe@xmath2as@xmath2 is zero at least at one @xmath60-point of one fermi surface sheet . the data hence imply nodal superconductivity , i.e. either @xmath5- or @xmath61-wave symmetry of the pairing state . for quantitative analysis of the low - temperature specific heat , the specific heat is fitted by @xmath62 which yields the residual sommerfeld - coefficient @xmath63 , the quasi - particle contribution @xmath64 and the lattice contribution @xmath65 . note , that the lattice contribution was defined from the normal state behaviour ( for details see the supplement of ref . the large @xmath4-contribution arises mainly from quasi - particle excitations near line node(s ) in k@xmath0na@xmath1fe@xmath2as@xmath2 . a quantitative estimate in ref . @xcite suggests @xmath5-wave superconductivity in k@xmath0na@xmath1fe@xmath2as@xmath2 . the device discussed so far has limitations at temperatures below @xmath66 . firstly , the observed schottky - contribution to the addenda heat capacity yields a lower limit to the resolution of the obtained specific heat . secondly , in order to avoid self - heating only low voltages can be applied to the ruox thermometer resulting in a nonsatisfying temperature resolution . a third problem arises when the @xmath27 effect @xcite becomes increasingly dominant at low temperatures . the model applied for data analysis assumes a temperature - independent thermal link @xmath67 during each heat pulse . however , this precondition is not met anymore when , due to a large @xmath27 effect , the sample platform temperature difference between the heating and cooling process is too large . to address these issues , a calorimeter with paramagnetic temperature sensor and squid - based readout is suggested as shown in figs . [ heatcapchip ] and [ meander_scheme ] . paramagnetic metallic sensors are well established low - temperature thermometers for micro - calorimeters ( see ref . these devices have been developed and utilized for high resolution particle detection . basic elements of magnetic calorimetry - based detectors are absorbers for the particles and paramagnetic sensors made of er - doped au ( * au*:er ) whose magnetisation obeys a curie - like behaviour . in a small magnetic field , the magnetisation strongly depends on temperature . temperature changes due to absorption of particles can be detected by a highly sensitive squid - based read - out . the details of the underlying physics of the sensor material and the detection scheme have been investigated thoroughly in the last decade @xcite . the new calorimeter design depicted in fig . [ heatcapchip ] features a * au*:er - sensor with an internal signal rise time as fast as @xmath68 at @xmath69 @xcite as a central element . the samples will be placed on a @xmath70 thin gold layer ( sample platform ) with a usable area of @xmath71 micro - fabricated on a sapphire substrate ( @xmath72 ) . a aupd film resistor ( c in fig . [ heatcapchip ] ) placed on the sapphire is used to generate a heat pulse . the heater is in electronic contact with the sample platform ( d ) to allow for fast heat flow . the gold layer may also be used to attach samples via ultrasonic bonding rather than apiezon n grease . this will strongly decrease the undesired @xmath27 effect . thermal equilibration is achieved via au bonds between the sample platform and the thermal bath . by varying the number of au bonds , we can adjust the thermal link @xmath67 depending on the expected heat capacity of the sample under investigation . to detect the temperature response of a heat pulse , the sample platform is electronically coupled to the sensor material ( a ) of the thermometer . this sensor material is placed in a small magnetic field produced by a persistent current in meandering superconducting coils ( b and underneath a ) with inductance @xmath73 . together with the input coil of a dc squid , these coils form a superconducting flux transformer . the temperature read - out scheme is depicted in fig . [ meander_scheme ] . the squid itself is placed next to the calorimeter chip and the coils are connected via superconducting niobium bonds . a gradiometric read out scheme of the paramagnetic temperature sensor is employed in order to minimize signal fluctuations caused by external magnetic disturbances . based on experience with cryogenic particle detectors @xcite , thermodynamic properties of the presented setup including the specific heat and the magnetisation of the sensor material can be predicted reliably using numerical methods . assuming typical values for the persistent current ( @xmath74 ) and the doping level of er ( @xmath75 ) , the contribution of the sensor material * au*:er to the addenda heat capacity is estimated to be @xmath76 in the temperatures range @xmath77 . furthermore , the sample platform has an electronic specific heat of @xmath78 @xcite . other specific heat contributions can be neglected due to the small amount of material used ( aupd ) or their small intrinsic heat capacities ( nb , phonons in the sapphire substrate ) . in total , we expect a 40-fold smaller addenda heat capacity compared to the discussed commercial device . the temperature resolution of the new calorimeter can be predicted taking into account geometric factors ( sensor volume @xmath79 , meander - pitch @xmath80 ) , the circuitry of the meanders and squid performance parameters . with the presented design we expect a temperature resolution of @xmath81 with an integration time of several @xmath82 , which is far better and faster than the ruox resistance thermometer of the presently used commercial device . a commercially available calorimeter was calibrated and used for specific heat studies of k@xmath0na@xmath1fe@xmath2as@xmath2 down to 20mk in a dilution refrigerator . a large @xmath4-term in the specific heat implies nodal superconductivity whose nature is found to be consistent with @xmath5-wave by quantitative analysis in ref . @xcite . at lowest temperatures , the commercial device is of limited use only and the design for a calorimeter based on magnetic thermometry is presented . the micro - fabricated magnetic calorimeter promises temperature resolution of @xmath6 and addenda heat capacity less than @xmath7 for @xmath16 . valuable discussions with s. kempf , c. pies , and a. reiser are gratefully acknowledged . this work was supported by the dfg through projects kl1824/6 , wu595/3 - 1 , bu887/15 - 1 , en299/5 - 1 and by the european community research infrastructures under the fp7 capacities specific program , microkelvin project number 228464 . g. r. stewart , rev . instr . * 54 * , 1 ( 1983 ) m. abdel - hafiez et al . , b * 87 * , 180507(r ) ( 2013 ) m. abdel - hafiez et al . , phys . rev . b * 85 * , 134533 ( 2012 ) quantum design , inc . , 6325 lusk boulevard , san diego , ca 92121 - 3733 , usa apiezon products , mi materials ltd , hibernia way , trafford park , manchester m32 0zd , united kingdom j. s. hwang , k. j. lin and c. tien , rev . inst . * 68 * , 94 ( 1997 ) national instruments corp . , 11500 n mopac expwy , austin , tx 78759 - 3504 , usa hightech development leiden , zeeforel 4 , 2318 mp leiden , the netherlands a. netsch et al . , aip conference proceedings * 850 * , 1593 ( 2006 ) j. m. ziman , adv . phys . * 10 * , 1 - 56 ( 1961 ) d. smith and f. fickett , j. res . natl . stand . technol . * 100 * , 119 ( 1995 ) h. j. schink and h. v. lohneysen , cryogenics * 21 * , 591 - 592 ( 1981 ) y. e. volokitin et al . , cryogenics * 34 * , 771 - 773 ( 1994 ) j. ho et al . , sci instrum . * 36 * , 1382 ( 1965 ) i. i. mazin , nature * 464 * , 183 ( 2010 ) l. ding et al . b * 77 * , 180510(r ) ( 2008 ) u. welp et al . , b * 79 * , 094505 ( 2009 ) u. stockert et al . b * 83 * , 224512 ( 2011 ) m. abdel - hafiez et al . , 391 , 012120 ( 2012 ) f. hardy et al . lett . * 111 * , 27002 ( 2013 ) r. e. schwall et al . , * 46 * , 1054 ( 1975 ) c. enss et al . , j. low temp . 121 , 137 - 177 ( 2000 ) a. fleischmann , c. enss , g. m. seidel , topics of applied physics * 99 * , cryogenic particle detection ( c. enss ed . ) , 151 - 216 , springer berlin heidelberg ( 2005 ) a. fleischmann et al . , aip conference proceedings 1185 , 571 - 78 ( 2009 ) c. pies et al . , j. low temp * 167 * , 269 ( 2012 ) d. l. martin , phys . b * 8 * , 5357 - 6360 ( 1973 )
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a commercially available calorimeter has been used to investigate the specific heat of a high - quality k@xmath0na@xmath1fe@xmath2as@xmath2 single crystal .
the addenda heat capacity of the calorimeter is determined in the temperature range @xmath3 .
the data of the k@xmath0na@xmath1fe@xmath2as@xmath2 crystal imply the presence of a large @xmath4 contribution to the specific heat which gives evidence of @xmath5-wave order parameter symmetry in the superconducting state . to improve the measurements ,
a novel design for a calorimeter with a paramagnetic temperature sensor is presented .
it promises a temperature resolution of @xmath6 and an addenda heat capacity less than @xmath7 at @xmath8 .
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